%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP027^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n017.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 08:54:58 AM UTC 2026
% Result : Theorem 12.99s 3.04s
% Output : Refutation 12.99s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : ITP027^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.07 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.06/0.27 % Computer : n017.cluster.edu
% 0.06/0.27 % Model : x86_64 x86_64
% 0.06/0.27 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.06/0.27 % Memory : 8042.1875MB
% 0.06/0.27 % OS : Linux 3.10.0-693.el7.x86_64
% 0.06/0.27 % CPULimit : 300
% 0.06/0.27 % WCLimit : 300
% 0.06/0.27 % DateTime : Tue Feb 24 08:22:26 EST 2026
% 0.06/0.27 % CPUTime :
% 0.06/0.27 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.36 ---- Original DTF file ---
% 0.15/0.36 thf(spec,logic,$$dhol).
% 0.15/0.36 %------------------------------------------------------------------------------
% 0.15/0.36 % File : ITP027^2 : TPTP v9.2.1. Released v7.5.0.
% 0.15/0.36 % Domain : Interactive Theorem Proving
% 0.15/0.36 % Problem : Sledgehammer Algebra8 problem prob_2341__6471840_1
% 0.15/0.36 % Version : Especial.
% 0.15/0.36 % English :
% 0.15/0.36
% 0.15/0.36 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.15/0.36 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.15/0.36 % Source : [Des21]
% 0.15/0.36 % Names : Algebra8/prob_2341__6471840_1 [Des21]
% 0.15/0.36
% 0.15/0.36 % Status : Theorem
% 0.15/0.36 % Rating : 0.67 v9.1.0, 0.33 v9.0.0, 0.67 v8.2.0, 0.33 v8.1.0, 0.50 v7.5.0
% 0.15/0.36 % Syntax : Number of formulae : 332 ( 27 unt; 61 typ; 0 def)
% 0.15/0.36 % Number of atoms : 1000 ( 202 equ; 0 cnn)
% 0.15/0.36 % Maximal formula atoms : 14 ( 3 avg)
% 0.15/0.36 % Number of connectives : 10303 ( 28 ~; 3 |; 27 &;9574 @)
% 0.15/0.36 % ( 0 <=>; 671 =>; 0 <=; 0 <~>)
% 0.15/0.36 % Maximal formula depth : 27 ( 13 avg)
% 0.15/0.36 % Number of types : 8 ( 7 usr)
% 0.15/0.36 % Number of type conns : 400 ( 400 >; 0 *; 0 +; 0 <<)
% 0.15/0.36 % Number of symbols : 55 ( 54 usr; 2 con; 0-11 aty)
% 0.15/0.36 % Number of variables : 1403 ( 59 ^;1232 !; 13 ?;1403 :)
% 0.15/0.36 % ( 99 !>; 0 ?*; 0 @-; 0 @+)
% 0.15/0.36 % SPC : TH1_THM_EQU_NAR
% 0.15/0.36
% 0.15/0.36 % Comments : This file was generated by Sledgehammer 2021-02-23 16:32:56.184
% 0.15/0.36 %------------------------------------------------------------------------------
% 0.15/0.36 % Could-be-implicit typings (12)
% 0.15/0.36 thf(ty_t_Algebra1_Ocarrier_Ocarrier__ext,type,
% 0.15/0.36 carrier_ext: $tType > $tType > $tType ).
% 0.15/0.36
% 0.15/0.36 thf(ty_t_Algebra7_OModule_OModule__ext,type,
% 0.15/0.36 module_ext: $tType > $tType > $tType > $tType ).
% 0.15/0.36
% 0.15/0.36 thf(ty_t_Algebra4_OaGroup_OaGroup__ext,type,
% 0.15/0.36 aGroup_ext: $tType > $tType > $tType ).
% 0.15/0.36
% 0.15/0.36 thf(ty_t_Algebra4_ORing_ORing__ext,type,
% 0.15/0.36 ring_ext: $tType > $tType > $tType ).
% 0.15/0.36
% 0.15/0.36 thf(ty_t_Product__Type_Ounit,type,
% 0.15/0.36 product_unit: $tType ).
% 0.15/0.36
% 0.15/0.36 thf(ty_t_Set_Oset,type,
% 0.15/0.36 set: $tType > $tType ).
% 0.15/0.36
% 0.15/0.36 thf(ty_t_Nat_Onat,type,
% 0.15/0.36 nat: $tType ).
% 0.15/0.36
% 0.15/0.36 thf(ty_tf_e,type,
% 0.15/0.36 e: $tType ).
% 0.15/0.36
% 0.15/0.36 thf(ty_tf_d,type,
% 0.15/0.36 d: $tType ).
% 0.15/0.36
% 0.15/0.36 thf(ty_tf_c,type,
% 0.15/0.36 c: $tType ).
% 0.15/0.36
% 0.15/0.36 thf(ty_tf_b,type,
% 0.15/0.36 b: $tType ).
% 0.15/0.36
% 0.15/0.36 thf(ty_tf_a,type,
% 0.15/0.36 a: $tType ).
% 0.15/0.36
% 0.15/0.36 % Explicit typings (49)
% 0.15/0.36 thf(sy_cl_HOL_Otype,type,
% 0.15/0.36 type:
% 0.15/0.36 !>[A: $tType] : $o ).
% 0.15/0.36
% 0.15/0.36 thf(sy_cl_Groups_Oplus,type,
% 0.15/0.36 plus:
% 0.15/0.36 !>[A: $tType] : $o ).
% 0.15/0.36
% 0.15/0.36 thf(sy_cl_Orderings_Oord,type,
% 0.15/0.36 ord:
% 0.15/0.36 !>[A: $tType] : $o ).
% 0.15/0.36
% 0.15/0.36 thf(sy_cl_Groups_Ogroup__add,type,
% 0.15/0.36 group_add:
% 0.15/0.36 !>[A: $tType] : $o ).
% 0.15/0.36
% 0.15/0.36 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.15/0.36 semigroup_add:
% 0.15/0.36 !>[A: $tType] : $o ).
% 0.15/0.36
% 0.15/0.36 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.15/0.36 comm_monoid_add:
% 0.15/0.36 !>[A: $tType] : $o ).
% 0.15/0.36
% 0.15/0.36 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.15/0.36 ab_semigroup_add:
% 0.15/0.36 !>[A: $tType] : $o ).
% 0.15/0.36
% 0.15/0.36 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.15/0.36 cancel_semigroup_add:
% 0.15/0.36 !>[A: $tType] : $o ).
% 0.15/0.36
% 0.15/0.36 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.15/0.36 ordere779506340up_add:
% 0.15/0.36 !>[A: $tType] : $o ).
% 0.15/0.36
% 0.15/0.36 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.15/0.36 ordere236663937imp_le:
% 0.15/0.36 !>[A: $tType] : $o ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra1_Ocarrier_Ocarrier,type,
% 0.15/0.36 carrier:
% 0.15/0.36 !>[A: $tType,Z: $tType] : ( ( carrier_ext @ A @ Z ) > ( set @ A ) ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_OJ__rad,type,
% 0.15/0.36 j_rad:
% 0.15/0.36 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_ORing,type,
% 0.15/0.36 ring:
% 0.15/0.36 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_ORing_Otp,type,
% 0.15/0.36 tp:
% 0.15/0.36 !>[A: $tType,Z: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ Z ) ) ) > A > A > A ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_ORing_Oun,type,
% 0.15/0.36 un:
% 0.15/0.36 !>[A: $tType,Z: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ Z ) ) ) > A ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_OSr,type,
% 0.15/0.36 sr:
% 0.15/0.36 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_OSubring,type,
% 0.15/0.36 subring:
% 0.15/0.36 !>[A: $tType,M: $tType,M1: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ M1 ) ) ) > $o ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_OUnit,type,
% 0.15/0.36 unit:
% 0.15/0.36 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > A > $o ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_OaGroup,type,
% 0.15/0.36 aGroup:
% 0.15/0.36 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ B ) ) > $o ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_OaGroup_Omop,type,
% 0.15/0.36 mop:
% 0.15/0.36 !>[A: $tType,Z: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ Z ) ) > A > A ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_OaGroup_Opop,type,
% 0.15/0.36 pop:
% 0.15/0.36 !>[A: $tType,Z: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ Z ) ) > A > A > A ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_Oaset__sum,type,
% 0.15/0.36 aset_sum:
% 0.15/0.36 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ B ) ) > ( set @ A ) > ( set @ A ) > ( set @ A ) ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_Ocarr__dsumag,type,
% 0.15/0.36 carr_dsumag:
% 0.15/0.36 !>[I: $tType,A: $tType,More: $tType] : ( ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ More ) ) ) > ( set @ ( I > A ) ) ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_Ocarr__prodag,type,
% 0.15/0.36 carr_prodag:
% 0.15/0.36 !>[I: $tType,A: $tType,More: $tType] : ( ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ More ) ) ) > ( set @ ( I > A ) ) ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_Olocal__ring,type,
% 0.15/0.36 local_ring:
% 0.15/0.36 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_Omaximal__ideal,type,
% 0.15/0.36 maximal_ideal:
% 0.15/0.36 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_Omul__closed__set,type,
% 0.15/0.36 mul_closed_set:
% 0.15/0.36 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_Onscal,type,
% 0.15/0.36 nscal:
% 0.15/0.36 !>[A: $tType,More: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ More ) ) ) > A > nat > A ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_Oprimary__ideal,type,
% 0.15/0.36 primary_ideal:
% 0.15/0.36 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_Oprod__mOp,type,
% 0.15/0.36 prod_mOp:
% 0.15/0.36 !>[I: $tType,A: $tType,More: $tType] : ( ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ More ) ) ) > ( I > A ) > I > A ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_Oprod__pOp,type,
% 0.15/0.36 prod_pOp:
% 0.15/0.36 !>[I: $tType,A: $tType,More: $tType] : ( ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ More ) ) ) > ( I > A ) > ( I > A ) > I > A ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_Oprod__zero,type,
% 0.15/0.36 prod_zero:
% 0.15/0.36 !>[I: $tType,A: $tType,More: $tType] : ( ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ More ) ) ) > I > A ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_OrInvim,type,
% 0.15/0.36 rInvim:
% 0.15/0.36 !>[A: $tType,M: $tType,B: $tType,M1: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ M ) ) ) > ( carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ M1 ) ) ) > ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_Osr,type,
% 0.15/0.36 sr2:
% 0.15/0.36 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra4_Ozeroring,type,
% 0.15/0.36 zeroring:
% 0.15/0.36 !>[A: $tType,More: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ More ) ) ) > $o ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra5_Osum__mult,type,
% 0.15/0.36 sum_mult:
% 0.15/0.36 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > ( set @ A ) > ( set @ A ) ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra7_OModule,type,
% 0.15/0.36 module:
% 0.15/0.36 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) ) > ( carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) ) > $o ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra7_OModule_Osprod,type,
% 0.15/0.36 sprod:
% 0.15/0.36 !>[A: $tType,B: $tType,Z: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ Z ) ) ) > B > A > A ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra7_OmHom,type,
% 0.15/0.36 mHom:
% 0.15/0.36 !>[B: $tType,M: $tType,A: $tType,M1: $tType,C: $tType,M2: $tType] : ( ( carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ M1 ) ) ) > ( carrier_ext @ C @ ( aGroup_ext @ C @ ( module_ext @ C @ B @ M2 ) ) ) > ( set @ ( A > C ) ) ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra8__Mirabelle__lwvimexpoc_OprodM,type,
% 0.15/0.36 algebr1000837334_prodM:
% 0.15/0.36 !>[R: $tType,M: $tType,I: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) ) > ( carrier_ext @ ( I > A ) @ ( aGroup_ext @ ( I > A ) @ ( module_ext @ ( I > A ) @ R @ product_unit ) ) ) ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Algebra8__Mirabelle__lwvimexpoc_OprodM__sprod,type,
% 0.15/0.36 algebr142713364_sprod:
% 0.15/0.36 !>[R: $tType,M: $tType,I: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) ) > R > ( I > A ) > I > A ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_FuncSet_OPi,type,
% 0.15/0.36 pi:
% 0.15/0.36 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( A > B ) ) ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.15/0.36 plus_plus:
% 0.15/0.36 !>[A: $tType] : ( A > A > A ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.15/0.36 ord_less_eq:
% 0.15/0.36 !>[A: $tType] : ( A > A > $o ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_Set_OCollect,type,
% 0.15/0.36 collect:
% 0.15/0.36 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_c_member,type,
% 0.15/0.36 member:
% 0.15/0.36 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_v_I,type,
% 0.15/0.36 i: set @ c ).
% 0.15/0.36
% 0.15/0.36 thf(sy_v_M,type,
% 0.15/0.36 m: c > ( carrier_ext @ d @ ( aGroup_ext @ d @ ( module_ext @ d @ a @ e ) ) ) ).
% 0.15/0.36
% 0.15/0.36 thf(sy_v_R,type,
% 0.15/0.36 r: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) ).
% 0.15/0.36
% 0.15/0.36 % Relevant facts (256)
% 0.15/0.36 thf(fact_0_Ring__axioms,axiom,
% 0.15/0.36 ring @ a @ b @ r ).
% 0.15/0.36
% 0.15/0.36 % Ring_axioms
% 0.15/0.36 thf(fact_1_bivar__fun,axiom,
% 0.15/0.36 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,A2: set @ A,B2: set @ B,C2: set @ C,A3: A] :
% 0.15/0.36 ( ( member @ ( A > B > C ) @ F
% 0.15/0.36 @ ( pi @ A @ ( B > C ) @ A2
% 0.15/0.36 @ ^ [Uu: A] :
% 0.15/0.36 ( pi @ B @ C @ B2
% 0.15/0.36 @ ^ [Uv: B] : C2 ) ) )
% 0.15/0.36 => ( ( member @ A @ A3 @ A2 )
% 0.15/0.36 => ( member @ ( B > C ) @ ( F @ A3 )
% 0.15/0.36 @ ( pi @ B @ C @ B2
% 0.15/0.36 @ ^ [Uu: B] : C2 ) ) ) ) ).
% 0.15/0.36
% 0.15/0.36 % bivar_fun
% 0.15/0.36 thf(fact_2_bivar__fun__mem,axiom,
% 0.15/0.36 ! [B: $tType,A: $tType,C: $tType,F: A > B > C,A2: set @ A,B2: set @ B,C2: set @ C,A3: A,B3: B] :
% 0.15/0.36 ( ( member @ ( A > B > C ) @ F
% 0.15/0.36 @ ( pi @ A @ ( B > C ) @ A2
% 0.15/0.36 @ ^ [Uu: A] :
% 0.15/0.36 ( pi @ B @ C @ B2
% 0.15/0.36 @ ^ [Uv: B] : C2 ) ) )
% 0.15/0.36 => ( ( member @ A @ A3 @ A2 )
% 0.15/0.36 => ( ( member @ B @ B3 @ B2 )
% 0.15/0.36 => ( member @ C @ ( F @ A3 @ B3 ) @ C2 ) ) ) ) ).
% 0.15/0.36
% 0.15/0.36 % bivar_fun_mem
% 0.15/0.36 thf(fact_3_Pi__I,axiom,
% 0.15/0.36 ! [B: $tType,A: $tType,A2: set @ A,F: A > B,B2: A > ( set @ B )] :
% 0.15/0.36 ( ! [X: A] :
% 0.15/0.36 ( ( member @ A @ X @ A2 )
% 0.15/0.36 => ( member @ B @ ( F @ X ) @ ( B2 @ X ) ) )
% 0.15/0.36 => ( member @ ( A > B ) @ F @ ( pi @ A @ B @ A2 @ B2 ) ) ) ).
% 0.15/0.36
% 0.15/0.36 % Pi_I
% 0.15/0.36 thf(fact_4_prodM__sprod__mem,axiom,
% 0.15/0.36 ! [E: $tType,D: $tType,C: $tType,I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ a @ E ) ) ),A3: a,M4: C > D] :
% 0.15/0.36 ( ! [X: C] :
% 0.15/0.36 ( ( member @ C @ X @ I2 )
% 0.15/0.36 => ( module @ D @ a @ E @ b @ ( M3 @ X ) @ r ) )
% 0.15/0.36 => ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.36 => ( ( member @ ( C > D ) @ M4 @ ( carr_prodag @ C @ D @ ( module_ext @ D @ a @ E ) @ I2 @ M3 ) )
% 0.15/0.36 => ( member @ ( C > D ) @ ( algebr142713364_sprod @ a @ b @ C @ D @ E @ r @ I2 @ M3 @ A3 @ M4 ) @ ( carr_prodag @ C @ D @ ( module_ext @ D @ a @ E ) @ I2 @ M3 ) ) ) ) ) ).
% 0.15/0.36
% 0.15/0.36 % prodM_sprod_mem
% 0.15/0.36 thf(fact_5_prodM__carr,axiom,
% 0.15/0.36 ! [E: $tType,D: $tType,C: $tType,I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ a @ E ) ) )] :
% 0.15/0.36 ( ! [X: C] :
% 0.15/0.36 ( ( member @ C @ X @ I2 )
% 0.15/0.36 => ( module @ D @ a @ E @ b @ ( M3 @ X ) @ r ) )
% 0.15/0.36 => ( ( carrier @ ( C > D ) @ ( aGroup_ext @ ( C > D ) @ ( module_ext @ ( C > D ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ C @ D @ E @ r @ I2 @ M3 ) )
% 0.15/0.36 = ( carr_prodag @ C @ D @ ( module_ext @ D @ a @ E ) @ I2 @ M3 ) ) ) ).
% 0.15/0.36
% 0.15/0.36 % prodM_carr
% 0.15/0.36 thf(fact_6_eq__fun,axiom,
% 0.15/0.36 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ B,G: A > B] :
% 0.15/0.36 ( ( member @ ( A > B ) @ F
% 0.15/0.36 @ ( pi @ A @ B @ A2
% 0.15/0.36 @ ^ [Uu: A] : B2 ) )
% 0.15/0.36 => ( ( F = G )
% 0.15/0.36 => ( member @ ( A > B ) @ G
% 0.15/0.36 @ ( pi @ A @ B @ A2
% 0.15/0.36 @ ^ [Uu: A] : B2 ) ) ) ) ).
% 0.15/0.36
% 0.15/0.36 % eq_fun
% 0.15/0.36 thf(fact_7_funcTr,axiom,
% 0.15/0.36 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ B,G: A > B,A3: A] :
% 0.15/0.36 ( ( member @ ( A > B ) @ F
% 0.15/0.36 @ ( pi @ A @ B @ A2
% 0.15/0.36 @ ^ [Uu: A] : B2 ) )
% 0.15/0.36 => ( ( member @ ( A > B ) @ G
% 0.15/0.36 @ ( pi @ A @ B @ A2
% 0.15/0.36 @ ^ [Uu: A] : B2 ) )
% 0.15/0.36 => ( ( F = G )
% 0.15/0.36 => ( ( member @ A @ A3 @ A2 )
% 0.15/0.36 => ( ( F @ A3 )
% 0.15/0.36 = ( G @ A3 ) ) ) ) ) ) ).
% 0.15/0.36
% 0.15/0.36 % funcTr
% 0.15/0.36 thf(fact_8_funcsetI,axiom,
% 0.15/0.36 ! [B: $tType,A: $tType,A2: set @ A,F: A > B,B2: set @ B] :
% 0.15/0.36 ( ! [X: A] :
% 0.15/0.36 ( ( member @ A @ X @ A2 )
% 0.15/0.36 => ( member @ B @ ( F @ X ) @ B2 ) )
% 0.15/0.36 => ( member @ ( A > B ) @ F
% 0.15/0.36 @ ( pi @ A @ B @ A2
% 0.15/0.36 @ ^ [Uu: A] : B2 ) ) ) ).
% 0.15/0.36
% 0.15/0.36 % funcsetI
% 0.15/0.36 thf(fact_9_eq__funcs,axiom,
% 0.15/0.36 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ B,G: A > B,X2: A] :
% 0.15/0.36 ( ( member @ ( A > B ) @ F
% 0.15/0.36 @ ( pi @ A @ B @ A2
% 0.15/0.36 @ ^ [Uu: A] : B2 ) )
% 0.15/0.36 => ( ( member @ ( A > B ) @ G
% 0.15/0.36 @ ( pi @ A @ B @ A2
% 0.15/0.36 @ ^ [Uu: A] : B2 ) )
% 0.15/0.36 => ( ( F = G )
% 0.15/0.36 => ( ( member @ A @ X2 @ A2 )
% 0.15/0.36 => ( ( F @ X2 )
% 0.15/0.36 = ( G @ X2 ) ) ) ) ) ) ).
% 0.15/0.36
% 0.15/0.36 % eq_funcs
% 0.15/0.36 thf(fact_10_funcset__id,axiom,
% 0.15/0.36 ! [A: $tType,A2: set @ A] :
% 0.15/0.36 ( member @ ( A > A )
% 0.15/0.36 @ ^ [X3: A] : X3
% 0.15/0.36 @ ( pi @ A @ A @ A2
% 0.15/0.36 @ ^ [Uu: A] : A2 ) ) ).
% 0.15/0.36
% 0.15/0.36 % funcset_id
% 0.15/0.36 thf(fact_11_Algebra1_Ofuncset__mem,axiom,
% 0.15/0.36 ! [A: $tType,B: $tType,F: A > B,A2: set @ A,B2: set @ B,X2: A] :
% 0.15/0.36 ( ( member @ ( A > B ) @ F
% 0.15/0.36 @ ( pi @ A @ B @ A2
% 0.15/0.36 @ ^ [Uu: A] : B2 ) )
% 0.15/0.36 => ( ( member @ A @ X2 @ A2 )
% 0.15/0.36 => ( member @ B @ ( F @ X2 ) @ B2 ) ) ) ).
% 0.15/0.36
% 0.15/0.36 % Algebra1.funcset_mem
% 0.15/0.36 thf(fact_12_prod__pOp__func,axiom,
% 0.15/0.36 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 0.15/0.36 ( ! [X: A] :
% 0.15/0.36 ( ( member @ A @ X @ I2 )
% 0.15/0.36 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 0.15/0.36 => ( member @ ( ( A > B ) > ( A > B ) > A > B ) @ ( prod_pOp @ A @ B @ C @ I2 @ A2 )
% 0.15/0.36 @ ( pi @ ( A > B ) @ ( ( A > B ) > A > B ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 )
% 0.15/0.36 @ ^ [Uu: A > B] :
% 0.15/0.36 ( pi @ ( A > B ) @ ( A > B ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 )
% 0.15/0.36 @ ^ [Uv: A > B] : ( carr_prodag @ A @ B @ C @ I2 @ A2 ) ) ) ) ) ).
% 0.15/0.36
% 0.15/0.36 % prod_pOp_func
% 0.15/0.36 thf(fact_13_ring__is__ag,axiom,
% 0.15/0.36 aGroup @ a @ ( ring_ext @ a @ b ) @ r ).
% 0.15/0.36
% 0.15/0.36 % ring_is_ag
% 0.15/0.36 thf(fact_14_prodM__mem__eq,axiom,
% 0.15/0.37 ! [E: $tType,D: $tType,C: $tType,I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ a @ E ) ) ),M12: C > D,M22: C > D] :
% 0.15/0.37 ( ! [X: C] :
% 0.15/0.37 ( ( member @ C @ X @ I2 )
% 0.15/0.37 => ( module @ D @ a @ E @ b @ ( M3 @ X ) @ r ) )
% 0.15/0.37 => ( ( member @ ( C > D ) @ M12 @ ( carrier @ ( C > D ) @ ( aGroup_ext @ ( C > D ) @ ( module_ext @ ( C > D ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ C @ D @ E @ r @ I2 @ M3 ) ) )
% 0.15/0.37 => ( ( member @ ( C > D ) @ M22 @ ( carrier @ ( C > D ) @ ( aGroup_ext @ ( C > D ) @ ( module_ext @ ( C > D ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ C @ D @ E @ r @ I2 @ M3 ) ) )
% 0.15/0.37 => ( ! [X: C] :
% 0.15/0.37 ( ( member @ C @ X @ I2 )
% 0.15/0.37 => ( ( M12 @ X )
% 0.15/0.37 = ( M22 @ X ) ) )
% 0.15/0.37 => ( M12 = M22 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % prodM_mem_eq
% 0.15/0.37 thf(fact_15_J__rad__mem,axiom,
% 0.15/0.37 ! [X2: a] :
% 0.15/0.37 ( ( member @ a @ X2 @ ( j_rad @ a @ b @ r ) )
% 0.15/0.37 => ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % J_rad_mem
% 0.15/0.37 thf(fact_16_Ring_ORing,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ring @ A @ B @ R2 ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.Ring
% 0.15/0.37 thf(fact_17_Ring_Oring__is__ag,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( aGroup @ A @ ( ring_ext @ A @ B ) @ R2 ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.ring_is_ag
% 0.15/0.37 thf(fact_18_Ring_OprodM__mem__eq,axiom,
% 0.15/0.37 ! [A: $tType,E: $tType,B: $tType,D: $tType,C: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ A @ E ) ) ),M12: C > D,M22: C > D] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ! [X: C] :
% 0.15/0.37 ( ( member @ C @ X @ I2 )
% 0.15/0.37 => ( module @ D @ A @ E @ B @ ( M3 @ X ) @ R2 ) )
% 0.15/0.37 => ( ( member @ ( C > D ) @ M12 @ ( carrier @ ( C > D ) @ ( aGroup_ext @ ( C > D ) @ ( module_ext @ ( C > D ) @ A @ product_unit ) ) @ ( algebr1000837334_prodM @ A @ B @ C @ D @ E @ R2 @ I2 @ M3 ) ) )
% 0.15/0.37 => ( ( member @ ( C > D ) @ M22 @ ( carrier @ ( C > D ) @ ( aGroup_ext @ ( C > D ) @ ( module_ext @ ( C > D ) @ A @ product_unit ) ) @ ( algebr1000837334_prodM @ A @ B @ C @ D @ E @ R2 @ I2 @ M3 ) ) )
% 0.15/0.37 => ( ! [X: C] :
% 0.15/0.37 ( ( member @ C @ X @ I2 )
% 0.15/0.37 => ( ( M12 @ X )
% 0.15/0.37 = ( M22 @ X ) ) )
% 0.15/0.37 => ( M12 = M22 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.prodM_mem_eq
% 0.15/0.37 thf(fact_19_carr__prodag__mem__eq,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B,Y: A > B] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 0.15/0.37 => ( ( member @ ( A > B ) @ X4 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 0.15/0.37 => ( ( member @ ( A > B ) @ Y @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 0.15/0.37 => ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( ( X4 @ X )
% 0.15/0.37 = ( Y @ X ) ) )
% 0.15/0.37 => ( X4 = Y ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % carr_prodag_mem_eq
% 0.15/0.37 thf(fact_20_prodag__sameTr1,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),B2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 0.15/0.37 => ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( ( A2 @ X )
% 0.15/0.37 = ( B2 @ X ) ) )
% 0.15/0.37 => ( ( carr_prodag @ A @ B @ C @ I2 @ A2 )
% 0.15/0.37 = ( carr_prodag @ A @ B @ C @ I2 @ B2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % prodag_sameTr1
% 0.15/0.37 thf(fact_21_prodag__comp__i,axiom,
% 0.15/0.37 ! [B: $tType,C: $tType,A: $tType,A3: A > B,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),I3: A] :
% 0.15/0.37 ( ( member @ ( A > B ) @ A3 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ I3 @ I2 )
% 0.15/0.37 => ( member @ B @ ( A3 @ I3 ) @ ( carrier @ B @ ( aGroup_ext @ B @ C ) @ ( A2 @ I3 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % prodag_comp_i
% 0.15/0.37 thf(fact_22_prodag__sameTr2,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),B2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 0.15/0.37 => ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( ( A2 @ X )
% 0.15/0.37 = ( B2 @ X ) ) )
% 0.15/0.37 => ( ( prod_pOp @ A @ B @ C @ I2 @ A2 )
% 0.15/0.37 = ( prod_pOp @ A @ B @ C @ I2 @ B2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % prodag_sameTr2
% 0.15/0.37 thf(fact_23_prodag__sameTr3,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),B2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 0.15/0.37 => ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( ( A2 @ X )
% 0.15/0.37 = ( B2 @ X ) ) )
% 0.15/0.37 => ( ( prod_mOp @ A @ B @ C @ I2 @ A2 )
% 0.15/0.37 = ( prod_mOp @ A @ B @ C @ I2 @ B2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % prodag_sameTr3
% 0.15/0.37 thf(fact_24_prodag__sameTr4,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),B2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 0.15/0.37 => ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( ( A2 @ X )
% 0.15/0.37 = ( B2 @ X ) ) )
% 0.15/0.37 => ( ( prod_zero @ A @ B @ C @ I2 @ A2 )
% 0.15/0.37 = ( prod_zero @ A @ B @ C @ I2 @ B2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % prodag_sameTr4
% 0.15/0.37 thf(fact_25_Ring_OprodM__carr,axiom,
% 0.15/0.37 ! [B: $tType,E: $tType,A: $tType,D: $tType,C: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ A @ E ) ) )] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ! [X: C] :
% 0.15/0.37 ( ( member @ C @ X @ I2 )
% 0.15/0.37 => ( module @ D @ A @ E @ B @ ( M3 @ X ) @ R2 ) )
% 0.15/0.37 => ( ( carrier @ ( C > D ) @ ( aGroup_ext @ ( C > D ) @ ( module_ext @ ( C > D ) @ A @ product_unit ) ) @ ( algebr1000837334_prodM @ A @ B @ C @ D @ E @ R2 @ I2 @ M3 ) )
% 0.15/0.37 = ( carr_prodag @ C @ D @ ( module_ext @ D @ A @ E ) @ I2 @ M3 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.prodM_carr
% 0.15/0.37 thf(fact_26_Ring_OprodM__sprod__mem,axiom,
% 0.15/0.37 ! [B: $tType,E: $tType,A: $tType,D: $tType,C: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ A @ E ) ) ),A3: A,M4: C > D] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ! [X: C] :
% 0.15/0.37 ( ( member @ C @ X @ I2 )
% 0.15/0.37 => ( module @ D @ A @ E @ B @ ( M3 @ X ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ ( C > D ) @ M4 @ ( carr_prodag @ C @ D @ ( module_ext @ D @ A @ E ) @ I2 @ M3 ) )
% 0.15/0.37 => ( member @ ( C > D ) @ ( algebr142713364_sprod @ A @ B @ C @ D @ E @ R2 @ I2 @ M3 @ A3 @ M4 ) @ ( carr_prodag @ C @ D @ ( module_ext @ D @ A @ E ) @ I2 @ M3 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.prodM_sprod_mem
% 0.15/0.37 thf(fact_27_prod__pOp__commute,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),A3: A > B,B3: A > B] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 0.15/0.37 => ( ( member @ ( A > B ) @ A3 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 0.15/0.37 => ( ( member @ ( A > B ) @ B3 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 0.15/0.37 => ( ( prod_pOp @ A @ B @ C @ I2 @ A2 @ A3 @ B3 )
% 0.15/0.37 = ( prod_pOp @ A @ B @ C @ I2 @ A2 @ B3 @ A3 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % prod_pOp_commute
% 0.15/0.37 thf(fact_28_prod__pOp__assoc,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),A3: A > B,B3: A > B,C3: A > B] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 0.15/0.37 => ( ( member @ ( A > B ) @ A3 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 0.15/0.37 => ( ( member @ ( A > B ) @ B3 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 0.15/0.37 => ( ( member @ ( A > B ) @ C3 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 0.15/0.37 => ( ( prod_pOp @ A @ B @ C @ I2 @ A2 @ ( prod_pOp @ A @ B @ C @ I2 @ A2 @ A3 @ B3 ) @ C3 )
% 0.15/0.37 = ( prod_pOp @ A @ B @ C @ I2 @ A2 @ A3 @ ( prod_pOp @ A @ B @ C @ I2 @ A2 @ B3 @ C3 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % prod_pOp_assoc
% 0.15/0.37 thf(fact_29_prod__pOp__mem,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B,Y: A > B] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 0.15/0.37 => ( ( member @ ( A > B ) @ X4 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 0.15/0.37 => ( ( member @ ( A > B ) @ Y @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 0.15/0.37 => ( member @ ( A > B ) @ ( prod_pOp @ A @ B @ C @ I2 @ A2 @ X4 @ Y ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % prod_pOp_mem
% 0.15/0.37 thf(fact_30_prod__mOp__mem,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 0.15/0.37 => ( ( member @ ( A > B ) @ X4 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 0.15/0.37 => ( member @ ( A > B ) @ ( prod_mOp @ A @ B @ C @ I2 @ A2 @ X4 ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % prod_mOp_mem
% 0.15/0.37 thf(fact_31_prod__zero__func,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 0.15/0.37 => ( member @ ( A > B ) @ ( prod_zero @ A @ B @ C @ I2 @ A2 ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % prod_zero_func
% 0.15/0.37 thf(fact_32_prod__mOp__func,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 0.15/0.37 => ( member @ ( ( A > B ) > A > B ) @ ( prod_mOp @ A @ B @ C @ I2 @ A2 )
% 0.15/0.37 @ ( pi @ ( A > B ) @ ( A > B ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 )
% 0.15/0.37 @ ^ [Uu: A > B] : ( carr_prodag @ A @ B @ C @ I2 @ A2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % prod_mOp_func
% 0.15/0.37 thf(fact_33_Pi__cong,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: set @ A,F: A > B,G: A > B,B2: A > ( set @ B )] :
% 0.15/0.37 ( ! [W: A] :
% 0.15/0.37 ( ( member @ A @ W @ A2 )
% 0.15/0.37 => ( ( F @ W )
% 0.15/0.37 = ( G @ W ) ) )
% 0.15/0.37 => ( ( member @ ( A > B ) @ F @ ( pi @ A @ B @ A2 @ B2 ) )
% 0.15/0.37 = ( member @ ( A > B ) @ G @ ( pi @ A @ B @ A2 @ B2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Pi_cong
% 0.15/0.37 thf(fact_34_Pi__mem,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: A > ( set @ B ),X2: A] :
% 0.15/0.37 ( ( member @ ( A > B ) @ F @ ( pi @ A @ B @ A2 @ B2 ) )
% 0.15/0.37 => ( ( member @ A @ X2 @ A2 )
% 0.15/0.37 => ( member @ B @ ( F @ X2 ) @ ( B2 @ X2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Pi_mem
% 0.15/0.37 thf(fact_35_Pi__iff,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,F: A > B,I2: set @ A,X4: A > ( set @ B )] :
% 0.15/0.37 ( ( member @ ( A > B ) @ F @ ( pi @ A @ B @ I2 @ X4 ) )
% 0.15/0.37 = ( ! [X3: A] :
% 0.15/0.37 ( ( member @ A @ X3 @ I2 )
% 0.15/0.37 => ( member @ B @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Pi_iff
% 0.15/0.37 thf(fact_36_Pi__I_H,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: set @ A,F: A > B,B2: A > ( set @ B )] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ A2 )
% 0.15/0.37 => ( member @ B @ ( F @ X ) @ ( B2 @ X ) ) )
% 0.15/0.37 => ( member @ ( A > B ) @ F @ ( pi @ A @ B @ A2 @ B2 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Pi_I'
% 0.15/0.37 thf(fact_37_PiE,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: A > ( set @ B ),X2: A] :
% 0.15/0.37 ( ( member @ ( A > B ) @ F @ ( pi @ A @ B @ A2 @ B2 ) )
% 0.15/0.37 => ( ~ ( member @ B @ ( F @ X2 ) @ ( B2 @ X2 ) )
% 0.15/0.37 => ~ ( member @ A @ X2 @ A2 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % PiE
% 0.15/0.37 thf(fact_38_mem__subring__mem__ring,axiom,
% 0.15/0.37 ! [C: $tType,S: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ),X2: a] :
% 0.15/0.37 ( ( subring @ a @ b @ C @ r @ S )
% 0.15/0.37 => ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ) @ S ) )
% 0.15/0.37 => ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % mem_subring_mem_ring
% 0.15/0.37 thf(fact_39_subring__Ring,axiom,
% 0.15/0.37 ! [C: $tType,S: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) )] :
% 0.15/0.37 ( ( subring @ a @ b @ C @ r @ S )
% 0.15/0.37 => ( ring @ a @ C @ S ) ) ).
% 0.15/0.37
% 0.15/0.37 % subring_Ring
% 0.15/0.37 thf(fact_40_prodM__sprod__val,axiom,
% 0.15/0.37 ! [E: $tType,D: $tType,C: $tType,I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ a @ E ) ) ),A3: a,M4: C > D,J: C] :
% 0.15/0.37 ( ! [X: C] :
% 0.15/0.37 ( ( member @ C @ X @ I2 )
% 0.15/0.37 => ( module @ D @ a @ E @ b @ ( M3 @ X ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ ( C > D ) @ M4 @ ( carr_prodag @ C @ D @ ( module_ext @ D @ a @ E ) @ I2 @ M3 ) )
% 0.15/0.37 => ( ( member @ C @ J @ I2 )
% 0.15/0.37 => ( ( algebr142713364_sprod @ a @ b @ C @ D @ E @ r @ I2 @ M3 @ A3 @ M4 @ J )
% 0.15/0.37 = ( sprod @ D @ a @ E @ ( M3 @ J ) @ A3 @ ( M4 @ J ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % prodM_sprod_val
% 0.15/0.37 thf(fact_41_Sr__ring,axiom,
% 0.15/0.37 ! [S: set @ a] :
% 0.15/0.37 ( ( sr2 @ a @ b @ r @ S )
% 0.15/0.37 => ( ring @ a @ b @ ( sr @ a @ b @ r @ S ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Sr_ring
% 0.15/0.37 thf(fact_42_mHom__func,axiom,
% 0.15/0.37 ! [E: $tType,D: $tType,F2: $tType,C: $tType,F: C > D,M3: carrier_ext @ C @ ( aGroup_ext @ C @ ( module_ext @ C @ a @ E ) ),N: carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ a @ F2 ) )] :
% 0.15/0.37 ( ( member @ ( C > D ) @ F @ ( mHom @ a @ b @ C @ E @ D @ F2 @ r @ M3 @ N ) )
% 0.15/0.37 => ( member @ ( C > D ) @ F
% 0.15/0.37 @ ( pi @ C @ D @ ( carrier @ C @ ( aGroup_ext @ C @ ( module_ext @ C @ a @ E ) ) @ M3 )
% 0.15/0.37 @ ^ [Uu: C] : ( carrier @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ a @ F2 ) ) @ N ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % mHom_func
% 0.15/0.37 thf(fact_43_rInvim__def,axiom,
% 0.15/0.37 ! [B: $tType,M1: $tType,M: $tType,A: $tType] :
% 0.15/0.37 ( ( rInvim @ A @ M @ B @ M1 )
% 0.15/0.37 = ( ^ [A4: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ M ) ),R3: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ M1 ) ),F3: A > B,K: set @ B] :
% 0.15/0.37 ( collect @ A
% 0.15/0.37 @ ^ [A5: A] :
% 0.15/0.37 ( ( member @ A @ A5 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ M ) ) @ A4 ) )
% 0.15/0.37 & ( member @ B @ ( F3 @ A5 ) @ K ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % rInvim_def
% 0.15/0.37 thf(fact_44_dsum__pOp__func,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 0.15/0.37 => ( member @ ( ( A > B ) > ( A > B ) > A > B ) @ ( prod_pOp @ A @ B @ C @ I2 @ A2 )
% 0.15/0.37 @ ( pi @ ( A > B ) @ ( ( A > B ) > A > B ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 )
% 0.15/0.37 @ ^ [Uu: A > B] :
% 0.15/0.37 ( pi @ ( A > B ) @ ( A > B ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 )
% 0.15/0.37 @ ^ [Uv: A > B] : ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % dsum_pOp_func
% 0.15/0.37 thf(fact_45_mem__Collect__eq,axiom,
% 0.15/0.37 ! [A: $tType,A3: A,P: A > $o] :
% 0.15/0.37 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.15/0.37 = ( P @ A3 ) ) ).
% 0.15/0.37
% 0.15/0.37 % mem_Collect_eq
% 0.15/0.37 thf(fact_46_Collect__mem__eq,axiom,
% 0.15/0.37 ! [A: $tType,A2: set @ A] :
% 0.15/0.37 ( ( collect @ A
% 0.15/0.37 @ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
% 0.15/0.37 = A2 ) ).
% 0.15/0.37
% 0.15/0.37 % Collect_mem_eq
% 0.15/0.37 thf(fact_47_Collect__cong,axiom,
% 0.15/0.37 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( P @ X )
% 0.15/0.37 = ( Q @ X ) )
% 0.15/0.37 => ( ( collect @ A @ P )
% 0.15/0.37 = ( collect @ A @ Q ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Collect_cong
% 0.15/0.37 thf(fact_48_ext,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( F @ X )
% 0.15/0.37 = ( G @ X ) )
% 0.15/0.37 => ( F = G ) ) ).
% 0.15/0.37
% 0.15/0.37 % ext
% 0.15/0.37 thf(fact_49_nsEqElm,axiom,
% 0.15/0.37 ! [X2: a,Y2: a,N2: nat] :
% 0.15/0.37 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( X2 = Y2 )
% 0.15/0.37 => ( ( nscal @ a @ b @ r @ X2 @ N2 )
% 0.15/0.37 = ( nscal @ a @ b @ r @ Y2 @ N2 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % nsEqElm
% 0.15/0.37 thf(fact_50_nsClose,axiom,
% 0.15/0.37 ! [X2: a,N2: nat] :
% 0.15/0.37 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( member @ a @ ( nscal @ a @ b @ r @ X2 @ N2 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % nsClose
% 0.15/0.37 thf(fact_51_tp__closed,axiom,
% 0.15/0.37 ( member @ ( a > a > a ) @ ( tp @ a @ b @ r )
% 0.15/0.37 @ ( pi @ a @ ( a > a ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 0.15/0.37 @ ^ [Uu: a] :
% 0.15/0.37 ( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 0.15/0.37 @ ^ [Uv: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % tp_closed
% 0.15/0.37 thf(fact_52_pop__closed,axiom,
% 0.15/0.37 ( member @ ( a > a > a ) @ ( pop @ a @ ( ring_ext @ a @ b ) @ r )
% 0.15/0.37 @ ( pi @ a @ ( a > a ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 0.15/0.37 @ ^ [Uu: a] :
% 0.15/0.37 ( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 0.15/0.37 @ ^ [Uv: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % pop_closed
% 0.15/0.37 thf(fact_53_rEQMulR,axiom,
% 0.15/0.37 ! [X2: a,Y2: a,Z2: a] :
% 0.15/0.37 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Z2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( X2 = Y2 )
% 0.15/0.37 => ( ( tp @ a @ b @ r @ X2 @ Z2 )
% 0.15/0.37 = ( tp @ a @ b @ r @ Y2 @ Z2 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % rEQMulR
% 0.15/0.37 thf(fact_54_rMulLC,axiom,
% 0.15/0.37 ! [X2: a,Y2: a,Z2: a] :
% 0.15/0.37 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Z2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( tp @ a @ b @ r @ X2 @ ( tp @ a @ b @ r @ Y2 @ Z2 ) )
% 0.15/0.37 = ( tp @ a @ b @ r @ Y2 @ ( tp @ a @ b @ r @ X2 @ Z2 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % rMulLC
% 0.15/0.37 thf(fact_55_ring__tOp__assoc,axiom,
% 0.15/0.37 ! [X2: a,Y2: a,Z2: a] :
% 0.15/0.37 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Z2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ X2 @ Y2 ) @ Z2 )
% 0.15/0.37 = ( tp @ a @ b @ r @ X2 @ ( tp @ a @ b @ r @ Y2 @ Z2 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % ring_tOp_assoc
% 0.15/0.37 thf(fact_56_ring__tOp__closed,axiom,
% 0.15/0.37 ! [X2: a,Y2: a] :
% 0.15/0.37 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( member @ a @ ( tp @ a @ b @ r @ X2 @ Y2 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % ring_tOp_closed
% 0.15/0.37 thf(fact_57_ring__tOp__commute,axiom,
% 0.15/0.37 ! [X2: a,Y2: a] :
% 0.15/0.37 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( tp @ a @ b @ r @ X2 @ Y2 )
% 0.15/0.37 = ( tp @ a @ b @ r @ Y2 @ X2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % ring_tOp_commute
% 0.15/0.37 thf(fact_58_ring__tOp__rel,axiom,
% 0.15/0.37 ! [X2: a,Xa: a,Y2: a,Ya: a] :
% 0.15/0.37 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Xa @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Ya @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ X2 @ Xa ) @ ( tp @ a @ b @ r @ Y2 @ Ya ) )
% 0.15/0.37 = ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ X2 @ Y2 ) @ ( tp @ a @ b @ r @ Xa @ Ya ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % ring_tOp_rel
% 0.15/0.37 thf(fact_59_tp__assoc,axiom,
% 0.15/0.37 ! [A3: a,B3: a,C3: a] :
% 0.15/0.37 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ C3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) @ C3 )
% 0.15/0.37 = ( tp @ a @ b @ r @ A3 @ ( tp @ a @ b @ r @ B3 @ C3 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % tp_assoc
% 0.15/0.37 thf(fact_60_tp__commute,axiom,
% 0.15/0.37 ! [A3: a,B3: a] :
% 0.15/0.37 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( tp @ a @ b @ r @ A3 @ B3 )
% 0.15/0.37 = ( tp @ a @ b @ r @ B3 @ A3 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % tp_commute
% 0.15/0.37 thf(fact_61_pop__aassoc,axiom,
% 0.15/0.37 ! [A3: a,B3: a,C3: a] :
% 0.15/0.37 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ C3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ B3 ) @ C3 )
% 0.15/0.37 = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B3 @ C3 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % pop_aassoc
% 0.15/0.37 thf(fact_62_pop__commute,axiom,
% 0.15/0.37 ! [A3: a,B3: a] :
% 0.15/0.37 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ B3 )
% 0.15/0.37 = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B3 @ A3 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % pop_commute
% 0.15/0.37 thf(fact_63_Sr__tOp__closed,axiom,
% 0.15/0.37 ! [S: set @ a,X2: a,Y2: a] :
% 0.15/0.37 ( ( sr2 @ a @ b @ r @ S )
% 0.15/0.37 => ( ( member @ a @ X2 @ S )
% 0.15/0.37 => ( ( member @ a @ Y2 @ S )
% 0.15/0.37 => ( member @ a @ ( tp @ a @ b @ r @ X2 @ Y2 ) @ S ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Sr_tOp_closed
% 0.15/0.37 thf(fact_64_Sr__pOp__closed,axiom,
% 0.15/0.37 ! [S: set @ a,X2: a,Y2: a] :
% 0.15/0.37 ( ( sr2 @ a @ b @ r @ S )
% 0.15/0.37 => ( ( member @ a @ X2 @ S )
% 0.15/0.37 => ( ( member @ a @ Y2 @ S )
% 0.15/0.37 => ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ X2 @ Y2 ) @ S ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Sr_pOp_closed
% 0.15/0.37 thf(fact_65_rg__distrib,axiom,
% 0.15/0.37 ! [A3: a,B3: a,C3: a] :
% 0.15/0.37 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ C3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( tp @ a @ b @ r @ A3 @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B3 @ C3 ) )
% 0.15/0.37 = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) @ ( tp @ a @ b @ r @ A3 @ C3 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % rg_distrib
% 0.15/0.37 thf(fact_66_ring__distrib1,axiom,
% 0.15/0.37 ! [X2: a,Y2: a,Z2: a] :
% 0.15/0.37 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Z2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( tp @ a @ b @ r @ X2 @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ Y2 @ Z2 ) )
% 0.15/0.37 = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ X2 @ Y2 ) @ ( tp @ a @ b @ r @ X2 @ Z2 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % ring_distrib1
% 0.15/0.37 thf(fact_67_ring__distrib2,axiom,
% 0.15/0.37 ! [X2: a,Y2: a,Z2: a] :
% 0.15/0.37 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Z2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ Y2 @ Z2 ) @ X2 )
% 0.15/0.37 = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ Y2 @ X2 ) @ ( tp @ a @ b @ r @ Z2 @ X2 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % ring_distrib2
% 0.15/0.37 thf(fact_68_ring__distrib3,axiom,
% 0.15/0.37 ! [A3: a,B3: a,X2: a,Y2: a] :
% 0.15/0.37 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ B3 ) @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ X2 @ Y2 ) )
% 0.15/0.37 = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ X2 ) @ ( tp @ a @ b @ r @ A3 @ Y2 ) ) @ ( tp @ a @ b @ r @ B3 @ X2 ) ) @ ( tp @ a @ b @ r @ B3 @ Y2 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % ring_distrib3
% 0.15/0.37 thf(fact_69_nsMulDistrL,axiom,
% 0.15/0.37 ! [X2: a,Y2: a,N2: nat] :
% 0.15/0.37 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( tp @ a @ b @ r @ X2 @ ( nscal @ a @ b @ r @ Y2 @ N2 ) )
% 0.15/0.37 = ( nscal @ a @ b @ r @ ( tp @ a @ b @ r @ X2 @ Y2 ) @ N2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % nsMulDistrL
% 0.15/0.37 thf(fact_70_nsMulDistrR,axiom,
% 0.15/0.37 ! [X2: a,Y2: a,N2: nat] :
% 0.15/0.37 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( tp @ a @ b @ r @ ( nscal @ a @ b @ r @ Y2 @ N2 ) @ X2 )
% 0.15/0.37 = ( nscal @ a @ b @ r @ ( tp @ a @ b @ r @ Y2 @ X2 ) @ N2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % nsMulDistrR
% 0.15/0.37 thf(fact_71_nsDistrL,axiom,
% 0.15/0.37 ! [X2: a,Y2: a,N2: nat] :
% 0.15/0.37 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( nscal @ a @ b @ r @ X2 @ N2 ) @ ( nscal @ a @ b @ r @ Y2 @ N2 ) )
% 0.15/0.37 = ( nscal @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ X2 @ Y2 ) @ N2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % nsDistrL
% 0.15/0.37 thf(fact_72_Subring__tOp__ring__tOp,axiom,
% 0.15/0.37 ! [C: $tType,S: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ),A3: a,B3: a] :
% 0.15/0.37 ( ( subring @ a @ b @ C @ r @ S )
% 0.15/0.37 => ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ) @ S ) )
% 0.15/0.37 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ) @ S ) )
% 0.15/0.37 => ( ( tp @ a @ C @ S @ A3 @ B3 )
% 0.15/0.37 = ( tp @ a @ b @ r @ A3 @ B3 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Subring_tOp_ring_tOp
% 0.15/0.37 thf(fact_73_Subring__pOp__ring__pOp,axiom,
% 0.15/0.37 ! [C: $tType,S: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ),A3: a,B3: a] :
% 0.15/0.37 ( ( subring @ a @ b @ C @ r @ S )
% 0.15/0.37 => ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ) @ S ) )
% 0.15/0.37 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ) @ S ) )
% 0.15/0.37 => ( ( pop @ a @ ( ring_ext @ a @ C ) @ S @ A3 @ B3 )
% 0.15/0.37 = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ B3 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Subring_pOp_ring_pOp
% 0.15/0.37 thf(fact_74_mul__closed__set__tOp__closed,axiom,
% 0.15/0.37 ! [S: set @ a,S2: a,T: a] :
% 0.15/0.37 ( ( mul_closed_set @ a @ b @ r @ S )
% 0.15/0.37 => ( ( member @ a @ S2 @ S )
% 0.15/0.37 => ( ( member @ a @ T @ S )
% 0.15/0.37 => ( member @ a @ ( tp @ a @ b @ r @ S2 @ T ) @ S ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % mul_closed_set_tOp_closed
% 0.15/0.37 thf(fact_75_Ring_OSr__tOp__closed,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A,X2: A,Y2: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( sr2 @ A @ B @ R2 @ S )
% 0.15/0.37 => ( ( member @ A @ X2 @ S )
% 0.15/0.37 => ( ( member @ A @ Y2 @ S )
% 0.15/0.37 => ( member @ A @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) @ S ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.Sr_tOp_closed
% 0.15/0.37 thf(fact_76_Ring_OSr__pOp__closed,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A,X2: A,Y2: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( sr2 @ A @ B @ R2 @ S )
% 0.15/0.37 => ( ( member @ A @ X2 @ S )
% 0.15/0.37 => ( ( member @ A @ Y2 @ S )
% 0.15/0.37 => ( member @ A @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ X2 @ Y2 ) @ S ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.Sr_pOp_closed
% 0.15/0.37 thf(fact_77_Ring_Oring__distrib3,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A,X2: A,Y2: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( tp @ A @ B @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ B3 ) @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ X2 @ Y2 ) )
% 0.15/0.37 = ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ X2 ) @ ( tp @ A @ B @ R2 @ A3 @ Y2 ) ) @ ( tp @ A @ B @ R2 @ B3 @ X2 ) ) @ ( tp @ A @ B @ R2 @ B3 @ Y2 ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.ring_distrib3
% 0.15/0.37 thf(fact_78_Ring_Oring__distrib2,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,Z2: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Z2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( tp @ A @ B @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ Y2 @ Z2 ) @ X2 )
% 0.15/0.37 = ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ Y2 @ X2 ) @ ( tp @ A @ B @ R2 @ Z2 @ X2 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.ring_distrib2
% 0.15/0.37 thf(fact_79_Ring_Oring__distrib1,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,Z2: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Z2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( tp @ A @ B @ R2 @ X2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ Y2 @ Z2 ) )
% 0.15/0.37 = ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) @ ( tp @ A @ B @ R2 @ X2 @ Z2 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.ring_distrib1
% 0.15/0.37 thf(fact_80_Ring_Org__distrib,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A,C3: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( tp @ A @ B @ R2 @ A3 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 @ C3 ) )
% 0.15/0.37 = ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) @ ( tp @ A @ B @ R2 @ A3 @ C3 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.rg_distrib
% 0.15/0.37 thf(fact_81_Ring_OnsMulDistrR,axiom,
% 0.15/0.37 ! [A: $tType,B: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,N2: nat] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( tp @ A @ B @ R2 @ ( nscal @ A @ B @ R2 @ Y2 @ N2 ) @ X2 )
% 0.15/0.37 = ( nscal @ A @ B @ R2 @ ( tp @ A @ B @ R2 @ Y2 @ X2 ) @ N2 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.nsMulDistrR
% 0.15/0.37 thf(fact_82_Ring_OnsMulDistrL,axiom,
% 0.15/0.37 ! [A: $tType,B: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,N2: nat] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( tp @ A @ B @ R2 @ X2 @ ( nscal @ A @ B @ R2 @ Y2 @ N2 ) )
% 0.15/0.37 = ( nscal @ A @ B @ R2 @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) @ N2 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.nsMulDistrL
% 0.15/0.37 thf(fact_83_Ring_OnsDistrL,axiom,
% 0.15/0.37 ! [A: $tType,B: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,N2: nat] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( nscal @ A @ B @ R2 @ X2 @ N2 ) @ ( nscal @ A @ B @ R2 @ Y2 @ N2 ) )
% 0.15/0.37 = ( nscal @ A @ B @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ X2 @ Y2 ) @ N2 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.nsDistrL
% 0.15/0.37 thf(fact_84_Ring_OSr__ring,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( sr2 @ A @ B @ R2 @ S )
% 0.15/0.37 => ( ring @ A @ B @ ( sr @ A @ B @ R2 @ S ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.Sr_ring
% 0.15/0.37 thf(fact_85_aGroup_Oag__pOp__commute,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A,Y2: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( pop @ A @ B @ A2 @ X2 @ Y2 )
% 0.15/0.37 = ( pop @ A @ B @ A2 @ Y2 @ X2 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.ag_pOp_commute
% 0.15/0.37 thf(fact_86_aGroup_Oag__add__commute,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( pop @ A @ B @ A2 @ A3 @ B3 )
% 0.15/0.37 = ( pop @ A @ B @ A2 @ B3 @ A3 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.ag_add_commute
% 0.15/0.37 thf(fact_87_aGroup_OpOp__assocTr43,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A,D2: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ D2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ B3 ) @ ( pop @ A @ B @ A2 @ C3 @ D2 ) )
% 0.15/0.37 = ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ ( pop @ A @ B @ A2 @ B3 @ C3 ) ) @ D2 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.pOp_assocTr43
% 0.15/0.37 thf(fact_88_aGroup_OpOp__assocTr42,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A,D2: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ D2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ B3 ) @ C3 ) @ D2 )
% 0.15/0.37 = ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ ( pop @ A @ B @ A2 @ B3 @ C3 ) ) @ D2 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.pOp_assocTr42
% 0.15/0.37 thf(fact_89_aGroup_OpOp__assocTr41,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A,D2: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ D2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ B3 ) @ C3 ) @ D2 )
% 0.15/0.37 = ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ B3 ) @ ( pop @ A @ B @ A2 @ C3 @ D2 ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.pOp_assocTr41
% 0.15/0.37 thf(fact_90_aGroup_Oag__pOp__closed,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A,Y2: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( member @ A @ ( pop @ A @ B @ A2 @ X2 @ Y2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.ag_pOp_closed
% 0.15/0.37 thf(fact_91_aGroup_OpOp__cancel__r,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( ( pop @ A @ B @ A2 @ A3 @ C3 )
% 0.15/0.37 = ( pop @ A @ B @ A2 @ B3 @ C3 ) )
% 0.15/0.37 => ( A3 = B3 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.pOp_cancel_r
% 0.15/0.37 thf(fact_92_aGroup_OpOp__cancel__l,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( ( pop @ A @ B @ A2 @ C3 @ A3 )
% 0.15/0.37 = ( pop @ A @ B @ A2 @ C3 @ B3 ) )
% 0.15/0.37 => ( A3 = B3 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.pOp_cancel_l
% 0.15/0.37 thf(fact_93_aGroup_Oag__pOp__assoc,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A,Y2: A,Z2: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ Z2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ X2 @ Y2 ) @ Z2 )
% 0.15/0.37 = ( pop @ A @ B @ A2 @ X2 @ ( pop @ A @ B @ A2 @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.ag_pOp_assoc
% 0.15/0.37 thf(fact_94_aGroup_Oag__pOp__add__r,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( A3 = B3 )
% 0.15/0.37 => ( ( pop @ A @ B @ A2 @ A3 @ C3 )
% 0.15/0.37 = ( pop @ A @ B @ A2 @ B3 @ C3 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.ag_pOp_add_r
% 0.15/0.37 thf(fact_95_aGroup_Oag__pOp__add__l,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( A3 = B3 )
% 0.15/0.37 => ( ( pop @ A @ B @ A2 @ C3 @ A3 )
% 0.15/0.37 = ( pop @ A @ B @ A2 @ C3 @ B3 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.ag_pOp_add_l
% 0.15/0.37 thf(fact_96_aGroup_OgEQAddcross,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),L1: A,L2: A,R1: A,R22: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ L1 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ L2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ R1 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ R1 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( L1 = R22 )
% 0.15/0.37 => ( ( L2 = R1 )
% 0.15/0.37 => ( ( pop @ A @ B @ A2 @ L1 @ L2 )
% 0.15/0.37 = ( pop @ A @ B @ A2 @ R1 @ R22 ) ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.gEQAddcross
% 0.15/0.37 thf(fact_97_aGroup_Oag__add4__rel,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A,D2: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ D2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ B3 ) @ ( pop @ A @ B @ A2 @ C3 @ D2 ) )
% 0.15/0.37 = ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ C3 ) @ ( pop @ A @ B @ A2 @ B3 @ D2 ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.ag_add4_rel
% 0.15/0.37 thf(fact_98_aGroup_Oaassoc,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ B3 ) @ C3 )
% 0.15/0.37 = ( pop @ A @ B @ A2 @ A3 @ ( pop @ A @ B @ A2 @ B3 @ C3 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.aassoc
% 0.15/0.37 thf(fact_99_Ring_Oring__tOp__commute,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( tp @ A @ B @ R2 @ X2 @ Y2 )
% 0.15/0.37 = ( tp @ A @ B @ R2 @ Y2 @ X2 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.ring_tOp_commute
% 0.15/0.37 thf(fact_100_Ring_Oring__tOp__closed,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( member @ A @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.ring_tOp_closed
% 0.15/0.37 thf(fact_101_Ring_Oring__tOp__assoc,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,Z2: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Z2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( tp @ A @ B @ R2 @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) @ Z2 )
% 0.15/0.37 = ( tp @ A @ B @ R2 @ X2 @ ( tp @ A @ B @ R2 @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.ring_tOp_assoc
% 0.15/0.37 thf(fact_102_Ring_Oring__tOp__rel,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Xa: A,Y2: A,Ya: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Xa @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Ya @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( tp @ A @ B @ R2 @ ( tp @ A @ B @ R2 @ X2 @ Xa ) @ ( tp @ A @ B @ R2 @ Y2 @ Ya ) )
% 0.15/0.37 = ( tp @ A @ B @ R2 @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) @ ( tp @ A @ B @ R2 @ Xa @ Ya ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.ring_tOp_rel
% 0.15/0.37 thf(fact_103_Ring_Otp__commute,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( tp @ A @ B @ R2 @ A3 @ B3 )
% 0.15/0.37 = ( tp @ A @ B @ R2 @ B3 @ A3 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.tp_commute
% 0.15/0.37 thf(fact_104_Ring_Otp__assoc,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A,C3: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( tp @ A @ B @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) @ C3 )
% 0.15/0.37 = ( tp @ A @ B @ R2 @ A3 @ ( tp @ A @ B @ R2 @ B3 @ C3 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.tp_assoc
% 0.15/0.37 thf(fact_105_Ring_OrEQMulR,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,Z2: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Z2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( X2 = Y2 )
% 0.15/0.37 => ( ( tp @ A @ B @ R2 @ X2 @ Z2 )
% 0.15/0.37 = ( tp @ A @ B @ R2 @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.rEQMulR
% 0.15/0.37 thf(fact_106_Ring_OrMulLC,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,Z2: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Z2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( tp @ A @ B @ R2 @ X2 @ ( tp @ A @ B @ R2 @ Y2 @ Z2 ) )
% 0.15/0.37 = ( tp @ A @ B @ R2 @ Y2 @ ( tp @ A @ B @ R2 @ X2 @ Z2 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.rMulLC
% 0.15/0.37 thf(fact_107_Ring_Opop__commute,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ B3 )
% 0.15/0.37 = ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 @ A3 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.pop_commute
% 0.15/0.37 thf(fact_108_Ring_Opop__aassoc,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A,C3: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ B3 ) @ C3 )
% 0.15/0.37 = ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 @ C3 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.pop_aassoc
% 0.15/0.37 thf(fact_109_Ring_OnsEqElm,axiom,
% 0.15/0.37 ! [A: $tType,B: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,N2: nat] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( X2 = Y2 )
% 0.15/0.37 => ( ( nscal @ A @ B @ R2 @ X2 @ N2 )
% 0.15/0.37 = ( nscal @ A @ B @ R2 @ Y2 @ N2 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.nsEqElm
% 0.15/0.37 thf(fact_110_Ring_OnsClose,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,N2: nat] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( member @ A @ ( nscal @ A @ B @ R2 @ X2 @ N2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.nsClose
% 0.15/0.37 thf(fact_111_prod__pOp__mem__i,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B,Y: A > B,I3: A] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 0.15/0.37 => ( ( member @ ( A > B ) @ X4 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 0.15/0.37 => ( ( member @ ( A > B ) @ Y @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ I3 @ I2 )
% 0.15/0.37 => ( ( prod_pOp @ A @ B @ C @ I2 @ A2 @ X4 @ Y @ I3 )
% 0.15/0.37 = ( pop @ B @ C @ ( A2 @ I3 ) @ ( X4 @ I3 ) @ ( Y @ I3 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % prod_pOp_mem_i
% 0.15/0.37 thf(fact_112_dsum__pOp__mem,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B,Y: A > B] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 0.15/0.37 => ( ( member @ ( A > B ) @ X4 @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) )
% 0.15/0.37 => ( ( member @ ( A > B ) @ Y @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) )
% 0.15/0.37 => ( member @ ( A > B ) @ ( prod_pOp @ A @ B @ C @ I2 @ A2 @ X4 @ Y ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % dsum_pOp_mem
% 0.15/0.37 thf(fact_113_dsum__iOp__mem,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 0.15/0.37 => ( ( member @ ( A > B ) @ X4 @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) )
% 0.15/0.37 => ( member @ ( A > B ) @ ( prod_mOp @ A @ B @ C @ I2 @ A2 @ X4 ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % dsum_iOp_mem
% 0.15/0.37 thf(fact_114_dsum__zero__func,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 0.15/0.37 => ( member @ ( A > B ) @ ( prod_zero @ A @ B @ C @ I2 @ A2 ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % dsum_zero_func
% 0.15/0.37 thf(fact_115_aGroup_Opop__closed,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B )] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( member @ ( A > A > A ) @ ( pop @ A @ B @ A2 )
% 0.15/0.37 @ ( pi @ A @ ( A > A ) @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 )
% 0.15/0.37 @ ^ [Uu: A] :
% 0.15/0.37 ( pi @ A @ A @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 )
% 0.15/0.37 @ ^ [Uv: A] : ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.pop_closed
% 0.15/0.37 thf(fact_116_Ring_OJ__rad__mem,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( j_rad @ A @ B @ R2 ) )
% 0.15/0.37 => ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.J_rad_mem
% 0.15/0.37 thf(fact_117_Ring_Otp__closed,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( member @ ( A > A > A ) @ ( tp @ A @ B @ R2 )
% 0.15/0.37 @ ( pi @ A @ ( A > A ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 )
% 0.15/0.37 @ ^ [Uu: A] :
% 0.15/0.37 ( pi @ A @ A @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 )
% 0.15/0.37 @ ^ [Uv: A] : ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.tp_closed
% 0.15/0.37 thf(fact_118_Ring_Opop__closed,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( member @ ( A > A > A ) @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 )
% 0.15/0.37 @ ( pi @ A @ ( A > A ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 )
% 0.15/0.37 @ ^ [Uu: A] :
% 0.15/0.37 ( pi @ A @ A @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 )
% 0.15/0.37 @ ^ [Uv: A] : ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.pop_closed
% 0.15/0.37 thf(fact_119_Ring_OprodM__sprod__val,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,E: $tType,D: $tType,C: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ A @ E ) ) ),A3: A,M4: C > D,J: C] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ! [X: C] :
% 0.15/0.37 ( ( member @ C @ X @ I2 )
% 0.15/0.37 => ( module @ D @ A @ E @ B @ ( M3 @ X ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ ( C > D ) @ M4 @ ( carr_prodag @ C @ D @ ( module_ext @ D @ A @ E ) @ I2 @ M3 ) )
% 0.15/0.37 => ( ( member @ C @ J @ I2 )
% 0.15/0.37 => ( ( algebr142713364_sprod @ A @ B @ C @ D @ E @ R2 @ I2 @ M3 @ A3 @ M4 @ J )
% 0.15/0.37 = ( sprod @ D @ A @ E @ ( M3 @ J ) @ A3 @ ( M4 @ J ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.prodM_sprod_val
% 0.15/0.37 thf(fact_120_dsum__iOp__func,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 0.15/0.37 => ( member @ ( ( A > B ) > A > B ) @ ( prod_mOp @ A @ B @ C @ I2 @ A2 )
% 0.15/0.37 @ ( pi @ ( A > B ) @ ( A > B ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 )
% 0.15/0.37 @ ^ [Uu: A > B] : ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % dsum_iOp_func
% 0.15/0.37 thf(fact_121_zeroring__J__rad__empty,axiom,
% 0.15/0.37 ( ( zeroring @ a @ b @ r )
% 0.15/0.37 => ( ( j_rad @ a @ b @ r )
% 0.15/0.37 = ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % zeroring_J_rad_empty
% 0.15/0.37 thf(fact_122_Module_OmHom__lin,axiom,
% 0.15/0.37 ! [D: $tType,C: $tType,B: $tType,F2: $tType,E: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ),M4: A,F: A > E,A3: B] :
% 0.15/0.37 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 0.15/0.37 => ( ( module @ E @ B @ F2 @ D @ N @ R2 )
% 0.15/0.37 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 0.15/0.37 => ( ( member @ ( A > E ) @ F @ ( mHom @ B @ D @ A @ C @ E @ F2 @ R2 @ M3 @ N ) )
% 0.15/0.37 => ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
% 0.15/0.37 => ( ( F @ ( sprod @ A @ B @ C @ M3 @ A3 @ M4 ) )
% 0.15/0.37 = ( sprod @ E @ B @ F2 @ N @ A3 @ ( F @ M4 ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Module.mHom_lin
% 0.15/0.37 thf(fact_123_Module_Osprod__assoc,axiom,
% 0.15/0.37 ! [D: $tType,B: $tType,C: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: B,B3: B,M4: A] :
% 0.15/0.37 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 0.15/0.37 => ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ B @ B3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 0.15/0.37 => ( ( sprod @ A @ B @ C @ M3 @ ( tp @ B @ D @ R2 @ A3 @ B3 ) @ M4 )
% 0.15/0.37 = ( sprod @ A @ B @ C @ M3 @ A3 @ ( sprod @ A @ B @ C @ M3 @ B3 @ M4 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Module.sprod_assoc
% 0.15/0.37 thf(fact_124_Ring_OmHom__func,axiom,
% 0.15/0.37 ! [B: $tType,E: $tType,D: $tType,A: $tType,F2: $tType,C: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),F: C > D,M3: carrier_ext @ C @ ( aGroup_ext @ C @ ( module_ext @ C @ A @ E ) ),N: carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ A @ F2 ) )] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ ( C > D ) @ F @ ( mHom @ A @ B @ C @ E @ D @ F2 @ R2 @ M3 @ N ) )
% 0.15/0.37 => ( member @ ( C > D ) @ F
% 0.15/0.37 @ ( pi @ C @ D @ ( carrier @ C @ ( aGroup_ext @ C @ ( module_ext @ C @ A @ E ) ) @ M3 )
% 0.15/0.37 @ ^ [Uu: C] : ( carrier @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ A @ F2 ) ) @ N ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.mHom_func
% 0.15/0.37 thf(fact_125_Ring_OSubring__pOp__ring__pOp,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ),A3: A,B3: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( subring @ A @ B @ C @ R2 @ S )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) )
% 0.15/0.37 => ( ( pop @ A @ ( ring_ext @ A @ C ) @ S @ A3 @ B3 )
% 0.15/0.37 = ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ B3 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.Subring_pOp_ring_pOp
% 0.15/0.37 thf(fact_126_Ring_OSubring__tOp__ring__tOp,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ),A3: A,B3: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( subring @ A @ B @ C @ R2 @ S )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) )
% 0.15/0.37 => ( ( tp @ A @ C @ S @ A3 @ B3 )
% 0.15/0.37 = ( tp @ A @ B @ R2 @ A3 @ B3 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.Subring_tOp_ring_tOp
% 0.15/0.37 thf(fact_127_Module_Osprod__closed,axiom,
% 0.15/0.37 ! [D: $tType,C: $tType,B: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: B,M4: A] :
% 0.15/0.37 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 0.15/0.37 => ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 0.15/0.37 => ( member @ A @ ( sprod @ A @ B @ C @ M3 @ A3 @ M4 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Module.sprod_closed
% 0.15/0.37 thf(fact_128_nsDistr,axiom,
% 0.15/0.37 ! [X2: a,N2: nat,M4: nat] :
% 0.15/0.37 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( nscal @ a @ b @ r @ X2 @ N2 ) @ ( nscal @ a @ b @ r @ X2 @ M4 ) )
% 0.15/0.37 = ( nscal @ a @ b @ r @ X2 @ ( plus_plus @ nat @ N2 @ M4 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % nsDistr
% 0.15/0.37 thf(fact_129_add__bothl,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( plus @ A )
% 0.15/0.37 => ! [A3: A,B3: A,C3: A] :
% 0.15/0.37 ( ( A3 = B3 )
% 0.15/0.37 => ( ( plus_plus @ A @ C3 @ A3 )
% 0.15/0.37 = ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % add_bothl
% 0.15/0.37 thf(fact_130_add__both,axiom,
% 0.15/0.37 ! [A3: nat,B3: nat,C3: nat] :
% 0.15/0.37 ( ( A3 = B3 )
% 0.15/0.37 => ( ( plus_plus @ nat @ A3 @ C3 )
% 0.15/0.37 = ( plus_plus @ nat @ B3 @ C3 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % add_both
% 0.15/0.37 thf(fact_131_Module_OmHom__add,axiom,
% 0.15/0.37 ! [D: $tType,C: $tType,F2: $tType,B: $tType,E: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ),F: A > E,M4: A,N2: A] :
% 0.15/0.37 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 0.15/0.37 => ( ( module @ E @ B @ F2 @ D @ N @ R2 )
% 0.15/0.37 => ( ( member @ ( A > E ) @ F @ ( mHom @ B @ D @ A @ C @ E @ F2 @ R2 @ M3 @ N ) )
% 0.15/0.37 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 0.15/0.37 => ( ( member @ A @ N2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 0.15/0.37 => ( ( F @ ( pop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ M4 @ N2 ) )
% 0.15/0.37 = ( pop @ E @ ( module_ext @ E @ B @ F2 ) @ N @ ( F @ M4 ) @ ( F @ N2 ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Module.mHom_add
% 0.15/0.37 thf(fact_132_Module_Osprod__r__distr,axiom,
% 0.15/0.37 ! [D: $tType,B: $tType,C: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: B,M4: A,N2: A] :
% 0.15/0.37 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 0.15/0.37 => ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 0.15/0.37 => ( ( member @ A @ N2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 0.15/0.37 => ( ( sprod @ A @ B @ C @ M3 @ A3 @ ( pop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ M4 @ N2 ) )
% 0.15/0.37 = ( pop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ ( sprod @ A @ B @ C @ M3 @ A3 @ M4 ) @ ( sprod @ A @ B @ C @ M3 @ A3 @ N2 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Module.sprod_r_distr
% 0.15/0.37 thf(fact_133_Module_Omodule__is__ag,axiom,
% 0.15/0.37 ! [D: $tType,C: $tType,B: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) )] :
% 0.15/0.37 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 0.15/0.37 => ( aGroup @ A @ ( module_ext @ A @ B @ C ) @ M3 ) ) ).
% 0.15/0.37
% 0.15/0.37 % Module.module_is_ag
% 0.15/0.37 thf(fact_134_Module_Osprod__l__distr,axiom,
% 0.15/0.37 ! [D: $tType,B: $tType,C: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: B,B3: B,M4: A] :
% 0.15/0.37 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 0.15/0.37 => ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ B @ B3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 0.15/0.37 => ( ( sprod @ A @ B @ C @ M3 @ ( pop @ B @ ( ring_ext @ B @ D ) @ R2 @ A3 @ B3 ) @ M4 )
% 0.15/0.37 = ( pop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ ( sprod @ A @ B @ C @ M3 @ A3 @ M4 ) @ ( sprod @ A @ B @ C @ M3 @ B3 @ M4 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Module.sprod_l_distr
% 0.15/0.37 thf(fact_135_Ring_OnsDistr,axiom,
% 0.15/0.37 ! [A: $tType,B: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,N2: nat,M4: nat] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( nscal @ A @ B @ R2 @ X2 @ N2 ) @ ( nscal @ A @ B @ R2 @ X2 @ M4 ) )
% 0.15/0.37 = ( nscal @ A @ B @ R2 @ X2 @ ( plus_plus @ nat @ N2 @ M4 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.nsDistr
% 0.15/0.37 thf(fact_136_Ring_Omul__closed__set__tOp__closed,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A,S2: A,T: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( mul_closed_set @ A @ B @ R2 @ S )
% 0.15/0.37 => ( ( member @ A @ S2 @ S )
% 0.15/0.37 => ( ( member @ A @ T @ S )
% 0.15/0.37 => ( member @ A @ ( tp @ A @ B @ R2 @ S2 @ T ) @ S ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.mul_closed_set_tOp_closed
% 0.15/0.37 thf(fact_137_Ring_Ozeroring__J__rad__empty,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( zeroring @ A @ B @ R2 )
% 0.15/0.37 => ( ( j_rad @ A @ B @ R2 )
% 0.15/0.37 = ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.zeroring_J_rad_empty
% 0.15/0.37 thf(fact_138_Module_Osc__Ring,axiom,
% 0.15/0.37 ! [C: $tType,A: $tType,D: $tType,B: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) )] :
% 0.15/0.37 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 0.15/0.37 => ( ring @ B @ D @ R2 ) ) ).
% 0.15/0.37
% 0.15/0.37 % Module.sc_Ring
% 0.15/0.37 thf(fact_139_Ring_Osubring__Ring,axiom,
% 0.15/0.37 ! [B: $tType,C: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) )] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( subring @ A @ B @ C @ R2 @ S )
% 0.15/0.37 => ( ring @ A @ C @ S ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.subring_Ring
% 0.15/0.37 thf(fact_140_Module_OmHom__eq,axiom,
% 0.15/0.37 ! [F2: $tType,D: $tType,C: $tType,B: $tType,E: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ),F: A > E,G: A > E] :
% 0.15/0.37 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 0.15/0.37 => ( ( module @ E @ B @ F2 @ D @ N @ R2 )
% 0.15/0.37 => ( ( member @ ( A > E ) @ F @ ( mHom @ B @ D @ A @ C @ E @ F2 @ R2 @ M3 @ N ) )
% 0.15/0.37 => ( ( member @ ( A > E ) @ G @ ( mHom @ B @ D @ A @ C @ E @ F2 @ R2 @ M3 @ N ) )
% 0.15/0.37 => ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 0.15/0.37 => ( ( F @ X )
% 0.15/0.37 = ( G @ X ) ) )
% 0.15/0.37 => ( F = G ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Module.mHom_eq
% 0.15/0.37 thf(fact_141_Module_OmHom__mem,axiom,
% 0.15/0.37 ! [C: $tType,D: $tType,A: $tType,F2: $tType,B: $tType,E: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ),F: A > E,M4: A] :
% 0.15/0.37 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 0.15/0.37 => ( ( module @ E @ B @ F2 @ D @ N @ R2 )
% 0.15/0.37 => ( ( member @ ( A > E ) @ F @ ( mHom @ B @ D @ A @ C @ E @ F2 @ R2 @ M3 @ N ) )
% 0.15/0.37 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 0.15/0.37 => ( member @ E @ ( F @ M4 ) @ ( carrier @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ) @ N ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Module.mHom_mem
% 0.15/0.37 thf(fact_142_Ring_Omem__subring__mem__ring,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ),X2: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( subring @ A @ B @ C @ R2 @ S )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) )
% 0.15/0.37 => ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.mem_subring_mem_ring
% 0.15/0.37 thf(fact_143_mul__closed__set__sub,axiom,
% 0.15/0.37 ! [S: set @ a] :
% 0.15/0.37 ( ( mul_closed_set @ a @ b @ r @ S )
% 0.15/0.37 => ( ord_less_eq @ ( set @ a ) @ S @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % mul_closed_set_sub
% 0.15/0.37 thf(fact_144_zeroring__no__maximal,axiom,
% 0.15/0.37 ( ( zeroring @ a @ b @ r )
% 0.15/0.37 => ~ ? [X_1: set @ a] : ( maximal_ideal @ a @ b @ r @ X_1 ) ) ).
% 0.15/0.37
% 0.15/0.37 % zeroring_no_maximal
% 0.15/0.37 thf(fact_145_id__maximal__Exist,axiom,
% 0.15/0.37 ( ~ ( zeroring @ a @ b @ r )
% 0.15/0.37 => ? [X_12: set @ a] : ( maximal_ideal @ a @ b @ r @ X_12 ) ) ).
% 0.15/0.37
% 0.15/0.37 % id_maximal_Exist
% 0.15/0.37 thf(fact_146_mop__closed,axiom,
% 0.15/0.37 ( member @ ( a > a ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r )
% 0.15/0.37 @ ( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 0.15/0.37 @ ^ [Uu: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % mop_closed
% 0.15/0.37 thf(fact_147_Sr__mOp__closed,axiom,
% 0.15/0.37 ! [S: set @ a,X2: a] :
% 0.15/0.37 ( ( sr2 @ a @ b @ r @ S )
% 0.15/0.37 => ( ( member @ a @ X2 @ S )
% 0.15/0.37 => ( member @ a @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ X2 ) @ S ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Sr_mOp_closed
% 0.15/0.37 thf(fact_148_ring__inv1,axiom,
% 0.15/0.37 ! [A3: a,B3: a] :
% 0.15/0.37 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) )
% 0.15/0.37 = ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A3 ) @ B3 ) )
% 0.15/0.37 & ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) )
% 0.15/0.37 = ( tp @ a @ b @ r @ A3 @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B3 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % ring_inv1
% 0.15/0.37 thf(fact_149_ring__inv1__1,axiom,
% 0.15/0.37 ! [A3: a,B3: a] :
% 0.15/0.37 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) )
% 0.15/0.37 = ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A3 ) @ B3 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % ring_inv1_1
% 0.15/0.37 thf(fact_150_ring__inv1__2,axiom,
% 0.15/0.37 ! [A3: a,B3: a] :
% 0.15/0.37 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) )
% 0.15/0.37 = ( tp @ a @ b @ r @ A3 @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B3 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % ring_inv1_2
% 0.15/0.37 thf(fact_151_ring__inv1__3,axiom,
% 0.15/0.37 ! [A3: a,B3: a] :
% 0.15/0.37 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( tp @ a @ b @ r @ A3 @ B3 )
% 0.15/0.37 = ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A3 ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B3 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % ring_inv1_3
% 0.15/0.37 thf(fact_152_ring__distrib4,axiom,
% 0.15/0.37 ! [A3: a,B3: a,X2: a,Y2: a] :
% 0.15/0.37 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ X2 @ Y2 ) ) )
% 0.15/0.37 = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B3 @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ Y2 ) ) ) @ ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ X2 ) ) @ Y2 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % ring_distrib4
% 0.15/0.37 thf(fact_153_Subring__minus__ring__minus,axiom,
% 0.15/0.37 ! [C: $tType,S: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ),X2: a] :
% 0.15/0.37 ( ( subring @ a @ b @ C @ r @ S )
% 0.15/0.37 => ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ) @ S ) )
% 0.15/0.37 => ( ( mop @ a @ ( ring_ext @ a @ C ) @ S @ X2 )
% 0.15/0.37 = ( mop @ a @ ( ring_ext @ a @ b ) @ r @ X2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Subring_minus_ring_minus
% 0.15/0.37 thf(fact_154_not__sub,axiom,
% 0.15/0.37 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.15/0.37 ( ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.15/0.37 => ? [A6: A] :
% 0.15/0.37 ( ( member @ A @ A6 @ A2 )
% 0.15/0.37 & ~ ( member @ A @ A6 @ B2 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % not_sub
% 0.15/0.37 thf(fact_155_eqsets__sub,axiom,
% 0.15/0.37 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.15/0.37 ( ( A2 = B2 )
% 0.15/0.37 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.15/0.37
% 0.15/0.37 % eqsets_sub
% 0.15/0.37 thf(fact_156_ge__convert,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( ord @ A )
% 0.15/0.37 => ! [A3: A,B3: A,C3: A] :
% 0.15/0.37 ( ( A3 = B3 )
% 0.15/0.37 => ( ( ord_less_eq @ A @ C3 @ A3 )
% 0.15/0.37 => ( ord_less_eq @ A @ C3 @ B3 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % ge_convert
% 0.15/0.37 thf(fact_157_le__convert,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( ord @ A )
% 0.15/0.37 => ! [A3: A,B3: A,C3: A] :
% 0.15/0.37 ( ( A3 = B3 )
% 0.15/0.37 => ( ( ord_less_eq @ A @ A3 @ C3 )
% 0.15/0.37 => ( ord_less_eq @ A @ B3 @ C3 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % le_convert
% 0.15/0.37 thf(fact_158_sub__which1,axiom,
% 0.15/0.37 ! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
% 0.15/0.37 ( ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.15/0.37 | ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ X2 @ A2 )
% 0.15/0.37 => ( ~ ( member @ A @ X2 @ B2 )
% 0.15/0.37 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % sub_which1
% 0.15/0.37 thf(fact_159_sub__which2,axiom,
% 0.15/0.37 ! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
% 0.15/0.37 ( ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.15/0.37 | ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) )
% 0.15/0.37 => ( ~ ( member @ A @ X2 @ A2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ B2 )
% 0.15/0.37 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % sub_which2
% 0.15/0.37 thf(fact_160_sets__not__eq,axiom,
% 0.15/0.37 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.15/0.37 ( ( A2 != B2 )
% 0.15/0.37 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.15/0.37 => ? [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ A2 )
% 0.15/0.37 & ~ ( member @ A @ X @ B2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % sets_not_eq
% 0.15/0.37 thf(fact_161_subset__self,axiom,
% 0.15/0.37 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 0.15/0.37
% 0.15/0.37 % subset_self
% 0.15/0.37 thf(fact_162_not__subseteq,axiom,
% 0.15/0.37 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.15/0.37 ( ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.15/0.37 => ? [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ A2 )
% 0.15/0.37 & ~ ( member @ A @ X @ B2 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % not_subseteq
% 0.15/0.37 thf(fact_163_proper__subset,axiom,
% 0.15/0.37 ! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
% 0.15/0.37 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.15/0.37 => ( ~ ( member @ A @ X2 @ A2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ B2 )
% 0.15/0.37 => ( A2 != B2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % proper_subset
% 0.15/0.37 thf(fact_164_conditional__subset,axiom,
% 0.15/0.37 ! [A: $tType,A2: set @ A,P: A > $o] :
% 0.15/0.37 ( ord_less_eq @ ( set @ A )
% 0.15/0.37 @ ( collect @ A
% 0.15/0.37 @ ^ [X3: A] :
% 0.15/0.37 ( ( member @ A @ X3 @ A2 )
% 0.15/0.37 & ( P @ X3 ) ) )
% 0.15/0.37 @ A2 ) ).
% 0.15/0.37
% 0.15/0.37 % conditional_subset
% 0.15/0.37 thf(fact_165_Nset__le,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( ord @ A )
% 0.15/0.37 => ! [X2: A,N2: A] :
% 0.15/0.37 ( ( member @ A @ X2
% 0.15/0.37 @ ( collect @ A
% 0.15/0.37 @ ^ [I4: A] : ( ord_less_eq @ A @ I4 @ N2 ) ) )
% 0.15/0.37 => ( ord_less_eq @ A @ X2 @ N2 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Nset_le
% 0.15/0.37 thf(fact_166_Pi__mono,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: set @ A,B2: A > ( set @ B ),C2: A > ( set @ B )] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ A2 )
% 0.15/0.37 => ( ord_less_eq @ ( set @ B ) @ ( B2 @ X ) @ ( C2 @ X ) ) )
% 0.15/0.37 => ( ord_less_eq @ ( set @ ( A > B ) ) @ ( pi @ A @ B @ A2 @ B2 ) @ ( pi @ A @ B @ A2 @ C2 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Pi_mono
% 0.15/0.37 thf(fact_167_Pi__anti__mono,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A7: set @ A,A2: set @ A,B2: A > ( set @ B )] :
% 0.15/0.37 ( ( ord_less_eq @ ( set @ A ) @ A7 @ A2 )
% 0.15/0.37 => ( ord_less_eq @ ( set @ ( A > B ) ) @ ( pi @ A @ B @ A2 @ B2 ) @ ( pi @ A @ B @ A7 @ B2 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Pi_anti_mono
% 0.15/0.37 thf(fact_168_extend__fun,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ B,B1: set @ B] :
% 0.15/0.37 ( ( member @ ( A > B ) @ F
% 0.15/0.37 @ ( pi @ A @ B @ A2
% 0.15/0.37 @ ^ [Uu: A] : B2 ) )
% 0.15/0.37 => ( ( ord_less_eq @ ( set @ B ) @ B2 @ B1 )
% 0.15/0.37 => ( member @ ( A > B ) @ F
% 0.15/0.37 @ ( pi @ A @ B @ A2
% 0.15/0.37 @ ^ [Uu: A] : B1 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % extend_fun
% 0.15/0.37 thf(fact_169_aGroup_Oag__mOp__closed,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( member @ A @ ( mop @ A @ B @ A2 @ X2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.ag_mOp_closed
% 0.15/0.37 thf(fact_170_aGroup_Oag__inv__inv,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( mop @ A @ B @ A2 @ ( mop @ A @ B @ A2 @ X2 ) )
% 0.15/0.37 = X2 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.ag_inv_inv
% 0.15/0.37 thf(fact_171_aGroup_Oag__inv__inj,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A,Y2: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( X2 != Y2 )
% 0.15/0.37 => ( ( mop @ A @ B @ A2 @ X2 )
% 0.15/0.37 != ( mop @ A @ B @ A2 @ Y2 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.ag_inv_inj
% 0.15/0.37 thf(fact_172_Ring_OSr__mOp__closed,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A,X2: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( sr2 @ A @ B @ R2 @ S )
% 0.15/0.37 => ( ( member @ A @ X2 @ S )
% 0.15/0.37 => ( member @ A @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ X2 ) @ S ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.Sr_mOp_closed
% 0.15/0.37 thf(fact_173_Subring__subset,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) )] :
% 0.15/0.37 ( ( subring @ A @ B @ C @ R2 @ S )
% 0.15/0.37 => ( ord_less_eq @ ( set @ A ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Subring_subset
% 0.15/0.37 thf(fact_174_Ring_Oring__inv1__3,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( tp @ A @ B @ R2 @ A3 @ B3 )
% 0.15/0.37 = ( tp @ A @ B @ R2 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 ) @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.ring_inv1_3
% 0.15/0.37 thf(fact_175_Ring_Oring__inv1__2,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) )
% 0.15/0.37 = ( tp @ A @ B @ R2 @ A3 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.ring_inv1_2
% 0.15/0.37 thf(fact_176_Ring_Oring__inv1__1,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) )
% 0.15/0.37 = ( tp @ A @ B @ R2 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 ) @ B3 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.ring_inv1_1
% 0.15/0.37 thf(fact_177_Ring_Oring__inv1,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) )
% 0.15/0.37 = ( tp @ A @ B @ R2 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 ) @ B3 ) )
% 0.15/0.37 & ( ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) )
% 0.15/0.37 = ( tp @ A @ B @ R2 @ A3 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.ring_inv1
% 0.15/0.37 thf(fact_178_Ring_Oid__maximal__Exist,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ~ ( zeroring @ A @ B @ R2 )
% 0.15/0.37 => ? [X_12: set @ A] : ( maximal_ideal @ A @ B @ R2 @ X_12 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.id_maximal_Exist
% 0.15/0.37 thf(fact_179_Ring_Ozeroring__no__maximal,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( zeroring @ A @ B @ R2 )
% 0.15/0.37 => ~ ? [X_1: set @ A] : ( maximal_ideal @ A @ B @ R2 @ X_1 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.zeroring_no_maximal
% 0.15/0.37 thf(fact_180_Ring_OSubring__minus__ring__minus,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ),X2: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( subring @ A @ B @ C @ R2 @ S )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) )
% 0.15/0.37 => ( ( mop @ A @ ( ring_ext @ A @ C ) @ S @ X2 )
% 0.15/0.37 = ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ X2 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.Subring_minus_ring_minus
% 0.15/0.37 thf(fact_181_Ring_Omop__closed,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( member @ ( A > A ) @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 )
% 0.15/0.37 @ ( pi @ A @ A @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 )
% 0.15/0.37 @ ^ [Uu: A] : ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.mop_closed
% 0.15/0.37 thf(fact_182_aGroup_OpOp__assoc__cancel,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ ( mop @ A @ B @ A2 @ B3 ) ) @ ( pop @ A @ B @ A2 @ B3 @ ( mop @ A @ B @ A2 @ C3 ) ) )
% 0.15/0.37 = ( pop @ A @ B @ A2 @ A3 @ ( mop @ A @ B @ A2 @ C3 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.pOp_assoc_cancel
% 0.15/0.37 thf(fact_183_aGroup_Oag__diff__minus,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( ( pop @ A @ B @ A2 @ A3 @ ( mop @ A @ B @ A2 @ B3 ) )
% 0.15/0.37 = C3 )
% 0.15/0.37 => ( ( pop @ A @ B @ A2 @ B3 @ ( mop @ A @ B @ A2 @ A3 ) )
% 0.15/0.37 = ( mop @ A @ B @ A2 @ C3 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.ag_diff_minus
% 0.15/0.37 thf(fact_184_aGroup_Oag__eq__sol2,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,X2: A,B3: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( ( pop @ A @ B @ A2 @ X2 @ A3 )
% 0.15/0.37 = B3 )
% 0.15/0.37 => ( X2
% 0.15/0.37 = ( pop @ A @ B @ A2 @ B3 @ ( mop @ A @ B @ A2 @ A3 ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.ag_eq_sol2
% 0.15/0.37 thf(fact_185_aGroup_Oag__eq__sol1,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,X2: A,B3: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( ( pop @ A @ B @ A2 @ A3 @ X2 )
% 0.15/0.37 = B3 )
% 0.15/0.37 => ( X2
% 0.15/0.37 = ( pop @ A @ B @ A2 @ ( mop @ A @ B @ A2 @ A3 ) @ B3 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.ag_eq_sol1
% 0.15/0.37 thf(fact_186_aGroup_Oag__p__inv,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A,Y2: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( mop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ X2 @ Y2 ) )
% 0.15/0.37 = ( pop @ A @ B @ A2 @ ( mop @ A @ B @ A2 @ X2 ) @ ( mop @ A @ B @ A2 @ Y2 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.ag_p_inv
% 0.15/0.37 thf(fact_187_Ring_Omul__closed__set__sub,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( mul_closed_set @ A @ B @ R2 @ S )
% 0.15/0.37 => ( ord_less_eq @ ( set @ A ) @ S @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.mul_closed_set_sub
% 0.15/0.37 thf(fact_188_mul__closed__set__def,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType] :
% 0.15/0.37 ( ( mul_closed_set @ A @ B )
% 0.15/0.37 = ( ^ [R3: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
% 0.15/0.37 ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.15/0.37 & ! [X3: A] :
% 0.15/0.37 ( ( member @ A @ X3 @ S3 )
% 0.15/0.37 => ! [Y3: A] :
% 0.15/0.37 ( ( member @ A @ Y3 @ S3 )
% 0.15/0.37 => ( member @ A @ ( tp @ A @ B @ R3 @ X3 @ Y3 ) @ S3 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % mul_closed_set_def
% 0.15/0.37 thf(fact_189_aGroup_Omop__closed,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B )] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( member @ ( A > A ) @ ( mop @ A @ B @ A2 )
% 0.15/0.37 @ ( pi @ A @ A @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 )
% 0.15/0.37 @ ^ [Uu: A] : ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.mop_closed
% 0.15/0.37 thf(fact_190_prod__mOp__mem__i,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B,I3: A] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 0.15/0.37 => ( ( member @ ( A > B ) @ X4 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ I3 @ I2 )
% 0.15/0.37 => ( ( prod_mOp @ A @ B @ C @ I2 @ A2 @ X4 @ I3 )
% 0.15/0.37 = ( mop @ B @ C @ ( A2 @ I3 ) @ ( X4 @ I3 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % prod_mOp_mem_i
% 0.15/0.37 thf(fact_191_Ring_Oring__distrib4,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A,X2: A,Y2: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) ) )
% 0.15/0.37 = ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ Y2 ) ) ) @ ( tp @ A @ B @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ X2 ) ) @ Y2 ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.ring_distrib4
% 0.15/0.37 thf(fact_192_nonunit__contained__maxid,axiom,
% 0.15/0.37 ! [A3: a] :
% 0.15/0.37 ( ~ ( zeroring @ a @ b @ r )
% 0.15/0.37 => ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ~ ( unit @ a @ b @ r @ A3 )
% 0.15/0.37 => ? [Mx: set @ a] :
% 0.15/0.37 ( ( maximal_ideal @ a @ b @ r @ Mx )
% 0.15/0.37 & ( member @ a @ A3 @ Mx ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % nonunit_contained_maxid
% 0.15/0.37 thf(fact_193_set__sum__mem,axiom,
% 0.15/0.37 ! [A3: a,I2: set @ a,B3: a,J2: set @ a] :
% 0.15/0.37 ( ( member @ a @ A3 @ I2 )
% 0.15/0.37 => ( ( member @ a @ B3 @ J2 )
% 0.15/0.37 => ( ( ord_less_eq @ ( set @ a ) @ I2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( ord_less_eq @ ( set @ a ) @ J2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ B3 ) @ ( aset_sum @ a @ ( ring_ext @ a @ b ) @ r @ I2 @ J2 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % set_sum_mem
% 0.15/0.37 thf(fact_194_sum__mult__pOp__closed,axiom,
% 0.15/0.37 ! [A2: set @ a,B2: set @ a,A3: a,B3: a] :
% 0.15/0.37 ( ( ord_less_eq @ ( set @ a ) @ A2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( ord_less_eq @ ( set @ a ) @ B2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ A3 @ ( sum_mult @ a @ b @ r @ A2 @ B2 ) )
% 0.15/0.37 => ( ( member @ a @ B3 @ ( sum_mult @ a @ b @ r @ A2 @ B2 ) )
% 0.15/0.37 => ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ B3 ) @ ( sum_mult @ a @ b @ r @ A2 @ B2 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % sum_mult_pOp_closed
% 0.15/0.37 thf(fact_195_sum__mult__subR,axiom,
% 0.15/0.37 ! [A2: set @ a,B2: set @ a] :
% 0.15/0.37 ( ( ord_less_eq @ ( set @ a ) @ A2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( ord_less_eq @ ( set @ a ) @ B2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ord_less_eq @ ( set @ a ) @ ( sum_mult @ a @ b @ r @ A2 @ B2 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % sum_mult_subR
% 0.15/0.37 thf(fact_196_times__mem__sum__mult,axiom,
% 0.15/0.37 ! [A2: set @ a,B2: set @ a,A3: a,B3: a] :
% 0.15/0.37 ( ( ord_less_eq @ ( set @ a ) @ A2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( ord_less_eq @ ( set @ a ) @ B2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( member @ a @ A3 @ A2 )
% 0.15/0.37 => ( ( member @ a @ B3 @ B2 )
% 0.15/0.37 => ( member @ a @ ( tp @ a @ b @ r @ A3 @ B3 ) @ ( sum_mult @ a @ b @ r @ A2 @ B2 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % times_mem_sum_mult
% 0.15/0.37 thf(fact_197_le__imp__add__int,axiom,
% 0.15/0.37 ! [I3: nat,J: nat] :
% 0.15/0.37 ( ( ord_less_eq @ nat @ I3 @ J )
% 0.15/0.37 => ? [K2: nat] :
% 0.15/0.37 ( J
% 0.15/0.37 = ( plus_plus @ nat @ I3 @ K2 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % le_imp_add_int
% 0.15/0.37 thf(fact_198_Module_OmHom__inv,axiom,
% 0.15/0.37 ! [D: $tType,C: $tType,B: $tType,F2: $tType,E: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ),M4: A,F: A > E] :
% 0.15/0.37 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 0.15/0.37 => ( ( module @ E @ B @ F2 @ D @ N @ R2 )
% 0.15/0.37 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 0.15/0.37 => ( ( member @ ( A > E ) @ F @ ( mHom @ B @ D @ A @ C @ E @ F2 @ R2 @ M3 @ N ) )
% 0.15/0.37 => ( ( F @ ( mop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ M4 ) )
% 0.15/0.37 = ( mop @ E @ ( module_ext @ E @ B @ F2 ) @ N @ ( F @ M4 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Module.mHom_inv
% 0.15/0.37 thf(fact_199_dsumag__sub__prodag,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 0.15/0.37 ( ! [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ I2 )
% 0.15/0.37 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 0.15/0.37 => ( ord_less_eq @ ( set @ ( A > B ) ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % dsumag_sub_prodag
% 0.15/0.37 thf(fact_200_Module_Osc__minus__am,axiom,
% 0.15/0.37 ! [D: $tType,B: $tType,C: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: B,M4: A] :
% 0.15/0.37 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 0.15/0.37 => ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 0.15/0.37 => ( ( mop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ ( sprod @ A @ B @ C @ M3 @ A3 @ M4 ) )
% 0.15/0.37 = ( sprod @ A @ B @ C @ M3 @ A3 @ ( mop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ M4 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Module.sc_minus_am
% 0.15/0.37 thf(fact_201_Ring_Osum__mult__subR,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A2: set @ A,B2: set @ A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ord_less_eq @ ( set @ A ) @ ( sum_mult @ A @ B @ R2 @ A2 @ B2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.sum_mult_subR
% 0.15/0.37 thf(fact_202_Module_Osc__minus__am1,axiom,
% 0.15/0.37 ! [B: $tType,D: $tType,C: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: B,M4: A] :
% 0.15/0.37 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 0.15/0.37 => ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 0.15/0.37 => ( ( mop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ ( sprod @ A @ B @ C @ M3 @ A3 @ M4 ) )
% 0.15/0.37 = ( sprod @ A @ B @ C @ M3 @ ( mop @ B @ ( ring_ext @ B @ D ) @ R2 @ A3 ) @ M4 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Module.sc_minus_am1
% 0.15/0.37 thf(fact_203_Ring_Otimes__mem__sum__mult,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A2: set @ A,B2: set @ A,A3: A,B3: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ A3 @ A2 )
% 0.15/0.37 => ( ( member @ A @ B3 @ B2 )
% 0.15/0.37 => ( member @ A @ ( tp @ A @ B @ R2 @ A3 @ B3 ) @ ( sum_mult @ A @ B @ R2 @ A2 @ B2 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.times_mem_sum_mult
% 0.15/0.37 thf(fact_204_Ring_Osum__mult__pOp__closed,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A2: set @ A,B2: set @ A,A3: A,B3: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( sum_mult @ A @ B @ R2 @ A2 @ B2 ) )
% 0.15/0.37 => ( ( member @ A @ B3 @ ( sum_mult @ A @ B @ R2 @ A2 @ B2 ) )
% 0.15/0.37 => ( member @ A @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ B3 ) @ ( sum_mult @ A @ B @ R2 @ A2 @ B2 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.sum_mult_pOp_closed
% 0.15/0.37 thf(fact_205_Ring_Oset__sum__mem,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,I2: set @ A,B3: A,J2: set @ A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ I2 )
% 0.15/0.37 => ( ( member @ A @ B3 @ J2 )
% 0.15/0.37 => ( ( ord_less_eq @ ( set @ A ) @ I2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( ord_less_eq @ ( set @ A ) @ J2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( member @ A @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ B3 ) @ ( aset_sum @ A @ ( ring_ext @ A @ B ) @ R2 @ I2 @ J2 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.set_sum_mem
% 0.15/0.37 thf(fact_206_Ring_Ononunit__contained__maxid,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ~ ( zeroring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ~ ( unit @ A @ B @ R2 @ A3 )
% 0.15/0.37 => ? [Mx: set @ A] :
% 0.15/0.37 ( ( maximal_ideal @ A @ B @ R2 @ Mx )
% 0.15/0.37 & ( member @ A @ A3 @ Mx ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.nonunit_contained_maxid
% 0.15/0.37 thf(fact_207_aGroup_Omem__set__sum,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),H: set @ A,K3: set @ A,X2: A] :
% 0.15/0.37 ( ( aGroup @ A @ B @ A2 )
% 0.15/0.37 => ( ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( ord_less_eq @ ( set @ A ) @ K3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( aset_sum @ A @ B @ A2 @ H @ K3 ) )
% 0.15/0.37 => ? [X: A] :
% 0.15/0.37 ( ( member @ A @ X @ H )
% 0.15/0.37 & ? [Xa2: A] :
% 0.15/0.37 ( ( member @ A @ Xa2 @ K3 )
% 0.15/0.37 & ( X2
% 0.15/0.37 = ( pop @ A @ B @ A2 @ X @ Xa2 ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % aGroup.mem_set_sum
% 0.15/0.37 thf(fact_208_J__rad__unit,axiom,
% 0.15/0.37 ! [X2: a] :
% 0.15/0.37 ( ~ ( zeroring @ a @ b @ r )
% 0.15/0.37 => ( ( member @ a @ X2 @ ( j_rad @ a @ b @ r ) )
% 0.15/0.37 => ! [Y4: a] :
% 0.15/0.37 ( ( member @ a @ Y4 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( unit @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) @ ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ X2 ) @ Y4 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % J_rad_unit
% 0.15/0.37 thf(fact_209_add__le__cancel__right,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( ordere236663937imp_le @ A )
% 0.15/0.37 => ! [A3: A,C3: A,B3: A] :
% 0.15/0.37 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
% 0.15/0.37 = ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % add_le_cancel_right
% 0.15/0.37 thf(fact_210_un__closed,axiom,
% 0.15/0.37 member @ a @ ( un @ a @ b @ r ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.15/0.37
% 0.15/0.37 % un_closed
% 0.15/0.37 thf(fact_211_maximal__ideal__proper,axiom,
% 0.15/0.37 ! [Mx2: set @ a] :
% 0.15/0.37 ( ( maximal_ideal @ a @ b @ r @ Mx2 )
% 0.15/0.37 => ~ ( member @ a @ ( un @ a @ b @ r ) @ Mx2 ) ) ).
% 0.15/0.37
% 0.15/0.37 % maximal_ideal_proper
% 0.15/0.37 thf(fact_212_Sr__one,axiom,
% 0.15/0.37 ! [S: set @ a] :
% 0.15/0.37 ( ( sr2 @ a @ b @ r @ S )
% 0.15/0.37 => ( member @ a @ ( un @ a @ b @ r ) @ S ) ) ).
% 0.15/0.37
% 0.15/0.37 % Sr_one
% 0.15/0.37 thf(fact_213_add__right__cancel,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( cancel_semigroup_add @ A )
% 0.15/0.37 => ! [B3: A,A3: A,C3: A] :
% 0.15/0.37 ( ( ( plus_plus @ A @ B3 @ A3 )
% 0.15/0.37 = ( plus_plus @ A @ C3 @ A3 ) )
% 0.15/0.37 = ( B3 = C3 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % add_right_cancel
% 0.15/0.37 thf(fact_214_add__left__cancel,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( cancel_semigroup_add @ A )
% 0.15/0.37 => ! [A3: A,B3: A,C3: A] :
% 0.15/0.37 ( ( ( plus_plus @ A @ A3 @ B3 )
% 0.15/0.37 = ( plus_plus @ A @ A3 @ C3 ) )
% 0.15/0.37 = ( B3 = C3 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % add_left_cancel
% 0.15/0.37 thf(fact_215_rg__l__unit,axiom,
% 0.15/0.37 ! [A3: a] :
% 0.15/0.37 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( tp @ a @ b @ r @ ( un @ a @ b @ r ) @ A3 )
% 0.15/0.37 = A3 ) ) ).
% 0.15/0.37
% 0.15/0.37 % rg_l_unit
% 0.15/0.37 thf(fact_216_ring__l__one,axiom,
% 0.15/0.37 ! [X2: a] :
% 0.15/0.37 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( tp @ a @ b @ r @ ( un @ a @ b @ r ) @ X2 )
% 0.15/0.37 = X2 ) ) ).
% 0.15/0.37
% 0.15/0.37 % ring_l_one
% 0.15/0.37 thf(fact_217_ring__r__one,axiom,
% 0.15/0.37 ! [X2: a] :
% 0.15/0.37 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( tp @ a @ b @ r @ X2 @ ( un @ a @ b @ r ) )
% 0.15/0.37 = X2 ) ) ).
% 0.15/0.37
% 0.15/0.37 % ring_r_one
% 0.15/0.37 thf(fact_218_Subring__one__ring__one,axiom,
% 0.15/0.37 ! [C: $tType,S: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) )] :
% 0.15/0.37 ( ( subring @ a @ b @ C @ r @ S )
% 0.15/0.37 => ( ( un @ a @ C @ S )
% 0.15/0.37 = ( un @ a @ b @ r ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Subring_one_ring_one
% 0.15/0.37 thf(fact_219_ring__times__minusr,axiom,
% 0.15/0.37 ! [A3: a] :
% 0.15/0.37 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A3 )
% 0.15/0.37 = ( tp @ a @ b @ r @ A3 @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % ring_times_minusr
% 0.15/0.37 thf(fact_220_ring__times__minusl,axiom,
% 0.15/0.37 ! [A3: a] :
% 0.15/0.37 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.15/0.37 => ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A3 )
% 0.15/0.37 = ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) ) @ A3 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % ring_times_minusl
% 0.15/0.37 thf(fact_221_add__le__cancel__left,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( ordere236663937imp_le @ A )
% 0.15/0.37 => ! [C3: A,A3: A,B3: A] :
% 0.15/0.37 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) )
% 0.15/0.37 = ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % add_le_cancel_left
% 0.15/0.37 thf(fact_222_localring__unit,axiom,
% 0.15/0.37 ! [Mx2: set @ a] :
% 0.15/0.37 ( ~ ( zeroring @ a @ b @ r )
% 0.15/0.37 => ( ( maximal_ideal @ a @ b @ r @ Mx2 )
% 0.15/0.37 => ( ! [X: a] :
% 0.15/0.37 ( ( member @ a @ X @ Mx2 )
% 0.15/0.37 => ( unit @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( un @ a @ b @ r ) ) ) )
% 0.15/0.37 => ( local_ring @ a @ b @ r ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % localring_unit
% 0.15/0.37 thf(fact_223_primary__ideal__proper1,axiom,
% 0.15/0.37 ! [Q2: set @ a] :
% 0.15/0.37 ( ( primary_ideal @ a @ b @ r @ Q2 )
% 0.15/0.37 => ~ ( member @ a @ ( un @ a @ b @ r ) @ Q2 ) ) ).
% 0.15/0.37
% 0.15/0.37 % primary_ideal_proper1
% 0.15/0.37 thf(fact_224_self__le,axiom,
% 0.15/0.37 ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% 0.15/0.37
% 0.15/0.37 % self_le
% 0.15/0.37 thf(fact_225_nat__eq__le,axiom,
% 0.15/0.37 ! [M4: nat,N2: nat] :
% 0.15/0.37 ( ( M4 = N2 )
% 0.15/0.37 => ( ord_less_eq @ nat @ M4 @ N2 ) ) ).
% 0.15/0.37
% 0.15/0.37 % nat_eq_le
% 0.15/0.37 thf(fact_226_n__in__Nsetn,axiom,
% 0.15/0.37 ! [N2: nat] :
% 0.15/0.37 ( member @ nat @ N2
% 0.15/0.37 @ ( collect @ nat
% 0.15/0.37 @ ^ [I4: nat] : ( ord_less_eq @ nat @ I4 @ N2 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % n_in_Nsetn
% 0.15/0.37 thf(fact_227_mem__of__Nset,axiom,
% 0.15/0.37 ! [X2: nat,N2: nat] :
% 0.15/0.37 ( ( ord_less_eq @ nat @ X2 @ N2 )
% 0.15/0.37 => ( member @ nat @ X2
% 0.15/0.37 @ ( collect @ nat
% 0.15/0.37 @ ^ [I4: nat] : ( ord_less_eq @ nat @ I4 @ N2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % mem_of_Nset
% 0.15/0.37 thf(fact_228_Ring_Oun__closed,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( member @ A @ ( un @ A @ B @ R2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.un_closed
% 0.15/0.37 thf(fact_229_Ring_Omaximal__ideal__proper,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),Mx2: set @ A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( maximal_ideal @ A @ B @ R2 @ Mx2 )
% 0.15/0.37 => ~ ( member @ A @ ( un @ A @ B @ R2 ) @ Mx2 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.maximal_ideal_proper
% 0.15/0.37 thf(fact_230_Ring_OSubring__one__ring__one,axiom,
% 0.15/0.37 ! [C: $tType,B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) )] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( subring @ A @ B @ C @ R2 @ S )
% 0.15/0.37 => ( ( un @ A @ C @ S )
% 0.15/0.37 = ( un @ A @ B @ R2 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.Subring_one_ring_one
% 0.15/0.37 thf(fact_231_Ring_OSr__one,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( sr2 @ A @ B @ R2 @ S )
% 0.15/0.37 => ( member @ A @ ( un @ A @ B @ R2 ) @ S ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.Sr_one
% 0.15/0.37 thf(fact_232_Ring_Oring__r__one,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( tp @ A @ B @ R2 @ X2 @ ( un @ A @ B @ R2 ) )
% 0.15/0.37 = X2 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.ring_r_one
% 0.15/0.37 thf(fact_233_Ring_Oring__l__one,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( tp @ A @ B @ R2 @ ( un @ A @ B @ R2 ) @ X2 )
% 0.15/0.37 = X2 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.ring_l_one
% 0.15/0.37 thf(fact_234_Ring_Org__l__unit,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( tp @ A @ B @ R2 @ ( un @ A @ B @ R2 ) @ A3 )
% 0.15/0.37 = A3 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.rg_l_unit
% 0.15/0.37 thf(fact_235_Module_Osprod__one,axiom,
% 0.15/0.37 ! [C: $tType,D: $tType,B: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),M4: A] :
% 0.15/0.37 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 0.15/0.37 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 0.15/0.37 => ( ( sprod @ A @ B @ C @ M3 @ ( un @ B @ D @ R2 ) @ M4 )
% 0.15/0.37 = M4 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Module.sprod_one
% 0.15/0.37 thf(fact_236_Unit__def,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType] :
% 0.15/0.37 ( ( unit @ A @ B )
% 0.15/0.37 = ( ^ [R3: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A5: A] :
% 0.15/0.37 ( ( member @ A @ A5 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.15/0.37 & ? [X3: A] :
% 0.15/0.37 ( ( member @ A @ X3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.15/0.37 & ( ( tp @ A @ B @ R3 @ A5 @ X3 )
% 0.15/0.37 = ( un @ A @ B @ R3 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Unit_def
% 0.15/0.37 thf(fact_237_add__right__imp__eq,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( cancel_semigroup_add @ A )
% 0.15/0.37 => ! [B3: A,A3: A,C3: A] :
% 0.15/0.37 ( ( ( plus_plus @ A @ B3 @ A3 )
% 0.15/0.37 = ( plus_plus @ A @ C3 @ A3 ) )
% 0.15/0.37 => ( B3 = C3 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % add_right_imp_eq
% 0.15/0.37 thf(fact_238_add__left__imp__eq,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( cancel_semigroup_add @ A )
% 0.15/0.37 => ! [A3: A,B3: A,C3: A] :
% 0.15/0.37 ( ( ( plus_plus @ A @ A3 @ B3 )
% 0.15/0.37 = ( plus_plus @ A @ A3 @ C3 ) )
% 0.15/0.37 => ( B3 = C3 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % add_left_imp_eq
% 0.15/0.37 thf(fact_239_add_Oleft__commute,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( ab_semigroup_add @ A )
% 0.15/0.37 => ! [B3: A,A3: A,C3: A] :
% 0.15/0.37 ( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A3 @ C3 ) )
% 0.15/0.37 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % add.left_commute
% 0.15/0.37 thf(fact_240_add_Ocommute,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( ab_semigroup_add @ A )
% 0.15/0.37 => ( ( plus_plus @ A )
% 0.15/0.37 = ( ^ [A5: A,B4: A] : ( plus_plus @ A @ B4 @ A5 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % add.commute
% 0.15/0.37 thf(fact_241_add_Oright__cancel,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( group_add @ A )
% 0.15/0.37 => ! [B3: A,A3: A,C3: A] :
% 0.15/0.37 ( ( ( plus_plus @ A @ B3 @ A3 )
% 0.15/0.37 = ( plus_plus @ A @ C3 @ A3 ) )
% 0.15/0.37 = ( B3 = C3 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % add.right_cancel
% 0.15/0.37 thf(fact_242_add_Oleft__cancel,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( group_add @ A )
% 0.15/0.37 => ! [A3: A,B3: A,C3: A] :
% 0.15/0.37 ( ( ( plus_plus @ A @ A3 @ B3 )
% 0.15/0.37 = ( plus_plus @ A @ A3 @ C3 ) )
% 0.15/0.37 = ( B3 = C3 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % add.left_cancel
% 0.15/0.37 thf(fact_243_add_Oassoc,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( semigroup_add @ A )
% 0.15/0.37 => ! [A3: A,B3: A,C3: A] :
% 0.15/0.37 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
% 0.15/0.37 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % add.assoc
% 0.15/0.37 thf(fact_244_group__cancel_Oadd2,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( comm_monoid_add @ A )
% 0.15/0.37 => ! [B2: A,K4: A,B3: A,A3: A] :
% 0.15/0.37 ( ( B2
% 0.15/0.37 = ( plus_plus @ A @ K4 @ B3 ) )
% 0.15/0.37 => ( ( plus_plus @ A @ A3 @ B2 )
% 0.15/0.37 = ( plus_plus @ A @ K4 @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % group_cancel.add2
% 0.15/0.37 thf(fact_245_group__cancel_Oadd1,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( comm_monoid_add @ A )
% 0.15/0.37 => ! [A2: A,K4: A,A3: A,B3: A] :
% 0.15/0.37 ( ( A2
% 0.15/0.37 = ( plus_plus @ A @ K4 @ A3 ) )
% 0.15/0.37 => ( ( plus_plus @ A @ A2 @ B3 )
% 0.15/0.37 = ( plus_plus @ A @ K4 @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % group_cancel.add1
% 0.15/0.37 thf(fact_246_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( ordere779506340up_add @ A )
% 0.15/0.37 => ! [I3: A,J: A,K4: A,L: A] :
% 0.15/0.37 ( ( ( I3 = J )
% 0.15/0.37 & ( K4 = L ) )
% 0.15/0.37 => ( ( plus_plus @ A @ I3 @ K4 )
% 0.15/0.37 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % add_mono_thms_linordered_semiring(4)
% 0.15/0.37 thf(fact_247_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( ab_semigroup_add @ A )
% 0.15/0.37 => ! [A3: A,B3: A,C3: A] :
% 0.15/0.37 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
% 0.15/0.37 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % ab_semigroup_add_class.add_ac(1)
% 0.15/0.37 thf(fact_248_Ring_Oring__times__minusl,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 )
% 0.15/0.37 = ( tp @ A @ B @ R2 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( un @ A @ B @ R2 ) ) @ A3 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.ring_times_minusl
% 0.15/0.37 thf(fact_249_Ring_Oring__times__minusr,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A] :
% 0.15/0.37 ( ( ring @ A @ B @ R2 )
% 0.15/0.37 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 0.15/0.37 => ( ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 )
% 0.15/0.37 = ( tp @ A @ B @ R2 @ A3 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( un @ A @ B @ R2 ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % Ring.ring_times_minusr
% 0.15/0.37 thf(fact_250_sr__def,axiom,
% 0.15/0.37 ! [B: $tType,A: $tType] :
% 0.15/0.37 ( ( sr2 @ A @ B )
% 0.15/0.37 = ( ^ [R3: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
% 0.15/0.37 ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.15/0.37 & ( member @ A @ ( un @ A @ B @ R3 ) @ S3 )
% 0.15/0.37 & ! [X3: A] :
% 0.15/0.37 ( ( member @ A @ X3 @ S3 )
% 0.15/0.37 => ! [Y3: A] :
% 0.15/0.37 ( ( member @ A @ Y3 @ S3 )
% 0.15/0.37 => ( ( member @ A @ ( pop @ A @ ( ring_ext @ A @ B ) @ R3 @ X3 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R3 @ Y3 ) ) @ S3 )
% 0.15/0.37 & ( member @ A @ ( tp @ A @ B @ R3 @ X3 @ Y3 ) @ S3 ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % sr_def
% 0.15/0.37 thf(fact_251_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( ordere779506340up_add @ A )
% 0.15/0.37 => ! [I3: A,J: A,K4: A,L: A] :
% 0.15/0.37 ( ( ( ord_less_eq @ A @ I3 @ J )
% 0.15/0.37 & ( K4 = L ) )
% 0.15/0.37 => ( ord_less_eq @ A @ ( plus_plus @ A @ I3 @ K4 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % add_mono_thms_linordered_semiring(3)
% 0.15/0.37 thf(fact_252_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( ordere779506340up_add @ A )
% 0.15/0.37 => ! [I3: A,J: A,K4: A,L: A] :
% 0.15/0.37 ( ( ( I3 = J )
% 0.15/0.37 & ( ord_less_eq @ A @ K4 @ L ) )
% 0.15/0.37 => ( ord_less_eq @ A @ ( plus_plus @ A @ I3 @ K4 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % add_mono_thms_linordered_semiring(2)
% 0.15/0.37 thf(fact_253_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( ordere779506340up_add @ A )
% 0.15/0.37 => ! [I3: A,J: A,K4: A,L: A] :
% 0.15/0.37 ( ( ( ord_less_eq @ A @ I3 @ J )
% 0.15/0.37 & ( ord_less_eq @ A @ K4 @ L ) )
% 0.15/0.37 => ( ord_less_eq @ A @ ( plus_plus @ A @ I3 @ K4 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % add_mono_thms_linordered_semiring(1)
% 0.15/0.37 thf(fact_254_add__mono,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( ordere779506340up_add @ A )
% 0.15/0.37 => ! [A3: A,B3: A,C3: A,D2: A] :
% 0.15/0.37 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.15/0.37 => ( ( ord_less_eq @ A @ C3 @ D2 )
% 0.15/0.37 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % add_mono
% 0.15/0.37 thf(fact_255_add__left__mono,axiom,
% 0.15/0.37 ! [A: $tType] :
% 0.15/0.37 ( ( ordere779506340up_add @ A )
% 0.15/0.37 => ! [A3: A,B3: A,C3: A] :
% 0.15/0.37 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.15/0.37 => ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 % add_left_mono
% 0.15/0.37
% 0.15/0.37 % Type constructors (12)
% 0.15/0.37 thf(tcon_fun___Orderings_Oord,axiom,
% 0.15/0.37 ! [A8: $tType,A9: $tType] :
% 0.15/0.37 ( ( ord @ A9 )
% 0.15/0.37 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.15/0.37 ordere236663937imp_le @ nat ).
% 0.15/0.37
% 0.15/0.37 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 0.15/0.37 ordere779506340up_add @ nat ).
% 0.15/0.37
% 0.15/0.37 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 0.15/0.37 cancel_semigroup_add @ nat ).
% 0.15/0.37
% 0.15/0.37 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 0.15/0.37 ab_semigroup_add @ nat ).
% 0.15/0.37
% 0.15/0.37 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 0.15/0.37 comm_monoid_add @ nat ).
% 0.15/0.37
% 0.15/0.37 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 0.15/0.37 semigroup_add @ nat ).
% 0.15/0.37
% 0.15/0.37 thf(tcon_Nat_Onat___Orderings_Oord_1,axiom,
% 0.15/0.37 ord @ nat ).
% 0.15/0.37
% 0.15/0.37 thf(tcon_Nat_Onat___Groups_Oplus,axiom,
% 0.15/0.37 plus @ nat ).
% 0.15/0.37
% 0.15/0.37 thf(tcon_Set_Oset___Orderings_Oord_2,axiom,
% 0.15/0.37 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(tcon_HOL_Obool___Orderings_Oord_3,axiom,
% 0.15/0.37 ord @ $o ).
% 0.15/0.37
% 0.15/0.37 thf(tcon_Product__Type_Ounit___Orderings_Oord_4,axiom,
% 0.15/0.37 ord @ product_unit ).
% 0.15/0.37
% 0.15/0.37 % Conjectures (3)
% 0.15/0.37 thf(conj_0,hypothesis,
% 0.15/0.37 ! [X5: c] :
% 0.15/0.37 ( ( member @ c @ X5 @ i )
% 0.15/0.37 => ( module @ d @ a @ e @ b @ ( m @ X5 ) @ r ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(conj_1,hypothesis,
% 0.15/0.37 ( ( member @ ( ( c > d ) > ( c > d ) > c > d ) @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
% 0.15/0.37 @ ( pi @ ( c > d ) @ ( ( c > d ) > c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
% 0.15/0.37 @ ^ [Uu: c > d] :
% 0.15/0.37 ( pi @ ( c > d ) @ ( c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
% 0.15/0.37 @ ^ [Uv: c > d] : ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) ) )
% 0.15/0.37 & ! [A10: c > d] :
% 0.15/0.37 ( ( member @ ( c > d ) @ A10 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
% 0.15/0.37 => ! [B5: c > d] :
% 0.15/0.37 ( ( member @ ( c > d ) @ B5 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
% 0.15/0.37 => ! [C4: c > d] :
% 0.15/0.37 ( ( member @ ( c > d ) @ C4 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
% 0.15/0.37 => ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A10 @ B5 ) @ C4 )
% 0.15/0.37 = ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A10 @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ B5 @ C4 ) ) ) ) ) )
% 0.15/0.37 & ! [A10: c > d] :
% 0.15/0.37 ( ( member @ ( c > d ) @ A10 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
% 0.15/0.37 => ! [B5: c > d] :
% 0.15/0.37 ( ( member @ ( c > d ) @ B5 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
% 0.15/0.37 => ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A10 @ B5 )
% 0.15/0.37 = ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ B5 @ A10 ) ) ) )
% 0.15/0.37 & ( member @ ( ( c > d ) > c > d ) @ ( prod_mOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
% 0.15/0.37 @ ( pi @ ( c > d ) @ ( c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
% 0.15/0.37 @ ^ [Uu: c > d] : ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) )
% 0.15/0.37 & ! [A10: c > d] :
% 0.15/0.37 ( ( member @ ( c > d ) @ A10 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
% 0.15/0.37 => ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_mOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A10 ) @ A10 )
% 0.15/0.37 = ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) )
% 0.15/0.37 & ( member @ ( c > d ) @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
% 0.15/0.37 & ! [A10: c > d] :
% 0.15/0.37 ( ( member @ ( c > d ) @ A10 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
% 0.15/0.37 => ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) @ A10 )
% 0.15/0.37 = A10 ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(conj_2,conjecture,
% 0.15/0.37 ! [A6: c > d] :
% 0.15/0.37 ( ~ ( member @ ( c > d ) @ A6 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
% 0.15/0.37 | ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A6 @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
% 0.15/0.37 = A6 ) ) ).
% 0.15/0.37
% 0.15/0.37 %------------------------------------------------------------------------------
% 0.15/0.37 ------------------------
% 2.88/1.45 ---- Embedded in THF ---
% 2.88/1.45 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.88/1.45 %%% Generated on Tue Feb 24 08:22:27 EST 2026
% 2.88/1.45 %%% using '$$dhol' embedding, version 1.3.0.
% 2.88/1.45 %%% Logic specification used:
% 2.88/1.45 %%% thf(spec, logic, $$dhol).
% 2.88/1.45
% 2.88/1.45 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.uhnk0bOYnS/DTF2DTF31945.p
% 2.88/1.45 thf(ty_t_Algebra1_Ocarrier_Ocarrier__ext, type, carrier_ext: ($tType > ($tType > $tType))).
% 2.88/1.45 thf(ty_t_Algebra7_OModule_OModule__ext, type, module_ext: ($tType > ($tType > ($tType > $tType)))).
% 2.88/1.45 thf(ty_t_Algebra4_OaGroup_OaGroup__ext, type, aGroup_ext: ($tType > ($tType > $tType))).
% 2.88/1.45 thf(ty_t_Algebra4_ORing_ORing__ext, type, ring_ext: ($tType > ($tType > $tType))).
% 2.88/1.45 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 2.88/1.45 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.88/1.45 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.88/1.45 thf(ty_tf_e, type, e: $tType).
% 2.88/1.45 thf(ty_tf_d, type, d: $tType).
% 2.88/1.45 thf(ty_tf_c, type, c: $tType).
% 2.88/1.45 thf(ty_tf_b, type, b: $tType).
% 2.88/1.45 thf(ty_tf_a, type, a: $tType).
% 2.88/1.45 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.88/1.45 thf(sy_cl_Groups_Oplus, type, plus: (!> [A:$tType]: ($o))).
% 2.88/1.45 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.88/1.45 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.88/1.45 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: ($o))).
% 2.88/1.45 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: ($o))).
% 2.88/1.45 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: ($o))).
% 2.88/1.45 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: ($o))).
% 2.88/1.45 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: ($o))).
% 2.88/1.45 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere236663937imp_le: (!> [A:$tType]: ($o))).
% 2.88/1.45 thf(sy_c_Algebra1_Ocarrier_Ocarrier, type, carrier: (!> [A:$tType,Z:$tType]: ((((carrier_ext @ A) @ Z) > (set @ A))))).
% 2.88/1.45 thf(sy_c_Algebra4_OJ__rad, type, j_rad: (!> [A:$tType,B:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) > (set @ A))))).
% 2.88/1.45 thf(sy_c_Algebra4_ORing, type, ring: (!> [A:$tType,B:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) > $o)))).
% 2.88/1.45 thf(sy_c_Algebra4_ORing_Otp, type, tp: (!> [A:$tType,Z:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ Z))) > (A > (A > A)))))).
% 2.88/1.45 thf(sy_c_Algebra4_ORing_Oun, type, un: (!> [A:$tType,Z:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ Z))) > A)))).
% 2.88/1.45 thf(sy_c_Algebra4_OSr, type, sr: (!> [A:$tType,B:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) > ((set @ A) > ((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B)))))))).
% 2.88/1.45 thf(sy_c_Algebra4_OSubring, type, subring: (!> [A:$tType,M:$tType,M1:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ M))) > (((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ M1))) > $o))))).
% 2.88/1.45 thf(sy_c_Algebra4_OUnit, type, unit: (!> [A:$tType,B:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) > (A > $o))))).
% 2.88/1.45 thf(sy_c_Algebra4_OaGroup, type, aGroup: (!> [A:$tType,B:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)) > $o)))).
% 2.88/1.45 thf(sy_c_Algebra4_OaGroup_Omop, type, mop: (!> [A:$tType,Z:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ Z)) > (A > A))))).
% 2.88/1.45 thf(sy_c_Algebra4_OaGroup_Opop, type, pop: (!> [A:$tType,Z:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ Z)) > (A > (A > A)))))).
% 2.88/1.45 thf(sy_c_Algebra4_Oaset__sum, type, aset_sum: (!> [A:$tType,B:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)) > ((set @ A) > ((set @ A) > (set @ A))))))).
% 2.88/1.45 thf(sy_c_Algebra4_Ocarr__dsumag, type, carr_dsumag: (!> [I:$tType,A:$tType,More:$tType]: (((set @ I) > ((I > ((carrier_ext @ A) @ ((aGroup_ext @ A) @ More))) > (set @ (I > A))))))).
% 2.88/1.45 thf(sy_c_Algebra4_Ocarr__prodag, type, carr_prodag: (!> [I:$tType,A:$tType,More:$tType]: (((set @ I) > ((I > ((carrier_ext @ A) @ ((aGroup_ext @ A) @ More))) > (set @ (I > A))))))).
% 2.88/1.45 thf(sy_c_Algebra4_Olocal__ring, type, local_ring: (!> [A:$tType,B:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) > $o)))).
% 2.88/1.45 thf(sy_c_Algebra4_Omaximal__ideal, type, maximal_ideal: (!> [A:$tType,B:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) > ((set @ A) > $o))))).
% 2.88/1.45 thf(sy_c_Algebra4_Omul__closed__set, type, mul_closed_set: (!> [A:$tType,B:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) > ((set @ A) > $o))))).
% 2.88/1.45 thf(sy_c_Algebra4_Onscal, type, nscal: (!> [A:$tType,More:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ More))) > (A > (nat > A)))))).
% 2.88/1.45 thf(sy_c_Algebra4_Oprimary__ideal, type, primary_ideal: (!> [A:$tType,B:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) > ((set @ A) > $o))))).
% 2.88/1.45 thf(sy_c_Algebra4_Oprod__mOp, type, prod_mOp: (!> [I:$tType,A:$tType,More:$tType]: (((set @ I) > ((I > ((carrier_ext @ A) @ ((aGroup_ext @ A) @ More))) > ((I > A) > (I > A))))))).
% 2.88/1.45 thf(sy_c_Algebra4_Oprod__pOp, type, prod_pOp: (!> [I:$tType,A:$tType,More:$tType]: (((set @ I) > ((I > ((carrier_ext @ A) @ ((aGroup_ext @ A) @ More))) > ((I > A) > ((I > A) > (I > A)))))))).
% 2.88/1.45 thf(sy_c_Algebra4_Oprod__zero, type, prod_zero: (!> [I:$tType,A:$tType,More:$tType]: (((set @ I) > ((I > ((carrier_ext @ A) @ ((aGroup_ext @ A) @ More))) > (I > A)))))).
% 2.88/1.45 thf(sy_c_Algebra4_OrInvim, type, rInvim: (!> [A:$tType,M:$tType,B:$tType,M1:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ M))) > (((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ M1))) > ((A > B) > ((set @ B) > (set @ A)))))))).
% 2.88/1.45 thf(sy_c_Algebra4_Osr, type, sr2: (!> [A:$tType,B:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) > ((set @ A) > $o))))).
% 2.88/1.45 thf(sy_c_Algebra4_Ozeroring, type, zeroring: (!> [A:$tType,More:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ More))) > $o)))).
% 2.88/1.45 thf(sy_c_Algebra5_Osum__mult, type, sum_mult: (!> [A:$tType,B:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) > ((set @ A) > ((set @ A) > (set @ A))))))).
% 2.88/1.45 thf(sy_c_Algebra7_OModule, type, module: (!> [A:$tType,B:$tType,C:$tType,D:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) > (((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))) > $o))))).
% 2.88/1.45 thf(sy_c_Algebra7_OModule_Osprod, type, sprod: (!> [A:$tType,B:$tType,Z:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ Z))) > (B > (A > A)))))).
% 2.88/1.45 thf(sy_c_Algebra7_OmHom, type, mHom: (!> [B:$tType,M:$tType,A:$tType,M1:$tType,C:$tType,M2:$tType]: ((((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ M))) > (((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ M1))) > (((carrier_ext @ C) @ ((aGroup_ext @ C) @ (((module_ext @ C) @ B) @ M2))) > (set @ (A > C)))))))).
% 2.88/1.45 thf(sy_c_Algebra8__Mirabelle__lwvimexpoc_OprodM, type, algebr1000837334_prodM: (!> [R:$tType,M:$tType,I:$tType,A:$tType,M1:$tType]: ((((carrier_ext @ R) @ ((aGroup_ext @ R) @ ((ring_ext @ R) @ M))) > ((set @ I) > ((I > ((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ R) @ M1)))) > ((carrier_ext @ (I > A)) @ ((aGroup_ext @ (I > A)) @ (((module_ext @ (I > A)) @ R) @ product_unit))))))))).
% 2.88/1.45 thf(sy_c_Algebra8__Mirabelle__lwvimexpoc_OprodM__sprod, type, algebr142713364_sprod: (!> [R:$tType,M:$tType,I:$tType,A:$tType,M1:$tType]: ((((carrier_ext @ R) @ ((aGroup_ext @ R) @ ((ring_ext @ R) @ M))) > ((set @ I) > ((I > ((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ R) @ M1)))) > (R > ((I > A) > (I > A))))))))).
% 2.88/1.45 thf(sy_c_FuncSet_OPi, type, pi: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (set @ B)) > (set @ (A > B))))))).
% 2.88/1.45 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 2.88/1.45 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.88/1.45 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.88/1.45 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.88/1.45 thf(sy_v_I, type, i: (set @ c)).
% 2.88/1.45 thf(sy_v_M, type, m: (c > ((carrier_ext @ d) @ ((aGroup_ext @ d) @ (((module_ext @ d) @ a) @ e))))).
% 2.88/1.45 thf(sy_v_R, type, r: ((carrier_ext @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b)))).
% 2.88/1.45 thf(fact_0_Ring__axioms, axiom, (((ring @ a) @ b) @ r)).
% 2.88/1.45 thf(fact_1_bivar__fun, axiom, (! [A:$tType,C:$tType,B:$tType,F:(A > (B > C)),A2:(set @ A),B2:(set @ B),C2:(set @ C),A3:A]: (((((member @ (A > (B > C))) @ F) @ ((((pi @ A) @ (B > C)) @ A2) @ (^ [Uu:A]: (((((pi @ B) @ C) @ B2) @ (^ [Uv:B]: (C2))))))) => ((((member @ A) @ A3) @ A2) => (((member @ (B > C)) @ (F @ A3)) @ ((((pi @ B) @ C) @ B2) @ (^ [Uu:B]: (C2))))))))).
% 2.88/1.45 thf(fact_2_bivar__fun__mem, axiom, (! [B:$tType,A:$tType,C:$tType,F:(A > (B > C)),A2:(set @ A),B2:(set @ B),C2:(set @ C),A3:A,B3:B]: (((((member @ (A > (B > C))) @ F) @ ((((pi @ A) @ (B > C)) @ A2) @ (^ [Uu:A]: (((((pi @ B) @ C) @ B2) @ (^ [Uv:B]: (C2))))))) => ((((member @ A) @ A3) @ A2) => ((((member @ B) @ B3) @ B2) => (((member @ C) @ ((F @ A3) @ B3)) @ C2))))))).
% 2.88/1.45 thf(fact_3_Pi__I, axiom, (! [B:$tType,A:$tType,A2:(set @ A),F:(A > B),B2:(A > (set @ B))]: (((! [X:A]: (((((member @ A) @ X) @ A2) => (((member @ B) @ (F @ X)) @ (B2 @ X))))) => (((member @ (A > B)) @ F) @ ((((pi @ A) @ B) @ A2) @ B2)))))).
% 2.88/1.45 thf(fact_4_prodM__sprod__mem, axiom, (! [E:$tType,D:$tType,C:$tType,I2:(set @ C),M3:(C > ((carrier_ext @ D) @ ((aGroup_ext @ D) @ (((module_ext @ D) @ a) @ E)))),A3:a,M4:(C > D)]: (((! [X:C]: (((((member @ C) @ X) @ I2) => ((((((module @ D) @ a) @ E) @ b) @ (M3 @ X)) @ r)))) => ((((member @ a) @ A3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ (C > D)) @ M4) @ (((((carr_prodag @ C) @ D) @ (((module_ext @ D) @ a) @ E)) @ I2) @ M3)) => (((member @ (C > D)) @ ((((((((((algebr142713364_sprod @ a) @ b) @ C) @ D) @ E) @ r) @ I2) @ M3) @ A3) @ M4)) @ (((((carr_prodag @ C) @ D) @ (((module_ext @ D) @ a) @ E)) @ I2) @ M3)))))))).
% 2.88/1.45 thf(fact_5_prodM__carr, axiom, (! [E:$tType,D:$tType,C:$tType,I2:(set @ C),M3:(C > ((carrier_ext @ D) @ ((aGroup_ext @ D) @ (((module_ext @ D) @ a) @ E))))]: (((! [X:C]: (((((member @ C) @ X) @ I2) => ((((((module @ D) @ a) @ E) @ b) @ (M3 @ X)) @ r)))) => ((((carrier @ (C > D)) @ ((aGroup_ext @ (C > D)) @ (((module_ext @ (C > D)) @ a) @ product_unit))) @ ((((((((algebr1000837334_prodM @ a) @ b) @ C) @ D) @ E) @ r) @ I2) @ M3)) = (((((carr_prodag @ C) @ D) @ (((module_ext @ D) @ a) @ E)) @ I2) @ M3)))))).
% 2.88/1.45 thf(fact_6_eq__fun, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),B2:(set @ B),G:(A > B)]: (((((member @ (A > B)) @ F) @ ((((pi @ A) @ B) @ A2) @ (^ [Uu:A]: (B2)))) => ((F = G) => (((member @ (A > B)) @ G) @ ((((pi @ A) @ B) @ A2) @ (^ [Uu:A]: (B2))))))))).
% 2.88/1.45 thf(fact_7_funcTr, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),B2:(set @ B),G:(A > B),A3:A]: (((((member @ (A > B)) @ F) @ ((((pi @ A) @ B) @ A2) @ (^ [Uu:A]: (B2)))) => ((((member @ (A > B)) @ G) @ ((((pi @ A) @ B) @ A2) @ (^ [Uu:A]: (B2)))) => ((F = G) => ((((member @ A) @ A3) @ A2) => ((F @ A3) = (G @ A3))))))))).
% 2.88/1.45 thf(fact_8_funcsetI, axiom, (! [B:$tType,A:$tType,A2:(set @ A),F:(A > B),B2:(set @ B)]: (((! [X:A]: (((((member @ A) @ X) @ A2) => (((member @ B) @ (F @ X)) @ B2)))) => (((member @ (A > B)) @ F) @ ((((pi @ A) @ B) @ A2) @ (^ [Uu:A]: (B2)))))))).
% 2.88/1.45 thf(fact_9_eq__funcs, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),B2:(set @ B),G:(A > B),X2:A]: (((((member @ (A > B)) @ F) @ ((((pi @ A) @ B) @ A2) @ (^ [Uu:A]: (B2)))) => ((((member @ (A > B)) @ G) @ ((((pi @ A) @ B) @ A2) @ (^ [Uu:A]: (B2)))) => ((F = G) => ((((member @ A) @ X2) @ A2) => ((F @ X2) = (G @ X2))))))))).
% 2.88/1.45 thf(fact_10_funcset__id, axiom, (! [A:$tType,A2:(set @ A)]: ((((member @ (A > A)) @ (^ [X3:A]: (X3))) @ ((((pi @ A) @ A) @ A2) @ (^ [Uu:A]: (A2))))))).
% 2.88/1.45 thf(fact_11_Algebra1_Ofuncset__mem, axiom, (! [A:$tType,B:$tType,F:(A > B),A2:(set @ A),B2:(set @ B),X2:A]: (((((member @ (A > B)) @ F) @ ((((pi @ A) @ B) @ A2) @ (^ [Uu:A]: (B2)))) => ((((member @ A) @ X2) @ A2) => (((member @ B) @ (F @ X2)) @ B2)))))).
% 2.88/1.45 thf(fact_12_prod__pOp__func, axiom, (! [C:$tType,B:$tType,A:$tType,I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C)))]: (((! [X:A]: (((((member @ A) @ X) @ I2) => (((aGroup @ B) @ C) @ (A2 @ X))))) => (((member @ ((A > B) > ((A > B) > (A > B)))) @ (((((prod_pOp @ A) @ B) @ C) @ I2) @ A2)) @ ((((pi @ (A > B)) @ ((A > B) > (A > B))) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)) @ (^ [Uu:(A > B)]: (((((pi @ (A > B)) @ (A > B)) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)) @ (^ [Uv:(A > B)]: ((((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)))))))))))).
% 2.88/1.45 thf(fact_13_ring__is__ag, axiom, (((aGroup @ a) @ ((ring_ext @ a) @ b)) @ r)).
% 2.88/1.45 thf(fact_14_prodM__mem__eq, axiom, (! [E:$tType,D:$tType,C:$tType,I2:(set @ C),M3:(C > ((carrier_ext @ D) @ ((aGroup_ext @ D) @ (((module_ext @ D) @ a) @ E)))),M12:(C > D),M22:(C > D)]: (((! [X:C]: (((((member @ C) @ X) @ I2) => ((((((module @ D) @ a) @ E) @ b) @ (M3 @ X)) @ r)))) => ((((member @ (C > D)) @ M12) @ (((carrier @ (C > D)) @ ((aGroup_ext @ (C > D)) @ (((module_ext @ (C > D)) @ a) @ product_unit))) @ ((((((((algebr1000837334_prodM @ a) @ b) @ C) @ D) @ E) @ r) @ I2) @ M3))) => ((((member @ (C > D)) @ M22) @ (((carrier @ (C > D)) @ ((aGroup_ext @ (C > D)) @ (((module_ext @ (C > D)) @ a) @ product_unit))) @ ((((((((algebr1000837334_prodM @ a) @ b) @ C) @ D) @ E) @ r) @ I2) @ M3))) => ((! [X:C]: (((((member @ C) @ X) @ I2) => ((M12 @ X) = (M22 @ X))))) => (M12 = M22)))))))).
% 2.88/1.45 thf(fact_15_J__rad__mem, axiom, (! [X2:a]: (((((member @ a) @ X2) @ (((j_rad @ a) @ b) @ r)) => (((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)))))).
% 2.88/1.45 thf(fact_16_Ring_ORing, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((ring @ A) @ B) @ R2) => (((ring @ A) @ B) @ R2))))).
% 2.88/1.45 thf(fact_17_Ring_Oring__is__ag, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((ring @ A) @ B) @ R2) => (((aGroup @ A) @ ((ring_ext @ A) @ B)) @ R2))))).
% 2.88/1.45 thf(fact_18_Ring_OprodM__mem__eq, axiom, (! [A:$tType,E:$tType,B:$tType,D:$tType,C:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),I2:(set @ C),M3:(C > ((carrier_ext @ D) @ ((aGroup_ext @ D) @ (((module_ext @ D) @ A) @ E)))),M12:(C > D),M22:(C > D)]: (((((ring @ A) @ B) @ R2) => ((! [X:C]: (((((member @ C) @ X) @ I2) => ((((((module @ D) @ A) @ E) @ B) @ (M3 @ X)) @ R2)))) => ((((member @ (C > D)) @ M12) @ (((carrier @ (C > D)) @ ((aGroup_ext @ (C > D)) @ (((module_ext @ (C > D)) @ A) @ product_unit))) @ ((((((((algebr1000837334_prodM @ A) @ B) @ C) @ D) @ E) @ R2) @ I2) @ M3))) => ((((member @ (C > D)) @ M22) @ (((carrier @ (C > D)) @ ((aGroup_ext @ (C > D)) @ (((module_ext @ (C > D)) @ A) @ product_unit))) @ ((((((((algebr1000837334_prodM @ A) @ B) @ C) @ D) @ E) @ R2) @ I2) @ M3))) => ((! [X:C]: (((((member @ C) @ X) @ I2) => ((M12 @ X) = (M22 @ X))))) => (M12 = M22))))))))).
% 2.88/1.45 thf(fact_19_carr__prodag__mem__eq, axiom, (! [C:$tType,B:$tType,A:$tType,I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C))),X4:(A > B),Y:(A > B)]: (((! [X:A]: (((((member @ A) @ X) @ I2) => (((aGroup @ B) @ C) @ (A2 @ X))))) => ((((member @ (A > B)) @ X4) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)) => ((((member @ (A > B)) @ Y) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)) => ((! [X:A]: (((((member @ A) @ X) @ I2) => ((X4 @ X) = (Y @ X))))) => (X4 = Y)))))))).
% 2.88/1.45 thf(fact_20_prodag__sameTr1, axiom, (! [C:$tType,B:$tType,A:$tType,I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C))),B2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C)))]: (((! [X:A]: (((((member @ A) @ X) @ I2) => (((aGroup @ B) @ C) @ (A2 @ X))))) => ((! [X:A]: (((((member @ A) @ X) @ I2) => ((A2 @ X) = (B2 @ X))))) => ((((((carr_prodag @ A) @ B) @ C) @ I2) @ A2) = (((((carr_prodag @ A) @ B) @ C) @ I2) @ B2))))))).
% 2.88/1.45 thf(fact_21_prodag__comp__i, axiom, (! [B:$tType,C:$tType,A:$tType,A3:(A > B),I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C))),I3:A]: (((((member @ (A > B)) @ A3) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)) => ((((member @ A) @ I3) @ I2) => (((member @ B) @ (A3 @ I3)) @ (((carrier @ B) @ ((aGroup_ext @ B) @ C)) @ (A2 @ I3)))))))).
% 2.88/1.45 thf(fact_22_prodag__sameTr2, axiom, (! [C:$tType,B:$tType,A:$tType,I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C))),B2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C)))]: (((! [X:A]: (((((member @ A) @ X) @ I2) => (((aGroup @ B) @ C) @ (A2 @ X))))) => ((! [X:A]: (((((member @ A) @ X) @ I2) => ((A2 @ X) = (B2 @ X))))) => ((((((prod_pOp @ A) @ B) @ C) @ I2) @ A2) = (((((prod_pOp @ A) @ B) @ C) @ I2) @ B2))))))).
% 2.88/1.45 thf(fact_23_prodag__sameTr3, axiom, (! [C:$tType,B:$tType,A:$tType,I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C))),B2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C)))]: (((! [X:A]: (((((member @ A) @ X) @ I2) => (((aGroup @ B) @ C) @ (A2 @ X))))) => ((! [X:A]: (((((member @ A) @ X) @ I2) => ((A2 @ X) = (B2 @ X))))) => ((((((prod_mOp @ A) @ B) @ C) @ I2) @ A2) = (((((prod_mOp @ A) @ B) @ C) @ I2) @ B2))))))).
% 2.88/1.45 thf(fact_24_prodag__sameTr4, axiom, (! [C:$tType,B:$tType,A:$tType,I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C))),B2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C)))]: (((! [X:A]: (((((member @ A) @ X) @ I2) => (((aGroup @ B) @ C) @ (A2 @ X))))) => ((! [X:A]: (((((member @ A) @ X) @ I2) => ((A2 @ X) = (B2 @ X))))) => ((((((prod_zero @ A) @ B) @ C) @ I2) @ A2) = (((((prod_zero @ A) @ B) @ C) @ I2) @ B2))))))).
% 2.88/1.45 thf(fact_25_Ring_OprodM__carr, axiom, (! [B:$tType,E:$tType,A:$tType,D:$tType,C:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),I2:(set @ C),M3:(C > ((carrier_ext @ D) @ ((aGroup_ext @ D) @ (((module_ext @ D) @ A) @ E))))]: (((((ring @ A) @ B) @ R2) => ((! [X:C]: (((((member @ C) @ X) @ I2) => ((((((module @ D) @ A) @ E) @ B) @ (M3 @ X)) @ R2)))) => ((((carrier @ (C > D)) @ ((aGroup_ext @ (C > D)) @ (((module_ext @ (C > D)) @ A) @ product_unit))) @ ((((((((algebr1000837334_prodM @ A) @ B) @ C) @ D) @ E) @ R2) @ I2) @ M3)) = (((((carr_prodag @ C) @ D) @ (((module_ext @ D) @ A) @ E)) @ I2) @ M3))))))).
% 2.88/1.45 thf(fact_26_Ring_OprodM__sprod__mem, axiom, (! [B:$tType,E:$tType,A:$tType,D:$tType,C:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),I2:(set @ C),M3:(C > ((carrier_ext @ D) @ ((aGroup_ext @ D) @ (((module_ext @ D) @ A) @ E)))),A3:A,M4:(C > D)]: (((((ring @ A) @ B) @ R2) => ((! [X:C]: (((((member @ C) @ X) @ I2) => ((((((module @ D) @ A) @ E) @ B) @ (M3 @ X)) @ R2)))) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ (C > D)) @ M4) @ (((((carr_prodag @ C) @ D) @ (((module_ext @ D) @ A) @ E)) @ I2) @ M3)) => (((member @ (C > D)) @ ((((((((((algebr142713364_sprod @ A) @ B) @ C) @ D) @ E) @ R2) @ I2) @ M3) @ A3) @ M4)) @ (((((carr_prodag @ C) @ D) @ (((module_ext @ D) @ A) @ E)) @ I2) @ M3))))))))).
% 2.88/1.45 thf(fact_27_prod__pOp__commute, axiom, (! [C:$tType,B:$tType,A:$tType,I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C))),A3:(A > B),B3:(A > B)]: (((! [X:A]: (((((member @ A) @ X) @ I2) => (((aGroup @ B) @ C) @ (A2 @ X))))) => ((((member @ (A > B)) @ A3) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)) => ((((member @ (A > B)) @ B3) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)) => ((((((((prod_pOp @ A) @ B) @ C) @ I2) @ A2) @ A3) @ B3) = (((((((prod_pOp @ A) @ B) @ C) @ I2) @ A2) @ B3) @ A3)))))))).
% 2.88/1.45 thf(fact_28_prod__pOp__assoc, axiom, (! [C:$tType,B:$tType,A:$tType,I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C))),A3:(A > B),B3:(A > B),C3:(A > B)]: (((! [X:A]: (((((member @ A) @ X) @ I2) => (((aGroup @ B) @ C) @ (A2 @ X))))) => ((((member @ (A > B)) @ A3) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)) => ((((member @ (A > B)) @ B3) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)) => ((((member @ (A > B)) @ C3) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)) => ((((((((prod_pOp @ A) @ B) @ C) @ I2) @ A2) @ (((((((prod_pOp @ A) @ B) @ C) @ I2) @ A2) @ A3) @ B3)) @ C3) = (((((((prod_pOp @ A) @ B) @ C) @ I2) @ A2) @ A3) @ (((((((prod_pOp @ A) @ B) @ C) @ I2) @ A2) @ B3) @ C3)))))))))).
% 2.88/1.45 thf(fact_29_prod__pOp__mem, axiom, (! [C:$tType,B:$tType,A:$tType,I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C))),X4:(A > B),Y:(A > B)]: (((! [X:A]: (((((member @ A) @ X) @ I2) => (((aGroup @ B) @ C) @ (A2 @ X))))) => ((((member @ (A > B)) @ X4) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)) => ((((member @ (A > B)) @ Y) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)) => (((member @ (A > B)) @ (((((((prod_pOp @ A) @ B) @ C) @ I2) @ A2) @ X4) @ Y)) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)))))))).
% 2.88/1.45 thf(fact_30_prod__mOp__mem, axiom, (! [C:$tType,B:$tType,A:$tType,I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C))),X4:(A > B)]: (((! [X:A]: (((((member @ A) @ X) @ I2) => (((aGroup @ B) @ C) @ (A2 @ X))))) => ((((member @ (A > B)) @ X4) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)) => (((member @ (A > B)) @ ((((((prod_mOp @ A) @ B) @ C) @ I2) @ A2) @ X4)) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2))))))).
% 2.88/1.45 thf(fact_31_prod__zero__func, axiom, (! [C:$tType,B:$tType,A:$tType,I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C)))]: (((! [X:A]: (((((member @ A) @ X) @ I2) => (((aGroup @ B) @ C) @ (A2 @ X))))) => (((member @ (A > B)) @ (((((prod_zero @ A) @ B) @ C) @ I2) @ A2)) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)))))).
% 2.88/1.45 thf(fact_32_prod__mOp__func, axiom, (! [C:$tType,B:$tType,A:$tType,I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C)))]: (((! [X:A]: (((((member @ A) @ X) @ I2) => (((aGroup @ B) @ C) @ (A2 @ X))))) => (((member @ ((A > B) > (A > B))) @ (((((prod_mOp @ A) @ B) @ C) @ I2) @ A2)) @ ((((pi @ (A > B)) @ (A > B)) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)) @ (^ [Uu:(A > B)]: ((((((carr_prodag @ A) @ B) @ C) @ I2) @ A2))))))))).
% 2.88/1.45 thf(fact_33_Pi__cong, axiom, (! [B:$tType,A:$tType,A2:(set @ A),F:(A > B),G:(A > B),B2:(A > (set @ B))]: (((! [W:A]: (((((member @ A) @ W) @ A2) => ((F @ W) = (G @ W))))) => ((((member @ (A > B)) @ F) @ ((((pi @ A) @ B) @ A2) @ B2)) = (((member @ (A > B)) @ G) @ ((((pi @ A) @ B) @ A2) @ B2))))))).
% 2.88/1.45 thf(fact_34_Pi__mem, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),B2:(A > (set @ B)),X2:A]: (((((member @ (A > B)) @ F) @ ((((pi @ A) @ B) @ A2) @ B2)) => ((((member @ A) @ X2) @ A2) => (((member @ B) @ (F @ X2)) @ (B2 @ X2))))))).
% 2.88/1.45 thf(fact_35_Pi__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),I2:(set @ A),X4:(A > (set @ B))]: (((((member @ (A > B)) @ F) @ ((((pi @ A) @ B) @ I2) @ X4)) = (! [X3:A]: (((((member @ A) @ X3) @ I2) => (((member @ B) @ (F @ X3)) @ (X4 @ X3))))))))).
% 2.88/1.45 thf(fact_36_Pi__I_H, axiom, (! [B:$tType,A:$tType,A2:(set @ A),F:(A > B),B2:(A > (set @ B))]: (((! [X:A]: (((((member @ A) @ X) @ A2) => (((member @ B) @ (F @ X)) @ (B2 @ X))))) => (((member @ (A > B)) @ F) @ ((((pi @ A) @ B) @ A2) @ B2)))))).
% 2.88/1.45 thf(fact_37_PiE, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),B2:(A > (set @ B)),X2:A]: (((((member @ (A > B)) @ F) @ ((((pi @ A) @ B) @ A2) @ B2)) => ((~ ((((member @ B) @ (F @ X2)) @ (B2 @ X2)))) => (~ ((((member @ A) @ X2) @ A2)))))))).
% 2.88/1.45 thf(fact_38_mem__subring__mem__ring, axiom, (! [C:$tType,S:((carrier_ext @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ C))),X2:a]: (((((((subring @ a) @ b) @ C) @ r) @ S) => ((((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ C))) @ S)) => (((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r))))))).
% 2.88/1.45 thf(fact_39_subring__Ring, axiom, (! [C:$tType,S:((carrier_ext @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ C)))]: (((((((subring @ a) @ b) @ C) @ r) @ S) => (((ring @ a) @ C) @ S))))).
% 2.88/1.45 thf(fact_40_prodM__sprod__val, axiom, (! [E:$tType,D:$tType,C:$tType,I2:(set @ C),M3:(C > ((carrier_ext @ D) @ ((aGroup_ext @ D) @ (((module_ext @ D) @ a) @ E)))),A3:a,M4:(C > D),J:C]: (((! [X:C]: (((((member @ C) @ X) @ I2) => ((((((module @ D) @ a) @ E) @ b) @ (M3 @ X)) @ r)))) => ((((member @ a) @ A3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ (C > D)) @ M4) @ (((((carr_prodag @ C) @ D) @ (((module_ext @ D) @ a) @ E)) @ I2) @ M3)) => ((((member @ C) @ J) @ I2) => ((((((((((((algebr142713364_sprod @ a) @ b) @ C) @ D) @ E) @ r) @ I2) @ M3) @ A3) @ M4) @ J) = ((((((sprod @ D) @ a) @ E) @ (M3 @ J)) @ A3) @ (M4 @ J)))))))))).
% 2.88/1.45 thf(fact_41_Sr__ring, axiom, (! [S:(set @ a)]: ((((((sr2 @ a) @ b) @ r) @ S) => (((ring @ a) @ b) @ ((((sr @ a) @ b) @ r) @ S)))))).
% 2.88/1.45 thf(fact_42_mHom__func, axiom, (! [E:$tType,D:$tType,F2:$tType,C:$tType,F:(C > D),M3:((carrier_ext @ C) @ ((aGroup_ext @ C) @ (((module_ext @ C) @ a) @ E))),N:((carrier_ext @ D) @ ((aGroup_ext @ D) @ (((module_ext @ D) @ a) @ F2)))]: (((((member @ (C > D)) @ F) @ (((((((((mHom @ a) @ b) @ C) @ E) @ D) @ F2) @ r) @ M3) @ N)) => (((member @ (C > D)) @ F) @ ((((pi @ C) @ D) @ (((carrier @ C) @ ((aGroup_ext @ C) @ (((module_ext @ C) @ a) @ E))) @ M3)) @ (^ [Uu:C]: ((((carrier @ D) @ ((aGroup_ext @ D) @ (((module_ext @ D) @ a) @ F2))) @ N))))))))).
% 2.88/1.45 thf(fact_43_rInvim__def, axiom, (! [B:$tType,M1:$tType,M:$tType,A:$tType]: ((((((rInvim @ A) @ M) @ B) @ M1) = (^ [A4:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ M))),R3:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ M1))),F3:(A > B),K:(set @ B)]: (((collect @ A) @ (^ [A5:A]: (((((member @ A) @ A5) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ M))) @ A4)) & (((member @ B) @ (F3 @ A5)) @ K))))))))))).
% 2.88/1.45 thf(fact_44_dsum__pOp__func, axiom, (! [C:$tType,B:$tType,A:$tType,I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C)))]: (((! [X:A]: (((((member @ A) @ X) @ I2) => (((aGroup @ B) @ C) @ (A2 @ X))))) => (((member @ ((A > B) > ((A > B) > (A > B)))) @ (((((prod_pOp @ A) @ B) @ C) @ I2) @ A2)) @ ((((pi @ (A > B)) @ ((A > B) > (A > B))) @ (((((carr_dsumag @ A) @ B) @ C) @ I2) @ A2)) @ (^ [Uu:(A > B)]: (((((pi @ (A > B)) @ (A > B)) @ (((((carr_dsumag @ A) @ B) @ C) @ I2) @ A2)) @ (^ [Uv:(A > B)]: ((((((carr_dsumag @ A) @ B) @ C) @ I2) @ A2)))))))))))).
% 2.88/1.45 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 2.88/1.45 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A2)))) = A2)))).
% 2.88/1.45 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X:A]: (((P @ X) = (Q @ X)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.88/1.45 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X:A]: (((F @ X) = (G @ X)))) => (F = G))))).
% 2.88/1.45 thf(fact_49_nsEqElm, axiom, (! [X2:a,Y2:a,N2:nat]: (((((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((X2 = Y2) => ((((((nscal @ a) @ b) @ r) @ X2) @ N2) = (((((nscal @ a) @ b) @ r) @ Y2) @ N2)))))))).
% 2.88/1.45 thf(fact_50_nsClose, axiom, (! [X2:a,N2:nat]: (((((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((member @ a) @ (((((nscal @ a) @ b) @ r) @ X2) @ N2)) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)))))).
% 2.88/1.45 thf(fact_51_tp__closed, axiom, (((member @ (a > (a > a))) @ (((tp @ a) @ b) @ r)) @ ((((pi @ a) @ (a > a)) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) @ (^ [Uu:a]: (((((pi @ a) @ a) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) @ (^ [Uv:a]: ((((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r))))))))).
% 2.88/1.45 thf(fact_52_pop__closed, axiom, (((member @ (a > (a > a))) @ (((pop @ a) @ ((ring_ext @ a) @ b)) @ r)) @ ((((pi @ a) @ (a > a)) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) @ (^ [Uu:a]: (((((pi @ a) @ a) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) @ (^ [Uv:a]: ((((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r))))))))).
% 2.88/1.45 thf(fact_53_rEQMulR, axiom, (! [X2:a,Y2:a,Z2:a]: (((((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((X2 = Y2) => ((((((tp @ a) @ b) @ r) @ X2) @ Z2) = (((((tp @ a) @ b) @ r) @ Y2) @ Z2))))))))).
% 2.88/1.45 thf(fact_54_rMulLC, axiom, (! [X2:a,Y2:a,Z2:a]: (((((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((tp @ a) @ b) @ r) @ X2) @ (((((tp @ a) @ b) @ r) @ Y2) @ Z2)) = (((((tp @ a) @ b) @ r) @ Y2) @ (((((tp @ a) @ b) @ r) @ X2) @ Z2))))))))).
% 2.88/1.45 thf(fact_55_ring__tOp__assoc, axiom, (! [X2:a,Y2:a,Z2:a]: (((((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((tp @ a) @ b) @ r) @ (((((tp @ a) @ b) @ r) @ X2) @ Y2)) @ Z2) = (((((tp @ a) @ b) @ r) @ X2) @ (((((tp @ a) @ b) @ r) @ Y2) @ Z2))))))))).
% 2.88/1.45 thf(fact_56_ring__tOp__closed, axiom, (! [X2:a,Y2:a]: (((((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((member @ a) @ (((((tp @ a) @ b) @ r) @ X2) @ Y2)) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r))))))).
% 2.88/1.45 thf(fact_57_ring__tOp__commute, axiom, (! [X2:a,Y2:a]: (((((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((tp @ a) @ b) @ r) @ X2) @ Y2) = (((((tp @ a) @ b) @ r) @ Y2) @ X2))))))).
% 2.88/1.45 thf(fact_58_ring__tOp__rel, axiom, (! [X2:a,Xa:a,Y2:a,Ya:a]: (((((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Xa) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Ya) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((tp @ a) @ b) @ r) @ (((((tp @ a) @ b) @ r) @ X2) @ Xa)) @ (((((tp @ a) @ b) @ r) @ Y2) @ Ya)) = (((((tp @ a) @ b) @ r) @ (((((tp @ a) @ b) @ r) @ X2) @ Y2)) @ (((((tp @ a) @ b) @ r) @ Xa) @ Ya)))))))))).
% 2.88/1.45 thf(fact_59_tp__assoc, axiom, (! [A3:a,B3:a,C3:a]: (((((member @ a) @ A3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ C3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((tp @ a) @ b) @ r) @ (((((tp @ a) @ b) @ r) @ A3) @ B3)) @ C3) = (((((tp @ a) @ b) @ r) @ A3) @ (((((tp @ a) @ b) @ r) @ B3) @ C3))))))))).
% 2.88/1.45 thf(fact_60_tp__commute, axiom, (! [A3:a,B3:a]: (((((member @ a) @ A3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((tp @ a) @ b) @ r) @ A3) @ B3) = (((((tp @ a) @ b) @ r) @ B3) @ A3))))))).
% 2.88/1.45 thf(fact_61_pop__aassoc, axiom, (! [A3:a,B3:a,C3:a]: (((((member @ a) @ A3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ C3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ A3) @ B3)) @ C3) = (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ A3) @ (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ B3) @ C3))))))))).
% 2.88/1.45 thf(fact_62_pop__commute, axiom, (! [A3:a,B3:a]: (((((member @ a) @ A3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ A3) @ B3) = (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ B3) @ A3))))))).
% 2.88/1.45 thf(fact_63_Sr__tOp__closed, axiom, (! [S:(set @ a),X2:a,Y2:a]: ((((((sr2 @ a) @ b) @ r) @ S) => ((((member @ a) @ X2) @ S) => ((((member @ a) @ Y2) @ S) => (((member @ a) @ (((((tp @ a) @ b) @ r) @ X2) @ Y2)) @ S))))))).
% 2.88/1.45 thf(fact_64_Sr__pOp__closed, axiom, (! [S:(set @ a),X2:a,Y2:a]: ((((((sr2 @ a) @ b) @ r) @ S) => ((((member @ a) @ X2) @ S) => ((((member @ a) @ Y2) @ S) => (((member @ a) @ (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ X2) @ Y2)) @ S))))))).
% 2.88/1.45 thf(fact_65_rg__distrib, axiom, (! [A3:a,B3:a,C3:a]: (((((member @ a) @ A3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ C3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((tp @ a) @ b) @ r) @ A3) @ (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ B3) @ C3)) = (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((tp @ a) @ b) @ r) @ A3) @ B3)) @ (((((tp @ a) @ b) @ r) @ A3) @ C3))))))))).
% 2.88/1.45 thf(fact_66_ring__distrib1, axiom, (! [X2:a,Y2:a,Z2:a]: (((((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((tp @ a) @ b) @ r) @ X2) @ (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ Y2) @ Z2)) = (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((tp @ a) @ b) @ r) @ X2) @ Y2)) @ (((((tp @ a) @ b) @ r) @ X2) @ Z2))))))))).
% 2.88/1.45 thf(fact_67_ring__distrib2, axiom, (! [X2:a,Y2:a,Z2:a]: (((((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((tp @ a) @ b) @ r) @ (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ Y2) @ Z2)) @ X2) = (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((tp @ a) @ b) @ r) @ Y2) @ X2)) @ (((((tp @ a) @ b) @ r) @ Z2) @ X2))))))))).
% 2.88/1.45 thf(fact_68_ring__distrib3, axiom, (! [A3:a,B3:a,X2:a,Y2:a]: (((((member @ a) @ A3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((tp @ a) @ b) @ r) @ (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ A3) @ B3)) @ (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ X2) @ Y2)) = (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((tp @ a) @ b) @ r) @ A3) @ X2)) @ (((((tp @ a) @ b) @ r) @ A3) @ Y2))) @ (((((tp @ a) @ b) @ r) @ B3) @ X2))) @ (((((tp @ a) @ b) @ r) @ B3) @ Y2)))))))))).
% 2.88/1.45 thf(fact_69_nsMulDistrL, axiom, (! [X2:a,Y2:a,N2:nat]: (((((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((tp @ a) @ b) @ r) @ X2) @ (((((nscal @ a) @ b) @ r) @ Y2) @ N2)) = (((((nscal @ a) @ b) @ r) @ (((((tp @ a) @ b) @ r) @ X2) @ Y2)) @ N2))))))).
% 2.88/1.45 thf(fact_70_nsMulDistrR, axiom, (! [X2:a,Y2:a,N2:nat]: (((((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((tp @ a) @ b) @ r) @ (((((nscal @ a) @ b) @ r) @ Y2) @ N2)) @ X2) = (((((nscal @ a) @ b) @ r) @ (((((tp @ a) @ b) @ r) @ Y2) @ X2)) @ N2))))))).
% 2.88/1.45 thf(fact_71_nsDistrL, axiom, (! [X2:a,Y2:a,N2:nat]: (((((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((nscal @ a) @ b) @ r) @ X2) @ N2)) @ (((((nscal @ a) @ b) @ r) @ Y2) @ N2)) = (((((nscal @ a) @ b) @ r) @ (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ X2) @ Y2)) @ N2))))))).
% 2.88/1.45 thf(fact_72_Subring__tOp__ring__tOp, axiom, (! [C:$tType,S:((carrier_ext @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ C))),A3:a,B3:a]: (((((((subring @ a) @ b) @ C) @ r) @ S) => ((((member @ a) @ A3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ C))) @ S)) => ((((member @ a) @ B3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ C))) @ S)) => ((((((tp @ a) @ C) @ S) @ A3) @ B3) = (((((tp @ a) @ b) @ r) @ A3) @ B3)))))))).
% 2.88/1.45 thf(fact_73_Subring__pOp__ring__pOp, axiom, (! [C:$tType,S:((carrier_ext @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ C))),A3:a,B3:a]: (((((((subring @ a) @ b) @ C) @ r) @ S) => ((((member @ a) @ A3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ C))) @ S)) => ((((member @ a) @ B3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ C))) @ S)) => ((((((pop @ a) @ ((ring_ext @ a) @ C)) @ S) @ A3) @ B3) = (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ A3) @ B3)))))))).
% 2.88/1.45 thf(fact_74_mul__closed__set__tOp__closed, axiom, (! [S:(set @ a),S2:a,T:a]: ((((((mul_closed_set @ a) @ b) @ r) @ S) => ((((member @ a) @ S2) @ S) => ((((member @ a) @ T) @ S) => (((member @ a) @ (((((tp @ a) @ b) @ r) @ S2) @ T)) @ S))))))).
% 2.88/1.45 thf(fact_75_Ring_OSr__tOp__closed, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),S:(set @ A),X2:A,Y2:A]: (((((ring @ A) @ B) @ R2) => (((((sr2 @ A) @ B) @ R2) @ S) => ((((member @ A) @ X2) @ S) => ((((member @ A) @ Y2) @ S) => (((member @ A) @ (((((tp @ A) @ B) @ R2) @ X2) @ Y2)) @ S)))))))).
% 2.88/1.45 thf(fact_76_Ring_OSr__pOp__closed, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),S:(set @ A),X2:A,Y2:A]: (((((ring @ A) @ B) @ R2) => (((((sr2 @ A) @ B) @ R2) @ S) => ((((member @ A) @ X2) @ S) => ((((member @ A) @ Y2) @ S) => (((member @ A) @ (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ X2) @ Y2)) @ S)))))))).
% 2.88/1.45 thf(fact_77_Ring_Oring__distrib3, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),A3:A,B3:A,X2:A,Y2:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Y2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((tp @ A) @ B) @ R2) @ (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ A3) @ B3)) @ (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ X2) @ Y2)) = (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ (((((tp @ A) @ B) @ R2) @ A3) @ X2)) @ (((((tp @ A) @ B) @ R2) @ A3) @ Y2))) @ (((((tp @ A) @ B) @ R2) @ B3) @ X2))) @ (((((tp @ A) @ B) @ R2) @ B3) @ Y2))))))))))).
% 2.88/1.45 thf(fact_78_Ring_Oring__distrib2, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),X2:A,Y2:A,Z2:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Y2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Z2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((tp @ A) @ B) @ R2) @ (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ Y2) @ Z2)) @ X2) = (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ (((((tp @ A) @ B) @ R2) @ Y2) @ X2)) @ (((((tp @ A) @ B) @ R2) @ Z2) @ X2)))))))))).
% 2.88/1.45 thf(fact_79_Ring_Oring__distrib1, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),X2:A,Y2:A,Z2:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Y2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Z2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((tp @ A) @ B) @ R2) @ X2) @ (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ Y2) @ Z2)) = (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ (((((tp @ A) @ B) @ R2) @ X2) @ Y2)) @ (((((tp @ A) @ B) @ R2) @ X2) @ Z2)))))))))).
% 2.88/1.45 thf(fact_80_Ring_Org__distrib, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),A3:A,B3:A,C3:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ C3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((tp @ A) @ B) @ R2) @ A3) @ (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ B3) @ C3)) = (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ (((((tp @ A) @ B) @ R2) @ A3) @ B3)) @ (((((tp @ A) @ B) @ R2) @ A3) @ C3)))))))))).
% 2.88/1.45 thf(fact_81_Ring_OnsMulDistrR, axiom, (! [A:$tType,B:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),X2:A,Y2:A,N2:nat]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Y2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((tp @ A) @ B) @ R2) @ (((((nscal @ A) @ B) @ R2) @ Y2) @ N2)) @ X2) = (((((nscal @ A) @ B) @ R2) @ (((((tp @ A) @ B) @ R2) @ Y2) @ X2)) @ N2)))))))).
% 2.88/1.45 thf(fact_82_Ring_OnsMulDistrL, axiom, (! [A:$tType,B:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),X2:A,Y2:A,N2:nat]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Y2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((tp @ A) @ B) @ R2) @ X2) @ (((((nscal @ A) @ B) @ R2) @ Y2) @ N2)) = (((((nscal @ A) @ B) @ R2) @ (((((tp @ A) @ B) @ R2) @ X2) @ Y2)) @ N2)))))))).
% 2.88/1.45 thf(fact_83_Ring_OnsDistrL, axiom, (! [A:$tType,B:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),X2:A,Y2:A,N2:nat]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Y2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ (((((nscal @ A) @ B) @ R2) @ X2) @ N2)) @ (((((nscal @ A) @ B) @ R2) @ Y2) @ N2)) = (((((nscal @ A) @ B) @ R2) @ (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ X2) @ Y2)) @ N2)))))))).
% 2.88/1.45 thf(fact_84_Ring_OSr__ring, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),S:(set @ A)]: (((((ring @ A) @ B) @ R2) => (((((sr2 @ A) @ B) @ R2) @ S) => (((ring @ A) @ B) @ ((((sr @ A) @ B) @ R2) @ S))))))).
% 2.88/1.45 thf(fact_85_aGroup_Oag__pOp__commute, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),X2:A,Y2:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ Y2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((((pop @ A) @ B) @ A2) @ X2) @ Y2) = (((((pop @ A) @ B) @ A2) @ Y2) @ X2)))))))).
% 2.88/1.45 thf(fact_86_aGroup_Oag__add__commute, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),A3:A,B3:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((((pop @ A) @ B) @ A2) @ A3) @ B3) = (((((pop @ A) @ B) @ A2) @ B3) @ A3)))))))).
% 2.88/1.45 thf(fact_87_aGroup_OpOp__assocTr43, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),A3:A,B3:A,C3:A,D2:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ C3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ D2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((((pop @ A) @ B) @ A2) @ (((((pop @ A) @ B) @ A2) @ A3) @ B3)) @ (((((pop @ A) @ B) @ A2) @ C3) @ D2)) = (((((pop @ A) @ B) @ A2) @ (((((pop @ A) @ B) @ A2) @ A3) @ (((((pop @ A) @ B) @ A2) @ B3) @ C3))) @ D2)))))))))).
% 2.88/1.45 thf(fact_88_aGroup_OpOp__assocTr42, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),A3:A,B3:A,C3:A,D2:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ C3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ D2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((((pop @ A) @ B) @ A2) @ (((((pop @ A) @ B) @ A2) @ (((((pop @ A) @ B) @ A2) @ A3) @ B3)) @ C3)) @ D2) = (((((pop @ A) @ B) @ A2) @ (((((pop @ A) @ B) @ A2) @ A3) @ (((((pop @ A) @ B) @ A2) @ B3) @ C3))) @ D2)))))))))).
% 2.88/1.45 thf(fact_89_aGroup_OpOp__assocTr41, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),A3:A,B3:A,C3:A,D2:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ C3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ D2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((((pop @ A) @ B) @ A2) @ (((((pop @ A) @ B) @ A2) @ (((((pop @ A) @ B) @ A2) @ A3) @ B3)) @ C3)) @ D2) = (((((pop @ A) @ B) @ A2) @ (((((pop @ A) @ B) @ A2) @ A3) @ B3)) @ (((((pop @ A) @ B) @ A2) @ C3) @ D2))))))))))).
% 2.88/1.45 thf(fact_90_aGroup_Oag__pOp__closed, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),X2:A,Y2:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ Y2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => (((member @ A) @ (((((pop @ A) @ B) @ A2) @ X2) @ Y2)) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)))))))).
% 2.88/1.45 thf(fact_91_aGroup_OpOp__cancel__r, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),A3:A,B3:A,C3:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ C3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => (((((((pop @ A) @ B) @ A2) @ A3) @ C3) = (((((pop @ A) @ B) @ A2) @ B3) @ C3)) => (A3 = B3))))))))).
% 2.88/1.45 thf(fact_92_aGroup_OpOp__cancel__l, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),A3:A,B3:A,C3:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ C3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => (((((((pop @ A) @ B) @ A2) @ C3) @ A3) = (((((pop @ A) @ B) @ A2) @ C3) @ B3)) => (A3 = B3))))))))).
% 2.88/1.45 thf(fact_93_aGroup_Oag__pOp__assoc, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),X2:A,Y2:A,Z2:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ Y2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ Z2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((((pop @ A) @ B) @ A2) @ (((((pop @ A) @ B) @ A2) @ X2) @ Y2)) @ Z2) = (((((pop @ A) @ B) @ A2) @ X2) @ (((((pop @ A) @ B) @ A2) @ Y2) @ Z2)))))))))).
% 2.88/1.45 thf(fact_94_aGroup_Oag__pOp__add__r, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),A3:A,B3:A,C3:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ C3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((A3 = B3) => ((((((pop @ A) @ B) @ A2) @ A3) @ C3) = (((((pop @ A) @ B) @ A2) @ B3) @ C3)))))))))).
% 2.88/1.45 thf(fact_95_aGroup_Oag__pOp__add__l, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),A3:A,B3:A,C3:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ C3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((A3 = B3) => ((((((pop @ A) @ B) @ A2) @ C3) @ A3) = (((((pop @ A) @ B) @ A2) @ C3) @ B3)))))))))).
% 2.88/1.45 thf(fact_96_aGroup_OgEQAddcross, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),L1:A,L2:A,R1:A,R22:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ L1) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ L2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ R1) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ R1) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((L1 = R22) => ((L2 = R1) => ((((((pop @ A) @ B) @ A2) @ L1) @ L2) = (((((pop @ A) @ B) @ A2) @ R1) @ R22)))))))))))).
% 2.88/1.45 thf(fact_97_aGroup_Oag__add4__rel, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),A3:A,B3:A,C3:A,D2:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ C3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ D2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((((pop @ A) @ B) @ A2) @ (((((pop @ A) @ B) @ A2) @ A3) @ B3)) @ (((((pop @ A) @ B) @ A2) @ C3) @ D2)) = (((((pop @ A) @ B) @ A2) @ (((((pop @ A) @ B) @ A2) @ A3) @ C3)) @ (((((pop @ A) @ B) @ A2) @ B3) @ D2))))))))))).
% 2.88/1.45 thf(fact_98_aGroup_Oaassoc, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),A3:A,B3:A,C3:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ C3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((((pop @ A) @ B) @ A2) @ (((((pop @ A) @ B) @ A2) @ A3) @ B3)) @ C3) = (((((pop @ A) @ B) @ A2) @ A3) @ (((((pop @ A) @ B) @ A2) @ B3) @ C3)))))))))).
% 2.88/1.45 thf(fact_99_Ring_Oring__tOp__commute, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),X2:A,Y2:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Y2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((tp @ A) @ B) @ R2) @ X2) @ Y2) = (((((tp @ A) @ B) @ R2) @ Y2) @ X2)))))))).
% 2.88/1.45 thf(fact_100_Ring_Oring__tOp__closed, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),X2:A,Y2:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Y2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => (((member @ A) @ (((((tp @ A) @ B) @ R2) @ X2) @ Y2)) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)))))))).
% 2.88/1.45 thf(fact_101_Ring_Oring__tOp__assoc, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),X2:A,Y2:A,Z2:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Y2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Z2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((tp @ A) @ B) @ R2) @ (((((tp @ A) @ B) @ R2) @ X2) @ Y2)) @ Z2) = (((((tp @ A) @ B) @ R2) @ X2) @ (((((tp @ A) @ B) @ R2) @ Y2) @ Z2)))))))))).
% 2.88/1.45 thf(fact_102_Ring_Oring__tOp__rel, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),X2:A,Xa:A,Y2:A,Ya:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Xa) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Y2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Ya) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((tp @ A) @ B) @ R2) @ (((((tp @ A) @ B) @ R2) @ X2) @ Xa)) @ (((((tp @ A) @ B) @ R2) @ Y2) @ Ya)) = (((((tp @ A) @ B) @ R2) @ (((((tp @ A) @ B) @ R2) @ X2) @ Y2)) @ (((((tp @ A) @ B) @ R2) @ Xa) @ Ya))))))))))).
% 2.88/1.45 thf(fact_103_Ring_Otp__commute, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),A3:A,B3:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((tp @ A) @ B) @ R2) @ A3) @ B3) = (((((tp @ A) @ B) @ R2) @ B3) @ A3)))))))).
% 2.88/1.45 thf(fact_104_Ring_Otp__assoc, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),A3:A,B3:A,C3:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ C3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((tp @ A) @ B) @ R2) @ (((((tp @ A) @ B) @ R2) @ A3) @ B3)) @ C3) = (((((tp @ A) @ B) @ R2) @ A3) @ (((((tp @ A) @ B) @ R2) @ B3) @ C3)))))))))).
% 2.88/1.45 thf(fact_105_Ring_OrEQMulR, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),X2:A,Y2:A,Z2:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Y2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Z2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((X2 = Y2) => ((((((tp @ A) @ B) @ R2) @ X2) @ Z2) = (((((tp @ A) @ B) @ R2) @ Y2) @ Z2)))))))))).
% 2.88/1.45 thf(fact_106_Ring_OrMulLC, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),X2:A,Y2:A,Z2:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Y2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Z2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((tp @ A) @ B) @ R2) @ X2) @ (((((tp @ A) @ B) @ R2) @ Y2) @ Z2)) = (((((tp @ A) @ B) @ R2) @ Y2) @ (((((tp @ A) @ B) @ R2) @ X2) @ Z2)))))))))).
% 2.88/1.45 thf(fact_107_Ring_Opop__commute, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),A3:A,B3:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ A3) @ B3) = (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ B3) @ A3)))))))).
% 2.88/1.45 thf(fact_108_Ring_Opop__aassoc, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),A3:A,B3:A,C3:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ C3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ A3) @ B3)) @ C3) = (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ A3) @ (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ B3) @ C3)))))))))).
% 2.88/1.45 thf(fact_109_Ring_OnsEqElm, axiom, (! [A:$tType,B:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),X2:A,Y2:A,N2:nat]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Y2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((X2 = Y2) => ((((((nscal @ A) @ B) @ R2) @ X2) @ N2) = (((((nscal @ A) @ B) @ R2) @ Y2) @ N2))))))))).
% 2.88/1.45 thf(fact_110_Ring_OnsClose, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),X2:A,N2:nat]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => (((member @ A) @ (((((nscal @ A) @ B) @ R2) @ X2) @ N2)) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2))))))).
% 2.88/1.45 thf(fact_111_prod__pOp__mem__i, axiom, (! [C:$tType,B:$tType,A:$tType,I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C))),X4:(A > B),Y:(A > B),I3:A]: (((! [X:A]: (((((member @ A) @ X) @ I2) => (((aGroup @ B) @ C) @ (A2 @ X))))) => ((((member @ (A > B)) @ X4) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)) => ((((member @ (A > B)) @ Y) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)) => ((((member @ A) @ I3) @ I2) => (((((((((prod_pOp @ A) @ B) @ C) @ I2) @ A2) @ X4) @ Y) @ I3) = (((((pop @ B) @ C) @ (A2 @ I3)) @ (X4 @ I3)) @ (Y @ I3)))))))))).
% 2.88/1.45 thf(fact_112_dsum__pOp__mem, axiom, (! [C:$tType,B:$tType,A:$tType,I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C))),X4:(A > B),Y:(A > B)]: (((! [X:A]: (((((member @ A) @ X) @ I2) => (((aGroup @ B) @ C) @ (A2 @ X))))) => ((((member @ (A > B)) @ X4) @ (((((carr_dsumag @ A) @ B) @ C) @ I2) @ A2)) => ((((member @ (A > B)) @ Y) @ (((((carr_dsumag @ A) @ B) @ C) @ I2) @ A2)) => (((member @ (A > B)) @ (((((((prod_pOp @ A) @ B) @ C) @ I2) @ A2) @ X4) @ Y)) @ (((((carr_dsumag @ A) @ B) @ C) @ I2) @ A2)))))))).
% 2.88/1.45 thf(fact_113_dsum__iOp__mem, axiom, (! [C:$tType,B:$tType,A:$tType,I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C))),X4:(A > B)]: (((! [X:A]: (((((member @ A) @ X) @ I2) => (((aGroup @ B) @ C) @ (A2 @ X))))) => ((((member @ (A > B)) @ X4) @ (((((carr_dsumag @ A) @ B) @ C) @ I2) @ A2)) => (((member @ (A > B)) @ ((((((prod_mOp @ A) @ B) @ C) @ I2) @ A2) @ X4)) @ (((((carr_dsumag @ A) @ B) @ C) @ I2) @ A2))))))).
% 2.88/1.45 thf(fact_114_dsum__zero__func, axiom, (! [C:$tType,B:$tType,A:$tType,I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C)))]: (((! [X:A]: (((((member @ A) @ X) @ I2) => (((aGroup @ B) @ C) @ (A2 @ X))))) => (((member @ (A > B)) @ (((((prod_zero @ A) @ B) @ C) @ I2) @ A2)) @ (((((carr_dsumag @ A) @ B) @ C) @ I2) @ A2)))))).
% 2.88/1.45 thf(fact_115_aGroup_Opop__closed, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B))]: (((((aGroup @ A) @ B) @ A2) => (((member @ (A > (A > A))) @ (((pop @ A) @ B) @ A2)) @ ((((pi @ A) @ (A > A)) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) @ (^ [Uu:A]: (((((pi @ A) @ A) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) @ (^ [Uv:A]: ((((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)))))))))))).
% 2.88/1.45 thf(fact_116_Ring_OJ__rad__mem, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),X2:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ X2) @ (((j_rad @ A) @ B) @ R2)) => (((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2))))))).
% 2.88/1.45 thf(fact_117_Ring_Otp__closed, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((ring @ A) @ B) @ R2) => (((member @ (A > (A > A))) @ (((tp @ A) @ B) @ R2)) @ ((((pi @ A) @ (A > A)) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) @ (^ [Uu:A]: (((((pi @ A) @ A) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) @ (^ [Uv:A]: ((((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)))))))))))).
% 2.88/1.45 thf(fact_118_Ring_Opop__closed, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((ring @ A) @ B) @ R2) => (((member @ (A > (A > A))) @ (((pop @ A) @ ((ring_ext @ A) @ B)) @ R2)) @ ((((pi @ A) @ (A > A)) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) @ (^ [Uu:A]: (((((pi @ A) @ A) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) @ (^ [Uv:A]: ((((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)))))))))))).
% 2.88/1.45 thf(fact_119_Ring_OprodM__sprod__val, axiom, (! [B:$tType,A:$tType,E:$tType,D:$tType,C:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),I2:(set @ C),M3:(C > ((carrier_ext @ D) @ ((aGroup_ext @ D) @ (((module_ext @ D) @ A) @ E)))),A3:A,M4:(C > D),J:C]: (((((ring @ A) @ B) @ R2) => ((! [X:C]: (((((member @ C) @ X) @ I2) => ((((((module @ D) @ A) @ E) @ B) @ (M3 @ X)) @ R2)))) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ (C > D)) @ M4) @ (((((carr_prodag @ C) @ D) @ (((module_ext @ D) @ A) @ E)) @ I2) @ M3)) => ((((member @ C) @ J) @ I2) => ((((((((((((algebr142713364_sprod @ A) @ B) @ C) @ D) @ E) @ R2) @ I2) @ M3) @ A3) @ M4) @ J) = ((((((sprod @ D) @ A) @ E) @ (M3 @ J)) @ A3) @ (M4 @ J))))))))))).
% 2.88/1.45 thf(fact_120_dsum__iOp__func, axiom, (! [C:$tType,B:$tType,A:$tType,I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C)))]: (((! [X:A]: (((((member @ A) @ X) @ I2) => (((aGroup @ B) @ C) @ (A2 @ X))))) => (((member @ ((A > B) > (A > B))) @ (((((prod_mOp @ A) @ B) @ C) @ I2) @ A2)) @ ((((pi @ (A > B)) @ (A > B)) @ (((((carr_dsumag @ A) @ B) @ C) @ I2) @ A2)) @ (^ [Uu:(A > B)]: ((((((carr_dsumag @ A) @ B) @ C) @ I2) @ A2))))))))).
% 2.88/1.45 thf(fact_121_zeroring__J__rad__empty, axiom, ((((zeroring @ a) @ b) @ r) => ((((j_rad @ a) @ b) @ r) = (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)))).
% 2.88/1.45 thf(fact_122_Module_OmHom__lin, axiom, (! [D:$tType,C:$tType,B:$tType,F2:$tType,E:$tType,A:$tType,M3:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),N:((carrier_ext @ E) @ ((aGroup_ext @ E) @ (((module_ext @ E) @ B) @ F2))),M4:A,F:(A > E),A3:B]: ((((((((module @ A) @ B) @ C) @ D) @ M3) @ R2) => (((((((module @ E) @ B) @ F2) @ D) @ N) @ R2) => ((((member @ A) @ M4) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M3)) => ((((member @ (A > E)) @ F) @ (((((((((mHom @ B) @ D) @ A) @ C) @ E) @ F2) @ R2) @ M3) @ N)) => ((((member @ B) @ A3) @ (((carrier @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))) @ R2)) => ((F @ ((((((sprod @ A) @ B) @ C) @ M3) @ A3) @ M4)) = ((((((sprod @ E) @ B) @ F2) @ N) @ A3) @ (F @ M4))))))))))).
% 2.88/1.45 thf(fact_123_Module_Osprod__assoc, axiom, (! [D:$tType,B:$tType,C:$tType,A:$tType,M3:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),A3:B,B3:B,M4:A]: ((((((((module @ A) @ B) @ C) @ D) @ M3) @ R2) => ((((member @ B) @ A3) @ (((carrier @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))) @ R2)) => ((((member @ B) @ B3) @ (((carrier @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))) @ R2)) => ((((member @ A) @ M4) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M3)) => (((((((sprod @ A) @ B) @ C) @ M3) @ (((((tp @ B) @ D) @ R2) @ A3) @ B3)) @ M4) = ((((((sprod @ A) @ B) @ C) @ M3) @ A3) @ ((((((sprod @ A) @ B) @ C) @ M3) @ B3) @ M4)))))))))).
% 2.88/1.45 thf(fact_124_Ring_OmHom__func, axiom, (! [B:$tType,E:$tType,D:$tType,A:$tType,F2:$tType,C:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),F:(C > D),M3:((carrier_ext @ C) @ ((aGroup_ext @ C) @ (((module_ext @ C) @ A) @ E))),N:((carrier_ext @ D) @ ((aGroup_ext @ D) @ (((module_ext @ D) @ A) @ F2)))]: (((((ring @ A) @ B) @ R2) => ((((member @ (C > D)) @ F) @ (((((((((mHom @ A) @ B) @ C) @ E) @ D) @ F2) @ R2) @ M3) @ N)) => (((member @ (C > D)) @ F) @ ((((pi @ C) @ D) @ (((carrier @ C) @ ((aGroup_ext @ C) @ (((module_ext @ C) @ A) @ E))) @ M3)) @ (^ [Uu:C]: ((((carrier @ D) @ ((aGroup_ext @ D) @ (((module_ext @ D) @ A) @ F2))) @ N)))))))))).
% 2.88/1.45 thf(fact_125_Ring_OSubring__pOp__ring__pOp, axiom, (! [C:$tType,B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),S:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ C))),A3:A,B3:A]: (((((ring @ A) @ B) @ R2) => ((((((subring @ A) @ B) @ C) @ R2) @ S) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ C))) @ S)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ C))) @ S)) => ((((((pop @ A) @ ((ring_ext @ A) @ C)) @ S) @ A3) @ B3) = (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ A3) @ B3))))))))).
% 2.88/1.45 thf(fact_126_Ring_OSubring__tOp__ring__tOp, axiom, (! [C:$tType,B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),S:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ C))),A3:A,B3:A]: (((((ring @ A) @ B) @ R2) => ((((((subring @ A) @ B) @ C) @ R2) @ S) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ C))) @ S)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ C))) @ S)) => ((((((tp @ A) @ C) @ S) @ A3) @ B3) = (((((tp @ A) @ B) @ R2) @ A3) @ B3))))))))).
% 2.88/1.45 thf(fact_127_Module_Osprod__closed, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,M3:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),A3:B,M4:A]: ((((((((module @ A) @ B) @ C) @ D) @ M3) @ R2) => ((((member @ B) @ A3) @ (((carrier @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))) @ R2)) => ((((member @ A) @ M4) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M3)) => (((member @ A) @ ((((((sprod @ A) @ B) @ C) @ M3) @ A3) @ M4)) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M3)))))))).
% 2.88/1.45 thf(fact_128_nsDistr, axiom, (! [X2:a,N2:nat,M4:nat]: (((((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((nscal @ a) @ b) @ r) @ X2) @ N2)) @ (((((nscal @ a) @ b) @ r) @ X2) @ M4)) = (((((nscal @ a) @ b) @ r) @ X2) @ (((plus_plus @ nat) @ N2) @ M4))))))).
% 2.88/1.45 thf(fact_129_add__bothl, axiom, (! [A:$tType]: (((plus @ A) => (! [A3:A,B3:A,C3:A]: (((A3 = B3) => ((((plus_plus @ A) @ C3) @ A3) = (((plus_plus @ A) @ C3) @ B3))))))))).
% 2.88/1.45 thf(fact_130_add__both, axiom, (! [A3:nat,B3:nat,C3:nat]: (((A3 = B3) => ((((plus_plus @ nat) @ A3) @ C3) = (((plus_plus @ nat) @ B3) @ C3)))))).
% 2.88/1.45 thf(fact_131_Module_OmHom__add, axiom, (! [D:$tType,C:$tType,F2:$tType,B:$tType,E:$tType,A:$tType,M3:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),N:((carrier_ext @ E) @ ((aGroup_ext @ E) @ (((module_ext @ E) @ B) @ F2))),F:(A > E),M4:A,N2:A]: ((((((((module @ A) @ B) @ C) @ D) @ M3) @ R2) => (((((((module @ E) @ B) @ F2) @ D) @ N) @ R2) => ((((member @ (A > E)) @ F) @ (((((((((mHom @ B) @ D) @ A) @ C) @ E) @ F2) @ R2) @ M3) @ N)) => ((((member @ A) @ M4) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M3)) => ((((member @ A) @ N2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M3)) => ((F @ (((((pop @ A) @ (((module_ext @ A) @ B) @ C)) @ M3) @ M4) @ N2)) = (((((pop @ E) @ (((module_ext @ E) @ B) @ F2)) @ N) @ (F @ M4)) @ (F @ N2))))))))))).
% 2.88/1.45 thf(fact_132_Module_Osprod__r__distr, axiom, (! [D:$tType,B:$tType,C:$tType,A:$tType,M3:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),A3:B,M4:A,N2:A]: ((((((((module @ A) @ B) @ C) @ D) @ M3) @ R2) => ((((member @ B) @ A3) @ (((carrier @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))) @ R2)) => ((((member @ A) @ M4) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M3)) => ((((member @ A) @ N2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M3)) => (((((((sprod @ A) @ B) @ C) @ M3) @ A3) @ (((((pop @ A) @ (((module_ext @ A) @ B) @ C)) @ M3) @ M4) @ N2)) = (((((pop @ A) @ (((module_ext @ A) @ B) @ C)) @ M3) @ ((((((sprod @ A) @ B) @ C) @ M3) @ A3) @ M4)) @ ((((((sprod @ A) @ B) @ C) @ M3) @ A3) @ N2)))))))))).
% 2.88/1.45 thf(fact_133_Module_Omodule__is__ag, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,M3:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D)))]: ((((((((module @ A) @ B) @ C) @ D) @ M3) @ R2) => (((aGroup @ A) @ (((module_ext @ A) @ B) @ C)) @ M3))))).
% 2.88/1.45 thf(fact_134_Module_Osprod__l__distr, axiom, (! [D:$tType,B:$tType,C:$tType,A:$tType,M3:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),A3:B,B3:B,M4:A]: ((((((((module @ A) @ B) @ C) @ D) @ M3) @ R2) => ((((member @ B) @ A3) @ (((carrier @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))) @ R2)) => ((((member @ B) @ B3) @ (((carrier @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))) @ R2)) => ((((member @ A) @ M4) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M3)) => (((((((sprod @ A) @ B) @ C) @ M3) @ (((((pop @ B) @ ((ring_ext @ B) @ D)) @ R2) @ A3) @ B3)) @ M4) = (((((pop @ A) @ (((module_ext @ A) @ B) @ C)) @ M3) @ ((((((sprod @ A) @ B) @ C) @ M3) @ A3) @ M4)) @ ((((((sprod @ A) @ B) @ C) @ M3) @ B3) @ M4)))))))))).
% 2.88/1.45 thf(fact_135_Ring_OnsDistr, axiom, (! [A:$tType,B:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),X2:A,N2:nat,M4:nat]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ (((((nscal @ A) @ B) @ R2) @ X2) @ N2)) @ (((((nscal @ A) @ B) @ R2) @ X2) @ M4)) = (((((nscal @ A) @ B) @ R2) @ X2) @ (((plus_plus @ nat) @ N2) @ M4)))))))).
% 2.88/1.45 thf(fact_136_Ring_Omul__closed__set__tOp__closed, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),S:(set @ A),S2:A,T:A]: (((((ring @ A) @ B) @ R2) => (((((mul_closed_set @ A) @ B) @ R2) @ S) => ((((member @ A) @ S2) @ S) => ((((member @ A) @ T) @ S) => (((member @ A) @ (((((tp @ A) @ B) @ R2) @ S2) @ T)) @ S)))))))).
% 2.88/1.45 thf(fact_137_Ring_Ozeroring__J__rad__empty, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((ring @ A) @ B) @ R2) => ((((zeroring @ A) @ B) @ R2) => ((((j_rad @ A) @ B) @ R2) = (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2))))))).
% 2.88/1.45 thf(fact_138_Module_Osc__Ring, axiom, (! [C:$tType,A:$tType,D:$tType,B:$tType,M3:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D)))]: ((((((((module @ A) @ B) @ C) @ D) @ M3) @ R2) => (((ring @ B) @ D) @ R2))))).
% 2.88/1.45 thf(fact_139_Ring_Osubring__Ring, axiom, (! [B:$tType,C:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),S:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ C)))]: (((((ring @ A) @ B) @ R2) => ((((((subring @ A) @ B) @ C) @ R2) @ S) => (((ring @ A) @ C) @ S)))))).
% 2.88/1.45 thf(fact_140_Module_OmHom__eq, axiom, (! [F2:$tType,D:$tType,C:$tType,B:$tType,E:$tType,A:$tType,M3:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),N:((carrier_ext @ E) @ ((aGroup_ext @ E) @ (((module_ext @ E) @ B) @ F2))),F:(A > E),G:(A > E)]: ((((((((module @ A) @ B) @ C) @ D) @ M3) @ R2) => (((((((module @ E) @ B) @ F2) @ D) @ N) @ R2) => ((((member @ (A > E)) @ F) @ (((((((((mHom @ B) @ D) @ A) @ C) @ E) @ F2) @ R2) @ M3) @ N)) => ((((member @ (A > E)) @ G) @ (((((((((mHom @ B) @ D) @ A) @ C) @ E) @ F2) @ R2) @ M3) @ N)) => ((! [X:A]: (((((member @ A) @ X) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M3)) => ((F @ X) = (G @ X))))) => (F = G))))))))).
% 2.88/1.45 thf(fact_141_Module_OmHom__mem, axiom, (! [C:$tType,D:$tType,A:$tType,F2:$tType,B:$tType,E:$tType,M3:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),N:((carrier_ext @ E) @ ((aGroup_ext @ E) @ (((module_ext @ E) @ B) @ F2))),F:(A > E),M4:A]: ((((((((module @ A) @ B) @ C) @ D) @ M3) @ R2) => (((((((module @ E) @ B) @ F2) @ D) @ N) @ R2) => ((((member @ (A > E)) @ F) @ (((((((((mHom @ B) @ D) @ A) @ C) @ E) @ F2) @ R2) @ M3) @ N)) => ((((member @ A) @ M4) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M3)) => (((member @ E) @ (F @ M4)) @ (((carrier @ E) @ ((aGroup_ext @ E) @ (((module_ext @ E) @ B) @ F2))) @ N))))))))).
% 2.88/1.45 thf(fact_142_Ring_Omem__subring__mem__ring, axiom, (! [C:$tType,B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),S:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ C))),X2:A]: (((((ring @ A) @ B) @ R2) => ((((((subring @ A) @ B) @ C) @ R2) @ S) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ C))) @ S)) => (((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)))))))).
% 2.88/1.45 thf(fact_143_mul__closed__set__sub, axiom, (! [S:(set @ a)]: ((((((mul_closed_set @ a) @ b) @ r) @ S) => (((ord_less_eq @ (set @ a)) @ S) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)))))).
% 2.88/1.45 thf(fact_144_zeroring__no__maximal, axiom, ((((zeroring @ a) @ b) @ r) => (~ ((? [X_1:(set @ a)]: (((((maximal_ideal @ a) @ b) @ r) @ X_1))))))).
% 2.88/1.45 thf(fact_145_id__maximal__Exist, axiom, ((~ ((((zeroring @ a) @ b) @ r))) => (? [X_12:(set @ a)]: (((((maximal_ideal @ a) @ b) @ r) @ X_12))))).
% 2.88/1.45 thf(fact_146_mop__closed, axiom, (((member @ (a > a)) @ (((mop @ a) @ ((ring_ext @ a) @ b)) @ r)) @ ((((pi @ a) @ a) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) @ (^ [Uu:a]: ((((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)))))).
% 2.88/1.45 thf(fact_147_Sr__mOp__closed, axiom, (! [S:(set @ a),X2:a]: ((((((sr2 @ a) @ b) @ r) @ S) => ((((member @ a) @ X2) @ S) => (((member @ a) @ ((((mop @ a) @ ((ring_ext @ a) @ b)) @ r) @ X2)) @ S)))))).
% 2.88/1.45 thf(fact_148_ring__inv1, axiom, (! [A3:a,B3:a]: (((((member @ a) @ A3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mop @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((tp @ a) @ b) @ r) @ A3) @ B3)) = (((((tp @ a) @ b) @ r) @ ((((mop @ a) @ ((ring_ext @ a) @ b)) @ r) @ A3)) @ B3)) & (((((mop @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((tp @ a) @ b) @ r) @ A3) @ B3)) = (((((tp @ a) @ b) @ r) @ A3) @ ((((mop @ a) @ ((ring_ext @ a) @ b)) @ r) @ B3))))))))).
% 2.88/1.45 thf(fact_149_ring__inv1__1, axiom, (! [A3:a,B3:a]: (((((member @ a) @ A3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((mop @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((tp @ a) @ b) @ r) @ A3) @ B3)) = (((((tp @ a) @ b) @ r) @ ((((mop @ a) @ ((ring_ext @ a) @ b)) @ r) @ A3)) @ B3))))))).
% 2.88/1.45 thf(fact_150_ring__inv1__2, axiom, (! [A3:a,B3:a]: (((((member @ a) @ A3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((mop @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((tp @ a) @ b) @ r) @ A3) @ B3)) = (((((tp @ a) @ b) @ r) @ A3) @ ((((mop @ a) @ ((ring_ext @ a) @ b)) @ r) @ B3)))))))).
% 2.88/1.45 thf(fact_151_ring__inv1__3, axiom, (! [A3:a,B3:a]: (((((member @ a) @ A3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((tp @ a) @ b) @ r) @ A3) @ B3) = (((((tp @ a) @ b) @ r) @ ((((mop @ a) @ ((ring_ext @ a) @ b)) @ r) @ A3)) @ ((((mop @ a) @ ((ring_ext @ a) @ b)) @ r) @ B3)))))))).
% 2.88/1.45 thf(fact_152_ring__distrib4, axiom, (! [A3:a,B3:a,X2:a,Y2:a]: (((((member @ a) @ A3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((tp @ a) @ b) @ r) @ A3) @ B3)) @ ((((mop @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((tp @ a) @ b) @ r) @ X2) @ Y2))) = (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((tp @ a) @ b) @ r) @ A3) @ (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ B3) @ ((((mop @ a) @ ((ring_ext @ a) @ b)) @ r) @ Y2)))) @ (((((tp @ a) @ b) @ r) @ (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ A3) @ ((((mop @ a) @ ((ring_ext @ a) @ b)) @ r) @ X2))) @ Y2)))))))))).
% 2.88/1.45 thf(fact_153_Subring__minus__ring__minus, axiom, (! [C:$tType,S:((carrier_ext @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ C))),X2:a]: (((((((subring @ a) @ b) @ C) @ r) @ S) => ((((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ C))) @ S)) => (((((mop @ a) @ ((ring_ext @ a) @ C)) @ S) @ X2) = ((((mop @ a) @ ((ring_ext @ a) @ b)) @ r) @ X2))))))).
% 2.88/1.45 thf(fact_154_not__sub, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((~ ((((ord_less_eq @ (set @ A)) @ A2) @ B2))) => (? [A6:A]: (((((member @ A) @ A6) @ A2) & (~ ((((member @ A) @ A6) @ B2)))))))))).
% 2.88/1.45 thf(fact_155_eqsets__sub, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.88/1.45 thf(fact_156_ge__convert, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C3:A]: (((A3 = B3) => ((((ord_less_eq @ A) @ C3) @ A3) => (((ord_less_eq @ A) @ C3) @ B3))))))))).
% 2.88/1.45 thf(fact_157_le__convert, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C3:A]: (((A3 = B3) => ((((ord_less_eq @ A) @ A3) @ C3) => (((ord_less_eq @ A) @ B3) @ C3))))))))).
% 2.88/1.45 thf(fact_158_sub__which1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X2:A]: ((((((ord_less_eq @ (set @ A)) @ A2) @ B2) | (((ord_less_eq @ (set @ A)) @ B2) @ A2)) => ((((member @ A) @ X2) @ A2) => ((~ ((((member @ A) @ X2) @ B2))) => (((ord_less_eq @ (set @ A)) @ B2) @ A2))))))).
% 2.88/1.45 thf(fact_159_sub__which2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X2:A]: ((((((ord_less_eq @ (set @ A)) @ A2) @ B2) | (((ord_less_eq @ (set @ A)) @ B2) @ A2)) => ((~ ((((member @ A) @ X2) @ A2))) => ((((member @ A) @ X2) @ B2) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))))).
% 2.88/1.45 thf(fact_160_sets__not__eq, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 != B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A2) => (? [X:A]: (((((member @ A) @ X) @ A2) & (~ ((((member @ A) @ X) @ B2))))))))))).
% 2.88/1.45 thf(fact_161_subset__self, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ A2)))).
% 2.88/1.45 thf(fact_162_not__subseteq, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((~ ((((ord_less_eq @ (set @ A)) @ A2) @ B2))) => (? [X:A]: (((((member @ A) @ X) @ A2) & (~ ((((member @ A) @ X) @ B2)))))))))).
% 2.88/1.45 thf(fact_163_proper__subset, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((~ ((((member @ A) @ X2) @ A2))) => ((((member @ A) @ X2) @ B2) => (A2 != B2))))))).
% 2.88/1.45 thf(fact_164_conditional__subset, axiom, (! [A:$tType,A2:(set @ A),P:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ A2) & (P @ X3)))))) @ A2)))).
% 2.88/1.45 thf(fact_165_Nset__le, axiom, (! [A:$tType]: (((ord @ A) => (! [X2:A,N2:A]: (((((member @ A) @ X2) @ ((collect @ A) @ (^ [I4:A]: ((((ord_less_eq @ A) @ I4) @ N2))))) => (((ord_less_eq @ A) @ X2) @ N2)))))))).
% 2.88/1.45 thf(fact_166_Pi__mono, axiom, (! [B:$tType,A:$tType,A2:(set @ A),B2:(A > (set @ B)),C2:(A > (set @ B))]: (((! [X:A]: (((((member @ A) @ X) @ A2) => (((ord_less_eq @ (set @ B)) @ (B2 @ X)) @ (C2 @ X))))) => (((ord_less_eq @ (set @ (A > B))) @ ((((pi @ A) @ B) @ A2) @ B2)) @ ((((pi @ A) @ B) @ A2) @ C2)))))).
% 2.88/1.45 thf(fact_167_Pi__anti__mono, axiom, (! [B:$tType,A:$tType,A7:(set @ A),A2:(set @ A),B2:(A > (set @ B))]: (((((ord_less_eq @ (set @ A)) @ A7) @ A2) => (((ord_less_eq @ (set @ (A > B))) @ ((((pi @ A) @ B) @ A2) @ B2)) @ ((((pi @ A) @ B) @ A7) @ B2)))))).
% 2.88/1.45 thf(fact_168_extend__fun, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),B2:(set @ B),B1:(set @ B)]: (((((member @ (A > B)) @ F) @ ((((pi @ A) @ B) @ A2) @ (^ [Uu:A]: (B2)))) => ((((ord_less_eq @ (set @ B)) @ B2) @ B1) => (((member @ (A > B)) @ F) @ ((((pi @ A) @ B) @ A2) @ (^ [Uu:A]: (B1))))))))).
% 2.88/1.45 thf(fact_169_aGroup_Oag__mOp__closed, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),X2:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => (((member @ A) @ ((((mop @ A) @ B) @ A2) @ X2)) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2))))))).
% 2.88/1.45 thf(fact_170_aGroup_Oag__inv__inv, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),X2:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => (((((mop @ A) @ B) @ A2) @ ((((mop @ A) @ B) @ A2) @ X2)) = X2)))))).
% 2.88/1.45 thf(fact_171_aGroup_Oag__inv__inj, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),X2:A,Y2:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ Y2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((X2 != Y2) => (((((mop @ A) @ B) @ A2) @ X2) != ((((mop @ A) @ B) @ A2) @ Y2))))))))).
% 2.88/1.45 thf(fact_172_Ring_OSr__mOp__closed, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),S:(set @ A),X2:A]: (((((ring @ A) @ B) @ R2) => (((((sr2 @ A) @ B) @ R2) @ S) => ((((member @ A) @ X2) @ S) => (((member @ A) @ ((((mop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ X2)) @ S))))))).
% 2.88/1.45 thf(fact_173_Subring__subset, axiom, (! [C:$tType,B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),S:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ C)))]: (((((((subring @ A) @ B) @ C) @ R2) @ S) => (((ord_less_eq @ (set @ A)) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ C))) @ S)) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)))))).
% 2.88/1.45 thf(fact_174_Ring_Oring__inv1__3, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),A3:A,B3:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((tp @ A) @ B) @ R2) @ A3) @ B3) = (((((tp @ A) @ B) @ R2) @ ((((mop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ A3)) @ ((((mop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ B3))))))))).
% 2.88/1.45 thf(fact_175_Ring_Oring__inv1__2, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),A3:A,B3:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => (((((mop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ (((((tp @ A) @ B) @ R2) @ A3) @ B3)) = (((((tp @ A) @ B) @ R2) @ A3) @ ((((mop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ B3))))))))).
% 2.88/1.45 thf(fact_176_Ring_Oring__inv1__1, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),A3:A,B3:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => (((((mop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ (((((tp @ A) @ B) @ R2) @ A3) @ B3)) = (((((tp @ A) @ B) @ R2) @ ((((mop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ A3)) @ B3)))))))).
% 2.88/1.45 thf(fact_177_Ring_Oring__inv1, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),A3:A,B3:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((mop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ (((((tp @ A) @ B) @ R2) @ A3) @ B3)) = (((((tp @ A) @ B) @ R2) @ ((((mop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ A3)) @ B3)) & (((((mop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ (((((tp @ A) @ B) @ R2) @ A3) @ B3)) = (((((tp @ A) @ B) @ R2) @ A3) @ ((((mop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ B3)))))))))).
% 2.88/1.45 thf(fact_178_Ring_Oid__maximal__Exist, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((ring @ A) @ B) @ R2) => ((~ ((((zeroring @ A) @ B) @ R2))) => (? [X_12:(set @ A)]: (((((maximal_ideal @ A) @ B) @ R2) @ X_12)))))))).
% 2.88/1.45 thf(fact_179_Ring_Ozeroring__no__maximal, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((ring @ A) @ B) @ R2) => ((((zeroring @ A) @ B) @ R2) => (~ ((? [X_1:(set @ A)]: (((((maximal_ideal @ A) @ B) @ R2) @ X_1)))))))))).
% 2.88/1.45 thf(fact_180_Ring_OSubring__minus__ring__minus, axiom, (! [C:$tType,B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),S:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ C))),X2:A]: (((((ring @ A) @ B) @ R2) => ((((((subring @ A) @ B) @ C) @ R2) @ S) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ C))) @ S)) => (((((mop @ A) @ ((ring_ext @ A) @ C)) @ S) @ X2) = ((((mop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ X2)))))))).
% 2.88/1.45 thf(fact_181_Ring_Omop__closed, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((ring @ A) @ B) @ R2) => (((member @ (A > A)) @ (((mop @ A) @ ((ring_ext @ A) @ B)) @ R2)) @ ((((pi @ A) @ A) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) @ (^ [Uu:A]: ((((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2))))))))).
% 2.88/1.45 thf(fact_182_aGroup_OpOp__assoc__cancel, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),A3:A,B3:A,C3:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ C3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((((pop @ A) @ B) @ A2) @ (((((pop @ A) @ B) @ A2) @ A3) @ ((((mop @ A) @ B) @ A2) @ B3))) @ (((((pop @ A) @ B) @ A2) @ B3) @ ((((mop @ A) @ B) @ A2) @ C3))) = (((((pop @ A) @ B) @ A2) @ A3) @ ((((mop @ A) @ B) @ A2) @ C3)))))))))).
% 2.88/1.45 thf(fact_183_aGroup_Oag__diff__minus, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),A3:A,B3:A,C3:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ C3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => (((((((pop @ A) @ B) @ A2) @ A3) @ ((((mop @ A) @ B) @ A2) @ B3)) = C3) => ((((((pop @ A) @ B) @ A2) @ B3) @ ((((mop @ A) @ B) @ A2) @ A3)) = ((((mop @ A) @ B) @ A2) @ C3)))))))))).
% 2.88/1.45 thf(fact_184_aGroup_Oag__eq__sol2, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),A3:A,X2:A,B3:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => (((((((pop @ A) @ B) @ A2) @ X2) @ A3) = B3) => (X2 = (((((pop @ A) @ B) @ A2) @ B3) @ ((((mop @ A) @ B) @ A2) @ A3))))))))))).
% 2.88/1.45 thf(fact_185_aGroup_Oag__eq__sol1, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),A3:A,X2:A,B3:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => (((((((pop @ A) @ B) @ A2) @ A3) @ X2) = B3) => (X2 = (((((pop @ A) @ B) @ A2) @ ((((mop @ A) @ B) @ A2) @ A3)) @ B3)))))))))).
% 2.88/1.45 thf(fact_186_aGroup_Oag__p__inv, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),X2:A,Y2:A]: (((((aGroup @ A) @ B) @ A2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ Y2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => (((((mop @ A) @ B) @ A2) @ (((((pop @ A) @ B) @ A2) @ X2) @ Y2)) = (((((pop @ A) @ B) @ A2) @ ((((mop @ A) @ B) @ A2) @ X2)) @ ((((mop @ A) @ B) @ A2) @ Y2))))))))).
% 2.88/1.45 thf(fact_187_Ring_Omul__closed__set__sub, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),S:(set @ A)]: (((((ring @ A) @ B) @ R2) => (((((mul_closed_set @ A) @ B) @ R2) @ S) => (((ord_less_eq @ (set @ A)) @ S) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2))))))).
% 2.88/1.45 thf(fact_188_mul__closed__set__def, axiom, (! [B:$tType,A:$tType]: ((((mul_closed_set @ A) @ B) = (^ [R3:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),S3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) & (! [X3:A]: (((((member @ A) @ X3) @ S3) => (! [Y3:A]: (((((member @ A) @ Y3) @ S3) => (((member @ A) @ (((((tp @ A) @ B) @ R3) @ X3) @ Y3)) @ S3)))))))))))))).
% 2.88/1.45 thf(fact_189_aGroup_Omop__closed, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B))]: (((((aGroup @ A) @ B) @ A2) => (((member @ (A > A)) @ (((mop @ A) @ B) @ A2)) @ ((((pi @ A) @ A) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) @ (^ [Uu:A]: ((((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2))))))))).
% 2.88/1.45 thf(fact_190_prod__mOp__mem__i, axiom, (! [C:$tType,B:$tType,A:$tType,I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C))),X4:(A > B),I3:A]: (((! [X:A]: (((((member @ A) @ X) @ I2) => (((aGroup @ B) @ C) @ (A2 @ X))))) => ((((member @ (A > B)) @ X4) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)) => ((((member @ A) @ I3) @ I2) => ((((((((prod_mOp @ A) @ B) @ C) @ I2) @ A2) @ X4) @ I3) = ((((mop @ B) @ C) @ (A2 @ I3)) @ (X4 @ I3))))))))).
% 2.88/1.45 thf(fact_191_Ring_Oring__distrib4, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),A3:A,B3:A,X2:A,Y2:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ B3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ Y2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ (((((tp @ A) @ B) @ R2) @ A3) @ B3)) @ ((((mop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ (((((tp @ A) @ B) @ R2) @ X2) @ Y2))) = (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ (((((tp @ A) @ B) @ R2) @ A3) @ (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ B3) @ ((((mop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ Y2)))) @ (((((tp @ A) @ B) @ R2) @ (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ A3) @ ((((mop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ X2))) @ Y2))))))))))).
% 2.88/1.45 thf(fact_192_nonunit__contained__maxid, axiom, (! [A3:a]: (((~ ((((zeroring @ a) @ b) @ r))) => ((((member @ a) @ A3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((~ (((((unit @ a) @ b) @ r) @ A3))) => (? [Mx:(set @ a)]: ((((((maximal_ideal @ a) @ b) @ r) @ Mx) & (((member @ a) @ A3) @ Mx)))))))))).
% 2.88/1.45 thf(fact_193_set__sum__mem, axiom, (! [A3:a,I2:(set @ a),B3:a,J2:(set @ a)]: (((((member @ a) @ A3) @ I2) => ((((member @ a) @ B3) @ J2) => ((((ord_less_eq @ (set @ a)) @ I2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((ord_less_eq @ (set @ a)) @ J2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((member @ a) @ (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ A3) @ B3)) @ (((((aset_sum @ a) @ ((ring_ext @ a) @ b)) @ r) @ I2) @ J2))))))))).
% 2.88/1.45 thf(fact_194_sum__mult__pOp__closed, axiom, (! [A2:(set @ a),B2:(set @ a),A3:a,B3:a]: (((((ord_less_eq @ (set @ a)) @ A2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((ord_less_eq @ (set @ a)) @ B2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ A3) @ (((((sum_mult @ a) @ b) @ r) @ A2) @ B2)) => ((((member @ a) @ B3) @ (((((sum_mult @ a) @ b) @ r) @ A2) @ B2)) => (((member @ a) @ (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ A3) @ B3)) @ (((((sum_mult @ a) @ b) @ r) @ A2) @ B2))))))))).
% 2.88/1.45 thf(fact_195_sum__mult__subR, axiom, (! [A2:(set @ a),B2:(set @ a)]: (((((ord_less_eq @ (set @ a)) @ A2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((ord_less_eq @ (set @ a)) @ B2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((ord_less_eq @ (set @ a)) @ (((((sum_mult @ a) @ b) @ r) @ A2) @ B2)) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r))))))).
% 2.88/1.45 thf(fact_196_times__mem__sum__mult, axiom, (! [A2:(set @ a),B2:(set @ a),A3:a,B3:a]: (((((ord_less_eq @ (set @ a)) @ A2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((ord_less_eq @ (set @ a)) @ B2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ A3) @ A2) => ((((member @ a) @ B3) @ B2) => (((member @ a) @ (((((tp @ a) @ b) @ r) @ A3) @ B3)) @ (((((sum_mult @ a) @ b) @ r) @ A2) @ B2))))))))).
% 2.88/1.45 thf(fact_197_le__imp__add__int, axiom, (! [I3:nat,J:nat]: (((((ord_less_eq @ nat) @ I3) @ J) => (? [K2:nat]: ((J = (((plus_plus @ nat) @ I3) @ K2)))))))).
% 2.88/1.45 thf(fact_198_Module_OmHom__inv, axiom, (! [D:$tType,C:$tType,B:$tType,F2:$tType,E:$tType,A:$tType,M3:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),N:((carrier_ext @ E) @ ((aGroup_ext @ E) @ (((module_ext @ E) @ B) @ F2))),M4:A,F:(A > E)]: ((((((((module @ A) @ B) @ C) @ D) @ M3) @ R2) => (((((((module @ E) @ B) @ F2) @ D) @ N) @ R2) => ((((member @ A) @ M4) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M3)) => ((((member @ (A > E)) @ F) @ (((((((((mHom @ B) @ D) @ A) @ C) @ E) @ F2) @ R2) @ M3) @ N)) => ((F @ ((((mop @ A) @ (((module_ext @ A) @ B) @ C)) @ M3) @ M4)) = ((((mop @ E) @ (((module_ext @ E) @ B) @ F2)) @ N) @ (F @ M4)))))))))).
% 2.88/1.45 thf(fact_199_dsumag__sub__prodag, axiom, (! [C:$tType,B:$tType,A:$tType,I2:(set @ A),A2:(A > ((carrier_ext @ B) @ ((aGroup_ext @ B) @ C)))]: (((! [X:A]: (((((member @ A) @ X) @ I2) => (((aGroup @ B) @ C) @ (A2 @ X))))) => (((ord_less_eq @ (set @ (A > B))) @ (((((carr_dsumag @ A) @ B) @ C) @ I2) @ A2)) @ (((((carr_prodag @ A) @ B) @ C) @ I2) @ A2)))))).
% 2.88/1.45 thf(fact_200_Module_Osc__minus__am, axiom, (! [D:$tType,B:$tType,C:$tType,A:$tType,M3:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),A3:B,M4:A]: ((((((((module @ A) @ B) @ C) @ D) @ M3) @ R2) => ((((member @ B) @ A3) @ (((carrier @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))) @ R2)) => ((((member @ A) @ M4) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M3)) => (((((mop @ A) @ (((module_ext @ A) @ B) @ C)) @ M3) @ ((((((sprod @ A) @ B) @ C) @ M3) @ A3) @ M4)) = ((((((sprod @ A) @ B) @ C) @ M3) @ A3) @ ((((mop @ A) @ (((module_ext @ A) @ B) @ C)) @ M3) @ M4))))))))).
% 2.88/1.45 thf(fact_201_Ring_Osum__mult__subR, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),A2:(set @ A),B2:(set @ A)]: (((((ring @ A) @ B) @ R2) => ((((ord_less_eq @ (set @ A)) @ A2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((ord_less_eq @ (set @ A)) @ B2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => (((ord_less_eq @ (set @ A)) @ (((((sum_mult @ A) @ B) @ R2) @ A2) @ B2)) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)))))))).
% 2.88/1.45 thf(fact_202_Module_Osc__minus__am1, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,M3:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),A3:B,M4:A]: ((((((((module @ A) @ B) @ C) @ D) @ M3) @ R2) => ((((member @ B) @ A3) @ (((carrier @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))) @ R2)) => ((((member @ A) @ M4) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M3)) => (((((mop @ A) @ (((module_ext @ A) @ B) @ C)) @ M3) @ ((((((sprod @ A) @ B) @ C) @ M3) @ A3) @ M4)) = ((((((sprod @ A) @ B) @ C) @ M3) @ ((((mop @ B) @ ((ring_ext @ B) @ D)) @ R2) @ A3)) @ M4)))))))).
% 2.88/1.45 thf(fact_203_Ring_Otimes__mem__sum__mult, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),A2:(set @ A),B2:(set @ A),A3:A,B3:A]: (((((ring @ A) @ B) @ R2) => ((((ord_less_eq @ (set @ A)) @ A2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((ord_less_eq @ (set @ A)) @ B2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ A3) @ A2) => ((((member @ A) @ B3) @ B2) => (((member @ A) @ (((((tp @ A) @ B) @ R2) @ A3) @ B3)) @ (((((sum_mult @ A) @ B) @ R2) @ A2) @ B2)))))))))).
% 2.88/1.45 thf(fact_204_Ring_Osum__mult__pOp__closed, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),A2:(set @ A),B2:(set @ A),A3:A,B3:A]: (((((ring @ A) @ B) @ R2) => ((((ord_less_eq @ (set @ A)) @ A2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((ord_less_eq @ (set @ A)) @ B2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((member @ A) @ A3) @ (((((sum_mult @ A) @ B) @ R2) @ A2) @ B2)) => ((((member @ A) @ B3) @ (((((sum_mult @ A) @ B) @ R2) @ A2) @ B2)) => (((member @ A) @ (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ A3) @ B3)) @ (((((sum_mult @ A) @ B) @ R2) @ A2) @ B2)))))))))).
% 2.88/1.45 thf(fact_205_Ring_Oset__sum__mem, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),A3:A,I2:(set @ A),B3:A,J2:(set @ A)]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ A3) @ I2) => ((((member @ A) @ B3) @ J2) => ((((ord_less_eq @ (set @ A)) @ I2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((ord_less_eq @ (set @ A)) @ J2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => (((member @ A) @ (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ A3) @ B3)) @ (((((aset_sum @ A) @ ((ring_ext @ A) @ B)) @ R2) @ I2) @ J2)))))))))).
% 2.88/1.45 thf(fact_206_Ring_Ononunit__contained__maxid, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),A3:A]: (((((ring @ A) @ B) @ R2) => ((~ ((((zeroring @ A) @ B) @ R2))) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((~ (((((unit @ A) @ B) @ R2) @ A3))) => (? [Mx:(set @ A)]: ((((((maximal_ideal @ A) @ B) @ R2) @ Mx) & (((member @ A) @ A3) @ Mx))))))))))).
% 2.88/1.45 thf(fact_207_aGroup_Omem__set__sum, axiom, (! [B:$tType,A:$tType,A2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ B)),H:(set @ A),K3:(set @ A),X2:A]: (((((aGroup @ A) @ B) @ A2) => ((((ord_less_eq @ (set @ A)) @ H) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((ord_less_eq @ (set @ A)) @ K3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ B)) @ A2)) => ((((member @ A) @ X2) @ (((((aset_sum @ A) @ B) @ A2) @ H) @ K3)) => (? [X:A]: (((((member @ A) @ X) @ H) & (? [Xa2:A]: (((((member @ A) @ Xa2) @ K3) & (X2 = (((((pop @ A) @ B) @ A2) @ X) @ Xa2))))))))))))))).
% 2.88/1.45 thf(fact_208_J__rad__unit, axiom, (! [X2:a]: (((~ ((((zeroring @ a) @ b) @ r))) => ((((member @ a) @ X2) @ (((j_rad @ a) @ b) @ r)) => (! [Y4:a]: (((((member @ a) @ Y4) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((unit @ a) @ b) @ r) @ (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((un @ a) @ b) @ r)) @ (((((tp @ a) @ b) @ r) @ ((((mop @ a) @ ((ring_ext @ a) @ b)) @ r) @ X2)) @ Y4))))))))))).
% 2.88/1.45 thf(fact_209_add__le__cancel__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A3:A,C3:A,B3:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A3) @ C3)) @ (((plus_plus @ A) @ B3) @ C3)) = (((ord_less_eq @ A) @ A3) @ B3)))))))).
% 2.88/1.45 thf(fact_210_un__closed, axiom, (((member @ a) @ (((un @ a) @ b) @ r)) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.88/1.45 thf(fact_211_maximal__ideal__proper, axiom, (! [Mx2:(set @ a)]: ((((((maximal_ideal @ a) @ b) @ r) @ Mx2) => (~ ((((member @ a) @ (((un @ a) @ b) @ r)) @ Mx2))))))).
% 2.88/1.45 thf(fact_212_Sr__one, axiom, (! [S:(set @ a)]: ((((((sr2 @ a) @ b) @ r) @ S) => (((member @ a) @ (((un @ a) @ b) @ r)) @ S))))).
% 2.88/1.45 thf(fact_213_add__right__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B3:A,A3:A,C3:A]: ((((((plus_plus @ A) @ B3) @ A3) = (((plus_plus @ A) @ C3) @ A3)) = (B3 = C3)))))))).
% 2.88/1.45 thf(fact_214_add__left__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A3:A,B3:A,C3:A]: ((((((plus_plus @ A) @ A3) @ B3) = (((plus_plus @ A) @ A3) @ C3)) = (B3 = C3)))))))).
% 2.88/1.45 thf(fact_215_rg__l__unit, axiom, (! [A3:a]: (((((member @ a) @ A3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((tp @ a) @ b) @ r) @ (((un @ a) @ b) @ r)) @ A3) = A3))))).
% 2.88/1.45 thf(fact_216_ring__l__one, axiom, (! [X2:a]: (((((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((tp @ a) @ b) @ r) @ (((un @ a) @ b) @ r)) @ X2) = X2))))).
% 2.88/1.45 thf(fact_217_ring__r__one, axiom, (! [X2:a]: (((((member @ a) @ X2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((tp @ a) @ b) @ r) @ X2) @ (((un @ a) @ b) @ r)) = X2))))).
% 2.88/1.45 thf(fact_218_Subring__one__ring__one, axiom, (! [C:$tType,S:((carrier_ext @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ C)))]: (((((((subring @ a) @ b) @ C) @ r) @ S) => ((((un @ a) @ C) @ S) = (((un @ a) @ b) @ r)))))).
% 2.88/1.45 thf(fact_219_ring__times__minusr, axiom, (! [A3:a]: (((((member @ a) @ A3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((mop @ a) @ ((ring_ext @ a) @ b)) @ r) @ A3) = (((((tp @ a) @ b) @ r) @ A3) @ ((((mop @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((un @ a) @ b) @ r)))))))).
% 2.88/1.45 thf(fact_220_ring__times__minusl, axiom, (! [A3:a]: (((((member @ a) @ A3) @ (((carrier @ a) @ ((aGroup_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((mop @ a) @ ((ring_ext @ a) @ b)) @ r) @ A3) = (((((tp @ a) @ b) @ r) @ ((((mop @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((un @ a) @ b) @ r))) @ A3)))))).
% 2.88/1.45 thf(fact_221_add__le__cancel__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C3:A,A3:A,B3:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C3) @ A3)) @ (((plus_plus @ A) @ C3) @ B3)) = (((ord_less_eq @ A) @ A3) @ B3)))))))).
% 2.88/1.45 thf(fact_222_localring__unit, axiom, (! [Mx2:(set @ a)]: (((~ ((((zeroring @ a) @ b) @ r))) => (((((maximal_ideal @ a) @ b) @ r) @ Mx2) => ((! [X:a]: (((((member @ a) @ X) @ Mx2) => ((((unit @ a) @ b) @ r) @ (((((pop @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ (((un @ a) @ b) @ r)))))) => (((local_ring @ a) @ b) @ r))))))).
% 2.88/1.45 thf(fact_223_primary__ideal__proper1, axiom, (! [Q2:(set @ a)]: ((((((primary_ideal @ a) @ b) @ r) @ Q2) => (~ ((((member @ a) @ (((un @ a) @ b) @ r)) @ Q2))))))).
% 2.88/1.45 thf(fact_224_self__le, axiom, (! [N2:nat]: ((((ord_less_eq @ nat) @ N2) @ N2)))).
% 2.88/1.45 thf(fact_225_nat__eq__le, axiom, (! [M4:nat,N2:nat]: (((M4 = N2) => (((ord_less_eq @ nat) @ M4) @ N2))))).
% 2.88/1.45 thf(fact_226_n__in__Nsetn, axiom, (! [N2:nat]: ((((member @ nat) @ N2) @ ((collect @ nat) @ (^ [I4:nat]: ((((ord_less_eq @ nat) @ I4) @ N2)))))))).
% 2.88/1.45 thf(fact_227_mem__of__Nset, axiom, (! [X2:nat,N2:nat]: (((((ord_less_eq @ nat) @ X2) @ N2) => (((member @ nat) @ X2) @ ((collect @ nat) @ (^ [I4:nat]: ((((ord_less_eq @ nat) @ I4) @ N2))))))))).
% 2.88/1.45 thf(fact_228_Ring_Oun__closed, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((ring @ A) @ B) @ R2) => (((member @ A) @ (((un @ A) @ B) @ R2)) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)))))).
% 2.88/1.45 thf(fact_229_Ring_Omaximal__ideal__proper, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),Mx2:(set @ A)]: (((((ring @ A) @ B) @ R2) => (((((maximal_ideal @ A) @ B) @ R2) @ Mx2) => (~ ((((member @ A) @ (((un @ A) @ B) @ R2)) @ Mx2)))))))).
% 2.88/1.45 thf(fact_230_Ring_OSubring__one__ring__one, axiom, (! [C:$tType,B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),S:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ C)))]: (((((ring @ A) @ B) @ R2) => ((((((subring @ A) @ B) @ C) @ R2) @ S) => ((((un @ A) @ C) @ S) = (((un @ A) @ B) @ R2))))))).
% 2.88/1.45 thf(fact_231_Ring_OSr__one, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),S:(set @ A)]: (((((ring @ A) @ B) @ R2) => (((((sr2 @ A) @ B) @ R2) @ S) => (((member @ A) @ (((un @ A) @ B) @ R2)) @ S)))))).
% 2.88/1.45 thf(fact_232_Ring_Oring__r__one, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),X2:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((tp @ A) @ B) @ R2) @ X2) @ (((un @ A) @ B) @ R2)) = X2)))))).
% 2.88/1.45 thf(fact_233_Ring_Oring__l__one, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),X2:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ X2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((tp @ A) @ B) @ R2) @ (((un @ A) @ B) @ R2)) @ X2) = X2)))))).
% 2.88/1.45 thf(fact_234_Ring_Org__l__unit, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),A3:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => ((((((tp @ A) @ B) @ R2) @ (((un @ A) @ B) @ R2)) @ A3) = A3)))))).
% 2.88/1.45 thf(fact_235_Module_Osprod__one, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,M3:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),M4:A]: ((((((((module @ A) @ B) @ C) @ D) @ M3) @ R2) => ((((member @ A) @ M4) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M3)) => (((((((sprod @ A) @ B) @ C) @ M3) @ (((un @ B) @ D) @ R2)) @ M4) = M4)))))).
% 2.88/1.45 thf(fact_236_Unit__def, axiom, (! [B:$tType,A:$tType]: ((((unit @ A) @ B) = (^ [R3:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),A5:A]: (((((member @ A) @ A5) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) & (? [X3:A]: (((((member @ A) @ X3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) & ((((((tp @ A) @ B) @ R3) @ A5) @ X3) = (((un @ A) @ B) @ R3)))))))))))).
% 2.88/1.45 thf(fact_237_add__right__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B3:A,A3:A,C3:A]: ((((((plus_plus @ A) @ B3) @ A3) = (((plus_plus @ A) @ C3) @ A3)) => (B3 = C3)))))))).
% 2.88/1.45 thf(fact_238_add__left__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A3:A,B3:A,C3:A]: ((((((plus_plus @ A) @ A3) @ B3) = (((plus_plus @ A) @ A3) @ C3)) => (B3 = C3)))))))).
% 2.88/1.45 thf(fact_239_add_Oleft__commute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [B3:A,A3:A,C3:A]: (((((plus_plus @ A) @ B3) @ (((plus_plus @ A) @ A3) @ C3)) = (((plus_plus @ A) @ A3) @ (((plus_plus @ A) @ B3) @ C3))))))))).
% 2.88/1.45 thf(fact_240_add_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => ((plus_plus @ A) = (^ [A5:A,B4:A]: ((((plus_plus @ A) @ B4) @ A5)))))))).
% 2.88/1.45 thf(fact_241_add_Oright__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [B3:A,A3:A,C3:A]: ((((((plus_plus @ A) @ B3) @ A3) = (((plus_plus @ A) @ C3) @ A3)) = (B3 = C3)))))))).
% 2.88/1.45 thf(fact_242_add_Oleft__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B3:A,C3:A]: ((((((plus_plus @ A) @ A3) @ B3) = (((plus_plus @ A) @ A3) @ C3)) = (B3 = C3)))))))).
% 2.88/1.45 thf(fact_243_add_Oassoc, axiom, (! [A:$tType]: (((semigroup_add @ A) => (! [A3:A,B3:A,C3:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A3) @ B3)) @ C3) = (((plus_plus @ A) @ A3) @ (((plus_plus @ A) @ B3) @ C3))))))))).
% 2.88/1.45 thf(fact_244_group__cancel_Oadd2, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [B2:A,K4:A,B3:A,A3:A]: (((B2 = (((plus_plus @ A) @ K4) @ B3)) => ((((plus_plus @ A) @ A3) @ B2) = (((plus_plus @ A) @ K4) @ (((plus_plus @ A) @ A3) @ B3)))))))))).
% 2.88/1.45 thf(fact_245_group__cancel_Oadd1, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A2:A,K4:A,A3:A,B3:A]: (((A2 = (((plus_plus @ A) @ K4) @ A3)) => ((((plus_plus @ A) @ A2) @ B3) = (((plus_plus @ A) @ K4) @ (((plus_plus @ A) @ A3) @ B3)))))))))).
% 2.88/1.45 thf(fact_246_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I3:A,J:A,K4:A,L:A]: ((((I3 = J) & (K4 = L)) => ((((plus_plus @ A) @ I3) @ K4) = (((plus_plus @ A) @ J) @ L))))))))).
% 2.88/1.45 thf(fact_247_ab__semigroup__add__class_Oadd__ac_I1_J, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [A3:A,B3:A,C3:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A3) @ B3)) @ C3) = (((plus_plus @ A) @ A3) @ (((plus_plus @ A) @ B3) @ C3))))))))).
% 2.88/1.45 thf(fact_248_Ring_Oring__times__minusl, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),A3:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => (((((mop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ A3) = (((((tp @ A) @ B) @ R2) @ ((((mop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ (((un @ A) @ B) @ R2))) @ A3))))))).
% 2.88/1.45 thf(fact_249_Ring_Oring__times__minusr, axiom, (! [B:$tType,A:$tType,R2:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),A3:A]: (((((ring @ A) @ B) @ R2) => ((((member @ A) @ A3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R2)) => (((((mop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ A3) = (((((tp @ A) @ B) @ R2) @ A3) @ ((((mop @ A) @ ((ring_ext @ A) @ B)) @ R2) @ (((un @ A) @ B) @ R2))))))))).
% 2.88/1.45 thf(fact_250_sr__def, axiom, (! [B:$tType,A:$tType]: ((((sr2 @ A) @ B) = (^ [R3:((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))),S3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S3) @ (((carrier @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) & ((((member @ A) @ (((un @ A) @ B) @ R3)) @ S3) & (! [X3:A]: (((((member @ A) @ X3) @ S3) => (! [Y3:A]: (((((member @ A) @ Y3) @ S3) => ((((member @ A) @ (((((pop @ A) @ ((ring_ext @ A) @ B)) @ R3) @ X3) @ ((((mop @ A) @ ((ring_ext @ A) @ B)) @ R3) @ Y3))) @ S3) & (((member @ A) @ (((((tp @ A) @ B) @ R3) @ X3) @ Y3)) @ S3)))))))))))))))).
% 2.88/1.45 thf(fact_251_add__mono__thms__linordered__semiring_I3_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I3:A,J:A,K4:A,L:A]: ((((((ord_less_eq @ A) @ I3) @ J) & (K4 = L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I3) @ K4)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.88/1.45 thf(fact_252_add__mono__thms__linordered__semiring_I2_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I3:A,J:A,K4:A,L:A]: ((((I3 = J) & (((ord_less_eq @ A) @ K4) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I3) @ K4)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.88/1.45 thf(fact_253_add__mono__thms__linordered__semiring_I1_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I3:A,J:A,K4:A,L:A]: ((((((ord_less_eq @ A) @ I3) @ J) & (((ord_less_eq @ A) @ K4) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I3) @ K4)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.88/1.45 thf(fact_254_add__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A3:A,B3:A,C3:A,D2:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ C3) @ D2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A3) @ C3)) @ (((plus_plus @ A) @ B3) @ D2)))))))))).
% 2.88/1.45 thf(fact_255_add__left__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A3:A,B3:A,C3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ C3) @ A3)) @ (((plus_plus @ A) @ C3) @ B3))))))))).
% 2.88/1.45 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: (((ord @ A9) => (ord @ (A8 > A9)))))).
% 2.88/1.45 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le, axiom, (ordere236663937imp_le @ nat)).
% 2.88/1.45 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add, axiom, (ordere779506340up_add @ nat)).
% 2.88/1.45 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add, axiom, (cancel_semigroup_add @ nat)).
% 2.88/1.45 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add, axiom, (ab_semigroup_add @ nat)).
% 2.88/1.45 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add, axiom, (comm_monoid_add @ nat)).
% 2.88/1.45 thf(tcon_Nat_Onat___Groups_Osemigroup__add, axiom, (semigroup_add @ nat)).
% 2.88/1.45 thf(tcon_Nat_Onat___Orderings_Oord_1, axiom, (ord @ nat)).
% 2.88/1.45 thf(tcon_Nat_Onat___Groups_Oplus, axiom, (plus @ nat)).
% 2.88/1.45 thf(tcon_Set_Oset___Orderings_Oord_2, axiom, (! [A8:$tType]: ((ord @ (set @ A8))))).
% 2.88/1.45 thf(tcon_HOL_Obool___Orderings_Oord_3, axiom, (ord @ $o)).
% 2.88/1.45 thf(tcon_Product__Type_Ounit___Orderings_Oord_4, axiom, (ord @ product_unit)).
% 2.88/1.45 thf(conj_0, hypothesis, (! [X5:c]: (((((member @ c) @ X5) @ i) => ((((((module @ d) @ a) @ e) @ b) @ (m @ X5)) @ r))))).
% 2.88/1.45 thf(conj_1, hypothesis, ((((member @ ((c > d) > ((c > d) > (c > d)))) @ (((((prod_pOp @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m)) @ ((((pi @ (c > d)) @ ((c > d) > (c > d))) @ (((((carr_prodag @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m)) @ (^ [Uu:(c > d)]: (((((pi @ (c > d)) @ (c > d)) @ (((((carr_prodag @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m)) @ (^ [Uv:(c > d)]: ((((((carr_prodag @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m)))))))) & ((! [A10:(c > d)]: (((((member @ (c > d)) @ A10) @ (((((carr_prodag @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m)) => (! [B5:(c > d)]: (((((member @ (c > d)) @ B5) @ (((((carr_prodag @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m)) => (! [C4:(c > d)]: (((((member @ (c > d)) @ C4) @ (((((carr_prodag @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m)) => ((((((((prod_pOp @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m) @ (((((((prod_pOp @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m) @ A10) @ B5)) @ C4) = (((((((prod_pOp @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m) @ A10) @ (((((((prod_pOp @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m) @ B5) @ C4)))))))))))) & ((! [A10:(c > d)]: (((((member @ (c > d)) @ A10) @ (((((carr_prodag @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m)) => (! [B5:(c > d)]: (((((member @ (c > d)) @ B5) @ (((((carr_prodag @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m)) => ((((((((prod_pOp @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m) @ A10) @ B5) = (((((((prod_pOp @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m) @ B5) @ A10)))))))) & ((((member @ ((c > d) > (c > d))) @ (((((prod_mOp @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m)) @ ((((pi @ (c > d)) @ (c > d)) @ (((((carr_prodag @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m)) @ (^ [Uu:(c > d)]: ((((((carr_prodag @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m))))) & ((! [A10:(c > d)]: (((((member @ (c > d)) @ A10) @ (((((carr_prodag @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m)) => ((((((((prod_pOp @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m) @ ((((((prod_mOp @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m) @ A10)) @ A10) = (((((prod_zero @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m))))) & ((((member @ (c > d)) @ (((((prod_zero @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m)) @ (((((carr_prodag @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m)) & (! [A10:(c > d)]: (((((member @ (c > d)) @ A10) @ (((((carr_prodag @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m)) => ((((((((prod_pOp @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m) @ (((((prod_zero @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m)) @ A10) = A10))))))))))).
% 2.88/1.45 thf(conj_2, conjecture, (! [A6:(c > d)]: (((~ ((((member @ (c > d)) @ A6) @ (((((carr_prodag @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m)))) | ((((((((prod_pOp @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m) @ A6) @ (((((prod_zero @ c) @ d) @ (((module_ext @ d) @ a) @ e)) @ i) @ m)) = A6))))).
% 2.88/1.45 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.uhnk0bOYnS/DTF2DTF31945.p
% 2.88/1.45 ------------------------
% 3.01/1.53 ---- Cleaned THF ---
% 3.01/1.53 thf(ty_t_Algebra1_Ocarrier_Ocarrier__ext,type,
% 3.01/1.53 carrier_ext: $tType > $tType > $tType ).
% 3.01/1.53
% 3.01/1.53 thf(ty_t_Algebra7_OModule_OModule__ext,type,
% 3.01/1.53 module_ext: $tType > $tType > $tType > $tType ).
% 3.01/1.53
% 3.01/1.53 thf(ty_t_Algebra4_OaGroup_OaGroup__ext,type,
% 3.01/1.53 aGroup_ext: $tType > $tType > $tType ).
% 3.01/1.53
% 3.01/1.53 thf(ty_t_Algebra4_ORing_ORing__ext,type,
% 3.01/1.53 ring_ext: $tType > $tType > $tType ).
% 3.01/1.53
% 3.01/1.53 thf(ty_t_Product__Type_Ounit,type,
% 3.01/1.53 product_unit: $tType ).
% 3.01/1.53
% 3.01/1.53 thf(ty_t_Set_Oset,type,
% 3.01/1.53 set: $tType > $tType ).
% 3.01/1.53
% 3.01/1.53 thf(ty_t_Nat_Onat,type,
% 3.01/1.53 nat: $tType ).
% 3.01/1.53
% 3.01/1.53 thf(ty_tf_e,type,
% 3.01/1.53 e: $tType ).
% 3.01/1.53
% 3.01/1.53 thf(ty_tf_d,type,
% 3.01/1.53 d: $tType ).
% 3.01/1.53
% 3.01/1.53 thf(ty_tf_c,type,
% 3.01/1.53 c: $tType ).
% 3.01/1.53
% 3.01/1.53 thf(ty_tf_b,type,
% 3.01/1.53 b: $tType ).
% 3.01/1.53
% 3.01/1.53 thf(ty_tf_a,type,
% 3.01/1.53 a: $tType ).
% 3.01/1.53
% 3.01/1.53 thf(sy_cl_HOL_Otype,type,
% 3.01/1.53 type:
% 3.01/1.53 !>[A: $tType] : $o ).
% 3.01/1.53
% 3.01/1.53 thf(sy_cl_Groups_Oplus,type,
% 3.01/1.53 plus:
% 3.01/1.53 !>[A: $tType] : $o ).
% 3.01/1.53
% 3.01/1.53 thf(sy_cl_Orderings_Oord,type,
% 3.01/1.53 ord:
% 3.01/1.53 !>[A: $tType] : $o ).
% 3.01/1.53
% 3.01/1.53 thf(sy_cl_Groups_Ogroup__add,type,
% 3.01/1.53 group_add:
% 3.01/1.53 !>[A: $tType] : $o ).
% 3.01/1.53
% 3.01/1.53 thf(sy_cl_Groups_Osemigroup__add,type,
% 3.01/1.53 semigroup_add:
% 3.01/1.53 !>[A: $tType] : $o ).
% 3.01/1.53
% 3.01/1.53 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 3.01/1.53 comm_monoid_add:
% 3.01/1.53 !>[A: $tType] : $o ).
% 3.01/1.53
% 3.01/1.53 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 3.01/1.53 ab_semigroup_add:
% 3.01/1.53 !>[A: $tType] : $o ).
% 3.01/1.53
% 3.01/1.53 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 3.01/1.53 cancel_semigroup_add:
% 3.01/1.53 !>[A: $tType] : $o ).
% 3.01/1.53
% 3.01/1.53 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 3.01/1.53 ordere779506340up_add:
% 3.01/1.53 !>[A: $tType] : $o ).
% 3.01/1.53
% 3.01/1.53 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 3.01/1.53 ordere236663937imp_le:
% 3.01/1.53 !>[A: $tType] : $o ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra1_Ocarrier_Ocarrier,type,
% 3.01/1.53 carrier:
% 3.01/1.53 !>[A: $tType,Z: $tType] : ( ( carrier_ext @ A @ Z ) > ( set @ A ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_OJ__rad,type,
% 3.01/1.53 j_rad:
% 3.01/1.53 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_ORing,type,
% 3.01/1.53 ring:
% 3.01/1.53 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_ORing_Otp,type,
% 3.01/1.53 tp:
% 3.01/1.53 !>[A: $tType,Z: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ Z ) ) ) > A > A > A ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_ORing_Oun,type,
% 3.01/1.53 un:
% 3.01/1.53 !>[A: $tType,Z: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ Z ) ) ) > A ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_OSr,type,
% 3.01/1.53 sr:
% 3.01/1.53 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_OSubring,type,
% 3.01/1.53 subring:
% 3.01/1.53 !>[A: $tType,M: $tType,M1: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ M1 ) ) ) > $o ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_OUnit,type,
% 3.01/1.53 unit:
% 3.01/1.53 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > A > $o ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_OaGroup,type,
% 3.01/1.53 aGroup:
% 3.01/1.53 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ B ) ) > $o ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_OaGroup_Omop,type,
% 3.01/1.53 mop:
% 3.01/1.53 !>[A: $tType,Z: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ Z ) ) > A > A ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_OaGroup_Opop,type,
% 3.01/1.53 pop:
% 3.01/1.53 !>[A: $tType,Z: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ Z ) ) > A > A > A ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_Oaset__sum,type,
% 3.01/1.53 aset_sum:
% 3.01/1.53 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ B ) ) > ( set @ A ) > ( set @ A ) > ( set @ A ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_Ocarr__dsumag,type,
% 3.01/1.53 carr_dsumag:
% 3.01/1.53 !>[I: $tType,A: $tType,More: $tType] : ( ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ More ) ) ) > ( set @ ( I > A ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_Ocarr__prodag,type,
% 3.01/1.53 carr_prodag:
% 3.01/1.53 !>[I: $tType,A: $tType,More: $tType] : ( ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ More ) ) ) > ( set @ ( I > A ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_Olocal__ring,type,
% 3.01/1.53 local_ring:
% 3.01/1.53 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_Omaximal__ideal,type,
% 3.01/1.53 maximal_ideal:
% 3.01/1.53 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_Omul__closed__set,type,
% 3.01/1.53 mul_closed_set:
% 3.01/1.53 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_Onscal,type,
% 3.01/1.53 nscal:
% 3.01/1.53 !>[A: $tType,More: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ More ) ) ) > A > nat > A ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_Oprimary__ideal,type,
% 3.01/1.53 primary_ideal:
% 3.01/1.53 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_Oprod__mOp,type,
% 3.01/1.53 prod_mOp:
% 3.01/1.53 !>[I: $tType,A: $tType,More: $tType] : ( ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ More ) ) ) > ( I > A ) > I > A ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_Oprod__pOp,type,
% 3.01/1.53 prod_pOp:
% 3.01/1.53 !>[I: $tType,A: $tType,More: $tType] : ( ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ More ) ) ) > ( I > A ) > ( I > A ) > I > A ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_Oprod__zero,type,
% 3.01/1.53 prod_zero:
% 3.01/1.53 !>[I: $tType,A: $tType,More: $tType] : ( ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ More ) ) ) > I > A ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_OrInvim,type,
% 3.01/1.53 rInvim:
% 3.01/1.53 !>[A: $tType,M: $tType,B: $tType,M1: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ M ) ) ) > ( carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ M1 ) ) ) > ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_Osr,type,
% 3.01/1.53 sr2:
% 3.01/1.53 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra4_Ozeroring,type,
% 3.01/1.53 zeroring:
% 3.01/1.53 !>[A: $tType,More: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ More ) ) ) > $o ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra5_Osum__mult,type,
% 3.01/1.53 sum_mult:
% 3.01/1.53 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > ( set @ A ) > ( set @ A ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra7_OModule,type,
% 3.01/1.53 module:
% 3.01/1.53 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) ) > ( carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) ) > $o ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra7_OModule_Osprod,type,
% 3.01/1.53 sprod:
% 3.01/1.53 !>[A: $tType,B: $tType,Z: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ Z ) ) ) > B > A > A ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra7_OmHom,type,
% 3.01/1.53 mHom:
% 3.01/1.53 !>[B: $tType,M: $tType,A: $tType,M1: $tType,C: $tType,M2: $tType] : ( ( carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ M1 ) ) ) > ( carrier_ext @ C @ ( aGroup_ext @ C @ ( module_ext @ C @ B @ M2 ) ) ) > ( set @ ( A > C ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra8__Mirabelle__lwvimexpoc_OprodM,type,
% 3.01/1.53 algebr1000837334_prodM:
% 3.01/1.53 !>[R: $tType,M: $tType,I: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) ) > ( carrier_ext @ ( I > A ) @ ( aGroup_ext @ ( I > A ) @ ( module_ext @ ( I > A ) @ R @ product_unit ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Algebra8__Mirabelle__lwvimexpoc_OprodM__sprod,type,
% 3.01/1.53 algebr142713364_sprod:
% 3.01/1.53 !>[R: $tType,M: $tType,I: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) ) > R > ( I > A ) > I > A ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_FuncSet_OPi,type,
% 3.01/1.53 pi:
% 3.01/1.53 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( A > B ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 3.01/1.53 plus_plus:
% 3.01/1.53 !>[A: $tType] : ( A > A > A ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.01/1.53 ord_less_eq:
% 3.01/1.53 !>[A: $tType] : ( A > A > $o ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_Set_OCollect,type,
% 3.01/1.53 collect:
% 3.01/1.53 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_c_member,type,
% 3.01/1.53 member:
% 3.01/1.53 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_v_I,type,
% 3.01/1.53 i: set @ c ).
% 3.01/1.53
% 3.01/1.53 thf(sy_v_M,type,
% 3.01/1.53 m: c > ( carrier_ext @ d @ ( aGroup_ext @ d @ ( module_ext @ d @ a @ e ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(sy_v_R,type,
% 3.01/1.53 r: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_0_Ring__axioms,axiom,
% 3.01/1.53 ring @ a @ b @ r ).
% 3.01/1.53
% 3.01/1.53 thf(fact_1_bivar__fun,axiom,
% 3.01/1.53 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,A2: set @ A,B2: set @ B,C2: set @ C,A3: A] :
% 3.01/1.53 ( ( member @ ( A > B > C ) @ F
% 3.01/1.53 @ ( pi @ A @ ( B > C ) @ A2
% 3.01/1.53 @ ^ [Uu: A] :
% 3.01/1.53 ( pi @ B @ C @ B2
% 3.01/1.53 @ ^ [Uv: B] : C2 ) ) )
% 3.01/1.53 => ( ( member @ A @ A3 @ A2 )
% 3.01/1.53 => ( member @ ( B > C ) @ ( F @ A3 )
% 3.01/1.53 @ ( pi @ B @ C @ B2
% 3.01/1.53 @ ^ [Uu: B] : C2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_2_bivar__fun__mem,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,C: $tType,F: A > B > C,A2: set @ A,B2: set @ B,C2: set @ C,A3: A,B3: B] :
% 3.01/1.53 ( ( member @ ( A > B > C ) @ F
% 3.01/1.53 @ ( pi @ A @ ( B > C ) @ A2
% 3.01/1.53 @ ^ [Uu: A] :
% 3.01/1.53 ( pi @ B @ C @ B2
% 3.01/1.53 @ ^ [Uv: B] : C2 ) ) )
% 3.01/1.53 => ( ( member @ A @ A3 @ A2 )
% 3.01/1.53 => ( ( member @ B @ B3 @ B2 )
% 3.01/1.53 => ( member @ C @ ( F @ A3 @ B3 ) @ C2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_3_Pi__I,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: set @ A,F: A > B,B2: A > ( set @ B )] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ A2 )
% 3.01/1.53 => ( member @ B @ ( F @ X ) @ ( B2 @ X ) ) )
% 3.01/1.53 => ( member @ ( A > B ) @ F @ ( pi @ A @ B @ A2 @ B2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_4_prodM__sprod__mem,axiom,
% 3.01/1.53 ! [E: $tType,D: $tType,C: $tType,I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ a @ E ) ) ),A3: a,M4: C > D] :
% 3.01/1.53 ( ! [X: C] :
% 3.01/1.53 ( ( member @ C @ X @ I2 )
% 3.01/1.53 => ( module @ D @ a @ E @ b @ ( M3 @ X ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ ( C > D ) @ M4 @ ( carr_prodag @ C @ D @ ( module_ext @ D @ a @ E ) @ I2 @ M3 ) )
% 3.01/1.53 => ( member @ ( C > D ) @ ( algebr142713364_sprod @ a @ b @ C @ D @ E @ r @ I2 @ M3 @ A3 @ M4 ) @ ( carr_prodag @ C @ D @ ( module_ext @ D @ a @ E ) @ I2 @ M3 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_5_prodM__carr,axiom,
% 3.01/1.53 ! [E: $tType,D: $tType,C: $tType,I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ a @ E ) ) )] :
% 3.01/1.53 ( ! [X: C] :
% 3.01/1.53 ( ( member @ C @ X @ I2 )
% 3.01/1.53 => ( module @ D @ a @ E @ b @ ( M3 @ X ) @ r ) )
% 3.01/1.53 => ( ( carrier @ ( C > D ) @ ( aGroup_ext @ ( C > D ) @ ( module_ext @ ( C > D ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ C @ D @ E @ r @ I2 @ M3 ) )
% 3.01/1.53 = ( carr_prodag @ C @ D @ ( module_ext @ D @ a @ E ) @ I2 @ M3 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_6_eq__fun,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ B,G: A > B] :
% 3.01/1.53 ( ( member @ ( A > B ) @ F
% 3.01/1.53 @ ( pi @ A @ B @ A2
% 3.01/1.53 @ ^ [Uu: A] : B2 ) )
% 3.01/1.53 => ( ( F = G )
% 3.01/1.53 => ( member @ ( A > B ) @ G
% 3.01/1.53 @ ( pi @ A @ B @ A2
% 3.01/1.53 @ ^ [Uu: A] : B2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_7_funcTr,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ B,G: A > B,A3: A] :
% 3.01/1.53 ( ( member @ ( A > B ) @ F
% 3.01/1.53 @ ( pi @ A @ B @ A2
% 3.01/1.53 @ ^ [Uu: A] : B2 ) )
% 3.01/1.53 => ( ( member @ ( A > B ) @ G
% 3.01/1.53 @ ( pi @ A @ B @ A2
% 3.01/1.53 @ ^ [Uu: A] : B2 ) )
% 3.01/1.53 => ( ( F = G )
% 3.01/1.53 => ( ( member @ A @ A3 @ A2 )
% 3.01/1.53 => ( ( F @ A3 )
% 3.01/1.53 = ( G @ A3 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_8_funcsetI,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: set @ A,F: A > B,B2: set @ B] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ A2 )
% 3.01/1.53 => ( member @ B @ ( F @ X ) @ B2 ) )
% 3.01/1.53 => ( member @ ( A > B ) @ F
% 3.01/1.53 @ ( pi @ A @ B @ A2
% 3.01/1.53 @ ^ [Uu: A] : B2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_9_eq__funcs,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ B,G: A > B,X2: A] :
% 3.01/1.53 ( ( member @ ( A > B ) @ F
% 3.01/1.53 @ ( pi @ A @ B @ A2
% 3.01/1.53 @ ^ [Uu: A] : B2 ) )
% 3.01/1.53 => ( ( member @ ( A > B ) @ G
% 3.01/1.53 @ ( pi @ A @ B @ A2
% 3.01/1.53 @ ^ [Uu: A] : B2 ) )
% 3.01/1.53 => ( ( F = G )
% 3.01/1.53 => ( ( member @ A @ X2 @ A2 )
% 3.01/1.53 => ( ( F @ X2 )
% 3.01/1.53 = ( G @ X2 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_10_funcset__id,axiom,
% 3.01/1.53 ! [A: $tType,A2: set @ A] :
% 3.01/1.53 ( member @ ( A > A )
% 3.01/1.53 @ ^ [X3: A] : X3
% 3.01/1.53 @ ( pi @ A @ A @ A2
% 3.01/1.53 @ ^ [Uu: A] : A2 ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_11_Algebra1_Ofuncset__mem,axiom,
% 3.01/1.53 ! [A: $tType,B: $tType,F: A > B,A2: set @ A,B2: set @ B,X2: A] :
% 3.01/1.53 ( ( member @ ( A > B ) @ F
% 3.01/1.53 @ ( pi @ A @ B @ A2
% 3.01/1.53 @ ^ [Uu: A] : B2 ) )
% 3.01/1.53 => ( ( member @ A @ X2 @ A2 )
% 3.01/1.53 => ( member @ B @ ( F @ X2 ) @ B2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_12_prod__pOp__func,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 3.01/1.53 => ( member @ ( ( A > B ) > ( A > B ) > A > B ) @ ( prod_pOp @ A @ B @ C @ I2 @ A2 )
% 3.01/1.53 @ ( pi @ ( A > B ) @ ( ( A > B ) > A > B ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 )
% 3.01/1.53 @ ^ [Uu: A > B] :
% 3.01/1.53 ( pi @ ( A > B ) @ ( A > B ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 )
% 3.01/1.53 @ ^ [Uv: A > B] : ( carr_prodag @ A @ B @ C @ I2 @ A2 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_13_ring__is__ag,axiom,
% 3.01/1.53 aGroup @ a @ ( ring_ext @ a @ b ) @ r ).
% 3.01/1.53
% 3.01/1.53 thf(fact_14_prodM__mem__eq,axiom,
% 3.01/1.53 ! [E: $tType,D: $tType,C: $tType,I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ a @ E ) ) ),M12: C > D,M22: C > D] :
% 3.01/1.53 ( ! [X: C] :
% 3.01/1.53 ( ( member @ C @ X @ I2 )
% 3.01/1.53 => ( module @ D @ a @ E @ b @ ( M3 @ X ) @ r ) )
% 3.01/1.53 => ( ( member @ ( C > D ) @ M12 @ ( carrier @ ( C > D ) @ ( aGroup_ext @ ( C > D ) @ ( module_ext @ ( C > D ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ C @ D @ E @ r @ I2 @ M3 ) ) )
% 3.01/1.53 => ( ( member @ ( C > D ) @ M22 @ ( carrier @ ( C > D ) @ ( aGroup_ext @ ( C > D ) @ ( module_ext @ ( C > D ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ C @ D @ E @ r @ I2 @ M3 ) ) )
% 3.01/1.53 => ( ! [X: C] :
% 3.01/1.53 ( ( member @ C @ X @ I2 )
% 3.01/1.53 => ( ( M12 @ X )
% 3.01/1.53 = ( M22 @ X ) ) )
% 3.01/1.53 => ( M12 = M22 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_15_J__rad__mem,axiom,
% 3.01/1.53 ! [X2: a] :
% 3.01/1.53 ( ( member @ a @ X2 @ ( j_rad @ a @ b @ r ) )
% 3.01/1.53 => ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_16_Ring_ORing,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ring @ A @ B @ R2 ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_17_Ring_Oring__is__ag,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( aGroup @ A @ ( ring_ext @ A @ B ) @ R2 ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_18_Ring_OprodM__mem__eq,axiom,
% 3.01/1.53 ! [A: $tType,E: $tType,B: $tType,D: $tType,C: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ A @ E ) ) ),M12: C > D,M22: C > D] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ! [X: C] :
% 3.01/1.53 ( ( member @ C @ X @ I2 )
% 3.01/1.53 => ( module @ D @ A @ E @ B @ ( M3 @ X ) @ R2 ) )
% 3.01/1.53 => ( ( member @ ( C > D ) @ M12 @ ( carrier @ ( C > D ) @ ( aGroup_ext @ ( C > D ) @ ( module_ext @ ( C > D ) @ A @ product_unit ) ) @ ( algebr1000837334_prodM @ A @ B @ C @ D @ E @ R2 @ I2 @ M3 ) ) )
% 3.01/1.53 => ( ( member @ ( C > D ) @ M22 @ ( carrier @ ( C > D ) @ ( aGroup_ext @ ( C > D ) @ ( module_ext @ ( C > D ) @ A @ product_unit ) ) @ ( algebr1000837334_prodM @ A @ B @ C @ D @ E @ R2 @ I2 @ M3 ) ) )
% 3.01/1.53 => ( ! [X: C] :
% 3.01/1.53 ( ( member @ C @ X @ I2 )
% 3.01/1.53 => ( ( M12 @ X )
% 3.01/1.53 = ( M22 @ X ) ) )
% 3.01/1.53 => ( M12 = M22 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_19_carr__prodag__mem__eq,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B,Y: A > B] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 3.01/1.53 => ( ( member @ ( A > B ) @ X4 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 3.01/1.53 => ( ( member @ ( A > B ) @ Y @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 3.01/1.53 => ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( ( X4 @ X )
% 3.01/1.53 = ( Y @ X ) ) )
% 3.01/1.53 => ( X4 = Y ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_20_prodag__sameTr1,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),B2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 3.01/1.53 => ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( ( A2 @ X )
% 3.01/1.53 = ( B2 @ X ) ) )
% 3.01/1.53 => ( ( carr_prodag @ A @ B @ C @ I2 @ A2 )
% 3.01/1.53 = ( carr_prodag @ A @ B @ C @ I2 @ B2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_21_prodag__comp__i,axiom,
% 3.01/1.53 ! [B: $tType,C: $tType,A: $tType,A3: A > B,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),I3: A] :
% 3.01/1.53 ( ( member @ ( A > B ) @ A3 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ I3 @ I2 )
% 3.01/1.53 => ( member @ B @ ( A3 @ I3 ) @ ( carrier @ B @ ( aGroup_ext @ B @ C ) @ ( A2 @ I3 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_22_prodag__sameTr2,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),B2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 3.01/1.53 => ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( ( A2 @ X )
% 3.01/1.53 = ( B2 @ X ) ) )
% 3.01/1.53 => ( ( prod_pOp @ A @ B @ C @ I2 @ A2 )
% 3.01/1.53 = ( prod_pOp @ A @ B @ C @ I2 @ B2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_23_prodag__sameTr3,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),B2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 3.01/1.53 => ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( ( A2 @ X )
% 3.01/1.53 = ( B2 @ X ) ) )
% 3.01/1.53 => ( ( prod_mOp @ A @ B @ C @ I2 @ A2 )
% 3.01/1.53 = ( prod_mOp @ A @ B @ C @ I2 @ B2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_24_prodag__sameTr4,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),B2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 3.01/1.53 => ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( ( A2 @ X )
% 3.01/1.53 = ( B2 @ X ) ) )
% 3.01/1.53 => ( ( prod_zero @ A @ B @ C @ I2 @ A2 )
% 3.01/1.53 = ( prod_zero @ A @ B @ C @ I2 @ B2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_25_Ring_OprodM__carr,axiom,
% 3.01/1.53 ! [B: $tType,E: $tType,A: $tType,D: $tType,C: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ A @ E ) ) )] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ! [X: C] :
% 3.01/1.53 ( ( member @ C @ X @ I2 )
% 3.01/1.53 => ( module @ D @ A @ E @ B @ ( M3 @ X ) @ R2 ) )
% 3.01/1.53 => ( ( carrier @ ( C > D ) @ ( aGroup_ext @ ( C > D ) @ ( module_ext @ ( C > D ) @ A @ product_unit ) ) @ ( algebr1000837334_prodM @ A @ B @ C @ D @ E @ R2 @ I2 @ M3 ) )
% 3.01/1.53 = ( carr_prodag @ C @ D @ ( module_ext @ D @ A @ E ) @ I2 @ M3 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_26_Ring_OprodM__sprod__mem,axiom,
% 3.01/1.53 ! [B: $tType,E: $tType,A: $tType,D: $tType,C: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ A @ E ) ) ),A3: A,M4: C > D] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ! [X: C] :
% 3.01/1.53 ( ( member @ C @ X @ I2 )
% 3.01/1.53 => ( module @ D @ A @ E @ B @ ( M3 @ X ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ ( C > D ) @ M4 @ ( carr_prodag @ C @ D @ ( module_ext @ D @ A @ E ) @ I2 @ M3 ) )
% 3.01/1.53 => ( member @ ( C > D ) @ ( algebr142713364_sprod @ A @ B @ C @ D @ E @ R2 @ I2 @ M3 @ A3 @ M4 ) @ ( carr_prodag @ C @ D @ ( module_ext @ D @ A @ E ) @ I2 @ M3 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_27_prod__pOp__commute,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),A3: A > B,B3: A > B] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 3.01/1.53 => ( ( member @ ( A > B ) @ A3 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 3.01/1.53 => ( ( member @ ( A > B ) @ B3 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 3.01/1.53 => ( ( prod_pOp @ A @ B @ C @ I2 @ A2 @ A3 @ B3 )
% 3.01/1.53 = ( prod_pOp @ A @ B @ C @ I2 @ A2 @ B3 @ A3 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_28_prod__pOp__assoc,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),A3: A > B,B3: A > B,C3: A > B] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 3.01/1.53 => ( ( member @ ( A > B ) @ A3 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 3.01/1.53 => ( ( member @ ( A > B ) @ B3 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 3.01/1.53 => ( ( member @ ( A > B ) @ C3 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 3.01/1.53 => ( ( prod_pOp @ A @ B @ C @ I2 @ A2 @ ( prod_pOp @ A @ B @ C @ I2 @ A2 @ A3 @ B3 ) @ C3 )
% 3.01/1.53 = ( prod_pOp @ A @ B @ C @ I2 @ A2 @ A3 @ ( prod_pOp @ A @ B @ C @ I2 @ A2 @ B3 @ C3 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_29_prod__pOp__mem,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B,Y: A > B] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 3.01/1.53 => ( ( member @ ( A > B ) @ X4 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 3.01/1.53 => ( ( member @ ( A > B ) @ Y @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 3.01/1.53 => ( member @ ( A > B ) @ ( prod_pOp @ A @ B @ C @ I2 @ A2 @ X4 @ Y ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_30_prod__mOp__mem,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 3.01/1.53 => ( ( member @ ( A > B ) @ X4 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 3.01/1.53 => ( member @ ( A > B ) @ ( prod_mOp @ A @ B @ C @ I2 @ A2 @ X4 ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_31_prod__zero__func,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 3.01/1.53 => ( member @ ( A > B ) @ ( prod_zero @ A @ B @ C @ I2 @ A2 ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_32_prod__mOp__func,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 3.01/1.53 => ( member @ ( ( A > B ) > A > B ) @ ( prod_mOp @ A @ B @ C @ I2 @ A2 )
% 3.01/1.53 @ ( pi @ ( A > B ) @ ( A > B ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 )
% 3.01/1.53 @ ^ [Uu: A > B] : ( carr_prodag @ A @ B @ C @ I2 @ A2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_33_Pi__cong,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: set @ A,F: A > B,G: A > B,B2: A > ( set @ B )] :
% 3.01/1.53 ( ! [W: A] :
% 3.01/1.53 ( ( member @ A @ W @ A2 )
% 3.01/1.53 => ( ( F @ W )
% 3.01/1.53 = ( G @ W ) ) )
% 3.01/1.53 => ( ( member @ ( A > B ) @ F @ ( pi @ A @ B @ A2 @ B2 ) )
% 3.01/1.53 = ( member @ ( A > B ) @ G @ ( pi @ A @ B @ A2 @ B2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_34_Pi__mem,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: A > ( set @ B ),X2: A] :
% 3.01/1.53 ( ( member @ ( A > B ) @ F @ ( pi @ A @ B @ A2 @ B2 ) )
% 3.01/1.53 => ( ( member @ A @ X2 @ A2 )
% 3.01/1.53 => ( member @ B @ ( F @ X2 ) @ ( B2 @ X2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_35_Pi__iff,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,F: A > B,I2: set @ A,X4: A > ( set @ B )] :
% 3.01/1.53 ( ( member @ ( A > B ) @ F @ ( pi @ A @ B @ I2 @ X4 ) )
% 3.01/1.53 = ( ! [X3: A] :
% 3.01/1.53 ( ( member @ A @ X3 @ I2 )
% 3.01/1.53 => ( member @ B @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_36_Pi__I_H,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: set @ A,F: A > B,B2: A > ( set @ B )] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ A2 )
% 3.01/1.53 => ( member @ B @ ( F @ X ) @ ( B2 @ X ) ) )
% 3.01/1.53 => ( member @ ( A > B ) @ F @ ( pi @ A @ B @ A2 @ B2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_37_PiE,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: A > ( set @ B ),X2: A] :
% 3.01/1.53 ( ( member @ ( A > B ) @ F @ ( pi @ A @ B @ A2 @ B2 ) )
% 3.01/1.53 => ( ~ ( member @ B @ ( F @ X2 ) @ ( B2 @ X2 ) )
% 3.01/1.53 => ~ ( member @ A @ X2 @ A2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_38_mem__subring__mem__ring,axiom,
% 3.01/1.53 ! [C: $tType,S: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ),X2: a] :
% 3.01/1.53 ( ( subring @ a @ b @ C @ r @ S )
% 3.01/1.53 => ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ) @ S ) )
% 3.01/1.53 => ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_39_subring__Ring,axiom,
% 3.01/1.53 ! [C: $tType,S: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) )] :
% 3.01/1.53 ( ( subring @ a @ b @ C @ r @ S )
% 3.01/1.53 => ( ring @ a @ C @ S ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_40_prodM__sprod__val,axiom,
% 3.01/1.53 ! [E: $tType,D: $tType,C: $tType,I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ a @ E ) ) ),A3: a,M4: C > D,J: C] :
% 3.01/1.53 ( ! [X: C] :
% 3.01/1.53 ( ( member @ C @ X @ I2 )
% 3.01/1.53 => ( module @ D @ a @ E @ b @ ( M3 @ X ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ ( C > D ) @ M4 @ ( carr_prodag @ C @ D @ ( module_ext @ D @ a @ E ) @ I2 @ M3 ) )
% 3.01/1.53 => ( ( member @ C @ J @ I2 )
% 3.01/1.53 => ( ( algebr142713364_sprod @ a @ b @ C @ D @ E @ r @ I2 @ M3 @ A3 @ M4 @ J )
% 3.01/1.53 = ( sprod @ D @ a @ E @ ( M3 @ J ) @ A3 @ ( M4 @ J ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_41_Sr__ring,axiom,
% 3.01/1.53 ! [S: set @ a] :
% 3.01/1.53 ( ( sr2 @ a @ b @ r @ S )
% 3.01/1.53 => ( ring @ a @ b @ ( sr @ a @ b @ r @ S ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_42_mHom__func,axiom,
% 3.01/1.53 ! [E: $tType,D: $tType,F2: $tType,C: $tType,F: C > D,M3: carrier_ext @ C @ ( aGroup_ext @ C @ ( module_ext @ C @ a @ E ) ),N: carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ a @ F2 ) )] :
% 3.01/1.53 ( ( member @ ( C > D ) @ F @ ( mHom @ a @ b @ C @ E @ D @ F2 @ r @ M3 @ N ) )
% 3.01/1.53 => ( member @ ( C > D ) @ F
% 3.01/1.53 @ ( pi @ C @ D @ ( carrier @ C @ ( aGroup_ext @ C @ ( module_ext @ C @ a @ E ) ) @ M3 )
% 3.01/1.53 @ ^ [Uu: C] : ( carrier @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ a @ F2 ) ) @ N ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_43_rInvim__def,axiom,
% 3.01/1.53 ! [B: $tType,M1: $tType,M: $tType,A: $tType] :
% 3.01/1.53 ( ( rInvim @ A @ M @ B @ M1 )
% 3.01/1.53 = ( ^ [A4: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ M ) ),R3: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ M1 ) ),F3: A > B,K: set @ B] :
% 3.01/1.53 ( collect @ A
% 3.01/1.53 @ ^ [A5: A] :
% 3.01/1.53 ( ( member @ A @ A5 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ M ) ) @ A4 ) )
% 3.01/1.53 & ( member @ B @ ( F3 @ A5 ) @ K ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_44_dsum__pOp__func,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 3.01/1.53 => ( member @ ( ( A > B ) > ( A > B ) > A > B ) @ ( prod_pOp @ A @ B @ C @ I2 @ A2 )
% 3.01/1.53 @ ( pi @ ( A > B ) @ ( ( A > B ) > A > B ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 )
% 3.01/1.53 @ ^ [Uu: A > B] :
% 3.01/1.53 ( pi @ ( A > B ) @ ( A > B ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 )
% 3.01/1.53 @ ^ [Uv: A > B] : ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_45_mem__Collect__eq,axiom,
% 3.01/1.53 ! [A: $tType,A3: A,P: A > $o] :
% 3.01/1.53 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 3.01/1.53 = ( P @ A3 ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_46_Collect__mem__eq,axiom,
% 3.01/1.53 ! [A: $tType,A2: set @ A] :
% 3.01/1.53 ( ( collect @ A
% 3.01/1.53 @ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
% 3.01/1.53 = A2 ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_47_Collect__cong,axiom,
% 3.01/1.53 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( P @ X )
% 3.01/1.53 = ( Q @ X ) )
% 3.01/1.53 => ( ( collect @ A @ P )
% 3.01/1.53 = ( collect @ A @ Q ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_48_ext,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( F @ X )
% 3.01/1.53 = ( G @ X ) )
% 3.01/1.53 => ( F = G ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_49_nsEqElm,axiom,
% 3.01/1.53 ! [X2: a,Y2: a,N2: nat] :
% 3.01/1.53 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( X2 = Y2 )
% 3.01/1.53 => ( ( nscal @ a @ b @ r @ X2 @ N2 )
% 3.01/1.53 = ( nscal @ a @ b @ r @ Y2 @ N2 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_50_nsClose,axiom,
% 3.01/1.53 ! [X2: a,N2: nat] :
% 3.01/1.53 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( member @ a @ ( nscal @ a @ b @ r @ X2 @ N2 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_51_tp__closed,axiom,
% 3.01/1.53 ( member @ ( a > a > a ) @ ( tp @ a @ b @ r )
% 3.01/1.53 @ ( pi @ a @ ( a > a ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 3.01/1.53 @ ^ [Uu: a] :
% 3.01/1.53 ( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 3.01/1.53 @ ^ [Uv: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_52_pop__closed,axiom,
% 3.01/1.53 ( member @ ( a > a > a ) @ ( pop @ a @ ( ring_ext @ a @ b ) @ r )
% 3.01/1.53 @ ( pi @ a @ ( a > a ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 3.01/1.53 @ ^ [Uu: a] :
% 3.01/1.53 ( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 3.01/1.53 @ ^ [Uv: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_53_rEQMulR,axiom,
% 3.01/1.53 ! [X2: a,Y2: a,Z2: a] :
% 3.01/1.53 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Z2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( X2 = Y2 )
% 3.01/1.53 => ( ( tp @ a @ b @ r @ X2 @ Z2 )
% 3.01/1.53 = ( tp @ a @ b @ r @ Y2 @ Z2 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_54_rMulLC,axiom,
% 3.01/1.53 ! [X2: a,Y2: a,Z2: a] :
% 3.01/1.53 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Z2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( tp @ a @ b @ r @ X2 @ ( tp @ a @ b @ r @ Y2 @ Z2 ) )
% 3.01/1.53 = ( tp @ a @ b @ r @ Y2 @ ( tp @ a @ b @ r @ X2 @ Z2 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_55_ring__tOp__assoc,axiom,
% 3.01/1.53 ! [X2: a,Y2: a,Z2: a] :
% 3.01/1.53 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Z2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ X2 @ Y2 ) @ Z2 )
% 3.01/1.53 = ( tp @ a @ b @ r @ X2 @ ( tp @ a @ b @ r @ Y2 @ Z2 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_56_ring__tOp__closed,axiom,
% 3.01/1.53 ! [X2: a,Y2: a] :
% 3.01/1.53 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( member @ a @ ( tp @ a @ b @ r @ X2 @ Y2 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_57_ring__tOp__commute,axiom,
% 3.01/1.53 ! [X2: a,Y2: a] :
% 3.01/1.53 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( tp @ a @ b @ r @ X2 @ Y2 )
% 3.01/1.53 = ( tp @ a @ b @ r @ Y2 @ X2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_58_ring__tOp__rel,axiom,
% 3.01/1.53 ! [X2: a,Xa: a,Y2: a,Ya: a] :
% 3.01/1.53 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Xa @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Ya @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ X2 @ Xa ) @ ( tp @ a @ b @ r @ Y2 @ Ya ) )
% 3.01/1.53 = ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ X2 @ Y2 ) @ ( tp @ a @ b @ r @ Xa @ Ya ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_59_tp__assoc,axiom,
% 3.01/1.53 ! [A3: a,B3: a,C3: a] :
% 3.01/1.53 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ C3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) @ C3 )
% 3.01/1.53 = ( tp @ a @ b @ r @ A3 @ ( tp @ a @ b @ r @ B3 @ C3 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_60_tp__commute,axiom,
% 3.01/1.53 ! [A3: a,B3: a] :
% 3.01/1.53 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( tp @ a @ b @ r @ A3 @ B3 )
% 3.01/1.53 = ( tp @ a @ b @ r @ B3 @ A3 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_61_pop__aassoc,axiom,
% 3.01/1.53 ! [A3: a,B3: a,C3: a] :
% 3.01/1.53 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ C3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ B3 ) @ C3 )
% 3.01/1.53 = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B3 @ C3 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_62_pop__commute,axiom,
% 3.01/1.53 ! [A3: a,B3: a] :
% 3.01/1.53 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ B3 )
% 3.01/1.53 = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B3 @ A3 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_63_Sr__tOp__closed,axiom,
% 3.01/1.53 ! [S: set @ a,X2: a,Y2: a] :
% 3.01/1.53 ( ( sr2 @ a @ b @ r @ S )
% 3.01/1.53 => ( ( member @ a @ X2 @ S )
% 3.01/1.53 => ( ( member @ a @ Y2 @ S )
% 3.01/1.53 => ( member @ a @ ( tp @ a @ b @ r @ X2 @ Y2 ) @ S ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_64_Sr__pOp__closed,axiom,
% 3.01/1.53 ! [S: set @ a,X2: a,Y2: a] :
% 3.01/1.53 ( ( sr2 @ a @ b @ r @ S )
% 3.01/1.53 => ( ( member @ a @ X2 @ S )
% 3.01/1.53 => ( ( member @ a @ Y2 @ S )
% 3.01/1.53 => ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ X2 @ Y2 ) @ S ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_65_rg__distrib,axiom,
% 3.01/1.53 ! [A3: a,B3: a,C3: a] :
% 3.01/1.53 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ C3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( tp @ a @ b @ r @ A3 @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B3 @ C3 ) )
% 3.01/1.53 = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) @ ( tp @ a @ b @ r @ A3 @ C3 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_66_ring__distrib1,axiom,
% 3.01/1.53 ! [X2: a,Y2: a,Z2: a] :
% 3.01/1.53 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Z2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( tp @ a @ b @ r @ X2 @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ Y2 @ Z2 ) )
% 3.01/1.53 = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ X2 @ Y2 ) @ ( tp @ a @ b @ r @ X2 @ Z2 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_67_ring__distrib2,axiom,
% 3.01/1.53 ! [X2: a,Y2: a,Z2: a] :
% 3.01/1.53 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Z2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ Y2 @ Z2 ) @ X2 )
% 3.01/1.53 = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ Y2 @ X2 ) @ ( tp @ a @ b @ r @ Z2 @ X2 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_68_ring__distrib3,axiom,
% 3.01/1.53 ! [A3: a,B3: a,X2: a,Y2: a] :
% 3.01/1.53 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ B3 ) @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ X2 @ Y2 ) )
% 3.01/1.53 = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ X2 ) @ ( tp @ a @ b @ r @ A3 @ Y2 ) ) @ ( tp @ a @ b @ r @ B3 @ X2 ) ) @ ( tp @ a @ b @ r @ B3 @ Y2 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_69_nsMulDistrL,axiom,
% 3.01/1.53 ! [X2: a,Y2: a,N2: nat] :
% 3.01/1.53 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( tp @ a @ b @ r @ X2 @ ( nscal @ a @ b @ r @ Y2 @ N2 ) )
% 3.01/1.53 = ( nscal @ a @ b @ r @ ( tp @ a @ b @ r @ X2 @ Y2 ) @ N2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_70_nsMulDistrR,axiom,
% 3.01/1.53 ! [X2: a,Y2: a,N2: nat] :
% 3.01/1.53 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( tp @ a @ b @ r @ ( nscal @ a @ b @ r @ Y2 @ N2 ) @ X2 )
% 3.01/1.53 = ( nscal @ a @ b @ r @ ( tp @ a @ b @ r @ Y2 @ X2 ) @ N2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_71_nsDistrL,axiom,
% 3.01/1.53 ! [X2: a,Y2: a,N2: nat] :
% 3.01/1.53 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( nscal @ a @ b @ r @ X2 @ N2 ) @ ( nscal @ a @ b @ r @ Y2 @ N2 ) )
% 3.01/1.53 = ( nscal @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ X2 @ Y2 ) @ N2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_72_Subring__tOp__ring__tOp,axiom,
% 3.01/1.53 ! [C: $tType,S: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ),A3: a,B3: a] :
% 3.01/1.53 ( ( subring @ a @ b @ C @ r @ S )
% 3.01/1.53 => ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ) @ S ) )
% 3.01/1.53 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ) @ S ) )
% 3.01/1.53 => ( ( tp @ a @ C @ S @ A3 @ B3 )
% 3.01/1.53 = ( tp @ a @ b @ r @ A3 @ B3 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_73_Subring__pOp__ring__pOp,axiom,
% 3.01/1.53 ! [C: $tType,S: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ),A3: a,B3: a] :
% 3.01/1.53 ( ( subring @ a @ b @ C @ r @ S )
% 3.01/1.53 => ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ) @ S ) )
% 3.01/1.53 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ) @ S ) )
% 3.01/1.53 => ( ( pop @ a @ ( ring_ext @ a @ C ) @ S @ A3 @ B3 )
% 3.01/1.53 = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ B3 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_74_mul__closed__set__tOp__closed,axiom,
% 3.01/1.53 ! [S: set @ a,S2: a,T: a] :
% 3.01/1.53 ( ( mul_closed_set @ a @ b @ r @ S )
% 3.01/1.53 => ( ( member @ a @ S2 @ S )
% 3.01/1.53 => ( ( member @ a @ T @ S )
% 3.01/1.53 => ( member @ a @ ( tp @ a @ b @ r @ S2 @ T ) @ S ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_75_Ring_OSr__tOp__closed,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A,X2: A,Y2: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( sr2 @ A @ B @ R2 @ S )
% 3.01/1.53 => ( ( member @ A @ X2 @ S )
% 3.01/1.53 => ( ( member @ A @ Y2 @ S )
% 3.01/1.53 => ( member @ A @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) @ S ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_76_Ring_OSr__pOp__closed,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A,X2: A,Y2: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( sr2 @ A @ B @ R2 @ S )
% 3.01/1.53 => ( ( member @ A @ X2 @ S )
% 3.01/1.53 => ( ( member @ A @ Y2 @ S )
% 3.01/1.53 => ( member @ A @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ X2 @ Y2 ) @ S ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_77_Ring_Oring__distrib3,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A,X2: A,Y2: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( tp @ A @ B @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ B3 ) @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ X2 @ Y2 ) )
% 3.01/1.53 = ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ X2 ) @ ( tp @ A @ B @ R2 @ A3 @ Y2 ) ) @ ( tp @ A @ B @ R2 @ B3 @ X2 ) ) @ ( tp @ A @ B @ R2 @ B3 @ Y2 ) ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_78_Ring_Oring__distrib2,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,Z2: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Z2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( tp @ A @ B @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ Y2 @ Z2 ) @ X2 )
% 3.01/1.53 = ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ Y2 @ X2 ) @ ( tp @ A @ B @ R2 @ Z2 @ X2 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_79_Ring_Oring__distrib1,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,Z2: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Z2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( tp @ A @ B @ R2 @ X2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ Y2 @ Z2 ) )
% 3.01/1.53 = ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) @ ( tp @ A @ B @ R2 @ X2 @ Z2 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_80_Ring_Org__distrib,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A,C3: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( tp @ A @ B @ R2 @ A3 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 @ C3 ) )
% 3.01/1.53 = ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) @ ( tp @ A @ B @ R2 @ A3 @ C3 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_81_Ring_OnsMulDistrR,axiom,
% 3.01/1.53 ! [A: $tType,B: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,N2: nat] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( tp @ A @ B @ R2 @ ( nscal @ A @ B @ R2 @ Y2 @ N2 ) @ X2 )
% 3.01/1.53 = ( nscal @ A @ B @ R2 @ ( tp @ A @ B @ R2 @ Y2 @ X2 ) @ N2 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_82_Ring_OnsMulDistrL,axiom,
% 3.01/1.53 ! [A: $tType,B: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,N2: nat] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( tp @ A @ B @ R2 @ X2 @ ( nscal @ A @ B @ R2 @ Y2 @ N2 ) )
% 3.01/1.53 = ( nscal @ A @ B @ R2 @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) @ N2 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_83_Ring_OnsDistrL,axiom,
% 3.01/1.53 ! [A: $tType,B: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,N2: nat] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( nscal @ A @ B @ R2 @ X2 @ N2 ) @ ( nscal @ A @ B @ R2 @ Y2 @ N2 ) )
% 3.01/1.53 = ( nscal @ A @ B @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ X2 @ Y2 ) @ N2 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_84_Ring_OSr__ring,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( sr2 @ A @ B @ R2 @ S )
% 3.01/1.53 => ( ring @ A @ B @ ( sr @ A @ B @ R2 @ S ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_85_aGroup_Oag__pOp__commute,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A,Y2: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( pop @ A @ B @ A2 @ X2 @ Y2 )
% 3.01/1.53 = ( pop @ A @ B @ A2 @ Y2 @ X2 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_86_aGroup_Oag__add__commute,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( pop @ A @ B @ A2 @ A3 @ B3 )
% 3.01/1.53 = ( pop @ A @ B @ A2 @ B3 @ A3 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_87_aGroup_OpOp__assocTr43,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A,D2: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ D2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ B3 ) @ ( pop @ A @ B @ A2 @ C3 @ D2 ) )
% 3.01/1.53 = ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ ( pop @ A @ B @ A2 @ B3 @ C3 ) ) @ D2 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_88_aGroup_OpOp__assocTr42,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A,D2: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ D2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ B3 ) @ C3 ) @ D2 )
% 3.01/1.53 = ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ ( pop @ A @ B @ A2 @ B3 @ C3 ) ) @ D2 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_89_aGroup_OpOp__assocTr41,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A,D2: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ D2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ B3 ) @ C3 ) @ D2 )
% 3.01/1.53 = ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ B3 ) @ ( pop @ A @ B @ A2 @ C3 @ D2 ) ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_90_aGroup_Oag__pOp__closed,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A,Y2: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( member @ A @ ( pop @ A @ B @ A2 @ X2 @ Y2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_91_aGroup_OpOp__cancel__r,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( ( pop @ A @ B @ A2 @ A3 @ C3 )
% 3.01/1.53 = ( pop @ A @ B @ A2 @ B3 @ C3 ) )
% 3.01/1.53 => ( A3 = B3 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_92_aGroup_OpOp__cancel__l,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( ( pop @ A @ B @ A2 @ C3 @ A3 )
% 3.01/1.53 = ( pop @ A @ B @ A2 @ C3 @ B3 ) )
% 3.01/1.53 => ( A3 = B3 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_93_aGroup_Oag__pOp__assoc,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A,Y2: A,Z2: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ Z2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ X2 @ Y2 ) @ Z2 )
% 3.01/1.53 = ( pop @ A @ B @ A2 @ X2 @ ( pop @ A @ B @ A2 @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_94_aGroup_Oag__pOp__add__r,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( A3 = B3 )
% 3.01/1.53 => ( ( pop @ A @ B @ A2 @ A3 @ C3 )
% 3.01/1.53 = ( pop @ A @ B @ A2 @ B3 @ C3 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_95_aGroup_Oag__pOp__add__l,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( A3 = B3 )
% 3.01/1.53 => ( ( pop @ A @ B @ A2 @ C3 @ A3 )
% 3.01/1.53 = ( pop @ A @ B @ A2 @ C3 @ B3 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_96_aGroup_OgEQAddcross,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),L1: A,L2: A,R1: A,R22: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ L1 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ L2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ R1 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ R1 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( L1 = R22 )
% 3.01/1.53 => ( ( L2 = R1 )
% 3.01/1.53 => ( ( pop @ A @ B @ A2 @ L1 @ L2 )
% 3.01/1.53 = ( pop @ A @ B @ A2 @ R1 @ R22 ) ) ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_97_aGroup_Oag__add4__rel,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A,D2: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ D2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ B3 ) @ ( pop @ A @ B @ A2 @ C3 @ D2 ) )
% 3.01/1.53 = ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ C3 ) @ ( pop @ A @ B @ A2 @ B3 @ D2 ) ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_98_aGroup_Oaassoc,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ B3 ) @ C3 )
% 3.01/1.53 = ( pop @ A @ B @ A2 @ A3 @ ( pop @ A @ B @ A2 @ B3 @ C3 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_99_Ring_Oring__tOp__commute,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( tp @ A @ B @ R2 @ X2 @ Y2 )
% 3.01/1.53 = ( tp @ A @ B @ R2 @ Y2 @ X2 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_100_Ring_Oring__tOp__closed,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( member @ A @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_101_Ring_Oring__tOp__assoc,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,Z2: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Z2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( tp @ A @ B @ R2 @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) @ Z2 )
% 3.01/1.53 = ( tp @ A @ B @ R2 @ X2 @ ( tp @ A @ B @ R2 @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_102_Ring_Oring__tOp__rel,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Xa: A,Y2: A,Ya: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Xa @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Ya @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( tp @ A @ B @ R2 @ ( tp @ A @ B @ R2 @ X2 @ Xa ) @ ( tp @ A @ B @ R2 @ Y2 @ Ya ) )
% 3.01/1.53 = ( tp @ A @ B @ R2 @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) @ ( tp @ A @ B @ R2 @ Xa @ Ya ) ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_103_Ring_Otp__commute,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( tp @ A @ B @ R2 @ A3 @ B3 )
% 3.01/1.53 = ( tp @ A @ B @ R2 @ B3 @ A3 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_104_Ring_Otp__assoc,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A,C3: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( tp @ A @ B @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) @ C3 )
% 3.01/1.53 = ( tp @ A @ B @ R2 @ A3 @ ( tp @ A @ B @ R2 @ B3 @ C3 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_105_Ring_OrEQMulR,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,Z2: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Z2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( X2 = Y2 )
% 3.01/1.53 => ( ( tp @ A @ B @ R2 @ X2 @ Z2 )
% 3.01/1.53 = ( tp @ A @ B @ R2 @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_106_Ring_OrMulLC,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,Z2: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Z2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( tp @ A @ B @ R2 @ X2 @ ( tp @ A @ B @ R2 @ Y2 @ Z2 ) )
% 3.01/1.53 = ( tp @ A @ B @ R2 @ Y2 @ ( tp @ A @ B @ R2 @ X2 @ Z2 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_107_Ring_Opop__commute,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ B3 )
% 3.01/1.53 = ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 @ A3 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_108_Ring_Opop__aassoc,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A,C3: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ B3 ) @ C3 )
% 3.01/1.53 = ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 @ C3 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_109_Ring_OnsEqElm,axiom,
% 3.01/1.53 ! [A: $tType,B: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,N2: nat] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( X2 = Y2 )
% 3.01/1.53 => ( ( nscal @ A @ B @ R2 @ X2 @ N2 )
% 3.01/1.53 = ( nscal @ A @ B @ R2 @ Y2 @ N2 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_110_Ring_OnsClose,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,N2: nat] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( member @ A @ ( nscal @ A @ B @ R2 @ X2 @ N2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_111_prod__pOp__mem__i,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B,Y: A > B,I3: A] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 3.01/1.53 => ( ( member @ ( A > B ) @ X4 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 3.01/1.53 => ( ( member @ ( A > B ) @ Y @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ I3 @ I2 )
% 3.01/1.53 => ( ( prod_pOp @ A @ B @ C @ I2 @ A2 @ X4 @ Y @ I3 )
% 3.01/1.53 = ( pop @ B @ C @ ( A2 @ I3 ) @ ( X4 @ I3 ) @ ( Y @ I3 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_112_dsum__pOp__mem,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B,Y: A > B] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 3.01/1.53 => ( ( member @ ( A > B ) @ X4 @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) )
% 3.01/1.53 => ( ( member @ ( A > B ) @ Y @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) )
% 3.01/1.53 => ( member @ ( A > B ) @ ( prod_pOp @ A @ B @ C @ I2 @ A2 @ X4 @ Y ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_113_dsum__iOp__mem,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 3.01/1.53 => ( ( member @ ( A > B ) @ X4 @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) )
% 3.01/1.53 => ( member @ ( A > B ) @ ( prod_mOp @ A @ B @ C @ I2 @ A2 @ X4 ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_114_dsum__zero__func,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 3.01/1.53 => ( member @ ( A > B ) @ ( prod_zero @ A @ B @ C @ I2 @ A2 ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_115_aGroup_Opop__closed,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B )] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( member @ ( A > A > A ) @ ( pop @ A @ B @ A2 )
% 3.01/1.53 @ ( pi @ A @ ( A > A ) @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 )
% 3.01/1.53 @ ^ [Uu: A] :
% 3.01/1.53 ( pi @ A @ A @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 )
% 3.01/1.53 @ ^ [Uv: A] : ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_116_Ring_OJ__rad__mem,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( j_rad @ A @ B @ R2 ) )
% 3.01/1.53 => ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_117_Ring_Otp__closed,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( member @ ( A > A > A ) @ ( tp @ A @ B @ R2 )
% 3.01/1.53 @ ( pi @ A @ ( A > A ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 )
% 3.01/1.53 @ ^ [Uu: A] :
% 3.01/1.53 ( pi @ A @ A @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 )
% 3.01/1.53 @ ^ [Uv: A] : ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_118_Ring_Opop__closed,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( member @ ( A > A > A ) @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 )
% 3.01/1.53 @ ( pi @ A @ ( A > A ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 )
% 3.01/1.53 @ ^ [Uu: A] :
% 3.01/1.53 ( pi @ A @ A @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 )
% 3.01/1.53 @ ^ [Uv: A] : ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_119_Ring_OprodM__sprod__val,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,E: $tType,D: $tType,C: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ A @ E ) ) ),A3: A,M4: C > D,J: C] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ! [X: C] :
% 3.01/1.53 ( ( member @ C @ X @ I2 )
% 3.01/1.53 => ( module @ D @ A @ E @ B @ ( M3 @ X ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ ( C > D ) @ M4 @ ( carr_prodag @ C @ D @ ( module_ext @ D @ A @ E ) @ I2 @ M3 ) )
% 3.01/1.53 => ( ( member @ C @ J @ I2 )
% 3.01/1.53 => ( ( algebr142713364_sprod @ A @ B @ C @ D @ E @ R2 @ I2 @ M3 @ A3 @ M4 @ J )
% 3.01/1.53 = ( sprod @ D @ A @ E @ ( M3 @ J ) @ A3 @ ( M4 @ J ) ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_120_dsum__iOp__func,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 3.01/1.53 => ( member @ ( ( A > B ) > A > B ) @ ( prod_mOp @ A @ B @ C @ I2 @ A2 )
% 3.01/1.53 @ ( pi @ ( A > B ) @ ( A > B ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 )
% 3.01/1.53 @ ^ [Uu: A > B] : ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_121_zeroring__J__rad__empty,axiom,
% 3.01/1.53 ( ( zeroring @ a @ b @ r )
% 3.01/1.53 => ( ( j_rad @ a @ b @ r )
% 3.01/1.53 = ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_122_Module_OmHom__lin,axiom,
% 3.01/1.53 ! [D: $tType,C: $tType,B: $tType,F2: $tType,E: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ),M4: A,F: A > E,A3: B] :
% 3.01/1.53 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 3.01/1.53 => ( ( module @ E @ B @ F2 @ D @ N @ R2 )
% 3.01/1.53 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 3.01/1.53 => ( ( member @ ( A > E ) @ F @ ( mHom @ B @ D @ A @ C @ E @ F2 @ R2 @ M3 @ N ) )
% 3.01/1.53 => ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
% 3.01/1.53 => ( ( F @ ( sprod @ A @ B @ C @ M3 @ A3 @ M4 ) )
% 3.01/1.53 = ( sprod @ E @ B @ F2 @ N @ A3 @ ( F @ M4 ) ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_123_Module_Osprod__assoc,axiom,
% 3.01/1.53 ! [D: $tType,B: $tType,C: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: B,B3: B,M4: A] :
% 3.01/1.53 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 3.01/1.53 => ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ B @ B3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 3.01/1.53 => ( ( sprod @ A @ B @ C @ M3 @ ( tp @ B @ D @ R2 @ A3 @ B3 ) @ M4 )
% 3.01/1.53 = ( sprod @ A @ B @ C @ M3 @ A3 @ ( sprod @ A @ B @ C @ M3 @ B3 @ M4 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_124_Ring_OmHom__func,axiom,
% 3.01/1.53 ! [B: $tType,E: $tType,D: $tType,A: $tType,F2: $tType,C: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),F: C > D,M3: carrier_ext @ C @ ( aGroup_ext @ C @ ( module_ext @ C @ A @ E ) ),N: carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ A @ F2 ) )] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ ( C > D ) @ F @ ( mHom @ A @ B @ C @ E @ D @ F2 @ R2 @ M3 @ N ) )
% 3.01/1.53 => ( member @ ( C > D ) @ F
% 3.01/1.53 @ ( pi @ C @ D @ ( carrier @ C @ ( aGroup_ext @ C @ ( module_ext @ C @ A @ E ) ) @ M3 )
% 3.01/1.53 @ ^ [Uu: C] : ( carrier @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ A @ F2 ) ) @ N ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_125_Ring_OSubring__pOp__ring__pOp,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ),A3: A,B3: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( subring @ A @ B @ C @ R2 @ S )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) )
% 3.01/1.53 => ( ( pop @ A @ ( ring_ext @ A @ C ) @ S @ A3 @ B3 )
% 3.01/1.53 = ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ B3 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_126_Ring_OSubring__tOp__ring__tOp,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ),A3: A,B3: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( subring @ A @ B @ C @ R2 @ S )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) )
% 3.01/1.53 => ( ( tp @ A @ C @ S @ A3 @ B3 )
% 3.01/1.53 = ( tp @ A @ B @ R2 @ A3 @ B3 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_127_Module_Osprod__closed,axiom,
% 3.01/1.53 ! [D: $tType,C: $tType,B: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: B,M4: A] :
% 3.01/1.53 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 3.01/1.53 => ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 3.01/1.53 => ( member @ A @ ( sprod @ A @ B @ C @ M3 @ A3 @ M4 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_128_nsDistr,axiom,
% 3.01/1.53 ! [X2: a,N2: nat,M4: nat] :
% 3.01/1.53 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( nscal @ a @ b @ r @ X2 @ N2 ) @ ( nscal @ a @ b @ r @ X2 @ M4 ) )
% 3.01/1.53 = ( nscal @ a @ b @ r @ X2 @ ( plus_plus @ nat @ N2 @ M4 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_129_add__bothl,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( plus @ A )
% 3.01/1.53 => ! [A3: A,B3: A,C3: A] :
% 3.01/1.53 ( ( A3 = B3 )
% 3.01/1.53 => ( ( plus_plus @ A @ C3 @ A3 )
% 3.01/1.53 = ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_130_add__both,axiom,
% 3.01/1.53 ! [A3: nat,B3: nat,C3: nat] :
% 3.01/1.53 ( ( A3 = B3 )
% 3.01/1.53 => ( ( plus_plus @ nat @ A3 @ C3 )
% 3.01/1.53 = ( plus_plus @ nat @ B3 @ C3 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_131_Module_OmHom__add,axiom,
% 3.01/1.53 ! [D: $tType,C: $tType,F2: $tType,B: $tType,E: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ),F: A > E,M4: A,N2: A] :
% 3.01/1.53 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 3.01/1.53 => ( ( module @ E @ B @ F2 @ D @ N @ R2 )
% 3.01/1.53 => ( ( member @ ( A > E ) @ F @ ( mHom @ B @ D @ A @ C @ E @ F2 @ R2 @ M3 @ N ) )
% 3.01/1.53 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 3.01/1.53 => ( ( member @ A @ N2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 3.01/1.53 => ( ( F @ ( pop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ M4 @ N2 ) )
% 3.01/1.53 = ( pop @ E @ ( module_ext @ E @ B @ F2 ) @ N @ ( F @ M4 ) @ ( F @ N2 ) ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_132_Module_Osprod__r__distr,axiom,
% 3.01/1.53 ! [D: $tType,B: $tType,C: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: B,M4: A,N2: A] :
% 3.01/1.53 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 3.01/1.53 => ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 3.01/1.53 => ( ( member @ A @ N2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 3.01/1.53 => ( ( sprod @ A @ B @ C @ M3 @ A3 @ ( pop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ M4 @ N2 ) )
% 3.01/1.53 = ( pop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ ( sprod @ A @ B @ C @ M3 @ A3 @ M4 ) @ ( sprod @ A @ B @ C @ M3 @ A3 @ N2 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_133_Module_Omodule__is__ag,axiom,
% 3.01/1.53 ! [D: $tType,C: $tType,B: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) )] :
% 3.01/1.53 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 3.01/1.53 => ( aGroup @ A @ ( module_ext @ A @ B @ C ) @ M3 ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_134_Module_Osprod__l__distr,axiom,
% 3.01/1.53 ! [D: $tType,B: $tType,C: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: B,B3: B,M4: A] :
% 3.01/1.53 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 3.01/1.53 => ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ B @ B3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 3.01/1.53 => ( ( sprod @ A @ B @ C @ M3 @ ( pop @ B @ ( ring_ext @ B @ D ) @ R2 @ A3 @ B3 ) @ M4 )
% 3.01/1.53 = ( pop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ ( sprod @ A @ B @ C @ M3 @ A3 @ M4 ) @ ( sprod @ A @ B @ C @ M3 @ B3 @ M4 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_135_Ring_OnsDistr,axiom,
% 3.01/1.53 ! [A: $tType,B: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,N2: nat,M4: nat] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( nscal @ A @ B @ R2 @ X2 @ N2 ) @ ( nscal @ A @ B @ R2 @ X2 @ M4 ) )
% 3.01/1.53 = ( nscal @ A @ B @ R2 @ X2 @ ( plus_plus @ nat @ N2 @ M4 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_136_Ring_Omul__closed__set__tOp__closed,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A,S2: A,T: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( mul_closed_set @ A @ B @ R2 @ S )
% 3.01/1.53 => ( ( member @ A @ S2 @ S )
% 3.01/1.53 => ( ( member @ A @ T @ S )
% 3.01/1.53 => ( member @ A @ ( tp @ A @ B @ R2 @ S2 @ T ) @ S ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_137_Ring_Ozeroring__J__rad__empty,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( zeroring @ A @ B @ R2 )
% 3.01/1.53 => ( ( j_rad @ A @ B @ R2 )
% 3.01/1.53 = ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_138_Module_Osc__Ring,axiom,
% 3.01/1.53 ! [C: $tType,A: $tType,D: $tType,B: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) )] :
% 3.01/1.53 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 3.01/1.53 => ( ring @ B @ D @ R2 ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_139_Ring_Osubring__Ring,axiom,
% 3.01/1.53 ! [B: $tType,C: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) )] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( subring @ A @ B @ C @ R2 @ S )
% 3.01/1.53 => ( ring @ A @ C @ S ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_140_Module_OmHom__eq,axiom,
% 3.01/1.53 ! [F2: $tType,D: $tType,C: $tType,B: $tType,E: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ),F: A > E,G: A > E] :
% 3.01/1.53 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 3.01/1.53 => ( ( module @ E @ B @ F2 @ D @ N @ R2 )
% 3.01/1.53 => ( ( member @ ( A > E ) @ F @ ( mHom @ B @ D @ A @ C @ E @ F2 @ R2 @ M3 @ N ) )
% 3.01/1.53 => ( ( member @ ( A > E ) @ G @ ( mHom @ B @ D @ A @ C @ E @ F2 @ R2 @ M3 @ N ) )
% 3.01/1.53 => ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 3.01/1.53 => ( ( F @ X )
% 3.01/1.53 = ( G @ X ) ) )
% 3.01/1.53 => ( F = G ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_141_Module_OmHom__mem,axiom,
% 3.01/1.53 ! [C: $tType,D: $tType,A: $tType,F2: $tType,B: $tType,E: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ),F: A > E,M4: A] :
% 3.01/1.53 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 3.01/1.53 => ( ( module @ E @ B @ F2 @ D @ N @ R2 )
% 3.01/1.53 => ( ( member @ ( A > E ) @ F @ ( mHom @ B @ D @ A @ C @ E @ F2 @ R2 @ M3 @ N ) )
% 3.01/1.53 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 3.01/1.53 => ( member @ E @ ( F @ M4 ) @ ( carrier @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ) @ N ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_142_Ring_Omem__subring__mem__ring,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ),X2: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( subring @ A @ B @ C @ R2 @ S )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) )
% 3.01/1.53 => ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_143_mul__closed__set__sub,axiom,
% 3.01/1.53 ! [S: set @ a] :
% 3.01/1.53 ( ( mul_closed_set @ a @ b @ r @ S )
% 3.01/1.53 => ( ord_less_eq @ ( set @ a ) @ S @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_144_zeroring__no__maximal,axiom,
% 3.01/1.53 ( ( zeroring @ a @ b @ r )
% 3.01/1.53 => ~ ? [X_1: set @ a] : ( maximal_ideal @ a @ b @ r @ X_1 ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_145_id__maximal__Exist,axiom,
% 3.01/1.53 ( ~ ( zeroring @ a @ b @ r )
% 3.01/1.53 => ? [X_12: set @ a] : ( maximal_ideal @ a @ b @ r @ X_12 ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_146_mop__closed,axiom,
% 3.01/1.53 ( member @ ( a > a ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r )
% 3.01/1.53 @ ( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 3.01/1.53 @ ^ [Uu: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_147_Sr__mOp__closed,axiom,
% 3.01/1.53 ! [S: set @ a,X2: a] :
% 3.01/1.53 ( ( sr2 @ a @ b @ r @ S )
% 3.01/1.53 => ( ( member @ a @ X2 @ S )
% 3.01/1.53 => ( member @ a @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ X2 ) @ S ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_148_ring__inv1,axiom,
% 3.01/1.53 ! [A3: a,B3: a] :
% 3.01/1.53 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) )
% 3.01/1.53 = ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A3 ) @ B3 ) )
% 3.01/1.53 & ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) )
% 3.01/1.53 = ( tp @ a @ b @ r @ A3 @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B3 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_149_ring__inv1__1,axiom,
% 3.01/1.53 ! [A3: a,B3: a] :
% 3.01/1.53 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) )
% 3.01/1.53 = ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A3 ) @ B3 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_150_ring__inv1__2,axiom,
% 3.01/1.53 ! [A3: a,B3: a] :
% 3.01/1.53 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) )
% 3.01/1.53 = ( tp @ a @ b @ r @ A3 @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B3 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_151_ring__inv1__3,axiom,
% 3.01/1.53 ! [A3: a,B3: a] :
% 3.01/1.53 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( tp @ a @ b @ r @ A3 @ B3 )
% 3.01/1.53 = ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A3 ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B3 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_152_ring__distrib4,axiom,
% 3.01/1.53 ! [A3: a,B3: a,X2: a,Y2: a] :
% 3.01/1.53 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ X2 @ Y2 ) ) )
% 3.01/1.53 = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B3 @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ Y2 ) ) ) @ ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ X2 ) ) @ Y2 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_153_Subring__minus__ring__minus,axiom,
% 3.01/1.53 ! [C: $tType,S: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ),X2: a] :
% 3.01/1.53 ( ( subring @ a @ b @ C @ r @ S )
% 3.01/1.53 => ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ) @ S ) )
% 3.01/1.53 => ( ( mop @ a @ ( ring_ext @ a @ C ) @ S @ X2 )
% 3.01/1.53 = ( mop @ a @ ( ring_ext @ a @ b ) @ r @ X2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_154_not__sub,axiom,
% 3.01/1.53 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.01/1.53 ( ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.01/1.53 => ? [A6: A] :
% 3.01/1.53 ( ( member @ A @ A6 @ A2 )
% 3.01/1.53 & ~ ( member @ A @ A6 @ B2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_155_eqsets__sub,axiom,
% 3.01/1.53 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.01/1.53 ( ( A2 = B2 )
% 3.01/1.53 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_156_ge__convert,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( ord @ A )
% 3.01/1.53 => ! [A3: A,B3: A,C3: A] :
% 3.01/1.53 ( ( A3 = B3 )
% 3.01/1.53 => ( ( ord_less_eq @ A @ C3 @ A3 )
% 3.01/1.53 => ( ord_less_eq @ A @ C3 @ B3 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_157_le__convert,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( ord @ A )
% 3.01/1.53 => ! [A3: A,B3: A,C3: A] :
% 3.01/1.53 ( ( A3 = B3 )
% 3.01/1.53 => ( ( ord_less_eq @ A @ A3 @ C3 )
% 3.01/1.53 => ( ord_less_eq @ A @ B3 @ C3 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_158_sub__which1,axiom,
% 3.01/1.53 ! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
% 3.01/1.53 ( ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.01/1.53 | ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ X2 @ A2 )
% 3.01/1.53 => ( ~ ( member @ A @ X2 @ B2 )
% 3.01/1.53 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_159_sub__which2,axiom,
% 3.01/1.53 ! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
% 3.01/1.53 ( ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.01/1.53 | ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) )
% 3.01/1.53 => ( ~ ( member @ A @ X2 @ A2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ B2 )
% 3.01/1.53 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_160_sets__not__eq,axiom,
% 3.01/1.53 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.01/1.53 ( ( A2 != B2 )
% 3.01/1.53 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 3.01/1.53 => ? [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ A2 )
% 3.01/1.53 & ~ ( member @ A @ X @ B2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_161_subset__self,axiom,
% 3.01/1.53 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_162_not__subseteq,axiom,
% 3.01/1.53 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.01/1.53 ( ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.01/1.53 => ? [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ A2 )
% 3.01/1.53 & ~ ( member @ A @ X @ B2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_163_proper__subset,axiom,
% 3.01/1.53 ! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
% 3.01/1.53 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.01/1.53 => ( ~ ( member @ A @ X2 @ A2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ B2 )
% 3.01/1.53 => ( A2 != B2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_164_conditional__subset,axiom,
% 3.01/1.53 ! [A: $tType,A2: set @ A,P: A > $o] :
% 3.01/1.53 ( ord_less_eq @ ( set @ A )
% 3.01/1.53 @ ( collect @ A
% 3.01/1.53 @ ^ [X3: A] :
% 3.01/1.53 ( ( member @ A @ X3 @ A2 )
% 3.01/1.53 & ( P @ X3 ) ) )
% 3.01/1.53 @ A2 ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_165_Nset__le,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( ord @ A )
% 3.01/1.53 => ! [X2: A,N2: A] :
% 3.01/1.53 ( ( member @ A @ X2
% 3.01/1.53 @ ( collect @ A
% 3.01/1.53 @ ^ [I4: A] : ( ord_less_eq @ A @ I4 @ N2 ) ) )
% 3.01/1.53 => ( ord_less_eq @ A @ X2 @ N2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_166_Pi__mono,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: set @ A,B2: A > ( set @ B ),C2: A > ( set @ B )] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ A2 )
% 3.01/1.53 => ( ord_less_eq @ ( set @ B ) @ ( B2 @ X ) @ ( C2 @ X ) ) )
% 3.01/1.53 => ( ord_less_eq @ ( set @ ( A > B ) ) @ ( pi @ A @ B @ A2 @ B2 ) @ ( pi @ A @ B @ A2 @ C2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_167_Pi__anti__mono,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A7: set @ A,A2: set @ A,B2: A > ( set @ B )] :
% 3.01/1.53 ( ( ord_less_eq @ ( set @ A ) @ A7 @ A2 )
% 3.01/1.53 => ( ord_less_eq @ ( set @ ( A > B ) ) @ ( pi @ A @ B @ A2 @ B2 ) @ ( pi @ A @ B @ A7 @ B2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_168_extend__fun,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ B,B1: set @ B] :
% 3.01/1.53 ( ( member @ ( A > B ) @ F
% 3.01/1.53 @ ( pi @ A @ B @ A2
% 3.01/1.53 @ ^ [Uu: A] : B2 ) )
% 3.01/1.53 => ( ( ord_less_eq @ ( set @ B ) @ B2 @ B1 )
% 3.01/1.53 => ( member @ ( A > B ) @ F
% 3.01/1.53 @ ( pi @ A @ B @ A2
% 3.01/1.53 @ ^ [Uu: A] : B1 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_169_aGroup_Oag__mOp__closed,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( member @ A @ ( mop @ A @ B @ A2 @ X2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_170_aGroup_Oag__inv__inv,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( mop @ A @ B @ A2 @ ( mop @ A @ B @ A2 @ X2 ) )
% 3.01/1.53 = X2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_171_aGroup_Oag__inv__inj,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A,Y2: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( X2 != Y2 )
% 3.01/1.53 => ( ( mop @ A @ B @ A2 @ X2 )
% 3.01/1.53 != ( mop @ A @ B @ A2 @ Y2 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_172_Ring_OSr__mOp__closed,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A,X2: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( sr2 @ A @ B @ R2 @ S )
% 3.01/1.53 => ( ( member @ A @ X2 @ S )
% 3.01/1.53 => ( member @ A @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ X2 ) @ S ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_173_Subring__subset,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) )] :
% 3.01/1.53 ( ( subring @ A @ B @ C @ R2 @ S )
% 3.01/1.53 => ( ord_less_eq @ ( set @ A ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_174_Ring_Oring__inv1__3,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( tp @ A @ B @ R2 @ A3 @ B3 )
% 3.01/1.53 = ( tp @ A @ B @ R2 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 ) @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_175_Ring_Oring__inv1__2,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) )
% 3.01/1.53 = ( tp @ A @ B @ R2 @ A3 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_176_Ring_Oring__inv1__1,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) )
% 3.01/1.53 = ( tp @ A @ B @ R2 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 ) @ B3 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_177_Ring_Oring__inv1,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) )
% 3.01/1.53 = ( tp @ A @ B @ R2 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 ) @ B3 ) )
% 3.01/1.53 & ( ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) )
% 3.01/1.53 = ( tp @ A @ B @ R2 @ A3 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_178_Ring_Oid__maximal__Exist,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ~ ( zeroring @ A @ B @ R2 )
% 3.01/1.53 => ? [X_12: set @ A] : ( maximal_ideal @ A @ B @ R2 @ X_12 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_179_Ring_Ozeroring__no__maximal,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( zeroring @ A @ B @ R2 )
% 3.01/1.53 => ~ ? [X_1: set @ A] : ( maximal_ideal @ A @ B @ R2 @ X_1 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_180_Ring_OSubring__minus__ring__minus,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ),X2: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( subring @ A @ B @ C @ R2 @ S )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) )
% 3.01/1.53 => ( ( mop @ A @ ( ring_ext @ A @ C ) @ S @ X2 )
% 3.01/1.53 = ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ X2 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_181_Ring_Omop__closed,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( member @ ( A > A ) @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 )
% 3.01/1.53 @ ( pi @ A @ A @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 )
% 3.01/1.53 @ ^ [Uu: A] : ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_182_aGroup_OpOp__assoc__cancel,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ ( mop @ A @ B @ A2 @ B3 ) ) @ ( pop @ A @ B @ A2 @ B3 @ ( mop @ A @ B @ A2 @ C3 ) ) )
% 3.01/1.53 = ( pop @ A @ B @ A2 @ A3 @ ( mop @ A @ B @ A2 @ C3 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_183_aGroup_Oag__diff__minus,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( ( pop @ A @ B @ A2 @ A3 @ ( mop @ A @ B @ A2 @ B3 ) )
% 3.01/1.53 = C3 )
% 3.01/1.53 => ( ( pop @ A @ B @ A2 @ B3 @ ( mop @ A @ B @ A2 @ A3 ) )
% 3.01/1.53 = ( mop @ A @ B @ A2 @ C3 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_184_aGroup_Oag__eq__sol2,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,X2: A,B3: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( ( pop @ A @ B @ A2 @ X2 @ A3 )
% 3.01/1.53 = B3 )
% 3.01/1.53 => ( X2
% 3.01/1.53 = ( pop @ A @ B @ A2 @ B3 @ ( mop @ A @ B @ A2 @ A3 ) ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_185_aGroup_Oag__eq__sol1,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,X2: A,B3: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( ( pop @ A @ B @ A2 @ A3 @ X2 )
% 3.01/1.53 = B3 )
% 3.01/1.53 => ( X2
% 3.01/1.53 = ( pop @ A @ B @ A2 @ ( mop @ A @ B @ A2 @ A3 ) @ B3 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_186_aGroup_Oag__p__inv,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A,Y2: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( mop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ X2 @ Y2 ) )
% 3.01/1.53 = ( pop @ A @ B @ A2 @ ( mop @ A @ B @ A2 @ X2 ) @ ( mop @ A @ B @ A2 @ Y2 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_187_Ring_Omul__closed__set__sub,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( mul_closed_set @ A @ B @ R2 @ S )
% 3.01/1.53 => ( ord_less_eq @ ( set @ A ) @ S @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_188_mul__closed__set__def,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType] :
% 3.01/1.53 ( ( mul_closed_set @ A @ B )
% 3.01/1.53 = ( ^ [R3: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
% 3.01/1.53 ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 3.01/1.53 & ! [X3: A] :
% 3.01/1.53 ( ( member @ A @ X3 @ S3 )
% 3.01/1.53 => ! [Y3: A] :
% 3.01/1.53 ( ( member @ A @ Y3 @ S3 )
% 3.01/1.53 => ( member @ A @ ( tp @ A @ B @ R3 @ X3 @ Y3 ) @ S3 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_189_aGroup_Omop__closed,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B )] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( member @ ( A > A ) @ ( mop @ A @ B @ A2 )
% 3.01/1.53 @ ( pi @ A @ A @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 )
% 3.01/1.53 @ ^ [Uu: A] : ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_190_prod__mOp__mem__i,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B,I3: A] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 3.01/1.53 => ( ( member @ ( A > B ) @ X4 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ I3 @ I2 )
% 3.01/1.53 => ( ( prod_mOp @ A @ B @ C @ I2 @ A2 @ X4 @ I3 )
% 3.01/1.53 = ( mop @ B @ C @ ( A2 @ I3 ) @ ( X4 @ I3 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_191_Ring_Oring__distrib4,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A,X2: A,Y2: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) ) )
% 3.01/1.53 = ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ Y2 ) ) ) @ ( tp @ A @ B @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ X2 ) ) @ Y2 ) ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_192_nonunit__contained__maxid,axiom,
% 3.01/1.53 ! [A3: a] :
% 3.01/1.53 ( ~ ( zeroring @ a @ b @ r )
% 3.01/1.53 => ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ~ ( unit @ a @ b @ r @ A3 )
% 3.01/1.53 => ? [Mx: set @ a] :
% 3.01/1.53 ( ( maximal_ideal @ a @ b @ r @ Mx )
% 3.01/1.53 & ( member @ a @ A3 @ Mx ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_193_set__sum__mem,axiom,
% 3.01/1.53 ! [A3: a,I2: set @ a,B3: a,J2: set @ a] :
% 3.01/1.53 ( ( member @ a @ A3 @ I2 )
% 3.01/1.53 => ( ( member @ a @ B3 @ J2 )
% 3.01/1.53 => ( ( ord_less_eq @ ( set @ a ) @ I2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( ord_less_eq @ ( set @ a ) @ J2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ B3 ) @ ( aset_sum @ a @ ( ring_ext @ a @ b ) @ r @ I2 @ J2 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_194_sum__mult__pOp__closed,axiom,
% 3.01/1.53 ! [A2: set @ a,B2: set @ a,A3: a,B3: a] :
% 3.01/1.53 ( ( ord_less_eq @ ( set @ a ) @ A2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( ord_less_eq @ ( set @ a ) @ B2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ A3 @ ( sum_mult @ a @ b @ r @ A2 @ B2 ) )
% 3.01/1.53 => ( ( member @ a @ B3 @ ( sum_mult @ a @ b @ r @ A2 @ B2 ) )
% 3.01/1.53 => ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ B3 ) @ ( sum_mult @ a @ b @ r @ A2 @ B2 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_195_sum__mult__subR,axiom,
% 3.01/1.53 ! [A2: set @ a,B2: set @ a] :
% 3.01/1.53 ( ( ord_less_eq @ ( set @ a ) @ A2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( ord_less_eq @ ( set @ a ) @ B2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ord_less_eq @ ( set @ a ) @ ( sum_mult @ a @ b @ r @ A2 @ B2 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_196_times__mem__sum__mult,axiom,
% 3.01/1.53 ! [A2: set @ a,B2: set @ a,A3: a,B3: a] :
% 3.01/1.53 ( ( ord_less_eq @ ( set @ a ) @ A2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( ord_less_eq @ ( set @ a ) @ B2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( member @ a @ A3 @ A2 )
% 3.01/1.53 => ( ( member @ a @ B3 @ B2 )
% 3.01/1.53 => ( member @ a @ ( tp @ a @ b @ r @ A3 @ B3 ) @ ( sum_mult @ a @ b @ r @ A2 @ B2 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_197_le__imp__add__int,axiom,
% 3.01/1.53 ! [I3: nat,J: nat] :
% 3.01/1.53 ( ( ord_less_eq @ nat @ I3 @ J )
% 3.01/1.53 => ? [K2: nat] :
% 3.01/1.53 ( J
% 3.01/1.53 = ( plus_plus @ nat @ I3 @ K2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_198_Module_OmHom__inv,axiom,
% 3.01/1.53 ! [D: $tType,C: $tType,B: $tType,F2: $tType,E: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ),M4: A,F: A > E] :
% 3.01/1.53 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 3.01/1.53 => ( ( module @ E @ B @ F2 @ D @ N @ R2 )
% 3.01/1.53 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 3.01/1.53 => ( ( member @ ( A > E ) @ F @ ( mHom @ B @ D @ A @ C @ E @ F2 @ R2 @ M3 @ N ) )
% 3.01/1.53 => ( ( F @ ( mop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ M4 ) )
% 3.01/1.53 = ( mop @ E @ ( module_ext @ E @ B @ F2 ) @ N @ ( F @ M4 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_199_dsumag__sub__prodag,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
% 3.01/1.53 ( ! [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ I2 )
% 3.01/1.53 => ( aGroup @ B @ C @ ( A2 @ X ) ) )
% 3.01/1.53 => ( ord_less_eq @ ( set @ ( A > B ) ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_200_Module_Osc__minus__am,axiom,
% 3.01/1.53 ! [D: $tType,B: $tType,C: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: B,M4: A] :
% 3.01/1.53 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 3.01/1.53 => ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 3.01/1.53 => ( ( mop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ ( sprod @ A @ B @ C @ M3 @ A3 @ M4 ) )
% 3.01/1.53 = ( sprod @ A @ B @ C @ M3 @ A3 @ ( mop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ M4 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_201_Ring_Osum__mult__subR,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A2: set @ A,B2: set @ A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ord_less_eq @ ( set @ A ) @ ( sum_mult @ A @ B @ R2 @ A2 @ B2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_202_Module_Osc__minus__am1,axiom,
% 3.01/1.53 ! [B: $tType,D: $tType,C: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: B,M4: A] :
% 3.01/1.53 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 3.01/1.53 => ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 3.01/1.53 => ( ( mop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ ( sprod @ A @ B @ C @ M3 @ A3 @ M4 ) )
% 3.01/1.53 = ( sprod @ A @ B @ C @ M3 @ ( mop @ B @ ( ring_ext @ B @ D ) @ R2 @ A3 ) @ M4 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_203_Ring_Otimes__mem__sum__mult,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A2: set @ A,B2: set @ A,A3: A,B3: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ A3 @ A2 )
% 3.01/1.53 => ( ( member @ A @ B3 @ B2 )
% 3.01/1.53 => ( member @ A @ ( tp @ A @ B @ R2 @ A3 @ B3 ) @ ( sum_mult @ A @ B @ R2 @ A2 @ B2 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_204_Ring_Osum__mult__pOp__closed,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A2: set @ A,B2: set @ A,A3: A,B3: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( sum_mult @ A @ B @ R2 @ A2 @ B2 ) )
% 3.01/1.53 => ( ( member @ A @ B3 @ ( sum_mult @ A @ B @ R2 @ A2 @ B2 ) )
% 3.01/1.53 => ( member @ A @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ B3 ) @ ( sum_mult @ A @ B @ R2 @ A2 @ B2 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_205_Ring_Oset__sum__mem,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,I2: set @ A,B3: A,J2: set @ A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ I2 )
% 3.01/1.53 => ( ( member @ A @ B3 @ J2 )
% 3.01/1.53 => ( ( ord_less_eq @ ( set @ A ) @ I2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( ord_less_eq @ ( set @ A ) @ J2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( member @ A @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ B3 ) @ ( aset_sum @ A @ ( ring_ext @ A @ B ) @ R2 @ I2 @ J2 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_206_Ring_Ononunit__contained__maxid,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ~ ( zeroring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ~ ( unit @ A @ B @ R2 @ A3 )
% 3.01/1.53 => ? [Mx: set @ A] :
% 3.01/1.53 ( ( maximal_ideal @ A @ B @ R2 @ Mx )
% 3.01/1.53 & ( member @ A @ A3 @ Mx ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_207_aGroup_Omem__set__sum,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),H: set @ A,K3: set @ A,X2: A] :
% 3.01/1.53 ( ( aGroup @ A @ B @ A2 )
% 3.01/1.53 => ( ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( ord_less_eq @ ( set @ A ) @ K3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( aset_sum @ A @ B @ A2 @ H @ K3 ) )
% 3.01/1.53 => ? [X: A] :
% 3.01/1.53 ( ( member @ A @ X @ H )
% 3.01/1.53 & ? [Xa2: A] :
% 3.01/1.53 ( ( member @ A @ Xa2 @ K3 )
% 3.01/1.53 & ( X2
% 3.01/1.53 = ( pop @ A @ B @ A2 @ X @ Xa2 ) ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_208_J__rad__unit,axiom,
% 3.01/1.53 ! [X2: a] :
% 3.01/1.53 ( ~ ( zeroring @ a @ b @ r )
% 3.01/1.53 => ( ( member @ a @ X2 @ ( j_rad @ a @ b @ r ) )
% 3.01/1.53 => ! [Y4: a] :
% 3.01/1.53 ( ( member @ a @ Y4 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( unit @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) @ ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ X2 ) @ Y4 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_209_add__le__cancel__right,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( ordere236663937imp_le @ A )
% 3.01/1.53 => ! [A3: A,C3: A,B3: A] :
% 3.01/1.53 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
% 3.01/1.53 = ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_210_un__closed,axiom,
% 3.01/1.53 member @ a @ ( un @ a @ b @ r ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_211_maximal__ideal__proper,axiom,
% 3.01/1.53 ! [Mx2: set @ a] :
% 3.01/1.53 ( ( maximal_ideal @ a @ b @ r @ Mx2 )
% 3.01/1.53 => ~ ( member @ a @ ( un @ a @ b @ r ) @ Mx2 ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_212_Sr__one,axiom,
% 3.01/1.53 ! [S: set @ a] :
% 3.01/1.53 ( ( sr2 @ a @ b @ r @ S )
% 3.01/1.53 => ( member @ a @ ( un @ a @ b @ r ) @ S ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_213_add__right__cancel,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( cancel_semigroup_add @ A )
% 3.01/1.53 => ! [B3: A,A3: A,C3: A] :
% 3.01/1.53 ( ( ( plus_plus @ A @ B3 @ A3 )
% 3.01/1.53 = ( plus_plus @ A @ C3 @ A3 ) )
% 3.01/1.53 = ( B3 = C3 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_214_add__left__cancel,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( cancel_semigroup_add @ A )
% 3.01/1.53 => ! [A3: A,B3: A,C3: A] :
% 3.01/1.53 ( ( ( plus_plus @ A @ A3 @ B3 )
% 3.01/1.53 = ( plus_plus @ A @ A3 @ C3 ) )
% 3.01/1.53 = ( B3 = C3 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_215_rg__l__unit,axiom,
% 3.01/1.53 ! [A3: a] :
% 3.01/1.53 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( tp @ a @ b @ r @ ( un @ a @ b @ r ) @ A3 )
% 3.01/1.53 = A3 ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_216_ring__l__one,axiom,
% 3.01/1.53 ! [X2: a] :
% 3.01/1.53 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( tp @ a @ b @ r @ ( un @ a @ b @ r ) @ X2 )
% 3.01/1.53 = X2 ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_217_ring__r__one,axiom,
% 3.01/1.53 ! [X2: a] :
% 3.01/1.53 ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( tp @ a @ b @ r @ X2 @ ( un @ a @ b @ r ) )
% 3.01/1.53 = X2 ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_218_Subring__one__ring__one,axiom,
% 3.01/1.53 ! [C: $tType,S: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) )] :
% 3.01/1.53 ( ( subring @ a @ b @ C @ r @ S )
% 3.01/1.53 => ( ( un @ a @ C @ S )
% 3.01/1.53 = ( un @ a @ b @ r ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_219_ring__times__minusr,axiom,
% 3.01/1.53 ! [A3: a] :
% 3.01/1.53 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A3 )
% 3.01/1.53 = ( tp @ a @ b @ r @ A3 @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_220_ring__times__minusl,axiom,
% 3.01/1.53 ! [A3: a] :
% 3.01/1.53 ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 3.01/1.53 => ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A3 )
% 3.01/1.53 = ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) ) @ A3 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_221_add__le__cancel__left,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( ordere236663937imp_le @ A )
% 3.01/1.53 => ! [C3: A,A3: A,B3: A] :
% 3.01/1.53 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) )
% 3.01/1.53 = ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_222_localring__unit,axiom,
% 3.01/1.53 ! [Mx2: set @ a] :
% 3.01/1.53 ( ~ ( zeroring @ a @ b @ r )
% 3.01/1.53 => ( ( maximal_ideal @ a @ b @ r @ Mx2 )
% 3.01/1.53 => ( ! [X: a] :
% 3.01/1.53 ( ( member @ a @ X @ Mx2 )
% 3.01/1.53 => ( unit @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( un @ a @ b @ r ) ) ) )
% 3.01/1.53 => ( local_ring @ a @ b @ r ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_223_primary__ideal__proper1,axiom,
% 3.01/1.53 ! [Q2: set @ a] :
% 3.01/1.53 ( ( primary_ideal @ a @ b @ r @ Q2 )
% 3.01/1.53 => ~ ( member @ a @ ( un @ a @ b @ r ) @ Q2 ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_224_self__le,axiom,
% 3.01/1.53 ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_225_nat__eq__le,axiom,
% 3.01/1.53 ! [M4: nat,N2: nat] :
% 3.01/1.53 ( ( M4 = N2 )
% 3.01/1.53 => ( ord_less_eq @ nat @ M4 @ N2 ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_226_n__in__Nsetn,axiom,
% 3.01/1.53 ! [N2: nat] :
% 3.01/1.53 ( member @ nat @ N2
% 3.01/1.53 @ ( collect @ nat
% 3.01/1.53 @ ^ [I4: nat] : ( ord_less_eq @ nat @ I4 @ N2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_227_mem__of__Nset,axiom,
% 3.01/1.53 ! [X2: nat,N2: nat] :
% 3.01/1.53 ( ( ord_less_eq @ nat @ X2 @ N2 )
% 3.01/1.53 => ( member @ nat @ X2
% 3.01/1.53 @ ( collect @ nat
% 3.01/1.53 @ ^ [I4: nat] : ( ord_less_eq @ nat @ I4 @ N2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_228_Ring_Oun__closed,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( member @ A @ ( un @ A @ B @ R2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_229_Ring_Omaximal__ideal__proper,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),Mx2: set @ A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( maximal_ideal @ A @ B @ R2 @ Mx2 )
% 3.01/1.53 => ~ ( member @ A @ ( un @ A @ B @ R2 ) @ Mx2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_230_Ring_OSubring__one__ring__one,axiom,
% 3.01/1.53 ! [C: $tType,B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) )] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( subring @ A @ B @ C @ R2 @ S )
% 3.01/1.53 => ( ( un @ A @ C @ S )
% 3.01/1.53 = ( un @ A @ B @ R2 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_231_Ring_OSr__one,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( sr2 @ A @ B @ R2 @ S )
% 3.01/1.53 => ( member @ A @ ( un @ A @ B @ R2 ) @ S ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_232_Ring_Oring__r__one,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( tp @ A @ B @ R2 @ X2 @ ( un @ A @ B @ R2 ) )
% 3.01/1.53 = X2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_233_Ring_Oring__l__one,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( tp @ A @ B @ R2 @ ( un @ A @ B @ R2 ) @ X2 )
% 3.01/1.53 = X2 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_234_Ring_Org__l__unit,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( tp @ A @ B @ R2 @ ( un @ A @ B @ R2 ) @ A3 )
% 3.01/1.53 = A3 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_235_Module_Osprod__one,axiom,
% 3.01/1.53 ! [C: $tType,D: $tType,B: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),M4: A] :
% 3.01/1.53 ( ( module @ A @ B @ C @ D @ M3 @ R2 )
% 3.01/1.53 => ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
% 3.01/1.53 => ( ( sprod @ A @ B @ C @ M3 @ ( un @ B @ D @ R2 ) @ M4 )
% 3.01/1.53 = M4 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_236_Unit__def,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType] :
% 3.01/1.53 ( ( unit @ A @ B )
% 3.01/1.53 = ( ^ [R3: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A5: A] :
% 3.01/1.53 ( ( member @ A @ A5 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 3.01/1.53 & ? [X3: A] :
% 3.01/1.53 ( ( member @ A @ X3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 3.01/1.53 & ( ( tp @ A @ B @ R3 @ A5 @ X3 )
% 3.01/1.53 = ( un @ A @ B @ R3 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_237_add__right__imp__eq,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( cancel_semigroup_add @ A )
% 3.01/1.53 => ! [B3: A,A3: A,C3: A] :
% 3.01/1.53 ( ( ( plus_plus @ A @ B3 @ A3 )
% 3.01/1.53 = ( plus_plus @ A @ C3 @ A3 ) )
% 3.01/1.53 => ( B3 = C3 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_238_add__left__imp__eq,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( cancel_semigroup_add @ A )
% 3.01/1.53 => ! [A3: A,B3: A,C3: A] :
% 3.01/1.53 ( ( ( plus_plus @ A @ A3 @ B3 )
% 3.01/1.53 = ( plus_plus @ A @ A3 @ C3 ) )
% 3.01/1.53 => ( B3 = C3 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_239_add_Oleft__commute,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( ab_semigroup_add @ A )
% 3.01/1.53 => ! [B3: A,A3: A,C3: A] :
% 3.01/1.53 ( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A3 @ C3 ) )
% 3.01/1.53 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_240_add_Ocommute,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( ab_semigroup_add @ A )
% 3.01/1.53 => ( ( plus_plus @ A )
% 3.01/1.53 = ( ^ [A5: A,B4: A] : ( plus_plus @ A @ B4 @ A5 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_241_add_Oright__cancel,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( group_add @ A )
% 3.01/1.53 => ! [B3: A,A3: A,C3: A] :
% 3.01/1.53 ( ( ( plus_plus @ A @ B3 @ A3 )
% 3.01/1.53 = ( plus_plus @ A @ C3 @ A3 ) )
% 3.01/1.53 = ( B3 = C3 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_242_add_Oleft__cancel,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( group_add @ A )
% 3.01/1.53 => ! [A3: A,B3: A,C3: A] :
% 3.01/1.53 ( ( ( plus_plus @ A @ A3 @ B3 )
% 3.01/1.53 = ( plus_plus @ A @ A3 @ C3 ) )
% 3.01/1.53 = ( B3 = C3 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_243_add_Oassoc,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( semigroup_add @ A )
% 3.01/1.53 => ! [A3: A,B3: A,C3: A] :
% 3.01/1.53 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
% 3.01/1.53 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_244_group__cancel_Oadd2,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( comm_monoid_add @ A )
% 3.01/1.53 => ! [B2: A,K4: A,B3: A,A3: A] :
% 3.01/1.53 ( ( B2
% 3.01/1.53 = ( plus_plus @ A @ K4 @ B3 ) )
% 3.01/1.53 => ( ( plus_plus @ A @ A3 @ B2 )
% 3.01/1.53 = ( plus_plus @ A @ K4 @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_245_group__cancel_Oadd1,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( comm_monoid_add @ A )
% 3.01/1.53 => ! [A2: A,K4: A,A3: A,B3: A] :
% 3.01/1.53 ( ( A2
% 3.01/1.53 = ( plus_plus @ A @ K4 @ A3 ) )
% 3.01/1.53 => ( ( plus_plus @ A @ A2 @ B3 )
% 3.01/1.53 = ( plus_plus @ A @ K4 @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_246_add__mono__thms__linordered__semiring_I4_J,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( ordere779506340up_add @ A )
% 3.01/1.53 => ! [I3: A,J: A,K4: A,L: A] :
% 3.01/1.53 ( ( ( I3 = J )
% 3.01/1.53 & ( K4 = L ) )
% 3.01/1.53 => ( ( plus_plus @ A @ I3 @ K4 )
% 3.01/1.53 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_247_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( ab_semigroup_add @ A )
% 3.01/1.53 => ! [A3: A,B3: A,C3: A] :
% 3.01/1.53 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
% 3.01/1.53 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_248_Ring_Oring__times__minusl,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 )
% 3.01/1.53 = ( tp @ A @ B @ R2 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( un @ A @ B @ R2 ) ) @ A3 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_249_Ring_Oring__times__minusr,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A] :
% 3.01/1.53 ( ( ring @ A @ B @ R2 )
% 3.01/1.53 => ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
% 3.01/1.53 => ( ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 )
% 3.01/1.53 = ( tp @ A @ B @ R2 @ A3 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( un @ A @ B @ R2 ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_250_sr__def,axiom,
% 3.01/1.53 ! [B: $tType,A: $tType] :
% 3.01/1.53 ( ( sr2 @ A @ B )
% 3.01/1.53 = ( ^ [R3: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
% 3.01/1.53 ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 3.01/1.53 & ( member @ A @ ( un @ A @ B @ R3 ) @ S3 )
% 3.01/1.53 & ! [X3: A] :
% 3.01/1.53 ( ( member @ A @ X3 @ S3 )
% 3.01/1.53 => ! [Y3: A] :
% 3.01/1.53 ( ( member @ A @ Y3 @ S3 )
% 3.01/1.53 => ( ( member @ A @ ( pop @ A @ ( ring_ext @ A @ B ) @ R3 @ X3 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R3 @ Y3 ) ) @ S3 )
% 3.01/1.53 & ( member @ A @ ( tp @ A @ B @ R3 @ X3 @ Y3 ) @ S3 ) ) ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_251_add__mono__thms__linordered__semiring_I3_J,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( ordere779506340up_add @ A )
% 3.01/1.53 => ! [I3: A,J: A,K4: A,L: A] :
% 3.01/1.53 ( ( ( ord_less_eq @ A @ I3 @ J )
% 3.01/1.53 & ( K4 = L ) )
% 3.01/1.53 => ( ord_less_eq @ A @ ( plus_plus @ A @ I3 @ K4 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_252_add__mono__thms__linordered__semiring_I2_J,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( ordere779506340up_add @ A )
% 3.01/1.53 => ! [I3: A,J: A,K4: A,L: A] :
% 3.01/1.53 ( ( ( I3 = J )
% 3.01/1.53 & ( ord_less_eq @ A @ K4 @ L ) )
% 3.01/1.53 => ( ord_less_eq @ A @ ( plus_plus @ A @ I3 @ K4 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_253_add__mono__thms__linordered__semiring_I1_J,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( ordere779506340up_add @ A )
% 3.01/1.53 => ! [I3: A,J: A,K4: A,L: A] :
% 3.01/1.53 ( ( ( ord_less_eq @ A @ I3 @ J )
% 3.01/1.53 & ( ord_less_eq @ A @ K4 @ L ) )
% 3.01/1.53 => ( ord_less_eq @ A @ ( plus_plus @ A @ I3 @ K4 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_254_add__mono,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( ordere779506340up_add @ A )
% 3.01/1.53 => ! [A3: A,B3: A,C3: A,D2: A] :
% 3.01/1.53 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.01/1.53 => ( ( ord_less_eq @ A @ C3 @ D2 )
% 3.01/1.53 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(fact_255_add__left__mono,axiom,
% 3.01/1.53 ! [A: $tType] :
% 3.01/1.53 ( ( ordere779506340up_add @ A )
% 3.01/1.53 => ! [A3: A,B3: A,C3: A] :
% 3.01/1.53 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.01/1.53 => ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(tcon_fun___Orderings_Oord,axiom,
% 3.01/1.53 ! [A8: $tType,A9: $tType] :
% 3.01/1.53 ( ( ord @ A9 )
% 3.01/1.53 => ( ord @ ( A8 > A9 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 3.01/1.53 ordere236663937imp_le @ nat ).
% 3.01/1.53
% 3.01/1.53 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 3.01/1.53 ordere779506340up_add @ nat ).
% 3.01/1.53
% 3.01/1.53 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 3.01/1.53 cancel_semigroup_add @ nat ).
% 3.01/1.53
% 3.01/1.53 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 3.01/1.53 ab_semigroup_add @ nat ).
% 3.01/1.53
% 3.01/1.53 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 3.01/1.53 comm_monoid_add @ nat ).
% 3.01/1.53
% 3.01/1.53 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 3.01/1.53 semigroup_add @ nat ).
% 3.01/1.53
% 3.01/1.53 thf(tcon_Nat_Onat___Orderings_Oord_1,axiom,
% 3.01/1.53 ord @ nat ).
% 3.01/1.53
% 3.01/1.53 thf(tcon_Nat_Onat___Groups_Oplus,axiom,
% 3.01/1.53 plus @ nat ).
% 3.01/1.53
% 3.01/1.53 thf(tcon_Set_Oset___Orderings_Oord_2,axiom,
% 3.01/1.53 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(tcon_HOL_Obool___Orderings_Oord_3,axiom,
% 3.01/1.53 ord @ $o ).
% 3.01/1.53
% 3.01/1.53 thf(tcon_Product__Type_Ounit___Orderings_Oord_4,axiom,
% 3.01/1.53 ord @ product_unit ).
% 3.01/1.53
% 3.01/1.53 thf(conj_0,hypothesis,
% 3.01/1.53 ! [X5: c] :
% 3.01/1.53 ( ( member @ c @ X5 @ i )
% 3.01/1.53 => ( module @ d @ a @ e @ b @ ( m @ X5 ) @ r ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(conj_1,hypothesis,
% 3.01/1.53 ( ( member @ ( ( c > d ) > ( c > d ) > c > d ) @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
% 3.01/1.53 @ ( pi @ ( c > d ) @ ( ( c > d ) > c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
% 3.01/1.53 @ ^ [Uu: c > d] :
% 3.01/1.53 ( pi @ ( c > d ) @ ( c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
% 3.01/1.53 @ ^ [Uv: c > d] : ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) ) )
% 3.01/1.53 & ! [A10: c > d] :
% 3.01/1.53 ( ( member @ ( c > d ) @ A10 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
% 3.01/1.53 => ! [B5: c > d] :
% 3.01/1.53 ( ( member @ ( c > d ) @ B5 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
% 3.01/1.53 => ! [C4: c > d] :
% 3.01/1.53 ( ( member @ ( c > d ) @ C4 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
% 3.01/1.53 => ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A10 @ B5 ) @ C4 )
% 3.01/1.53 = ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A10 @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ B5 @ C4 ) ) ) ) ) )
% 3.01/1.53 & ! [A10: c > d] :
% 3.01/1.53 ( ( member @ ( c > d ) @ A10 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
% 3.01/1.53 => ! [B5: c > d] :
% 3.01/1.53 ( ( member @ ( c > d ) @ B5 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
% 3.01/1.53 => ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A10 @ B5 )
% 3.01/1.53 = ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ B5 @ A10 ) ) ) )
% 3.01/1.53 & ( member @ ( ( c > d ) > c > d ) @ ( prod_mOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
% 3.01/1.53 @ ( pi @ ( c > d ) @ ( c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
% 3.01/1.53 @ ^ [Uu: c > d] : ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) )
% 3.01/1.53 & ! [A10: c > d] :
% 3.01/1.53 ( ( member @ ( c > d ) @ A10 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
% 3.01/1.53 => ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_mOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A10 ) @ A10 )
% 3.01/1.53 = ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) )
% 3.01/1.53 & ( member @ ( c > d ) @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
% 3.01/1.53 & ! [A10: c > d] :
% 3.01/1.53 ( ( member @ ( c > d ) @ A10 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
% 3.01/1.53 => ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) @ A10 )
% 3.01/1.53 = A10 ) ) ) ).
% 3.01/1.53
% 3.01/1.53 thf(conj_2,conjecture,
% 3.01/1.53 ! [A6: c > d] :
% 3.01/1.53 ( ~ ( member @ ( c > d ) @ A6 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
% 3.01/1.53 | ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A6 @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
% 3.01/1.53 = A6 ) ) ).
% 3.01/1.53 ------------------------
% 3.01/1.56 % (32052)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_31945 for (2999ds/183Mi)
% 3.01/1.57 % (32054)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_31945 for (2999ds/27Mi)
% 3.01/1.57 % (32055)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_31945 for (2999ds/2Mi)
% 3.01/1.57 % (32053)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_31945 for (2999ds/4Mi)
% 3.01/1.57 % (32057)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_31945 for (2999ds/275Mi)
% 3.01/1.57 % (32056)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_31945 for (2999ds/2Mi)
% 3.01/1.57 % (32058)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_31945 for (2999ds/18Mi)
% 3.01/1.57 % (32055)Instruction limit reached!
% 3.01/1.57 % (32055)------------------------------
% 3.01/1.57 % (32055)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.57 % (32055)Termination reason: Unknown
% 3.01/1.57 % (32056)Instruction limit reached!
% 3.01/1.57 % (32056)------------------------------
% 3.01/1.57 % (32056)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.57 % (32055)Termination phase: shuffling
% 3.01/1.57
% 3.01/1.57 % (32055)Memory used [KB]: 1407
% 3.01/1.57 % (32055)Time elapsed: 0.003 s
% 3.01/1.57 % (32055)Instructions burned: 3 (million)
% 3.01/1.57 % (32055)------------------------------
% 3.01/1.57 % (32055)------------------------------
% 3.01/1.57 % (32056)Termination reason: Unknown
% 3.01/1.57 % (32056)Termination phase: shuffling
% 3.01/1.57
% 3.01/1.57 % (32056)Memory used [KB]: 1407
% 3.01/1.57 % (32056)Time elapsed: 0.003 s
% 3.01/1.57 % (32056)Instructions burned: 3 (million)
% 3.01/1.57 % (32056)------------------------------
% 3.01/1.57 % (32056)------------------------------
% 3.01/1.57 % (32053)Instruction limit reached!
% 3.01/1.57 % (32053)------------------------------
% 3.01/1.57 % (32053)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.57 % (32053)Termination reason: Unknown
% 3.01/1.57 % (32053)Termination phase: shuffling
% 3.01/1.57
% 3.01/1.57 % (32053)Memory used [KB]: 1535
% 3.01/1.57 % (32053)Time elapsed: 0.004 s
% 3.01/1.57 % (32053)Instructions burned: 5 (million)
% 3.01/1.57 % (32053)------------------------------
% 3.01/1.57 % (32053)------------------------------
% 3.01/1.58 % (32058)Instruction limit reached!
% 3.01/1.58 % (32058)------------------------------
% 3.01/1.58 % (32058)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.58 % (32058)Termination reason: Unknown
% 3.01/1.58 % (32058)Termination phase: shuffling
% 3.01/1.58
% 3.01/1.58 % (32058)Memory used [KB]: 1791
% 3.01/1.58 % (32058)Time elapsed: 0.011 s
% 3.01/1.58 % (32058)Instructions burned: 19 (million)
% 3.01/1.58 % (32058)------------------------------
% 3.01/1.58 % (32058)------------------------------
% 3.01/1.58 % (32054)Instruction limit reached!
% 3.01/1.58 % (32054)------------------------------
% 3.01/1.58 % (32054)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.58 % (32054)Termination reason: Unknown
% 3.01/1.58 % (32054)Termination phase: shuffling
% 3.01/1.58
% 3.01/1.58 % (32054)Memory used [KB]: 1918
% 3.01/1.58 % (32054)Time elapsed: 0.015 s
% 3.01/1.58 % (32054)Instructions burned: 27 (million)
% 3.01/1.58 % (32054)------------------------------
% 3.01/1.58 % (32054)------------------------------
% 3.01/1.59 % (32059)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_31945 for (2999ds/3Mi)
% 3.01/1.59 % (32060)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_31945 for (2999ds/37Mi)
% 3.01/1.59 % (32059)Instruction limit reached!
% 3.01/1.59 % (32059)------------------------------
% 3.01/1.59 % (32059)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.59 % (32059)Termination reason: Unknown
% 3.01/1.59 % (32059)Termination phase: shuffling
% 3.01/1.59
% 3.01/1.59 % (32059)Memory used [KB]: 1407
% 3.01/1.59 % (32059)Time elapsed: 0.003 s
% 3.01/1.59 % (32059)Instructions burned: 4 (million)
% 3.01/1.59 % (32059)------------------------------
% 3.01/1.59 % (32059)------------------------------
% 3.01/1.59 % (32061)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_31945 for (2999ds/15Mi)
% 3.01/1.60 % (32062)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_31945 for (2999ds/3Mi)
% 3.01/1.60 % (32061)Instruction limit reached!
% 3.01/1.60 % (32061)------------------------------
% 3.01/1.60 % (32061)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.60 % (32061)Termination reason: Unknown
% 3.01/1.60 % (32061)Termination phase: shuffling
% 3.01/1.60
% 3.01/1.60 % (32061)Memory used [KB]: 1663
% 3.01/1.60 % (32061)Time elapsed: 0.009 s
% 3.01/1.60 % (32061)Instructions burned: 16 (million)
% 3.01/1.60 % (32061)------------------------------
% 3.01/1.60 % (32061)------------------------------
% 3.08/1.60 % (32062)Instruction limit reached!
% 3.08/1.60 % (32062)------------------------------
% 3.08/1.60 % (32062)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.60 % (32062)Termination reason: Unknown
% 3.08/1.60 % (32062)Termination phase: shuffling
% 3.08/1.60
% 3.08/1.60 % (32062)Memory used [KB]: 1407
% 3.08/1.60 % (32062)Time elapsed: 0.003 s
% 3.08/1.60 % (32062)Instructions burned: 3 (million)
% 3.08/1.60 % (32062)------------------------------
% 3.08/1.60 % (32062)------------------------------
% 3.08/1.60 % (32060)Instruction limit reached!
% 3.08/1.60 % (32060)------------------------------
% 3.08/1.60 % (32060)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.60 % (32060)Termination reason: Unknown
% 3.08/1.60 % (32060)Termination phase: shuffling
% 3.08/1.60
% 3.08/1.60 % (32060)Memory used [KB]: 2046
% 3.08/1.60 % (32060)Time elapsed: 0.018 s
% 3.08/1.60 % (32060)Instructions burned: 37 (million)
% 3.08/1.60 % (32060)------------------------------
% 3.08/1.60 % (32060)------------------------------
% 3.08/1.60 % (32064)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_31945 for (2999ds/7Mi)
% 3.08/1.60 % (32063)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_31945 for (2999ds/1041Mi)
% 3.08/1.61 % (32064)Instruction limit reached!
% 3.08/1.61 % (32064)------------------------------
% 3.08/1.61 % (32064)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.61 % (32064)Termination reason: Unknown
% 3.08/1.61 % (32064)Termination phase: shuffling
% 3.08/1.61
% 3.08/1.61 % (32064)Memory used [KB]: 1535
% 3.08/1.61 % (32064)Time elapsed: 0.005 s
% 3.08/1.61 % (32064)Instructions burned: 8 (million)
% 3.08/1.61 % (32064)------------------------------
% 3.08/1.61 % (32064)------------------------------
% 3.08/1.61 % (32065)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_31945 for (2999ds/16Mi)
% 3.08/1.61 % (32066)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_31945 for (2999ds/3Mi)
% 3.08/1.61 % (32066)Instruction limit reached!
% 3.08/1.61 % (32066)------------------------------
% 3.08/1.61 % (32066)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.61 % (32066)Termination reason: Unknown
% 3.08/1.61 % (32066)Termination phase: shuffling
% 3.08/1.61
% 3.08/1.61 % (32066)Memory used [KB]: 1407
% 3.08/1.61 % (32066)Time elapsed: 0.003 s
% 3.08/1.61 % (32066)Instructions burned: 3 (million)
% 3.08/1.61 % (32066)------------------------------
% 3.08/1.61 % (32066)------------------------------
% 3.08/1.62 % (32067)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_31945 for (2999ds/3Mi)
% 3.08/1.62 % (32065)Instruction limit reached!
% 3.08/1.62 % (32065)------------------------------
% 3.08/1.62 % (32065)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.62 % (32065)Termination reason: Unknown
% 3.08/1.62 % (32065)Termination phase: shuffling
% 3.08/1.62
% 3.08/1.62 % (32065)Memory used [KB]: 1663
% 3.08/1.62 % (32065)Time elapsed: 0.009 s
% 3.08/1.62 % (32065)Instructions burned: 16 (million)
% 3.08/1.62 % (32065)------------------------------
% 3.08/1.62 % (32065)------------------------------
% 3.08/1.62 % (32067)Instruction limit reached!
% 3.08/1.62 % (32067)------------------------------
% 3.08/1.62 % (32067)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.62 % (32067)Termination reason: Unknown
% 3.08/1.62 % (32067)Termination phase: shuffling
% 3.08/1.62
% 3.08/1.62 % (32067)Memory used [KB]: 1407
% 3.08/1.62 % (32067)Time elapsed: 0.003 s
% 3.08/1.62 % (32067)Instructions burned: 3 (million)
% 3.08/1.62 % (32067)------------------------------
% 3.08/1.62 % (32067)------------------------------
% 3.08/1.62 % (32068)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_31945 for (2999ds/7Mi)
% 3.08/1.63 % (32068)Instruction limit reached!
% 3.08/1.63 % (32068)------------------------------
% 3.08/1.63 % (32068)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.63 % (32068)Termination reason: Unknown
% 3.08/1.63 % (32068)Termination phase: shuffling
% 3.08/1.63
% 3.08/1.63 % (32068)Memory used [KB]: 1535
% 3.08/1.63 % (32068)Time elapsed: 0.005 s
% 3.08/1.63 % (32068)Instructions burned: 8 (million)
% 3.08/1.63 % (32068)------------------------------
% 3.08/1.63 % (32068)------------------------------
% 3.08/1.63 % (32052)Instruction limit reached!
% 3.08/1.63 % (32052)------------------------------
% 3.08/1.63 % (32052)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.63 % (32052)Termination reason: Unknown
% 3.08/1.63 % (32052)Termination phase: Saturation
% 3.08/1.63
% 3.08/1.63 % (32052)Memory used [KB]: 7291
% 3.08/1.63 % (32052)Time elapsed: 0.056 s
% 3.08/1.63 % (32052)Instructions burned: 183 (million)
% 3.08/1.63 % (32052)------------------------------
% 3.08/1.63 % (32052)------------------------------
% 3.08/1.63 % (32069)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_31945 for (2999ds/3Mi)
% 3.08/1.63 % (32069)Instruction limit reached!
% 3.08/1.63 % (32069)------------------------------
% 3.08/1.63 % (32069)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.63 % (32069)Termination reason: Unknown
% 3.08/1.63 % (32069)Termination phase: shuffling
% 3.08/1.63
% 3.08/1.63 % (32069)Memory used [KB]: 1407
% 3.08/1.63 % (32069)Time elapsed: 0.003 s
% 3.08/1.63 % (32069)Instructions burned: 4 (million)
% 3.08/1.63 % (32069)------------------------------
% 3.08/1.63 % (32069)------------------------------
% 3.08/1.64 % (32070)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_31945 for (2999ds/4Mi)
% 3.08/1.64 % (32071)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_31945 for (2999ds/18Mi)
% 3.08/1.64 % (32070)Instruction limit reached!
% 3.08/1.64 % (32070)------------------------------
% 3.08/1.64 % (32070)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.64 % (32070)Termination reason: Unknown
% 3.08/1.64 % (32070)Termination phase: shuffling
% 3.08/1.64
% 3.08/1.64 % (32070)Memory used [KB]: 1407
% 3.08/1.64 % (32070)Time elapsed: 0.004 s
% 3.08/1.64 % (32070)Instructions burned: 5 (million)
% 3.08/1.64 % (32070)------------------------------
% 3.08/1.64 % (32070)------------------------------
% 3.08/1.64 % (32073)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_31945 for (2999ds/6Mi)
% 3.08/1.64 % (32073)Instruction limit reached!
% 3.08/1.64 % (32073)------------------------------
% 3.08/1.64 % (32073)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.64 % (32073)Termination reason: Unknown
% 3.08/1.64 % (32073)Termination phase: shuffling
% 3.08/1.64
% 3.08/1.64 % (32073)Memory used [KB]: 1535
% 3.08/1.64 % (32073)Time elapsed: 0.003 s
% 3.08/1.64 % (32073)Instructions burned: 6 (million)
% 3.08/1.64 % (32073)------------------------------
% 3.08/1.64 % (32073)------------------------------
% 3.08/1.64 % (32072)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_31945 for (2999ds/710Mi)
% 3.08/1.64 % (32071)Instruction limit reached!
% 3.08/1.64 % (32071)------------------------------
% 3.08/1.64 % (32071)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.64 % (32071)Termination reason: Unknown
% 3.08/1.64 % (32071)Termination phase: shuffling
% 3.08/1.64
% 3.08/1.64 % (32071)Memory used [KB]: 1791
% 3.08/1.64 % (32071)Time elapsed: 0.010 s
% 3.08/1.65 % (32071)Instructions burned: 18 (million)
% 3.08/1.65 % (32071)------------------------------
% 3.08/1.65 % (32071)------------------------------
% 3.08/1.65 % (32074)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_31945 for (2999ds/902Mi)
% 3.08/1.65 % (32075)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_31945 for (2999ds/21Mi)
% 3.08/1.66 % (32076)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_31945 for (2999ds/5Mi)
% 3.08/1.66 % (32075)Instruction limit reached!
% 3.08/1.66 % (32075)------------------------------
% 3.08/1.66 % (32075)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.66 % (32075)Termination reason: Unknown
% 3.08/1.66 % (32075)Termination phase: shuffling
% 3.08/1.66
% 3.08/1.66 % (32075)Memory used [KB]: 1791
% 3.08/1.66 % (32075)Time elapsed: 0.008 s
% 3.08/1.66 % (32075)Instructions burned: 21 (million)
% 3.08/1.66 % (32075)------------------------------
% 3.08/1.66 % (32075)------------------------------
% 3.08/1.66 % (32076)Instruction limit reached!
% 3.08/1.66 % (32076)------------------------------
% 3.08/1.66 % (32076)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.66 % (32076)Termination reason: Unknown
% 3.08/1.66 % (32076)Termination phase: shuffling
% 3.08/1.66
% 3.08/1.66 % (32076)Memory used [KB]: 1535
% 3.08/1.66 % (32076)Time elapsed: 0.004 s
% 3.08/1.66 % (32076)Instructions burned: 5 (million)
% 3.08/1.66 % (32076)------------------------------
% 3.08/1.66 % (32076)------------------------------
% 3.08/1.66 % (32077)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_31945 for (2998ds/6Mi)
% 3.08/1.66 % (32077)Instruction limit reached!
% 3.08/1.66 % (32077)------------------------------
% 3.08/1.66 % (32077)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.66 % (32077)Termination reason: Unknown
% 3.08/1.66 % (32077)Termination phase: shuffling
% 3.08/1.66
% 3.08/1.66 % (32077)Memory used [KB]: 1535
% 3.08/1.66 % (32077)Time elapsed: 0.004 s
% 3.08/1.66 % (32077)Instructions burned: 6 (million)
% 3.08/1.66 % (32077)------------------------------
% 3.08/1.66 % (32077)------------------------------
% 3.08/1.67 % (32078)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_31945 for (2998ds/377Mi)
% 3.08/1.68 % (32080)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_31945 for (2998ds/19Mi)
% 3.08/1.69 % (32080)Instruction limit reached!
% 3.08/1.69 % (32080)------------------------------
% 3.08/1.69 % (32080)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.69 % (32080)Termination reason: Unknown
% 3.08/1.69 % (32080)Termination phase: shuffling
% 3.08/1.69
% 3.08/1.69 % (32080)Memory used [KB]: 1791
% 3.08/1.69 % (32080)Time elapsed: 0.010 s
% 3.08/1.69 % (32080)Instructions burned: 19 (million)
% 3.08/1.69 % (32080)------------------------------
% 3.08/1.69 % (32080)------------------------------
% 3.08/1.69 % (32079)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_31945 for (2998ds/779Mi)
% 3.08/1.70 % (32057)Instruction limit reached!
% 3.08/1.70 % (32057)------------------------------
% 3.08/1.70 % (32057)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.70 % (32057)Termination reason: Unknown
% 3.08/1.70 % (32057)Termination phase: Saturation
% 3.08/1.70
% 3.08/1.70 % (32057)Memory used [KB]: 7803
% 3.08/1.70 % (32057)Time elapsed: 0.131 s
% 3.08/1.70 % (32057)Instructions burned: 275 (million)
% 3.08/1.70 % (32057)------------------------------
% 3.08/1.70 % (32057)------------------------------
% 3.08/1.70 % (32081)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_31945 for (2998ds/879Mi)
% 3.08/1.71 % (32082)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_31945 for (2998ds/17Mi)
% 3.08/1.71 % (32082)Instruction limit reached!
% 3.08/1.71 % (32082)------------------------------
% 3.08/1.71 % (32082)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.71 % (32082)Termination reason: Unknown
% 3.08/1.71 % (32082)Termination phase: shuffling
% 3.08/1.71
% 3.08/1.71 % (32082)Memory used [KB]: 1791
% 3.08/1.71 % (32082)Time elapsed: 0.007 s
% 3.08/1.71 % (32082)Instructions burned: 18 (million)
% 3.08/1.71 % (32082)------------------------------
% 3.08/1.71 % (32082)------------------------------
% 3.08/1.73 % (32084)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_31945 for (2998ds/3Mi)
% 3.08/1.73 % (32084)Instruction limit reached!
% 3.08/1.73 % (32084)------------------------------
% 3.08/1.73 % (32084)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.73 % (32084)Termination reason: Unknown
% 3.08/1.73 % (32084)Termination phase: shuffling
% 3.08/1.73
% 3.08/1.73 % (32084)Memory used [KB]: 1407
% 3.08/1.73 % (32084)Time elapsed: 0.004 s
% 3.08/1.73 % (32084)Instructions burned: 4 (million)
% 3.08/1.73 % (32084)------------------------------
% 3.08/1.73 % (32084)------------------------------
% 3.08/1.75 % (32085)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_31945 for (2998ds/30Mi)
% 3.08/1.76 % (32085)Instruction limit reached!
% 3.08/1.76 % (32085)------------------------------
% 3.08/1.76 % (32085)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.76 % (32085)Termination reason: Unknown
% 3.08/1.76 % (32085)Termination phase: shuffling
% 3.08/1.76
% 3.08/1.76 % (32085)Memory used [KB]: 1918
% 3.08/1.76 % (32085)Time elapsed: 0.016 s
% 3.08/1.76 % (32085)Instructions burned: 30 (million)
% 3.08/1.76 % (32085)------------------------------
% 3.08/1.76 % (32085)------------------------------
% 3.08/1.78 % (32086)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_31945 for (2997ds/127Mi)
% 3.08/1.83 % (32086)Instruction limit reached!
% 3.08/1.83 % (32086)------------------------------
% 3.08/1.83 % (32086)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.83 % (32086)Termination reason: Unknown
% 3.08/1.83 % (32086)Termination phase: Property scanning
% 3.08/1.83
% 3.08/1.83 % (32086)Memory used [KB]: 2686
% 3.08/1.83 % (32086)Time elapsed: 0.049 s
% 3.08/1.83 % (32086)Instructions burned: 128 (million)
% 3.08/1.83 % (32086)------------------------------
% 3.08/1.83 % (32086)------------------------------
% 3.08/1.83 % (32078)Instruction limit reached!
% 3.08/1.83 % (32078)------------------------------
% 3.08/1.83 % (32078)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.83 % (32078)Termination reason: Unknown
% 3.08/1.83 % (32078)Termination phase: Saturation
% 3.08/1.83
% 3.08/1.83 % (32078)Memory used [KB]: 8699
% 3.08/1.83 % (32078)Time elapsed: 0.162 s
% 3.08/1.83 % (32078)Instructions burned: 377 (million)
% 3.08/1.83 % (32078)------------------------------
% 3.08/1.83 % (32078)------------------------------
% 3.08/1.84 % (32087)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_31945 for (2997ds/100Mi)
% 3.08/1.85 % (32088)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_31945 for (2997ds/3Mi)
% 3.08/1.85 % (32088)Instruction limit reached!
% 3.08/1.85 % (32088)------------------------------
% 3.08/1.85 % (32088)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.85 % (32088)Termination reason: Unknown
% 3.08/1.85 % (32088)Termination phase: shuffling
% 3.08/1.85
% 3.08/1.85 % (32088)Memory used [KB]: 1407
% 3.08/1.85 % (32088)Time elapsed: 0.003 s
% 3.08/1.85 % (32088)Instructions burned: 3 (million)
% 3.08/1.85 % (32088)------------------------------
% 3.08/1.85 % (32088)------------------------------
% 3.08/1.86 % (32087)Instruction limit reached!
% 3.08/1.86 % (32087)------------------------------
% 3.08/1.86 % (32087)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.86 % (32087)Termination reason: Unknown
% 3.08/1.86 % (32087)Termination phase: Property scanning
% 3.08/1.86
% 3.08/1.86 % (32087)Memory used [KB]: 2430
% 3.08/1.86 % (32087)Time elapsed: 0.028 s
% 3.08/1.86 % (32087)Instructions burned: 100 (million)
% 3.08/1.86 % (32087)------------------------------
% 3.08/1.86 % (32087)------------------------------
% 3.08/1.86 % (32089)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_31945 for (2997ds/20Mi)
% 3.08/1.87 % (32089)Instruction limit reached!
% 3.08/1.87 % (32089)------------------------------
% 3.08/1.87 % (32089)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.87 % (32089)Termination reason: Unknown
% 3.08/1.87 % (32089)Termination phase: shuffling
% 3.08/1.87
% 3.08/1.87 % (32089)Memory used [KB]: 1791
% 3.08/1.87 % (32089)Time elapsed: 0.008 s
% 3.08/1.87 % (32089)Instructions burned: 20 (million)
% 3.08/1.87 % (32089)------------------------------
% 3.08/1.87 % (32089)------------------------------
% 3.08/1.88 % (32091)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_31945 for (2996ds/107Mi)
% 3.08/1.88 % (32090)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_31945 for (2996ds/86Mi)
% 3.08/1.92 % (32090)Instruction limit reached!
% 3.08/1.92 % (32090)------------------------------
% 3.08/1.92 % (32090)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.92 % (32090)Termination reason: Unknown
% 3.08/1.92 % (32090)Termination phase: Preprocessing 1
% 3.08/1.92
% 3.08/1.92 % (32090)Memory used [KB]: 2302
% 3.08/1.92 % (32090)Time elapsed: 0.043 s
% 3.08/1.92 % (32090)Instructions burned: 86 (million)
% 3.08/1.92 % (32090)------------------------------
% 3.08/1.92 % (32090)------------------------------
% 3.08/1.93 % (32091)Instruction limit reached!
% 3.08/1.93 % (32091)------------------------------
% 3.08/1.93 % (32091)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.93 % (32091)Termination reason: Unknown
% 3.08/1.93 % (32091)Termination phase: Preprocessing 3
% 3.08/1.93
% 3.08/1.93 % (32091)Memory used [KB]: 2558
% 3.08/1.93 % (32091)Time elapsed: 0.051 s
% 3.08/1.93 % (32091)Instructions burned: 107 (million)
% 3.08/1.93 % (32091)------------------------------
% 3.08/1.93 % (32091)------------------------------
% 3.08/1.94 % (32092)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_31945 for (2996ds/39Mi)
% 3.08/1.94 % (32093)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_31945 for (2996ds/448Mi)
% 3.08/1.97 % (32092)Instruction limit reached!
% 3.08/1.97 % (32092)------------------------------
% 3.08/1.97 % (32092)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.97 % (32092)Termination reason: Unknown
% 3.08/1.97 % (32092)Termination phase: shuffling
% 3.08/1.97
% 3.08/1.97 % (32092)Memory used [KB]: 2046
% 3.08/1.97 % (32092)Time elapsed: 0.013 s
% 3.08/1.97 % (32092)Instructions burned: 39 (million)
% 3.08/1.97 % (32092)------------------------------
% 3.08/1.97 % (32092)------------------------------
% 3.08/1.98 % (32072)Instruction limit reached!
% 3.08/1.98 % (32072)------------------------------
% 3.08/1.98 % (32072)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.98 % (32072)Termination reason: Unknown
% 3.08/1.98 % (32072)Termination phase: Saturation
% 3.08/1.98
% 3.08/1.98 % (32072)Memory used [KB]: 8699
% 3.08/1.98 % (32072)Time elapsed: 0.340 s
% 3.08/1.98 % (32072)Instructions burned: 711 (million)
% 3.08/1.98 % (32072)------------------------------
% 3.08/1.98 % (32072)------------------------------
% 3.08/1.98 % (32095)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_31945 for (2995ds/46Mi)
% 3.08/2.00 % (32096)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_31945 for (2995ds/98Mi)
% 3.08/2.01 % (32095)Instruction limit reached!
% 3.08/2.01 % (32095)------------------------------
% 3.08/2.01 % (32095)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/2.01 % (32095)Termination reason: Unknown
% 3.08/2.01 % (32095)Termination phase: shuffling
% 3.08/2.01
% 3.08/2.01 % (32095)Memory used [KB]: 2174
% 3.08/2.01 % (32095)Time elapsed: 0.025 s
% 3.08/2.01 % (32095)Instructions burned: 47 (million)
% 3.08/2.01 % (32095)------------------------------
% 3.08/2.01 % (32095)------------------------------
% 3.08/2.02 % (32079)Instruction limit reached!
% 3.08/2.02 % (32079)------------------------------
% 3.08/2.02 % (32079)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/2.02 % (32079)Termination reason: Unknown
% 3.08/2.02 % (32079)Termination phase: Saturation
% 3.08/2.02
% 3.08/2.02 % (32079)Memory used [KB]: 8955
% 3.08/2.02 % (32079)Time elapsed: 0.355 s
% 3.08/2.02 % (32079)Instructions burned: 779 (million)
% 3.08/2.02 % (32079)------------------------------
% 3.08/2.02 % (32079)------------------------------
% 3.08/2.02 % (32098)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_31945 for (2995ds/507Mi)
% 6.20/2.03 % (32099)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_31945 for (2995ds/149Mi)
% 6.20/2.04 % (32096)Instruction limit reached!
% 6.20/2.04 % (32096)------------------------------
% 6.20/2.04 % (32096)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.04 % (32096)Termination reason: Unknown
% 6.20/2.04 % (32096)Termination phase: Property scanning
% 6.20/2.04
% 6.20/2.04 % (32096)Memory used [KB]: 2430
% 6.20/2.04 % (32096)Time elapsed: 0.046 s
% 6.20/2.04 % (32096)Instructions burned: 98 (million)
% 6.20/2.04 % (32096)------------------------------
% 6.20/2.04 % (32096)------------------------------
% 6.20/2.06 % (32100)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_31945 for (2995ds/39Mi)
% 6.20/2.06 % (32074)Instruction limit reached!
% 6.20/2.06 % (32074)------------------------------
% 6.20/2.06 % (32074)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.06 % (32074)Termination reason: Unknown
% 6.20/2.06 % (32074)Termination phase: Saturation
% 6.20/2.06
% 6.20/2.06 % (32074)Memory used [KB]: 8571
% 6.20/2.06 % (32074)Time elapsed: 0.414 s
% 6.20/2.06 % (32074)Instructions burned: 903 (million)
% 6.20/2.06 % (32074)------------------------------
% 6.20/2.06 % (32074)------------------------------
% 6.20/2.06 % (32063)Instruction limit reached!
% 6.20/2.06 % (32063)------------------------------
% 6.20/2.06 % (32063)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.06 % (32063)Termination reason: Unknown
% 6.20/2.06 % (32063)Termination phase: Saturation
% 6.20/2.06
% 6.20/2.06 % (32063)Memory used [KB]: 10234
% 6.20/2.06 % (32063)Time elapsed: 0.456 s
% 6.20/2.06 % (32063)Instructions burned: 1042 (million)
% 6.20/2.06 % (32063)------------------------------
% 6.20/2.06 % (32063)------------------------------
% 6.20/2.07 % (32100)Instruction limit reached!
% 6.20/2.07 % (32100)------------------------------
% 6.20/2.07 % (32100)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.07 % (32100)Termination reason: Unknown
% 6.20/2.07 % (32100)Termination phase: shuffling
% 6.20/2.07
% 6.20/2.07 % (32100)Memory used [KB]: 2046
% 6.20/2.07 % (32100)Time elapsed: 0.016 s
% 6.20/2.07 % (32100)Instructions burned: 41 (million)
% 6.20/2.07 % (32100)------------------------------
% 6.20/2.07 % (32100)------------------------------
% 6.20/2.08 % (32101)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_31945 for (2994ds/6Mi)
% 6.20/2.08 % (32101)Instruction limit reached!
% 6.20/2.08 % (32101)------------------------------
% 6.20/2.08 % (32101)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.08 % (32101)Termination reason: Unknown
% 6.20/2.08 % (32101)Termination phase: shuffling
% 6.20/2.08
% 6.20/2.08 % (32101)Memory used [KB]: 1535
% 6.20/2.08 % (32101)Time elapsed: 0.004 s
% 6.20/2.08 % (32101)Instructions burned: 6 (million)
% 6.20/2.08 % (32101)------------------------------
% 6.20/2.08 % (32101)------------------------------
% 6.20/2.08 % (32102)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_31945 for (2994ds/1087Mi)
% 6.20/2.08 % (32103)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_31945 for (2994ds/47Mi)
% 6.20/2.09 % (32099)Instruction limit reached!
% 6.20/2.09 % (32099)------------------------------
% 6.20/2.09 % (32099)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.09 % (32099)Termination reason: Unknown
% 6.20/2.09 % (32099)Termination phase: Property scanning
% 6.20/2.09
% 6.20/2.09 % (32099)Memory used [KB]: 2686
% 6.20/2.09 % (32099)Time elapsed: 0.055 s
% 6.20/2.09 % (32099)Instructions burned: 149 (million)
% 6.20/2.09 % (32099)------------------------------
% 6.20/2.09 % (32099)------------------------------
% 6.20/2.10 % (32104)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_31945 for (2994ds/44Mi)
% 6.20/2.10 % (32103)Instruction limit reached!
% 6.20/2.10 % (32103)------------------------------
% 6.20/2.10 % (32103)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.10 % (32103)Termination reason: Unknown
% 6.20/2.10 % (32103)Termination phase: shuffling
% 6.20/2.10
% 6.20/2.10 % (32103)Memory used [KB]: 2174
% 6.20/2.10 % (32103)Time elapsed: 0.017 s
% 6.20/2.10 % (32103)Instructions burned: 49 (million)
% 6.20/2.10 % (32103)------------------------------
% 6.20/2.10 % (32103)------------------------------
% 6.20/2.10 % (32081)Instruction limit reached!
% 6.20/2.10 % (32081)------------------------------
% 6.20/2.10 % (32081)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.10 % (32081)Termination reason: Unknown
% 6.20/2.10 % (32081)Termination phase: Saturation
% 6.20/2.10
% 6.20/2.10 % (32081)Memory used [KB]: 9594
% 6.20/2.10 % (32081)Time elapsed: 0.396 s
% 6.20/2.10 % (32081)Instructions burned: 880 (million)
% 6.20/2.10 % (32081)------------------------------
% 6.20/2.10 % (32081)------------------------------
% 6.20/2.10 % (32105)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_31945 for (2994ds/432Mi)
% 6.20/2.11 % (32106)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_31945 for (2994ds/62Mi)
% 6.20/2.12 % (32104)Instruction limit reached!
% 6.20/2.12 % (32104)------------------------------
% 6.20/2.12 % (32104)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.12 % (32104)Termination reason: Unknown
% 6.20/2.12 % (32104)Termination phase: shuffling
% 6.20/2.12
% 6.20/2.12 % (32104)Memory used [KB]: 2174
% 6.20/2.12 % (32104)Time elapsed: 0.022 s
% 6.20/2.12 % (32104)Instructions burned: 46 (million)
% 6.20/2.12 % (32104)------------------------------
% 6.20/2.12 % (32104)------------------------------
% 6.20/2.12 % (32107)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_31945 for (2994ds/31Mi)
% 6.68/2.13 % (32107)Instruction limit reached!
% 6.68/2.13 % (32107)------------------------------
% 6.68/2.13 % (32107)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.68/2.13 % (32107)Termination reason: Unknown
% 6.68/2.13 % (32107)Termination phase: shuffling
% 6.68/2.13
% 6.68/2.13 % (32107)Memory used [KB]: 1918
% 6.68/2.13 % (32107)Time elapsed: 0.017 s
% 6.68/2.13 % (32107)Instructions burned: 31 (million)
% 6.68/2.13 % (32107)------------------------------
% 6.68/2.13 % (32107)------------------------------
% 6.68/2.13 % (32108)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_31945 for (2994ds/96Mi)
% 6.68/2.13 % (32106)Instruction limit reached!
% 6.68/2.13 % (32106)------------------------------
% 6.68/2.13 % (32106)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.68/2.13 % (32106)Termination reason: Unknown
% 6.68/2.13 % (32106)Termination phase: shuffling
% 6.68/2.13
% 6.68/2.13 % (32106)Memory used [KB]: 2302
% 6.68/2.14 % (32106)Time elapsed: 0.028 s
% 6.68/2.14 % (32106)Instructions burned: 63 (million)
% 6.68/2.14 % (32106)------------------------------
% 6.68/2.14 % (32106)------------------------------
% 6.68/2.14 % (32093)Instruction limit reached!
% 6.68/2.14 % (32093)------------------------------
% 6.68/2.14 % (32093)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.68/2.14 % (32093)Termination reason: Unknown
% 6.68/2.14 % (32093)Termination phase: Saturation
% 6.68/2.14
% 6.68/2.14 % (32093)Memory used [KB]: 8699
% 6.68/2.14 % (32093)Time elapsed: 0.193 s
% 6.68/2.14 % (32093)Instructions burned: 451 (million)
% 6.68/2.14 % (32093)------------------------------
% 6.68/2.14 % (32093)------------------------------
% 6.68/2.15 % (32111)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_31945 for (2994ds/83Mi)
% 6.68/2.15 % (32109)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_31945 for (2994ds/453Mi)
% 6.68/2.15 % (32110)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_31945 for (2994ds/71Mi)
% 6.68/2.18 % (32110)Instruction limit reached!
% 6.68/2.18 % (32110)------------------------------
% 6.68/2.18 % (32110)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.68/2.18 % (32110)Termination reason: Unknown
% 6.68/2.18 % (32110)Termination phase: Property scanning
% 6.68/2.18
% 6.68/2.18 % (32110)Memory used [KB]: 2302
% 6.68/2.18 % (32110)Time elapsed: 0.028 s
% 6.68/2.18 % (32110)Instructions burned: 73 (million)
% 6.68/2.18 % (32110)------------------------------
% 6.68/2.18 % (32110)------------------------------
% 6.68/2.18 % (32108)Instruction limit reached!
% 6.68/2.18 % (32108)------------------------------
% 6.68/2.18 % (32108)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.68/2.18 % (32108)Termination reason: Unknown
% 6.68/2.18 % (32108)Termination phase: Preprocessing 3
% 6.68/2.18
% 6.68/2.18 % (32108)Memory used [KB]: 2558
% 6.68/2.18 % (32108)Time elapsed: 0.048 s
% 6.68/2.18 % (32108)Instructions burned: 97 (million)
% 6.68/2.18 % (32108)------------------------------
% 6.68/2.18 % (32108)------------------------------
% 6.68/2.18 % (32111)Instruction limit reached!
% 6.68/2.18 % (32111)------------------------------
% 6.68/2.18 % (32111)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.68/2.18 % (32111)Termination reason: Unknown
% 6.68/2.18 % (32111)Termination phase: Property scanning
% 6.68/2.18
% 6.68/2.18 % (32111)Memory used [KB]: 2302
% 6.68/2.18 % (32111)Time elapsed: 0.034 s
% 6.68/2.18 % (32111)Instructions burned: 84 (million)
% 6.68/2.18 % (32111)------------------------------
% 6.68/2.18 % (32111)------------------------------
% 6.68/2.19 % (32112)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_31945 for (2993ds/11Mi)
% 6.68/2.19 % (32112)Instruction limit reached!
% 6.68/2.19 % (32112)------------------------------
% 6.68/2.19 % (32112)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.68/2.19 % (32112)Termination reason: Unknown
% 6.68/2.19 % (32112)Termination phase: shuffling
% 6.68/2.19
% 6.68/2.19 % (32112)Memory used [KB]: 1663
% 6.68/2.19 % (32112)Time elapsed: 0.005 s
% 6.68/2.19 % (32112)Instructions burned: 13 (million)
% 6.68/2.19 % (32112)------------------------------
% 6.68/2.19 % (32112)------------------------------
% 6.68/2.19 % (32113)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_31945 for (2993ds/41Mi)
% 6.68/2.19 % (32114)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_31945 for (2993ds/6Mi)
% 6.93/2.20 % (32114)Instruction limit reached!
% 6.93/2.20 % (32114)------------------------------
% 6.93/2.20 % (32114)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.93/2.20 % (32114)Termination reason: Unknown
% 6.93/2.20 % (32114)Termination phase: shuffling
% 6.93/2.20
% 6.93/2.20 % (32114)Memory used [KB]: 1535
% 6.93/2.20 % (32114)Time elapsed: 0.005 s
% 6.93/2.20 % (32114)Instructions burned: 7 (million)
% 6.93/2.20 % (32114)------------------------------
% 6.93/2.20 % (32114)------------------------------
% 6.93/2.20 % (32115)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_31945 for (2993ds/150Mi)
% 6.93/2.21 % (32113)Instruction limit reached!
% 6.93/2.21 % (32113)------------------------------
% 6.93/2.21 % (32113)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.93/2.21 % (32113)Termination reason: Unknown
% 6.93/2.21 % (32113)Termination phase: shuffling
% 6.93/2.21
% 6.93/2.21 % (32113)Memory used [KB]: 2174
% 6.93/2.21 % (32113)Time elapsed: 0.020 s
% 6.93/2.21 % (32113)Instructions burned: 42 (million)
% 6.93/2.21 % (32113)------------------------------
% 6.93/2.21 % (32113)------------------------------
% 6.93/2.22 % (32116)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_31945 for (2993ds/343Mi)
% 6.93/2.23 % (32117)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_31945 for (2993ds/109Mi)
% 6.93/2.25 % (32115)Instruction limit reached!
% 6.93/2.25 % (32115)------------------------------
% 6.93/2.25 % (32115)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.93/2.25 % (32115)Termination reason: Unknown
% 6.93/2.25 % (32115)Termination phase: Property scanning
% 6.93/2.25
% 6.93/2.25 % (32115)Memory used [KB]: 2814
% 6.93/2.25 % (32115)Time elapsed: 0.046 s
% 6.93/2.25 % (32115)Instructions burned: 151 (million)
% 6.93/2.25 % (32115)------------------------------
% 6.93/2.25 % (32115)------------------------------
% 6.93/2.26 % (32098)Instruction limit reached!
% 6.93/2.26 % (32098)------------------------------
% 6.93/2.26 % (32098)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.93/2.26 % (32098)Termination reason: Unknown
% 6.93/2.26 % (32098)Termination phase: Saturation
% 6.93/2.26
% 6.93/2.26 % (32098)Memory used [KB]: 8315
% 6.93/2.26 % (32098)Time elapsed: 0.241 s
% 6.93/2.26 % (32098)Instructions burned: 509 (million)
% 6.93/2.26 % (32098)------------------------------
% 6.93/2.26 % (32098)------------------------------
% 6.93/2.27 % (32118)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_31945 for (2992ds/253Mi)
% 6.93/2.28 % (32119)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_31945 for (2992ds/140Mi)
% 6.93/2.28 % (32117)Instruction limit reached!
% 6.93/2.28 % (32117)------------------------------
% 6.93/2.28 % (32117)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.93/2.28 % (32117)Termination reason: Unknown
% 6.93/2.28 % (32117)Termination phase: Preprocessing 3
% 6.93/2.28
% 6.93/2.28 % (32117)Memory used [KB]: 2558
% 6.93/2.28 % (32117)Time elapsed: 0.054 s
% 6.93/2.28 % (32117)Instructions burned: 111 (million)
% 6.93/2.28 % (32117)------------------------------
% 6.93/2.28 % (32117)------------------------------
% 7.38/2.30 % (32120)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_31945 for (2992ds/614Mi)
% 7.38/2.31 % (32105)Instruction limit reached!
% 7.38/2.31 % (32105)------------------------------
% 7.38/2.31 % (32105)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.31 % (32105)Termination reason: Unknown
% 7.38/2.31 % (32105)Termination phase: Saturation
% 7.38/2.31
% 7.38/2.31 % (32105)Memory used [KB]: 4861
% 7.38/2.31 % (32105)Time elapsed: 0.206 s
% 7.38/2.31 % (32105)Instructions burned: 433 (million)
% 7.38/2.31 % (32105)------------------------------
% 7.38/2.31 % (32105)------------------------------
% 7.38/2.31 % (32109)Instruction limit reached!
% 7.38/2.31 % (32109)------------------------------
% 7.38/2.31 % (32109)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.31 % (32109)Termination reason: Unknown
% 7.38/2.31 % (32109)Termination phase: Saturation
% 7.38/2.31
% 7.38/2.31 % (32109)Memory used [KB]: 8571
% 7.38/2.31 % (32109)Time elapsed: 0.162 s
% 7.38/2.31 % (32109)Instructions burned: 455 (million)
% 7.38/2.31 % (32109)------------------------------
% 7.38/2.31 % (32109)------------------------------
% 7.38/2.32 % (32122)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_31945 for (2992ds/339Mi)
% 7.38/2.32 % (32121)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_31945 for (2992ds/202Mi)
% 7.38/2.35 % (32119)Instruction limit reached!
% 7.38/2.35 % (32119)------------------------------
% 7.38/2.35 % (32119)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.35 % (32119)Termination reason: Unknown
% 7.38/2.35 % (32119)Termination phase: Saturation
% 7.38/2.35
% 7.38/2.35 % (32119)Memory used [KB]: 7291
% 7.38/2.35 % (32119)Time elapsed: 0.068 s
% 7.38/2.35 % (32119)Instructions burned: 140 (million)
% 7.38/2.35 % (32119)------------------------------
% 7.38/2.35 % (32119)------------------------------
% 7.38/2.36 % (32123)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_31945 for (2991ds/283Mi)
% 7.38/2.38 % (32116)Instruction limit reached!
% 7.38/2.38 % (32116)------------------------------
% 7.38/2.38 % (32116)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.38 % (32116)Termination reason: Unknown
% 7.38/2.38 % (32116)Termination phase: Saturation
% 7.38/2.38
% 7.38/2.38 % (32116)Memory used [KB]: 7931
% 7.38/2.38 % (32116)Time elapsed: 0.166 s
% 7.38/2.38 % (32116)Instructions burned: 343 (million)
% 7.38/2.38 % (32116)------------------------------
% 7.38/2.38 % (32116)------------------------------
% 7.38/2.38 % (32118)Instruction limit reached!
% 7.38/2.38 % (32118)------------------------------
% 7.38/2.38 % (32118)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.38 % (32118)Termination reason: Unknown
% 7.38/2.38 % (32118)Termination phase: Saturation
% 7.38/2.38
% 7.38/2.38 % (32118)Memory used [KB]: 7931
% 7.38/2.38 % (32118)Time elapsed: 0.121 s
% 7.38/2.38 % (32118)Instructions burned: 253 (million)
% 7.38/2.38 % (32118)------------------------------
% 7.38/2.38 % (32118)------------------------------
% 7.38/2.39 % (32124)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_31945 for (2991ds/134Mi)
% 8.17/2.40 % (32125)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_31945 for (2991ds/298Mi)
% 8.17/2.42 % (32121)Instruction limit reached!
% 8.17/2.42 % (32121)------------------------------
% 8.17/2.42 % (32121)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.17/2.42 % (32121)Termination reason: Unknown
% 8.17/2.42 % (32121)Termination phase: Saturation
% 8.17/2.42
% 8.17/2.42 % (32121)Memory used [KB]: 7803
% 8.17/2.42 % (32121)Time elapsed: 0.098 s
% 8.17/2.42 % (32121)Instructions burned: 203 (million)
% 8.17/2.42 % (32121)------------------------------
% 8.17/2.42 % (32121)------------------------------
% 8.17/2.43 % (32124)Instruction limit reached!
% 8.17/2.43 % (32124)------------------------------
% 8.17/2.43 % (32124)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.17/2.43 % (32124)Termination reason: Unknown
% 8.17/2.43 % (32124)Termination phase: Property scanning
% 8.17/2.43
% 8.17/2.43 % (32124)Memory used [KB]: 2686
% 8.17/2.43 % (32124)Time elapsed: 0.042 s
% 8.17/2.43 % (32124)Instructions burned: 136 (million)
% 8.17/2.43 % (32124)------------------------------
% 8.17/2.43 % (32124)------------------------------
% 8.17/2.44 % (32126)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_31945 for (2991ds/192Mi)
% 8.17/2.44 % (32127)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_31945 for (2991ds/258Mi)
% 8.17/2.45 % (32122)Instruction limit reached!
% 8.17/2.45 % (32122)------------------------------
% 8.17/2.45 % (32122)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.17/2.46 % (32122)Termination reason: Unknown
% 8.17/2.46 % (32122)Termination phase: Saturation
% 8.17/2.46
% 8.17/2.46 % (32122)Memory used [KB]: 7803
% 8.17/2.46 % (32122)Time elapsed: 0.136 s
% 8.17/2.46 % (32122)Instructions burned: 339 (million)
% 8.17/2.46 % (32122)------------------------------
% 8.17/2.46 % (32122)------------------------------
% 8.17/2.48 % (32128)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_31945 for (2990ds/90Mi)
% 8.17/2.49 % (32123)Instruction limit reached!
% 8.17/2.49 % (32123)------------------------------
% 8.17/2.49 % (32123)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.17/2.49 % (32123)Termination reason: Unknown
% 8.17/2.49 % (32123)Termination phase: Saturation
% 8.17/2.49
% 8.17/2.49 % (32123)Memory used [KB]: 7803
% 8.17/2.49 % (32123)Time elapsed: 0.133 s
% 8.17/2.49 % (32123)Instructions burned: 283 (million)
% 8.17/2.49 % (32123)------------------------------
% 8.17/2.49 % (32123)------------------------------
% 8.62/2.51 % (32128)Instruction limit reached!
% 8.62/2.51 % (32128)------------------------------
% 8.62/2.51 % (32128)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.62/2.51 % (32128)Termination reason: Unknown
% 8.62/2.51 % (32128)Termination phase: Preprocessing 2
% 8.62/2.51
% 8.62/2.51 % (32128)Memory used [KB]: 2430
% 8.62/2.51 % (32128)Time elapsed: 0.030 s
% 8.62/2.51 % (32128)Instructions burned: 91 (million)
% 8.62/2.51 % (32128)------------------------------
% 8.62/2.51 % (32128)------------------------------
% 8.62/2.51 % (32129)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_31945 for (2990ds/231Mi)
% 8.62/2.52 % (32130)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_31945 for (2990ds/458Mi)
% 8.62/2.53 % (32126)Instruction limit reached!
% 8.62/2.53 % (32126)------------------------------
% 8.62/2.53 % (32126)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.62/2.53 % (32126)Termination reason: Unknown
% 8.62/2.53 % (32126)Termination phase: Saturation
% 8.62/2.53
% 8.62/2.53 % (32126)Memory used [KB]: 3198
% 8.62/2.53 % (32126)Time elapsed: 0.091 s
% 8.62/2.53 % (32126)Instructions burned: 192 (million)
% 8.62/2.53 % (32126)------------------------------
% 8.62/2.53 % (32126)------------------------------
% 8.62/2.54 % (32125)Instruction limit reached!
% 8.62/2.54 % (32125)------------------------------
% 8.62/2.54 % (32125)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.62/2.54 % (32125)Termination reason: Unknown
% 8.62/2.54 % (32125)Termination phase: Saturation
% 8.62/2.54
% 8.62/2.54 % (32125)Memory used [KB]: 7803
% 8.62/2.54 % (32125)Time elapsed: 0.135 s
% 8.62/2.54 % (32125)Instructions burned: 299 (million)
% 8.62/2.54 % (32125)------------------------------
% 8.62/2.54 % (32125)------------------------------
% 8.62/2.55 % (32131)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_31945 for (2990ds/31Mi)
% 8.62/2.55 % (32127)Instruction limit reached!
% 8.62/2.55 % (32127)------------------------------
% 8.62/2.55 % (32127)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.62/2.55 % (32127)Termination reason: Unknown
% 8.62/2.55 % (32127)Termination phase: Saturation
% 8.62/2.55
% 8.62/2.55 % (32127)Memory used [KB]: 8059
% 8.62/2.55 % (32127)Time elapsed: 0.110 s
% 8.62/2.55 % (32127)Instructions burned: 258 (million)
% 8.62/2.55 % (32127)------------------------------
% 8.62/2.55 % (32127)------------------------------
% 8.62/2.55 % (32132)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_31945 for (2990ds/297Mi)
% 8.62/2.57 % (32133)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_31945 for (2989ds/987Mi)
% 8.62/2.57 % (32131)Instruction limit reached!
% 8.62/2.57 % (32131)------------------------------
% 8.62/2.57 % (32131)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.62/2.57 % (32131)Termination reason: Unknown
% 8.62/2.57 % (32131)Termination phase: shuffling
% 8.62/2.57
% 8.62/2.57 % (32131)Memory used [KB]: 1918
% 8.62/2.57 % (32131)Time elapsed: 0.013 s
% 8.62/2.57 % (32131)Instructions burned: 32 (million)
% 8.62/2.57 % (32131)------------------------------
% 8.62/2.57 % (32131)------------------------------
% 8.62/2.58 % (32134)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_31945 for (2989ds/182Mi)
% 8.62/2.59 % (32102)Instruction limit reached!
% 8.62/2.59 % (32102)------------------------------
% 8.62/2.59 % (32102)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.62/2.59 % (32102)Termination reason: Unknown
% 8.62/2.59 % (32102)Termination phase: Saturation
% 8.62/2.59
% 8.62/2.59 % (32102)Memory used [KB]: 8443
% 8.62/2.59 % (32102)Time elapsed: 0.529 s
% 8.62/2.59 % (32102)Instructions burned: 1088 (million)
% 8.62/2.59 % (32102)------------------------------
% 8.62/2.59 % (32102)------------------------------
% 8.62/2.59 % (32120)Instruction limit reached!
% 8.62/2.59 % (32120)------------------------------
% 8.62/2.59 % (32120)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.62/2.59 % (32120)Termination reason: Unknown
% 8.62/2.59 % (32120)Termination phase: Saturation
% 8.62/2.59
% 8.62/2.59 % (32120)Memory used [KB]: 9083
% 8.62/2.59 % (32120)Time elapsed: 0.293 s
% 8.62/2.59 % (32120)Instructions burned: 616 (million)
% 8.62/2.59 % (32120)------------------------------
% 8.62/2.59 % (32120)------------------------------
% 9.88/2.60 % (32135)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_31945 for (2989ds/182Mi)
% 9.88/2.61 % (32136)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_31945 for (2989ds/154Mi)
% 9.88/2.62 % (32129)Instruction limit reached!
% 9.88/2.62 % (32129)------------------------------
% 9.88/2.62 % (32129)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.88/2.62 % (32129)Termination reason: Unknown
% 9.88/2.62 % (32129)Termination phase: Twee Goal Transformation
% 9.88/2.62
% 9.88/2.62 % (32129)Memory used [KB]: 2942
% 9.88/2.62 % (32129)Time elapsed: 0.104 s
% 9.88/2.62 % (32129)Instructions burned: 231 (million)
% 9.88/2.62 % (32129)------------------------------
% 9.88/2.62 % (32129)------------------------------
% 9.88/2.64 % (32137)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_31945 for (2989ds/79Mi)
% 9.88/2.64 % (32134)Instruction limit reached!
% 9.88/2.64 % (32134)------------------------------
% 9.88/2.64 % (32134)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.88/2.64 % (32134)Termination reason: Unknown
% 9.88/2.64 % (32134)Termination phase: Saturation
% 9.88/2.64
% 9.88/2.64 % (32134)Memory used [KB]: 7419
% 9.88/2.64 % (32134)Time elapsed: 0.058 s
% 9.88/2.64 % (32134)Instructions burned: 182 (million)
% 9.88/2.64 % (32134)------------------------------
% 9.88/2.64 % (32134)------------------------------
% 9.88/2.66 % (32139)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_31945 for (2988ds/216Mi)
% 9.88/2.66 % (32137)Instruction limit reached!
% 9.88/2.66 % (32137)------------------------------
% 9.88/2.66 % (32137)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.88/2.66 % (32137)Termination reason: Unknown
% 9.88/2.66 % (32137)Termination phase: Property scanning
% 9.88/2.66
% 9.88/2.66 % (32137)Memory used [KB]: 2302
% 9.88/2.66 % (32137)Time elapsed: 0.027 s
% 9.88/2.66 % (32137)Instructions burned: 82 (million)
% 9.88/2.66 % (32137)------------------------------
% 9.88/2.66 % (32137)------------------------------
% 9.88/2.67 % (32140)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_31945 for (2988ds/114Mi)
% 9.88/2.68 % (32136)Instruction limit reached!
% 9.88/2.68 % (32136)------------------------------
% 9.88/2.68 % (32136)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.88/2.68 % (32136)Termination reason: Unknown
% 9.88/2.68 % (32136)Termination phase: Property scanning
% 9.88/2.68
% 9.88/2.68 % (32136)Memory used [KB]: 2942
% 9.88/2.68 % (32136)Time elapsed: 0.071 s
% 9.88/2.68 % (32136)Instructions burned: 155 (million)
% 9.88/2.68 % (32136)------------------------------
% 9.88/2.68 % (32136)------------------------------
% 9.88/2.69 % (32135)Instruction limit reached!
% 9.88/2.69 % (32135)------------------------------
% 9.88/2.69 % (32135)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.88/2.69 % (32135)Termination reason: Unknown
% 9.88/2.69 % (32135)Termination phase: Property scanning
% 9.88/2.69
% 9.88/2.69 % (32135)Memory used [KB]: 2814
% 9.88/2.69 % (32135)Time elapsed: 0.085 s
% 9.88/2.69 % (32135)Instructions burned: 182 (million)
% 9.88/2.69 % (32135)------------------------------
% 9.88/2.69 % (32135)------------------------------
% 9.88/2.69 % (32132)Instruction limit reached!
% 9.88/2.69 % (32132)------------------------------
% 9.88/2.69 % (32132)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.88/2.69 % (32132)Termination reason: Unknown
% 9.88/2.69 % (32132)Termination phase: Saturation
% 9.88/2.69
% 9.88/2.69 % (32132)Memory used [KB]: 7547
% 9.88/2.69 % (32132)Time elapsed: 0.135 s
% 9.88/2.69 % (32132)Instructions burned: 299 (million)
% 9.88/2.69 % (32132)------------------------------
% 9.88/2.69 % (32132)------------------------------
% 9.88/2.69 % (32141)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_31945 for (2988ds/922Mi)
% 9.88/2.70 % (32143)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_31945 for (2988ds/164Mi)
% 10.44/2.70 % (32142)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_31945 for (2988ds/182Mi)
% 10.44/2.72 % (32140)Instruction limit reached!
% 10.44/2.72 % (32140)------------------------------
% 10.44/2.72 % (32140)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.44/2.72 % (32140)Termination reason: Unknown
% 10.44/2.72 % (32140)Termination phase: Property scanning
% 10.44/2.72
% 10.44/2.72 % (32140)Memory used [KB]: 2686
% 10.44/2.72 % (32140)Time elapsed: 0.050 s
% 10.44/2.72 % (32140)Instructions burned: 116 (million)
% 10.44/2.72 % (32140)------------------------------
% 10.44/2.72 % (32140)------------------------------
% 10.44/2.73 % (32130)Instruction limit reached!
% 10.44/2.73 % (32130)------------------------------
% 10.44/2.73 % (32130)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.44/2.73 % (32130)Termination reason: Unknown
% 10.44/2.73 % (32130)Termination phase: Saturation
% 10.44/2.73
% 10.44/2.73 % (32130)Memory used [KB]: 8315
% 10.44/2.73 % (32130)Time elapsed: 0.201 s
% 10.44/2.73 % (32130)Instructions burned: 459 (million)
% 10.44/2.73 % (32130)------------------------------
% 10.44/2.73 % (32130)------------------------------
% 10.44/2.75 % (32144)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_31945 for (2988ds/50Mi)
% 10.44/2.75 % (32145)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_31945 for (2988ds/368Mi)
% 10.44/2.76 % (32139)Instruction limit reached!
% 10.44/2.76 % (32139)------------------------------
% 10.44/2.76 % (32139)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.44/2.76 % (32139)Termination reason: Unknown
% 10.44/2.76 % (32139)Termination phase: Saturation
% 10.44/2.76
% 10.44/2.76 % (32139)Memory used [KB]: 7164
% 10.44/2.76 % (32139)Time elapsed: 0.099 s
% 10.44/2.76 % (32139)Instructions burned: 217 (million)
% 10.44/2.76 % (32139)------------------------------
% 10.44/2.76 % (32139)------------------------------
% 10.44/2.76 % (32143)Instruction limit reached!
% 10.44/2.76 % (32143)------------------------------
% 10.44/2.76 % (32143)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.44/2.76 % (32143)Termination reason: Unknown
% 10.44/2.76 % (32143)Termination phase: Property scanning
% 10.44/2.76
% 10.44/2.76 % (32143)Memory used [KB]: 2686
% 10.44/2.76 % (32143)Time elapsed: 0.062 s
% 10.44/2.76 % (32143)Instructions burned: 164 (million)
% 10.44/2.76 % (32143)------------------------------
% 10.44/2.76 % (32143)------------------------------
% 10.44/2.77 % (32146)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_31945 for (2987ds/1021Mi)
% 10.44/2.77 % (32147)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_31945 for (2987ds/267Mi)
% 10.44/2.77 % (32144)Instruction limit reached!
% 10.44/2.77 % (32144)------------------------------
% 10.44/2.77 % (32144)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.44/2.77 % (32144)Termination reason: Unknown
% 10.44/2.77 % (32144)Termination phase: shuffling
% 10.44/2.77
% 10.44/2.77 % (32144)Memory used [KB]: 2174
% 10.44/2.77 % (32144)Time elapsed: 0.022 s
% 10.44/2.77 % (32144)Instructions burned: 50 (million)
% 10.44/2.77 % (32144)------------------------------
% 10.44/2.77 % (32144)------------------------------
% 10.44/2.78 % (32142)Instruction limit reached!
% 10.44/2.78 % (32142)------------------------------
% 10.44/2.78 % (32142)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.44/2.78 % (32142)Termination reason: Unknown
% 10.44/2.78 % (32142)Termination phase: Property scanning
% 10.44/2.78
% 10.44/2.78 % (32142)Memory used [KB]: 2814
% 10.44/2.78 % (32142)Time elapsed: 0.082 s
% 10.44/2.78 % (32142)Instructions burned: 182 (million)
% 10.44/2.78 % (32142)------------------------------
% 10.44/2.78 % (32142)------------------------------
% 10.44/2.79 % (32148)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_31945 for (2987ds/204Mi)
% 11.02/2.80 % (32149)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_31945 for (2987ds/26Mi)
% 11.02/2.81 % (32149)Instruction limit reached!
% 11.02/2.81 % (32149)------------------------------
% 11.02/2.81 % (32149)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.02/2.81 % (32149)Termination reason: Unknown
% 11.02/2.81 % (32149)Termination phase: shuffling
% 11.02/2.81
% 11.02/2.81 % (32149)Memory used [KB]: 1918
% 11.02/2.81 % (32149)Time elapsed: 0.014 s
% 11.02/2.81 % (32149)Instructions burned: 28 (million)
% 11.02/2.81 % (32149)------------------------------
% 11.02/2.81 % (32149)------------------------------
% 11.02/2.83 % (32150)dis+1002_1:1_bd=preordered:c=on:cbe=off:cnfonf=lazy_not_gen_be_off:hud=10:ixr=off:prag=on:i=53:si=on:rtra=on_0 on DTF2THF_31945 for (2987ds/53Mi)
% 11.02/2.85 % (32150)Instruction limit reached!
% 11.02/2.85 % (32150)------------------------------
% 11.02/2.85 % (32150)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.02/2.85 % (32150)Termination reason: Unknown
% 11.02/2.85 % (32150)Termination phase: shuffling
% 11.02/2.85
% 11.02/2.85 % (32150)Memory used [KB]: 2302
% 11.02/2.85 % (32150)Time elapsed: 0.025 s
% 11.02/2.85 % (32150)Instructions burned: 53 (million)
% 11.02/2.85 % (32150)------------------------------
% 11.02/2.85 % (32150)------------------------------
% 11.02/2.87 % (32151)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=4436:si=on:rtra=on_0 on DTF2THF_31945 for (2986ds/4436Mi)
% 11.02/2.88 % (32148)Instruction limit reached!
% 11.02/2.88 % (32148)------------------------------
% 11.02/2.88 % (32148)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.02/2.88 % (32148)Termination reason: Unknown
% 11.02/2.88 % (32148)Termination phase: Saturation
% 11.02/2.88
% 11.02/2.88 % (32148)Memory used [KB]: 7419
% 11.02/2.88 % (32148)Time elapsed: 0.096 s
% 11.02/2.88 % (32148)Instructions burned: 205 (million)
% 11.02/2.88 % (32148)------------------------------
% 11.02/2.88 % (32148)------------------------------
% 11.19/2.90 % (32147)Instruction limit reached!
% 11.19/2.90 % (32147)------------------------------
% 11.19/2.90 % (32147)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.19/2.90 % (32147)Termination reason: Unknown
% 11.19/2.90 % (32147)Termination phase: Saturation
% 11.19/2.90
% 11.19/2.90 % (32147)Memory used [KB]: 7803
% 11.19/2.90 % (32147)Time elapsed: 0.125 s
% 11.19/2.90 % (32147)Instructions burned: 268 (million)
% 11.19/2.90 % (32147)------------------------------
% 11.19/2.90 % (32147)------------------------------
% 11.19/2.90 % (32152)dis+1010_1:1_acc=model:avsq=on:ixr=off:prag=on:s2a=on:sac=on:i=1539:si=on:rtra=on_0 on DTF2THF_31945 for (2986ds/1539Mi)
% 11.19/2.91 % (32153)lrs+2_1:1_bd=off:e2e=on:fde=none:kws=inv_precedence:plsq=on:plsqr=32,1:sos=on:i=408:si=on:rtra=on_0 on DTF2THF_31945 for (2986ds/408Mi)
% 11.19/2.91 % (32145)Instruction limit reached!
% 11.19/2.91 % (32145)------------------------------
% 11.19/2.91 % (32145)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.19/2.92 % (32145)Termination reason: Unknown
% 11.19/2.92 % (32145)Termination phase: Saturation
% 11.19/2.92
% 11.19/2.92 % (32145)Memory used [KB]: 8699
% 11.19/2.92 % (32145)Time elapsed: 0.168 s
% 11.19/2.92 % (32145)Instructions burned: 368 (million)
% 11.19/2.92 % (32145)------------------------------
% 11.19/2.92 % (32145)------------------------------
% 11.19/2.92 % (32154)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=5205:si=on:rtra=on_0 on DTF2THF_31945 for (2986ds/5205Mi)
% 11.19/2.98 % (32133)Instruction limit reached!
% 11.19/2.98 % (32133)------------------------------
% 11.19/2.98 % (32133)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.19/2.98 % (32133)Termination reason: Unknown
% 11.19/2.98 % (32133)Termination phase: Saturation
% 11.19/2.98
% 11.19/2.98 % (32133)Memory used [KB]: 10618
% 11.19/2.98 % (32133)Time elapsed: 0.427 s
% 11.19/2.98 % (32133)Instructions burned: 987 (million)
% 11.19/2.98 % (32133)------------------------------
% 11.19/2.98 % (32133)------------------------------
% 11.19/3.00 % (32155)dis+1010_1:4_au=on:av=off:cbe=off:nwc=10.0:s2a=on:s2at=3.0:sd=2:ss=axioms:st=5.0:i=1913:si=on:rtra=on_0 on DTF2THF_31945 for (2985ds/1913Mi)
% 12.99/3.04 % (32152)First to succeed.
% 12.99/3.04 % (32152)Refutation found. Thanks to Tanya!
% 12.99/3.04 % SZS status Theorem for DTF2THF_31945
% 12.99/3.04 % SZS output start Proof for DTF2THF_31945
% 12.99/3.04 thf(type_def_5, type, carrier_ext: ($tType * $tType) > $tType).
% 12.99/3.04 thf(type_def_6, type, set: ($tType) > $tType).
% 12.99/3.04 thf(type_def_8, type, ring_ext: ($tType * $tType) > $tType).
% 12.99/3.04 thf(type_def_9, type, aGroup_ext: ($tType * $tType) > $tType).
% 12.99/3.04 thf(type_def_10, type, nat: $tType).
% 12.99/3.04 thf(type_def_11, type, module_ext: ($tType * $tType * $tType) > $tType).
% 12.99/3.04 thf(type_def_12, type, product_unit: $tType).
% 12.99/3.04 thf(type_def_13, type, c: $tType).
% 12.99/3.04 thf(type_def_14, type, d: $tType).
% 12.99/3.04 thf(type_def_15, type, a: $tType).
% 12.99/3.04 thf(type_def_16, type, e: $tType).
% 12.99/3.04 thf(type_def_17, type, b: $tType).
% 12.99/3.04 thf(func_def_0, type, carrier_ext: ($tType * $tType) > $tType).
% 12.99/3.04 thf(func_def_1, type, module_ext: ($tType * $tType * $tType) > $tType).
% 12.99/3.04 thf(func_def_2, type, aGroup_ext: ($tType * $tType) > $tType).
% 12.99/3.04 thf(func_def_3, type, ring_ext: ($tType * $tType) > $tType).
% 12.99/3.04 thf(func_def_4, type, product_unit: $tType).
% 12.99/3.04 thf(func_def_5, type, set: ($tType) > $tType).
% 12.99/3.04 thf(func_def_6, type, nat: $tType).
% 12.99/3.04 thf(func_def_7, type, e: $tType).
% 12.99/3.04 thf(func_def_8, type, d: $tType).
% 12.99/3.04 thf(func_def_9, type, c: $tType).
% 12.99/3.04 thf(func_def_10, type, b: $tType).
% 12.99/3.04 thf(func_def_11, type, a: $tType).
% 12.99/3.04 thf(func_def_12, type, type: !>[X0: $tType]:($o)).
% 12.99/3.04 thf(func_def_13, type, plus: !>[X0: $tType]:($o)).
% 12.99/3.04 thf(func_def_14, type, ord: !>[X0: $tType]:($o)).
% 12.99/3.04 thf(func_def_15, type, group_add: !>[X0: $tType]:($o)).
% 12.99/3.04 thf(func_def_16, type, semigroup_add: !>[X0: $tType]:($o)).
% 12.99/3.04 thf(func_def_17, type, comm_monoid_add: !>[X0: $tType]:($o)).
% 12.99/3.04 thf(func_def_18, type, ab_semigroup_add: !>[X0: $tType]:($o)).
% 12.99/3.04 thf(func_def_19, type, cancel_semigroup_add: !>[X0: $tType]:($o)).
% 12.99/3.04 thf(func_def_20, type, ordere779506340up_add: !>[X0: $tType]:($o)).
% 12.99/3.04 thf(func_def_21, type, ordere236663937imp_le: !>[X0: $tType]:($o)).
% 12.99/3.04 thf(func_def_22, type, carrier: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ X1 > set @ X0)).
% 12.99/3.04 thf(func_def_23, type, j_rad: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X0)).
% 12.99/3.04 thf(func_def_24, type, ring: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > $o)).
% 12.99/3.04 thf(func_def_25, type, tp: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > X0 > X0 > X0)).
% 12.99/3.04 thf(func_def_26, type, un: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > X0)).
% 12.99/3.04 thf(func_def_27, type, sr: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X0 > carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1)).
% 12.99/3.04 thf(func_def_28, type, subring: !>[X0: $tType, X1: $tType, X2: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X2 > $o)).
% 12.99/3.04 thf(func_def_29, type, unit: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > X0 > $o)).
% 12.99/3.04 thf(func_def_30, type, aGroup: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ X1 > $o)).
% 12.99/3.04 thf(func_def_31, type, mop: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ X1 > X0 > X0)).
% 12.99/3.04 thf(func_def_32, type, pop: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ X1 > X0 > X0 > X0)).
% 12.99/3.04 thf(func_def_33, type, aset_sum: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ X1 > set @ X0 > set @ X0 > set @ X0)).
% 12.99/3.04 thf(func_def_34, type, carr_dsumag: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ X0 > (X0 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X2) > set @ (X0 > X1))).
% 12.99/3.04 thf(func_def_35, type, carr_prodag: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ X0 > (X0 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X2) > set @ (X0 > X1))).
% 12.99/3.04 thf(func_def_36, type, local_ring: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > $o)).
% 12.99/3.04 thf(func_def_37, type, maximal_ideal: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X0 > $o)).
% 12.99/3.04 thf(func_def_38, type, mul_closed_set: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X0 > $o)).
% 12.99/3.04 thf(func_def_39, type, nscal: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > X0 > nat > X0)).
% 12.99/3.04 thf(func_def_40, type, primary_ideal: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X0 > $o)).
% 12.99/3.04 thf(func_def_41, type, prod_mOp: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ X0 > (X0 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X2) > (X0 > X1) > X0 > X1)).
% 12.99/3.04 thf(func_def_42, type, prod_pOp: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ X0 > (X0 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X2) > (X0 > X1) > (X0 > X1) > X0 > X1)).
% 12.99/3.04 thf(func_def_43, type, prod_zero: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ X0 > (X0 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X2) > X0 > X1)).
% 12.99/3.04 thf(func_def_44, type, rInvim: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > carrier_ext @ X2 @ aGroup_ext @ X2 @ ring_ext @ X2 @ X3 > (X0 > X2) > set @ X2 > set @ X0)).
% 12.99/3.04 thf(func_def_45, type, sr2: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X0 > $o)).
% 12.99/3.04 thf(func_def_46, type, zeroring: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > $o)).
% 12.99/3.04 thf(func_def_47, type, sum_mult: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X0 > set @ X0 > set @ X0)).
% 12.99/3.04 thf(func_def_48, type, module: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ module_ext @ X0 @ X1 @ X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X3 > $o)).
% 12.99/3.04 thf(func_def_49, type, sprod: !>[X0: $tType, X1: $tType, X2: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ module_ext @ X0 @ X1 @ X2 > X1 > X0 > X0)).
% 12.99/3.04 thf(func_def_50, type, mHom: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > carrier_ext @ X2 @ aGroup_ext @ X2 @ module_ext @ X2 @ X0 @ X3 > carrier_ext @ X4 @ aGroup_ext @ X4 @ module_ext @ X4 @ X0 @ X5 > set @ (X2 > X4))).
% 12.99/3.04 thf(func_def_51, type, algebr1000837334_prodM: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X2 > (X2 > carrier_ext @ X3 @ aGroup_ext @ X3 @ module_ext @ X3 @ X0 @ X4) > carrier_ext @ (X2 > X3) @ aGroup_ext @ (X2 > X3) @ module_ext @ (X2 > X3) @ X0 @ product_unit)).
% 12.99/3.04 thf(func_def_52, type, algebr142713364_sprod: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X2 > (X2 > carrier_ext @ X3 @ aGroup_ext @ X3 @ module_ext @ X3 @ X0 @ X4) > X0 > (X2 > X3) > X2 > X3)).
% 12.99/3.04 thf(func_def_53, type, pi: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > set @ X1) > set @ (X0 > X1))).
% 12.99/3.04 thf(func_def_54, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 12.99/3.04 thf(func_def_55, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 12.99/3.04 thf(func_def_56, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 12.99/3.04 thf(func_def_57, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 12.99/3.04 thf(func_def_58, type, i: set @ c).
% 12.99/3.04 thf(func_def_59, type, m: c > carrier_ext @ d @ aGroup_ext @ d @ module_ext @ d @ a @ e).
% 12.99/3.04 thf(func_def_60, type, r: carrier_ext @ a @ aGroup_ext @ a @ ring_ext @ a @ b).
% 12.99/3.04 thf(func_def_75, type, sK0: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((X4 > carrier_ext @ X3 @ aGroup_ext @ X3 @ module_ext @ X3 @ X1 @ X2) > set @ X4 > carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X0 > X4)).
% 12.99/3.04 thf(func_def_76, type, sK1: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 12.99/3.04 thf(func_def_77, type, sK2: !>[X0: $tType, X1: $tType]:((X1 > set @ X0) > (X1 > X0) > set @ X1 > X1)).
% 12.99/3.04 thf(func_def_78, type, sK3: !>[X0: $tType, X1: $tType]:(carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X0 > set @ X1)).
% 12.99/3.04 thf(func_def_79, type, sK4: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_80, type, sK5: set @ a).
% 12.99/3.04 thf(func_def_81, type, sK6: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 12.99/3.04 thf(func_def_82, type, sK7: set @ a > a).
% 12.99/3.04 thf(func_def_83, type, sK8: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_84, type, sK9: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > (X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_85, type, sK10: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_86, type, sK11: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > (X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_87, type, sK12: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_88, type, sK13: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_89, type, sK14: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > (X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_90, type, sK15: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_91, type, sK16: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_92, type, sK17: nat > nat > nat).
% 12.99/3.04 thf(func_def_93, type, sK18: c > d).
% 12.99/3.04 thf(func_def_94, type, sK19: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_95, type, sK20: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_96, type, sK21: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_97, type, sK22: !>[X0: $tType, X1: $tType]:(set @ X0 > (X1 > X0) > set @ X1 > X1)).
% 12.99/3.04 thf(func_def_98, type, sK23: !>[X0: $tType, X1: $tType]:(X1 > carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X0 > set @ X1)).
% 12.99/3.04 thf(func_def_99, type, sK24: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_100, type, sK25: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > (X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_101, type, sK26: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_102, type, sK27: a > set @ a).
% 12.99/3.04 thf(func_def_103, type, sK28: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ module_ext @ X1 @ a @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_104, type, sK29: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ module_ext @ X1 @ a @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_105, type, sK30: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 12.99/3.04 thf(func_def_106, type, sK31: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((X4 > carrier_ext @ X3 @ aGroup_ext @ X3 @ module_ext @ X3 @ X0 @ X1) > set @ X4 > carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X2 > X4)).
% 12.99/3.04 thf(func_def_107, type, sK32: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_108, type, sK33: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 12.99/3.04 thf(func_def_109, type, sK34: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_110, type, sK35: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 12.99/3.04 thf(func_def_111, type, sK36: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 12.99/3.04 thf(func_def_112, type, sK37: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_113, type, sK38: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_114, type, sK39: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ module_ext @ X1 @ a @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_115, type, sK40: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X3 > X2) > (X3 > X2) > carrier_ext @ X3 @ aGroup_ext @ X3 @ module_ext @ X3 @ X1 @ X0 > X3)).
% 12.99/3.04 thf(func_def_116, type, sK41: !>[X0: $tType, X1: $tType]:((X1 > set @ X0) > (X1 > X0) > set @ X1 > X1)).
% 12.99/3.04 thf(func_def_117, type, sK42: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 12.99/3.04 thf(func_def_118, type, sK43: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((X4 > carrier_ext @ X3 @ aGroup_ext @ X3 @ module_ext @ X3 @ X2 @ X1) > set @ X4 > carrier_ext @ X2 @ aGroup_ext @ X2 @ ring_ext @ X2 @ X0 > X4)).
% 12.99/3.04 thf(func_def_119, type, sK44: !>[X0: $tType, X1: $tType]:(X1 > set @ X1 > set @ X1 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0 > X1)).
% 12.99/3.04 thf(func_def_120, type, sK45: !>[X0: $tType, X1: $tType]:(X1 > set @ X1 > set @ X1 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0 > X1)).
% 12.99/3.04 thf(func_def_121, type, sK46: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ module_ext @ X1 @ a @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_122, type, sK47: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 12.99/3.04 thf(func_def_123, type, sK48: !>[X0: $tType, X1: $tType]:((X1 > set @ X0) > (X1 > set @ X0) > set @ X1 > X1)).
% 12.99/3.04 thf(func_def_124, type, sK49: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_125, type, sK50: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_126, type, sK51: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_127, type, sK52: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 12.99/3.04 thf(func_def_128, type, sK53: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((X4 > carrier_ext @ X3 @ aGroup_ext @ X3 @ module_ext @ X3 @ X2 @ X1) > set @ X4 > carrier_ext @ X2 @ aGroup_ext @ X2 @ ring_ext @ X2 @ X0 > X4)).
% 12.99/3.04 thf(func_def_129, type, sK54: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ X0) > set @ X2 > X2)).
% 12.99/3.04 thf(func_def_131, type, sK56: !>[X0: $tType, X1: $tType]:(carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X0 > set @ X1 > X1)).
% 12.99/3.04 thf(func_def_132, type, sK57: !>[X0: $tType, X1: $tType]:(set @ X1 > carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 12.99/3.04 thf(func_def_133, type, sK58: set @ a > a).
% 12.99/3.04 thf(func_def_134, type, sK59: set @ a > a).
% 12.99/3.04 thf(func_def_135, type, sK60: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > set @ X0) > (X1 > X0) > X1)).
% 12.99/3.04 thf(func_def_136, type, sK61: !>[X0: $tType, X1: $tType]:(X1 > carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 12.99/3.04 thf(func_def_137, type, sK62: a > a > a).
% 12.99/3.04 thf(func_def_138, type, sK63: set @ a > a).
% 12.99/3.04 thf(func_def_139, type, sK64: set @ a > a).
% 12.99/3.04 thf(func_def_140, type, sK65: !>[X0: $tType, X1: $tType]:(set @ X1 > carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 12.99/3.04 thf(func_def_141, type, sK66: !>[X0: $tType, X1: $tType]:(set @ X1 > carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 12.99/3.04 thf(func_def_142, type, sK67: !>[X0: $tType, X1: $tType]:(set @ X1 > carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 12.99/3.04 thf(func_def_143, type, sK68: !>[X0: $tType, X1: $tType]:(set @ X1 > carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 12.99/3.04 thf(func_def_144, type, sK69: !>[X0: $tType, X1: $tType]:(set @ X1 > carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 12.99/3.04 thf(func_def_145, type, sK70: !>[X0: $tType, X1: $tType]:(set @ X1 > carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 12.99/3.04 thf(func_def_146, type, sK71: !>[X0: $tType, X1: $tType]:(set @ X1 > carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 12.99/3.04 thf(func_def_147, type, sK72: !>[X0: $tType, X1: $tType]:(set @ X1 > carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 12.99/3.04 thf(func_def_148, type, sK73: set @ a > a).
% 12.99/3.04 thf(func_def_149, type, sK74: set @ a > a).
% 12.99/3.04 thf(func_def_150, type, sK75: !>[X0: $tType, X1: $tType]:(set @ X1 > carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 12.99/3.04 thf(func_def_151, type, sK76: !>[X0: $tType, X1: $tType]:(set @ X1 > carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 12.99/3.04 thf(func_def_152, type, sK77: set @ a > a).
% 12.99/3.04 thf(func_def_153, type, sK78: set @ a > a).
% 12.99/3.04 thf(func_def_154, type, sK79: a > a).
% 12.99/3.04 thf(func_def_155, type, sK80: a > a).
% 12.99/3.04 thf(func_def_156, type, sK81: set @ a > a).
% 12.99/3.04 thf(func_def_157, type, sK82: set @ a > a).
% 12.99/3.04 thf(func_def_158, type, sK83: set @ a > a).
% 12.99/3.04 thf(func_def_159, type, sK84: set @ a > a).
% 12.99/3.04 thf(func_def_160, type, sK85: !>[X0: $tType, X1: $tType]:(X1 > carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 12.99/3.04 thf(func_def_161, type, sK86: !>[X0: $tType, X1: $tType]:(set @ X1 > carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 12.99/3.04 thf(func_def_162, type, sK87: !>[X0: $tType, X1: $tType]:(set @ X1 > carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 12.99/3.04 thf(func_def_163, type, sK88: set @ a > a).
% 12.99/3.04 thf(func_def_164, type, sK89: set @ a > a).
% 12.99/3.04 thf(func_def_166, type, ph90: !>[X0: $tType]:(X0)).
% 12.99/3.04 thf(func_def_167, type, sK91: c).
% 12.99/3.04 thf(f2185,plain,(
% 12.99/3.04 $false),
% 12.99/3.04 inference(avatar_sat_refutation,[status(thm)],[f2007,f2017,f2027,f2042,f2184])).
% 12.99/3.04 thf(f2184,plain,(
% 12.99/3.04 ~spl55_3 | ~spl55_5 | spl55_7 | ~spl55_9),
% 12.99/3.04 inference(avatar_contradiction_clause,[status(thm)],[f2183])).
% 12.99/3.04 thf(f2183,plain,(
% 12.99/3.04 $false | (~spl55_3 | ~spl55_5 | spl55_7 | ~spl55_9)),
% 12.99/3.04 inference(subsumption_resolution,[status(thm)],[f2182,f2026])).
% 12.99/3.04 thf(f2026,plain,(
% 12.99/3.04 ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ sK18 @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != sK18) | spl55_7),
% 12.99/3.04 inference(avatar_component_clause,[status(thm)],[f2024])).
% 12.99/3.04 thf(f2024,definition,(
% 12.99/3.04 spl55_7 <=> ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ sK18 @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = sK18)),
% 12.99/3.04 introduced(definition,[new_symbols(naming,[spl55_7])],[avatar_definition])).
% 12.99/3.04 thf(f2182,plain,(
% 12.99/3.04 ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ sK18 @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = sK18) | (~spl55_3 | ~spl55_5 | ~spl55_9)),
% 12.99/3.04 inference(forward_demodulation,[status(thm)],[f2181,f2041])).
% 12.99/3.04 thf(f2041,plain,(
% 12.99/3.04 (sK18 = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ sK18)) | ~spl55_9),
% 12.99/3.04 inference(avatar_component_clause,[status(thm)],[f2039])).
% 12.99/3.04 thf(f2039,definition,(
% 12.99/3.04 spl55_9 <=> (sK18 = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ sK18))),
% 12.99/3.04 introduced(definition,[new_symbols(naming,[spl55_9])],[avatar_definition])).
% 12.99/3.04 thf(f2181,plain,(
% 12.99/3.04 ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ sK18 @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ sK18)) | (~spl55_3 | ~spl55_5)),
% 12.99/3.04 inference(trivial_inequality_removal,[status(thm)],[f2179])).
% 12.99/3.04 thf(f2179,plain,(
% 12.99/3.04 ($true != $true) | ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ sK18 @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ sK18)) | (~spl55_3 | ~spl55_5)),
% 12.99/3.04 inference(superposition,[status(thm)],[f2079,f2016])).
% 12.99/3.04 thf(f2016,plain,(
% 12.99/3.04 ((member @ (c > d) @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true) | ~spl55_5),
% 12.99/3.04 inference(avatar_component_clause,[status(thm)],[f2014])).
% 12.99/3.04 thf(f2014,definition,(
% 12.99/3.04 spl55_5 <=> ((member @ (c > d) @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true)),
% 12.99/3.04 introduced(definition,[new_symbols(naming,[spl55_5])],[avatar_definition])).
% 12.99/3.04 thf(f2079,plain,(
% 12.99/3.04 ( ! [X0 : c > d] : (((member @ (c > d) @ X0 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != $true) | ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X0 @ sK18) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ sK18 @ X0))) ) | ~spl55_3),
% 12.99/3.04 inference(trivial_inequality_removal,[status(thm)],[f2077])).
% 12.99/3.04 thf(f2077,plain,(
% 12.99/3.04 ( ! [X0 : c > d] : (($true != $true) | ((member @ (c > d) @ X0 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != $true) | ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X0 @ sK18) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ sK18 @ X0))) ) | ~spl55_3),
% 12.99/3.04 inference(superposition,[status(thm)],[f1628,f2006])).
% 12.99/3.04 thf(f2006,plain,(
% 12.99/3.04 ((member @ (c > d) @ sK18 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true) | ~spl55_3),
% 12.99/3.04 inference(avatar_component_clause,[status(thm)],[f2004])).
% 12.99/3.04 thf(f2004,definition,(
% 12.99/3.04 spl55_3 <=> ((member @ (c > d) @ sK18 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true)),
% 12.99/3.04 introduced(definition,[new_symbols(naming,[spl55_3])],[avatar_definition])).
% 12.99/3.04 thf(f1628,plain,(
% 12.99/3.04 ( ! [X4 : c > d,X5 : c > d] : (((member @ (c > d) @ X5 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != $true) | ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X4 @ X5) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X5 @ X4)) | ((member @ (c > d) @ X4 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != $true)) )),
% 12.99/3.04 inference(cnf_transformation,[status(thm)],[f1335])).
% 12.99/3.04 thf(f1335,plain,(
% 12.99/3.04 ($true = (member @ ((c > d) > (c > d) > c > d) @ (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (pi @ (c > d) @ ((c > d) > c > d) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (^[Y0 : c > d]: (pi @ (c > d) @ (c > d) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (^[Y1 : c > d]: (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m))))))) & ($true = (member @ ((c > d) > c > d) @ (prod_mOp @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (pi @ (c > d) @ (c > d) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (^[Y0 : c > d]: (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m))))) & ! [X0 : c > d] : (((member @ (c > d) @ X0 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != $true) | ! [X1 : c > d] : (((member @ (c > d) @ X1 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != $true) | ! [X2 : c > d] : (((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X0 @ X1) @ X2) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X0 @ (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X1 @ X2))) | ((member @ (c > d) @ X2 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != $true)))) & ! [X3 : c > d] : (((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ X3) = X3) | ((member @ (c > d) @ X3 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != $true)) & ! [X4 : c > d] : (! [X5 : c > d] : (((member @ (c > d) @ X5 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != $true) | ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X4 @ X5) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X5 @ X4))) | ((member @ (c > d) @ X4 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != $true)) & ! [X6 : c > d] : (((prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_mOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X6) @ X6)) | ((member @ (c > d) @ X6 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != $true)) & ((member @ (c > d) @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true)),
% 12.99/3.04 inference(rectify,[status(thm)],[f945])).
% 12.99/3.04 thf(f945,plain,(
% 12.99/3.04 ($true = (member @ ((c > d) > (c > d) > c > d) @ (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (pi @ (c > d) @ ((c > d) > c > d) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (^[Y0 : c > d]: (pi @ (c > d) @ (c > d) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (^[Y1 : c > d]: (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m))))))) & ($true = (member @ ((c > d) > c > d) @ (prod_mOp @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (pi @ (c > d) @ (c > d) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (^[Y0 : c > d]: (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m))))) & ! [X3 : c > d] : (((member @ (c > d) @ X3 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != $true) | ! [X4 : c > d] : (((member @ (c > d) @ X4 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != $true) | ! [X5 : c > d] : (((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X3 @ X4) @ X5) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X3 @ (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X4 @ X5))) | ((member @ (c > d) @ X5 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != $true)))) & ! [X6 : c > d] : (((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ X6) = X6) | ((member @ (c > d) @ X6 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != $true)) & ! [X1 : c > d] : (! [X2 : c > d] : (((member @ (c > d) @ X2 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != $true) | ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X1 @ X2) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X2 @ X1))) | ((member @ (c > d) @ X1 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != $true)) & ! [X0 : c > d] : (((prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_mOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X0) @ X0)) | ((member @ (c > d) @ X0 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != $true)) & ((member @ (c > d) @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true)),
% 12.99/3.04 inference(ennf_transformation,[status(thm)],[f818])).
% 12.99/3.04 thf(f818,plain,(
% 12.99/3.04 ((member @ (c > d) @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true) & ! [X3 : c > d] : (((member @ (c > d) @ X3 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true) => ! [X4 : c > d] : (((member @ (c > d) @ X4 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true) => ! [X5 : c > d] : (((member @ (c > d) @ X5 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true) => ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X3 @ X4) @ X5) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X3 @ (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X4 @ X5)))))) & ! [X6 : c > d] : (((member @ (c > d) @ X6 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true) => ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ X6) = X6)) & ! [X1 : c > d] : (((member @ (c > d) @ X1 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true) => ! [X2 : c > d] : (((member @ (c > d) @ X2 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true) => ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X1 @ X2) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X2 @ X1)))) & ($true = (member @ ((c > d) > (c > d) > c > d) @ (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (pi @ (c > d) @ ((c > d) > c > d) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (^[Y0 : c > d]: (pi @ (c > d) @ (c > d) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (^[Y1 : c > d]: (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m))))))) & ! [X0 : c > d] : (((member @ (c > d) @ X0 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true) => ((prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_mOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X0) @ X0))) & ($true = (member @ ((c > d) > c > d) @ (prod_mOp @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (pi @ (c > d) @ (c > d) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (^[Y0 : c > d]: (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)))))),
% 12.99/3.04 inference(rectify,[status(thm)],[f605])).
% 12.99/3.04 thf(f605,plain,(
% 12.99/3.04 ! [X0 : c > d] : (((member @ (c > d) @ X0 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true) => ((prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_mOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X0) @ X0))) & ($true = (member @ ((c > d) > (c > d) > c > d) @ (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (pi @ (c > d) @ ((c > d) > c > d) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (^[Y0 : c > d]: (pi @ (c > d) @ (c > d) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (^[Y1 : c > d]: (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m))))))) & ! [X3 : c > d] : (((member @ (c > d) @ X3 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true) => ! [X4 : c > d] : (((member @ (c > d) @ X4 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true) => ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X3 @ X4) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X4 @ X3)))) & ($true = (member @ ((c > d) > c > d) @ (prod_mOp @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (pi @ (c > d) @ (c > d) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (^[Y0 : c > d]: (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m))))) & ((member @ (c > d) @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true) & ! [X6 : c > d] : (((member @ (c > d) @ X6 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true) => ! [X7 : c > d] : (((member @ (c > d) @ X7 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true) => ! [X8 : c > d] : (((member @ (c > d) @ X8 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true) => ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X6 @ X7) @ X8) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X6 @ (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X7 @ X8)))))) & ! [X9 : c > d] : (($true = (member @ (c > d) @ X9 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m))) => ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ X9) = X9))),
% 12.99/3.04 inference(fool_elimination,[status(thm)],[f604])).
% 12.99/3.04 thf(f604,plain,(
% 12.99/3.04 ! [X0 : c > d] : ((member @ (c > d) @ X0 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) => ((prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_mOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X0) @ X0))) & (member @ ((c > d) > (c > d) > c > d) @ (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (pi @ (c > d) @ ((c > d) > c > d) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (^[X1 : c > d] : (pi @ (c > d) @ (c > d) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (^[X2 : c > d] : (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)))))) & ! [X3 : c > d] : ((member @ (c > d) @ X3 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) => ! [X4 : c > d] : ((member @ (c > d) @ X4 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) => ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X3 @ X4) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X4 @ X3)))) & (member @ ((c > d) > c > d) @ (prod_mOp @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (pi @ (c > d) @ (c > d) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (^[X5 : c > d] : (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)))) & (member @ (c > d) @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) & ! [X6 : c > d] : ((member @ (c > d) @ X6 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) => ! [X7 : c > d] : ((member @ (c > d) @ X7 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) => ! [X8 : c > d] : ((member @ (c > d) @ X8 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) => ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X6 @ X7) @ X8) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X6 @ (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X7 @ X8)))))) & ! [X9 : c > d] : ((member @ (c > d) @ X9 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) => ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ X9) = X9))),
% 12.99/3.04 inference(rectify,[status(thm)],[f270])).
% 12.99/3.04 thf(f270,axiom,(
% 12.99/3.04 ! [X81 : c > d] : ((member @ (c > d) @ X81 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) => ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_mOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X81) @ X81) = (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m))) & (member @ ((c > d) > (c > d) > c > d) @ (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (pi @ (c > d) @ ((c > d) > c > d) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (^[X16 : c > d] : (pi @ (c > d) @ (c > d) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (^[X17 : c > d] : (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)))))) & ! [X81 : c > d] : ((member @ (c > d) @ X81 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) => ! [X82 : c > d] : ((member @ (c > d) @ X82 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) => ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X81 @ X82) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X82 @ X81)))) & (member @ ((c > d) > c > d) @ (prod_mOp @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (pi @ (c > d) @ (c > d) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (^[X16 : c > d] : (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)))) & (member @ (c > d) @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) & ! [X81 : c > d] : ((member @ (c > d) @ X81 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) => ! [X82 : c > d] : ((member @ (c > d) @ X82 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) => ! [X83 : c > d] : ((member @ (c > d) @ X83 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) => ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X81 @ X82) @ X83) = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X81 @ (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X82 @ X83)))))) & ! [X81 : c > d] : ((member @ (c > d) @ X81 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) => ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ X81) = X81))),
% 12.99/3.04 file('/export/starexec/sandbox2/tmp/tmp.uhnk0bOYnS/DTF2THF_31945.p',conj_1)).
% 12.99/3.04 thf(f2042,plain,(
% 12.99/3.04 spl55_9 | ~spl55_3),
% 12.99/3.04 inference(avatar_split_clause,[status(thm)],[f2036,f2004,f2039])).
% 12.99/3.04 thf(f2036,plain,(
% 12.99/3.04 (sK18 = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ sK18)) | ~spl55_3),
% 12.99/3.04 inference(trivial_inequality_removal,[status(thm)],[f2035])).
% 12.99/3.04 thf(f2035,plain,(
% 12.99/3.04 (sK18 = (prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ sK18)) | ($true != $true) | ~spl55_3),
% 12.99/3.04 inference(superposition,[status(thm)],[f1629,f2006])).
% 12.99/3.04 thf(f1629,plain,(
% 12.99/3.04 ( ! [X3 : c > d] : (((member @ (c > d) @ X3 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != $true) | ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ X3) = X3)) )),
% 12.99/3.04 inference(cnf_transformation,[status(thm)],[f1335])).
% 12.99/3.04 thf(f2027,plain,(
% 12.99/3.04 ~spl55_7),
% 12.99/3.04 inference(avatar_split_clause,[status(thm)],[f1428,f2024])).
% 12.99/3.04 thf(f1428,plain,(
% 12.99/3.04 ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ sK18 @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != sK18)),
% 12.99/3.04 inference(cnf_transformation,[status(thm)],[f1276])).
% 12.99/3.04 thf(f1276,plain,(
% 12.99/3.04 ((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ sK18 @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != sK18) & ((member @ (c > d) @ sK18 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true)),
% 12.99/3.04 inference(skolemisation,[status(esa),new_symbols(skolem,[sK18])],[f960,f1275])).
% 12.99/3.04 thf(f1275,plain,(
% 12.99/3.04 ? [X0 : c > d] : (((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X0 @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != X0) & ((member @ (c > d) @ X0 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true)) => (((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ sK18 @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != sK18) & ((member @ (c > d) @ sK18 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true))),
% 12.99/3.04 introduced(definition,[],[choice_axiom])).
% 12.99/3.04 thf(f960,plain,(
% 12.99/3.04 ? [X0 : c > d] : (((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X0 @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != X0) & ((member @ (c > d) @ X0 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true))),
% 12.99/3.04 inference(ennf_transformation,[status(thm)],[f810])).
% 12.99/3.04 thf(f810,plain,(
% 12.99/3.04 ~! [X0 : c > d] : (((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X0 @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = X0) | ((member @ (c > d) @ X0 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) != $true))),
% 12.99/3.04 inference(flattening,[status(thm)],[f337])).
% 12.99/3.04 thf(f337,plain,(
% 12.99/3.04 ~! [X0 : c > d] : (((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X0 @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = X0) | ~((member @ (c > d) @ X0 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true))),
% 12.99/3.04 inference(fool_elimination,[status(thm)],[f336])).
% 12.99/3.04 thf(f336,plain,(
% 12.99/3.04 ~! [X0 : c > d] : (((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X0 @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = X0) | ~(member @ (c > d) @ X0 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)))),
% 12.99/3.04 inference(rectify,[status(thm)],[f272])).
% 12.99/3.04 thf(f272,negated_conjecture,(
% 12.99/3.04 ~! [X60 : c > d] : (((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X60 @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = X60) | ~(member @ (c > d) @ X60 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)))),
% 12.99/3.04 inference(negated_conjecture,[status(cth)],[f271])).
% 12.99/3.04 thf(f271,conjecture,(
% 12.99/3.04 ! [X60 : c > d] : (((prod_pOp @ c @ d @ module_ext @ d @ a @ e @ i @ m @ X60 @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = X60) | ~(member @ (c > d) @ X60 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)))),
% 12.99/3.04 file('/export/starexec/sandbox2/tmp/tmp.uhnk0bOYnS/DTF2THF_31945.p',conj_2)).
% 12.99/3.04 thf(f2017,plain,(
% 12.99/3.04 spl55_5),
% 12.99/3.04 inference(avatar_split_clause,[status(thm)],[f1626,f2014])).
% 12.99/3.04 thf(f1626,plain,(
% 12.99/3.04 ((member @ (c > d) @ (prod_zero @ c @ d @ module_ext @ d @ a @ e @ i @ m) @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true)),
% 12.99/3.04 inference(cnf_transformation,[status(thm)],[f1335])).
% 12.99/3.04 thf(f2007,plain,(
% 12.99/3.04 spl55_3),
% 12.99/3.04 inference(avatar_split_clause,[status(thm)],[f1427,f2004])).
% 12.99/3.04 thf(f1427,plain,(
% 12.99/3.04 ((member @ (c > d) @ sK18 @ (carr_prodag @ c @ d @ module_ext @ d @ a @ e @ i @ m)) = $true)),
% 12.99/3.04 inference(cnf_transformation,[status(thm)],[f1276])).
% 12.99/3.04 % SZS output end Proof for DTF2THF_31945
% 12.99/3.04 % (32152)------------------------------
% 12.99/3.04 % (32152)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.99/3.04 % (32152)Termination reason: Refutation
% 12.99/3.04
% 12.99/3.04 % (32152)Memory used [KB]: 8187
% 12.99/3.04 % (32152)Time elapsed: 0.151 s
% 12.99/3.04 % (32152)Instructions burned: 280 (million)
% 12.99/3.04 % (32152)------------------------------
% 12.99/3.04 % (32152)------------------------------
% 12.99/3.04 % (32051)Success in time 1.5 s
%------------------------------------------------------------------------------