%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP108^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n018.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:37 AM UTC 2026
% Result : Theorem 7.86s 2.43s
% Output : Refutation 7.86s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP108^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n018.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:26:01 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : ITP108^2 : TPTP v9.2.1. Released v7.5.0.
% 0.30/0.46 % Domain : Interactive Theorem Proving
% 0.30/0.46 % Problem : Sledgehammer Localization problem prob_758__8980450_1
% 0.30/0.46 % Version : Especial.
% 0.30/0.46 % English :
% 0.30/0.46
% 0.30/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.30/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.30/0.46 % Source : [Des21]
% 0.30/0.46 % Names : Localization/prob_758__8980450_1 [Des21]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.00 v7.5.0
% 0.30/0.46 % Syntax : Number of formulae : 325 ( 88 unt; 56 typ; 0 def)
% 0.30/0.46 % Number of atoms : 692 ( 264 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 8 ( 2 avg)
% 0.30/0.46 % Number of connectives : 9320 ( 20 ~; 0 |; 8 &;8955 @)
% 0.30/0.46 % ( 0 <=>; 337 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 17 ( 8 avg)
% 0.30/0.46 % Number of types : 4 ( 3 usr)
% 0.30/0.46 % Number of type conns : 62 ( 62 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 54 ( 53 usr; 10 con; 0-6 aty)
% 0.30/0.46 % Number of variables : 709 ( 18 ^; 631 !; 4 ?; 709 :)
% 0.30/0.46 % ( 56 !>; 0 ?*; 0 @-; 0 @+)
% 0.30/0.46 % SPC : TH1_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:24:15.941
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % Could-be-implicit typings (8)
% 0.30/0.46 thf(ty_t_Congruence_Opartial__object_Opartial__object__ext,type,
% 0.30/0.46 partia1265057091ct_ext: $tType > $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_Group_Omonoid_Omonoid__ext,type,
% 0.30/0.46 monoid_ext: $tType > $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_Ring_Oring_Oring__ext,type,
% 0.30/0.46 ring_ext: $tType > $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_Product__Type_Oprod,type,
% 0.30/0.46 product_prod: $tType > $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_Set_Oset,type,
% 0.30/0.46 set: $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_Int_Oint,type,
% 0.30/0.46 int: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_tf_b,type,
% 0.30/0.46 b: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_tf_a,type,
% 0.30/0.46 a: $tType ).
% 0.30/0.46
% 0.30/0.46 % Explicit typings (48)
% 0.30/0.46 thf(sy_cl_Groups_Ogroup__add,type,
% 0.30/0.46 group_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Oab__group__add,type,
% 0.30/0.46 ab_group_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.30/0.46 semigroup_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.30/0.46 comm_monoid_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.30/0.46 ab_semigroup_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.30/0.46 linordered_semidom:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.30/0.46 cancel_semigroup_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.30/0.46 ordered_ab_group_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.30/0.46 cancel1352612707id_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.30/0.46 cancel146912293up_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.30/0.46 ordere779506340up_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.30/0.46 canoni770627133id_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.30/0.46 ordere236663937imp_le:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
% 0.30/0.46 ordere623563068d_diff:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_AbelCoset_Oa__l__coset,type,
% 0.30/0.46 a_l_coset:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > A > ( set @ A ) > ( set @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_AbelCoset_Oadditive__subgroup,type,
% 0.30/0.46 additive_subgroup:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_AbelCoset_Oset__add,type,
% 0.30/0.46 set_add:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > ( set @ A ) > ( set @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Congruence_Opartial__object_Ocarrier,type,
% 0.30/0.46 partial_carrier:
% 0.30/0.46 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ Z ) > ( set @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Group_Omonoid_Omult,type,
% 0.30/0.46 mult:
% 0.30/0.46 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ Z ) ) > A > A > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.30/0.46 minus_minus:
% 0.30/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.30/0.46 plus_plus:
% 0.30/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.30/0.46 uminus_uminus:
% 0.30/0.46 !>[A: $tType] : ( A > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Oeq__obj__rng__of__frac,type,
% 0.30/0.46 locali1648044335f_frac:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Omult__submonoid__of__crng,type,
% 0.30/0.46 locali7914339f_crng:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Omult__submonoid__of__rng,type,
% 0.30/0.46 locali1402578320of_rng:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Osubmonoid,type,
% 0.30/0.46 locali1441642228monoid:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > ( set @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.30/0.46 bot_bot:
% 0.30/0.46 !>[A: $tType] : A ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.30/0.46 ord_less_eq:
% 0.30/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.30/0.46 product_fst:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 0.30/0.46 product_snd:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Ring_Oa__inv,type,
% 0.30/0.46 a_inv:
% 0.30/0.46 !>[A: $tType,M: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ M ) ) ) > A > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Ring_Oa__minus,type,
% 0.30/0.46 a_minus:
% 0.30/0.46 !>[A: $tType,M: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ M ) ) ) > A > A > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Ring_Oadd__pow,type,
% 0.30/0.46 add_pow:
% 0.30/0.46 !>[A: $tType,C: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ C ) ) ) > B > A > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Ring_Oring_Oadd,type,
% 0.30/0.46 add:
% 0.30/0.46 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ) ) > A > A > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Ring_Oring_Ozero,type,
% 0.30/0.46 zero:
% 0.30/0.46 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ) ) > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Ring_Osemiring,type,
% 0.30/0.46 semiring:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_OCollect,type,
% 0.30/0.46 collect:
% 0.30/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_member,type,
% 0.30/0.47 member:
% 0.30/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_R,type,
% 0.30/0.47 r: partia1265057091ct_ext @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_S,type,
% 0.30/0.47 s: set @ a ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_r,type,
% 0.30/0.47 r2: a ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_r_H,type,
% 0.30/0.47 r3: a ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_s,type,
% 0.30/0.47 s2: a ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_s_H,type,
% 0.30/0.47 s3: a ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_t_H____,type,
% 0.30/0.47 t: a ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_t____,type,
% 0.30/0.47 t2: a ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_x_H____,type,
% 0.30/0.47 x: product_prod @ a @ a ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_y_H____,type,
% 0.30/0.47 y: product_prod @ a @ a ).
% 0.30/0.47
% 0.30/0.47 % Relevant facts (256)
% 0.30/0.47 thf(fact_0_f18,axiom,
% 0.30/0.47 member @ a @ t @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % f18
% 0.30/0.47 thf(fact_1_f17,axiom,
% 0.30/0.47 member @ a @ t2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % f17
% 0.30/0.47 thf(fact_2_right__add__eq,axiom,
% 0.30/0.47 ! [A2: a,B2: a,C2: a] :
% 0.30/0.47 ( ( A2 = B2 )
% 0.30/0.47 => ( ( add @ a @ b @ r @ C2 @ A2 )
% 0.30/0.47 = ( add @ a @ b @ r @ C2 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % right_add_eq
% 0.30/0.47 thf(fact_3_local_Oright__minus__eq,axiom,
% 0.30/0.47 ! [A2: a,B2: a,C2: a] :
% 0.30/0.47 ( ( A2 = B2 )
% 0.30/0.47 => ( ( a_minus @ a @ b @ r @ C2 @ A2 )
% 0.30/0.47 = ( a_minus @ a @ b @ r @ C2 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % local.right_minus_eq
% 0.30/0.47 thf(fact_4_m__assoc,axiom,
% 0.30/0.47 ! [X: a,Y: a,Z2: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ Z2 )
% 0.30/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % m_assoc
% 0.30/0.47 thf(fact_5_m__comm,axiom,
% 0.30/0.47 ! [X: a,Y: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y )
% 0.30/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % m_comm
% 0.30/0.47 thf(fact_6_m__lcomm,axiom,
% 0.30/0.47 ! [X: a,Y: a,Z2: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) )
% 0.30/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Z2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % m_lcomm
% 0.30/0.47 thf(fact_7_add_Om__assoc,axiom,
% 0.30/0.47 ! [X: a,Y: a,Z2: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( add @ a @ b @ r @ ( add @ a @ b @ r @ X @ Y ) @ Z2 )
% 0.30/0.47 = ( add @ a @ b @ r @ X @ ( add @ a @ b @ r @ Y @ Z2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.m_assoc
% 0.30/0.47 thf(fact_8_add_Om__comm,axiom,
% 0.30/0.47 ! [X: a,Y: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( add @ a @ b @ r @ X @ Y )
% 0.30/0.47 = ( add @ a @ b @ r @ Y @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.m_comm
% 0.30/0.47 thf(fact_9_add_Om__lcomm,axiom,
% 0.30/0.47 ! [X: a,Y: a,Z2: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( add @ a @ b @ r @ X @ ( add @ a @ b @ r @ Y @ Z2 ) )
% 0.30/0.47 = ( add @ a @ b @ r @ Y @ ( add @ a @ b @ r @ X @ Z2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.m_lcomm
% 0.30/0.47 thf(fact_10_f20,axiom,
% 0.30/0.47 member @ a @ s3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % f20
% 0.30/0.47 thf(fact_11_f19,axiom,
% 0.30/0.47 member @ a @ s2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % f19
% 0.30/0.47 thf(fact_12_f41,axiom,
% 0.30/0.47 member @ a @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % f41
% 0.30/0.47 thf(fact_13_f42,axiom,
% 0.30/0.47 member @ a @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % f42
% 0.30/0.47 thf(fact_14_l__distr,axiom,
% 0.30/0.47 ! [X: a,Y: a,Z2: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add @ a @ b @ r @ X @ Y ) @ Z2 )
% 0.30/0.47 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Z2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % l_distr
% 0.30/0.47 thf(fact_15_r__distr,axiom,
% 0.30/0.47 ! [X: a,Y: a,Z2: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Z2 @ ( add @ a @ b @ r @ X @ Y ) )
% 0.30/0.47 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Z2 @ X ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Z2 @ Y ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % r_distr
% 0.30/0.47 thf(fact_16_four__elem__comm,axiom,
% 0.30/0.47 ! [A2: a,B2: a,C2: a,D: a] :
% 0.30/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ D @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( a_minus @ a @ b @ r @ ( add @ a @ b @ r @ ( a_minus @ a @ b @ r @ A2 @ C2 ) @ B2 ) @ D )
% 0.30/0.47 = ( a_minus @ a @ b @ r @ ( a_minus @ a @ b @ r @ ( add @ a @ b @ r @ A2 @ B2 ) @ C2 ) @ D ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % four_elem_comm
% 0.30/0.47 thf(fact_17_right__inv__add,axiom,
% 0.30/0.47 ! [A2: a,B2: a,C2: a] :
% 0.30/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( a_minus @ a @ b @ r @ ( a_minus @ a @ b @ r @ C2 @ A2 ) @ B2 )
% 0.30/0.47 = ( a_minus @ a @ b @ r @ C2 @ ( add @ a @ b @ r @ A2 @ B2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % right_inv_add
% 0.30/0.47 thf(fact_18_f21,axiom,
% 0.30/0.47 member @ a @ ( product_snd @ a @ a @ y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % f21
% 0.30/0.47 thf(fact_19_f40,axiom,
% 0.30/0.47 ( ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) ) )
% 0.30/0.47 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % f40
% 0.30/0.47 thf(fact_20_f22,axiom,
% 0.30/0.47 member @ a @ ( product_fst @ a @ a @ x ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % f22
% 0.30/0.47 thf(fact_21__092_060open_062_092_060lbrakk_062snd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_A_092_060in_062_Acarrier_AR_059_Asnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_A_092_060in_062_Acarrier_AR_059_At_A_092_060otimes_062_At_H_A_092_060in_062_Acarrier_AR_092_060rbrakk_062_A_092_060Longrightarrow_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_A_092_060oplus_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_092_060close_062,axiom,
% 0.30/0.47 ( ( member @ a @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( add @ a @ b @ r @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) )
% 0.30/0.47 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>\<lbrakk>snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r) \<ominus> s \<otimes> s' \<otimes> (snd y' \<otimes> fst x') \<in> carrier R; snd x' \<otimes> snd y' \<otimes> (s \<otimes> r') \<ominus> s \<otimes> s' \<otimes> (snd x' \<otimes> fst y') \<in> carrier R; t \<otimes> t' \<in> carrier R\<rbrakk> \<Longrightarrow> t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r) \<ominus> s \<otimes> s' \<otimes> (snd y' \<otimes> fst x') \<oplus> (snd x' \<otimes> snd y' \<otimes> (s \<otimes> r') \<ominus> s \<otimes> s' \<otimes> (snd x' \<otimes> fst y'))) = t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r) \<ominus> s \<otimes> s' \<otimes> (snd y' \<otimes> fst x')) \<oplus> t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s \<otimes> r') \<ominus> s \<otimes> s' \<otimes> (snd x' \<otimes> fst y'))\<close>
% 0.30/0.47 thf(fact_22_f15,axiom,
% 0.30/0.47 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % f15
% 0.30/0.47 thf(fact_23__092_060open_062snd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060in_062_Acarrier_AR_092_060close_062,axiom,
% 0.30/0.47 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>snd x' \<otimes> snd y' \<in> carrier R\<close>
% 0.30/0.47 thf(fact_24_f28,axiom,
% 0.30/0.47 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % f28
% 0.30/0.47 thf(fact_25_f11,axiom,
% 0.30/0.47 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % f11
% 0.30/0.47 thf(fact_26_f10,axiom,
% 0.30/0.47 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % f10
% 0.30/0.47 thf(fact_27_f27,axiom,
% 0.30/0.47 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % f27
% 0.30/0.47 thf(fact_28_f35,axiom,
% 0.30/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) )
% 0.30/0.47 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % f35
% 0.30/0.47 thf(fact_29__092_060open_062t_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_A_092_060ominus_062_As_A_092_060otimes_062_Afst_Ax_H_J_A_061_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_At_A_092_060otimes_062_A_Is_A_092_060otimes_062_Afst_Ax_H_J_092_060close_062,axiom,
% 0.30/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) )
% 0.30/0.47 = ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>t \<otimes> (snd x' \<otimes> r \<ominus> s \<otimes> fst x') = t \<otimes> (snd x' \<otimes> r) \<ominus> t \<otimes> (s \<otimes> fst x')\<close>
% 0.30/0.47 thf(fact_30__092_060open_062t_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_A_092_060ominus_062_As_H_A_092_060otimes_062_Afst_Ay_H_J_A_061_At_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_At_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Afst_Ay_H_J_092_060close_062,axiom,
% 0.30/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) )
% 0.30/0.47 = ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>t' \<otimes> (snd y' \<otimes> r' \<ominus> s' \<otimes> fst y') = t' \<otimes> (snd y' \<otimes> r') \<ominus> t' \<otimes> (s' \<otimes> fst y')\<close>
% 0.30/0.47 thf(fact_31_f16,axiom,
% 0.30/0.47 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % f16
% 0.30/0.47 thf(fact_32_f24,axiom,
% 0.30/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) )
% 0.30/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ t2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % f24
% 0.30/0.47 thf(fact_33_f9,axiom,
% 0.30/0.47 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % f9
% 0.30/0.47 thf(fact_34__092_060open_062t_A_092_060otimes_062_As_A_092_060otimes_062_Asnd_Ax_H_A_092_060in_062_Acarrier_AR_092_060close_062,axiom,
% 0.30/0.47 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>t \<otimes> s \<otimes> snd x' \<in> carrier R\<close>
% 0.30/0.47 thf(fact_35_f23,axiom,
% 0.30/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) )
% 0.30/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ t2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % f23
% 0.30/0.47 thf(fact_36_f34,axiom,
% 0.30/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) )
% 0.30/0.47 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % f34
% 0.30/0.47 thf(fact_37_f36,axiom,
% 0.30/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) )
% 0.30/0.47 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % f36
% 0.30/0.47 thf(fact_38_f13,axiom,
% 0.30/0.47 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % f13
% 0.30/0.47 thf(fact_39__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_061_At_H_A_092_060otimes_062_As_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_At_H_A_092_060otimes_062_As_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_At_A_092_060otimes_062_A_Is_A_092_060otimes_062_Afst_Ax_H_J_092_060close_062,axiom,
% 0.30/0.47 ( ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) )
% 0.30/0.47 = ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ t2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ t2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r)) \<ominus> t \<otimes> t' \<otimes> (s \<otimes> s' \<otimes> (snd y' \<otimes> fst x')) = t' \<otimes> s' \<otimes> snd y' \<otimes> t \<otimes> (snd x' \<otimes> r) \<ominus> t' \<otimes> s' \<otimes> snd y' \<otimes> t \<otimes> (s \<otimes> fst x')\<close>
% 0.30/0.47 thf(fact_40_f12,axiom,
% 0.30/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) )
% 0.30/0.47 = ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ t2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ t2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % f12
% 0.30/0.47 thf(fact_41_f25,axiom,
% 0.30/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) )
% 0.30/0.47 = ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % f25
% 0.30/0.47 thf(fact_42_f32,axiom,
% 0.30/0.47 ( ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) )
% 0.30/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % f32
% 0.30/0.47 thf(fact_43_f26,axiom,
% 0.30/0.47 ( ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) )
% 0.30/0.47 = ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % f26
% 0.30/0.47 thf(fact_44_f29,axiom,
% 0.30/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) )
% 0.30/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % f29
% 0.30/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,P: A > $o] :
% 0.30/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.30/0.47 = ( P @ A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % mem_Collect_eq
% 0.30/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A] :
% 0.30/0.47 ( ( collect @ A
% 0.30/0.47 @ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
% 0.30/0.47 = A3 ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_mem_eq
% 0.30/0.47 thf(fact_47_Collect__cong,axiom,
% 0.30/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.47 ( ! [X3: A] :
% 0.30/0.47 ( ( P @ X3 )
% 0.30/0.47 = ( Q @ X3 ) )
% 0.30/0.47 => ( ( collect @ A @ P )
% 0.30/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_cong
% 0.30/0.47 thf(fact_48_f30,axiom,
% 0.30/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) )
% 0.30/0.47 = ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % f30
% 0.30/0.47 thf(fact_49_f33,axiom,
% 0.30/0.47 ( ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) )
% 0.30/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % f33
% 0.30/0.47 thf(fact_50__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_J_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_J_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_092_060close_062,axiom,
% 0.30/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) )
% 0.30/0.47 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r \<oplus> s \<otimes> r')) = t \<otimes> t' \<otimes> (snd x' \<otimes> snd y') \<otimes> (s' \<otimes> r) \<oplus> t \<otimes> t' \<otimes> (snd x' \<otimes> snd y') \<otimes> (s \<otimes> r')\<close>
% 0.30/0.47 thf(fact_51_f14,axiom,
% 0.30/0.47 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % f14
% 0.30/0.47 thf(fact_52__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_092_060close_062,axiom,
% 0.30/0.47 ( ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) )
% 0.30/0.47 = ( a_minus @ a @ b @ r @ ( add @ a @ b @ r @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r) \<ominus> s \<otimes> s' \<otimes> (snd y' \<otimes> fst x')) \<oplus> t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s \<otimes> r') \<ominus> s \<otimes> s' \<otimes> (snd x' \<otimes> fst y')) = t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r)) \<ominus> t \<otimes> t' \<otimes> (s \<otimes> s' \<otimes> (snd y' \<otimes> fst x')) \<oplus> t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s \<otimes> r')) \<ominus> t \<otimes> t' \<otimes> (s \<otimes> s' \<otimes> (snd x' \<otimes> fst y'))\<close>
% 0.30/0.47 thf(fact_53_f31,axiom,
% 0.30/0.47 ( ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) ) )
% 0.30/0.47 = ( add @ a @ b @ r @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % f31
% 0.30/0.47 thf(fact_54_f37,axiom,
% 0.30/0.47 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % f37
% 0.30/0.47 thf(fact_55_f39,axiom,
% 0.30/0.47 member @ a @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % f39
% 0.30/0.47 thf(fact_56_f38,axiom,
% 0.30/0.47 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % f38
% 0.30/0.47 thf(fact_57_f5,axiom,
% 0.30/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) )
% 0.30/0.47 = ( zero @ a @ b @ r ) ) ).
% 0.30/0.47
% 0.30/0.47 % f5
% 0.30/0.47 thf(fact_58_f7,axiom,
% 0.30/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) )
% 0.30/0.47 = ( zero @ a @ b @ r ) ) ).
% 0.30/0.47
% 0.30/0.47 % f7
% 0.30/0.47 thf(fact_59_minus__closed,axiom,
% 0.30/0.47 ! [X: a,Y: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( member @ a @ ( a_minus @ a @ b @ r @ X @ Y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minus_closed
% 0.30/0.47 thf(fact_60_add_Om__closed,axiom,
% 0.30/0.47 ! [X: a,Y: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( member @ a @ ( add @ a @ b @ r @ X @ Y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.m_closed
% 0.30/0.47 thf(fact_61_local_Oadd_Oright__cancel,axiom,
% 0.30/0.47 ! [X: a,Y: a,Z2: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( ( add @ a @ b @ r @ Y @ X )
% 0.30/0.47 = ( add @ a @ b @ r @ Z2 @ X ) )
% 0.30/0.47 = ( Y = Z2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % local.add.right_cancel
% 0.30/0.47 thf(fact_62_semiring__simprules_I3_J,axiom,
% 0.30/0.47 ! [X: a,Y: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % semiring_simprules(3)
% 0.30/0.47 thf(fact_63_f4,axiom,
% 0.30/0.47 member @ a @ t2 @ s ).
% 0.30/0.47
% 0.30/0.47 % f4
% 0.30/0.47 thf(fact_64_f6,axiom,
% 0.30/0.47 member @ a @ t @ s ).
% 0.30/0.47
% 0.30/0.47 % f6
% 0.30/0.47 thf(fact_65_f8,axiom,
% 0.30/0.47 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ s ).
% 0.30/0.47
% 0.30/0.47 % f8
% 0.30/0.47 thf(fact_66_local_Osemiring__axioms,axiom,
% 0.30/0.47 semiring @ a @ b @ r ).
% 0.30/0.47
% 0.30/0.47 % local.semiring_axioms
% 0.30/0.47 thf(fact_67_minus__eq,axiom,
% 0.30/0.47 ! [X: a,Y: a] :
% 0.30/0.47 ( ( a_minus @ a @ b @ r @ X @ Y )
% 0.30/0.47 = ( add @ a @ b @ r @ X @ ( a_inv @ a @ b @ r @ Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minus_eq
% 0.30/0.47 thf(fact_68_prod__eqI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 0.30/0.47 ( ( ( product_fst @ A @ B @ P2 )
% 0.30/0.47 = ( product_fst @ A @ B @ Q2 ) )
% 0.30/0.47 => ( ( ( product_snd @ A @ B @ P2 )
% 0.30/0.47 = ( product_snd @ A @ B @ Q2 ) )
% 0.30/0.47 => ( P2 = Q2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_eqI
% 0.30/0.47 thf(fact_69_exE__realizer_H,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,P: A > B > $o,P2: product_prod @ B @ A] :
% 0.30/0.47 ( ( P @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
% 0.30/0.47 => ~ ! [X3: B,Y2: A] :
% 0.30/0.47 ~ ( P @ Y2 @ X3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % exE_realizer'
% 0.30/0.47 thf(fact_70_prod_Oexpand,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 0.30/0.47 ( ( ( ( product_fst @ A @ B @ Prod )
% 0.30/0.47 = ( product_fst @ A @ B @ Prod2 ) )
% 0.30/0.47 & ( ( product_snd @ A @ B @ Prod )
% 0.30/0.47 = ( product_snd @ A @ B @ Prod2 ) ) )
% 0.30/0.47 => ( Prod = Prod2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod.expand
% 0.30/0.47 thf(fact_71_local_Ominus__unique,axiom,
% 0.30/0.47 ! [Y: a,X: a,Y3: a] :
% 0.30/0.47 ( ( ( add @ a @ b @ r @ Y @ X )
% 0.30/0.47 = ( zero @ a @ b @ r ) )
% 0.30/0.47 => ( ( ( add @ a @ b @ r @ X @ Y3 )
% 0.30/0.47 = ( zero @ a @ b @ r ) )
% 0.30/0.47 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( Y = Y3 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % local.minus_unique
% 0.30/0.47 thf(fact_72_add_Or__inv__ex,axiom,
% 0.30/0.47 ! [X: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ? [X3: a] :
% 0.30/0.47 ( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 & ( ( add @ a @ b @ r @ X @ X3 )
% 0.30/0.47 = ( zero @ a @ b @ r ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.r_inv_ex
% 0.30/0.47 thf(fact_73_add_Oone__unique,axiom,
% 0.30/0.47 ! [U: a] :
% 0.30/0.47 ( ( member @ a @ U @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ! [X3: a] :
% 0.30/0.47 ( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( add @ a @ b @ r @ U @ X3 )
% 0.30/0.47 = X3 ) )
% 0.30/0.47 => ( U
% 0.30/0.47 = ( zero @ a @ b @ r ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.one_unique
% 0.30/0.47 thf(fact_74_add_Ol__inv__ex,axiom,
% 0.30/0.47 ! [X: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ? [X3: a] :
% 0.30/0.47 ( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 & ( ( add @ a @ b @ r @ X3 @ X )
% 0.30/0.47 = ( zero @ a @ b @ r ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.l_inv_ex
% 0.30/0.47 thf(fact_75_add_Oinv__comm,axiom,
% 0.30/0.47 ! [X: a,Y: a] :
% 0.30/0.47 ( ( ( add @ a @ b @ r @ X @ Y )
% 0.30/0.47 = ( zero @ a @ b @ r ) )
% 0.30/0.47 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( add @ a @ b @ r @ Y @ X )
% 0.30/0.47 = ( zero @ a @ b @ r ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.inv_comm
% 0.30/0.47 thf(fact_76_r__minus,axiom,
% 0.30/0.47 ! [X: a,Y: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( a_inv @ a @ b @ r @ Y ) )
% 0.30/0.47 = ( a_inv @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % r_minus
% 0.30/0.47 thf(fact_77_l__minus,axiom,
% 0.30/0.47 ! [X: a,Y: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( a_inv @ a @ b @ r @ X ) @ Y )
% 0.30/0.47 = ( a_inv @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % l_minus
% 0.30/0.47 thf(fact_78_r__neg2,axiom,
% 0.30/0.47 ! [X: a,Y: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( add @ a @ b @ r @ X @ ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ Y ) )
% 0.30/0.47 = Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % r_neg2
% 0.30/0.47 thf(fact_79_r__neg1,axiom,
% 0.30/0.47 ! [X: a,Y: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ ( add @ a @ b @ r @ X @ Y ) )
% 0.30/0.47 = Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % r_neg1
% 0.30/0.47 thf(fact_80_local_Ominus__add,axiom,
% 0.30/0.47 ! [X: a,Y: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( a_inv @ a @ b @ r @ ( add @ a @ b @ r @ X @ Y ) )
% 0.30/0.47 = ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ ( a_inv @ a @ b @ r @ Y ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % local.minus_add
% 0.30/0.47 thf(fact_81_add_Oinv__solve__right_H,axiom,
% 0.30/0.47 ! [A2: a,B2: a,C2: a] :
% 0.30/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( ( add @ a @ b @ r @ B2 @ ( a_inv @ a @ b @ r @ C2 ) )
% 0.30/0.47 = A2 )
% 0.30/0.47 = ( B2
% 0.30/0.47 = ( add @ a @ b @ r @ A2 @ C2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.inv_solve_right'
% 0.30/0.47 thf(fact_82_add_Oinv__solve__right,axiom,
% 0.30/0.47 ! [A2: a,B2: a,C2: a] :
% 0.30/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( A2
% 0.30/0.47 = ( add @ a @ b @ r @ B2 @ ( a_inv @ a @ b @ r @ C2 ) ) )
% 0.30/0.47 = ( B2
% 0.30/0.47 = ( add @ a @ b @ r @ A2 @ C2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.inv_solve_right
% 0.30/0.47 thf(fact_83_add_Oinv__solve__left_H,axiom,
% 0.30/0.47 ! [A2: a,B2: a,C2: a] :
% 0.30/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ B2 ) @ C2 )
% 0.30/0.47 = A2 )
% 0.30/0.47 = ( C2
% 0.30/0.47 = ( add @ a @ b @ r @ B2 @ A2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.inv_solve_left'
% 0.30/0.47 thf(fact_84_add_Oinv__solve__left,axiom,
% 0.30/0.47 ! [A2: a,B2: a,C2: a] :
% 0.30/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( A2
% 0.30/0.47 = ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ B2 ) @ C2 ) )
% 0.30/0.47 = ( C2
% 0.30/0.47 = ( add @ a @ b @ r @ B2 @ A2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.inv_solve_left
% 0.30/0.47 thf(fact_85_add_Oinv__mult__group,axiom,
% 0.30/0.47 ! [X: a,Y: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( a_inv @ a @ b @ r @ ( add @ a @ b @ r @ X @ Y ) )
% 0.30/0.47 = ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ Y ) @ ( a_inv @ a @ b @ r @ X ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.inv_mult_group
% 0.30/0.47 thf(fact_86_a__transpose__inv,axiom,
% 0.30/0.47 ! [X: a,Y: a,Z2: a] :
% 0.30/0.47 ( ( ( add @ a @ b @ r @ X @ Y )
% 0.30/0.47 = Z2 )
% 0.30/0.47 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ Z2 )
% 0.30/0.47 = Y ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % a_transpose_inv
% 0.30/0.47 thf(fact_87_eq__obj__rng__of__frac__axioms,axiom,
% 0.30/0.47 locali1648044335f_frac @ a @ b @ r @ s ).
% 0.30/0.47
% 0.30/0.47 % eq_obj_rng_of_frac_axioms
% 0.30/0.47 thf(fact_88_sum__zero__eq__neg,axiom,
% 0.30/0.47 ! [X: a,Y: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( ( add @ a @ b @ r @ X @ Y )
% 0.30/0.47 = ( zero @ a @ b @ r ) )
% 0.30/0.47 => ( X
% 0.30/0.47 = ( a_inv @ a @ b @ r @ Y ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sum_zero_eq_neg
% 0.30/0.47 thf(fact_89_r__neg,axiom,
% 0.30/0.47 ! [X: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( add @ a @ b @ r @ X @ ( a_inv @ a @ b @ r @ X ) )
% 0.30/0.47 = ( zero @ a @ b @ r ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % r_neg
% 0.30/0.47 thf(fact_90_minus__equality,axiom,
% 0.30/0.47 ! [Y: a,X: a] :
% 0.30/0.47 ( ( ( add @ a @ b @ r @ Y @ X )
% 0.30/0.47 = ( zero @ a @ b @ r ) )
% 0.30/0.47 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( a_inv @ a @ b @ r @ X )
% 0.30/0.47 = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minus_equality
% 0.30/0.47 thf(fact_91_l__neg,axiom,
% 0.30/0.47 ! [X: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ X )
% 0.30/0.47 = ( zero @ a @ b @ r ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % l_neg
% 0.30/0.47 thf(fact_92_inv__add,axiom,
% 0.30/0.47 ! [A2: a,B2: a] :
% 0.30/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( a_inv @ a @ b @ r @ ( add @ a @ b @ r @ A2 @ B2 ) )
% 0.30/0.47 = ( a_minus @ a @ b @ r @ ( a_inv @ a @ b @ r @ A2 ) @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % inv_add
% 0.30/0.47 thf(fact_93__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_O_A_092_060lbrakk_062t_A_092_060in_062_AS_059_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_A_092_060ominus_062_As_A_092_060otimes_062_Afst_Ax_H_J_A_061_A_092_060zero_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.30/0.47 ~ ! [T: a] :
% 0.30/0.47 ( ( member @ a @ T @ s )
% 0.30/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ T @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) )
% 0.30/0.47 != ( zero @ a @ b @ r ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>\<And>thesis. (\<And>t. \<lbrakk>t \<in> S; t \<otimes> (snd x' \<otimes> r \<ominus> s \<otimes> fst x') = \<zero>\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.30/0.47 thf(fact_94__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_H_O_A_092_060lbrakk_062t_H_A_092_060in_062_AS_059_At_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_A_092_060ominus_062_As_H_A_092_060otimes_062_Afst_Ay_H_J_A_061_A_092_060zero_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.30/0.47 ~ ! [T2: a] :
% 0.30/0.47 ( ( member @ a @ T2 @ s )
% 0.30/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ T2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) )
% 0.30/0.47 != ( zero @ a @ b @ r ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>\<And>thesis. (\<And>t'. \<lbrakk>t' \<in> S; t' \<otimes> (snd y' \<otimes> r' \<ominus> s' \<otimes> fst y') = \<zero>\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.30/0.47 thf(fact_95_zero__closed,axiom,
% 0.30/0.47 member @ a @ ( zero @ a @ b @ r ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % zero_closed
% 0.30/0.47 thf(fact_96_local_Ominus__minus,axiom,
% 0.30/0.47 ! [X: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( a_inv @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) )
% 0.30/0.47 = X ) ) ).
% 0.30/0.47
% 0.30/0.47 % local.minus_minus
% 0.30/0.47 thf(fact_97_add_Oinv__closed,axiom,
% 0.30/0.47 ! [X: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( member @ a @ ( a_inv @ a @ b @ r @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.inv_closed
% 0.30/0.47 thf(fact_98_local_Ominus__zero,axiom,
% 0.30/0.47 ( ( a_inv @ a @ b @ r @ ( zero @ a @ b @ r ) )
% 0.30/0.47 = ( zero @ a @ b @ r ) ) ).
% 0.30/0.47
% 0.30/0.47 % local.minus_zero
% 0.30/0.47 thf(fact_99_m__closed,axiom,
% 0.30/0.47 ! [X: a,Y: a] :
% 0.30/0.47 ( ( member @ a @ X @ s )
% 0.30/0.47 => ( ( member @ a @ Y @ s )
% 0.30/0.47 => ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ s ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % m_closed
% 0.30/0.47 thf(fact_100_r__null,axiom,
% 0.30/0.47 ! [X: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( zero @ a @ b @ r ) )
% 0.30/0.47 = ( zero @ a @ b @ r ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % r_null
% 0.30/0.47 thf(fact_101_l__null,axiom,
% 0.30/0.47 ! [X: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( zero @ a @ b @ r ) @ X )
% 0.30/0.47 = ( zero @ a @ b @ r ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % l_null
% 0.30/0.47 thf(fact_102_r__zero,axiom,
% 0.30/0.47 ! [X: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( add @ a @ b @ r @ X @ ( zero @ a @ b @ r ) )
% 0.30/0.47 = X ) ) ).
% 0.30/0.47
% 0.30/0.47 % r_zero
% 0.30/0.47 thf(fact_103_l__zero,axiom,
% 0.30/0.47 ! [X: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( add @ a @ b @ r @ ( zero @ a @ b @ r ) @ X )
% 0.30/0.47 = X ) ) ).
% 0.30/0.47
% 0.30/0.47 % l_zero
% 0.30/0.47 thf(fact_104_add_Or__cancel__one_H,axiom,
% 0.30/0.47 ! [X: a,A2: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( X
% 0.30/0.47 = ( add @ a @ b @ r @ A2 @ X ) )
% 0.30/0.47 = ( A2
% 0.30/0.47 = ( zero @ a @ b @ r ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.r_cancel_one'
% 0.30/0.47 thf(fact_105_add_Or__cancel__one,axiom,
% 0.30/0.47 ! [X: a,A2: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( ( add @ a @ b @ r @ A2 @ X )
% 0.30/0.47 = X )
% 0.30/0.47 = ( A2
% 0.30/0.47 = ( zero @ a @ b @ r ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.r_cancel_one
% 0.30/0.47 thf(fact_106_add_Ol__cancel__one_H,axiom,
% 0.30/0.47 ! [X: a,A2: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( X
% 0.30/0.47 = ( add @ a @ b @ r @ X @ A2 ) )
% 0.30/0.47 = ( A2
% 0.30/0.47 = ( zero @ a @ b @ r ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.l_cancel_one'
% 0.30/0.47 thf(fact_107_add_Ol__cancel__one,axiom,
% 0.30/0.47 ! [X: a,A2: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( ( add @ a @ b @ r @ X @ A2 )
% 0.30/0.47 = X )
% 0.30/0.47 = ( A2
% 0.30/0.47 = ( zero @ a @ b @ r ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.l_cancel_one
% 0.30/0.47 thf(fact_108_add_Oinv__eq__1__iff,axiom,
% 0.30/0.47 ! [X: a] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( ( a_inv @ a @ b @ r @ X )
% 0.30/0.47 = ( zero @ a @ b @ r ) )
% 0.30/0.47 = ( X
% 0.30/0.47 = ( zero @ a @ b @ r ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.inv_eq_1_iff
% 0.30/0.47 thf(fact_109_mult__submonoid__of__rng__axioms,axiom,
% 0.30/0.47 locali1402578320of_rng @ a @ b @ r @ s ).
% 0.30/0.47
% 0.30/0.47 % mult_submonoid_of_rng_axioms
% 0.30/0.47 thf(fact_110_mult__submonoid__of__crng__axioms,axiom,
% 0.30/0.47 locali7914339f_crng @ a @ b @ r @ s ).
% 0.30/0.47
% 0.30/0.47 % mult_submonoid_of_crng_axioms
% 0.30/0.47 thf(fact_111_prod__eq__iff,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ^ [Y4: product_prod @ A @ B,Z3: product_prod @ A @ B] : ( Y4 = Z3 ) )
% 0.30/0.47 = ( ^ [S: product_prod @ A @ B,T3: product_prod @ A @ B] :
% 0.30/0.47 ( ( ( product_fst @ A @ B @ S )
% 0.30/0.47 = ( product_fst @ A @ B @ T3 ) )
% 0.30/0.47 & ( ( product_snd @ A @ B @ S )
% 0.30/0.47 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_eq_iff
% 0.30/0.47 thf(fact_112_submonoid__axioms,axiom,
% 0.30/0.47 locali1441642228monoid @ a @ ( ring_ext @ a @ b ) @ r @ s ).
% 0.30/0.47
% 0.30/0.47 % submonoid_axioms
% 0.30/0.47 thf(fact_113_add_Oint__pow__inv,axiom,
% 0.30/0.47 ! [X: a,I: int] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( add_pow @ a @ b @ int @ r @ I @ ( a_inv @ a @ b @ r @ X ) )
% 0.30/0.47 = ( a_inv @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.int_pow_inv
% 0.30/0.47 thf(fact_114_semiring_Osemiring__simprules_I11_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.30/0.47 ( ( semiring @ A @ B @ R )
% 0.30/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( add @ A @ B @ R @ X @ ( zero @ A @ B @ R ) )
% 0.30/0.47 = X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % semiring.semiring_simprules(11)
% 0.30/0.47 thf(fact_115_semiring_Osemiring__simprules_I6_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.30/0.47 ( ( semiring @ A @ B @ R )
% 0.30/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( add @ A @ B @ R @ ( zero @ A @ B @ R ) @ X )
% 0.30/0.47 = X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % semiring.semiring_simprules(6)
% 0.30/0.47 thf(fact_116_semiring_Ol__null,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.30/0.47 ( ( semiring @ A @ B @ R )
% 0.30/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R @ ( zero @ A @ B @ R ) @ X )
% 0.30/0.47 = ( zero @ A @ B @ R ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % semiring.l_null
% 0.30/0.47 thf(fact_117_semiring_Or__null,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.30/0.47 ( ( semiring @ A @ B @ R )
% 0.30/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R @ X @ ( zero @ A @ B @ R ) )
% 0.30/0.47 = ( zero @ A @ B @ R ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % semiring.r_null
% 0.30/0.47 thf(fact_118_semiring_Or__distr,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 0.30/0.47 ( ( semiring @ A @ B @ R )
% 0.30/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R @ Z2 @ ( add @ A @ B @ R @ X @ Y ) )
% 0.30/0.47 = ( add @ A @ B @ R @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ Z2 @ X ) @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ Z2 @ Y ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % semiring.r_distr
% 0.30/0.47 thf(fact_119_add__pow__ldistr__int,axiom,
% 0.30/0.47 ! [A2: a,B2: a,K: int] :
% 0.30/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add_pow @ a @ b @ int @ r @ K @ A2 ) @ B2 )
% 0.30/0.47 = ( add_pow @ a @ b @ int @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_pow_ldistr_int
% 0.30/0.47 thf(fact_120_add__pow__rdistr__int,axiom,
% 0.30/0.47 ! [A2: a,B2: a,K: int] :
% 0.30/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ ( add_pow @ a @ b @ int @ r @ K @ B2 ) )
% 0.30/0.47 = ( add_pow @ a @ b @ int @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_pow_rdistr_int
% 0.30/0.47 thf(fact_121_add_Oint__pow__distrib,axiom,
% 0.30/0.47 ! [X: a,Y: a,I: int] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( add_pow @ a @ b @ int @ r @ I @ ( add @ a @ b @ r @ X @ Y ) )
% 0.30/0.47 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( add_pow @ a @ b @ int @ r @ I @ Y ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.int_pow_distrib
% 0.30/0.47 thf(fact_122_add_Oint__pow__mult__distrib,axiom,
% 0.30/0.47 ! [X: a,Y: a,I: int] :
% 0.30/0.47 ( ( ( add @ a @ b @ r @ X @ Y )
% 0.30/0.47 = ( add @ a @ b @ r @ Y @ X ) )
% 0.30/0.47 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( add_pow @ a @ b @ int @ r @ I @ ( add @ a @ b @ r @ X @ Y ) )
% 0.30/0.47 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( add_pow @ a @ b @ int @ r @ I @ Y ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.int_pow_mult_distrib
% 0.30/0.47 thf(fact_123_add_Oint__pow__closed,axiom,
% 0.30/0.47 ! [X: a,I: int] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( member @ a @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.int_pow_closed
% 0.30/0.47 thf(fact_124_add_Oint__pow__one,axiom,
% 0.30/0.47 ! [Z2: int] :
% 0.30/0.47 ( ( add_pow @ a @ b @ int @ r @ Z2 @ ( zero @ a @ b @ r ) )
% 0.30/0.47 = ( zero @ a @ b @ r ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.int_pow_one
% 0.30/0.47 thf(fact_125_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Ois__submonoid,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,M2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S2: set @ A] :
% 0.30/0.47 ( ( locali1441642228monoid @ A @ B @ M2 @ S2 )
% 0.30/0.47 => ( locali1441642228monoid @ A @ B @ M2 @ S2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Localization_Mirabelle_ojffvtlkod.submonoid.is_submonoid
% 0.30/0.47 thf(fact_126_mult__submonoid__of__crng_Oaxioms_I2_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S2: set @ A] :
% 0.30/0.47 ( ( locali7914339f_crng @ A @ B @ R @ S2 )
% 0.30/0.47 => ( locali1402578320of_rng @ A @ B @ R @ S2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_submonoid_of_crng.axioms(2)
% 0.30/0.47 thf(fact_127_mult__submonoid__of__rng_Oaxioms_I2_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S2: set @ A] :
% 0.30/0.47 ( ( locali1402578320of_rng @ A @ B @ R @ S2 )
% 0.30/0.47 => ( locali1441642228monoid @ A @ ( ring_ext @ A @ B ) @ R @ S2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_submonoid_of_rng.axioms(2)
% 0.30/0.47 thf(fact_128_eq__obj__rng__of__frac_Oaxioms_I2_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S2: set @ A] :
% 0.30/0.47 ( ( locali1648044335f_frac @ A @ B @ R @ S2 )
% 0.30/0.47 => ( locali7914339f_crng @ A @ B @ R @ S2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % eq_obj_rng_of_frac.axioms(2)
% 0.30/0.47 thf(fact_129_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Om__closed,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,M2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S2: set @ A,X: A,Y: A] :
% 0.30/0.47 ( ( locali1441642228monoid @ A @ B @ M2 @ S2 )
% 0.30/0.47 => ( ( member @ A @ X @ S2 )
% 0.30/0.47 => ( ( member @ A @ Y @ S2 )
% 0.30/0.47 => ( member @ A @ ( mult @ A @ B @ M2 @ X @ Y ) @ S2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Localization_Mirabelle_ojffvtlkod.submonoid.m_closed
% 0.30/0.47 thf(fact_130_a__minus__def,axiom,
% 0.30/0.47 ! [M: $tType,A: $tType] :
% 0.30/0.47 ( ( a_minus @ A @ M )
% 0.30/0.47 = ( ^ [R2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ M ) ),X2: A,Y5: A] : ( add @ A @ M @ R2 @ X2 @ ( a_inv @ A @ M @ R2 @ Y5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % a_minus_def
% 0.30/0.47 thf(fact_131_semiring_Osemiring__simprules_I8_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 0.30/0.47 ( ( semiring @ A @ B @ R )
% 0.30/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ X @ Y ) @ Z2 )
% 0.30/0.47 = ( mult @ A @ ( ring_ext @ A @ B ) @ R @ X @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ Y @ Z2 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % semiring.semiring_simprules(8)
% 0.30/0.47 thf(fact_132_semiring_Osemiring__simprules_I3_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 0.30/0.47 ( ( semiring @ A @ B @ R )
% 0.30/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( member @ A @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % semiring.semiring_simprules(3)
% 0.30/0.47 thf(fact_133_semiring_Osemiring__simprules_I12_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 0.30/0.47 ( ( semiring @ A @ B @ R )
% 0.30/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( add @ A @ B @ R @ X @ ( add @ A @ B @ R @ Y @ Z2 ) )
% 0.30/0.47 = ( add @ A @ B @ R @ Y @ ( add @ A @ B @ R @ X @ Z2 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % semiring.semiring_simprules(12)
% 0.30/0.47 thf(fact_134_semiring_Osemiring__simprules_I7_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 0.30/0.47 ( ( semiring @ A @ B @ R )
% 0.30/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( add @ A @ B @ R @ X @ Y )
% 0.30/0.47 = ( add @ A @ B @ R @ Y @ X ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % semiring.semiring_simprules(7)
% 0.30/0.47 thf(fact_135_semiring_Osemiring__simprules_I5_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 0.30/0.47 ( ( semiring @ A @ B @ R )
% 0.30/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( add @ A @ B @ R @ ( add @ A @ B @ R @ X @ Y ) @ Z2 )
% 0.30/0.47 = ( add @ A @ B @ R @ X @ ( add @ A @ B @ R @ Y @ Z2 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % semiring.semiring_simprules(5)
% 0.30/0.47 thf(fact_136_semiring_Osemiring__simprules_I1_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 0.30/0.47 ( ( semiring @ A @ B @ R )
% 0.30/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( member @ A @ ( add @ A @ B @ R @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % semiring.semiring_simprules(1)
% 0.30/0.47 thf(fact_137_semiring_Osemiring__simprules_I2_J,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.30/0.47 ( ( semiring @ A @ B @ R )
% 0.30/0.47 => ( member @ A @ ( zero @ A @ B @ R ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % semiring.semiring_simprules(2)
% 0.30/0.47 thf(fact_138_semiring_Ol__distr,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 0.30/0.47 ( ( semiring @ A @ B @ R )
% 0.30/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 0.30/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R @ ( add @ A @ B @ R @ X @ Y ) @ Z2 )
% 0.30/0.47 = ( add @ A @ B @ R @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ X @ Z2 ) @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ Y @ Z2 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % semiring.l_distr
% 0.30/0.47 thf(fact_139_add_Oint__pow__diff,axiom,
% 0.30/0.47 ! [X: a,N: int,M3: int] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( add_pow @ a @ b @ int @ r @ ( minus_minus @ int @ N @ M3 ) @ X )
% 0.30/0.47 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ N @ X ) @ ( a_inv @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ M3 @ X ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.int_pow_diff
% 0.30/0.47 thf(fact_140_add_Oint__pow__neg,axiom,
% 0.30/0.47 ! [X: a,I: int] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( add_pow @ a @ b @ int @ r @ ( uminus_uminus @ int @ I ) @ X )
% 0.30/0.47 = ( a_inv @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.int_pow_neg
% 0.30/0.47 thf(fact_141_add_Oint__pow__mult,axiom,
% 0.30/0.47 ! [X: a,I: int,J: int] :
% 0.30/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( add_pow @ a @ b @ int @ r @ ( plus_plus @ int @ I @ J ) @ X )
% 0.30/0.47 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( add_pow @ a @ b @ int @ r @ J @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.int_pow_mult
% 0.30/0.47 thf(fact_142_subset,axiom,
% 0.30/0.47 ord_less_eq @ ( set @ a ) @ s @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.30/0.47
% 0.30/0.47 % subset
% 0.30/0.47 thf(fact_143_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Osubset,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,M2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S2: set @ A] :
% 0.30/0.47 ( ( locali1441642228monoid @ A @ B @ M2 @ S2 )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ S2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ M2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Localization_Mirabelle_ojffvtlkod.submonoid.subset
% 0.30/0.47 thf(fact_144_diff__minus__eq__add,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( minus_minus @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
% 0.30/0.47 = ( plus_plus @ A @ A2 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_minus_eq_add
% 0.30/0.47 thf(fact_145_uminus__add__conv__diff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ab_group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
% 0.30/0.47 = ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % uminus_add_conv_diff
% 0.30/0.47 thf(fact_146_le__add__diff__inverse,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linordered_semidom @ A )
% 0.30/0.47 => ! [B2: A,A2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.47 => ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) )
% 0.30/0.47 = A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_add_diff_inverse
% 0.30/0.47 thf(fact_147_le__add__diff__inverse2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linordered_semidom @ A )
% 0.30/0.47 => ! [B2: A,A2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.47 => ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
% 0.30/0.47 = A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_add_diff_inverse2
% 0.30/0.47 thf(fact_148_add__right__cancel,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( cancel_semigroup_add @ A )
% 0.30/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.30/0.47 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.30/0.47 = ( plus_plus @ A @ C2 @ A2 ) )
% 0.30/0.47 = ( B2 = C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_right_cancel
% 0.30/0.47 thf(fact_149_add__left__cancel,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( cancel_semigroup_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.30/0.47 = ( plus_plus @ A @ A2 @ C2 ) )
% 0.30/0.47 = ( B2 = C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_left_cancel
% 0.30/0.47 thf(fact_150_neg__equal__iff__equal,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( ( uminus_uminus @ A @ A2 )
% 0.30/0.47 = ( uminus_uminus @ A @ B2 ) )
% 0.30/0.47 = ( A2 = B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % neg_equal_iff_equal
% 0.30/0.47 thf(fact_151_add_Oinverse__inverse,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [A2: A] :
% 0.30/0.47 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
% 0.30/0.47 = A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.inverse_inverse
% 0.30/0.47 thf(fact_152_add__le__cancel__right,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere236663937imp_le @ A )
% 0.30/0.47 => ! [A2: A,C2: A,B2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.30/0.47 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_le_cancel_right
% 0.30/0.47 thf(fact_153_add__le__cancel__left,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere236663937imp_le @ A )
% 0.30/0.47 => ! [C2: A,A2: A,B2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 0.30/0.47 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_le_cancel_left
% 0.30/0.47 thf(fact_154_neg__le__iff__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [B2: A,A2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
% 0.30/0.47 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % neg_le_iff_le
% 0.30/0.47 thf(fact_155_add__diff__cancel__right_H,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( cancel146912293up_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 0.30/0.47 = A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_diff_cancel_right'
% 0.30/0.47 thf(fact_156_add__diff__cancel__right,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( cancel146912293up_add @ A )
% 0.30/0.47 => ! [A2: A,C2: A,B2: A] :
% 0.30/0.47 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.30/0.47 = ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_diff_cancel_right
% 0.30/0.47 thf(fact_157_add__diff__cancel__left_H,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( cancel146912293up_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 )
% 0.30/0.47 = B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_diff_cancel_left'
% 0.30/0.47 thf(fact_158_add__diff__cancel__left,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( cancel146912293up_add @ A )
% 0.30/0.47 => ! [C2: A,A2: A,B2: A] :
% 0.30/0.47 ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 0.30/0.47 = ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_diff_cancel_left
% 0.30/0.47 thf(fact_159_diff__add__cancel,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
% 0.30/0.47 = A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_add_cancel
% 0.30/0.47 thf(fact_160_add__diff__cancel,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 0.30/0.47 = A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_diff_cancel
% 0.30/0.47 thf(fact_161_minus__add__distrib,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ab_group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
% 0.30/0.47 = ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minus_add_distrib
% 0.30/0.47 thf(fact_162_minus__add__cancel,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) )
% 0.30/0.47 = B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % minus_add_cancel
% 0.30/0.47 thf(fact_163_add__minus__cancel,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) )
% 0.30/0.47 = B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_minus_cancel
% 0.30/0.47 thf(fact_164_minus__diff__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B2 ) )
% 0.30/0.47 = ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minus_diff_eq
% 0.30/0.47 thf(fact_165_add__right__imp__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( cancel_semigroup_add @ A )
% 0.30/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.30/0.47 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.30/0.47 = ( plus_plus @ A @ C2 @ A2 ) )
% 0.30/0.47 => ( B2 = C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_right_imp_eq
% 0.30/0.47 thf(fact_166_add__left__imp__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( cancel_semigroup_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.30/0.47 = ( plus_plus @ A @ A2 @ C2 ) )
% 0.30/0.47 => ( B2 = C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_left_imp_eq
% 0.30/0.47 thf(fact_167_add_Oleft__commute,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ab_semigroup_add @ A )
% 0.30/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.30/0.47 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
% 0.30/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.left_commute
% 0.30/0.47 thf(fact_168_add_Ocommute,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ab_semigroup_add @ A )
% 0.30/0.47 => ( ( plus_plus @ A )
% 0.30/0.47 = ( ^ [A4: A,B3: A] : ( plus_plus @ A @ B3 @ A4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.commute
% 0.30/0.47 thf(fact_169_group__add__class_Oadd_Oright__cancel,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.30/0.47 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.30/0.47 = ( plus_plus @ A @ C2 @ A2 ) )
% 0.30/0.47 = ( B2 = C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % group_add_class.add.right_cancel
% 0.30/0.47 thf(fact_170_add_Oleft__cancel,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.30/0.47 = ( plus_plus @ A @ A2 @ C2 ) )
% 0.30/0.47 = ( B2 = C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.left_cancel
% 0.30/0.47 thf(fact_171_add_Oassoc,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semigroup_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 0.30/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.assoc
% 0.30/0.47 thf(fact_172_group__cancel_Oadd2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( comm_monoid_add @ A )
% 0.30/0.47 => ! [B4: A,K: A,B2: A,A2: A] :
% 0.30/0.47 ( ( B4
% 0.30/0.47 = ( plus_plus @ A @ K @ B2 ) )
% 0.30/0.47 => ( ( plus_plus @ A @ A2 @ B4 )
% 0.30/0.47 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % group_cancel.add2
% 0.30/0.47 thf(fact_173_group__cancel_Oadd1,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( comm_monoid_add @ A )
% 0.30/0.47 => ! [A3: A,K: A,A2: A,B2: A] :
% 0.30/0.47 ( ( A3
% 0.30/0.47 = ( plus_plus @ A @ K @ A2 ) )
% 0.30/0.47 => ( ( plus_plus @ A @ A3 @ B2 )
% 0.30/0.47 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % group_cancel.add1
% 0.30/0.47 thf(fact_174_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere779506340up_add @ A )
% 0.30/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.30/0.47 ( ( ( I = J )
% 0.30/0.47 & ( K = L ) )
% 0.30/0.47 => ( ( plus_plus @ A @ I @ K )
% 0.30/0.47 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_mono_thms_linordered_semiring(4)
% 0.30/0.47 thf(fact_175_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ab_semigroup_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 0.30/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ab_semigroup_add_class.add_ac(1)
% 0.30/0.47 thf(fact_176_diff__right__commute,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( cancel146912293up_add @ A )
% 0.30/0.47 => ! [A2: A,C2: A,B2: A] :
% 0.30/0.47 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 )
% 0.30/0.47 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_right_commute
% 0.30/0.47 thf(fact_177_diff__eq__diff__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A,D: A] :
% 0.30/0.47 ( ( ( minus_minus @ A @ A2 @ B2 )
% 0.30/0.47 = ( minus_minus @ A @ C2 @ D ) )
% 0.30/0.47 => ( ( A2 = B2 )
% 0.30/0.47 = ( C2 = D ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_eq_diff_eq
% 0.30/0.47 thf(fact_178_minus__equation__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( ( uminus_uminus @ A @ A2 )
% 0.30/0.47 = B2 )
% 0.30/0.47 = ( ( uminus_uminus @ A @ B2 )
% 0.30/0.47 = A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minus_equation_iff
% 0.30/0.47 thf(fact_179_equation__minus__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( A2
% 0.30/0.47 = ( uminus_uminus @ A @ B2 ) )
% 0.30/0.47 = ( B2
% 0.30/0.47 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % equation_minus_iff
% 0.30/0.47 thf(fact_180_add__le__imp__le__right,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere236663937imp_le @ A )
% 0.30/0.47 => ! [A2: A,C2: A,B2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.30/0.47 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_le_imp_le_right
% 0.30/0.47 thf(fact_181_add__le__imp__le__left,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere236663937imp_le @ A )
% 0.30/0.47 => ! [C2: A,A2: A,B2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 0.30/0.47 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_le_imp_le_left
% 0.30/0.47 thf(fact_182_le__iff__add,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( canoni770627133id_add @ A )
% 0.30/0.47 => ( ( ord_less_eq @ A )
% 0.30/0.47 = ( ^ [A4: A,B3: A] :
% 0.30/0.47 ? [C3: A] :
% 0.30/0.47 ( B3
% 0.30/0.47 = ( plus_plus @ A @ A4 @ C3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_iff_add
% 0.30/0.47 thf(fact_183_add__right__mono,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere779506340up_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_right_mono
% 0.30/0.47 thf(fact_184_less__eqE,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( canoni770627133id_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ~ ! [C4: A] :
% 0.30/0.47 ( B2
% 0.30/0.47 != ( plus_plus @ A @ A2 @ C4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_eqE
% 0.30/0.47 thf(fact_185_add__left__mono,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere779506340up_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_left_mono
% 0.30/0.47 thf(fact_186_add__mono,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere779506340up_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A,D: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ C2 @ D )
% 0.30/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_mono
% 0.30/0.47 thf(fact_187_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere779506340up_add @ A )
% 0.30/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.30/0.47 ( ( ( ord_less_eq @ A @ I @ J )
% 0.30/0.47 & ( ord_less_eq @ A @ K @ L ) )
% 0.30/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_mono_thms_linordered_semiring(1)
% 0.30/0.47 thf(fact_188_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere779506340up_add @ A )
% 0.30/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.30/0.47 ( ( ( I = J )
% 0.30/0.47 & ( ord_less_eq @ A @ K @ L ) )
% 0.30/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_mono_thms_linordered_semiring(2)
% 0.30/0.47 thf(fact_189_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere779506340up_add @ A )
% 0.30/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.30/0.47 ( ( ( ord_less_eq @ A @ I @ J )
% 0.30/0.47 & ( K = L ) )
% 0.30/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_mono_thms_linordered_semiring(3)
% 0.30/0.47 thf(fact_190_diff__eq__diff__less__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A,D: A] :
% 0.30/0.47 ( ( ( minus_minus @ A @ A2 @ B2 )
% 0.30/0.47 = ( minus_minus @ A @ C2 @ D ) )
% 0.30/0.47 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 = ( ord_less_eq @ A @ C2 @ D ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_eq_diff_less_eq
% 0.30/0.47 thf(fact_191_diff__right__mono,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_right_mono
% 0.30/0.47 thf(fact_192_diff__left__mono,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_left_mono
% 0.30/0.47 thf(fact_193_diff__mono,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A,D: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ D @ C2 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_mono
% 0.30/0.47 thf(fact_194_le__imp__neg__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_imp_neg_le
% 0.30/0.47 thf(fact_195_minus__le__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
% 0.30/0.47 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minus_le_iff
% 0.30/0.47 thf(fact_196_le__minus__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
% 0.30/0.47 = ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_minus_iff
% 0.30/0.47 thf(fact_197_add__implies__diff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( cancel1352612707id_add @ A )
% 0.30/0.47 => ! [C2: A,B2: A,A2: A] :
% 0.30/0.47 ( ( ( plus_plus @ A @ C2 @ B2 )
% 0.30/0.47 = A2 )
% 0.30/0.47 => ( C2
% 0.30/0.47 = ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_implies_diff
% 0.30/0.47 thf(fact_198_diff__diff__add,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( cancel146912293up_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
% 0.30/0.47 = ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_diff_add
% 0.30/0.47 thf(fact_199_diff__add__eq__diff__diff__swap,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.30/0.47 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_add_eq_diff_diff_swap
% 0.30/0.47 thf(fact_200_diff__add__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ab_group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
% 0.30/0.47 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_add_eq
% 0.30/0.47 thf(fact_201_diff__diff__eq2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
% 0.30/0.47 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_diff_eq2
% 0.30/0.47 thf(fact_202_add__diff__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
% 0.30/0.47 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_diff_eq
% 0.30/0.47 thf(fact_203_eq__diff__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [A2: A,C2: A,B2: A] :
% 0.30/0.47 ( ( A2
% 0.30/0.47 = ( minus_minus @ A @ C2 @ B2 ) )
% 0.30/0.47 = ( ( plus_plus @ A @ A2 @ B2 )
% 0.30/0.47 = C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eq_diff_eq
% 0.30/0.47 thf(fact_204_diff__eq__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( ( minus_minus @ A @ A2 @ B2 )
% 0.30/0.47 = C2 )
% 0.30/0.47 = ( A2
% 0.30/0.47 = ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_eq_eq
% 0.30/0.47 thf(fact_205_group__cancel_Osub1,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ab_group_add @ A )
% 0.30/0.47 => ! [A3: A,K: A,A2: A,B2: A] :
% 0.30/0.47 ( ( A3
% 0.30/0.47 = ( plus_plus @ A @ K @ A2 ) )
% 0.30/0.47 => ( ( minus_minus @ A @ A3 @ B2 )
% 0.30/0.47 = ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % group_cancel.sub1
% 0.30/0.47 thf(fact_206_add_Oinverse__distrib__swap,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
% 0.30/0.47 = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.inverse_distrib_swap
% 0.30/0.47 thf(fact_207_group__cancel_Oneg1,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ab_group_add @ A )
% 0.30/0.47 => ! [A3: A,K: A,A2: A] :
% 0.30/0.47 ( ( A3
% 0.30/0.47 = ( plus_plus @ A @ K @ A2 ) )
% 0.30/0.47 => ( ( uminus_uminus @ A @ A3 )
% 0.30/0.47 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % group_cancel.neg1
% 0.30/0.47 thf(fact_208_minus__diff__commute,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ab_group_add @ A )
% 0.30/0.47 => ! [B2: A,A2: A] :
% 0.30/0.47 ( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A2 )
% 0.30/0.47 = ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minus_diff_commute
% 0.30/0.47 thf(fact_209_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere623563068d_diff @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( ( minus_minus @ A @ B2 @ A2 )
% 0.30/0.47 = C2 )
% 0.30/0.47 = ( B2
% 0.30/0.47 = ( plus_plus @ A @ C2 @ A2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.30/0.47 thf(fact_210_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere623563068d_diff @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ A2 ) )
% 0.30/0.47 = B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.30/0.47 thf(fact_211_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere623563068d_diff @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
% 0.30/0.47 = ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.30/0.47 thf(fact_212_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere623563068d_diff @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 )
% 0.30/0.47 = ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.30/0.47 thf(fact_213_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere623563068d_diff @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 )
% 0.30/0.47 = ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.30/0.47 thf(fact_214_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere623563068d_diff @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 )
% 0.30/0.47 = ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.30/0.47 thf(fact_215_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere623563068d_diff @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
% 0.30/0.47 = ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.30/0.47 thf(fact_216_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere623563068d_diff @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
% 0.30/0.47 = ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.30/0.47 thf(fact_217_le__add__diff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere623563068d_diff @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_add_diff
% 0.30/0.47 thf(fact_218_add__le__add__imp__diff__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linordered_semidom @ A )
% 0.30/0.47 => ! [I: A,K: A,N: A,J: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
% 0.30/0.47 => ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
% 0.30/0.47 => ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
% 0.30/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ N @ K ) @ J ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_le_add_imp_diff_le
% 0.30/0.47 thf(fact_219_diff__add,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere623563068d_diff @ A )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ A2 )
% 0.30/0.47 = B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_add
% 0.30/0.47 thf(fact_220_add__le__imp__le__diff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linordered_semidom @ A )
% 0.30/0.47 => ! [I: A,K: A,N: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
% 0.30/0.47 => ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N @ K ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_le_imp_le_diff
% 0.30/0.47 thf(fact_221_le__diff__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [A2: A,C2: A,B2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) )
% 0.30/0.47 = ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_diff_eq
% 0.30/0.47 thf(fact_222_diff__le__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
% 0.30/0.47 = ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_le_eq
% 0.30/0.47 thf(fact_223_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ab_group_add @ A )
% 0.30/0.47 => ( ( minus_minus @ A )
% 0.30/0.47 = ( ^ [A4: A,B3: A] : ( plus_plus @ A @ A4 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.30/0.47 thf(fact_224_diff__conv__add__uminus,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ( ( minus_minus @ A )
% 0.30/0.47 = ( ^ [A4: A,B3: A] : ( plus_plus @ A @ A4 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_conv_add_uminus
% 0.30/0.47 thf(fact_225_group__cancel_Osub2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ab_group_add @ A )
% 0.30/0.47 => ! [B4: A,K: A,B2: A,A2: A] :
% 0.30/0.47 ( ( B4
% 0.30/0.47 = ( plus_plus @ A @ K @ B2 ) )
% 0.30/0.47 => ( ( minus_minus @ A @ A2 @ B4 )
% 0.30/0.47 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % group_cancel.sub2
% 0.30/0.47 thf(fact_226_add_Oone__in__subset,axiom,
% 0.30/0.47 ! [H: set @ a] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( H
% 0.30/0.47 != ( bot_bot @ ( set @ a ) ) )
% 0.30/0.47 => ( ! [X3: a] :
% 0.30/0.47 ( ( member @ a @ X3 @ H )
% 0.30/0.47 => ( member @ a @ ( a_inv @ a @ b @ r @ X3 ) @ H ) )
% 0.30/0.47 => ( ! [X3: a] :
% 0.30/0.47 ( ( member @ a @ X3 @ H )
% 0.30/0.47 => ! [Xa: a] :
% 0.30/0.47 ( ( member @ a @ Xa @ H )
% 0.30/0.47 => ( member @ a @ ( add @ a @ b @ r @ X3 @ Xa ) @ H ) ) )
% 0.30/0.47 => ( member @ a @ ( zero @ a @ b @ r ) @ H ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.one_in_subset
% 0.30/0.47 thf(fact_227_a__lcos__mult__one,axiom,
% 0.30/0.47 ! [M2: set @ a] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ a ) @ M2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( a_l_coset @ a @ b @ r @ ( zero @ a @ b @ r ) @ M2 )
% 0.30/0.47 = M2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % a_lcos_mult_one
% 0.30/0.47 thf(fact_228_a__lcos__m__assoc,axiom,
% 0.30/0.47 ! [M2: set @ a,G: a,H2: a] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ a ) @ M2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ G @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ H2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( a_l_coset @ a @ b @ r @ G @ ( a_l_coset @ a @ b @ r @ H2 @ M2 ) )
% 0.30/0.47 = ( a_l_coset @ a @ b @ r @ ( add @ a @ b @ r @ G @ H2 ) @ M2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % a_lcos_m_assoc
% 0.30/0.47 thf(fact_229_carrier__not__empty,axiom,
% 0.30/0.47 ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 0.30/0.47 != ( bot_bot @ ( set @ a ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % carrier_not_empty
% 0.30/0.47 thf(fact_230_a__l__coset__subset__G,axiom,
% 0.30/0.47 ! [H: set @ a,X: a] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ord_less_eq @ ( set @ a ) @ ( a_l_coset @ a @ b @ r @ X @ H ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % a_l_coset_subset_G
% 0.30/0.47 thf(fact_231_setadd__subset__G,axiom,
% 0.30/0.47 ! [H: set @ a,K2: set @ a] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( ord_less_eq @ ( set @ a ) @ K2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ord_less_eq @ ( set @ a ) @ ( set_add @ a @ b @ r @ H @ K2 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % setadd_subset_G
% 0.30/0.47 thf(fact_232_set__add__closed,axiom,
% 0.30/0.47 ! [A3: set @ a,B4: set @ a] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ a ) @ A3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ( ord_less_eq @ ( set @ a ) @ B4 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.30/0.47 => ( ord_less_eq @ ( set @ a ) @ ( set_add @ a @ b @ r @ A3 @ B4 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % set_add_closed
% 0.30/0.47 thf(fact_233_add__additive__subgroups,axiom,
% 0.30/0.47 ! [H: set @ a,K2: set @ a] :
% 0.30/0.47 ( ( additive_subgroup @ a @ b @ H @ r )
% 0.30/0.47 => ( ( additive_subgroup @ a @ b @ K2 @ r )
% 0.30/0.47 => ( additive_subgroup @ a @ b @ ( set_add @ a @ b @ r @ H @ K2 ) @ r ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add_additive_subgroups
% 0.30/0.47 thf(fact_234_Diff__eq__empty__iff,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 0.30/0.47 ( ( ( minus_minus @ ( set @ A ) @ A3 @ B4 )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Diff_eq_empty_iff
% 0.30/0.47 thf(fact_235_empty__iff,axiom,
% 0.30/0.47 ! [A: $tType,C2: A] :
% 0.30/0.47 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % empty_iff
% 0.30/0.47 thf(fact_236_all__not__in__conv,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A] :
% 0.30/0.47 ( ( ! [X2: A] :
% 0.30/0.47 ~ ( member @ A @ X2 @ A3 ) )
% 0.30/0.47 = ( A3
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % all_not_in_conv
% 0.30/0.47 thf(fact_237_Collect__empty__eq,axiom,
% 0.30/0.47 ! [A: $tType,P: A > $o] :
% 0.30/0.47 ( ( ( collect @ A @ P )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( ! [X2: A] :
% 0.30/0.47 ~ ( P @ X2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_empty_eq
% 0.30/0.47 thf(fact_238_empty__Collect__eq,axiom,
% 0.30/0.47 ! [A: $tType,P: A > $o] :
% 0.30/0.47 ( ( ( bot_bot @ ( set @ A ) )
% 0.30/0.47 = ( collect @ A @ P ) )
% 0.30/0.47 = ( ! [X2: A] :
% 0.30/0.47 ~ ( P @ X2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % empty_Collect_eq
% 0.30/0.47 thf(fact_239_subsetI,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 0.30/0.47 ( ! [X3: A] :
% 0.30/0.47 ( ( member @ A @ X3 @ A3 )
% 0.30/0.47 => ( member @ A @ X3 @ B4 ) )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % subsetI
% 0.30/0.47 thf(fact_240_subset__antisym,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 0.30/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ A3 )
% 0.30/0.47 => ( A3 = B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_antisym
% 0.30/0.47 thf(fact_241_Compl__subset__Compl__iff,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) )
% 0.30/0.47 = ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Compl_subset_Compl_iff
% 0.30/0.47 thf(fact_242_Compl__anti__mono,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Compl_anti_mono
% 0.30/0.47 thf(fact_243_subset__empty,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( A3
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_empty
% 0.30/0.47 thf(fact_244_empty__subsetI,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% 0.30/0.47
% 0.30/0.47 % empty_subsetI
% 0.30/0.47 thf(fact_245_Diff__empty,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A] :
% 0.30/0.47 ( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = A3 ) ).
% 0.30/0.47
% 0.30/0.47 % Diff_empty
% 0.30/0.47 thf(fact_246_empty__Diff,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A] :
% 0.30/0.47 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % empty_Diff
% 0.30/0.47 thf(fact_247_Diff__cancel,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A] :
% 0.30/0.47 ( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Diff_cancel
% 0.30/0.47 thf(fact_248_emptyE,axiom,
% 0.30/0.47 ! [A: $tType,A2: A] :
% 0.30/0.47 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % emptyE
% 0.30/0.47 thf(fact_249_equals0D,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A,A2: A] :
% 0.30/0.47 ( ( A3
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % equals0D
% 0.30/0.47 thf(fact_250_equals0I,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A] :
% 0.30/0.47 ( ! [Y2: A] :
% 0.30/0.47 ~ ( member @ A @ Y2 @ A3 )
% 0.30/0.47 => ( A3
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % equals0I
% 0.30/0.47 thf(fact_251_ex__in__conv,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A] :
% 0.30/0.47 ( ( ? [X2: A] : ( member @ A @ X2 @ A3 ) )
% 0.30/0.47 = ( A3
% 0.30/0.47 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ex_in_conv
% 0.30/0.47 thf(fact_252_in__mono,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A,B4: set @ A,X: A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 0.30/0.47 => ( ( member @ A @ X @ A3 )
% 0.30/0.47 => ( member @ A @ X @ B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % in_mono
% 0.30/0.47 thf(fact_253_subsetD,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A,B4: set @ A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 0.30/0.47 => ( ( member @ A @ C2 @ A3 )
% 0.30/0.47 => ( member @ A @ C2 @ B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subsetD
% 0.30/0.47 thf(fact_254_equalityE,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 0.30/0.47 ( ( A3 = B4 )
% 0.30/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 0.30/0.47 => ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % equalityE
% 0.30/0.47 thf(fact_255_subset__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.47 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.30/0.47 ! [X2: A] :
% 0.30/0.47 ( ( member @ A @ X2 @ A5 )
% 0.30/0.47 => ( member @ A @ X2 @ B5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_eq
% 0.30/0.47
% 0.30/0.47 % Type constructors (12)
% 0.30/0.47 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.30/0.47 ordere236663937imp_le @ int ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
% 0.30/0.47 ordere779506340up_add @ int ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.30/0.47 cancel146912293up_add @ int ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 0.30/0.47 cancel1352612707id_add @ int ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 0.30/0.47 ordered_ab_group_add @ int ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
% 0.30/0.47 cancel_semigroup_add @ int ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 0.30/0.47 linordered_semidom @ int ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
% 0.30/0.47 ab_semigroup_add @ int ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
% 0.30/0.47 comm_monoid_add @ int ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
% 0.30/0.47 semigroup_add @ int ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
% 0.30/0.47 ab_group_add @ int ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 0.30/0.47 group_add @ int ).
% 0.30/0.47
% 0.30/0.47 % Conjectures (1)
% 0.30/0.47 thf(conj_0,conjecture,
% 0.30/0.47 ( ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) ) )
% 0.30/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( add @ a @ b @ r @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 ------------------------
% 2.58/1.54 ---- Embedded in THF ---
% 2.58/1.55 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.58/1.55 %%% Generated on Tue Feb 24 08:26:02 EST 2026
% 2.58/1.55 %%% using '$$dhol' embedding, version 1.3.0.
% 2.58/1.55 %%% Logic specification used:
% 2.58/1.55 %%% thf(spec, logic, $$dhol).
% 2.58/1.55
% 2.58/1.55 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.TcuWUcH1aD/DTF2DTF3078.p
% 2.58/1.55 thf(ty_t_Congruence_Opartial__object_Opartial__object__ext, type, partia1265057091ct_ext: ($tType > ($tType > $tType))).
% 2.58/1.55 thf(ty_t_Group_Omonoid_Omonoid__ext, type, monoid_ext: ($tType > ($tType > $tType))).
% 2.58/1.55 thf(ty_t_Ring_Oring_Oring__ext, type, ring_ext: ($tType > ($tType > $tType))).
% 2.58/1.55 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.58/1.55 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.58/1.55 thf(ty_t_Int_Oint, type, int: $tType).
% 2.58/1.55 thf(ty_tf_b, type, b: $tType).
% 2.58/1.55 thf(ty_tf_a, type, a: $tType).
% 2.58/1.55 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.58/1.55 thf(sy_cl_Groups_Oab__group__add, type, ab_group_add: (!> [A:$tType]: ($o))).
% 2.58/1.55 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: ($o))).
% 2.58/1.55 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: ($o))).
% 2.58/1.55 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: ($o))).
% 2.58/1.55 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: ($o))).
% 2.58/1.55 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: ($o))).
% 2.58/1.55 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 2.58/1.55 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: ($o))).
% 2.58/1.55 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: ($o))).
% 2.58/1.55 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: ($o))).
% 2.58/1.55 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.58/1.55 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere236663937imp_le: (!> [A:$tType]: ($o))).
% 2.58/1.55 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff, type, ordere623563068d_diff: (!> [A:$tType]: ($o))).
% 2.58/1.55 thf(sy_c_AbelCoset_Oa__l__coset, type, a_l_coset: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > (A > ((set @ A) > (set @ A))))))).
% 2.58/1.55 thf(sy_c_AbelCoset_Oadditive__subgroup, type, additive_subgroup: (!> [A:$tType,B:$tType]: (((set @ A) > (((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > $o))))).
% 2.58/1.55 thf(sy_c_AbelCoset_Oset__add, type, set_add: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > ((set @ A) > ((set @ A) > (set @ A))))))).
% 2.58/1.55 thf(sy_c_Congruence_Opartial__object_Ocarrier, type, partial_carrier: (!> [A:$tType,Z:$tType]: ((((partia1265057091ct_ext @ A) @ Z) > (set @ A))))).
% 2.58/1.55 thf(sy_c_Group_Omonoid_Omult, type, mult: (!> [A:$tType,Z:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ Z)) > (A > (A > A)))))).
% 2.58/1.55 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.58/1.55 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 2.58/1.55 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 2.58/1.55 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Oeq__obj__rng__of__frac, type, locali1648044335f_frac: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > ((set @ A) > $o))))).
% 2.58/1.55 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Omult__submonoid__of__crng, type, locali7914339f_crng: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > ((set @ A) > $o))))).
% 2.58/1.55 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Omult__submonoid__of__rng, type, locali1402578320of_rng: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > ((set @ A) > $o))))).
% 2.58/1.55 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Osubmonoid, type, locali1441642228monoid: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)) > ((set @ A) > $o))))).
% 2.58/1.55 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.58/1.55 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.58/1.55 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))).
% 2.58/1.55 thf(sy_c_Product__Type_Oprod_Osnd, type, product_snd: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > B)))).
% 2.58/1.55 thf(sy_c_Ring_Oa__inv, type, a_inv: (!> [A:$tType,M:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ M))) > (A > A))))).
% 2.58/1.55 thf(sy_c_Ring_Oa__minus, type, a_minus: (!> [A:$tType,M:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ M))) > (A > (A > A)))))).
% 2.58/1.55 thf(sy_c_Ring_Oadd__pow, type, add_pow: (!> [A:$tType,C:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ C))) > (B > (A > A)))))).
% 2.58/1.55 thf(sy_c_Ring_Oring_Oadd, type, add: (!> [A:$tType,Z:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ Z))) > (A > (A > A)))))).
% 2.58/1.55 thf(sy_c_Ring_Oring_Ozero, type, zero: (!> [A:$tType,Z:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ Z))) > A)))).
% 2.58/1.55 thf(sy_c_Ring_Osemiring, type, semiring: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > $o)))).
% 2.58/1.55 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.58/1.55 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.58/1.55 thf(sy_v_R, type, r: ((partia1265057091ct_ext @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b)))).
% 2.58/1.55 thf(sy_v_S, type, s: (set @ a)).
% 2.58/1.55 thf(sy_v_r, type, r2: a).
% 2.58/1.55 thf(sy_v_r_H, type, r3: a).
% 2.58/1.55 thf(sy_v_s, type, s2: a).
% 2.58/1.55 thf(sy_v_s_H, type, s3: a).
% 2.58/1.55 thf(sy_v_t_H____, type, t: a).
% 2.58/1.55 thf(sy_v_t____, type, t2: a).
% 2.58/1.55 thf(sy_v_x_H____, type, x: ((product_prod @ a) @ a)).
% 2.58/1.55 thf(sy_v_y_H____, type, y: ((product_prod @ a) @ a)).
% 2.58/1.55 thf(fact_0_f18, axiom, (((member @ a) @ t) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_1_f17, axiom, (((member @ a) @ t2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_2_right__add__eq, axiom, (! [A2:a,B2:a,C2:a]: (((A2 = B2) => ((((((add @ a) @ b) @ r) @ C2) @ A2) = (((((add @ a) @ b) @ r) @ C2) @ B2)))))).
% 2.58/1.55 thf(fact_3_local_Oright__minus__eq, axiom, (! [A2:a,B2:a,C2:a]: (((A2 = B2) => ((((((a_minus @ a) @ b) @ r) @ C2) @ A2) = (((((a_minus @ a) @ b) @ r) @ C2) @ B2)))))).
% 2.58/1.55 thf(fact_4_m__assoc, axiom, (! [X:a,Y:a,Z2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Y)) @ Z2) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Y) @ Z2))))))))).
% 2.58/1.55 thf(fact_5_m__comm, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Y) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Y) @ X))))))).
% 2.58/1.55 thf(fact_6_m__lcomm, axiom, (! [X:a,Y:a,Z2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Y) @ Z2)) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Y) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Z2))))))))).
% 2.58/1.55 thf(fact_7_add_Om__assoc, axiom, (! [X:a,Y:a,Z2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ (((((add @ a) @ b) @ r) @ X) @ Y)) @ Z2) = (((((add @ a) @ b) @ r) @ X) @ (((((add @ a) @ b) @ r) @ Y) @ Z2))))))))).
% 2.58/1.55 thf(fact_8_add_Om__comm, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ X) @ Y) = (((((add @ a) @ b) @ r) @ Y) @ X))))))).
% 2.58/1.55 thf(fact_9_add_Om__lcomm, axiom, (! [X:a,Y:a,Z2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ X) @ (((((add @ a) @ b) @ r) @ Y) @ Z2)) = (((((add @ a) @ b) @ r) @ Y) @ (((((add @ a) @ b) @ r) @ X) @ Z2))))))))).
% 2.58/1.55 thf(fact_10_f20, axiom, (((member @ a) @ s3) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_11_f19, axiom, (((member @ a) @ s2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_12_f41, axiom, (((member @ a) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x))))) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_13_f42, axiom, (((member @ a) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_fst @ a) @ a) @ y))))) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_14_l__distr, axiom, (! [X:a,Y:a,Z2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((add @ a) @ b) @ r) @ X) @ Y)) @ Z2) = (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Z2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Y) @ Z2))))))))).
% 2.58/1.55 thf(fact_15_r__distr, axiom, (! [X:a,Y:a,Z2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Z2) @ (((((add @ a) @ b) @ r) @ X) @ Y)) = (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Z2) @ X)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Z2) @ Y))))))))).
% 2.58/1.55 thf(fact_16_four__elem__comm, axiom, (! [A2:a,B2:a,C2:a,D:a]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ C2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ D) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((a_minus @ a) @ b) @ r) @ (((((add @ a) @ b) @ r) @ (((((a_minus @ a) @ b) @ r) @ A2) @ C2)) @ B2)) @ D) = (((((a_minus @ a) @ b) @ r) @ (((((a_minus @ a) @ b) @ r) @ (((((add @ a) @ b) @ r) @ A2) @ B2)) @ C2)) @ D))))))))).
% 2.58/1.55 thf(fact_17_right__inv__add, axiom, (! [A2:a,B2:a,C2:a]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ C2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((a_minus @ a) @ b) @ r) @ (((((a_minus @ a) @ b) @ r) @ C2) @ A2)) @ B2) = (((((a_minus @ a) @ b) @ r) @ C2) @ (((((add @ a) @ b) @ r) @ A2) @ B2))))))))).
% 2.58/1.55 thf(fact_18_f21, axiom, (((member @ a) @ (((product_snd @ a) @ a) @ y)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_19_f40, axiom, ((((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ s3)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ r2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ (((product_fst @ a) @ a) @ x)))))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ s2)) @ (((product_snd @ a) @ a) @ x))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ r3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ (((product_fst @ a) @ a) @ y)))))) = (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x)))))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_fst @ a) @ a) @ y)))))))).
% 2.58/1.55 thf(fact_20_f22, axiom, (((member @ a) @ (((product_fst @ a) @ a) @ x)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_21__092_060open_062_092_060lbrakk_062snd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_A_092_060in_062_Acarrier_AR_059_Asnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_A_092_060in_062_Acarrier_AR_059_At_A_092_060otimes_062_At_H_A_092_060in_062_Acarrier_AR_092_060rbrakk_062_A_092_060Longrightarrow_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_A_092_060oplus_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_092_060close_062, axiom, ((((member @ a) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x))))) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_fst @ a) @ a) @ y))))) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((add @ a) @ b) @ r) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x))))) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_fst @ a) @ a) @ y)))))) = (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x)))))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_fst @ a) @ a) @ y))))))))))).
% 2.58/1.55 thf(fact_22_f15, axiom, (((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_23__092_060open_062snd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060in_062_Acarrier_AR_092_060close_062, axiom, (((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_24_f28, axiom, (((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ (((product_fst @ a) @ a) @ y))) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_25_f11, axiom, (((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ (((product_fst @ a) @ a) @ x))) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_26_f10, axiom, (((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ r2)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_27_f27, axiom, (((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ r3)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_28_f35, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_fst @ a) @ a) @ y)))) = (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x)))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_fst @ a) @ a) @ y)))))).
% 2.58/1.55 thf(fact_29__092_060open_062t_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_A_092_060ominus_062_As_A_092_060otimes_062_Afst_Ax_H_J_A_061_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_At_A_092_060otimes_062_A_Is_A_092_060otimes_062_Afst_Ax_H_J_092_060close_062, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ r2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ (((product_fst @ a) @ a) @ x)))) = (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ r2))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ (((product_fst @ a) @ a) @ x)))))).
% 2.58/1.55 thf(fact_30__092_060open_062t_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_A_092_060ominus_062_As_H_A_092_060otimes_062_Afst_Ay_H_J_A_061_At_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_At_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Afst_Ay_H_J_092_060close_062, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ r3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ (((product_fst @ a) @ a) @ y)))) = (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ r3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ (((product_fst @ a) @ a) @ y)))))).
% 2.58/1.55 thf(fact_31_f16, axiom, (((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x))) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_32_f24, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2))) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ s3)) @ (((product_snd @ a) @ a) @ y))) @ t2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ r2)))).
% 2.58/1.55 thf(fact_33_f9, axiom, (((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ s3)) @ (((product_snd @ a) @ a) @ y))) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_34__092_060open_062t_A_092_060otimes_062_As_A_092_060otimes_062_Asnd_Ax_H_A_092_060in_062_Acarrier_AR_092_060close_062, axiom, (((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ s2)) @ (((product_snd @ a) @ a) @ x))) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_35_f23, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x)))) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ s3)) @ (((product_snd @ a) @ a) @ y))) @ t2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ (((product_fst @ a) @ a) @ x))))).
% 2.58/1.55 thf(fact_36_f34, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r3))) = (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r3))))).
% 2.58/1.55 thf(fact_37_f36, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3))) @ (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_fst @ a) @ a) @ y)))) = (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x)))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_fst @ a) @ a) @ y)))))).
% 2.58/1.55 thf(fact_38_f13, axiom, (((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2))) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_39__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_061_At_H_A_092_060otimes_062_As_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_At_H_A_092_060otimes_062_As_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_At_A_092_060otimes_062_A_Is_A_092_060otimes_062_Afst_Ax_H_J_092_060close_062, axiom, ((((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2)))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x))))) = (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ s3)) @ (((product_snd @ a) @ a) @ y))) @ t2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ r2))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ s3)) @ (((product_snd @ a) @ a) @ y))) @ t2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ (((product_fst @ a) @ a) @ x)))))).
% 2.58/1.55 thf(fact_40_f12, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ s3)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ r2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ (((product_fst @ a) @ a) @ x))))) = (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ s3)) @ (((product_snd @ a) @ a) @ y))) @ t2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ r2))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ s3)) @ (((product_snd @ a) @ a) @ y))) @ t2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ (((product_fst @ a) @ a) @ x)))))).
% 2.58/1.55 thf(fact_41_f25, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ s3)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ r2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ (((product_fst @ a) @ a) @ x))))) = (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2)))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x))))))).
% 2.58/1.55 thf(fact_42_f32, axiom, ((((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2)))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x))))) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x))))))).
% 2.58/1.55 thf(fact_43_f26, axiom, ((((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r3)))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_fst @ a) @ a) @ y))))) = (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ s2)) @ (((product_snd @ a) @ a) @ x))) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ r3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ s2)) @ (((product_snd @ a) @ a) @ x))) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ (((product_fst @ a) @ a) @ y)))))).
% 2.58/1.55 thf(fact_44_f29, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ s2)) @ (((product_snd @ a) @ a) @ x))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ r3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ (((product_fst @ a) @ a) @ y))))) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ s2)) @ (((product_snd @ a) @ a) @ x))) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ r3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ (((product_fst @ a) @ a) @ y))))))).
% 2.58/1.55 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.58/1.55 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A3)))) = A3)))).
% 2.58/1.55 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.58/1.55 thf(fact_48_f30, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ s2)) @ (((product_snd @ a) @ a) @ x))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ r3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ (((product_fst @ a) @ a) @ y))))) = (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r3)))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_fst @ a) @ a) @ y))))))).
% 2.58/1.55 thf(fact_49_f33, axiom, ((((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r3)))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_fst @ a) @ a) @ y))))) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_fst @ a) @ a) @ y))))))).
% 2.58/1.55 thf(fact_50__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_J_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_J_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_092_060close_062, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r3)))) = (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y)))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y)))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r3))))).
% 2.58/1.55 thf(fact_51_f14, axiom, (((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x)))) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_52__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_092_060close_062, axiom, ((((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x)))))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_fst @ a) @ a) @ y)))))) = (((((a_minus @ a) @ b) @ r) @ (((((add @ a) @ b) @ r) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2)))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x)))))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r3))))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_fst @ a) @ a) @ y))))))).
% 2.58/1.55 thf(fact_53_f31, axiom, ((((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ s3)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ r2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ (((product_fst @ a) @ a) @ x)))))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ s2)) @ (((product_snd @ a) @ a) @ x))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ r3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ (((product_fst @ a) @ a) @ y)))))) = (((((add @ a) @ b) @ r) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2)))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x)))))) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r3)))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_fst @ a) @ a) @ y)))))))).
% 2.58/1.55 thf(fact_54_f37, axiom, (((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x)))))) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_55_f39, axiom, (((member @ a) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2)))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x)))))) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_56_f38, axiom, (((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_fst @ a) @ a) @ y)))))) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_57_f5, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ r2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ (((product_fst @ a) @ a) @ x)))) = (((zero @ a) @ b) @ r))).
% 2.58/1.55 thf(fact_58_f7, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ r3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ (((product_fst @ a) @ a) @ y)))) = (((zero @ a) @ b) @ r))).
% 2.58/1.55 thf(fact_59_minus__closed, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((member @ a) @ (((((a_minus @ a) @ b) @ r) @ X) @ Y)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))))))).
% 2.58/1.55 thf(fact_60_add_Om__closed, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((member @ a) @ (((((add @ a) @ b) @ r) @ X) @ Y)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))))))).
% 2.58/1.55 thf(fact_61_local_Oadd_Oright__cancel, axiom, (! [X:a,Y:a,Z2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add @ a) @ b) @ r) @ Y) @ X) = (((((add @ a) @ b) @ r) @ Z2) @ X)) = (Y = Z2)))))))).
% 2.58/1.55 thf(fact_62_semiring__simprules_I3_J, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Y)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))))))).
% 2.58/1.55 thf(fact_63_f4, axiom, (((member @ a) @ t2) @ s)).
% 2.58/1.55 thf(fact_64_f6, axiom, (((member @ a) @ t) @ s)).
% 2.58/1.55 thf(fact_65_f8, axiom, (((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ s)).
% 2.58/1.55 thf(fact_66_local_Osemiring__axioms, axiom, (((semiring @ a) @ b) @ r)).
% 2.58/1.55 thf(fact_67_minus__eq, axiom, (! [X:a,Y:a]: (((((((a_minus @ a) @ b) @ r) @ X) @ Y) = (((((add @ a) @ b) @ r) @ X) @ ((((a_inv @ a) @ b) @ r) @ Y)))))).
% 2.58/1.55 thf(fact_68_prod__eqI, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B),Q2:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ P2) = (((product_fst @ A) @ B) @ Q2)) => (((((product_snd @ A) @ B) @ P2) = (((product_snd @ A) @ B) @ Q2)) => (P2 = Q2)))))).
% 2.58/1.55 thf(fact_69_exE__realizer_H, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),P2:((product_prod @ B) @ A)]: ((((P @ (((product_snd @ B) @ A) @ P2)) @ (((product_fst @ B) @ A) @ P2)) => (~ ((! [X3:B,Y2:A]: ((~ (((P @ Y2) @ X3))))))))))).
% 2.58/1.55 thf(fact_70_prod_Oexpand, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B),Prod2:((product_prod @ A) @ B)]: (((((((product_fst @ A) @ B) @ Prod) = (((product_fst @ A) @ B) @ Prod2)) & ((((product_snd @ A) @ B) @ Prod) = (((product_snd @ A) @ B) @ Prod2))) => (Prod = Prod2))))).
% 2.58/1.55 thf(fact_71_local_Ominus__unique, axiom, (! [Y:a,X:a,Y3:a]: ((((((((add @ a) @ b) @ r) @ Y) @ X) = (((zero @ a) @ b) @ r)) => (((((((add @ a) @ b) @ r) @ X) @ Y3) = (((zero @ a) @ b) @ r)) => ((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y3) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (Y = Y3))))))))).
% 2.58/1.55 thf(fact_72_add_Or__inv__ex, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (? [X3:a]: (((((member @ a) @ X3) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) & ((((((add @ a) @ b) @ r) @ X) @ X3) = (((zero @ a) @ b) @ r))))))))).
% 2.58/1.55 thf(fact_73_add_Oone__unique, axiom, (! [U:a]: (((((member @ a) @ U) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((! [X3:a]: (((((member @ a) @ X3) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ U) @ X3) = X3)))) => (U = (((zero @ a) @ b) @ r))))))).
% 2.58/1.55 thf(fact_74_add_Ol__inv__ex, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (? [X3:a]: (((((member @ a) @ X3) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) & ((((((add @ a) @ b) @ r) @ X3) @ X) = (((zero @ a) @ b) @ r))))))))).
% 2.58/1.55 thf(fact_75_add_Oinv__comm, axiom, (! [X:a,Y:a]: ((((((((add @ a) @ b) @ r) @ X) @ Y) = (((zero @ a) @ b) @ r)) => ((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ Y) @ X) = (((zero @ a) @ b) @ r)))))))).
% 2.58/1.55 thf(fact_76_r__minus, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ ((((a_inv @ a) @ b) @ r) @ Y)) = ((((a_inv @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Y)))))))).
% 2.58/1.55 thf(fact_77_l__minus, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ ((((a_inv @ a) @ b) @ r) @ X)) @ Y) = ((((a_inv @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Y)))))))).
% 2.58/1.55 thf(fact_78_r__neg2, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ X) @ (((((add @ a) @ b) @ r) @ ((((a_inv @ a) @ b) @ r) @ X)) @ Y)) = Y)))))).
% 2.58/1.55 thf(fact_79_r__neg1, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ ((((a_inv @ a) @ b) @ r) @ X)) @ (((((add @ a) @ b) @ r) @ X) @ Y)) = Y)))))).
% 2.58/1.55 thf(fact_80_local_Ominus__add, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((a_inv @ a) @ b) @ r) @ (((((add @ a) @ b) @ r) @ X) @ Y)) = (((((add @ a) @ b) @ r) @ ((((a_inv @ a) @ b) @ r) @ X)) @ ((((a_inv @ a) @ b) @ r) @ Y)))))))).
% 2.58/1.55 thf(fact_81_add_Oinv__solve__right_H, axiom, (! [A2:a,B2:a,C2:a]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ C2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add @ a) @ b) @ r) @ B2) @ ((((a_inv @ a) @ b) @ r) @ C2)) = A2) = (B2 = (((((add @ a) @ b) @ r) @ A2) @ C2))))))))).
% 2.58/1.55 thf(fact_82_add_Oinv__solve__right, axiom, (! [A2:a,B2:a,C2:a]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ C2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((A2 = (((((add @ a) @ b) @ r) @ B2) @ ((((a_inv @ a) @ b) @ r) @ C2))) = (B2 = (((((add @ a) @ b) @ r) @ A2) @ C2))))))))).
% 2.58/1.55 thf(fact_83_add_Oinv__solve__left_H, axiom, (! [A2:a,B2:a,C2:a]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ C2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add @ a) @ b) @ r) @ ((((a_inv @ a) @ b) @ r) @ B2)) @ C2) = A2) = (C2 = (((((add @ a) @ b) @ r) @ B2) @ A2))))))))).
% 2.58/1.55 thf(fact_84_add_Oinv__solve__left, axiom, (! [A2:a,B2:a,C2:a]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ C2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((A2 = (((((add @ a) @ b) @ r) @ ((((a_inv @ a) @ b) @ r) @ B2)) @ C2)) = (C2 = (((((add @ a) @ b) @ r) @ B2) @ A2))))))))).
% 2.58/1.55 thf(fact_85_add_Oinv__mult__group, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((a_inv @ a) @ b) @ r) @ (((((add @ a) @ b) @ r) @ X) @ Y)) = (((((add @ a) @ b) @ r) @ ((((a_inv @ a) @ b) @ r) @ Y)) @ ((((a_inv @ a) @ b) @ r) @ X)))))))).
% 2.58/1.55 thf(fact_86_a__transpose__inv, axiom, (! [X:a,Y:a,Z2:a]: ((((((((add @ a) @ b) @ r) @ X) @ Y) = Z2) => ((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ ((((a_inv @ a) @ b) @ r) @ X)) @ Z2) = Y)))))))).
% 2.58/1.55 thf(fact_87_eq__obj__rng__of__frac__axioms, axiom, ((((locali1648044335f_frac @ a) @ b) @ r) @ s)).
% 2.58/1.55 thf(fact_88_sum__zero__eq__neg, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add @ a) @ b) @ r) @ X) @ Y) = (((zero @ a) @ b) @ r)) => (X = ((((a_inv @ a) @ b) @ r) @ Y)))))))).
% 2.58/1.55 thf(fact_89_r__neg, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ X) @ ((((a_inv @ a) @ b) @ r) @ X)) = (((zero @ a) @ b) @ r)))))).
% 2.58/1.55 thf(fact_90_minus__equality, axiom, (! [Y:a,X:a]: ((((((((add @ a) @ b) @ r) @ Y) @ X) = (((zero @ a) @ b) @ r)) => ((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((a_inv @ a) @ b) @ r) @ X) = Y))))))).
% 2.58/1.55 thf(fact_91_l__neg, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ ((((a_inv @ a) @ b) @ r) @ X)) @ X) = (((zero @ a) @ b) @ r)))))).
% 2.58/1.55 thf(fact_92_inv__add, axiom, (! [A2:a,B2:a]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((a_inv @ a) @ b) @ r) @ (((((add @ a) @ b) @ r) @ A2) @ B2)) = (((((a_minus @ a) @ b) @ r) @ ((((a_inv @ a) @ b) @ r) @ A2)) @ B2))))))).
% 2.58/1.55 thf(fact_93__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_O_A_092_060lbrakk_062t_A_092_060in_062_AS_059_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_A_092_060ominus_062_As_A_092_060otimes_062_Afst_Ax_H_J_A_061_A_092_060zero_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom, (~ ((! [T:a]: (((((member @ a) @ T) @ s) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ T) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ r2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ (((product_fst @ a) @ a) @ x)))) != (((zero @ a) @ b) @ r)))))))).
% 2.58/1.55 thf(fact_94__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_H_O_A_092_060lbrakk_062t_H_A_092_060in_062_AS_059_At_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_A_092_060ominus_062_As_H_A_092_060otimes_062_Afst_Ay_H_J_A_061_A_092_060zero_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom, (~ ((! [T2:a]: (((((member @ a) @ T2) @ s) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ T2) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ r3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ (((product_fst @ a) @ a) @ y)))) != (((zero @ a) @ b) @ r)))))))).
% 2.58/1.55 thf(fact_95_zero__closed, axiom, (((member @ a) @ (((zero @ a) @ b) @ r)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_96_local_Ominus__minus, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((a_inv @ a) @ b) @ r) @ ((((a_inv @ a) @ b) @ r) @ X)) = X))))).
% 2.58/1.55 thf(fact_97_add_Oinv__closed, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((member @ a) @ ((((a_inv @ a) @ b) @ r) @ X)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)))))).
% 2.58/1.55 thf(fact_98_local_Ominus__zero, axiom, (((((a_inv @ a) @ b) @ r) @ (((zero @ a) @ b) @ r)) = (((zero @ a) @ b) @ r))).
% 2.58/1.55 thf(fact_99_m__closed, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ s) => ((((member @ a) @ Y) @ s) => (((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Y)) @ s)))))).
% 2.58/1.55 thf(fact_100_r__null, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ (((zero @ a) @ b) @ r)) = (((zero @ a) @ b) @ r)))))).
% 2.58/1.55 thf(fact_101_l__null, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((zero @ a) @ b) @ r)) @ X) = (((zero @ a) @ b) @ r)))))).
% 2.58/1.55 thf(fact_102_r__zero, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ X) @ (((zero @ a) @ b) @ r)) = X))))).
% 2.58/1.55 thf(fact_103_l__zero, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ (((zero @ a) @ b) @ r)) @ X) = X))))).
% 2.58/1.55 thf(fact_104_add_Or__cancel__one_H, axiom, (! [X:a,A2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((X = (((((add @ a) @ b) @ r) @ A2) @ X)) = (A2 = (((zero @ a) @ b) @ r)))))))).
% 2.58/1.55 thf(fact_105_add_Or__cancel__one, axiom, (! [X:a,A2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add @ a) @ b) @ r) @ A2) @ X) = X) = (A2 = (((zero @ a) @ b) @ r)))))))).
% 2.58/1.55 thf(fact_106_add_Ol__cancel__one_H, axiom, (! [X:a,A2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((X = (((((add @ a) @ b) @ r) @ X) @ A2)) = (A2 = (((zero @ a) @ b) @ r)))))))).
% 2.58/1.55 thf(fact_107_add_Ol__cancel__one, axiom, (! [X:a,A2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add @ a) @ b) @ r) @ X) @ A2) = X) = (A2 = (((zero @ a) @ b) @ r)))))))).
% 2.58/1.55 thf(fact_108_add_Oinv__eq__1__iff, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((a_inv @ a) @ b) @ r) @ X) = (((zero @ a) @ b) @ r)) = (X = (((zero @ a) @ b) @ r))))))).
% 2.58/1.55 thf(fact_109_mult__submonoid__of__rng__axioms, axiom, ((((locali1402578320of_rng @ a) @ b) @ r) @ s)).
% 2.58/1.55 thf(fact_110_mult__submonoid__of__crng__axioms, axiom, ((((locali7914339f_crng @ a) @ b) @ r) @ s)).
% 2.58/1.55 thf(fact_111_prod__eq__iff, axiom, (! [B:$tType,A:$tType]: (((^ [Y4:((product_prod @ A) @ B),Z3:((product_prod @ A) @ B)]: ((Y4 = Z3))) = (^ [S:((product_prod @ A) @ B),T3:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ S) = (((product_fst @ A) @ B) @ T3)) & ((((product_snd @ A) @ B) @ S) = (((product_snd @ A) @ B) @ T3))))))))).
% 2.58/1.55 thf(fact_112_submonoid__axioms, axiom, ((((locali1441642228monoid @ a) @ ((ring_ext @ a) @ b)) @ r) @ s)).
% 2.58/1.55 thf(fact_113_add_Oint__pow__inv, axiom, (! [X:a,I:int]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add_pow @ a) @ b) @ int) @ r) @ I) @ ((((a_inv @ a) @ b) @ r) @ X)) = ((((a_inv @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ int) @ r) @ I) @ X))))))).
% 2.58/1.55 thf(fact_114_semiring_Osemiring__simprules_I11_J, axiom, (! [B:$tType,A:$tType,R:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((semiring @ A) @ B) @ R) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((((add @ A) @ B) @ R) @ X) @ (((zero @ A) @ B) @ R)) = X)))))).
% 2.58/1.55 thf(fact_115_semiring_Osemiring__simprules_I6_J, axiom, (! [B:$tType,A:$tType,R:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((semiring @ A) @ B) @ R) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((((add @ A) @ B) @ R) @ (((zero @ A) @ B) @ R)) @ X) = X)))))).
% 2.58/1.55 thf(fact_116_semiring_Ol__null, axiom, (! [B:$tType,A:$tType,R:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((semiring @ A) @ B) @ R) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R) @ (((zero @ A) @ B) @ R)) @ X) = (((zero @ A) @ B) @ R))))))).
% 2.58/1.55 thf(fact_117_semiring_Or__null, axiom, (! [B:$tType,A:$tType,R:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((semiring @ A) @ B) @ R) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R) @ X) @ (((zero @ A) @ B) @ R)) = (((zero @ A) @ B) @ R))))))).
% 2.58/1.55 thf(fact_118_semiring_Or__distr, axiom, (! [B:$tType,A:$tType,R:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A,Z2:A]: (((((semiring @ A) @ B) @ R) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((member @ A) @ Z2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R) @ Z2) @ (((((add @ A) @ B) @ R) @ X) @ Y)) = (((((add @ A) @ B) @ R) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R) @ Z2) @ X)) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R) @ Z2) @ Y)))))))))).
% 2.58/1.55 thf(fact_119_add__pow__ldistr__int, axiom, (! [A2:a,B2:a,K:int]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ ((((((add_pow @ a) @ b) @ int) @ r) @ K) @ A2)) @ B2) = ((((((add_pow @ a) @ b) @ int) @ r) @ K) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ A2) @ B2)))))))).
% 2.58/1.55 thf(fact_120_add__pow__rdistr__int, axiom, (! [A2:a,B2:a,K:int]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ A2) @ ((((((add_pow @ a) @ b) @ int) @ r) @ K) @ B2)) = ((((((add_pow @ a) @ b) @ int) @ r) @ K) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ A2) @ B2)))))))).
% 2.58/1.55 thf(fact_121_add_Oint__pow__distrib, axiom, (! [X:a,Y:a,I:int]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add_pow @ a) @ b) @ int) @ r) @ I) @ (((((add @ a) @ b) @ r) @ X) @ Y)) = (((((add @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ int) @ r) @ I) @ X)) @ ((((((add_pow @ a) @ b) @ int) @ r) @ I) @ Y)))))))).
% 2.58/1.55 thf(fact_122_add_Oint__pow__mult__distrib, axiom, (! [X:a,Y:a,I:int]: ((((((((add @ a) @ b) @ r) @ X) @ Y) = (((((add @ a) @ b) @ r) @ Y) @ X)) => ((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add_pow @ a) @ b) @ int) @ r) @ I) @ (((((add @ a) @ b) @ r) @ X) @ Y)) = (((((add @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ int) @ r) @ I) @ X)) @ ((((((add_pow @ a) @ b) @ int) @ r) @ I) @ Y))))))))).
% 2.58/1.55 thf(fact_123_add_Oint__pow__closed, axiom, (! [X:a,I:int]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((member @ a) @ ((((((add_pow @ a) @ b) @ int) @ r) @ I) @ X)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)))))).
% 2.58/1.55 thf(fact_124_add_Oint__pow__one, axiom, (! [Z2:int]: ((((((((add_pow @ a) @ b) @ int) @ r) @ Z2) @ (((zero @ a) @ b) @ r)) = (((zero @ a) @ b) @ r))))).
% 2.58/1.55 thf(fact_125_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Ois__submonoid, axiom, (! [B:$tType,A:$tType,M2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),S2:(set @ A)]: ((((((locali1441642228monoid @ A) @ B) @ M2) @ S2) => ((((locali1441642228monoid @ A) @ B) @ M2) @ S2))))).
% 2.58/1.55 thf(fact_126_mult__submonoid__of__crng_Oaxioms_I2_J, axiom, (! [B:$tType,A:$tType,R:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),S2:(set @ A)]: ((((((locali7914339f_crng @ A) @ B) @ R) @ S2) => ((((locali1402578320of_rng @ A) @ B) @ R) @ S2))))).
% 2.58/1.55 thf(fact_127_mult__submonoid__of__rng_Oaxioms_I2_J, axiom, (! [B:$tType,A:$tType,R:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),S2:(set @ A)]: ((((((locali1402578320of_rng @ A) @ B) @ R) @ S2) => ((((locali1441642228monoid @ A) @ ((ring_ext @ A) @ B)) @ R) @ S2))))).
% 2.58/1.55 thf(fact_128_eq__obj__rng__of__frac_Oaxioms_I2_J, axiom, (! [B:$tType,A:$tType,R:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),S2:(set @ A)]: ((((((locali1648044335f_frac @ A) @ B) @ R) @ S2) => ((((locali7914339f_crng @ A) @ B) @ R) @ S2))))).
% 2.58/1.55 thf(fact_129_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Om__closed, axiom, (! [B:$tType,A:$tType,M2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),S2:(set @ A),X:A,Y:A]: ((((((locali1441642228monoid @ A) @ B) @ M2) @ S2) => ((((member @ A) @ X) @ S2) => ((((member @ A) @ Y) @ S2) => (((member @ A) @ (((((mult @ A) @ B) @ M2) @ X) @ Y)) @ S2))))))).
% 2.58/1.55 thf(fact_130_a__minus__def, axiom, (! [M:$tType,A:$tType]: ((((a_minus @ A) @ M) = (^ [R2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ M))),X2:A,Y5:A]: ((((((add @ A) @ M) @ R2) @ X2) @ ((((a_inv @ A) @ M) @ R2) @ Y5)))))))).
% 2.58/1.55 thf(fact_131_semiring_Osemiring__simprules_I8_J, axiom, (! [B:$tType,A:$tType,R:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A,Z2:A]: (((((semiring @ A) @ B) @ R) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((member @ A) @ Z2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R) @ X) @ Y)) @ Z2) = (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R) @ X) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R) @ Y) @ Z2)))))))))).
% 2.58/1.55 thf(fact_132_semiring_Osemiring__simprules_I3_J, axiom, (! [B:$tType,A:$tType,R:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A]: (((((semiring @ A) @ B) @ R) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => (((member @ A) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R) @ X) @ Y)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)))))))).
% 2.58/1.55 thf(fact_133_semiring_Osemiring__simprules_I12_J, axiom, (! [B:$tType,A:$tType,R:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A,Z2:A]: (((((semiring @ A) @ B) @ R) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((member @ A) @ Z2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((((add @ A) @ B) @ R) @ X) @ (((((add @ A) @ B) @ R) @ Y) @ Z2)) = (((((add @ A) @ B) @ R) @ Y) @ (((((add @ A) @ B) @ R) @ X) @ Z2)))))))))).
% 2.58/1.55 thf(fact_134_semiring_Osemiring__simprules_I7_J, axiom, (! [B:$tType,A:$tType,R:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A]: (((((semiring @ A) @ B) @ R) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((((add @ A) @ B) @ R) @ X) @ Y) = (((((add @ A) @ B) @ R) @ Y) @ X)))))))).
% 2.58/1.55 thf(fact_135_semiring_Osemiring__simprules_I5_J, axiom, (! [B:$tType,A:$tType,R:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A,Z2:A]: (((((semiring @ A) @ B) @ R) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((member @ A) @ Z2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((((add @ A) @ B) @ R) @ (((((add @ A) @ B) @ R) @ X) @ Y)) @ Z2) = (((((add @ A) @ B) @ R) @ X) @ (((((add @ A) @ B) @ R) @ Y) @ Z2)))))))))).
% 2.58/1.55 thf(fact_136_semiring_Osemiring__simprules_I1_J, axiom, (! [B:$tType,A:$tType,R:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A]: (((((semiring @ A) @ B) @ R) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => (((member @ A) @ (((((add @ A) @ B) @ R) @ X) @ Y)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)))))))).
% 2.58/1.55 thf(fact_137_semiring_Osemiring__simprules_I2_J, axiom, (! [B:$tType,A:$tType,R:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((semiring @ A) @ B) @ R) => (((member @ A) @ (((zero @ A) @ B) @ R)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)))))).
% 2.58/1.55 thf(fact_138_semiring_Ol__distr, axiom, (! [B:$tType,A:$tType,R:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A,Z2:A]: (((((semiring @ A) @ B) @ R) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((member @ A) @ Z2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R) @ (((((add @ A) @ B) @ R) @ X) @ Y)) @ Z2) = (((((add @ A) @ B) @ R) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R) @ X) @ Z2)) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R) @ Y) @ Z2)))))))))).
% 2.58/1.55 thf(fact_139_add_Oint__pow__diff, axiom, (! [X:a,N:int,M3:int]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add_pow @ a) @ b) @ int) @ r) @ (((minus_minus @ int) @ N) @ M3)) @ X) = (((((add @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ int) @ r) @ N) @ X)) @ ((((a_inv @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ int) @ r) @ M3) @ X)))))))).
% 2.58/1.55 thf(fact_140_add_Oint__pow__neg, axiom, (! [X:a,I:int]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add_pow @ a) @ b) @ int) @ r) @ ((uminus_uminus @ int) @ I)) @ X) = ((((a_inv @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ int) @ r) @ I) @ X))))))).
% 2.58/1.55 thf(fact_141_add_Oint__pow__mult, axiom, (! [X:a,I:int,J:int]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add_pow @ a) @ b) @ int) @ r) @ (((plus_plus @ int) @ I) @ J)) @ X) = (((((add @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ int) @ r) @ I) @ X)) @ ((((((add_pow @ a) @ b) @ int) @ r) @ J) @ X))))))).
% 2.58/1.55 thf(fact_142_subset, axiom, (((ord_less_eq @ (set @ a)) @ s) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.58/1.55 thf(fact_143_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Osubset, axiom, (! [B:$tType,A:$tType,M2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),S2:(set @ A)]: ((((((locali1441642228monoid @ A) @ B) @ M2) @ S2) => (((ord_less_eq @ (set @ A)) @ S2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ M2)))))).
% 2.58/1.55 thf(fact_144_diff__minus__eq__add, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ A2) @ ((uminus_uminus @ A) @ B2)) = (((plus_plus @ A) @ A2) @ B2)))))))).
% 2.58/1.55 thf(fact_145_uminus__add__conv__diff, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ B2) = (((minus_minus @ A) @ B2) @ A2)))))))).
% 2.58/1.55 thf(fact_146_le__add__diff__inverse, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((plus_plus @ A) @ B2) @ (((minus_minus @ A) @ A2) @ B2)) = A2)))))))).
% 2.58/1.55 thf(fact_147_le__add__diff__inverse2, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((plus_plus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ B2) = A2)))))))).
% 2.58/1.55 thf(fact_148_add__right__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A2:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C2) @ A2)) = (B2 = C2)))))))).
% 2.58/1.55 thf(fact_149_add__left__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C2)) = (B2 = C2)))))))).
% 2.58/1.55 thf(fact_150_neg__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((uminus_uminus @ A) @ A2) = ((uminus_uminus @ A) @ B2)) = (A2 = B2)))))))).
% 2.58/1.55 thf(fact_151_add_Oinverse__inverse, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ A2)) = A2))))))).
% 2.58/1.55 thf(fact_152_add__le__cancel__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.58/1.55 thf(fact_153_add__le__cancel__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.58/1.55 thf(fact_154_neg__le__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.58/1.55 thf(fact_155_add__diff__cancel__right_H, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = A2))))))).
% 2.58/1.55 thf(fact_156_add__diff__cancel__right, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,C2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ A2) @ B2)))))))).
% 2.58/1.55 thf(fact_157_add__diff__cancel__left_H, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ A2) = B2))))))).
% 2.58/1.55 thf(fact_158_add__diff__cancel__left, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [C2:A,A2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2)) = (((minus_minus @ A) @ A2) @ B2)))))))).
% 2.58/1.55 thf(fact_159_diff__add__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ B2) = A2))))))).
% 2.58/1.55 thf(fact_160_add__diff__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = A2))))))).
% 2.58/1.55 thf(fact_161_minus__add__distrib, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,B2:A]: ((((uminus_uminus @ A) @ (((plus_plus @ A) @ A2) @ B2)) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ ((uminus_uminus @ A) @ B2))))))))).
% 2.58/1.55 thf(fact_162_minus__add__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ (((plus_plus @ A) @ A2) @ B2)) = B2))))))).
% 2.58/1.55 thf(fact_163_add__minus__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ B2)) = B2))))))).
% 2.58/1.55 thf(fact_164_minus__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: ((((uminus_uminus @ A) @ (((minus_minus @ A) @ A2) @ B2)) = (((minus_minus @ A) @ B2) @ A2)))))))).
% 2.58/1.55 thf(fact_165_add__right__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A2:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C2) @ A2)) => (B2 = C2)))))))).
% 2.58/1.55 thf(fact_166_add__left__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C2)) => (B2 = C2)))))))).
% 2.58/1.55 thf(fact_167_add_Oleft__commute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [B2:A,A2:A,C2:A]: (((((plus_plus @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C2)) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 2.58/1.55 thf(fact_168_add_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => ((plus_plus @ A) = (^ [A4:A,B3:A]: ((((plus_plus @ A) @ B3) @ A4)))))))).
% 2.58/1.55 thf(fact_169_group__add__class_Oadd_Oright__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [B2:A,A2:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C2) @ A2)) = (B2 = C2)))))))).
% 2.58/1.55 thf(fact_170_add_Oleft__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C2)) = (B2 = C2)))))))).
% 2.58/1.55 thf(fact_171_add_Oassoc, axiom, (! [A:$tType]: (((semigroup_add @ A) => (! [A2:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 2.58/1.55 thf(fact_172_group__cancel_Oadd2, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [B4:A,K:A,B2:A,A2:A]: (((B4 = (((plus_plus @ A) @ K) @ B2)) => ((((plus_plus @ A) @ A2) @ B4) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A2) @ B2)))))))))).
% 2.58/1.55 thf(fact_173_group__cancel_Oadd1, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A3:A,K:A,A2:A,B2:A]: (((A3 = (((plus_plus @ A) @ K) @ A2)) => ((((plus_plus @ A) @ A3) @ B2) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A2) @ B2)))))))))).
% 2.58/1.55 thf(fact_174_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (K = L)) => ((((plus_plus @ A) @ I) @ K) = (((plus_plus @ A) @ J) @ L))))))))).
% 2.58/1.55 thf(fact_175_ab__semigroup__add__class_Oadd__ac_I1_J, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [A2:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 2.58/1.55 thf(fact_176_diff__right__commute, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,C2:A,B2:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ B2) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2)))))))).
% 2.58/1.55 thf(fact_177_diff__eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A,D:A]: ((((((minus_minus @ A) @ A2) @ B2) = (((minus_minus @ A) @ C2) @ D)) => ((A2 = B2) = (C2 = D))))))))).
% 2.58/1.55 thf(fact_178_minus__equation__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((uminus_uminus @ A) @ A2) = B2) = (((uminus_uminus @ A) @ B2) = A2)))))))).
% 2.58/1.55 thf(fact_179_equation__minus__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((A2 = ((uminus_uminus @ A) @ B2)) = (B2 = ((uminus_uminus @ A) @ A2))))))))).
% 2.58/1.55 thf(fact_180_add__le__imp__le__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.58/1.55 thf(fact_181_add__le__imp__le__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2)) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.58/1.55 thf(fact_182_le__iff__add, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => ((ord_less_eq @ A) = (^ [A4:A,B3:A]: ((? [C3:A]: ((B3 = (((plus_plus @ A) @ A4) @ C3))))))))))).
% 2.58/1.55 thf(fact_183_add__right__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 2.58/1.55 thf(fact_184_less__eqE, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (~ ((! [C4:A]: ((B2 != (((plus_plus @ A) @ A2) @ C4))))))))))))).
% 2.58/1.55 thf(fact_185_add__left__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2))))))))).
% 2.58/1.55 thf(fact_186_add__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A2:A,B2:A,C2:A,D:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ D) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ D)))))))))).
% 2.58/1.55 thf(fact_187_add__mono__thms__linordered__semiring_I1_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.58/1.55 thf(fact_188_add__mono__thms__linordered__semiring_I2_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.58/1.55 thf(fact_189_add__mono__thms__linordered__semiring_I3_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (K = L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.58/1.55 thf(fact_190_diff__eq__diff__less__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A,C2:A,D:A]: ((((((minus_minus @ A) @ A2) @ B2) = (((minus_minus @ A) @ C2) @ D)) => ((((ord_less_eq @ A) @ A2) @ B2) = (((ord_less_eq @ A) @ C2) @ D))))))))).
% 2.58/1.55 thf(fact_191_diff__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ (((minus_minus @ A) @ B2) @ C2))))))))).
% 2.58/1.55 thf(fact_192_diff__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C2) @ A2)) @ (((minus_minus @ A) @ C2) @ B2))))))))).
% 2.58/1.55 thf(fact_193_diff__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A,D:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ D) @ C2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ (((minus_minus @ A) @ B2) @ D)))))))))).
% 2.58/1.55 thf(fact_194_le__imp__neg__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2))))))))).
% 2.58/1.55 thf(fact_195_minus__le__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ B2) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ A2)))))))).
% 2.58/1.55 thf(fact_196_le__minus__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ B2)) = (((ord_less_eq @ A) @ B2) @ ((uminus_uminus @ A) @ A2))))))))).
% 2.58/1.55 thf(fact_197_add__implies__diff, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [C2:A,B2:A,A2:A]: ((((((plus_plus @ A) @ C2) @ B2) = A2) => (C2 = (((minus_minus @ A) @ A2) @ B2))))))))).
% 2.58/1.55 thf(fact_198_diff__diff__add, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,B2:A,C2:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2) = (((minus_minus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 2.58/1.55 thf(fact_199_diff__add__eq__diff__diff__swap, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((minus_minus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ B2)))))))).
% 2.58/1.55 thf(fact_200_diff__add__eq, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2) = (((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ B2)))))))).
% 2.58/1.55 thf(fact_201_diff__diff__eq2, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((minus_minus @ A) @ A2) @ (((minus_minus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ B2)))))))).
% 2.58/1.55 thf(fact_202_add__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((plus_plus @ A) @ A2) @ (((minus_minus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2)))))))).
% 2.58/1.55 thf(fact_203_eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,C2:A,B2:A]: (((A2 = (((minus_minus @ A) @ C2) @ B2)) = ((((plus_plus @ A) @ A2) @ B2) = C2)))))))).
% 2.58/1.55 thf(fact_204_diff__eq__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: ((((((minus_minus @ A) @ A2) @ B2) = C2) = (A2 = (((plus_plus @ A) @ C2) @ B2))))))))).
% 2.58/1.55 thf(fact_205_group__cancel_Osub1, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A,K:A,A2:A,B2:A]: (((A3 = (((plus_plus @ A) @ K) @ A2)) => ((((minus_minus @ A) @ A3) @ B2) = (((plus_plus @ A) @ K) @ (((minus_minus @ A) @ A2) @ B2)))))))))).
% 2.58/1.55 thf(fact_206_add_Oinverse__distrib__swap, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: ((((uminus_uminus @ A) @ (((plus_plus @ A) @ A2) @ B2)) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2))))))))).
% 2.58/1.55 thf(fact_207_group__cancel_Oneg1, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A,K:A,A2:A]: (((A3 = (((plus_plus @ A) @ K) @ A2)) => (((uminus_uminus @ A) @ A3) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ K)) @ ((uminus_uminus @ A) @ A2)))))))))).
% 2.58/1.55 thf(fact_208_minus__diff__commute, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [B2:A,A2:A]: (((((minus_minus @ A) @ ((uminus_uminus @ A) @ B2)) @ A2) = (((minus_minus @ A) @ ((uminus_uminus @ A) @ A2)) @ B2)))))))).
% 2.58/1.55 thf(fact_209_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add, axiom, (! [A:$tType]: (((ordere623563068d_diff @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ A2) @ B2) => (((((minus_minus @ A) @ B2) @ A2) = C2) = (B2 = (((plus_plus @ A) @ C2) @ A2))))))))))).
% 2.58/1.55 thf(fact_210_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse, axiom, (! [A:$tType]: (((ordere623563068d_diff @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((plus_plus @ A) @ A2) @ (((minus_minus @ A) @ B2) @ A2)) = B2)))))))).
% 2.58/1.55 thf(fact_211_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right, axiom, (! [A:$tType]: (((ordere623563068d_diff @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((minus_minus @ A) @ C2) @ (((minus_minus @ A) @ B2) @ A2)) = (((minus_minus @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ B2))))))))).
% 2.58/1.55 thf(fact_212_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2, axiom, (! [A:$tType]: (((ordere623563068d_diff @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((minus_minus @ A) @ (((plus_plus @ A) @ B2) @ C2)) @ A2) = (((plus_plus @ A) @ (((minus_minus @ A) @ B2) @ A2)) @ C2))))))))).
% 2.58/1.55 thf(fact_213_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2, axiom, (! [A:$tType]: (((ordere623563068d_diff @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((plus_plus @ A) @ (((minus_minus @ A) @ B2) @ A2)) @ C2) = (((minus_minus @ A) @ (((plus_plus @ A) @ B2) @ C2)) @ A2))))))))).
% 2.58/1.55 thf(fact_214_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc, axiom, (! [A:$tType]: (((ordere623563068d_diff @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((minus_minus @ A) @ (((plus_plus @ A) @ C2) @ B2)) @ A2) = (((plus_plus @ A) @ C2) @ (((minus_minus @ A) @ B2) @ A2)))))))))).
% 2.58/1.55 thf(fact_215_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc, axiom, (! [A:$tType]: (((ordere623563068d_diff @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((plus_plus @ A) @ C2) @ (((minus_minus @ A) @ B2) @ A2)) = (((minus_minus @ A) @ (((plus_plus @ A) @ C2) @ B2)) @ A2))))))))).
% 2.58/1.55 thf(fact_216_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2, axiom, (! [A:$tType]: (((ordere623563068d_diff @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ (((minus_minus @ A) @ B2) @ A2)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ B2))))))))).
% 2.58/1.55 thf(fact_217_le__add__diff, axiom, (! [A:$tType]: (((ordere623563068d_diff @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ C2) @ (((minus_minus @ A) @ (((plus_plus @ A) @ B2) @ C2)) @ A2))))))))).
% 2.58/1.55 thf(fact_218_add__le__add__imp__diff__le, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [I:A,K:A,N:A,J:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ N) => ((((ord_less_eq @ A) @ N) @ (((plus_plus @ A) @ J) @ K)) => ((((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ N) => ((((ord_less_eq @ A) @ N) @ (((plus_plus @ A) @ J) @ K)) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ N) @ K)) @ J))))))))))).
% 2.58/1.55 thf(fact_219_diff__add, axiom, (! [A:$tType]: (((ordere623563068d_diff @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((plus_plus @ A) @ (((minus_minus @ A) @ B2) @ A2)) @ A2) = B2)))))))).
% 2.58/1.55 thf(fact_220_add__le__imp__le__diff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [I:A,K:A,N:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ N) => (((ord_less_eq @ A) @ I) @ (((minus_minus @ A) @ N) @ K))))))))).
% 2.58/1.55 thf(fact_221_le__diff__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (((minus_minus @ A) @ C2) @ B2)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2)))))))).
% 2.58/1.55 thf(fact_222_diff__le__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2) = (((ord_less_eq @ A) @ A2) @ (((plus_plus @ A) @ C2) @ B2))))))))).
% 2.58/1.55 thf(fact_223_ab__group__add__class_Oab__diff__conv__add__uminus, axiom, (! [A:$tType]: (((ab_group_add @ A) => ((minus_minus @ A) = (^ [A4:A,B3:A]: ((((plus_plus @ A) @ A4) @ ((uminus_uminus @ A) @ B3))))))))).
% 2.58/1.55 thf(fact_224_diff__conv__add__uminus, axiom, (! [A:$tType]: (((group_add @ A) => ((minus_minus @ A) = (^ [A4:A,B3:A]: ((((plus_plus @ A) @ A4) @ ((uminus_uminus @ A) @ B3))))))))).
% 2.58/1.55 thf(fact_225_group__cancel_Osub2, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [B4:A,K:A,B2:A,A2:A]: (((B4 = (((plus_plus @ A) @ K) @ B2)) => ((((minus_minus @ A) @ A2) @ B4) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ K)) @ (((minus_minus @ A) @ A2) @ B2)))))))))).
% 2.58/1.55 thf(fact_226_add_Oone__in__subset, axiom, (! [H:(set @ a)]: (((((ord_less_eq @ (set @ a)) @ H) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((H != (bot_bot @ (set @ a))) => ((! [X3:a]: (((((member @ a) @ X3) @ H) => (((member @ a) @ ((((a_inv @ a) @ b) @ r) @ X3)) @ H)))) => ((! [X3:a]: (((((member @ a) @ X3) @ H) => (! [Xa:a]: (((((member @ a) @ Xa) @ H) => (((member @ a) @ (((((add @ a) @ b) @ r) @ X3) @ Xa)) @ H))))))) => (((member @ a) @ (((zero @ a) @ b) @ r)) @ H)))))))).
% 2.58/1.55 thf(fact_227_a__lcos__mult__one, axiom, (! [M2:(set @ a)]: (((((ord_less_eq @ (set @ a)) @ M2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((a_l_coset @ a) @ b) @ r) @ (((zero @ a) @ b) @ r)) @ M2) = M2))))).
% 2.58/1.55 thf(fact_228_a__lcos__m__assoc, axiom, (! [M2:(set @ a),G:a,H2:a]: (((((ord_less_eq @ (set @ a)) @ M2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ G) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ H2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((a_l_coset @ a) @ b) @ r) @ G) @ (((((a_l_coset @ a) @ b) @ r) @ H2) @ M2)) = (((((a_l_coset @ a) @ b) @ r) @ (((((add @ a) @ b) @ r) @ G) @ H2)) @ M2)))))))).
% 2.58/1.55 thf(fact_229_carrier__not__empty, axiom, ((((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r) != (bot_bot @ (set @ a)))).
% 2.58/1.55 thf(fact_230_a__l__coset__subset__G, axiom, (! [H:(set @ a),X:a]: (((((ord_less_eq @ (set @ a)) @ H) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((ord_less_eq @ (set @ a)) @ (((((a_l_coset @ a) @ b) @ r) @ X) @ H)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))))))).
% 2.58/1.55 thf(fact_231_setadd__subset__G, axiom, (! [H:(set @ a),K2:(set @ a)]: (((((ord_less_eq @ (set @ a)) @ H) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((ord_less_eq @ (set @ a)) @ K2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((ord_less_eq @ (set @ a)) @ (((((set_add @ a) @ b) @ r) @ H) @ K2)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))))))).
% 2.58/1.55 thf(fact_232_set__add__closed, axiom, (! [A3:(set @ a),B4:(set @ a)]: (((((ord_less_eq @ (set @ a)) @ A3) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((ord_less_eq @ (set @ a)) @ B4) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((ord_less_eq @ (set @ a)) @ (((((set_add @ a) @ b) @ r) @ A3) @ B4)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))))))).
% 2.58/1.55 thf(fact_233_add__additive__subgroups, axiom, (! [H:(set @ a),K2:(set @ a)]: ((((((additive_subgroup @ a) @ b) @ H) @ r) => (((((additive_subgroup @ a) @ b) @ K2) @ r) => ((((additive_subgroup @ a) @ b) @ (((((set_add @ a) @ b) @ r) @ H) @ K2)) @ r)))))).
% 2.58/1.55 thf(fact_234_Diff__eq__empty__iff, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A)]: ((((((minus_minus @ (set @ A)) @ A3) @ B4) = (bot_bot @ (set @ A))) = (((ord_less_eq @ (set @ A)) @ A3) @ B4))))).
% 2.58/1.55 thf(fact_235_empty__iff, axiom, (! [A:$tType,C2:A]: ((~ ((((member @ A) @ C2) @ (bot_bot @ (set @ A)))))))).
% 2.58/1.55 thf(fact_236_all__not__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X2:A]: ((~ ((((member @ A) @ X2) @ A3))))) = (A3 = (bot_bot @ (set @ A))))))).
% 2.58/1.55 thf(fact_237_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X2:A]: ((~ ((P @ X2))))))))).
% 2.58/1.55 thf(fact_238_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X2:A]: ((~ ((P @ X2))))))))).
% 2.58/1.55 thf(fact_239_subsetI, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A3) => (((member @ A) @ X3) @ B4)))) => (((ord_less_eq @ (set @ A)) @ A3) @ B4))))).
% 2.58/1.55 thf(fact_240_subset__antisym, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ A3) => (A3 = B4)))))).
% 2.58/1.55 thf(fact_241_Compl__subset__Compl__iff, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A3)) @ ((uminus_uminus @ (set @ A)) @ B4)) = (((ord_less_eq @ (set @ A)) @ B4) @ A3))))).
% 2.58/1.55 thf(fact_242_Compl__anti__mono, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B4) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ B4)) @ ((uminus_uminus @ (set @ A)) @ A3)))))).
% 2.58/1.55 thf(fact_243_subset__empty, axiom, (! [A:$tType,A3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = (A3 = (bot_bot @ (set @ A))))))).
% 2.58/1.55 thf(fact_244_empty__subsetI, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A3)))).
% 2.58/1.55 thf(fact_245_Diff__empty, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = A3)))).
% 2.58/1.55 thf(fact_246_empty__Diff, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ (bot_bot @ (set @ A))) @ A3) = (bot_bot @ (set @ A)))))).
% 2.58/1.55 thf(fact_247_Diff__cancel, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ A3) = (bot_bot @ (set @ A)))))).
% 2.58/1.55 thf(fact_248_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.58/1.55 thf(fact_249_equals0D, axiom, (! [A:$tType,A3:(set @ A),A2:A]: (((A3 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A3))))))).
% 2.58/1.55 thf(fact_250_equals0I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [Y2:A]: ((~ ((((member @ A) @ Y2) @ A3))))) => (A3 = (bot_bot @ (set @ A))))))).
% 2.58/1.55 thf(fact_251_ex__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((? [X2:A]: ((((member @ A) @ X2) @ A3))) = (A3 != (bot_bot @ (set @ A))))))).
% 2.58/1.55 thf(fact_252_in__mono, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B4) => ((((member @ A) @ X) @ A3) => (((member @ A) @ X) @ B4)))))).
% 2.58/1.55 thf(fact_253_subsetD, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B4) => ((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ B4)))))).
% 2.58/1.55 thf(fact_254_equalityE, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A)]: (((A3 = B4) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B4) => (~ ((((ord_less_eq @ (set @ A)) @ B4) @ A3)))))))))).
% 2.58/1.55 thf(fact_255_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: ((! [X2:A]: (((((member @ A) @ X2) @ A5) => (((member @ A) @ X2) @ B5)))))))))).
% 2.58/1.55 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le, axiom, (ordere236663937imp_le @ int)).
% 2.58/1.55 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add, axiom, (ordere779506340up_add @ int)).
% 2.58/1.55 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add, axiom, (cancel146912293up_add @ int)).
% 2.58/1.55 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add, axiom, (cancel1352612707id_add @ int)).
% 2.58/1.55 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add, axiom, (ordered_ab_group_add @ int)).
% 2.58/1.55 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add, axiom, (cancel_semigroup_add @ int)).
% 2.58/1.55 thf(tcon_Int_Oint___Rings_Olinordered__semidom, axiom, (linordered_semidom @ int)).
% 2.58/1.55 thf(tcon_Int_Oint___Groups_Oab__semigroup__add, axiom, (ab_semigroup_add @ int)).
% 2.58/1.55 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add, axiom, (comm_monoid_add @ int)).
% 2.58/1.55 thf(tcon_Int_Oint___Groups_Osemigroup__add, axiom, (semigroup_add @ int)).
% 2.58/1.55 thf(tcon_Int_Oint___Groups_Oab__group__add, axiom, (ab_group_add @ int)).
% 2.58/1.55 thf(tcon_Int_Oint___Groups_Ogroup__add, axiom, (group_add @ int)).
% 2.58/1.55 thf(conj_0, conjecture, ((((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ s3)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ r2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ (((product_fst @ a) @ a) @ x)))))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ s2)) @ (((product_snd @ a) @ a) @ x))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ r3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ (((product_fst @ a) @ a) @ y)))))) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ t2) @ t)) @ (((((add @ a) @ b) @ r) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ r2))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ y)) @ (((product_fst @ a) @ a) @ x))))) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_snd @ a) @ a) @ y))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((product_snd @ a) @ a) @ x)) @ (((product_fst @ a) @ a) @ y)))))))).
% 2.58/1.55 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.TcuWUcH1aD/DTF2DTF3078.p
% 2.58/1.55 ------------------------
% 2.58/1.58 ---- Cleaned THF ---
% 2.58/1.58 thf(ty_t_Congruence_Opartial__object_Opartial__object__ext,type,
% 2.58/1.58 partia1265057091ct_ext: $tType > $tType > $tType ).
% 2.58/1.58
% 2.58/1.58 thf(ty_t_Group_Omonoid_Omonoid__ext,type,
% 2.58/1.58 monoid_ext: $tType > $tType > $tType ).
% 2.58/1.58
% 2.58/1.58 thf(ty_t_Ring_Oring_Oring__ext,type,
% 2.58/1.58 ring_ext: $tType > $tType > $tType ).
% 2.58/1.58
% 2.58/1.58 thf(ty_t_Product__Type_Oprod,type,
% 2.58/1.58 product_prod: $tType > $tType > $tType ).
% 2.58/1.58
% 2.58/1.58 thf(ty_t_Set_Oset,type,
% 2.58/1.58 set: $tType > $tType ).
% 2.58/1.58
% 2.58/1.58 thf(ty_t_Int_Oint,type,
% 2.58/1.58 int: $tType ).
% 2.58/1.58
% 2.58/1.58 thf(ty_tf_b,type,
% 2.58/1.58 b: $tType ).
% 2.58/1.58
% 2.58/1.58 thf(ty_tf_a,type,
% 2.58/1.58 a: $tType ).
% 2.58/1.58
% 2.58/1.58 thf(sy_cl_Groups_Ogroup__add,type,
% 2.58/1.58 group_add:
% 2.58/1.58 !>[A: $tType] : $o ).
% 2.58/1.58
% 2.58/1.58 thf(sy_cl_Groups_Oab__group__add,type,
% 2.58/1.58 ab_group_add:
% 2.58/1.58 !>[A: $tType] : $o ).
% 2.58/1.58
% 2.58/1.58 thf(sy_cl_Groups_Osemigroup__add,type,
% 2.58/1.58 semigroup_add:
% 2.58/1.58 !>[A: $tType] : $o ).
% 2.58/1.58
% 2.58/1.58 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 2.58/1.58 comm_monoid_add:
% 2.58/1.58 !>[A: $tType] : $o ).
% 2.58/1.58
% 2.58/1.58 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 2.58/1.58 ab_semigroup_add:
% 2.58/1.58 !>[A: $tType] : $o ).
% 2.58/1.58
% 2.58/1.58 thf(sy_cl_Rings_Olinordered__semidom,type,
% 2.58/1.58 linordered_semidom:
% 2.58/1.58 !>[A: $tType] : $o ).
% 2.58/1.58
% 2.58/1.58 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 2.58/1.58 cancel_semigroup_add:
% 2.58/1.58 !>[A: $tType] : $o ).
% 2.58/1.58
% 2.58/1.58 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.58/1.58 ordered_ab_group_add:
% 2.58/1.58 !>[A: $tType] : $o ).
% 2.58/1.58
% 2.58/1.58 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 2.58/1.58 cancel1352612707id_add:
% 2.58/1.58 !>[A: $tType] : $o ).
% 2.58/1.58
% 2.58/1.58 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.58/1.58 cancel146912293up_add:
% 2.58/1.58 !>[A: $tType] : $o ).
% 2.58/1.58
% 2.58/1.58 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 2.58/1.58 ordere779506340up_add:
% 2.58/1.58 !>[A: $tType] : $o ).
% 2.58/1.58
% 2.58/1.58 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.58/1.58 canoni770627133id_add:
% 2.58/1.58 !>[A: $tType] : $o ).
% 2.58/1.58
% 2.58/1.58 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 2.58/1.58 ordere236663937imp_le:
% 2.58/1.58 !>[A: $tType] : $o ).
% 2.58/1.58
% 2.58/1.58 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
% 2.58/1.58 ordere623563068d_diff:
% 2.58/1.58 !>[A: $tType] : $o ).
% 2.58/1.58
% 2.58/1.58 thf(sy_c_AbelCoset_Oa__l__coset,type,
% 2.58/1.58 a_l_coset:
% 2.58/1.58 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > A > ( set @ A ) > ( set @ A ) ) ).
% 2.58/1.58
% 2.58/1.58 thf(sy_c_AbelCoset_Oadditive__subgroup,type,
% 2.58/1.58 additive_subgroup:
% 2.58/1.58 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 2.58/1.58
% 2.58/1.58 thf(sy_c_AbelCoset_Oset__add,type,
% 2.58/1.58 set_add:
% 2.58/1.58 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > ( set @ A ) > ( set @ A ) ) ).
% 2.58/1.58
% 2.58/1.58 thf(sy_c_Congruence_Opartial__object_Ocarrier,type,
% 2.58/1.58 partial_carrier:
% 2.58/1.58 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ Z ) > ( set @ A ) ) ).
% 2.58/1.58
% 2.58/1.58 thf(sy_c_Group_Omonoid_Omult,type,
% 2.58/1.58 mult:
% 2.58/1.58 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ Z ) ) > A > A > A ) ).
% 2.58/1.58
% 2.58/1.58 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.58/1.58 minus_minus:
% 2.58/1.58 !>[A: $tType] : ( A > A > A ) ).
% 2.58/1.58
% 2.58/1.58 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 2.58/1.58 plus_plus:
% 2.58/1.58 !>[A: $tType] : ( A > A > A ) ).
% 2.58/1.58
% 2.58/1.58 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 2.58/1.58 uminus_uminus:
% 2.58/1.58 !>[A: $tType] : ( A > A ) ).
% 2.58/1.58
% 2.58/1.58 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Oeq__obj__rng__of__frac,type,
% 2.58/1.58 locali1648044335f_frac:
% 2.58/1.58 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 2.58/1.58
% 2.58/1.58 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Omult__submonoid__of__crng,type,
% 2.58/1.58 locali7914339f_crng:
% 2.58/1.58 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 2.58/1.58
% 2.58/1.58 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Omult__submonoid__of__rng,type,
% 2.58/1.58 locali1402578320of_rng:
% 2.58/1.58 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 2.58/1.58
% 2.58/1.58 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Osubmonoid,type,
% 2.58/1.58 locali1441642228monoid:
% 2.58/1.58 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > ( set @ A ) > $o ) ).
% 2.58/1.58
% 2.58/1.58 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.58/1.58 bot_bot:
% 2.58/1.58 !>[A: $tType] : A ).
% 2.58/1.58
% 2.58/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.58/1.58 ord_less_eq:
% 2.58/1.58 !>[A: $tType] : ( A > A > $o ) ).
% 2.58/1.58
% 2.58/1.58 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 2.58/1.58 product_fst:
% 2.58/1.58 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 2.58/1.58
% 2.58/1.58 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 2.58/1.58 product_snd:
% 2.58/1.58 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 2.58/1.58
% 2.58/1.58 thf(sy_c_Ring_Oa__inv,type,
% 2.58/1.58 a_inv:
% 2.58/1.58 !>[A: $tType,M: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ M ) ) ) > A > A ) ).
% 2.58/1.58
% 2.58/1.58 thf(sy_c_Ring_Oa__minus,type,
% 2.58/1.58 a_minus:
% 2.58/1.58 !>[A: $tType,M: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ M ) ) ) > A > A > A ) ).
% 2.58/1.58
% 2.58/1.58 thf(sy_c_Ring_Oadd__pow,type,
% 2.58/1.58 add_pow:
% 2.58/1.58 !>[A: $tType,C: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ C ) ) ) > B > A > A ) ).
% 2.58/1.58
% 2.58/1.58 thf(sy_c_Ring_Oring_Oadd,type,
% 2.58/1.58 add:
% 2.58/1.58 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ) ) > A > A > A ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Ring_Oring_Ozero,type,
% 2.58/1.59 zero:
% 2.58/1.59 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ) ) > A ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Ring_Osemiring,type,
% 2.58/1.59 semiring:
% 2.58/1.59 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_Set_OCollect,type,
% 2.58/1.59 collect:
% 2.58/1.59 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_c_member,type,
% 2.58/1.59 member:
% 2.58/1.59 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_v_R,type,
% 2.58/1.59 r: partia1265057091ct_ext @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(sy_v_S,type,
% 2.58/1.59 s: set @ a ).
% 2.58/1.59
% 2.58/1.59 thf(sy_v_r,type,
% 2.58/1.59 r2: a ).
% 2.58/1.59
% 2.58/1.59 thf(sy_v_r_H,type,
% 2.58/1.59 r3: a ).
% 2.58/1.59
% 2.58/1.59 thf(sy_v_s,type,
% 2.58/1.59 s2: a ).
% 2.58/1.59
% 2.58/1.59 thf(sy_v_s_H,type,
% 2.58/1.59 s3: a ).
% 2.58/1.59
% 2.58/1.59 thf(sy_v_t_H____,type,
% 2.58/1.59 t: a ).
% 2.58/1.59
% 2.58/1.59 thf(sy_v_t____,type,
% 2.58/1.59 t2: a ).
% 2.58/1.59
% 2.58/1.59 thf(sy_v_x_H____,type,
% 2.58/1.59 x: product_prod @ a @ a ).
% 2.58/1.59
% 2.58/1.59 thf(sy_v_y_H____,type,
% 2.58/1.59 y: product_prod @ a @ a ).
% 2.58/1.59
% 2.58/1.59 thf(fact_0_f18,axiom,
% 2.58/1.59 member @ a @ t @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_1_f17,axiom,
% 2.58/1.59 member @ a @ t2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_2_right__add__eq,axiom,
% 2.58/1.59 ! [A2: a,B2: a,C2: a] :
% 2.58/1.59 ( ( A2 = B2 )
% 2.58/1.59 => ( ( add @ a @ b @ r @ C2 @ A2 )
% 2.58/1.59 = ( add @ a @ b @ r @ C2 @ B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_3_local_Oright__minus__eq,axiom,
% 2.58/1.59 ! [A2: a,B2: a,C2: a] :
% 2.58/1.59 ( ( A2 = B2 )
% 2.58/1.59 => ( ( a_minus @ a @ b @ r @ C2 @ A2 )
% 2.58/1.59 = ( a_minus @ a @ b @ r @ C2 @ B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_4_m__assoc,axiom,
% 2.58/1.59 ! [X: a,Y: a,Z2: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ Z2 )
% 2.58/1.59 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_5_m__comm,axiom,
% 2.58/1.59 ! [X: a,Y: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y )
% 2.58/1.59 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ X ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_6_m__lcomm,axiom,
% 2.58/1.59 ! [X: a,Y: a,Z2: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) )
% 2.58/1.59 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Z2 ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_7_add_Om__assoc,axiom,
% 2.58/1.59 ! [X: a,Y: a,Z2: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( add @ a @ b @ r @ ( add @ a @ b @ r @ X @ Y ) @ Z2 )
% 2.58/1.59 = ( add @ a @ b @ r @ X @ ( add @ a @ b @ r @ Y @ Z2 ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_8_add_Om__comm,axiom,
% 2.58/1.59 ! [X: a,Y: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( add @ a @ b @ r @ X @ Y )
% 2.58/1.59 = ( add @ a @ b @ r @ Y @ X ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_9_add_Om__lcomm,axiom,
% 2.58/1.59 ! [X: a,Y: a,Z2: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( add @ a @ b @ r @ X @ ( add @ a @ b @ r @ Y @ Z2 ) )
% 2.58/1.59 = ( add @ a @ b @ r @ Y @ ( add @ a @ b @ r @ X @ Z2 ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_10_f20,axiom,
% 2.58/1.59 member @ a @ s3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_11_f19,axiom,
% 2.58/1.59 member @ a @ s2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_12_f41,axiom,
% 2.58/1.59 member @ a @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_13_f42,axiom,
% 2.58/1.59 member @ a @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_14_l__distr,axiom,
% 2.58/1.59 ! [X: a,Y: a,Z2: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add @ a @ b @ r @ X @ Y ) @ Z2 )
% 2.58/1.59 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Z2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_15_r__distr,axiom,
% 2.58/1.59 ! [X: a,Y: a,Z2: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Z2 @ ( add @ a @ b @ r @ X @ Y ) )
% 2.58/1.59 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Z2 @ X ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Z2 @ Y ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_16_four__elem__comm,axiom,
% 2.58/1.59 ! [A2: a,B2: a,C2: a,D: a] :
% 2.58/1.59 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ D @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( a_minus @ a @ b @ r @ ( add @ a @ b @ r @ ( a_minus @ a @ b @ r @ A2 @ C2 ) @ B2 ) @ D )
% 2.58/1.59 = ( a_minus @ a @ b @ r @ ( a_minus @ a @ b @ r @ ( add @ a @ b @ r @ A2 @ B2 ) @ C2 ) @ D ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_17_right__inv__add,axiom,
% 2.58/1.59 ! [A2: a,B2: a,C2: a] :
% 2.58/1.59 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( a_minus @ a @ b @ r @ ( a_minus @ a @ b @ r @ C2 @ A2 ) @ B2 )
% 2.58/1.59 = ( a_minus @ a @ b @ r @ C2 @ ( add @ a @ b @ r @ A2 @ B2 ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_18_f21,axiom,
% 2.58/1.59 member @ a @ ( product_snd @ a @ a @ y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_19_f40,axiom,
% 2.58/1.59 ( ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) ) )
% 2.58/1.59 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_20_f22,axiom,
% 2.58/1.59 member @ a @ ( product_fst @ a @ a @ x ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_21__092_060open_062_092_060lbrakk_062snd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_A_092_060in_062_Acarrier_AR_059_Asnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_A_092_060in_062_Acarrier_AR_059_At_A_092_060otimes_062_At_H_A_092_060in_062_Acarrier_AR_092_060rbrakk_062_A_092_060Longrightarrow_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_A_092_060oplus_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_092_060close_062,axiom,
% 2.58/1.59 ( ( member @ a @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( add @ a @ b @ r @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) )
% 2.58/1.59 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_22_f15,axiom,
% 2.58/1.59 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_23__092_060open_062snd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060in_062_Acarrier_AR_092_060close_062,axiom,
% 2.58/1.59 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_24_f28,axiom,
% 2.58/1.59 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_25_f11,axiom,
% 2.58/1.59 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_26_f10,axiom,
% 2.58/1.59 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_27_f27,axiom,
% 2.58/1.59 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_28_f35,axiom,
% 2.58/1.59 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) )
% 2.58/1.59 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_29__092_060open_062t_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_A_092_060ominus_062_As_A_092_060otimes_062_Afst_Ax_H_J_A_061_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_At_A_092_060otimes_062_A_Is_A_092_060otimes_062_Afst_Ax_H_J_092_060close_062,axiom,
% 2.58/1.59 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) )
% 2.58/1.59 = ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_30__092_060open_062t_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_A_092_060ominus_062_As_H_A_092_060otimes_062_Afst_Ay_H_J_A_061_At_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_At_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Afst_Ay_H_J_092_060close_062,axiom,
% 2.58/1.59 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) )
% 2.58/1.59 = ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_31_f16,axiom,
% 2.58/1.59 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_32_f24,axiom,
% 2.58/1.59 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) )
% 2.58/1.59 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ t2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_33_f9,axiom,
% 2.58/1.59 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_34__092_060open_062t_A_092_060otimes_062_As_A_092_060otimes_062_Asnd_Ax_H_A_092_060in_062_Acarrier_AR_092_060close_062,axiom,
% 2.58/1.59 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_35_f23,axiom,
% 2.58/1.59 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) )
% 2.58/1.59 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ t2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_36_f34,axiom,
% 2.58/1.59 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) )
% 2.58/1.59 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_37_f36,axiom,
% 2.58/1.59 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) )
% 2.58/1.59 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_38_f13,axiom,
% 2.58/1.59 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_39__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_061_At_H_A_092_060otimes_062_As_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_At_H_A_092_060otimes_062_As_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_At_A_092_060otimes_062_A_Is_A_092_060otimes_062_Afst_Ax_H_J_092_060close_062,axiom,
% 2.58/1.59 ( ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) )
% 2.58/1.59 = ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ t2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ t2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_40_f12,axiom,
% 2.58/1.59 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) )
% 2.58/1.59 = ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ t2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ t2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_41_f25,axiom,
% 2.58/1.59 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) )
% 2.58/1.59 = ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_42_f32,axiom,
% 2.58/1.59 ( ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) )
% 2.58/1.59 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_43_f26,axiom,
% 2.58/1.59 ( ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) )
% 2.58/1.59 = ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_44_f29,axiom,
% 2.58/1.59 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) )
% 2.58/1.59 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_45_mem__Collect__eq,axiom,
% 2.58/1.59 ! [A: $tType,A2: A,P: A > $o] :
% 2.58/1.59 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.58/1.59 = ( P @ A2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_46_Collect__mem__eq,axiom,
% 2.58/1.59 ! [A: $tType,A3: set @ A] :
% 2.58/1.59 ( ( collect @ A
% 2.58/1.59 @ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
% 2.58/1.59 = A3 ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_47_Collect__cong,axiom,
% 2.58/1.59 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.58/1.59 ( ! [X3: A] :
% 2.58/1.59 ( ( P @ X3 )
% 2.58/1.59 = ( Q @ X3 ) )
% 2.58/1.59 => ( ( collect @ A @ P )
% 2.58/1.59 = ( collect @ A @ Q ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_48_f30,axiom,
% 2.58/1.59 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) )
% 2.58/1.59 = ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_49_f33,axiom,
% 2.58/1.59 ( ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) )
% 2.58/1.59 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_50__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_J_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_J_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_092_060close_062,axiom,
% 2.58/1.59 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) )
% 2.58/1.59 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_51_f14,axiom,
% 2.58/1.59 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_52__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_092_060close_062,axiom,
% 2.58/1.59 ( ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) )
% 2.58/1.59 = ( a_minus @ a @ b @ r @ ( add @ a @ b @ r @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_53_f31,axiom,
% 2.58/1.59 ( ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) ) )
% 2.58/1.59 = ( add @ a @ b @ r @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_54_f37,axiom,
% 2.58/1.59 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_55_f39,axiom,
% 2.58/1.59 member @ a @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_56_f38,axiom,
% 2.58/1.59 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_57_f5,axiom,
% 2.58/1.59 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) )
% 2.58/1.59 = ( zero @ a @ b @ r ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_58_f7,axiom,
% 2.58/1.59 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) )
% 2.58/1.59 = ( zero @ a @ b @ r ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_59_minus__closed,axiom,
% 2.58/1.59 ! [X: a,Y: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( member @ a @ ( a_minus @ a @ b @ r @ X @ Y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_60_add_Om__closed,axiom,
% 2.58/1.59 ! [X: a,Y: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( member @ a @ ( add @ a @ b @ r @ X @ Y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_61_local_Oadd_Oright__cancel,axiom,
% 2.58/1.59 ! [X: a,Y: a,Z2: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( ( add @ a @ b @ r @ Y @ X )
% 2.58/1.59 = ( add @ a @ b @ r @ Z2 @ X ) )
% 2.58/1.59 = ( Y = Z2 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_62_semiring__simprules_I3_J,axiom,
% 2.58/1.59 ! [X: a,Y: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_63_f4,axiom,
% 2.58/1.59 member @ a @ t2 @ s ).
% 2.58/1.59
% 2.58/1.59 thf(fact_64_f6,axiom,
% 2.58/1.59 member @ a @ t @ s ).
% 2.58/1.59
% 2.58/1.59 thf(fact_65_f8,axiom,
% 2.58/1.59 member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ s ).
% 2.58/1.59
% 2.58/1.59 thf(fact_66_local_Osemiring__axioms,axiom,
% 2.58/1.59 semiring @ a @ b @ r ).
% 2.58/1.59
% 2.58/1.59 thf(fact_67_minus__eq,axiom,
% 2.58/1.59 ! [X: a,Y: a] :
% 2.58/1.59 ( ( a_minus @ a @ b @ r @ X @ Y )
% 2.58/1.59 = ( add @ a @ b @ r @ X @ ( a_inv @ a @ b @ r @ Y ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_68_prod__eqI,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 2.58/1.59 ( ( ( product_fst @ A @ B @ P2 )
% 2.58/1.59 = ( product_fst @ A @ B @ Q2 ) )
% 2.58/1.59 => ( ( ( product_snd @ A @ B @ P2 )
% 2.58/1.59 = ( product_snd @ A @ B @ Q2 ) )
% 2.58/1.59 => ( P2 = Q2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_69_exE__realizer_H,axiom,
% 2.58/1.59 ! [A: $tType,B: $tType,P: A > B > $o,P2: product_prod @ B @ A] :
% 2.58/1.59 ( ( P @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
% 2.58/1.59 => ~ ! [X3: B,Y2: A] :
% 2.58/1.59 ~ ( P @ Y2 @ X3 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_70_prod_Oexpand,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 2.58/1.59 ( ( ( ( product_fst @ A @ B @ Prod )
% 2.58/1.59 = ( product_fst @ A @ B @ Prod2 ) )
% 2.58/1.59 & ( ( product_snd @ A @ B @ Prod )
% 2.58/1.59 = ( product_snd @ A @ B @ Prod2 ) ) )
% 2.58/1.59 => ( Prod = Prod2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_71_local_Ominus__unique,axiom,
% 2.58/1.59 ! [Y: a,X: a,Y3: a] :
% 2.58/1.59 ( ( ( add @ a @ b @ r @ Y @ X )
% 2.58/1.59 = ( zero @ a @ b @ r ) )
% 2.58/1.59 => ( ( ( add @ a @ b @ r @ X @ Y3 )
% 2.58/1.59 = ( zero @ a @ b @ r ) )
% 2.58/1.59 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( Y = Y3 ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_72_add_Or__inv__ex,axiom,
% 2.58/1.59 ! [X: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ? [X3: a] :
% 2.58/1.59 ( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 & ( ( add @ a @ b @ r @ X @ X3 )
% 2.58/1.59 = ( zero @ a @ b @ r ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_73_add_Oone__unique,axiom,
% 2.58/1.59 ! [U: a] :
% 2.58/1.59 ( ( member @ a @ U @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ! [X3: a] :
% 2.58/1.59 ( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( add @ a @ b @ r @ U @ X3 )
% 2.58/1.59 = X3 ) )
% 2.58/1.59 => ( U
% 2.58/1.59 = ( zero @ a @ b @ r ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_74_add_Ol__inv__ex,axiom,
% 2.58/1.59 ! [X: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ? [X3: a] :
% 2.58/1.59 ( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 & ( ( add @ a @ b @ r @ X3 @ X )
% 2.58/1.59 = ( zero @ a @ b @ r ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_75_add_Oinv__comm,axiom,
% 2.58/1.59 ! [X: a,Y: a] :
% 2.58/1.59 ( ( ( add @ a @ b @ r @ X @ Y )
% 2.58/1.59 = ( zero @ a @ b @ r ) )
% 2.58/1.59 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( add @ a @ b @ r @ Y @ X )
% 2.58/1.59 = ( zero @ a @ b @ r ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_76_r__minus,axiom,
% 2.58/1.59 ! [X: a,Y: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( a_inv @ a @ b @ r @ Y ) )
% 2.58/1.59 = ( a_inv @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_77_l__minus,axiom,
% 2.58/1.59 ! [X: a,Y: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( a_inv @ a @ b @ r @ X ) @ Y )
% 2.58/1.59 = ( a_inv @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_78_r__neg2,axiom,
% 2.58/1.59 ! [X: a,Y: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( add @ a @ b @ r @ X @ ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ Y ) )
% 2.58/1.59 = Y ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_79_r__neg1,axiom,
% 2.58/1.59 ! [X: a,Y: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ ( add @ a @ b @ r @ X @ Y ) )
% 2.58/1.59 = Y ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_80_local_Ominus__add,axiom,
% 2.58/1.59 ! [X: a,Y: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( a_inv @ a @ b @ r @ ( add @ a @ b @ r @ X @ Y ) )
% 2.58/1.59 = ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ ( a_inv @ a @ b @ r @ Y ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_81_add_Oinv__solve__right_H,axiom,
% 2.58/1.59 ! [A2: a,B2: a,C2: a] :
% 2.58/1.59 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( ( add @ a @ b @ r @ B2 @ ( a_inv @ a @ b @ r @ C2 ) )
% 2.58/1.59 = A2 )
% 2.58/1.59 = ( B2
% 2.58/1.59 = ( add @ a @ b @ r @ A2 @ C2 ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_82_add_Oinv__solve__right,axiom,
% 2.58/1.59 ! [A2: a,B2: a,C2: a] :
% 2.58/1.59 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( A2
% 2.58/1.59 = ( add @ a @ b @ r @ B2 @ ( a_inv @ a @ b @ r @ C2 ) ) )
% 2.58/1.59 = ( B2
% 2.58/1.59 = ( add @ a @ b @ r @ A2 @ C2 ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_83_add_Oinv__solve__left_H,axiom,
% 2.58/1.59 ! [A2: a,B2: a,C2: a] :
% 2.58/1.59 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ B2 ) @ C2 )
% 2.58/1.59 = A2 )
% 2.58/1.59 = ( C2
% 2.58/1.59 = ( add @ a @ b @ r @ B2 @ A2 ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_84_add_Oinv__solve__left,axiom,
% 2.58/1.59 ! [A2: a,B2: a,C2: a] :
% 2.58/1.59 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( A2
% 2.58/1.59 = ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ B2 ) @ C2 ) )
% 2.58/1.59 = ( C2
% 2.58/1.59 = ( add @ a @ b @ r @ B2 @ A2 ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_85_add_Oinv__mult__group,axiom,
% 2.58/1.59 ! [X: a,Y: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( a_inv @ a @ b @ r @ ( add @ a @ b @ r @ X @ Y ) )
% 2.58/1.59 = ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ Y ) @ ( a_inv @ a @ b @ r @ X ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_86_a__transpose__inv,axiom,
% 2.58/1.59 ! [X: a,Y: a,Z2: a] :
% 2.58/1.59 ( ( ( add @ a @ b @ r @ X @ Y )
% 2.58/1.59 = Z2 )
% 2.58/1.59 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ Z2 )
% 2.58/1.59 = Y ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_87_eq__obj__rng__of__frac__axioms,axiom,
% 2.58/1.59 locali1648044335f_frac @ a @ b @ r @ s ).
% 2.58/1.59
% 2.58/1.59 thf(fact_88_sum__zero__eq__neg,axiom,
% 2.58/1.59 ! [X: a,Y: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( ( add @ a @ b @ r @ X @ Y )
% 2.58/1.59 = ( zero @ a @ b @ r ) )
% 2.58/1.59 => ( X
% 2.58/1.59 = ( a_inv @ a @ b @ r @ Y ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_89_r__neg,axiom,
% 2.58/1.59 ! [X: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( add @ a @ b @ r @ X @ ( a_inv @ a @ b @ r @ X ) )
% 2.58/1.59 = ( zero @ a @ b @ r ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_90_minus__equality,axiom,
% 2.58/1.59 ! [Y: a,X: a] :
% 2.58/1.59 ( ( ( add @ a @ b @ r @ Y @ X )
% 2.58/1.59 = ( zero @ a @ b @ r ) )
% 2.58/1.59 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( a_inv @ a @ b @ r @ X )
% 2.58/1.59 = Y ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_91_l__neg,axiom,
% 2.58/1.59 ! [X: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ X )
% 2.58/1.59 = ( zero @ a @ b @ r ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_92_inv__add,axiom,
% 2.58/1.59 ! [A2: a,B2: a] :
% 2.58/1.59 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( a_inv @ a @ b @ r @ ( add @ a @ b @ r @ A2 @ B2 ) )
% 2.58/1.59 = ( a_minus @ a @ b @ r @ ( a_inv @ a @ b @ r @ A2 ) @ B2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_93__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_O_A_092_060lbrakk_062t_A_092_060in_062_AS_059_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_A_092_060ominus_062_As_A_092_060otimes_062_Afst_Ax_H_J_A_061_A_092_060zero_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 2.58/1.59 ~ ! [T: a] :
% 2.58/1.59 ( ( member @ a @ T @ s )
% 2.58/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ T @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) )
% 2.58/1.59 != ( zero @ a @ b @ r ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_94__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_H_O_A_092_060lbrakk_062t_H_A_092_060in_062_AS_059_At_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_A_092_060ominus_062_As_H_A_092_060otimes_062_Afst_Ay_H_J_A_061_A_092_060zero_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 2.58/1.59 ~ ! [T2: a] :
% 2.58/1.59 ( ( member @ a @ T2 @ s )
% 2.58/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ T2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) )
% 2.58/1.59 != ( zero @ a @ b @ r ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_95_zero__closed,axiom,
% 2.58/1.59 member @ a @ ( zero @ a @ b @ r ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_96_local_Ominus__minus,axiom,
% 2.58/1.59 ! [X: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( a_inv @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) )
% 2.58/1.59 = X ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_97_add_Oinv__closed,axiom,
% 2.58/1.59 ! [X: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( member @ a @ ( a_inv @ a @ b @ r @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_98_local_Ominus__zero,axiom,
% 2.58/1.59 ( ( a_inv @ a @ b @ r @ ( zero @ a @ b @ r ) )
% 2.58/1.59 = ( zero @ a @ b @ r ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_99_m__closed,axiom,
% 2.58/1.59 ! [X: a,Y: a] :
% 2.58/1.59 ( ( member @ a @ X @ s )
% 2.58/1.59 => ( ( member @ a @ Y @ s )
% 2.58/1.59 => ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ s ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_100_r__null,axiom,
% 2.58/1.59 ! [X: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( zero @ a @ b @ r ) )
% 2.58/1.59 = ( zero @ a @ b @ r ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_101_l__null,axiom,
% 2.58/1.59 ! [X: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( zero @ a @ b @ r ) @ X )
% 2.58/1.59 = ( zero @ a @ b @ r ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_102_r__zero,axiom,
% 2.58/1.59 ! [X: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( add @ a @ b @ r @ X @ ( zero @ a @ b @ r ) )
% 2.58/1.59 = X ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_103_l__zero,axiom,
% 2.58/1.59 ! [X: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( add @ a @ b @ r @ ( zero @ a @ b @ r ) @ X )
% 2.58/1.59 = X ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_104_add_Or__cancel__one_H,axiom,
% 2.58/1.59 ! [X: a,A2: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( X
% 2.58/1.59 = ( add @ a @ b @ r @ A2 @ X ) )
% 2.58/1.59 = ( A2
% 2.58/1.59 = ( zero @ a @ b @ r ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_105_add_Or__cancel__one,axiom,
% 2.58/1.59 ! [X: a,A2: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( ( add @ a @ b @ r @ A2 @ X )
% 2.58/1.59 = X )
% 2.58/1.59 = ( A2
% 2.58/1.59 = ( zero @ a @ b @ r ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_106_add_Ol__cancel__one_H,axiom,
% 2.58/1.59 ! [X: a,A2: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( X
% 2.58/1.59 = ( add @ a @ b @ r @ X @ A2 ) )
% 2.58/1.59 = ( A2
% 2.58/1.59 = ( zero @ a @ b @ r ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_107_add_Ol__cancel__one,axiom,
% 2.58/1.59 ! [X: a,A2: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( ( add @ a @ b @ r @ X @ A2 )
% 2.58/1.59 = X )
% 2.58/1.59 = ( A2
% 2.58/1.59 = ( zero @ a @ b @ r ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_108_add_Oinv__eq__1__iff,axiom,
% 2.58/1.59 ! [X: a] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( ( a_inv @ a @ b @ r @ X )
% 2.58/1.59 = ( zero @ a @ b @ r ) )
% 2.58/1.59 = ( X
% 2.58/1.59 = ( zero @ a @ b @ r ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_109_mult__submonoid__of__rng__axioms,axiom,
% 2.58/1.59 locali1402578320of_rng @ a @ b @ r @ s ).
% 2.58/1.59
% 2.58/1.59 thf(fact_110_mult__submonoid__of__crng__axioms,axiom,
% 2.58/1.59 locali7914339f_crng @ a @ b @ r @ s ).
% 2.58/1.59
% 2.58/1.59 thf(fact_111_prod__eq__iff,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType] :
% 2.58/1.59 ( ( ^ [Y4: product_prod @ A @ B,Z3: product_prod @ A @ B] : ( Y4 = Z3 ) )
% 2.58/1.59 = ( ^ [S: product_prod @ A @ B,T3: product_prod @ A @ B] :
% 2.58/1.59 ( ( ( product_fst @ A @ B @ S )
% 2.58/1.59 = ( product_fst @ A @ B @ T3 ) )
% 2.58/1.59 & ( ( product_snd @ A @ B @ S )
% 2.58/1.59 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_112_submonoid__axioms,axiom,
% 2.58/1.59 locali1441642228monoid @ a @ ( ring_ext @ a @ b ) @ r @ s ).
% 2.58/1.59
% 2.58/1.59 thf(fact_113_add_Oint__pow__inv,axiom,
% 2.58/1.59 ! [X: a,I: int] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( add_pow @ a @ b @ int @ r @ I @ ( a_inv @ a @ b @ r @ X ) )
% 2.58/1.59 = ( a_inv @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_114_semiring_Osemiring__simprules_I11_J,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 2.58/1.59 ( ( semiring @ A @ B @ R )
% 2.58/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( add @ A @ B @ R @ X @ ( zero @ A @ B @ R ) )
% 2.58/1.59 = X ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_115_semiring_Osemiring__simprules_I6_J,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 2.58/1.59 ( ( semiring @ A @ B @ R )
% 2.58/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( add @ A @ B @ R @ ( zero @ A @ B @ R ) @ X )
% 2.58/1.59 = X ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_116_semiring_Ol__null,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 2.58/1.59 ( ( semiring @ A @ B @ R )
% 2.58/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R @ ( zero @ A @ B @ R ) @ X )
% 2.58/1.59 = ( zero @ A @ B @ R ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_117_semiring_Or__null,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 2.58/1.59 ( ( semiring @ A @ B @ R )
% 2.58/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R @ X @ ( zero @ A @ B @ R ) )
% 2.58/1.59 = ( zero @ A @ B @ R ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_118_semiring_Or__distr,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 2.58/1.59 ( ( semiring @ A @ B @ R )
% 2.58/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R @ Z2 @ ( add @ A @ B @ R @ X @ Y ) )
% 2.58/1.59 = ( add @ A @ B @ R @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ Z2 @ X ) @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ Z2 @ Y ) ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_119_add__pow__ldistr__int,axiom,
% 2.58/1.59 ! [A2: a,B2: a,K: int] :
% 2.58/1.59 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add_pow @ a @ b @ int @ r @ K @ A2 ) @ B2 )
% 2.58/1.59 = ( add_pow @ a @ b @ int @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_120_add__pow__rdistr__int,axiom,
% 2.58/1.59 ! [A2: a,B2: a,K: int] :
% 2.58/1.59 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ ( add_pow @ a @ b @ int @ r @ K @ B2 ) )
% 2.58/1.59 = ( add_pow @ a @ b @ int @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_121_add_Oint__pow__distrib,axiom,
% 2.58/1.59 ! [X: a,Y: a,I: int] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( add_pow @ a @ b @ int @ r @ I @ ( add @ a @ b @ r @ X @ Y ) )
% 2.58/1.59 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( add_pow @ a @ b @ int @ r @ I @ Y ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_122_add_Oint__pow__mult__distrib,axiom,
% 2.58/1.59 ! [X: a,Y: a,I: int] :
% 2.58/1.59 ( ( ( add @ a @ b @ r @ X @ Y )
% 2.58/1.59 = ( add @ a @ b @ r @ Y @ X ) )
% 2.58/1.59 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( add_pow @ a @ b @ int @ r @ I @ ( add @ a @ b @ r @ X @ Y ) )
% 2.58/1.59 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( add_pow @ a @ b @ int @ r @ I @ Y ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_123_add_Oint__pow__closed,axiom,
% 2.58/1.59 ! [X: a,I: int] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( member @ a @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_124_add_Oint__pow__one,axiom,
% 2.58/1.59 ! [Z2: int] :
% 2.58/1.59 ( ( add_pow @ a @ b @ int @ r @ Z2 @ ( zero @ a @ b @ r ) )
% 2.58/1.59 = ( zero @ a @ b @ r ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_125_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Ois__submonoid,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,M2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S2: set @ A] :
% 2.58/1.59 ( ( locali1441642228monoid @ A @ B @ M2 @ S2 )
% 2.58/1.59 => ( locali1441642228monoid @ A @ B @ M2 @ S2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_126_mult__submonoid__of__crng_Oaxioms_I2_J,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S2: set @ A] :
% 2.58/1.59 ( ( locali7914339f_crng @ A @ B @ R @ S2 )
% 2.58/1.59 => ( locali1402578320of_rng @ A @ B @ R @ S2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_127_mult__submonoid__of__rng_Oaxioms_I2_J,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S2: set @ A] :
% 2.58/1.59 ( ( locali1402578320of_rng @ A @ B @ R @ S2 )
% 2.58/1.59 => ( locali1441642228monoid @ A @ ( ring_ext @ A @ B ) @ R @ S2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_128_eq__obj__rng__of__frac_Oaxioms_I2_J,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S2: set @ A] :
% 2.58/1.59 ( ( locali1648044335f_frac @ A @ B @ R @ S2 )
% 2.58/1.59 => ( locali7914339f_crng @ A @ B @ R @ S2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_129_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Om__closed,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,M2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S2: set @ A,X: A,Y: A] :
% 2.58/1.59 ( ( locali1441642228monoid @ A @ B @ M2 @ S2 )
% 2.58/1.59 => ( ( member @ A @ X @ S2 )
% 2.58/1.59 => ( ( member @ A @ Y @ S2 )
% 2.58/1.59 => ( member @ A @ ( mult @ A @ B @ M2 @ X @ Y ) @ S2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_130_a__minus__def,axiom,
% 2.58/1.59 ! [M: $tType,A: $tType] :
% 2.58/1.59 ( ( a_minus @ A @ M )
% 2.58/1.59 = ( ^ [R2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ M ) ),X2: A,Y5: A] : ( add @ A @ M @ R2 @ X2 @ ( a_inv @ A @ M @ R2 @ Y5 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_131_semiring_Osemiring__simprules_I8_J,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 2.58/1.59 ( ( semiring @ A @ B @ R )
% 2.58/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ X @ Y ) @ Z2 )
% 2.58/1.59 = ( mult @ A @ ( ring_ext @ A @ B ) @ R @ X @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ Y @ Z2 ) ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_132_semiring_Osemiring__simprules_I3_J,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 2.58/1.59 ( ( semiring @ A @ B @ R )
% 2.58/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( member @ A @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_133_semiring_Osemiring__simprules_I12_J,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 2.58/1.59 ( ( semiring @ A @ B @ R )
% 2.58/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( add @ A @ B @ R @ X @ ( add @ A @ B @ R @ Y @ Z2 ) )
% 2.58/1.59 = ( add @ A @ B @ R @ Y @ ( add @ A @ B @ R @ X @ Z2 ) ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_134_semiring_Osemiring__simprules_I7_J,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 2.58/1.59 ( ( semiring @ A @ B @ R )
% 2.58/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( add @ A @ B @ R @ X @ Y )
% 2.58/1.59 = ( add @ A @ B @ R @ Y @ X ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_135_semiring_Osemiring__simprules_I5_J,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 2.58/1.59 ( ( semiring @ A @ B @ R )
% 2.58/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( add @ A @ B @ R @ ( add @ A @ B @ R @ X @ Y ) @ Z2 )
% 2.58/1.59 = ( add @ A @ B @ R @ X @ ( add @ A @ B @ R @ Y @ Z2 ) ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_136_semiring_Osemiring__simprules_I1_J,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 2.58/1.59 ( ( semiring @ A @ B @ R )
% 2.58/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( member @ A @ ( add @ A @ B @ R @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_137_semiring_Osemiring__simprules_I2_J,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 2.58/1.59 ( ( semiring @ A @ B @ R )
% 2.58/1.59 => ( member @ A @ ( zero @ A @ B @ R ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_138_semiring_Ol__distr,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 2.58/1.59 ( ( semiring @ A @ B @ R )
% 2.58/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
% 2.58/1.59 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R @ ( add @ A @ B @ R @ X @ Y ) @ Z2 )
% 2.58/1.59 = ( add @ A @ B @ R @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ X @ Z2 ) @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ Y @ Z2 ) ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_139_add_Oint__pow__diff,axiom,
% 2.58/1.59 ! [X: a,N: int,M3: int] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( add_pow @ a @ b @ int @ r @ ( minus_minus @ int @ N @ M3 ) @ X )
% 2.58/1.59 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ N @ X ) @ ( a_inv @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ M3 @ X ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_140_add_Oint__pow__neg,axiom,
% 2.58/1.59 ! [X: a,I: int] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( add_pow @ a @ b @ int @ r @ ( uminus_uminus @ int @ I ) @ X )
% 2.58/1.59 = ( a_inv @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_141_add_Oint__pow__mult,axiom,
% 2.58/1.59 ! [X: a,I: int,J: int] :
% 2.58/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( add_pow @ a @ b @ int @ r @ ( plus_plus @ int @ I @ J ) @ X )
% 2.58/1.59 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( add_pow @ a @ b @ int @ r @ J @ X ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_142_subset,axiom,
% 2.58/1.59 ord_less_eq @ ( set @ a ) @ s @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_143_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Osubset,axiom,
% 2.58/1.59 ! [B: $tType,A: $tType,M2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S2: set @ A] :
% 2.58/1.59 ( ( locali1441642228monoid @ A @ B @ M2 @ S2 )
% 2.58/1.59 => ( ord_less_eq @ ( set @ A ) @ S2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ M2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_144_diff__minus__eq__add,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A] :
% 2.58/1.59 ( ( minus_minus @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
% 2.58/1.59 = ( plus_plus @ A @ A2 @ B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_145_uminus__add__conv__diff,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ab_group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A] :
% 2.58/1.59 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
% 2.58/1.59 = ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_146_le__add__diff__inverse,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( linordered_semidom @ A )
% 2.58/1.59 => ! [B2: A,A2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.58/1.59 => ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) )
% 2.58/1.59 = A2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_147_le__add__diff__inverse2,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( linordered_semidom @ A )
% 2.58/1.59 => ! [B2: A,A2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.58/1.59 => ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
% 2.58/1.59 = A2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_148_add__right__cancel,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( cancel_semigroup_add @ A )
% 2.58/1.59 => ! [B2: A,A2: A,C2: A] :
% 2.58/1.59 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.58/1.59 = ( plus_plus @ A @ C2 @ A2 ) )
% 2.58/1.59 = ( B2 = C2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_149_add__left__cancel,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( cancel_semigroup_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.58/1.59 = ( plus_plus @ A @ A2 @ C2 ) )
% 2.58/1.59 = ( B2 = C2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_150_neg__equal__iff__equal,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A] :
% 2.58/1.59 ( ( ( uminus_uminus @ A @ A2 )
% 2.58/1.59 = ( uminus_uminus @ A @ B2 ) )
% 2.58/1.59 = ( A2 = B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_151_add_Oinverse__inverse,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( group_add @ A )
% 2.58/1.59 => ! [A2: A] :
% 2.58/1.59 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
% 2.58/1.59 = A2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_152_add__le__cancel__right,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere236663937imp_le @ A )
% 2.58/1.59 => ! [A2: A,C2: A,B2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 2.58/1.59 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_153_add__le__cancel__left,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere236663937imp_le @ A )
% 2.58/1.59 => ! [C2: A,A2: A,B2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 2.58/1.59 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_154_neg__le__iff__le,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordered_ab_group_add @ A )
% 2.58/1.59 => ! [B2: A,A2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
% 2.58/1.59 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_155_add__diff__cancel__right_H,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( cancel146912293up_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A] :
% 2.58/1.59 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 2.58/1.59 = A2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_156_add__diff__cancel__right,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( cancel146912293up_add @ A )
% 2.58/1.59 => ! [A2: A,C2: A,B2: A] :
% 2.58/1.59 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 2.58/1.59 = ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_157_add__diff__cancel__left_H,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( cancel146912293up_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A] :
% 2.58/1.59 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 )
% 2.58/1.59 = B2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_158_add__diff__cancel__left,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( cancel146912293up_add @ A )
% 2.58/1.59 => ! [C2: A,A2: A,B2: A] :
% 2.58/1.59 ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 2.58/1.59 = ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_159_diff__add__cancel,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A] :
% 2.58/1.59 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
% 2.58/1.59 = A2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_160_add__diff__cancel,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A] :
% 2.58/1.59 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 2.58/1.59 = A2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_161_minus__add__distrib,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ab_group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A] :
% 2.58/1.59 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
% 2.58/1.59 = ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_162_minus__add__cancel,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A] :
% 2.58/1.59 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) )
% 2.58/1.59 = B2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_163_add__minus__cancel,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A] :
% 2.58/1.59 ( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) )
% 2.58/1.59 = B2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_164_minus__diff__eq,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A] :
% 2.58/1.59 ( ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B2 ) )
% 2.58/1.59 = ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_165_add__right__imp__eq,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( cancel_semigroup_add @ A )
% 2.58/1.59 => ! [B2: A,A2: A,C2: A] :
% 2.58/1.59 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.58/1.59 = ( plus_plus @ A @ C2 @ A2 ) )
% 2.58/1.59 => ( B2 = C2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_166_add__left__imp__eq,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( cancel_semigroup_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.58/1.59 = ( plus_plus @ A @ A2 @ C2 ) )
% 2.58/1.59 => ( B2 = C2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_167_add_Oleft__commute,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ab_semigroup_add @ A )
% 2.58/1.59 => ! [B2: A,A2: A,C2: A] :
% 2.58/1.59 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
% 2.58/1.59 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_168_add_Ocommute,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ab_semigroup_add @ A )
% 2.58/1.59 => ( ( plus_plus @ A )
% 2.58/1.59 = ( ^ [A4: A,B3: A] : ( plus_plus @ A @ B3 @ A4 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_169_group__add__class_Oadd_Oright__cancel,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( group_add @ A )
% 2.58/1.59 => ! [B2: A,A2: A,C2: A] :
% 2.58/1.59 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.58/1.59 = ( plus_plus @ A @ C2 @ A2 ) )
% 2.58/1.59 = ( B2 = C2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_170_add_Oleft__cancel,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.58/1.59 = ( plus_plus @ A @ A2 @ C2 ) )
% 2.58/1.59 = ( B2 = C2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_171_add_Oassoc,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( semigroup_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 2.58/1.59 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_172_group__cancel_Oadd2,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( comm_monoid_add @ A )
% 2.58/1.59 => ! [B4: A,K: A,B2: A,A2: A] :
% 2.58/1.59 ( ( B4
% 2.58/1.59 = ( plus_plus @ A @ K @ B2 ) )
% 2.58/1.59 => ( ( plus_plus @ A @ A2 @ B4 )
% 2.58/1.59 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_173_group__cancel_Oadd1,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( comm_monoid_add @ A )
% 2.58/1.59 => ! [A3: A,K: A,A2: A,B2: A] :
% 2.58/1.59 ( ( A3
% 2.58/1.59 = ( plus_plus @ A @ K @ A2 ) )
% 2.58/1.59 => ( ( plus_plus @ A @ A3 @ B2 )
% 2.58/1.59 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_174_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere779506340up_add @ A )
% 2.58/1.59 => ! [I: A,J: A,K: A,L: A] :
% 2.58/1.59 ( ( ( I = J )
% 2.58/1.59 & ( K = L ) )
% 2.58/1.59 => ( ( plus_plus @ A @ I @ K )
% 2.58/1.59 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_175_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ab_semigroup_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 2.58/1.59 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_176_diff__right__commute,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( cancel146912293up_add @ A )
% 2.58/1.59 => ! [A2: A,C2: A,B2: A] :
% 2.58/1.59 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 )
% 2.58/1.59 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_177_diff__eq__diff__eq,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A,D: A] :
% 2.58/1.59 ( ( ( minus_minus @ A @ A2 @ B2 )
% 2.58/1.59 = ( minus_minus @ A @ C2 @ D ) )
% 2.58/1.59 => ( ( A2 = B2 )
% 2.58/1.59 = ( C2 = D ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_178_minus__equation__iff,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A] :
% 2.58/1.59 ( ( ( uminus_uminus @ A @ A2 )
% 2.58/1.59 = B2 )
% 2.58/1.59 = ( ( uminus_uminus @ A @ B2 )
% 2.58/1.59 = A2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_179_equation__minus__iff,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A] :
% 2.58/1.59 ( ( A2
% 2.58/1.59 = ( uminus_uminus @ A @ B2 ) )
% 2.58/1.59 = ( B2
% 2.58/1.59 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_180_add__le__imp__le__right,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere236663937imp_le @ A )
% 2.58/1.59 => ! [A2: A,C2: A,B2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 2.58/1.59 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_181_add__le__imp__le__left,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere236663937imp_le @ A )
% 2.58/1.59 => ! [C2: A,A2: A,B2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 2.58/1.59 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_182_le__iff__add,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( canoni770627133id_add @ A )
% 2.58/1.59 => ( ( ord_less_eq @ A )
% 2.58/1.59 = ( ^ [A4: A,B3: A] :
% 2.58/1.59 ? [C3: A] :
% 2.58/1.59 ( B3
% 2.58/1.59 = ( plus_plus @ A @ A4 @ C3 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_183_add__right__mono,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere779506340up_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.58/1.59 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_184_less__eqE,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( canoni770627133id_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.58/1.59 => ~ ! [C4: A] :
% 2.58/1.59 ( B2
% 2.58/1.59 != ( plus_plus @ A @ A2 @ C4 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_185_add__left__mono,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere779506340up_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.58/1.59 => ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_186_add__mono,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere779506340up_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A,D: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.58/1.59 => ( ( ord_less_eq @ A @ C2 @ D )
% 2.58/1.59 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_187_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere779506340up_add @ A )
% 2.58/1.59 => ! [I: A,J: A,K: A,L: A] :
% 2.58/1.59 ( ( ( ord_less_eq @ A @ I @ J )
% 2.58/1.59 & ( ord_less_eq @ A @ K @ L ) )
% 2.58/1.59 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_188_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere779506340up_add @ A )
% 2.58/1.59 => ! [I: A,J: A,K: A,L: A] :
% 2.58/1.59 ( ( ( I = J )
% 2.58/1.59 & ( ord_less_eq @ A @ K @ L ) )
% 2.58/1.59 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_189_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere779506340up_add @ A )
% 2.58/1.59 => ! [I: A,J: A,K: A,L: A] :
% 2.58/1.59 ( ( ( ord_less_eq @ A @ I @ J )
% 2.58/1.59 & ( K = L ) )
% 2.58/1.59 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_190_diff__eq__diff__less__eq,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordered_ab_group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A,D: A] :
% 2.58/1.59 ( ( ( minus_minus @ A @ A2 @ B2 )
% 2.58/1.59 = ( minus_minus @ A @ C2 @ D ) )
% 2.58/1.59 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.58/1.59 = ( ord_less_eq @ A @ C2 @ D ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_191_diff__right__mono,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordered_ab_group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.58/1.59 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_192_diff__left__mono,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordered_ab_group_add @ A )
% 2.58/1.59 => ! [B2: A,A2: A,C2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.58/1.59 => ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_193_diff__mono,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordered_ab_group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A,D: A,C2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.58/1.59 => ( ( ord_less_eq @ A @ D @ C2 )
% 2.58/1.59 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_194_le__imp__neg__le,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordered_ab_group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.58/1.59 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_195_minus__le__iff,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordered_ab_group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
% 2.58/1.59 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_196_le__minus__iff,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordered_ab_group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
% 2.58/1.59 = ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_197_add__implies__diff,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( cancel1352612707id_add @ A )
% 2.58/1.59 => ! [C2: A,B2: A,A2: A] :
% 2.58/1.59 ( ( ( plus_plus @ A @ C2 @ B2 )
% 2.58/1.59 = A2 )
% 2.58/1.59 => ( C2
% 2.58/1.59 = ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_198_diff__diff__add,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( cancel146912293up_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
% 2.58/1.59 = ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_199_diff__add__eq__diff__diff__swap,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
% 2.58/1.59 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_200_diff__add__eq,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ab_group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
% 2.58/1.59 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_201_diff__diff__eq2,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
% 2.58/1.59 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_202_add__diff__eq,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
% 2.58/1.59 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_203_eq__diff__eq,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( group_add @ A )
% 2.58/1.59 => ! [A2: A,C2: A,B2: A] :
% 2.58/1.59 ( ( A2
% 2.58/1.59 = ( minus_minus @ A @ C2 @ B2 ) )
% 2.58/1.59 = ( ( plus_plus @ A @ A2 @ B2 )
% 2.58/1.59 = C2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_204_diff__eq__eq,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( ( minus_minus @ A @ A2 @ B2 )
% 2.58/1.59 = C2 )
% 2.58/1.59 = ( A2
% 2.58/1.59 = ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_205_group__cancel_Osub1,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ab_group_add @ A )
% 2.58/1.59 => ! [A3: A,K: A,A2: A,B2: A] :
% 2.58/1.59 ( ( A3
% 2.58/1.59 = ( plus_plus @ A @ K @ A2 ) )
% 2.58/1.59 => ( ( minus_minus @ A @ A3 @ B2 )
% 2.58/1.59 = ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_206_add_Oinverse__distrib__swap,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A] :
% 2.58/1.59 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
% 2.58/1.59 = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_207_group__cancel_Oneg1,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ab_group_add @ A )
% 2.58/1.59 => ! [A3: A,K: A,A2: A] :
% 2.58/1.59 ( ( A3
% 2.58/1.59 = ( plus_plus @ A @ K @ A2 ) )
% 2.58/1.59 => ( ( uminus_uminus @ A @ A3 )
% 2.58/1.59 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_208_minus__diff__commute,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ab_group_add @ A )
% 2.58/1.59 => ! [B2: A,A2: A] :
% 2.58/1.59 ( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A2 )
% 2.58/1.59 = ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_209_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere623563068d_diff @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.58/1.59 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.58/1.59 => ( ( ( minus_minus @ A @ B2 @ A2 )
% 2.58/1.59 = C2 )
% 2.58/1.59 = ( B2
% 2.58/1.59 = ( plus_plus @ A @ C2 @ A2 ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_210_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere623563068d_diff @ A )
% 2.58/1.59 => ! [A2: A,B2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.58/1.59 => ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ A2 ) )
% 2.58/1.59 = B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_211_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere623563068d_diff @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.58/1.59 => ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
% 2.58/1.59 = ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_212_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere623563068d_diff @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.58/1.59 => ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 )
% 2.58/1.59 = ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_213_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere623563068d_diff @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.58/1.59 => ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 )
% 2.58/1.59 = ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_214_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere623563068d_diff @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.58/1.59 => ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 )
% 2.58/1.59 = ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_215_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere623563068d_diff @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.58/1.59 => ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
% 2.58/1.59 = ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_216_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere623563068d_diff @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.58/1.59 => ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
% 2.58/1.59 = ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_217_le__add__diff,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere623563068d_diff @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.58/1.59 => ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_218_add__le__add__imp__diff__le,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( linordered_semidom @ A )
% 2.58/1.59 => ! [I: A,K: A,N: A,J: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
% 2.58/1.59 => ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
% 2.58/1.59 => ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
% 2.58/1.59 => ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
% 2.58/1.59 => ( ord_less_eq @ A @ ( minus_minus @ A @ N @ K ) @ J ) ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_219_diff__add,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordere623563068d_diff @ A )
% 2.58/1.59 => ! [A2: A,B2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.58/1.59 => ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ A2 )
% 2.58/1.59 = B2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_220_add__le__imp__le__diff,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( linordered_semidom @ A )
% 2.58/1.59 => ! [I: A,K: A,N: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
% 2.58/1.59 => ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N @ K ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_221_le__diff__eq,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordered_ab_group_add @ A )
% 2.58/1.59 => ! [A2: A,C2: A,B2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) )
% 2.58/1.59 = ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_222_diff__le__eq,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ordered_ab_group_add @ A )
% 2.58/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.58/1.59 ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
% 2.58/1.59 = ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_223_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ab_group_add @ A )
% 2.58/1.59 => ( ( minus_minus @ A )
% 2.58/1.59 = ( ^ [A4: A,B3: A] : ( plus_plus @ A @ A4 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_224_diff__conv__add__uminus,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( group_add @ A )
% 2.58/1.59 => ( ( minus_minus @ A )
% 2.58/1.59 = ( ^ [A4: A,B3: A] : ( plus_plus @ A @ A4 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_225_group__cancel_Osub2,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ab_group_add @ A )
% 2.58/1.59 => ! [B4: A,K: A,B2: A,A2: A] :
% 2.58/1.59 ( ( B4
% 2.58/1.59 = ( plus_plus @ A @ K @ B2 ) )
% 2.58/1.59 => ( ( minus_minus @ A @ A2 @ B4 )
% 2.58/1.59 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_226_add_Oone__in__subset,axiom,
% 2.58/1.59 ! [H: set @ a] :
% 2.58/1.59 ( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( H
% 2.58/1.59 != ( bot_bot @ ( set @ a ) ) )
% 2.58/1.59 => ( ! [X3: a] :
% 2.58/1.59 ( ( member @ a @ X3 @ H )
% 2.58/1.59 => ( member @ a @ ( a_inv @ a @ b @ r @ X3 ) @ H ) )
% 2.58/1.59 => ( ! [X3: a] :
% 2.58/1.59 ( ( member @ a @ X3 @ H )
% 2.58/1.59 => ! [Xa: a] :
% 2.58/1.59 ( ( member @ a @ Xa @ H )
% 2.58/1.59 => ( member @ a @ ( add @ a @ b @ r @ X3 @ Xa ) @ H ) ) )
% 2.58/1.59 => ( member @ a @ ( zero @ a @ b @ r ) @ H ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_227_a__lcos__mult__one,axiom,
% 2.58/1.59 ! [M2: set @ a] :
% 2.58/1.59 ( ( ord_less_eq @ ( set @ a ) @ M2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( a_l_coset @ a @ b @ r @ ( zero @ a @ b @ r ) @ M2 )
% 2.58/1.59 = M2 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_228_a__lcos__m__assoc,axiom,
% 2.58/1.59 ! [M2: set @ a,G: a,H2: a] :
% 2.58/1.59 ( ( ord_less_eq @ ( set @ a ) @ M2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ G @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ H2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( a_l_coset @ a @ b @ r @ G @ ( a_l_coset @ a @ b @ r @ H2 @ M2 ) )
% 2.58/1.59 = ( a_l_coset @ a @ b @ r @ ( add @ a @ b @ r @ G @ H2 ) @ M2 ) ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_229_carrier__not__empty,axiom,
% 2.58/1.59 ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 2.58/1.59 != ( bot_bot @ ( set @ a ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_230_a__l__coset__subset__G,axiom,
% 2.58/1.59 ! [H: set @ a,X: a] :
% 2.58/1.59 ( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ord_less_eq @ ( set @ a ) @ ( a_l_coset @ a @ b @ r @ X @ H ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_231_setadd__subset__G,axiom,
% 2.58/1.59 ! [H: set @ a,K2: set @ a] :
% 2.58/1.59 ( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( ord_less_eq @ ( set @ a ) @ K2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ord_less_eq @ ( set @ a ) @ ( set_add @ a @ b @ r @ H @ K2 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_232_set__add__closed,axiom,
% 2.58/1.59 ! [A3: set @ a,B4: set @ a] :
% 2.58/1.59 ( ( ord_less_eq @ ( set @ a ) @ A3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ( ord_less_eq @ ( set @ a ) @ B4 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.58/1.59 => ( ord_less_eq @ ( set @ a ) @ ( set_add @ a @ b @ r @ A3 @ B4 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_233_add__additive__subgroups,axiom,
% 2.58/1.59 ! [H: set @ a,K2: set @ a] :
% 2.58/1.59 ( ( additive_subgroup @ a @ b @ H @ r )
% 2.58/1.59 => ( ( additive_subgroup @ a @ b @ K2 @ r )
% 2.58/1.59 => ( additive_subgroup @ a @ b @ ( set_add @ a @ b @ r @ H @ K2 ) @ r ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_234_Diff__eq__empty__iff,axiom,
% 2.58/1.59 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 2.58/1.59 ( ( ( minus_minus @ ( set @ A ) @ A3 @ B4 )
% 2.58/1.59 = ( bot_bot @ ( set @ A ) ) )
% 2.58/1.59 = ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_235_empty__iff,axiom,
% 2.58/1.59 ! [A: $tType,C2: A] :
% 2.58/1.59 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_236_all__not__in__conv,axiom,
% 2.58/1.59 ! [A: $tType,A3: set @ A] :
% 2.58/1.59 ( ( ! [X2: A] :
% 2.58/1.59 ~ ( member @ A @ X2 @ A3 ) )
% 2.58/1.59 = ( A3
% 2.58/1.59 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_237_Collect__empty__eq,axiom,
% 2.58/1.59 ! [A: $tType,P: A > $o] :
% 2.58/1.59 ( ( ( collect @ A @ P )
% 2.58/1.59 = ( bot_bot @ ( set @ A ) ) )
% 2.58/1.59 = ( ! [X2: A] :
% 2.58/1.59 ~ ( P @ X2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_238_empty__Collect__eq,axiom,
% 2.58/1.59 ! [A: $tType,P: A > $o] :
% 2.58/1.59 ( ( ( bot_bot @ ( set @ A ) )
% 2.58/1.59 = ( collect @ A @ P ) )
% 2.58/1.59 = ( ! [X2: A] :
% 2.58/1.59 ~ ( P @ X2 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_239_subsetI,axiom,
% 2.58/1.59 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 2.58/1.59 ( ! [X3: A] :
% 2.58/1.59 ( ( member @ A @ X3 @ A3 )
% 2.58/1.59 => ( member @ A @ X3 @ B4 ) )
% 2.58/1.59 => ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_240_subset__antisym,axiom,
% 2.58/1.59 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 2.58/1.59 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 2.58/1.59 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ A3 )
% 2.58/1.59 => ( A3 = B4 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_241_Compl__subset__Compl__iff,axiom,
% 2.58/1.59 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 2.58/1.59 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) )
% 2.58/1.59 = ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_242_Compl__anti__mono,axiom,
% 2.58/1.59 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 2.58/1.59 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 2.58/1.59 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_243_subset__empty,axiom,
% 2.58/1.59 ! [A: $tType,A3: set @ A] :
% 2.58/1.59 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 2.58/1.59 = ( A3
% 2.58/1.59 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_244_empty__subsetI,axiom,
% 2.58/1.59 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_245_Diff__empty,axiom,
% 2.58/1.59 ! [A: $tType,A3: set @ A] :
% 2.58/1.59 ( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 2.58/1.59 = A3 ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_246_empty__Diff,axiom,
% 2.58/1.59 ! [A: $tType,A3: set @ A] :
% 2.58/1.59 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
% 2.58/1.59 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_247_Diff__cancel,axiom,
% 2.58/1.59 ! [A: $tType,A3: set @ A] :
% 2.58/1.59 ( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
% 2.58/1.59 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_248_emptyE,axiom,
% 2.58/1.59 ! [A: $tType,A2: A] :
% 2.58/1.59 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_249_equals0D,axiom,
% 2.58/1.59 ! [A: $tType,A3: set @ A,A2: A] :
% 2.58/1.59 ( ( A3
% 2.58/1.59 = ( bot_bot @ ( set @ A ) ) )
% 2.58/1.59 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_250_equals0I,axiom,
% 2.58/1.59 ! [A: $tType,A3: set @ A] :
% 2.58/1.59 ( ! [Y2: A] :
% 2.58/1.59 ~ ( member @ A @ Y2 @ A3 )
% 2.58/1.59 => ( A3
% 2.58/1.59 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_251_ex__in__conv,axiom,
% 2.58/1.59 ! [A: $tType,A3: set @ A] :
% 2.58/1.59 ( ( ? [X2: A] : ( member @ A @ X2 @ A3 ) )
% 2.58/1.59 = ( A3
% 2.58/1.59 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_252_in__mono,axiom,
% 2.58/1.59 ! [A: $tType,A3: set @ A,B4: set @ A,X: A] :
% 2.58/1.59 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 2.58/1.59 => ( ( member @ A @ X @ A3 )
% 2.58/1.59 => ( member @ A @ X @ B4 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_253_subsetD,axiom,
% 2.58/1.59 ! [A: $tType,A3: set @ A,B4: set @ A,C2: A] :
% 2.58/1.59 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 2.58/1.59 => ( ( member @ A @ C2 @ A3 )
% 2.58/1.59 => ( member @ A @ C2 @ B4 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_254_equalityE,axiom,
% 2.58/1.59 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 2.58/1.59 ( ( A3 = B4 )
% 2.58/1.59 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 2.58/1.59 => ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(fact_255_subset__eq,axiom,
% 2.58/1.59 ! [A: $tType] :
% 2.58/1.59 ( ( ord_less_eq @ ( set @ A ) )
% 2.58/1.59 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.58/1.59 ! [X2: A] :
% 2.58/1.59 ( ( member @ A @ X2 @ A5 )
% 2.58/1.59 => ( member @ A @ X2 @ B5 ) ) ) ) ).
% 2.58/1.59
% 2.58/1.59 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 2.58/1.59 ordere236663937imp_le @ int ).
% 2.58/1.59
% 2.58/1.59 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
% 2.58/1.59 ordere779506340up_add @ int ).
% 2.58/1.59
% 2.58/1.59 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
% 2.58/1.59 cancel146912293up_add @ int ).
% 2.58/1.59
% 2.58/1.59 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 2.58/1.59 cancel1352612707id_add @ int ).
% 2.58/1.59
% 2.58/1.59 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 2.58/1.59 ordered_ab_group_add @ int ).
% 2.58/1.59
% 2.58/1.59 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
% 2.58/1.59 cancel_semigroup_add @ int ).
% 2.58/1.59
% 2.58/1.59 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 2.58/1.59 linordered_semidom @ int ).
% 2.58/1.59
% 2.58/1.59 thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
% 2.58/1.59 ab_semigroup_add @ int ).
% 2.58/1.59
% 2.58/1.59 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
% 2.58/1.59 comm_monoid_add @ int ).
% 2.58/1.59
% 2.58/1.59 thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
% 2.58/1.59 semigroup_add @ int ).
% 2.58/1.59
% 2.58/1.59 thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
% 2.58/1.59 ab_group_add @ int ).
% 2.58/1.59
% 2.58/1.59 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 2.58/1.59 group_add @ int ).
% 2.58/1.59
% 2.58/1.59 thf(conj_0,conjecture,
% 2.58/1.59 ( ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) ) )
% 2.58/1.59 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( add @ a @ b @ r @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ) ).
% 2.58/1.59 ------------------------
% 2.75/1.61 % (3203)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/183Mi)
% 2.75/1.61 % (3205)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/27Mi)
% 2.75/1.61 % (3209)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/18Mi)
% 2.75/1.62 % (3206)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/2Mi)
% 2.75/1.62 % (3206)Instruction limit reached!
% 2.75/1.62 % (3206)------------------------------
% 2.75/1.62 % (3206)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.62 % (3206)Termination reason: Unknown
% 2.75/1.62 % (3206)Termination phase: shuffling
% 2.75/1.62
% 2.75/1.62 % (3206)Memory used [KB]: 1279
% 2.75/1.62 % (3206)Time elapsed: 0.002 s
% 2.75/1.62 % (3206)Instructions burned: 3 (million)
% 2.75/1.62 % (3206)------------------------------
% 2.75/1.62 % (3206)------------------------------
% 2.75/1.62 % (3209)Instruction limit reached!
% 2.75/1.62 % (3209)------------------------------
% 2.75/1.62 % (3209)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.62 % (3209)Termination reason: Unknown
% 2.75/1.62 % (3209)Termination phase: shuffling
% 2.75/1.62
% 2.75/1.62 % (3209)Memory used [KB]: 1407
% 2.75/1.62 % (3209)Time elapsed: 0.007 s
% 2.75/1.62 % (3209)Instructions burned: 18 (million)
% 2.75/1.62 % (3209)------------------------------
% 2.75/1.62 % (3209)------------------------------
% 2.75/1.62 % (3204)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/4Mi)
% 2.75/1.62 % (3207)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/2Mi)
% 2.75/1.62 % (3208)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/275Mi)
% 2.75/1.62 % (3205)Instruction limit reached!
% 2.75/1.62 % (3205)------------------------------
% 2.75/1.62 % (3205)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.62 % (3205)Termination reason: Unknown
% 2.75/1.62 % (3205)Termination phase: shuffling
% 2.75/1.62
% 2.75/1.62 % (3205)Memory used [KB]: 1535
% 2.75/1.62 % (3205)Time elapsed: 0.011 s
% 2.75/1.62 % (3205)Instructions burned: 30 (million)
% 2.75/1.62 % (3205)------------------------------
% 2.75/1.62 % (3205)------------------------------
% 2.75/1.62 % (3207)Instruction limit reached!
% 2.75/1.62 % (3207)------------------------------
% 2.75/1.62 % (3207)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.62 % (3207)Termination reason: Unknown
% 2.75/1.62 % (3207)Termination phase: shuffling
% 2.75/1.62
% 2.75/1.62 % (3207)Memory used [KB]: 1279
% 2.75/1.62 % (3207)Time elapsed: 0.004 s
% 2.75/1.62 % (3207)Instructions burned: 3 (million)
% 2.75/1.62 % (3207)------------------------------
% 2.75/1.62 % (3207)------------------------------
% 2.75/1.63 % (3204)Instruction limit reached!
% 2.75/1.63 % (3204)------------------------------
% 2.75/1.63 % (3204)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.63 % (3204)Termination reason: Unknown
% 2.75/1.63 % (3204)Termination phase: shuffling
% 2.75/1.63
% 2.75/1.63 % (3204)Memory used [KB]: 1279
% 2.75/1.63 % (3204)Time elapsed: 0.005 s
% 2.75/1.63 % (3204)Instructions burned: 5 (million)
% 2.75/1.63 % (3204)------------------------------
% 2.75/1.63 % (3204)------------------------------
% 2.75/1.63 % (3214)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/3Mi)
% 2.75/1.63 % (3214)Instruction limit reached!
% 2.75/1.63 % (3214)------------------------------
% 2.75/1.63 % (3214)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.63 % (3214)Termination reason: Unknown
% 2.75/1.63 % (3214)Termination phase: shuffling
% 2.75/1.63
% 2.75/1.63 % (3214)Memory used [KB]: 1279
% 2.75/1.63 % (3214)Time elapsed: 0.003 s
% 2.75/1.63 % (3214)Instructions burned: 4 (million)
% 2.75/1.63 % (3214)------------------------------
% 2.75/1.63 % (3214)------------------------------
% 2.75/1.63 % (3217)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/37Mi)
% 2.75/1.63 % (3220)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/15Mi)
% 2.75/1.64 % (3224)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/7Mi)
% 2.75/1.64 % (3220)Instruction limit reached!
% 2.75/1.64 % (3220)------------------------------
% 2.75/1.64 % (3220)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.64 % (3220)Termination reason: Unknown
% 2.75/1.64 % (3220)Termination phase: shuffling
% 2.75/1.64
% 2.75/1.64 % (3220)Memory used [KB]: 1407
% 2.75/1.64 % (3220)Time elapsed: 0.006 s
% 2.75/1.64 % (3220)Instructions burned: 15 (million)
% 2.75/1.64 % (3220)------------------------------
% 2.75/1.64 % (3220)------------------------------
% 2.75/1.64 % (3217)Instruction limit reached!
% 2.75/1.64 % (3217)------------------------------
% 2.75/1.64 % (3217)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.64 % (3217)Termination reason: Unknown
% 2.75/1.64 % (3217)Termination phase: shuffling
% 2.75/1.64
% 2.75/1.64 % (3217)Memory used [KB]: 1791
% 2.75/1.64 % (3217)Time elapsed: 0.011 s
% 2.75/1.64 % (3217)Instructions burned: 37 (million)
% 2.75/1.64 % (3217)------------------------------
% 2.75/1.64 % (3217)------------------------------
% 2.75/1.64 % (3224)Instruction limit reached!
% 2.75/1.64 % (3224)------------------------------
% 2.75/1.64 % (3224)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.64 % (3224)Termination reason: Unknown
% 2.75/1.64 % (3224)Termination phase: shuffling
% 2.75/1.64
% 2.75/1.64 % (3224)Memory used [KB]: 1407
% 2.75/1.64 % (3224)Time elapsed: 0.004 s
% 2.75/1.64 % (3224)Instructions burned: 9 (million)
% 2.75/1.64 % (3224)------------------------------
% 2.75/1.64 % (3224)------------------------------
% 2.75/1.64 % (3221)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/3Mi)
% 2.75/1.64 % (3222)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/1041Mi)
% 2.75/1.64 % (3221)Instruction limit reached!
% 2.75/1.64 % (3221)------------------------------
% 2.75/1.64 % (3221)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.64 % (3221)Termination reason: Unknown
% 2.75/1.64 % (3221)Termination phase: shuffling
% 2.75/1.64
% 2.75/1.64 % (3221)Memory used [KB]: 1279
% 2.75/1.64 % (3221)Time elapsed: 0.004 s
% 2.75/1.64 % (3221)Instructions burned: 3 (million)
% 2.75/1.64 % (3221)------------------------------
% 2.75/1.64 % (3221)------------------------------
% 2.75/1.65 % (3229)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/16Mi)
% 2.75/1.65 % (3231)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/3Mi)
% 2.75/1.65 % (3231)Instruction limit reached!
% 2.75/1.65 % (3231)------------------------------
% 2.75/1.65 % (3231)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.65 % (3231)Termination reason: Unknown
% 2.75/1.65 % (3231)Termination phase: shuffling
% 2.75/1.65
% 2.75/1.65 % (3231)Memory used [KB]: 1279
% 2.75/1.65 % (3231)Time elapsed: 0.003 s
% 2.75/1.65 % (3231)Instructions burned: 6 (million)
% 2.75/1.65 % (3231)------------------------------
% 2.75/1.65 % (3231)------------------------------
% 2.75/1.65 % (3232)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/3Mi)
% 2.75/1.65 % (3232)Instruction limit reached!
% 2.75/1.65 % (3232)------------------------------
% 2.75/1.65 % (3232)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.65 % (3232)Termination reason: Unknown
% 2.75/1.65 % (3232)Termination phase: shuffling
% 2.75/1.65
% 2.75/1.65 % (3232)Memory used [KB]: 1279
% 2.75/1.65 % (3232)Time elapsed: 0.002 s
% 2.75/1.65 % (3232)Instructions burned: 3 (million)
% 2.75/1.65 % (3232)------------------------------
% 2.75/1.65 % (3232)------------------------------
% 2.75/1.65 % (3229)Instruction limit reached!
% 2.75/1.65 % (3229)------------------------------
% 2.75/1.65 % (3229)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.65 % (3229)Termination reason: Unknown
% 2.75/1.65 % (3229)Termination phase: shuffling
% 2.75/1.65
% 2.75/1.65 % (3229)Memory used [KB]: 1407
% 2.75/1.65 % (3229)Time elapsed: 0.007 s
% 2.75/1.65 % (3229)Instructions burned: 18 (million)
% 2.75/1.65 % (3229)------------------------------
% 2.75/1.65 % (3229)------------------------------
% 2.75/1.66 % (3235)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/7Mi)
% 2.75/1.66 % (3235)Instruction limit reached!
% 2.75/1.66 % (3235)------------------------------
% 2.75/1.66 % (3235)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.66 % (3235)Termination reason: Unknown
% 2.75/1.66 % (3235)Termination phase: shuffling
% 2.75/1.66
% 2.75/1.66 % (3235)Memory used [KB]: 1279
% 2.75/1.66 % (3235)Time elapsed: 0.004 s
% 2.75/1.66 % (3235)Instructions burned: 8 (million)
% 2.75/1.66 % (3235)------------------------------
% 2.75/1.66 % (3235)------------------------------
% 2.75/1.66 % (3239)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/3Mi)
% 2.75/1.66 % (3239)Instruction limit reached!
% 2.75/1.66 % (3239)------------------------------
% 2.75/1.66 % (3239)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.66 % (3239)Termination reason: Unknown
% 2.75/1.66 % (3239)Termination phase: shuffling
% 2.75/1.66
% 2.75/1.66 % (3239)Memory used [KB]: 1279
% 2.75/1.66 % (3239)Time elapsed: 0.002 s
% 2.75/1.66 % (3239)Instructions burned: 3 (million)
% 2.75/1.66 % (3239)------------------------------
% 2.75/1.66 % (3239)------------------------------
% 2.75/1.66 % (3240)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/4Mi)
% 2.75/1.66 % (3241)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/18Mi)
% 2.75/1.66 % (3240)Instruction limit reached!
% 2.75/1.66 % (3240)------------------------------
% 2.75/1.66 % (3240)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.66 % (3240)Termination reason: Unknown
% 2.75/1.66 % (3240)Termination phase: shuffling
% 2.75/1.66
% 2.75/1.66 % (3240)Memory used [KB]: 1279
% 2.75/1.66 % (3240)Time elapsed: 0.003 s
% 2.75/1.66 % (3240)Instructions burned: 4 (million)
% 2.75/1.66 % (3240)------------------------------
% 2.75/1.66 % (3240)------------------------------
% 2.75/1.67 % (3203)Instruction limit reached!
% 2.75/1.67 % (3203)------------------------------
% 2.75/1.67 % (3203)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.67 % (3203)Termination reason: Unknown
% 2.75/1.67 % (3203)Termination phase: Saturation
% 2.75/1.67
% 2.75/1.67 % (3203)Memory used [KB]: 6652
% 2.75/1.67 % (3203)Time elapsed: 0.054 s
% 2.75/1.67 % (3203)Instructions burned: 183 (million)
% 2.75/1.67 % (3203)------------------------------
% 2.75/1.67 % (3203)------------------------------
% 2.75/1.67 % (3241)Instruction limit reached!
% 2.75/1.67 % (3241)------------------------------
% 2.75/1.67 % (3241)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.67 % (3241)Termination reason: Unknown
% 2.75/1.67 % (3241)Termination phase: shuffling
% 2.75/1.67
% 2.75/1.67 % (3241)Memory used [KB]: 1407
% 2.75/1.67 % (3241)Time elapsed: 0.007 s
% 2.75/1.67 % (3241)Instructions burned: 18 (million)
% 2.75/1.67 % (3241)------------------------------
% 2.75/1.67 % (3241)------------------------------
% 2.75/1.67 % (3245)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/710Mi)
% 2.75/1.67 % (3246)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/6Mi)
% 2.75/1.67 % (3246)Instruction limit reached!
% 2.75/1.67 % (3246)------------------------------
% 2.75/1.67 % (3246)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.67 % (3246)Termination reason: Unknown
% 2.75/1.67 % (3246)Termination phase: shuffling
% 2.75/1.67
% 2.75/1.67 % (3246)Memory used [KB]: 1279
% 2.75/1.67 % (3246)Time elapsed: 0.003 s
% 2.75/1.67 % (3246)Instructions burned: 6 (million)
% 2.75/1.67 % (3246)------------------------------
% 2.75/1.67 % (3246)------------------------------
% 2.75/1.67 % (3248)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/902Mi)
% 2.75/1.68 % (3250)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/21Mi)
% 2.75/1.68 % (3252)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/5Mi)
% 2.75/1.68 % (3252)Instruction limit reached!
% 2.75/1.68 % (3252)------------------------------
% 2.75/1.68 % (3252)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.68 % (3252)Termination reason: Unknown
% 2.75/1.68 % (3252)Termination phase: shuffling
% 2.75/1.68
% 2.75/1.68 % (3252)Memory used [KB]: 1407
% 2.75/1.68 % (3252)Time elapsed: 0.003 s
% 2.75/1.68 % (3252)Instructions burned: 7 (million)
% 2.75/1.68 % (3252)------------------------------
% 2.75/1.68 % (3252)------------------------------
% 2.75/1.68 % (3254)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/6Mi)
% 2.75/1.68 % (3250)Instruction limit reached!
% 2.75/1.68 % (3250)------------------------------
% 2.75/1.68 % (3250)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.68 % (3250)Termination reason: Unknown
% 2.75/1.68 % (3250)Termination phase: shuffling
% 2.75/1.68
% 2.75/1.68 % (3250)Memory used [KB]: 1407
% 2.75/1.68 % (3250)Time elapsed: 0.008 s
% 2.75/1.68 % (3250)Instructions burned: 21 (million)
% 2.75/1.68 % (3250)------------------------------
% 2.75/1.68 % (3250)------------------------------
% 2.75/1.68 % (3254)Instruction limit reached!
% 2.75/1.68 % (3254)------------------------------
% 2.75/1.68 % (3254)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.68 % (3254)Termination reason: Unknown
% 2.75/1.68 % (3254)Termination phase: shuffling
% 2.75/1.68
% 2.75/1.68 % (3254)Memory used [KB]: 1407
% 2.75/1.68 % (3254)Time elapsed: 0.003 s
% 2.75/1.68 % (3254)Instructions burned: 7 (million)
% 2.75/1.68 % (3254)------------------------------
% 2.75/1.68 % (3254)------------------------------
% 2.75/1.69 % (3260)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/377Mi)
% 2.75/1.69 % (3263)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/779Mi)
% 2.75/1.69 % (3264)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/19Mi)
% 2.75/1.70 % (3264)Instruction limit reached!
% 2.75/1.70 % (3264)------------------------------
% 2.75/1.70 % (3264)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.70 % (3264)Termination reason: Unknown
% 2.75/1.70 % (3264)Termination phase: shuffling
% 2.75/1.70
% 2.75/1.70 % (3264)Memory used [KB]: 1535
% 2.75/1.70 % (3264)Time elapsed: 0.007 s
% 2.75/1.70 % (3264)Instructions burned: 21 (million)
% 2.75/1.70 % (3264)------------------------------
% 2.75/1.70 % (3264)------------------------------
% 2.75/1.71 % (3271)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_3078 for (2999ds/879Mi)
% 2.75/1.73 % (3208)Instruction limit reached!
% 2.75/1.73 % (3208)------------------------------
% 2.75/1.73 % (3208)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.73 % (3208)Termination reason: Unknown
% 2.75/1.73 % (3208)Termination phase: Saturation
% 2.75/1.73
% 2.75/1.73 % (3208)Memory used [KB]: 6780
% 2.75/1.73 % (3208)Time elapsed: 0.105 s
% 2.75/1.73 % (3208)Instructions burned: 276 (million)
% 2.75/1.73 % (3208)------------------------------
% 2.75/1.73 % (3208)------------------------------
% 2.75/1.73 % (3281)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_3078 for (2998ds/17Mi)
% 2.75/1.74 % (3281)Instruction limit reached!
% 2.75/1.74 % (3281)------------------------------
% 2.75/1.74 % (3281)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.74 % (3281)Termination reason: Unknown
% 2.75/1.74 % (3281)Termination phase: shuffling
% 2.75/1.74
% 2.75/1.74 % (3281)Memory used [KB]: 1535
% 2.75/1.74 % (3281)Time elapsed: 0.007 s
% 2.75/1.74 % (3281)Instructions burned: 18 (million)
% 2.75/1.74 % (3281)------------------------------
% 2.75/1.74 % (3281)------------------------------
% 2.75/1.75 % (3288)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_3078 for (2998ds/3Mi)
% 2.75/1.75 % (3288)Instruction limit reached!
% 2.75/1.75 % (3288)------------------------------
% 2.75/1.75 % (3288)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.75 % (3288)Termination reason: Unknown
% 2.75/1.75 % (3288)Termination phase: shuffling
% 2.75/1.75
% 2.75/1.75 % (3288)Memory used [KB]: 1279
% 2.75/1.75 % (3288)Time elapsed: 0.003 s
% 2.75/1.75 % (3288)Instructions burned: 5 (million)
% 2.75/1.75 % (3288)------------------------------
% 2.75/1.75 % (3288)------------------------------
% 2.75/1.76 % (3291)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_3078 for (2998ds/30Mi)
% 2.75/1.77 % (3291)Instruction limit reached!
% 2.75/1.77 % (3291)------------------------------
% 2.75/1.77 % (3291)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.77 % (3291)Termination reason: Unknown
% 2.75/1.77 % (3291)Termination phase: shuffling
% 2.75/1.77
% 2.75/1.77 % (3291)Memory used [KB]: 1535
% 2.75/1.77 % (3291)Time elapsed: 0.011 s
% 2.75/1.77 % (3291)Instructions burned: 32 (million)
% 2.75/1.77 % (3291)------------------------------
% 2.75/1.77 % (3291)------------------------------
% 2.75/1.78 % (3301)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_3078 for (2998ds/127Mi)
% 2.75/1.80 % (3260)Instruction limit reached!
% 2.75/1.80 % (3260)------------------------------
% 2.75/1.80 % (3260)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.80 % (3260)Termination reason: Unknown
% 2.75/1.80 % (3260)Termination phase: Saturation
% 2.75/1.80
% 2.75/1.80 % (3260)Memory used [KB]: 7291
% 2.75/1.80 % (3260)Time elapsed: 0.116 s
% 2.75/1.80 % (3260)Instructions burned: 379 (million)
% 2.75/1.80 % (3260)------------------------------
% 2.75/1.80 % (3260)------------------------------
% 2.75/1.81 % (3315)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_3078 for (2997ds/100Mi)
% 2.75/1.82 % (3301)Instruction limit reached!
% 2.75/1.82 % (3301)------------------------------
% 2.75/1.82 % (3301)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.82 % (3301)Termination reason: Unknown
% 2.75/1.82 % (3301)Termination phase: Property scanning
% 2.75/1.82
% 2.75/1.82 % (3301)Memory used [KB]: 1791
% 2.75/1.82 % (3301)Time elapsed: 0.036 s
% 2.75/1.82 % (3301)Instructions burned: 127 (million)
% 2.75/1.82 % (3301)------------------------------
% 2.75/1.82 % (3301)------------------------------
% 2.75/1.83 % (3323)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_3078 for (2997ds/3Mi)
% 2.75/1.83 % (3323)Instruction limit reached!
% 2.75/1.83 % (3323)------------------------------
% 2.75/1.83 % (3323)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.83 % (3323)Termination reason: Unknown
% 2.75/1.83 % (3323)Termination phase: shuffling
% 2.75/1.83
% 2.75/1.83 % (3323)Memory used [KB]: 1407
% 2.75/1.83 % (3323)Time elapsed: 0.003 s
% 2.75/1.83 % (3323)Instructions burned: 6 (million)
% 2.75/1.83 % (3323)------------------------------
% 2.75/1.83 % (3323)------------------------------
% 2.75/1.84 % (3324)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_3078 for (2997ds/20Mi)
% 2.75/1.84 % (3315)Instruction limit reached!
% 2.75/1.84 % (3315)------------------------------
% 2.75/1.84 % (3315)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.84 % (3315)Termination reason: Unknown
% 2.75/1.84 % (3315)Termination phase: Property scanning
% 2.75/1.84
% 2.75/1.84 % (3315)Memory used [KB]: 2046
% 2.75/1.84 % (3315)Time elapsed: 0.028 s
% 2.75/1.84 % (3315)Instructions burned: 102 (million)
% 2.75/1.84 % (3315)------------------------------
% 2.75/1.84 % (3315)------------------------------
% 2.75/1.84 % (3324)Instruction limit reached!
% 2.75/1.84 % (3324)------------------------------
% 2.75/1.84 % (3324)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.84 % (3324)Termination reason: Unknown
% 2.75/1.84 % (3324)Termination phase: shuffling
% 2.75/1.84
% 2.75/1.84 % (3324)Memory used [KB]: 1535
% 2.75/1.84 % (3324)Time elapsed: 0.008 s
% 2.75/1.84 % (3324)Instructions burned: 22 (million)
% 2.75/1.84 % (3324)------------------------------
% 2.75/1.84 % (3324)------------------------------
% 2.75/1.85 % (3335)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_3078 for (2997ds/86Mi)
% 2.75/1.85 % (3337)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_3078 for (2997ds/107Mi)
% 2.75/1.87 % (3245)Refutation not found, non-redundant clauses discarded% (3245)------------------------------
% 2.75/1.87 % (3245)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.87 % (3245)Termination reason: Refutation not found, non-redundant clauses discarded
% 2.75/1.87
% 2.75/1.87 % (3245)Memory used [KB]: 7803
% 2.75/1.87 % (3245)Time elapsed: 0.197 s
% 2.75/1.87 % (3245)Instructions burned: 667 (million)
% 2.75/1.87 % (3245)------------------------------
% 2.75/1.87 % (3245)------------------------------
% 2.75/1.88 % (3335)Instruction limit reached!
% 2.75/1.88 % (3335)------------------------------
% 2.75/1.88 % (3335)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.88 % (3335)Termination reason: Unknown
% 2.75/1.88 % (3335)Termination phase: Property scanning
% 2.75/1.88
% 2.75/1.88 % (3335)Memory used [KB]: 1791
% 2.75/1.88 % (3335)Time elapsed: 0.026 s
% 2.75/1.88 % (3335)Instructions burned: 89 (million)
% 2.75/1.88 % (3335)------------------------------
% 2.75/1.88 % (3335)------------------------------
% 2.75/1.88 % (3351)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_3078 for (2997ds/39Mi)
% 2.75/1.88 % (3337)Instruction limit reached!
% 2.75/1.88 % (3337)------------------------------
% 2.75/1.88 % (3337)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.88 % (3337)Termination reason: Unknown
% 2.75/1.88 % (3337)Termination phase: Property scanning
% 2.75/1.88
% 2.75/1.88 % (3337)Memory used [KB]: 1791
% 2.75/1.88 % (3337)Time elapsed: 0.030 s
% 2.75/1.88 % (3337)Instructions burned: 107 (million)
% 2.75/1.88 % (3337)------------------------------
% 2.75/1.88 % (3337)------------------------------
% 2.75/1.89 % (3351)Instruction limit reached!
% 2.75/1.89 % (3351)------------------------------
% 2.75/1.89 % (3351)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.89 % (3351)Termination reason: Unknown
% 2.75/1.89 % (3351)Termination phase: Property scanning
% 2.75/1.89
% 2.75/1.89 % (3351)Memory used [KB]: 1918
% 2.75/1.89 % (3351)Time elapsed: 0.012 s
% 2.75/1.89 % (3351)Instructions burned: 41 (million)
% 2.75/1.89 % (3351)------------------------------
% 2.75/1.89 % (3351)------------------------------
% 2.75/1.89 % (3353)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_3078 for (2997ds/46Mi)
% 2.75/1.89 % (3352)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_3078 for (2997ds/448Mi)
% 2.75/1.90 % (3359)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_3078 for (2997ds/98Mi)
% 2.75/1.91 % (3353)Instruction limit reached!
% 2.75/1.91 % (3353)------------------------------
% 2.75/1.91 % (3353)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.91 % (3353)Termination reason: Unknown
% 2.75/1.91 % (3353)Termination phase: Property scanning
% 2.75/1.91
% 2.75/1.91 % (3353)Memory used [KB]: 1663
% 2.75/1.91 % (3353)Time elapsed: 0.016 s
% 2.75/1.91 % (3353)Instructions burned: 46 (million)
% 2.75/1.91 % (3353)------------------------------
% 2.75/1.91 % (3353)------------------------------
% 2.75/1.92 % (3370)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_3078 for (2997ds/507Mi)
% 2.75/1.92 % (3359)Instruction limit reached!
% 2.75/1.92 % (3359)------------------------------
% 2.75/1.92 % (3359)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.92 % (3359)Termination reason: Unknown
% 2.75/1.92 % (3359)Termination phase: Property scanning
% 2.75/1.92
% 2.75/1.92 % (3359)Memory used [KB]: 2046
% 2.75/1.92 % (3359)Time elapsed: 0.027 s
% 2.75/1.92 % (3359)Instructions burned: 98 (million)
% 2.75/1.92 % (3359)------------------------------
% 2.75/1.92 % (3359)------------------------------
% 2.75/1.93 % (3263)Instruction limit reached!
% 2.75/1.93 % (3263)------------------------------
% 2.75/1.93 % (3263)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.93 % (3263)Termination reason: Unknown
% 2.75/1.93 % (3263)Termination phase: Saturation
% 2.75/1.93
% 2.75/1.93 % (3263)Memory used [KB]: 7419
% 2.75/1.93 % (3263)Time elapsed: 0.236 s
% 2.75/1.93 % (3263)Instructions burned: 780 (million)
% 2.75/1.93 % (3263)------------------------------
% 2.75/1.93 % (3263)------------------------------
% 2.75/1.93 % (3378)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_3078 for (2996ds/149Mi)
% 2.75/1.94 % (3383)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_3078 for (2996ds/39Mi)
% 2.75/1.95 % (3248)Instruction limit reached!
% 2.75/1.95 % (3248)------------------------------
% 2.75/1.95 % (3248)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.95 % (3248)Termination reason: Unknown
% 2.75/1.95 % (3248)Termination phase: Saturation
% 2.75/1.95
% 2.75/1.95 % (3248)Memory used [KB]: 7291
% 2.75/1.95 % (3248)Time elapsed: 0.274 s
% 2.75/1.95 % (3248)Instructions burned: 904 (million)
% 2.75/1.95 % (3248)------------------------------
% 2.75/1.95 % (3248)------------------------------
% 2.75/1.95 % (3383)Instruction limit reached!
% 2.75/1.95 % (3383)------------------------------
% 2.75/1.95 % (3383)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.95 % (3383)Termination reason: Unknown
% 2.75/1.95 % (3383)Termination phase: Property scanning
% 2.75/1.95
% 2.75/1.95 % (3383)Memory used [KB]: 1663
% 2.75/1.95 % (3383)Time elapsed: 0.014 s
% 2.75/1.95 % (3383)Instructions burned: 41 (million)
% 2.75/1.95 % (3383)------------------------------
% 2.75/1.95 % (3383)------------------------------
% 2.75/1.96 % (3388)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_3078 for (2996ds/6Mi)
% 2.75/1.96 % (3271)Refutation not found, non-redundant clauses discarded% (3271)------------------------------
% 2.75/1.96 % (3271)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.96 % (3271)Termination reason: Refutation not found, non-redundant clauses discarded
% 2.75/1.96
% 2.75/1.96 % (3271)Memory used [KB]: 8059
% 2.75/1.96 % (3271)Time elapsed: 0.248 s
% 2.75/1.96 % (3271)Instructions burned: 852 (million)
% 2.75/1.96 % (3271)------------------------------
% 2.75/1.96 % (3271)------------------------------
% 2.75/1.96 % (3388)Instruction limit reached!
% 2.75/1.96 % (3388)------------------------------
% 2.75/1.96 % (3388)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.96 % (3388)Termination reason: Unknown
% 2.75/1.96 % (3388)Termination phase: shuffling
% 2.75/1.96
% 2.75/1.96 % (3388)Memory used [KB]: 1407
% 2.75/1.96 % (3388)Time elapsed: 0.003 s
% 2.75/1.96 % (3388)Instructions burned: 7 (million)
% 2.75/1.96 % (3388)------------------------------
% 2.75/1.96 % (3388)------------------------------
% 2.75/1.96 % (3392)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_3078 for (2996ds/1087Mi)
% 2.75/1.97 % (3222)Instruction limit reached!
% 2.75/1.97 % (3222)------------------------------
% 2.75/1.97 % (3222)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.97 % (3222)Termination reason: Unknown
% 2.75/1.97 % (3222)Termination phase: Saturation
% 2.75/1.97
% 2.75/1.97 % (3222)Memory used [KB]: 8443
% 2.75/1.97 % (3222)Time elapsed: 0.325 s
% 2.75/1.97 % (3222)Instructions burned: 1041 (million)
% 2.75/1.97 % (3222)------------------------------
% 2.75/1.97 % (3222)------------------------------
% 2.75/1.97 % (3396)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_3078 for (2996ds/47Mi)
% 2.75/1.97 % (3397)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_3078 for (2996ds/44Mi)
% 2.75/1.97 % (3378)Instruction limit reached!
% 2.75/1.97 % (3378)------------------------------
% 2.75/1.97 % (3378)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.97 % (3378)Termination reason: Unknown
% 2.75/1.97 % (3378)Termination phase: Saturation
% 2.75/1.97
% 2.75/1.97 % (3378)Memory used [KB]: 6524
% 2.75/1.97 % (3378)Time elapsed: 0.042 s
% 2.75/1.97 % (3378)Instructions burned: 149 (million)
% 2.75/1.97 % (3378)------------------------------
% 2.75/1.97 % (3378)------------------------------
% 2.75/1.98 % (3399)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_3078 for (2996ds/432Mi)
% 2.75/1.98 % (3396)Instruction limit reached!
% 2.75/1.98 % (3396)------------------------------
% 2.75/1.98 % (3396)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.98 % (3396)Termination reason: Unknown
% 2.75/1.98 % (3396)Termination phase: Property scanning
% 2.75/1.98
% 2.75/1.98 % (3396)Memory used [KB]: 1663
% 2.75/1.98 % (3396)Time elapsed: 0.016 s
% 2.75/1.98 % (3396)Instructions burned: 49 (million)
% 2.75/1.98 % (3396)------------------------------
% 2.75/1.98 % (3396)------------------------------
% 2.75/1.98 % (3397)Instruction limit reached!
% 2.75/1.98 % (3397)------------------------------
% 2.75/1.98 % (3397)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/1.98 % (3397)Termination reason: Unknown
% 2.75/1.98 % (3397)Termination phase: Property scanning
% 2.75/1.98
% 2.75/1.98 % (3397)Memory used [KB]: 1918
% 2.75/1.98 % (3397)Time elapsed: 0.014 s
% 2.75/1.98 % (3397)Instructions burned: 47 (million)
% 2.75/1.98 % (3397)------------------------------
% 2.75/1.98 % (3397)------------------------------
% 2.75/1.98 % (3403)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_3078 for (2996ds/62Mi)
% 2.75/1.99 % (3407)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_3078 for (2996ds/31Mi)
% 2.75/1.99 % (3408)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_3078 for (2996ds/96Mi)
% 2.75/2.00 % (3403)Instruction limit reached!
% 2.75/2.00 % (3403)------------------------------
% 2.75/2.00 % (3403)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/2.00 % (3403)Termination reason: Unknown
% 2.75/2.00 % (3403)Termination phase: Property scanning
% 2.75/2.00
% 2.75/2.00 % (3403)Memory used [KB]: 1918
% 2.75/2.00 % (3403)Time elapsed: 0.018 s
% 2.75/2.00 % (3403)Instructions burned: 63 (million)
% 2.75/2.00 % (3403)------------------------------
% 2.75/2.00 % (3403)------------------------------
% 2.75/2.00 % (3407)Instruction limit reached!
% 2.75/2.00 % (3407)------------------------------
% 2.75/2.00 % (3407)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.75/2.00 % (3407)Termination reason: Unknown
% 2.75/2.00 % (3407)Termination phase: shuffling
% 2.75/2.00
% 2.75/2.00 % (3407)Memory used [KB]: 1535
% 2.75/2.00 % (3407)Time elapsed: 0.012 s
% 2.75/2.00 % (3407)Instructions burned: 32 (million)
% 2.75/2.00 % (3407)------------------------------
% 2.75/2.00 % (3407)------------------------------
% 2.75/2.01 % (3415)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_3078 for (2995ds/453Mi)
% 2.75/2.01 % (3417)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_3078 for (2995ds/71Mi)
% 5.54/2.02 % (3408)Instruction limit reached!
% 5.54/2.02 % (3408)------------------------------
% 5.54/2.02 % (3408)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.02 % (3408)Termination reason: Unknown
% 5.54/2.02 % (3408)Termination phase: Function definition elimination
% 5.54/2.02
% 5.54/2.02 % (3408)Memory used [KB]: 1791
% 5.54/2.02 % (3408)Time elapsed: 0.031 s
% 5.54/2.02 % (3408)Instructions burned: 97 (million)
% 5.54/2.02 % (3408)------------------------------
% 5.54/2.02 % (3408)------------------------------
% 5.54/2.03 % (3417)Instruction limit reached!
% 5.54/2.03 % (3417)------------------------------
% 5.54/2.03 % (3417)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.03 % (3417)Termination reason: Unknown
% 5.54/2.03 % (3417)Termination phase: Property scanning
% 5.54/2.03
% 5.54/2.03 % (3417)Memory used [KB]: 1663
% 5.54/2.03 % (3417)Time elapsed: 0.022 s
% 5.54/2.03 % (3417)Instructions burned: 71 (million)
% 5.54/2.03 % (3417)------------------------------
% 5.54/2.03 % (3417)------------------------------
% 5.54/2.03 % (3425)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_3078 for (2995ds/83Mi)
% 5.54/2.04 % (3429)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_3078 for (2995ds/11Mi)
% 5.54/2.04 % (3429)Instruction limit reached!
% 5.54/2.04 % (3429)------------------------------
% 5.54/2.04 % (3429)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.04 % (3429)Termination reason: Unknown
% 5.54/2.04 % (3429)Termination phase: shuffling
% 5.54/2.04
% 5.54/2.04 % (3429)Memory used [KB]: 1407
% 5.54/2.04 % (3429)Time elapsed: 0.004 s
% 5.54/2.04 % (3429)Instructions burned: 11 (million)
% 5.54/2.04 % (3429)------------------------------
% 5.54/2.04 % (3429)------------------------------
% 5.54/2.05 % (3433)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_3078 for (2995ds/41Mi)
% 5.54/2.06 % (3425)Instruction limit reached!
% 5.54/2.06 % (3425)------------------------------
% 5.54/2.06 % (3425)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.06 % (3425)Termination reason: Unknown
% 5.54/2.06 % (3425)Termination phase: Property scanning
% 5.54/2.06
% 5.54/2.06 % (3425)Memory used [KB]: 1663
% 5.54/2.06 % (3425)Time elapsed: 0.025 s
% 5.54/2.06 % (3425)Instructions burned: 84 (million)
% 5.54/2.06 % (3425)------------------------------
% 5.54/2.06 % (3425)------------------------------
% 5.54/2.06 % (3352)Instruction limit reached!
% 5.54/2.06 % (3352)------------------------------
% 5.54/2.06 % (3352)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.06 % (3352)Termination reason: Unknown
% 5.54/2.06 % (3352)Termination phase: Saturation
% 5.54/2.06
% 5.54/2.06 % (3352)Memory used [KB]: 7291
% 5.54/2.06 % (3352)Time elapsed: 0.165 s
% 5.54/2.06 % (3352)Instructions burned: 450 (million)
% 5.54/2.06 % (3352)------------------------------
% 5.54/2.06 % (3352)------------------------------
% 5.54/2.06 % (3370)Instruction limit reached!
% 5.54/2.06 % (3370)------------------------------
% 5.54/2.06 % (3370)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.06 % (3370)Termination reason: Unknown
% 5.54/2.06 % (3370)Termination phase: Saturation
% 5.54/2.06
% 5.54/2.06 % (3370)Memory used [KB]: 7164
% 5.54/2.06 % (3370)Time elapsed: 0.145 s
% 5.54/2.06 % (3370)Instructions burned: 508 (million)
% 5.54/2.06 % (3370)------------------------------
% 5.54/2.06 % (3370)------------------------------
% 5.54/2.06 % (3435)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_3078 for (2995ds/6Mi)
% 5.54/2.06 % (3433)Instruction limit reached!
% 5.54/2.06 % (3433)------------------------------
% 5.54/2.06 % (3433)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.06 % (3433)Termination reason: Unknown
% 5.54/2.06 % (3433)Termination phase: Property scanning
% 5.54/2.06
% 5.54/2.06 % (3433)Memory used [KB]: 1918
% 5.54/2.06 % (3433)Time elapsed: 0.013 s
% 5.54/2.06 % (3433)Instructions burned: 44 (million)
% 5.54/2.06 % (3433)------------------------------
% 5.54/2.06 % (3433)------------------------------
% 5.54/2.07 % (3436)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_3078 for (2995ds/150Mi)
% 5.54/2.07 % (3435)Instruction limit reached!
% 5.54/2.07 % (3435)------------------------------
% 5.54/2.07 % (3435)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.07 % (3435)Termination reason: Unknown
% 5.54/2.07 % (3435)Termination phase: shuffling
% 5.54/2.07
% 5.54/2.07 % (3435)Memory used [KB]: 1407
% 5.54/2.07 % (3435)Time elapsed: 0.003 s
% 5.54/2.07 % (3435)Instructions burned: 6 (million)
% 5.54/2.07 % (3435)------------------------------
% 5.54/2.07 % (3435)------------------------------
% 5.54/2.07 % (3438)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_3078 for (2995ds/343Mi)
% 5.54/2.07 % (3440)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_3078 for (2995ds/109Mi)
% 5.54/2.08 % (3442)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_3078 for (2995ds/253Mi)
% 5.54/2.10 % (3399)Instruction limit reached!
% 5.54/2.10 % (3399)------------------------------
% 5.54/2.10 % (3399)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.10 % (3399)Termination reason: Unknown
% 5.54/2.10 % (3399)Termination phase: Saturation
% 5.54/2.10
% 5.54/2.10 % (3399)Memory used [KB]: 3326
% 5.54/2.10 % (3399)Time elapsed: 0.130 s
% 5.54/2.10 % (3399)Instructions burned: 433 (million)
% 5.54/2.10 % (3399)------------------------------
% 5.54/2.10 % (3399)------------------------------
% 5.54/2.11 % (3440)Instruction limit reached!
% 5.54/2.11 % (3440)------------------------------
% 5.54/2.11 % (3440)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.11 % (3440)Termination reason: Unknown
% 5.54/2.11 % (3440)Termination phase: Function definition elimination
% 5.54/2.11
% 5.54/2.11 % (3440)Memory used [KB]: 1791
% 5.54/2.11 % (3440)Time elapsed: 0.033 s
% 5.54/2.11 % (3440)Instructions burned: 111 (million)
% 5.54/2.11 % (3440)------------------------------
% 5.54/2.11 % (3440)------------------------------
% 5.54/2.11 % (3436)Instruction limit reached!
% 5.54/2.11 % (3436)------------------------------
% 5.54/2.11 % (3436)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.11 % (3436)Termination reason: Unknown
% 5.54/2.11 % (3436)Termination phase: Saturation
% 5.54/2.11
% 5.54/2.11 % (3436)Memory used [KB]: 6396
% 5.54/2.11 % (3436)Time elapsed: 0.044 s
% 5.54/2.11 % (3436)Instructions burned: 151 (million)
% 5.54/2.11 % (3436)------------------------------
% 5.54/2.11 % (3436)------------------------------
% 6.07/2.11 % (3452)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_3078 for (2994ds/140Mi)
% 6.07/2.12 % (3453)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_3078 for (2994ds/614Mi)
% 6.07/2.12 % (3454)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_3078 for (2994ds/202Mi)
% 6.07/2.14 % (3415)Instruction limit reached!
% 6.07/2.14 % (3415)------------------------------
% 6.07/2.14 % (3415)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.07/2.14 % (3415)Termination reason: Unknown
% 6.07/2.14 % (3415)Termination phase: Saturation
% 6.07/2.14
% 6.07/2.14 % (3415)Memory used [KB]: 7419
% 6.07/2.14 % (3415)Time elapsed: 0.131 s
% 6.07/2.14 % (3415)Instructions burned: 453 (million)
% 6.07/2.14 % (3415)------------------------------
% 6.07/2.14 % (3415)------------------------------
% 6.07/2.15 % (3461)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_3078 for (2994ds/339Mi)
% 6.07/2.15 % (3442)Instruction limit reached!
% 6.07/2.15 % (3442)------------------------------
% 6.07/2.15 % (3442)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.07/2.15 % (3442)Termination reason: Unknown
% 6.07/2.15 % (3442)Termination phase: Saturation
% 6.07/2.15
% 6.07/2.15 % (3442)Memory used [KB]: 7036
% 6.07/2.15 % (3442)Time elapsed: 0.073 s
% 6.07/2.15 % (3442)Instructions burned: 253 (million)
% 6.07/2.15 % (3442)------------------------------
% 6.07/2.15 % (3442)------------------------------
% 6.07/2.15 % (3452)Instruction limit reached!
% 6.07/2.15 % (3452)------------------------------
% 6.07/2.15 % (3452)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.07/2.15 % (3452)Termination reason: Unknown
% 6.07/2.15 % (3452)Termination phase: Saturation
% 6.07/2.15
% 6.07/2.15 % (3452)Memory used [KB]: 6268
% 6.07/2.15 % (3452)Time elapsed: 0.041 s
% 6.07/2.15 % (3452)Instructions burned: 142 (million)
% 6.07/2.15 % (3452)------------------------------
% 6.07/2.15 % (3452)------------------------------
% 6.07/2.16 % (3468)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=283:si=on:rtra=on_0 on DTF2THF_3078 for (2994ds/283Mi)
% 6.07/2.16 % (3473)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_3078 for (2994ds/134Mi)
% 6.07/2.17 % (3438)Instruction limit reached!
% 6.07/2.17 % (3438)------------------------------
% 6.07/2.17 % (3438)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.07/2.17 % (3438)Termination reason: Unknown
% 6.07/2.17 % (3438)Termination phase: Saturation
% 6.07/2.17
% 6.07/2.17 % (3438)Memory used [KB]: 6780
% 6.07/2.17 % (3438)Time elapsed: 0.105 s
% 6.07/2.17 % (3438)Instructions burned: 344 (million)
% 6.07/2.17 % (3438)------------------------------
% 6.07/2.17 % (3438)------------------------------
% 6.07/2.17 % (3454)Instruction limit reached!
% 6.07/2.17 % (3454)------------------------------
% 6.07/2.17 % (3454)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.07/2.17 % (3454)Termination reason: Unknown
% 6.07/2.17 % (3454)Termination phase: Saturation
% 6.07/2.17
% 6.07/2.17 % (3454)Memory used [KB]: 6652
% 6.07/2.17 % (3454)Time elapsed: 0.058 s
% 6.07/2.17 % (3454)Instructions burned: 204 (million)
% 6.07/2.17 % (3454)------------------------------
% 6.07/2.17 % (3454)------------------------------
% 6.07/2.18 % (3484)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_3078 for (2994ds/298Mi)
% 6.07/2.18 % (3486)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_3078 for (2994ds/192Mi)
% 6.61/2.20 % (3473)Instruction limit reached!
% 6.61/2.20 % (3473)------------------------------
% 6.61/2.20 % (3473)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.61/2.20 % (3473)Termination reason: Unknown
% 6.61/2.20 % (3473)Termination phase: Saturation
% 6.61/2.20
% 6.61/2.20 % (3473)Memory used [KB]: 6396
% 6.61/2.20 % (3473)Time elapsed: 0.040 s
% 6.61/2.20 % (3473)Instructions burned: 136 (million)
% 6.61/2.20 % (3473)------------------------------
% 6.61/2.20 % (3473)------------------------------
% 6.61/2.21 % (3499)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_3078 for (2993ds/258Mi)
% 6.61/2.24 % (3468)Instruction limit reached!
% 6.61/2.24 % (3468)------------------------------
% 6.61/2.24 % (3468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.61/2.24 % (3468)Termination reason: Unknown
% 6.61/2.24 % (3468)Termination phase: Saturation
% 6.61/2.24
% 6.61/2.24 % (3468)Memory used [KB]: 6908
% 6.61/2.24 % (3468)Time elapsed: 0.082 s
% 6.61/2.24 % (3468)Instructions burned: 285 (million)
% 6.61/2.24 % (3468)------------------------------
% 6.61/2.24 % (3468)------------------------------
% 6.61/2.24 % (3486)Instruction limit reached!
% 6.61/2.24 % (3486)------------------------------
% 6.61/2.24 % (3486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.61/2.24 % (3486)Termination reason: Unknown
% 6.61/2.24 % (3486)Termination phase: Saturation
% 6.61/2.24
% 6.61/2.24 % (3486)Memory used [KB]: 2686
% 6.61/2.24 % (3486)Time elapsed: 0.056 s
% 6.61/2.24 % (3486)Instructions burned: 192 (million)
% 6.61/2.24 % (3486)------------------------------
% 6.61/2.24 % (3486)------------------------------
% 6.61/2.25 % (3504)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_3078 for (2993ds/90Mi)
% 6.61/2.25 % (3505)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_3078 for (2993ds/231Mi)
% 6.61/2.25 % (3461)Instruction limit reached!
% 6.61/2.25 % (3461)------------------------------
% 6.61/2.25 % (3461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.61/2.25 % (3461)Termination reason: Unknown
% 6.61/2.25 % (3461)Termination phase: Saturation
% 6.61/2.25
% 6.61/2.25 % (3461)Memory used [KB]: 6908
% 6.61/2.25 % (3461)Time elapsed: 0.103 s
% 6.61/2.25 % (3461)Instructions burned: 339 (million)
% 6.61/2.25 % (3461)------------------------------
% 6.61/2.25 % (3461)------------------------------
% 6.61/2.26 % (3506)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_3078 for (2993ds/458Mi)
% 6.61/2.27 % (3484)Instruction limit reached!
% 6.61/2.27 % (3484)------------------------------
% 6.61/2.27 % (3484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.61/2.27 % (3484)Termination reason: Unknown
% 6.61/2.27 % (3484)Termination phase: Saturation
% 6.61/2.27
% 6.61/2.27 % (3484)Memory used [KB]: 7419
% 6.61/2.27 % (3484)Time elapsed: 0.086 s
% 6.61/2.27 % (3484)Instructions burned: 298 (million)
% 6.61/2.27 % (3484)------------------------------
% 6.61/2.27 % (3484)------------------------------
% 6.61/2.27 % (3504)Instruction limit reached!
% 6.61/2.27 % (3504)------------------------------
% 6.61/2.27 % (3504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.61/2.27 % (3504)Termination reason: Unknown
% 6.61/2.27 % (3504)Termination phase: Property scanning
% 6.61/2.27
% 6.61/2.27 % (3504)Memory used [KB]: 1791
% 6.61/2.27 % (3504)Time elapsed: 0.027 s
% 6.61/2.27 % (3504)Instructions burned: 90 (million)
% 6.61/2.27 % (3504)------------------------------
% 6.61/2.27 % (3504)------------------------------
% 6.61/2.28 % (3507)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_3078 for (2993ds/31Mi)
% 6.61/2.28 % (3508)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_3078 for (2993ds/297Mi)
% 6.61/2.28 % (3499)Instruction limit reached!
% 6.61/2.28 % (3499)------------------------------
% 6.61/2.28 % (3499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.61/2.28 % (3499)Termination reason: Unknown
% 6.61/2.28 % (3499)Termination phase: Saturation
% 6.61/2.28
% 6.61/2.28 % (3499)Memory used [KB]: 6908
% 6.61/2.28 % (3499)Time elapsed: 0.073 s
% 6.61/2.28 % (3499)Instructions burned: 259 (million)
% 6.61/2.28 % (3499)------------------------------
% 6.61/2.28 % (3499)------------------------------
% 6.61/2.29 % (3507)Instruction limit reached!
% 6.61/2.29 % (3507)------------------------------
% 6.61/2.29 % (3507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.61/2.29 % (3507)Termination reason: Unknown
% 6.61/2.29 % (3507)Termination phase: shuffling
% 6.61/2.29
% 6.61/2.29 % (3507)Memory used [KB]: 1535
% 6.61/2.29 % (3507)Time elapsed: 0.011 s
% 6.61/2.29 % (3507)Instructions burned: 31 (million)
% 6.61/2.29 % (3507)------------------------------
% 6.61/2.29 % (3507)------------------------------
% 6.61/2.29 % (3392)Instruction limit reached!
% 6.61/2.29 % (3392)------------------------------
% 6.61/2.29 % (3392)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.61/2.29 % (3392)Termination reason: Unknown
% 6.61/2.29 % (3392)Termination phase: Saturation
% 6.61/2.29
% 6.61/2.29 % (3392)Memory used [KB]: 7547
% 6.61/2.29 % (3392)Time elapsed: 0.353 s
% 6.61/2.29 % (3392)Instructions burned: 1087 (million)
% 6.61/2.29 % (3392)------------------------------
% 6.61/2.29 % (3392)------------------------------
% 6.61/2.29 % (3509)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_3078 for (2993ds/987Mi)
% 6.61/2.29 % (3453)Instruction limit reached!
% 6.61/2.29 % (3453)------------------------------
% 6.61/2.29 % (3453)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.61/2.29 % (3453)Termination reason: Unknown
% 6.61/2.29 % (3453)Termination phase: Saturation
% 6.61/2.29
% 6.61/2.29 % (3453)Memory used [KB]: 7675
% 6.61/2.29 % (3453)Time elapsed: 0.180 s
% 6.61/2.29 % (3453)Instructions burned: 615 (million)
% 6.61/2.29 % (3453)------------------------------
% 6.61/2.29 % (3453)------------------------------
% 7.45/2.30 % (3510)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_3078 for (2993ds/182Mi)
% 7.45/2.30 % (3511)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_3078 for (2993ds/182Mi)
% 7.45/2.30 % (3512)dis+10_1:8_cha=on:cnfonf=lazy_pi_sigma_gen:fe=axiom:hud=5:plsq=on:plsqr=32,1:prag=on:sos=all:i=154:si=on:rtra=on_0 on DTF2THF_3078 for (2993ds/154Mi)
% 7.45/2.31 % (3505)Instruction limit reached!
% 7.45/2.31 % (3505)------------------------------
% 7.45/2.31 % (3505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.45/2.31 % (3505)Termination reason: Unknown
% 7.45/2.31 % (3505)Termination phase: Saturation
% 7.45/2.31
% 7.45/2.31 % (3505)Memory used [KB]: 6652
% 7.45/2.31 % (3505)Time elapsed: 0.061 s
% 7.45/2.31 % (3505)Instructions burned: 232 (million)
% 7.45/2.31 % (3505)------------------------------
% 7.45/2.31 % (3505)------------------------------
% 7.45/2.32 % (3513)dis+1003_1:64_atotf=0.4:au=on:fsr=off:hflw=5:hfsq=on:hfsqc=20:hfsql=off:plsq=on:plsqc=1:plsqr=32,1:i=79:si=on:rtra=on_0 on DTF2THF_3078 for (2992ds/79Mi)
% 7.45/2.34 % (3513)Instruction limit reached!
% 7.45/2.34 % (3513)------------------------------
% 7.45/2.34 % (3513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.45/2.34 % (3513)Termination reason: Unknown
% 7.45/2.34 % (3513)Termination phase: Property scanning
% 7.45/2.34
% 7.45/2.34 % (3513)Memory used [KB]: 1791
% 7.45/2.34 % (3513)Time elapsed: 0.025 s
% 7.45/2.34 % (3513)Instructions burned: 79 (million)
% 7.45/2.34 % (3513)------------------------------
% 7.45/2.34 % (3513)------------------------------
% 7.45/2.35 % (3512)Instruction limit reached!
% 7.45/2.35 % (3512)------------------------------
% 7.45/2.35 % (3512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.45/2.35 % (3512)Termination reason: Unknown
% 7.45/2.35 % (3512)Termination phase: Saturation
% 7.45/2.35
% 7.45/2.35 % (3512)Memory used [KB]: 2430
% 7.45/2.35 % (3512)Time elapsed: 0.044 s
% 7.45/2.35 % (3512)Instructions burned: 156 (million)
% 7.45/2.35 % (3512)------------------------------
% 7.45/2.35 % (3512)------------------------------
% 7.45/2.35 % (3510)Instruction limit reached!
% 7.45/2.35 % (3510)------------------------------
% 7.45/2.35 % (3510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.45/2.35 % (3510)Termination reason: Unknown
% 7.45/2.35 % (3510)Termination phase: Saturation
% 7.45/2.35
% 7.45/2.35 % (3510)Memory used [KB]: 6652
% 7.45/2.35 % (3510)Time elapsed: 0.054 s
% 7.45/2.35 % (3510)Instructions burned: 182 (million)
% 7.45/2.35 % (3510)------------------------------
% 7.45/2.35 % (3510)------------------------------
% 7.45/2.35 % (3511)Instruction limit reached!
% 7.45/2.35 % (3511)------------------------------
% 7.45/2.35 % (3511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.45/2.35 % (3511)Termination reason: Unknown
% 7.45/2.35 % (3511)Termination phase: Saturation
% 7.45/2.35
% 7.45/2.35 % (3511)Memory used [KB]: 6652
% 7.45/2.35 % (3511)Time elapsed: 0.054 s
% 7.45/2.35 % (3511)Instructions burned: 182 (million)
% 7.45/2.35 % (3511)------------------------------
% 7.45/2.35 % (3511)------------------------------
% 7.45/2.35 % (3514)dis+1010_5:2_anc=all_dependent:awrs=converge:bd=off:bet=on:c=on:cha=on:cnfonf=lazy_not_gen_be_off:fe=axiom:hud=5:kws=precedence:piset=and:prag=on:s2a=on:sac=on:slsq=on:slsqc=1:sp=reverse_arity:ss=axioms:st=1.5:i=216:si=on:rtra=on_0 on DTF2THF_3078 for (2992ds/216Mi)
% 7.45/2.36 % (3515)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_3078 for (2992ds/114Mi)
% 7.45/2.36 % (3516)dis+1002_1:1_acc=on:au=on:bd=off:e2e=on:sgt=8:ss=axioms:i=922:si=on:rtra=on_0 on DTF2THF_3078 for (2992ds/922Mi)
% 7.45/2.36 % (3517)lrs+10_1:128_au=on:bd=off:cnfonf=lazy_not_gen_be_off:nwc=10.0:piset=and:i=182:si=on:rtra=on_0 on DTF2THF_3078 for (2992ds/182Mi)
% 7.45/2.37 % (3508)Instruction limit reached!
% 7.45/2.37 % (3508)------------------------------
% 7.45/2.37 % (3508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.45/2.37 % (3508)Termination reason: Unknown
% 7.45/2.37 % (3508)Termination phase: Saturation
% 7.45/2.37
% 7.45/2.37 % (3508)Memory used [KB]: 7036
% 7.45/2.37 % (3508)Time elapsed: 0.085 s
% 7.45/2.37 % (3508)Instructions burned: 298 (million)
% 7.45/2.37 % (3508)------------------------------
% 7.45/2.37 % (3508)------------------------------
% 7.45/2.38 % (3518)ott+1010_1:1024_au=on:av=off:c=on:cnfonf=conj_eager:piset=equals:s2a=on:sp=occurrence:spb=units:ss=axioms:i=164:si=on:rtra=on_0 on DTF2THF_3078 for (2992ds/164Mi)
% 7.45/2.39 % (3515)Instruction limit reached!
% 7.45/2.39 % (3515)------------------------------
% 7.45/2.39 % (3515)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.45/2.39 % (3515)Termination reason: Unknown
% 7.45/2.39 % (3515)Termination phase: Property scanning
% 7.45/2.39
% 7.45/2.39 % (3515)Memory used [KB]: 1791
% 7.45/2.39 % (3515)Time elapsed: 0.033 s
% 7.45/2.39 % (3515)Instructions burned: 116 (million)
% 7.45/2.39 % (3515)------------------------------
% 7.45/2.39 % (3515)------------------------------
% 7.86/2.40 % (3506)Instruction limit reached!
% 7.86/2.40 % (3506)------------------------------
% 7.86/2.40 % (3506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.40 % (3506)Termination reason: Unknown
% 7.86/2.40 % (3506)Termination phase: Saturation
% 7.86/2.40
% 7.86/2.40 % (3506)Memory used [KB]: 7291
% 7.86/2.40 % (3506)Time elapsed: 0.140 s
% 7.86/2.40 % (3506)Instructions burned: 460 (million)
% 7.86/2.40 % (3506)------------------------------
% 7.86/2.40 % (3506)------------------------------
% 7.86/2.40 % (3519)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=50:si=on:rtra=on_0 on DTF2THF_3078 for (2992ds/50Mi)
% 7.86/2.41 % (3514)Instruction limit reached!
% 7.86/2.41 % (3514)------------------------------
% 7.86/2.41 % (3514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.41 % (3514)Termination reason: Unknown
% 7.86/2.41 % (3514)Termination phase: Saturation
% 7.86/2.41
% 7.86/2.41 % (3514)Memory used [KB]: 6780
% 7.86/2.41 % (3514)Time elapsed: 0.059 s
% 7.86/2.41 % (3514)Instructions burned: 216 (million)
% 7.86/2.41 % (3514)------------------------------
% 7.86/2.41 % (3514)------------------------------
% 7.86/2.41 % (3517)Instruction limit reached!
% 7.86/2.41 % (3517)------------------------------
% 7.86/2.41 % (3517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.41 % (3517)Termination reason: Unknown
% 7.86/2.41 % (3517)Termination phase: Saturation
% 7.86/2.41
% 7.86/2.41 % (3517)Memory used [KB]: 6780
% 7.86/2.41 % (3517)Time elapsed: 0.051 s
% 7.86/2.41 % (3517)Instructions burned: 185 (million)
% 7.86/2.41 % (3517)------------------------------
% 7.86/2.41 % (3517)------------------------------
% 7.86/2.41 % (3519)Instruction limit reached!
% 7.86/2.41 % (3519)------------------------------
% 7.86/2.41 % (3519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.41 % (3519)Termination reason: Unknown
% 7.86/2.41 % (3519)Termination phase: Property scanning
% 7.86/2.41
% 7.86/2.41 % (3519)Memory used [KB]: 1918
% 7.86/2.41 % (3519)Time elapsed: 0.015 s
% 7.86/2.41 % (3519)Instructions burned: 51 (million)
% 7.86/2.41 % (3519)------------------------------
% 7.86/2.41 % (3519)------------------------------
% 7.86/2.42 % (3520)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_3078 for (2992ds/368Mi)
% 7.86/2.42 % (3521)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1021:si=on:rtra=on_0 on DTF2THF_3078 for (2991ds/1021Mi)
% 7.86/2.42 % (3522)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=267:si=on:rtra=on_0 on DTF2THF_3078 for (2991ds/267Mi)
% 7.86/2.42 % (3518)Instruction limit reached!
% 7.86/2.42 % (3518)------------------------------
% 7.86/2.42 % (3518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.42 % (3518)Termination reason: Unknown
% 7.86/2.42 % (3518)Termination phase: Saturation
% 7.86/2.42
% 7.86/2.42 % (3518)Memory used [KB]: 2174
% 7.86/2.42 % (3518)Time elapsed: 0.045 s
% 7.86/2.42 % (3518)Instructions burned: 164 (million)
% 7.86/2.42 % (3518)------------------------------
% 7.86/2.42 % (3518)------------------------------
% 7.86/2.42 % (3523)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=204:si=on:rtra=on_0 on DTF2THF_3078 for (2991ds/204Mi)
% 7.86/2.42 % (3516)First to succeed.
% 7.86/2.43 % (3516)Refutation found. Thanks to Tanya!
% 7.86/2.43 % SZS status Theorem for DTF2THF_3078
% 7.86/2.43 % SZS output start Proof for DTF2THF_3078
% 7.86/2.43 thf(type_def_5, type, ring_ext: ($tType * $tType) > $tType).
% 7.86/2.43 thf(type_def_6, type, monoid_ext: ($tType * $tType) > $tType).
% 7.86/2.43 thf(type_def_7, type, partia1265057091ct_ext: ($tType * $tType) > $tType).
% 7.86/2.43 thf(type_def_8, type, set: ($tType) > $tType).
% 7.86/2.43 thf(type_def_10, type, product_prod: ($tType * $tType) > $tType).
% 7.86/2.43 thf(type_def_11, type, a: $tType).
% 7.86/2.43 thf(type_def_12, type, b: $tType).
% 7.86/2.43 thf(type_def_13, type, int: $tType).
% 7.86/2.43 thf(func_def_0, type, partia1265057091ct_ext: ($tType * $tType) > $tType).
% 7.86/2.43 thf(func_def_1, type, monoid_ext: ($tType * $tType) > $tType).
% 7.86/2.43 thf(func_def_2, type, ring_ext: ($tType * $tType) > $tType).
% 7.86/2.43 thf(func_def_3, type, product_prod: ($tType * $tType) > $tType).
% 7.86/2.43 thf(func_def_4, type, set: ($tType) > $tType).
% 7.86/2.43 thf(func_def_5, type, int: $tType).
% 7.86/2.43 thf(func_def_6, type, b: $tType).
% 7.86/2.43 thf(func_def_7, type, a: $tType).
% 7.86/2.43 thf(func_def_8, type, group_add: !>[X0: $tType]:($o)).
% 7.86/2.43 thf(func_def_9, type, ab_group_add: !>[X0: $tType]:($o)).
% 7.86/2.43 thf(func_def_10, type, semigroup_add: !>[X0: $tType]:($o)).
% 7.86/2.43 thf(func_def_11, type, comm_monoid_add: !>[X0: $tType]:($o)).
% 7.86/2.43 thf(func_def_12, type, ab_semigroup_add: !>[X0: $tType]:($o)).
% 7.86/2.43 thf(func_def_13, type, linordered_semidom: !>[X0: $tType]:($o)).
% 7.86/2.43 thf(func_def_14, type, cancel_semigroup_add: !>[X0: $tType]:($o)).
% 7.86/2.43 thf(func_def_15, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 7.86/2.43 thf(func_def_16, type, cancel1352612707id_add: !>[X0: $tType]:($o)).
% 7.86/2.43 thf(func_def_17, type, cancel146912293up_add: !>[X0: $tType]:($o)).
% 7.86/2.43 thf(func_def_18, type, ordere779506340up_add: !>[X0: $tType]:($o)).
% 7.86/2.43 thf(func_def_19, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 7.86/2.43 thf(func_def_20, type, ordere236663937imp_le: !>[X0: $tType]:($o)).
% 7.86/2.43 thf(func_def_21, type, ordere623563068d_diff: !>[X0: $tType]:($o)).
% 7.86/2.43 thf(func_def_22, type, a_l_coset: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > X0 > set @ X0 > set @ X0)).
% 7.86/2.43 thf(func_def_23, type, additive_subgroup: !>[X0: $tType, X1: $tType]:(set @ X0 > partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > $o)).
% 7.86/2.43 thf(func_def_24, type, set_add: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X0 > set @ X0 > set @ X0)).
% 7.86/2.43 thf(func_def_25, type, partial_carrier: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ X1 > set @ X0)).
% 7.86/2.43 thf(func_def_26, type, mult: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ X1 > X0 > X0 > X0)).
% 7.86/2.43 thf(func_def_27, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 7.86/2.43 thf(func_def_28, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 7.86/2.43 thf(func_def_29, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 7.86/2.43 thf(func_def_30, type, locali1648044335f_frac: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X0 > $o)).
% 7.86/2.43 thf(func_def_31, type, locali7914339f_crng: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X0 > $o)).
% 7.86/2.43 thf(func_def_32, type, locali1402578320of_rng: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X0 > $o)).
% 7.86/2.43 thf(func_def_33, type, locali1441642228monoid: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ X1 > set @ X0 > $o)).
% 7.86/2.43 thf(func_def_34, type, bot_bot: !>[X0: $tType]:(X0)).
% 7.86/2.43 thf(func_def_35, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 7.86/2.43 thf(func_def_36, type, product_fst: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 7.86/2.43 thf(func_def_37, type, product_snd: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 7.86/2.43 thf(func_def_38, type, a_inv: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > X0 > X0)).
% 7.86/2.43 thf(func_def_39, type, a_minus: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > X0 > X0 > X0)).
% 7.86/2.43 thf(func_def_40, type, add_pow: !>[X0: $tType, X1: $tType, X2: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > X2 > X0 > X0)).
% 7.86/2.43 thf(func_def_41, type, add: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > X0 > X0 > X0)).
% 7.86/2.43 thf(func_def_42, type, zero: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > X0)).
% 7.86/2.43 thf(func_def_43, type, semiring: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > $o)).
% 7.86/2.43 thf(func_def_44, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 7.86/2.43 thf(func_def_45, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 7.86/2.43 thf(func_def_46, type, r: partia1265057091ct_ext @ a @ monoid_ext @ a @ ring_ext @ a @ b).
% 7.86/2.43 thf(func_def_47, type, s: set @ a).
% 7.86/2.43 thf(func_def_48, type, r2: a).
% 7.86/2.43 thf(func_def_49, type, r3: a).
% 7.86/2.43 thf(func_def_50, type, s2: a).
% 7.86/2.43 thf(func_def_51, type, s3: a).
% 7.86/2.43 thf(func_def_52, type, t: a).
% 7.86/2.43 thf(func_def_53, type, t2: a).
% 7.86/2.43 thf(func_def_54, type, x: product_prod @ a @ a).
% 7.86/2.43 thf(func_def_55, type, y: product_prod @ a @ a).
% 7.86/2.43 thf(func_def_68, type, sK0: a > a).
% 7.86/2.43 thf(func_def_69, type, sK1: a > a).
% 7.86/2.43 thf(func_def_70, type, sK2: a > a).
% 7.86/2.43 thf(func_def_71, type, sK3: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X1)).
% 7.86/2.43 thf(func_def_72, type, sK4: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X0)).
% 7.86/2.43 thf(func_def_75, type, ph6: !>[X0: $tType]:(X0)).
% 7.86/2.43 thf(f1730,plain,(
% 7.86/2.43 $false),
% 7.86/2.43 inference(avatar_sat_refutation,[status(thm)],[f942,f990,f1024,f1058,f1125,f1126,f1151,f1728,f1729])).
% 7.86/2.43 thf(f1729,plain,(
% 7.86/2.43 ((add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))) != (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (add @ a @ b @ r @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y))))))) | ((add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t @ s3) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ r2) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ s2) @ (product_snd @ a @ a @ x)) @ (mult @ a @ ring_ext @ a @ b @ r @ t @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ r3) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ (product_fst @ a @ a @ y)))))) != (add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y))))))) | ((add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t @ s3) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ r2) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ s2) @ (product_snd @ a @ a @ x)) @ (mult @ a @ ring_ext @ a @ b @ r @ t @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ r3) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ (product_fst @ a @ a @ y)))))) = (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (add @ a @ b @ r @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))))),
% 7.86/2.43 introduced(definition,[],[theory_tautology_sat_conflict])).
% 7.86/2.43 thf(f1728,plain,(
% 7.86/2.43 spl5_29 | ~spl5_44 | ~spl5_49),
% 7.86/2.43 inference(avatar_contradiction_clause,[status(thm)],[f1727])).
% 7.86/2.43 thf(f1727,plain,(
% 7.86/2.43 $false | (spl5_29 | ~spl5_44 | ~spl5_49)),
% 7.86/2.43 inference(subsumption_resolution,[status(thm)],[f1726,f1124])).
% 7.86/2.43 thf(f1124,plain,(
% 7.86/2.43 ((member @ a @ t @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true) | ~spl5_44),
% 7.86/2.43 inference(avatar_component_clause,[status(thm)],[f1122])).
% 7.86/2.43 thf(f1122,definition,(
% 7.86/2.43 spl5_44 <=> ((member @ a @ t @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true)),
% 7.86/2.43 introduced(definition,[new_symbols(naming,[spl5_44])],[avatar_definition])).
% 7.86/2.43 thf(f1726,plain,(
% 7.86/2.43 ((member @ a @ t @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) != $true) | (spl5_29 | ~spl5_49)),
% 7.86/2.43 inference(subsumption_resolution,[status(thm)],[f1724,f1150])).
% 7.86/2.43 thf(f1150,plain,(
% 7.86/2.43 ((member @ a @ t2 @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true) | ~spl5_49),
% 7.86/2.43 inference(avatar_component_clause,[status(thm)],[f1148])).
% 7.86/2.43 thf(f1148,definition,(
% 7.86/2.43 spl5_49 <=> ((member @ a @ t2 @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true)),
% 7.86/2.43 introduced(definition,[new_symbols(naming,[spl5_49])],[avatar_definition])).
% 7.86/2.43 thf(f1724,plain,(
% 7.86/2.43 ((member @ a @ t2 @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) != $true) | ((member @ a @ t @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) != $true) | spl5_29),
% 7.86/2.43 inference(trivial_inequality_removal,[status(thm)],[f1716])).
% 7.86/2.43 thf(f1716,plain,(
% 7.86/2.43 ($true != $true) | ((member @ a @ t2 @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) != $true) | ((member @ a @ t @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) != $true) | spl5_29),
% 7.86/2.43 inference(superposition,[status(thm)],[f1049,f833])).
% 7.86/2.43 thf(f833,plain,(
% 7.86/2.43 ( ! [X0 : a,X1 : a] : (((member @ a @ (mult @ a @ ring_ext @ a @ b @ r @ X0 @ X1) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true) | ((member @ a @ X0 @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) != $true) | ($true != (member @ a @ X1 @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)))) )),
% 7.86/2.43 inference(cnf_transformation,[status(thm)],[f751])).
% 7.86/2.43 thf(f751,plain,(
% 7.86/2.43 ! [X0 : a,X1 : a] : (($true != (member @ a @ X1 @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r))) | ((member @ a @ (mult @ a @ ring_ext @ a @ b @ r @ X0 @ X1) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true) | ((member @ a @ X0 @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) != $true))),
% 7.86/2.43 inference(flattening,[status(thm)],[f750])).
% 7.86/2.43 thf(f750,plain,(
% 7.86/2.43 ! [X0 : a,X1 : a] : ((((member @ a @ (mult @ a @ ring_ext @ a @ b @ r @ X0 @ X1) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true) | ($true != (member @ a @ X1 @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)))) | ((member @ a @ X0 @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) != $true))),
% 7.86/2.43 inference(ennf_transformation,[status(thm)],[f533])).
% 7.86/2.43 thf(f533,plain,(
% 7.86/2.43 ! [X0 : a,X1 : a] : (((member @ a @ X0 @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true) => (($true = (member @ a @ X1 @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r))) => ((member @ a @ (mult @ a @ ring_ext @ a @ b @ r @ X0 @ X1) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true)))),
% 7.86/2.43 inference(fool_elimination,[status(thm)],[f532])).
% 7.86/2.43 thf(f532,plain,(
% 7.86/2.43 ! [X0 : a,X1 : a] : ((member @ a @ X0 @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) => ((member @ a @ X1 @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) => (member @ a @ (mult @ a @ ring_ext @ a @ b @ r @ X0 @ X1) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r))))),
% 7.86/2.43 inference(rectify,[status(thm)],[f63])).
% 7.86/2.43 thf(f63,axiom,(
% 7.86/2.43 ! [X8 : a,X9 : a] : ((member @ a @ X8 @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) => ((member @ a @ X9 @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) => (member @ a @ (mult @ a @ ring_ext @ a @ b @ r @ X8 @ X9) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r))))),
% 7.86/2.43 file('/export/starexec/sandbox/tmp/tmp.TcuWUcH1aD/DTF2THF_3078.p',fact_62_semiring__simprules_I3_J)).
% 7.86/2.43 thf(f1049,plain,(
% 7.86/2.43 ((member @ a @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) != $true) | spl5_29),
% 7.86/2.43 inference(avatar_component_clause,[status(thm)],[f1047])).
% 7.86/2.43 thf(f1047,definition,(
% 7.86/2.43 spl5_29 <=> ((member @ a @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true)),
% 7.86/2.43 introduced(definition,[new_symbols(naming,[spl5_29])],[avatar_definition])).
% 7.86/2.43 thf(f1151,plain,(
% 7.86/2.43 spl5_49),
% 7.86/2.43 inference(avatar_split_clause,[status(thm)],[f861,f1148])).
% 7.86/2.43 thf(f861,plain,(
% 7.86/2.43 ((member @ a @ t2 @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true)),
% 7.86/2.43 inference(cnf_transformation,[status(thm)],[f575])).
% 7.86/2.43 thf(f575,plain,(
% 7.86/2.43 ((member @ a @ t2 @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true)),
% 7.86/2.43 inference(fool_elimination,[status(thm)],[f574])).
% 7.86/2.43 thf(f574,plain,(
% 7.86/2.43 (member @ a @ t2 @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r))),
% 7.86/2.43 inference(rectify,[status(thm)],[f2])).
% 7.86/2.43 thf(f2,axiom,(
% 7.86/2.43 (member @ a @ t2 @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r))),
% 7.86/2.43 file('/export/starexec/sandbox/tmp/tmp.TcuWUcH1aD/DTF2THF_3078.p',fact_1_f17)).
% 7.86/2.43 thf(f1126,plain,(
% 7.86/2.43 spl5_31),
% 7.86/2.43 inference(avatar_split_clause,[status(thm)],[f825,f1055])).
% 7.86/2.43 thf(f1055,definition,(
% 7.86/2.43 spl5_31 <=> ((member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true)),
% 7.86/2.43 introduced(definition,[new_symbols(naming,[spl5_31])],[avatar_definition])).
% 7.86/2.43 thf(f825,plain,(
% 7.86/2.43 ((member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true)),
% 7.86/2.43 inference(cnf_transformation,[status(thm)],[f343])).
% 7.86/2.43 thf(f343,plain,(
% 7.86/2.43 ((member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true)),
% 7.86/2.43 inference(fool_elimination,[status(thm)],[f342])).
% 7.86/2.43 thf(f342,plain,(
% 7.86/2.43 (member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r))),
% 7.86/2.43 inference(rectify,[status(thm)],[f13])).
% 7.86/2.43 thf(f13,axiom,(
% 7.86/2.43 (member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r))),
% 7.86/2.43 file('/export/starexec/sandbox/tmp/tmp.TcuWUcH1aD/DTF2THF_3078.p',fact_12_f41)).
% 7.86/2.43 thf(f1125,plain,(
% 7.86/2.43 spl5_44),
% 7.86/2.43 inference(avatar_split_clause,[status(thm)],[f843,f1122])).
% 7.86/2.43 thf(f843,plain,(
% 7.86/2.43 ((member @ a @ t @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true)),
% 7.86/2.43 inference(cnf_transformation,[status(thm)],[f283])).
% 7.86/2.43 thf(f283,plain,(
% 7.86/2.43 ((member @ a @ t @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true)),
% 7.86/2.43 inference(fool_elimination,[status(thm)],[f282])).
% 7.86/2.43 thf(f282,plain,(
% 7.86/2.43 (member @ a @ t @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r))),
% 7.86/2.43 inference(rectify,[status(thm)],[f1])).
% 7.86/2.43 thf(f1,axiom,(
% 7.86/2.43 (member @ a @ t @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r))),
% 7.86/2.43 file('/export/starexec/sandbox/tmp/tmp.TcuWUcH1aD/DTF2THF_3078.p',fact_0_f18)).
% 7.86/2.43 thf(f1058,plain,(
% 7.86/2.43 ~spl5_29 | spl5_30 | ~spl5_24 | ~spl5_31),
% 7.86/2.43 inference(avatar_split_clause,[status(thm)],[f867,f1055,f1021,f1051,f1047])).
% 7.86/2.43 thf(f1051,definition,(
% 7.86/2.43 spl5_30 <=> ((add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))) = (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (add @ a @ b @ r @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))))),
% 7.86/2.43 introduced(definition,[new_symbols(naming,[spl5_30])],[avatar_definition])).
% 7.86/2.43 thf(f1021,definition,(
% 7.86/2.43 spl5_24 <=> ((member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true)),
% 7.86/2.43 introduced(definition,[new_symbols(naming,[spl5_24])],[avatar_definition])).
% 7.86/2.43 thf(f867,plain,(
% 7.86/2.43 ((member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) != $true) | ((member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) != $true) | ((add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))) = (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (add @ a @ b @ r @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y))))))) | ((member @ a @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) != $true)),
% 7.86/2.43 inference(cnf_transformation,[status(thm)],[f789])).
% 7.86/2.43 thf(f789,plain,(
% 7.86/2.43 ((member @ a @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) != $true) | ((member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) != $true) | ((add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))) = (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (add @ a @ b @ r @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y))))))) | ((member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) != $true)),
% 7.86/2.43 inference(flattening,[status(thm)],[f788])).
% 7.86/2.43 thf(f788,plain,(
% 7.86/2.43 ((((add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))) = (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (add @ a @ b @ r @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y))))))) | ((member @ a @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) != $true)) | ((member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) != $true)) | ((member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) != $true)),
% 7.86/2.43 inference(ennf_transformation,[status(thm)],[f467])).
% 7.86/2.43 thf(f467,plain,(
% 7.86/2.43 ((member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true) => (((member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true) => (((member @ a @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true) => ((add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))) = (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (add @ a @ b @ r @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))))))),
% 7.86/2.43 inference(fool_elimination,[status(thm)],[f466])).
% 7.86/2.43 thf(f466,plain,(
% 7.86/2.43 (member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) => ((member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) => ((member @ a @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) => ((add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))) = (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (add @ a @ b @ r @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))))))),
% 7.86/2.43 inference(rectify,[status(thm)],[f22])).
% 7.86/2.43 thf(f22,axiom,(
% 7.86/2.43 (member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) => ((member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) => ((member @ a @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) => ((add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))) = (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (add @ a @ b @ r @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))))))),
% 7.86/2.43 file('/export/starexec/sandbox/tmp/tmp.TcuWUcH1aD/DTF2THF_3078.p',fact_21__092_060open_062_092_060lbrakk_062snd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_A_092_060in_062_Acarrier_AR_059_Asnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_A_092_060in_062_Acarrier_AR_059_At_A_092_060otimes_062_At_H_A_092_060in_062_Acarrier_AR_092_060rbrakk_062_A_092_060Longrightarrow_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_A_092_060oplus_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_092_060close_062)).
% 7.86/2.43 thf(f1024,plain,(
% 7.86/2.43 spl5_24),
% 7.86/2.43 inference(avatar_split_clause,[status(thm)],[f869,f1021])).
% 7.86/2.43 thf(f869,plain,(
% 7.86/2.43 ((member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true)),
% 7.86/2.43 inference(cnf_transformation,[status(thm)],[f591])).
% 7.86/2.43 thf(f591,plain,(
% 7.86/2.43 ((member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r)) = $true)),
% 7.86/2.43 inference(fool_elimination,[status(thm)],[f590])).
% 7.86/2.43 thf(f590,plain,(
% 7.86/2.43 (member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r))),
% 7.86/2.43 inference(rectify,[status(thm)],[f14])).
% 7.86/2.43 thf(f14,axiom,(
% 7.86/2.43 (member @ a @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))) @ (partial_carrier @ a @ monoid_ext @ a @ ring_ext @ a @ b @ r))),
% 7.86/2.43 file('/export/starexec/sandbox/tmp/tmp.TcuWUcH1aD/DTF2THF_3078.p',fact_13_f42)).
% 7.86/2.43 thf(f990,plain,(
% 7.86/2.43 spl5_17),
% 7.86/2.43 inference(avatar_split_clause,[status(thm)],[f860,f987])).
% 7.86/2.43 thf(f987,definition,(
% 7.86/2.43 spl5_17 <=> ((add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t @ s3) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ r2) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ s2) @ (product_snd @ a @ a @ x)) @ (mult @ a @ ring_ext @ a @ b @ r @ t @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ r3) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ (product_fst @ a @ a @ y)))))) = (add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))))),
% 7.86/2.43 introduced(definition,[new_symbols(naming,[spl5_17])],[avatar_definition])).
% 7.86/2.43 thf(f860,plain,(
% 7.86/2.43 ((add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t @ s3) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ r2) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ s2) @ (product_snd @ a @ a @ x)) @ (mult @ a @ ring_ext @ a @ b @ r @ t @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ r3) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ (product_fst @ a @ a @ y)))))) = (add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))))),
% 7.86/2.43 inference(cnf_transformation,[status(thm)],[f20])).
% 7.86/2.43 thf(f20,axiom,(
% 7.86/2.43 ((add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t @ s3) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ r2) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ s2) @ (product_snd @ a @ a @ x)) @ (mult @ a @ ring_ext @ a @ b @ r @ t @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ r3) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ (product_fst @ a @ a @ y)))))) = (add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))))),
% 7.86/2.43 file('/export/starexec/sandbox/tmp/tmp.TcuWUcH1aD/DTF2THF_3078.p',fact_19_f40)).
% 7.86/2.43 thf(f942,plain,(
% 7.86/2.43 ~spl5_7),
% 7.86/2.43 inference(avatar_split_clause,[status(thm)],[f816,f939])).
% 7.86/2.43 thf(f939,definition,(
% 7.86/2.43 spl5_7 <=> ((add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t @ s3) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ r2) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ s2) @ (product_snd @ a @ a @ x)) @ (mult @ a @ ring_ext @ a @ b @ r @ t @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ r3) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ (product_fst @ a @ a @ y)))))) = (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (add @ a @ b @ r @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))))),
% 7.86/2.43 introduced(definition,[new_symbols(naming,[spl5_7])],[avatar_definition])).
% 7.86/2.43 thf(f816,plain,(
% 7.86/2.43 ((add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t @ s3) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ r2) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ s2) @ (product_snd @ a @ a @ x)) @ (mult @ a @ ring_ext @ a @ b @ r @ t @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ r3) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ (product_fst @ a @ a @ y)))))) != (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (add @ a @ b @ r @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))))),
% 7.86/2.43 inference(cnf_transformation,[status(thm)],[f738])).
% 7.86/2.43 thf(f738,plain,(
% 7.86/2.43 ((add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t @ s3) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ r2) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ s2) @ (product_snd @ a @ a @ x)) @ (mult @ a @ ring_ext @ a @ b @ r @ t @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ r3) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ (product_fst @ a @ a @ y)))))) != (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (add @ a @ b @ r @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))))),
% 7.86/2.43 inference(flattening,[status(thm)],[f270])).
% 7.86/2.43 thf(f270,negated_conjecture,(
% 7.86/2.43 ~((add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t @ s3) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ r2) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ s2) @ (product_snd @ a @ a @ x)) @ (mult @ a @ ring_ext @ a @ b @ r @ t @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ r3) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ (product_fst @ a @ a @ y)))))) = (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (add @ a @ b @ r @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))))),
% 7.86/2.43 inference(negated_conjecture,[status(cth)],[f269])).
% 7.86/2.43 thf(f269,conjecture,(
% 7.86/2.43 ((add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t @ s3) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ r2) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ (product_fst @ a @ a @ x))))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ s2) @ (product_snd @ a @ a @ x)) @ (mult @ a @ ring_ext @ a @ b @ r @ t @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ r3) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ (product_fst @ a @ a @ y)))))) = (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ t2 @ t) @ (add @ a @ b @ r @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ r2)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ y) @ (product_fst @ a @ a @ x)))) @ (a_minus @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_snd @ a @ a @ y)) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ r3)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ (product_snd @ a @ a @ x) @ (product_fst @ a @ a @ y)))))))),
% 7.86/2.43 file('/export/starexec/sandbox/tmp/tmp.TcuWUcH1aD/DTF2THF_3078.p',conj_0)).
% 7.86/2.43 % SZS output end Proof for DTF2THF_3078
% 7.86/2.43 % (3516)------------------------------
% 7.86/2.43 % (3516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.43 % (3516)Termination reason: Refutation
% 7.86/2.43
% 7.86/2.43 % (3516)Memory used [KB]: 7164
% 7.86/2.43 % (3516)Time elapsed: 0.070 s
% 7.86/2.43 % (3516)Instructions burned: 221 (million)
% 7.86/2.43 % (3516)------------------------------
% 7.86/2.43 % (3516)------------------------------
% 7.86/2.43 % (3201)Success in time 0.82 s
%------------------------------------------------------------------------------