%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP107^2 : 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 : n011.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:36 AM UTC 2026
% Result : Theorem 2.97s 1.63s
% Output : Refutation 2.97s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP107^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n011.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:25:47 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.28/0.47 ---- Original DTF file ---
% 0.28/0.47 thf(spec,logic,$$dhol).
% 0.28/0.47 %------------------------------------------------------------------------------
% 0.28/0.47 % File : ITP107^2 : TPTP v9.2.1. Released v7.5.0.
% 0.28/0.47 % Domain : Interactive Theorem Proving
% 0.28/0.47 % Problem : Sledgehammer Localization problem prob_542__8966104_1
% 0.28/0.47 % Version : Especial.
% 0.28/0.47 % English :
% 0.28/0.47
% 0.28/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.28/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.28/0.47 % Source : [Des21]
% 0.28/0.47 % Names : Localization/prob_542__8966104_1 [Des21]
% 0.28/0.47
% 0.28/0.47 % Status : Theorem
% 0.28/0.47 % Rating : 0.33 v9.1.0, 0.00 v7.5.0
% 0.28/0.47 % Syntax : Number of formulae : 310 ( 88 unt; 53 typ; 0 def)
% 0.28/0.47 % Number of atoms : 712 ( 254 equ; 0 cnn)
% 0.28/0.47 % Maximal formula atoms : 14 ( 2 avg)
% 0.28/0.47 % Number of connectives : 7071 ( 57 ~; 5 |; 33 &;6619 @)
% 0.28/0.47 % ( 0 <=>; 357 =>; 0 <=; 0 <~>)
% 0.28/0.47 % Maximal formula depth : 28 ( 10 avg)
% 0.28/0.47 % Number of types : 6 ( 5 usr)
% 0.28/0.47 % Number of type conns : 93 ( 93 >; 0 *; 0 +; 0 <<)
% 0.28/0.47 % Number of symbols : 49 ( 48 usr; 7 con; 0-6 aty)
% 0.28/0.47 % Number of variables : 976 ( 12 ^; 882 !; 15 ?; 976 :)
% 0.28/0.47 % ( 67 !>; 0 ?*; 0 @-; 0 @+)
% 0.28/0.47 % SPC : TH1_THM_EQU_NAR
% 0.28/0.47
% 0.28/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:23:31.187
% 0.28/0.47 %------------------------------------------------------------------------------
% 0.28/0.47 % Could-be-implicit typings (11)
% 0.28/0.47 thf(ty_t_Congruence_Opartial__object_Opartial__object__ext,type,
% 0.28/0.47 partia1265057091ct_ext: $tType > $tType > $tType ).
% 0.28/0.47
% 0.28/0.47 thf(ty_t_Congruence_Oeq__object_Oeq__object__ext,type,
% 0.28/0.47 eq_eq_object_ext: $tType > $tType > $tType ).
% 0.28/0.47
% 0.28/0.47 thf(ty_t_Group_Omonoid_Omonoid__ext,type,
% 0.28/0.47 monoid_ext: $tType > $tType > $tType ).
% 0.28/0.47
% 0.28/0.47 thf(ty_t_Ring_Oring_Oring__ext,type,
% 0.28/0.47 ring_ext: $tType > $tType > $tType ).
% 0.28/0.47
% 0.28/0.47 thf(ty_t_Product__Type_Ounit,type,
% 0.28/0.47 product_unit: $tType ).
% 0.28/0.47
% 0.28/0.47 thf(ty_t_Product__Type_Oprod,type,
% 0.28/0.47 product_prod: $tType > $tType > $tType ).
% 0.28/0.47
% 0.28/0.47 thf(ty_t_Set_Oset,type,
% 0.28/0.47 set: $tType > $tType ).
% 0.28/0.47
% 0.28/0.47 thf(ty_t_Nat_Onat,type,
% 0.28/0.47 nat: $tType ).
% 0.28/0.47
% 0.28/0.47 thf(ty_t_Int_Oint,type,
% 0.28/0.47 int: $tType ).
% 0.28/0.47
% 0.28/0.47 thf(ty_tf_b,type,
% 0.28/0.47 b: $tType ).
% 0.28/0.47
% 0.28/0.47 thf(ty_tf_a,type,
% 0.28/0.47 a: $tType ).
% 0.28/0.47
% 0.28/0.47 % Explicit typings (42)
% 0.28/0.47 thf(sy_c_AbelCoset_OA__RCOSETS,type,
% 0.28/0.47 a_RCOSETS:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_AbelCoset_Oa__l__coset,type,
% 0.28/0.47 a_l_coset:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > A > ( set @ A ) > ( set @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_AbelCoset_Oa__r__coset,type,
% 0.28/0.47 a_r_coset:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > A > ( set @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_AbelCoset_Oadditive__subgroup,type,
% 0.28/0.47 additive_subgroup:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_AbelCoset_Oset__add,type,
% 0.28/0.47 set_add:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > ( set @ A ) > ( set @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Congruence_Oequivalence,type,
% 0.28/0.47 equivalence:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( eq_eq_object_ext @ A @ B ) ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Congruence_Opartial__object_Ocarrier,type,
% 0.28/0.47 partial_carrier:
% 0.28/0.47 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ Z ) > ( set @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Group_OUnits,type,
% 0.28/0.47 units:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > ( set @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Group_Ogroup,type,
% 0.28/0.47 group:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Group_Omonoid,type,
% 0.28/0.47 monoid:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Group_Omonoid_Omult,type,
% 0.28/0.47 mult:
% 0.28/0.47 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ Z ) ) > A > A > A ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Group_Omonoid_Oone,type,
% 0.28/0.47 one:
% 0.28/0.47 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ Z ) ) > A ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Group_Ounits__of,type,
% 0.28/0.47 units_of:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ product_unit ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.28/0.47 minus_minus:
% 0.28/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Oeq__class__of__rng__of__frac,type,
% 0.28/0.47 locali1641774487f_frac:
% 0.28/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( partia1265057091ct_ext @ ( product_prod @ A @ B ) @ ( eq_eq_object_ext @ ( product_prod @ A @ B ) @ C ) ) > A > B > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Oeq__obj__rng__of__frac,type,
% 0.28/0.47 locali1648044335f_frac:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Oeq__obj__rng__of__frac_Orec__monoid__rng__of__frac,type,
% 0.28/0.47 locali1768110497f_frac:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > ( partia1265057091ct_ext @ ( set @ ( product_prod @ A @ A ) ) @ ( monoid_ext @ ( set @ ( product_prod @ A @ A ) ) @ product_unit ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Omult__submonoid__of__crng,type,
% 0.28/0.47 locali7914339f_crng:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Omult__submonoid__of__rng,type,
% 0.28/0.47 locali1402578320of_rng:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Oset__eq__class__of__rng__of__frac,type,
% 0.28/0.47 locali990921413f_frac:
% 0.28/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( partia1265057091ct_ext @ ( product_prod @ A @ B ) @ ( eq_eq_object_ext @ ( product_prod @ A @ B ) @ C ) ) > ( set @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Osubmonoid,type,
% 0.28/0.47 locali1441642228monoid:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > ( set @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.28/0.47 bot_bot:
% 0.28/0.47 !>[A: $tType] : A ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.28/0.47 ord_less_eq:
% 0.28/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Product__Type_OPair,type,
% 0.28/0.47 product_Pair:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.28/0.47 product_rec_prod:
% 0.28/0.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Ring_Oabelian__monoid,type,
% 0.28/0.47 abelian_monoid:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Ring_Oadd__pow,type,
% 0.28/0.47 add_pow:
% 0.28/0.47 !>[A: $tType,C: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ C ) ) ) > B > A > A ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Ring_Ofield,type,
% 0.28/0.47 field:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Ring_Oring,type,
% 0.28/0.47 ring:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Ring_Oring_Ozero,type,
% 0.28/0.47 zero:
% 0.28/0.47 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ) ) > A ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Ring_Osemiring,type,
% 0.28/0.47 semiring:
% 0.28/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Set_OCollect,type,
% 0.28/0.47 collect:
% 0.28/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Set_OPow,type,
% 0.28/0.47 pow:
% 0.28/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Set_Oinsert,type,
% 0.28/0.47 insert:
% 0.28/0.47 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_member,type,
% 0.28/0.47 member:
% 0.28/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_v_R,type,
% 0.28/0.47 r: partia1265057091ct_ext @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_v_S,type,
% 0.28/0.47 s: set @ a ).
% 0.28/0.47
% 0.28/0.47 thf(sy_v_r,type,
% 0.28/0.47 r2: a ).
% 0.28/0.47
% 0.28/0.47 thf(sy_v_r_H,type,
% 0.28/0.47 r3: a ).
% 0.28/0.47
% 0.28/0.47 thf(sy_v_rel,type,
% 0.28/0.47 rel: partia1265057091ct_ext @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_v_s,type,
% 0.28/0.47 s2: a ).
% 0.28/0.47
% 0.28/0.47 thf(sy_v_s_H,type,
% 0.28/0.47 s3: a ).
% 0.28/0.47
% 0.28/0.47 % Relevant facts (256)
% 0.28/0.47 thf(fact_0_f3,axiom,
% 0.28/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r3 )
% 0.28/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % f3
% 0.28/0.47 thf(fact_1_f4,axiom,
% 0.28/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 )
% 0.28/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ s2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % f4
% 0.28/0.47 thf(fact_2_eq__obj__rng__of__frac__axioms,axiom,
% 0.28/0.47 locali1648044335f_frac @ a @ b @ r @ s ).
% 0.28/0.47
% 0.28/0.47 % eq_obj_rng_of_frac_axioms
% 0.28/0.47 thf(fact_3_f1,axiom,
% 0.28/0.47 ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) )
% 0.28/0.47 = ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % f1
% 0.28/0.47 thf(fact_4_f2,axiom,
% 0.28/0.47 ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) )
% 0.28/0.47 = ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ s2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % f2
% 0.28/0.47 thf(fact_5_mult__submonoid__of__crng__axioms,axiom,
% 0.28/0.47 locali7914339f_crng @ a @ b @ r @ s ).
% 0.28/0.47
% 0.28/0.47 % mult_submonoid_of_crng_axioms
% 0.28/0.47 thf(fact_6_mult__submonoid__of__rng__axioms,axiom,
% 0.28/0.47 locali1402578320of_rng @ a @ b @ r @ s ).
% 0.28/0.47
% 0.28/0.47 % mult_submonoid_of_rng_axioms
% 0.28/0.47 thf(fact_7_eq__obj__rng__of__frac_Orec__monoid__rng__of__frac_Ocong,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType] :
% 0.28/0.47 ( ( locali1768110497f_frac @ A @ B )
% 0.28/0.47 = ( locali1768110497f_frac @ A @ B ) ) ).
% 0.28/0.47
% 0.28/0.47 % eq_obj_rng_of_frac.rec_monoid_rng_of_frac.cong
% 0.28/0.47 thf(fact_8_submonoid__axioms,axiom,
% 0.28/0.47 locali1441642228monoid @ a @ ( ring_ext @ a @ b ) @ r @ s ).
% 0.28/0.47
% 0.28/0.47 % submonoid_axioms
% 0.28/0.47 thf(fact_9_m__closed,axiom,
% 0.28/0.47 ! [X: a,Y: a] :
% 0.28/0.47 ( ( member @ a @ X @ s )
% 0.28/0.47 => ( ( member @ a @ Y @ s )
% 0.28/0.47 => ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ s ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % m_closed
% 0.28/0.47 thf(fact_10_monoid__rng__of__frac,axiom,
% 0.28/0.47 monoid @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) ).
% 0.28/0.47
% 0.28/0.47 % monoid_rng_of_frac
% 0.28/0.47 thf(fact_11_local_Osemiring__axioms,axiom,
% 0.28/0.47 semiring @ a @ b @ r ).
% 0.28/0.47
% 0.28/0.47 % local.semiring_axioms
% 0.28/0.47 thf(fact_12_one__closed,axiom,
% 0.28/0.47 member @ a @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ s ).
% 0.28/0.47
% 0.28/0.47 % one_closed
% 0.28/0.47 thf(fact_13_equiv__obj__rng__of__frac,axiom,
% 0.28/0.47 equivalence @ ( product_prod @ a @ a ) @ product_unit @ rel ).
% 0.28/0.47
% 0.28/0.47 % equiv_obj_rng_of_frac
% 0.28/0.47 thf(fact_14_assms_I2_J,axiom,
% 0.28/0.47 member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ r3 @ s3 ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ).
% 0.28/0.47
% 0.28/0.47 % assms(2)
% 0.28/0.47 thf(fact_15_assms_I1_J,axiom,
% 0.28/0.47 member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ r2 @ s2 ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ).
% 0.28/0.47
% 0.28/0.47 % assms(1)
% 0.28/0.47 thf(fact_16_assoc__mult__rng__of__frac,axiom,
% 0.28/0.47 ! [R: a,S: a,T2: a,U: a,V: a,W: a] :
% 0.28/0.47 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.28/0.47 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ T2 @ U ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.28/0.47 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ V @ W ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.28/0.47 => ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ T2 @ U ) ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ V @ W ) )
% 0.28/0.47 = ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) @ ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ T2 @ U ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ V @ W ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % assoc_mult_rng_of_frac
% 0.28/0.47 thf(fact_17_abelian__monoid__axioms,axiom,
% 0.28/0.47 abelian_monoid @ a @ b @ r ).
% 0.28/0.47
% 0.28/0.47 % abelian_monoid_axioms
% 0.28/0.47 thf(fact_18_mult__rng__of__frac__fundamental__lemma,axiom,
% 0.28/0.47 ! [R: a,S: a,R2: a,S2: a] :
% 0.28/0.47 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.28/0.47 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R2 @ S2 ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.28/0.47 => ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R2 @ S2 ) )
% 0.28/0.47 = ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ R @ R2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ S @ S2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_rng_of_frac_fundamental_lemma
% 0.28/0.47 thf(fact_19_local_Oring__axioms,axiom,
% 0.28/0.47 ring @ a @ b @ r ).
% 0.28/0.47
% 0.28/0.47 % local.ring_axioms
% 0.28/0.47 thf(fact_20_closed__mult__rng__of__frac,axiom,
% 0.28/0.47 ! [R: a,S: a,T2: a,U: a] :
% 0.28/0.47 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.28/0.47 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ T2 @ U ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.28/0.47 => ( member @ ( set @ ( product_prod @ a @ a ) ) @ ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ T2 @ U ) ) @ ( locali990921413f_frac @ a @ a @ product_unit @ rel ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % closed_mult_rng_of_frac
% 0.28/0.47 thf(fact_21_submonoid_Oaxioms_I1_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,M: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S3: set @ A] :
% 0.28/0.47 ( ( locali1441642228monoid @ A @ B @ M @ S3 )
% 0.28/0.47 => ( monoid @ A @ B @ M ) ) ).
% 0.28/0.47
% 0.28/0.47 % submonoid.axioms(1)
% 0.28/0.47 thf(fact_22_eq__obj__rng__of__frac_Oaxioms_I2_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
% 0.28/0.47 ( ( locali1648044335f_frac @ A @ B @ R3 @ S3 )
% 0.28/0.47 => ( locali7914339f_crng @ A @ B @ R3 @ S3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % eq_obj_rng_of_frac.axioms(2)
% 0.28/0.47 thf(fact_23_mult__submonoid__of__rng_Oaxioms_I2_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
% 0.28/0.47 ( ( locali1402578320of_rng @ A @ B @ R3 @ S3 )
% 0.28/0.47 => ( locali1441642228monoid @ A @ ( ring_ext @ A @ B ) @ R3 @ S3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_submonoid_of_rng.axioms(2)
% 0.28/0.47 thf(fact_24_mult__submonoid__of__rng_Oaxioms_I1_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
% 0.28/0.47 ( ( locali1402578320of_rng @ A @ B @ R3 @ S3 )
% 0.28/0.47 => ( ring @ A @ B @ R3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_submonoid_of_rng.axioms(1)
% 0.28/0.47 thf(fact_25_mult__submonoid__of__crng_Oaxioms_I2_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
% 0.28/0.47 ( ( locali7914339f_crng @ A @ B @ R3 @ S3 )
% 0.28/0.47 => ( locali1402578320of_rng @ A @ B @ R3 @ S3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_submonoid_of_crng.axioms(2)
% 0.28/0.47 thf(fact_26_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Oone__closed,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,M: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S3: set @ A] :
% 0.28/0.47 ( ( locali1441642228monoid @ A @ B @ M @ S3 )
% 0.28/0.47 => ( member @ A @ ( one @ A @ B @ M ) @ S3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % Localization_Mirabelle_ojffvtlkod.submonoid.one_closed
% 0.28/0.47 thf(fact_27_mult__submonoid__of__rng__def,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType] :
% 0.28/0.47 ( ( locali1402578320of_rng @ A @ B )
% 0.28/0.47 = ( ^ [R4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S4: set @ A] :
% 0.28/0.47 ( ( ring @ A @ B @ R4 )
% 0.28/0.47 & ( locali1441642228monoid @ A @ ( ring_ext @ A @ B ) @ R4 @ S4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_submonoid_of_rng_def
% 0.28/0.47 thf(fact_28_mult__submonoid__of__rng_Ointro,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
% 0.28/0.47 ( ( ring @ A @ B @ R3 )
% 0.28/0.47 => ( ( locali1441642228monoid @ A @ ( ring_ext @ A @ B ) @ R3 @ S3 )
% 0.28/0.47 => ( locali1402578320of_rng @ A @ B @ R3 @ S3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_submonoid_of_rng.intro
% 0.28/0.47 thf(fact_29_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Ois__submonoid,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,M: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S3: set @ A] :
% 0.28/0.47 ( ( locali1441642228monoid @ A @ B @ M @ S3 )
% 0.28/0.47 => ( locali1441642228monoid @ A @ B @ M @ S3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % Localization_Mirabelle_ojffvtlkod.submonoid.is_submonoid
% 0.28/0.47 thf(fact_30_eq__obj__rng__of__frac_Omonoid__rng__of__frac,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
% 0.28/0.47 ( ( locali1648044335f_frac @ A @ B @ R3 @ S3 )
% 0.28/0.47 => ( monoid @ ( set @ ( product_prod @ A @ A ) ) @ product_unit @ ( locali1768110497f_frac @ A @ B @ R3 @ S3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % eq_obj_rng_of_frac.monoid_rng_of_frac
% 0.28/0.47 thf(fact_31_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Om__closed,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,M: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S3: set @ A,X: A,Y: A] :
% 0.28/0.47 ( ( locali1441642228monoid @ A @ B @ M @ S3 )
% 0.28/0.47 => ( ( member @ A @ X @ S3 )
% 0.28/0.47 => ( ( member @ A @ Y @ S3 )
% 0.28/0.47 => ( member @ A @ ( mult @ A @ B @ M @ X @ Y ) @ S3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Localization_Mirabelle_ojffvtlkod.submonoid.m_closed
% 0.28/0.47 thf(fact_32_right__unit__mult__rng__of__frac,axiom,
% 0.28/0.47 ! [R: a,S: a] :
% 0.28/0.47 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.28/0.47 => ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) @ ( one @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) ) )
% 0.28/0.47 = ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % right_unit_mult_rng_of_frac
% 0.28/0.47 thf(fact_33_left__unit__mult__rng__of__frac,axiom,
% 0.28/0.47 ! [R: a,S: a] :
% 0.28/0.47 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.28/0.47 => ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( one @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) )
% 0.28/0.47 = ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % left_unit_mult_rng_of_frac
% 0.28/0.47 thf(fact_34_zero__in__mult__submonoid,axiom,
% 0.28/0.47 ! [R: a,S: a,R2: a,S2: a] :
% 0.28/0.47 ( ( member @ a @ ( zero @ a @ b @ r ) @ s )
% 0.28/0.47 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.28/0.47 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R2 @ S2 ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.28/0.47 => ( ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S )
% 0.28/0.47 = ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R2 @ S2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_in_mult_submonoid
% 0.28/0.47 thf(fact_35_non__empty__class,axiom,
% 0.28/0.47 ! [R: a,S: a] :
% 0.28/0.47 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.28/0.47 => ( ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S )
% 0.28/0.47 != ( bot_bot @ ( set @ ( product_prod @ a @ a ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % non_empty_class
% 0.28/0.47 thf(fact_36_one__unique,axiom,
% 0.28/0.47 ! [U: a] :
% 0.28/0.47 ( ( member @ a @ U @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ! [X2: a] :
% 0.28/0.47 ( ( member @ a @ X2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ U @ X2 )
% 0.28/0.47 = X2 ) )
% 0.28/0.47 => ( U
% 0.28/0.47 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % one_unique
% 0.28/0.47 thf(fact_37_inv__unique,axiom,
% 0.28/0.47 ! [Y: a,X: a,Y2: a] :
% 0.28/0.47 ( ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ X )
% 0.28/0.47 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.28/0.47 => ( ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y2 )
% 0.28/0.47 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.28/0.47 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( member @ a @ Y2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( Y = Y2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inv_unique
% 0.28/0.47 thf(fact_38_Units__inv__comm,axiom,
% 0.28/0.47 ! [X: a,Y: a] :
% 0.28/0.47 ( ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y )
% 0.28/0.47 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.28/0.47 => ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.28/0.47 => ( ( member @ a @ Y @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.28/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ X )
% 0.28/0.47 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Units_inv_comm
% 0.28/0.47 thf(fact_39_prod_Oinject,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 0.28/0.47 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.28/0.47 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 0.28/0.47 = ( ( X1 = Y1 )
% 0.28/0.47 & ( X22 = Y22 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod.inject
% 0.28/0.47 thf(fact_40_old_Oprod_Oinject,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.28/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.28/0.47 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.28/0.47 = ( ( A2 = A3 )
% 0.28/0.47 & ( B2 = B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % old.prod.inject
% 0.28/0.47 thf(fact_41_semiring_Oaxioms_I1_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.28/0.47 ( ( semiring @ A @ B @ R3 )
% 0.28/0.47 => ( abelian_monoid @ A @ B @ R3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring.axioms(1)
% 0.28/0.47 thf(fact_42_monoid__axioms,axiom,
% 0.28/0.47 monoid @ a @ ( ring_ext @ a @ b ) @ r ).
% 0.28/0.47
% 0.28/0.47 % monoid_axioms
% 0.28/0.47 thf(fact_43_m__assoc,axiom,
% 0.28/0.47 ! [X: a,Y: a,Z2: a] :
% 0.28/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ Z2 )
% 0.28/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % m_assoc
% 0.28/0.47 thf(fact_44_m__comm,axiom,
% 0.28/0.47 ! [X: a,Y: a] :
% 0.28/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y )
% 0.28/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ X ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % m_comm
% 0.28/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.28/0.47 ! [A: $tType,A2: A,P: A > $o] :
% 0.28/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.28/0.47 = ( P @ A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % mem_Collect_eq
% 0.28/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.28/0.47 ! [A: $tType,A4: set @ A] :
% 0.28/0.47 ( ( collect @ A
% 0.28/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
% 0.28/0.47 = A4 ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_mem_eq
% 0.28/0.47 thf(fact_47_Collect__cong,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.28/0.47 ( ! [X2: A] :
% 0.28/0.47 ( ( P @ X2 )
% 0.28/0.47 = ( Q @ X2 ) )
% 0.28/0.47 => ( ( collect @ A @ P )
% 0.28/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_cong
% 0.28/0.47 thf(fact_48_ext,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.28/0.47 ( ! [X2: A] :
% 0.28/0.47 ( ( F @ X2 )
% 0.28/0.47 = ( G @ X2 ) )
% 0.28/0.47 => ( F = G ) ) ).
% 0.28/0.47
% 0.28/0.47 % ext
% 0.28/0.47 thf(fact_49_m__lcomm,axiom,
% 0.28/0.47 ! [X: a,Y: a,Z2: a] :
% 0.28/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) )
% 0.28/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Z2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % m_lcomm
% 0.28/0.47 thf(fact_50_Units__closed,axiom,
% 0.28/0.47 ! [X: a] :
% 0.28/0.47 ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.28/0.47 => ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Units_closed
% 0.28/0.47 thf(fact_51_Units__l__inv__ex,axiom,
% 0.28/0.47 ! [X: a] :
% 0.28/0.47 ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.28/0.47 => ? [X2: a] :
% 0.28/0.47 ( ( member @ a @ X2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 & ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X2 @ X )
% 0.28/0.47 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Units_l_inv_ex
% 0.28/0.47 thf(fact_52_Units__r__inv__ex,axiom,
% 0.28/0.47 ! [X: a] :
% 0.28/0.47 ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.28/0.47 => ? [X2: a] :
% 0.28/0.47 ( ( member @ a @ X2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 & ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ X2 )
% 0.28/0.47 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Units_r_inv_ex
% 0.28/0.47 thf(fact_53_zero__closed,axiom,
% 0.28/0.47 member @ a @ ( zero @ a @ b @ r ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.28/0.47
% 0.28/0.47 % zero_closed
% 0.28/0.47 thf(fact_54_semiring__simprules_I3_J,axiom,
% 0.28/0.47 ! [X: a,Y: a] :
% 0.28/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring_simprules(3)
% 0.28/0.47 thf(fact_55_semiring__simprules_I4_J,axiom,
% 0.28/0.47 member @ a @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.28/0.47
% 0.28/0.47 % semiring_simprules(4)
% 0.28/0.47 thf(fact_56_Units__m__closed,axiom,
% 0.28/0.47 ! [X: a,Y: a] :
% 0.28/0.47 ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.28/0.47 => ( ( member @ a @ Y @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.28/0.47 => ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Units_m_closed
% 0.28/0.47 thf(fact_57_Units__one__closed,axiom,
% 0.28/0.47 member @ a @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) ).
% 0.28/0.47
% 0.28/0.47 % Units_one_closed
% 0.28/0.47 thf(fact_58_l__null,axiom,
% 0.28/0.47 ! [X: a] :
% 0.28/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( zero @ a @ b @ r ) @ X )
% 0.28/0.47 = ( zero @ a @ b @ r ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % l_null
% 0.28/0.47 thf(fact_59_r__null,axiom,
% 0.28/0.47 ! [X: a] :
% 0.28/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( zero @ a @ b @ r ) )
% 0.28/0.47 = ( zero @ a @ b @ r ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % r_null
% 0.28/0.47 thf(fact_60_l__one,axiom,
% 0.28/0.47 ! [X: a] :
% 0.28/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ X )
% 0.28/0.47 = X ) ) ).
% 0.28/0.47
% 0.28/0.47 % l_one
% 0.28/0.47 thf(fact_61_r__one,axiom,
% 0.28/0.47 ! [X: a] :
% 0.28/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.28/0.47 = X ) ) ).
% 0.28/0.47
% 0.28/0.47 % r_one
% 0.28/0.47 thf(fact_62_Units__l__cancel,axiom,
% 0.28/0.47 ! [X: a,Y: a,Z2: a] :
% 0.28/0.47 ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.28/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y )
% 0.28/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Z2 ) )
% 0.28/0.47 = ( Y = Z2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Units_l_cancel
% 0.28/0.47 thf(fact_63_abelian__monoidE_I2_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.28/0.47 ( ( abelian_monoid @ A @ B @ R3 )
% 0.28/0.47 => ( member @ A @ ( zero @ A @ B @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abelian_monoidE(2)
% 0.28/0.47 thf(fact_64_ring_Oring__simprules_I2_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.28/0.47 ( ( ring @ A @ B @ R3 )
% 0.28/0.47 => ( member @ A @ ( zero @ A @ B @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ring.ring_simprules(2)
% 0.28/0.47 thf(fact_65_semiring_Osemiring__simprules_I2_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.28/0.47 ( ( semiring @ A @ B @ R3 )
% 0.28/0.47 => ( member @ A @ ( zero @ A @ B @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring.semiring_simprules(2)
% 0.28/0.47 thf(fact_66_abelian__monoid_Ozero__closed,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.28/0.47 ( ( abelian_monoid @ A @ B @ G2 )
% 0.28/0.47 => ( member @ A @ ( zero @ A @ B @ G2 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abelian_monoid.zero_closed
% 0.28/0.47 thf(fact_67_ring_Ois__monoid,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.28/0.47 ( ( ring @ A @ B @ R3 )
% 0.28/0.47 => ( monoid @ A @ ( ring_ext @ A @ B ) @ R3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % ring.is_monoid
% 0.28/0.47 thf(fact_68_semiring_Oaxioms_I2_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.28/0.47 ( ( semiring @ A @ B @ R3 )
% 0.28/0.47 => ( monoid @ A @ ( ring_ext @ A @ B ) @ R3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring.axioms(2)
% 0.28/0.47 thf(fact_69_ring_Oring__simprules_I24_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.28/0.47 ( ( ring @ A @ B @ R3 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( zero @ A @ B @ R3 ) @ X )
% 0.28/0.47 = ( zero @ A @ B @ R3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ring.ring_simprules(24)
% 0.28/0.47 thf(fact_70_ring_Oring__simprules_I25_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.28/0.47 ( ( ring @ A @ B @ R3 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ ( zero @ A @ B @ R3 ) )
% 0.28/0.47 = ( zero @ A @ B @ R3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ring.ring_simprules(25)
% 0.28/0.47 thf(fact_71_semiring_Or__null,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.28/0.47 ( ( semiring @ A @ B @ R3 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ ( zero @ A @ B @ R3 ) )
% 0.28/0.47 = ( zero @ A @ B @ R3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring.r_null
% 0.28/0.47 thf(fact_72_semiring_Ol__null,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.28/0.47 ( ( semiring @ A @ B @ R3 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( zero @ A @ B @ R3 ) @ X )
% 0.28/0.47 = ( zero @ A @ B @ R3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring.l_null
% 0.28/0.47 thf(fact_73_ring_Oring__simprules_I5_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 0.28/0.47 ( ( ring @ A @ B @ R3 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( member @ A @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ring.ring_simprules(5)
% 0.28/0.47 thf(fact_74_ring_Oring__simprules_I11_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 0.28/0.47 ( ( ring @ A @ B @ R3 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ Y ) @ Z2 )
% 0.28/0.47 = ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ Y @ Z2 ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ring.ring_simprules(11)
% 0.28/0.47 thf(fact_75_semiring_Osemiring__simprules_I3_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 0.28/0.47 ( ( semiring @ A @ B @ R3 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( member @ A @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring.semiring_simprules(3)
% 0.28/0.47 thf(fact_76_semiring_Osemiring__simprules_I8_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 0.28/0.47 ( ( semiring @ A @ B @ R3 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ Y ) @ Z2 )
% 0.28/0.47 = ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ Y @ Z2 ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring.semiring_simprules(8)
% 0.28/0.47 thf(fact_77_ring_Oring__simprules_I6_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.28/0.47 ( ( ring @ A @ B @ R3 )
% 0.28/0.47 => ( member @ A @ ( one @ A @ ( ring_ext @ A @ B ) @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ring.ring_simprules(6)
% 0.28/0.47 thf(fact_78_semiring_Osemiring__simprules_I4_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.28/0.47 ( ( semiring @ A @ B @ R3 )
% 0.28/0.47 => ( member @ A @ ( one @ A @ ( ring_ext @ A @ B ) @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring.semiring_simprules(4)
% 0.28/0.47 thf(fact_79_ring_Oring__simprules_I12_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.28/0.47 ( ( ring @ A @ B @ R3 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( one @ A @ ( ring_ext @ A @ B ) @ R3 ) @ X )
% 0.28/0.47 = X ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ring.ring_simprules(12)
% 0.28/0.47 thf(fact_80_semiring_Osemiring__simprules_I9_J,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.28/0.47 ( ( semiring @ A @ B @ R3 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( one @ A @ ( ring_ext @ A @ B ) @ R3 ) @ X )
% 0.28/0.47 = X ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring.semiring_simprules(9)
% 0.28/0.47 thf(fact_81_old_Oprod_Oinducts,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.28/0.47 ( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
% 0.28/0.47 => ( P @ Prod ) ) ).
% 0.28/0.47
% 0.28/0.47 % old.prod.inducts
% 0.28/0.47 thf(fact_82_old_Oprod_Oexhaust,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.28/0.47 ~ ! [A5: A,B4: B] :
% 0.28/0.47 ( Y
% 0.28/0.47 != ( product_Pair @ A @ B @ A5 @ B4 ) ) ).
% 0.28/0.47
% 0.28/0.47 % old.prod.exhaust
% 0.28/0.47 thf(fact_83_prod__induct7,axiom,
% 0.28/0.47 ! [G3: $tType,F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) )] :
% 0.28/0.47 ( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G3 ) @ E2 @ ( product_Pair @ F2 @ G3 @ F3 @ G4 ) ) ) ) ) ) )
% 0.28/0.47 => ( P @ X ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_induct7
% 0.28/0.47 thf(fact_84_prod__induct6,axiom,
% 0.28/0.47 ! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
% 0.28/0.47 ( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
% 0.28/0.47 => ( P @ X ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_induct6
% 0.28/0.47 thf(fact_85_prod__induct5,axiom,
% 0.28/0.47 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.28/0.47 ( ! [A5: A,B4: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.28/0.47 => ( P @ X ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_induct5
% 0.28/0.47 thf(fact_86_prod__induct4,axiom,
% 0.28/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.28/0.47 ( ! [A5: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 0.28/0.47 => ( P @ X ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_induct4
% 0.28/0.47 thf(fact_87_prod__induct3,axiom,
% 0.28/0.47 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.28/0.47 ( ! [A5: A,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
% 0.28/0.47 => ( P @ X ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_induct3
% 0.28/0.47 thf(fact_88_prod__cases7,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) )] :
% 0.28/0.47 ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2,G4: G3] :
% 0.28/0.47 ( Y
% 0.28/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G3 ) @ E2 @ ( product_Pair @ F2 @ G3 @ F3 @ G4 ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases7
% 0.28/0.47 thf(fact_89_prod__cases6,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
% 0.28/0.47 ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2] :
% 0.28/0.47 ( Y
% 0.28/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases6
% 0.28/0.47 thf(fact_90_prod__cases5,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.28/0.47 ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E] :
% 0.28/0.47 ( Y
% 0.28/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases5
% 0.28/0.47 thf(fact_91_prod__cases4,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.28/0.47 ~ ! [A5: A,B4: B,C2: C,D2: D] :
% 0.28/0.47 ( Y
% 0.28/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases4
% 0.28/0.47 thf(fact_92_prod__cases3,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.28/0.47 ~ ! [A5: A,B4: B,C2: C] :
% 0.28/0.47 ( Y
% 0.28/0.47 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases3
% 0.28/0.47 thf(fact_93_Pair__inject,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.28/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.28/0.47 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.28/0.47 => ~ ( ( A2 = A3 )
% 0.28/0.47 => ( B2 != B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Pair_inject
% 0.28/0.47 thf(fact_94_prod__cases,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.28/0.47 ( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
% 0.28/0.47 => ( P @ P2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases
% 0.28/0.47 thf(fact_95_surj__pair,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.28/0.47 ? [X2: A,Y3: B] :
% 0.28/0.47 ( P2
% 0.28/0.47 = ( product_Pair @ A @ B @ X2 @ Y3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % surj_pair
% 0.28/0.47 thf(fact_96_ring_Ois__ring,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.28/0.47 ( ( ring @ A @ B @ R3 )
% 0.28/0.47 => ( ring @ A @ B @ R3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % ring.is_ring
% 0.28/0.47 thf(fact_97_cring__fieldI2,axiom,
% 0.28/0.47 ( ( ( zero @ a @ b @ r )
% 0.28/0.47 != ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.28/0.47 => ( ! [A5: a] :
% 0.28/0.47 ( ( member @ a @ A5 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( A5
% 0.28/0.47 != ( zero @ a @ b @ r ) )
% 0.28/0.47 => ? [X4: a] :
% 0.28/0.47 ( ( member @ a @ X4 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 & ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A5 @ X4 )
% 0.28/0.47 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) )
% 0.28/0.47 => ( field @ a @ b @ r ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % cring_fieldI2
% 0.28/0.47 thf(fact_98_monoid_Or__one,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
% 0.28/0.47 ( ( monoid @ A @ B @ G2 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( mult @ A @ B @ G2 @ X @ ( one @ A @ B @ G2 ) )
% 0.28/0.47 = X ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % monoid.r_one
% 0.28/0.47 thf(fact_99_monoid_Ol__one,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
% 0.28/0.47 ( ( monoid @ A @ B @ G2 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( mult @ A @ B @ G2 @ ( one @ A @ B @ G2 ) @ X )
% 0.28/0.47 = X ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % monoid.l_one
% 0.28/0.47 thf(fact_100_monoid_Oone__closed,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.28/0.47 ( ( monoid @ A @ B @ G2 )
% 0.28/0.47 => ( member @ A @ ( one @ A @ B @ G2 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % monoid.one_closed
% 0.28/0.47 thf(fact_101_monoid_OUnits__r__inv__ex,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
% 0.28/0.47 ( ( monoid @ A @ B @ G2 )
% 0.28/0.47 => ( ( member @ A @ X @ ( units @ A @ B @ G2 ) )
% 0.28/0.47 => ? [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 & ( ( mult @ A @ B @ G2 @ X @ X2 )
% 0.28/0.47 = ( one @ A @ B @ G2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % monoid.Units_r_inv_ex
% 0.28/0.47 thf(fact_102_monoid_OUnits__l__inv__ex,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
% 0.28/0.47 ( ( monoid @ A @ B @ G2 )
% 0.28/0.47 => ( ( member @ A @ X @ ( units @ A @ B @ G2 ) )
% 0.28/0.47 => ? [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 & ( ( mult @ A @ B @ G2 @ X2 @ X )
% 0.28/0.47 = ( one @ A @ B @ G2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % monoid.Units_l_inv_ex
% 0.28/0.47 thf(fact_103_monoid_OUnits__inv__comm,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A] :
% 0.28/0.47 ( ( monoid @ A @ B @ G2 )
% 0.28/0.47 => ( ( ( mult @ A @ B @ G2 @ X @ Y )
% 0.28/0.47 = ( one @ A @ B @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ X @ ( units @ A @ B @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ Y @ ( units @ A @ B @ G2 ) )
% 0.28/0.47 => ( ( mult @ A @ B @ G2 @ Y @ X )
% 0.28/0.47 = ( one @ A @ B @ G2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % monoid.Units_inv_comm
% 0.28/0.47 thf(fact_104_add__pow__ldistr__int,axiom,
% 0.28/0.47 ! [A2: a,B2: a,K: int] :
% 0.28/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add_pow @ a @ b @ int @ r @ K @ A2 ) @ B2 )
% 0.28/0.47 = ( add_pow @ a @ b @ int @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_pow_ldistr_int
% 0.28/0.47 thf(fact_105_add__pow__rdistr__int,axiom,
% 0.28/0.47 ! [A2: a,B2: a,K: int] :
% 0.28/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ ( add_pow @ a @ b @ int @ r @ K @ B2 ) )
% 0.28/0.47 = ( add_pow @ a @ b @ int @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_pow_rdistr_int
% 0.28/0.47 thf(fact_106_add_Oint__pow__closed,axiom,
% 0.28/0.47 ! [X: a,I: int] :
% 0.28/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( member @ a @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add.int_pow_closed
% 0.28/0.47 thf(fact_107_add_Oint__pow__one,axiom,
% 0.28/0.47 ! [Z2: int] :
% 0.28/0.47 ( ( add_pow @ a @ b @ int @ r @ Z2 @ ( zero @ a @ b @ r ) )
% 0.28/0.47 = ( zero @ a @ b @ r ) ) ).
% 0.28/0.47
% 0.28/0.47 % add.int_pow_one
% 0.28/0.47 thf(fact_108_field_Ois__ring,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.28/0.47 ( ( field @ A @ B @ R3 )
% 0.28/0.47 => ( ring @ A @ B @ R3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % field.is_ring
% 0.28/0.47 thf(fact_109_one__not__zero,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.28/0.47 ( ( field @ A @ B @ R3 )
% 0.28/0.47 => ( ( one @ A @ ( ring_ext @ A @ B ) @ R3 )
% 0.28/0.47 != ( zero @ A @ B @ R3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % one_not_zero
% 0.28/0.47 thf(fact_110_integral,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A] :
% 0.28/0.47 ( ( field @ A @ B @ R3 )
% 0.28/0.47 => ( ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 )
% 0.28/0.47 = ( zero @ A @ B @ R3 ) )
% 0.28/0.47 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( A2
% 0.28/0.47 = ( zero @ A @ B @ R3 ) )
% 0.28/0.47 | ( B2
% 0.28/0.47 = ( zero @ A @ B @ R3 ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % integral
% 0.28/0.47 thf(fact_111_ring_Oadd__pow__ldistr__int,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A,K: int] :
% 0.28/0.47 ( ( ring @ A @ B @ R3 )
% 0.28/0.47 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( add_pow @ A @ B @ int @ R3 @ K @ A2 ) @ B2 )
% 0.28/0.47 = ( add_pow @ A @ B @ int @ R3 @ K @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ring.add_pow_ldistr_int
% 0.28/0.47 thf(fact_112_ring_Oadd__pow__rdistr__int,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A,K: int] :
% 0.28/0.47 ( ( ring @ A @ B @ R3 )
% 0.28/0.47 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ ( add_pow @ A @ B @ int @ R3 @ K @ B2 ) )
% 0.28/0.47 = ( add_pow @ A @ B @ int @ R3 @ K @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ring.add_pow_rdistr_int
% 0.28/0.47 thf(fact_113_monoid_Ocarrier__not__empty,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.28/0.47 ( ( monoid @ A @ B @ G2 )
% 0.28/0.47 => ( ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 )
% 0.28/0.47 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % monoid.carrier_not_empty
% 0.28/0.47 thf(fact_114_monoid_Om__assoc,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A,Z2: A] :
% 0.28/0.47 ( ( monoid @ A @ B @ G2 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( mult @ A @ B @ G2 @ ( mult @ A @ B @ G2 @ X @ Y ) @ Z2 )
% 0.28/0.47 = ( mult @ A @ B @ G2 @ X @ ( mult @ A @ B @ G2 @ Y @ Z2 ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % monoid.m_assoc
% 0.28/0.47 thf(fact_115_monoid_Om__closed,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A] :
% 0.28/0.47 ( ( monoid @ A @ B @ G2 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( member @ A @ ( mult @ A @ B @ G2 @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % monoid.m_closed
% 0.28/0.47 thf(fact_116_monoid_OUnits__closed,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
% 0.28/0.47 ( ( monoid @ A @ B @ G2 )
% 0.28/0.47 => ( ( member @ A @ X @ ( units @ A @ B @ G2 ) )
% 0.28/0.47 => ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % monoid.Units_closed
% 0.28/0.47 thf(fact_117_monoid_OUnits__m__closed,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A] :
% 0.28/0.47 ( ( monoid @ A @ B @ G2 )
% 0.28/0.47 => ( ( member @ A @ X @ ( units @ A @ B @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ Y @ ( units @ A @ B @ G2 ) )
% 0.28/0.47 => ( member @ A @ ( mult @ A @ B @ G2 @ X @ Y ) @ ( units @ A @ B @ G2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % monoid.Units_m_closed
% 0.28/0.47 thf(fact_118_monoid_OUnits__one__closed,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.28/0.47 ( ( monoid @ A @ B @ G2 )
% 0.28/0.47 => ( member @ A @ ( one @ A @ B @ G2 ) @ ( units @ A @ B @ G2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % monoid.Units_one_closed
% 0.28/0.47 thf(fact_119_monoidI,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.28/0.47 ( ! [X2: A,Y3: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ Y3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( member @ A @ ( mult @ A @ B @ G2 @ X2 @ Y3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) )
% 0.28/0.47 => ( ( member @ A @ ( one @ A @ B @ G2 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ! [X2: A,Y3: A,Z3: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ Y3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ Z3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( mult @ A @ B @ G2 @ ( mult @ A @ B @ G2 @ X2 @ Y3 ) @ Z3 )
% 0.28/0.47 = ( mult @ A @ B @ G2 @ X2 @ ( mult @ A @ B @ G2 @ Y3 @ Z3 ) ) ) ) ) )
% 0.28/0.47 => ( ! [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( mult @ A @ B @ G2 @ ( one @ A @ B @ G2 ) @ X2 )
% 0.28/0.47 = X2 ) )
% 0.28/0.47 => ( ! [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( mult @ A @ B @ G2 @ X2 @ ( one @ A @ B @ G2 ) )
% 0.28/0.47 = X2 ) )
% 0.28/0.47 => ( monoid @ A @ B @ G2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % monoidI
% 0.28/0.47 thf(fact_120_Group_Omonoid__def,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType] :
% 0.28/0.47 ( ( monoid @ A @ B )
% 0.28/0.47 = ( ^ [G5: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.28/0.47 ( ! [X3: A,Y4: A] :
% 0.28/0.47 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
% 0.28/0.47 => ( ( member @ A @ Y4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
% 0.28/0.47 => ( member @ A @ ( mult @ A @ B @ G5 @ X3 @ Y4 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) ) ) )
% 0.28/0.47 & ! [X3: A,Y4: A,Z4: A] :
% 0.28/0.47 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
% 0.28/0.47 => ( ( member @ A @ Y4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
% 0.28/0.47 => ( ( member @ A @ Z4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
% 0.28/0.47 => ( ( mult @ A @ B @ G5 @ ( mult @ A @ B @ G5 @ X3 @ Y4 ) @ Z4 )
% 0.28/0.47 = ( mult @ A @ B @ G5 @ X3 @ ( mult @ A @ B @ G5 @ Y4 @ Z4 ) ) ) ) ) )
% 0.28/0.47 & ( member @ A @ ( one @ A @ B @ G5 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
% 0.28/0.47 & ! [X3: A] :
% 0.28/0.47 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
% 0.28/0.47 => ( ( mult @ A @ B @ G5 @ ( one @ A @ B @ G5 ) @ X3 )
% 0.28/0.47 = X3 ) )
% 0.28/0.47 & ! [X3: A] :
% 0.28/0.47 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
% 0.28/0.47 => ( ( mult @ A @ B @ G5 @ X3 @ ( one @ A @ B @ G5 ) )
% 0.28/0.47 = X3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Group.monoid_def
% 0.28/0.47 thf(fact_121_Group_Omonoid_Ointro,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.28/0.47 ( ! [X2: A,Y3: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ Y3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( member @ A @ ( mult @ A @ B @ G2 @ X2 @ Y3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) )
% 0.28/0.47 => ( ! [X2: A,Y3: A,Z3: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ Y3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ Z3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( mult @ A @ B @ G2 @ ( mult @ A @ B @ G2 @ X2 @ Y3 ) @ Z3 )
% 0.28/0.47 = ( mult @ A @ B @ G2 @ X2 @ ( mult @ A @ B @ G2 @ Y3 @ Z3 ) ) ) ) ) )
% 0.28/0.47 => ( ( member @ A @ ( one @ A @ B @ G2 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ! [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( mult @ A @ B @ G2 @ ( one @ A @ B @ G2 ) @ X2 )
% 0.28/0.47 = X2 ) )
% 0.28/0.47 => ( ! [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( mult @ A @ B @ G2 @ X2 @ ( one @ A @ B @ G2 ) )
% 0.28/0.47 = X2 ) )
% 0.28/0.47 => ( monoid @ A @ B @ G2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Group.monoid.intro
% 0.28/0.47 thf(fact_122_monoid_Oinv__unique,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),Y: A,X: A,Y2: A] :
% 0.28/0.47 ( ( monoid @ A @ B @ G2 )
% 0.28/0.47 => ( ( ( mult @ A @ B @ G2 @ Y @ X )
% 0.28/0.47 = ( one @ A @ B @ G2 ) )
% 0.28/0.47 => ( ( ( mult @ A @ B @ G2 @ X @ Y2 )
% 0.28/0.47 = ( one @ A @ B @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ Y2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( Y = Y2 ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % monoid.inv_unique
% 0.28/0.47 thf(fact_123_monoid_Oone__unique,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),U: A] :
% 0.28/0.47 ( ( monoid @ A @ B @ G2 )
% 0.28/0.47 => ( ( member @ A @ U @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ! [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( mult @ A @ B @ G2 @ U @ X2 )
% 0.28/0.47 = X2 ) )
% 0.28/0.47 => ( U
% 0.28/0.47 = ( one @ A @ B @ G2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % monoid.one_unique
% 0.28/0.47 thf(fact_124_monoid_OUnits__l__cancel,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A,Z2: A] :
% 0.28/0.47 ( ( monoid @ A @ B @ G2 )
% 0.28/0.47 => ( ( member @ A @ X @ ( units @ A @ B @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( ( mult @ A @ B @ G2 @ X @ Y )
% 0.28/0.47 = ( mult @ A @ B @ G2 @ X @ Z2 ) )
% 0.28/0.47 = ( Y = Z2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % monoid.Units_l_cancel
% 0.28/0.47 thf(fact_125_subset,axiom,
% 0.28/0.47 ord_less_eq @ ( set @ a ) @ s @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.28/0.47
% 0.28/0.47 % subset
% 0.28/0.47 thf(fact_126_add__pow__rdistr,axiom,
% 0.28/0.47 ! [A2: a,B2: a,K: nat] :
% 0.28/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ ( add_pow @ a @ b @ nat @ r @ K @ B2 ) )
% 0.28/0.47 = ( add_pow @ a @ b @ nat @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_pow_rdistr
% 0.28/0.47 thf(fact_127_add__pow__ldistr,axiom,
% 0.28/0.47 ! [A2: a,B2: a,K: nat] :
% 0.28/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add_pow @ a @ b @ nat @ r @ K @ A2 ) @ B2 )
% 0.28/0.47 = ( add_pow @ a @ b @ nat @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_pow_ldistr
% 0.28/0.47 thf(fact_128_old_Oprod_Orec,axiom,
% 0.28/0.47 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.28/0.47 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.28/0.47 = ( F1 @ A2 @ B2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % old.prod.rec
% 0.28/0.47 thf(fact_129_group__l__invI,axiom,
% 0.28/0.47 ( ! [X2: a] :
% 0.28/0.47 ( ( member @ a @ X2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ? [Xa: a] :
% 0.28/0.47 ( ( member @ a @ Xa @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 & ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Xa @ X2 )
% 0.28/0.47 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) )
% 0.28/0.47 => ( group @ a @ ( ring_ext @ a @ b ) @ r ) ) ).
% 0.28/0.47
% 0.28/0.47 % group_l_invI
% 0.28/0.47 thf(fact_130_carrier__not__empty,axiom,
% 0.28/0.47 ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 0.28/0.47 != ( bot_bot @ ( set @ a ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % carrier_not_empty
% 0.28/0.47 thf(fact_131_add_Onat__pow__closed,axiom,
% 0.28/0.47 ! [X: a,N: nat] :
% 0.28/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.47 => ( member @ a @ ( add_pow @ a @ b @ nat @ r @ N @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add.nat_pow_closed
% 0.28/0.47 thf(fact_132_add_Onat__pow__one,axiom,
% 0.28/0.47 ! [N: nat] :
% 0.28/0.47 ( ( add_pow @ a @ b @ nat @ r @ N @ ( zero @ a @ b @ r ) )
% 0.28/0.47 = ( zero @ a @ b @ r ) ) ).
% 0.28/0.47
% 0.28/0.47 % add.nat_pow_one
% 0.28/0.47 thf(fact_133_group_Ois__group,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.28/0.47 ( ( group @ A @ B @ G2 )
% 0.28/0.47 => ( group @ A @ B @ G2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % group.is_group
% 0.28/0.47 thf(fact_134_group_OUnits,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.28/0.47 ( ( group @ A @ B @ G2 )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) @ ( units @ A @ B @ G2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % group.Units
% 0.28/0.47 thf(fact_135_group_Ois__monoid,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.28/0.47 ( ( group @ A @ B @ G2 )
% 0.28/0.47 => ( monoid @ A @ B @ G2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % group.is_monoid
% 0.28/0.47 thf(fact_136_Group_Ogroup_Oright__cancel,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A,Z2: A] :
% 0.28/0.47 ( ( group @ A @ B @ G2 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( ( mult @ A @ B @ G2 @ Y @ X )
% 0.28/0.47 = ( mult @ A @ B @ G2 @ Z2 @ X ) )
% 0.28/0.47 = ( Y = Z2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Group.group.right_cancel
% 0.28/0.47 thf(fact_137_group_OUnits__eq,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.28/0.47 ( ( group @ A @ B @ G2 )
% 0.28/0.47 => ( ( units @ A @ B @ G2 )
% 0.28/0.47 = ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % group.Units_eq
% 0.28/0.47 thf(fact_138_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Osubset,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,M: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S3: set @ A] :
% 0.28/0.47 ( ( locali1441642228monoid @ A @ B @ M @ S3 )
% 0.28/0.47 => ( ord_less_eq @ ( set @ A ) @ S3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ M ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Localization_Mirabelle_ojffvtlkod.submonoid.subset
% 0.28/0.47 thf(fact_139_group_Or__cancel__one_H,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,A2: A] :
% 0.28/0.47 ( ( group @ A @ B @ G2 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( X
% 0.28/0.47 = ( mult @ A @ B @ G2 @ A2 @ X ) )
% 0.28/0.47 = ( A2
% 0.28/0.47 = ( one @ A @ B @ G2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % group.r_cancel_one'
% 0.28/0.47 thf(fact_140_group_Ol__cancel__one_H,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,A2: A] :
% 0.28/0.47 ( ( group @ A @ B @ G2 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( X
% 0.28/0.47 = ( mult @ A @ B @ G2 @ X @ A2 ) )
% 0.28/0.47 = ( A2
% 0.28/0.47 = ( one @ A @ B @ G2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % group.l_cancel_one'
% 0.28/0.47 thf(fact_141_group_Or__cancel__one,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,A2: A] :
% 0.28/0.47 ( ( group @ A @ B @ G2 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( ( mult @ A @ B @ G2 @ A2 @ X )
% 0.28/0.47 = X )
% 0.28/0.47 = ( A2
% 0.28/0.47 = ( one @ A @ B @ G2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % group.r_cancel_one
% 0.28/0.47 thf(fact_142_group_Ol__cancel__one,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,A2: A] :
% 0.28/0.47 ( ( group @ A @ B @ G2 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( ( mult @ A @ B @ G2 @ X @ A2 )
% 0.28/0.47 = X )
% 0.28/0.47 = ( A2
% 0.28/0.47 = ( one @ A @ B @ G2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % group.l_cancel_one
% 0.28/0.47 thf(fact_143_group_Or__inv__ex,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
% 0.28/0.47 ( ( group @ A @ B @ G2 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ? [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 & ( ( mult @ A @ B @ G2 @ X @ X2 )
% 0.28/0.47 = ( one @ A @ B @ G2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % group.r_inv_ex
% 0.28/0.47 thf(fact_144_group_Ol__inv__ex,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
% 0.28/0.47 ( ( group @ A @ B @ G2 )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ? [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 & ( ( mult @ A @ B @ G2 @ X2 @ X )
% 0.28/0.47 = ( one @ A @ B @ G2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % group.l_inv_ex
% 0.28/0.47 thf(fact_145_group_Oinv__comm,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A] :
% 0.28/0.47 ( ( group @ A @ B @ G2 )
% 0.28/0.47 => ( ( ( mult @ A @ B @ G2 @ X @ Y )
% 0.28/0.47 = ( one @ A @ B @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( mult @ A @ B @ G2 @ Y @ X )
% 0.28/0.47 = ( one @ A @ B @ G2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % group.inv_comm
% 0.28/0.47 thf(fact_146_groupI,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.28/0.47 ( ! [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ! [Y3: A] :
% 0.28/0.47 ( ( member @ A @ Y3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( member @ A @ ( mult @ A @ B @ G2 @ X2 @ Y3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) )
% 0.28/0.47 => ( ( member @ A @ ( one @ A @ B @ G2 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ! [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ! [Y3: A] :
% 0.28/0.47 ( ( member @ A @ Y3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ! [Z3: A] :
% 0.28/0.47 ( ( member @ A @ Z3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( mult @ A @ B @ G2 @ ( mult @ A @ B @ G2 @ X2 @ Y3 ) @ Z3 )
% 0.28/0.47 = ( mult @ A @ B @ G2 @ X2 @ ( mult @ A @ B @ G2 @ Y3 @ Z3 ) ) ) ) ) )
% 0.28/0.47 => ( ! [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ( ( mult @ A @ B @ G2 @ ( one @ A @ B @ G2 ) @ X2 )
% 0.28/0.47 = X2 ) )
% 0.28/0.47 => ( ! [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ? [Xa: A] :
% 0.28/0.47 ( ( member @ A @ Xa @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 & ( ( mult @ A @ B @ G2 @ Xa @ X2 )
% 0.28/0.47 = ( one @ A @ B @ G2 ) ) ) )
% 0.28/0.47 => ( group @ A @ B @ G2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % groupI
% 0.28/0.47 thf(fact_147_monoid_Ogroup__l__invI,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.28/0.47 ( ( monoid @ A @ B @ G2 )
% 0.28/0.47 => ( ! [X2: A] :
% 0.28/0.47 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 => ? [Xa: A] :
% 0.28/0.47 ( ( member @ A @ Xa @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.28/0.47 & ( ( mult @ A @ B @ G2 @ Xa @ X2 )
% 0.28/0.47 = ( one @ A @ B @ G2 ) ) ) )
% 0.28/0.47 => ( group @ A @ B @ G2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % monoid.group_l_invI
% 0.28/0.47 thf(fact_148_semiring_Oadd__pow__rdistr,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A,K: nat] :
% 0.28/0.47 ( ( semiring @ A @ B @ R3 )
% 0.28/0.47 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ ( add_pow @ A @ B @ nat @ R3 @ K @ B2 ) )
% 0.28/0.47 = ( add_pow @ A @ B @ nat @ R3 @ K @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring.add_pow_rdistr
% 0.28/0.47 thf(fact_149_semiring_Oadd__pow__ldistr,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A,K: nat] :
% 0.28/0.47 ( ( semiring @ A @ B @ R3 )
% 0.28/0.47 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.28/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( add_pow @ A @ B @ nat @ R3 @ K @ A2 ) @ B2 )
% 0.28/0.47 = ( add_pow @ A @ B @ nat @ R3 @ K @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring.add_pow_ldistr
% 0.28/0.47 thf(fact_150_a__lcos__mult__one,axiom,
% 0.28/0.48 ! [M: set @ a] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ a ) @ M @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.48 => ( ( a_l_coset @ a @ b @ r @ ( zero @ a @ b @ r ) @ M )
% 0.28/0.48 = M ) ) ).
% 0.28/0.48
% 0.28/0.48 % a_lcos_mult_one
% 0.28/0.48 thf(fact_151_set__add__closed,axiom,
% 0.28/0.48 ! [A4: set @ a,B5: set @ a] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ a ) @ A4 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.48 => ( ( ord_less_eq @ ( set @ a ) @ B5 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.48 => ( ord_less_eq @ ( set @ a ) @ ( set_add @ a @ b @ r @ A4 @ B5 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % set_add_closed
% 0.28/0.48 thf(fact_152_setadd__subset__G,axiom,
% 0.28/0.48 ! [H: set @ a,K2: set @ a] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.48 => ( ( ord_less_eq @ ( set @ a ) @ K2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.48 => ( ord_less_eq @ ( set @ a ) @ ( set_add @ a @ b @ r @ H @ K2 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % setadd_subset_G
% 0.28/0.48 thf(fact_153_subset__empty,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 0.28/0.48 = ( A4
% 0.28/0.48 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_empty
% 0.28/0.48 thf(fact_154_empty__iff,axiom,
% 0.28/0.48 ! [A: $tType,C3: A] :
% 0.28/0.48 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % empty_iff
% 0.28/0.48 thf(fact_155_all__not__in__conv,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A] :
% 0.28/0.48 ( ( ! [X3: A] :
% 0.28/0.48 ~ ( member @ A @ X3 @ A4 ) )
% 0.28/0.48 = ( A4
% 0.28/0.48 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % all_not_in_conv
% 0.28/0.48 thf(fact_156_Collect__empty__eq,axiom,
% 0.28/0.48 ! [A: $tType,P: A > $o] :
% 0.28/0.48 ( ( ( collect @ A @ P )
% 0.28/0.48 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.48 = ( ! [X3: A] :
% 0.28/0.48 ~ ( P @ X3 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Collect_empty_eq
% 0.28/0.48 thf(fact_157_empty__Collect__eq,axiom,
% 0.28/0.48 ! [A: $tType,P: A > $o] :
% 0.28/0.48 ( ( ( bot_bot @ ( set @ A ) )
% 0.28/0.48 = ( collect @ A @ P ) )
% 0.28/0.48 = ( ! [X3: A] :
% 0.28/0.48 ~ ( P @ X3 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % empty_Collect_eq
% 0.28/0.48 thf(fact_158_subsetI,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.28/0.48 ( ! [X2: A] :
% 0.28/0.48 ( ( member @ A @ X2 @ A4 )
% 0.28/0.48 => ( member @ A @ X2 @ B5 ) )
% 0.28/0.48 => ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.28/0.48
% 0.28/0.48 % subsetI
% 0.28/0.48 thf(fact_159_subset__antisym,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.28/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ A4 )
% 0.28/0.48 => ( A4 = B5 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_antisym
% 0.28/0.48 thf(fact_160_a__l__coset__subset__G,axiom,
% 0.28/0.48 ! [H: set @ a,X: a] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.48 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.48 => ( ord_less_eq @ ( set @ a ) @ ( a_l_coset @ a @ b @ r @ X @ H ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % a_l_coset_subset_G
% 0.28/0.48 thf(fact_161_empty__subsetI,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% 0.28/0.48
% 0.28/0.48 % empty_subsetI
% 0.28/0.48 thf(fact_162_emptyE,axiom,
% 0.28/0.48 ! [A: $tType,A2: A] :
% 0.28/0.48 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % emptyE
% 0.28/0.48 thf(fact_163_equals0D,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,A2: A] :
% 0.28/0.48 ( ( A4
% 0.28/0.48 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.48 => ~ ( member @ A @ A2 @ A4 ) ) ).
% 0.28/0.48
% 0.28/0.48 % equals0D
% 0.28/0.48 thf(fact_164_equals0I,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A] :
% 0.28/0.48 ( ! [Y3: A] :
% 0.28/0.48 ~ ( member @ A @ Y3 @ A4 )
% 0.28/0.48 => ( A4
% 0.28/0.48 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % equals0I
% 0.28/0.48 thf(fact_165_ex__in__conv,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A] :
% 0.28/0.48 ( ( ? [X3: A] : ( member @ A @ X3 @ A4 ) )
% 0.28/0.48 = ( A4
% 0.28/0.48 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % ex_in_conv
% 0.28/0.48 thf(fact_166_in__mono,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,B5: set @ A,X: A] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.28/0.48 => ( ( member @ A @ X @ A4 )
% 0.28/0.48 => ( member @ A @ X @ B5 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % in_mono
% 0.28/0.48 thf(fact_167_subsetD,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,B5: set @ A,C3: A] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.28/0.48 => ( ( member @ A @ C3 @ A4 )
% 0.28/0.48 => ( member @ A @ C3 @ B5 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subsetD
% 0.28/0.48 thf(fact_168_equalityE,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.28/0.48 ( ( A4 = B5 )
% 0.28/0.48 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.28/0.48 => ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % equalityE
% 0.28/0.48 thf(fact_169_subset__eq,axiom,
% 0.28/0.48 ! [A: $tType] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ A ) )
% 0.28/0.48 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.28/0.48 ! [X3: A] :
% 0.28/0.48 ( ( member @ A @ X3 @ A6 )
% 0.28/0.48 => ( member @ A @ X3 @ B6 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_eq
% 0.28/0.48 thf(fact_170_equalityD1,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.28/0.48 ( ( A4 = B5 )
% 0.28/0.48 => ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.28/0.48
% 0.28/0.48 % equalityD1
% 0.28/0.48 thf(fact_171_equalityD2,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.28/0.48 ( ( A4 = B5 )
% 0.28/0.48 => ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ).
% 0.28/0.48
% 0.28/0.48 % equalityD2
% 0.28/0.48 thf(fact_172_subset__iff,axiom,
% 0.28/0.48 ! [A: $tType] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ A ) )
% 0.28/0.48 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.28/0.48 ! [T3: A] :
% 0.28/0.48 ( ( member @ A @ T3 @ A6 )
% 0.28/0.48 => ( member @ A @ T3 @ B6 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_iff
% 0.28/0.48 thf(fact_173_subset__refl,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% 0.28/0.48
% 0.28/0.48 % subset_refl
% 0.28/0.48 thf(fact_174_Collect__mono,axiom,
% 0.28/0.48 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.28/0.48 ( ! [X2: A] :
% 0.28/0.48 ( ( P @ X2 )
% 0.28/0.48 => ( Q @ X2 ) )
% 0.28/0.48 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Collect_mono
% 0.28/0.48 thf(fact_175_subset__trans,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,B5: set @ A,C4: set @ A] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.28/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
% 0.28/0.48 => ( ord_less_eq @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_trans
% 0.28/0.48 thf(fact_176_set__eq__subset,axiom,
% 0.28/0.48 ! [A: $tType] :
% 0.28/0.48 ( ( ^ [Y5: set @ A,Z5: set @ A] : ( Y5 = Z5 ) )
% 0.28/0.48 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.28/0.48 & ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % set_eq_subset
% 0.28/0.48 thf(fact_177_Collect__mono__iff,axiom,
% 0.28/0.48 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.28/0.48 = ( ! [X3: A] :
% 0.28/0.48 ( ( P @ X3 )
% 0.28/0.48 => ( Q @ X3 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Collect_mono_iff
% 0.28/0.48 thf(fact_178_units__group,axiom,
% 0.28/0.48 group @ a @ product_unit @ ( units_of @ a @ ( ring_ext @ a @ b ) @ r ) ).
% 0.28/0.48
% 0.28/0.48 % units_group
% 0.28/0.48 thf(fact_179_add__additive__subgroups,axiom,
% 0.28/0.48 ! [H: set @ a,K2: set @ a] :
% 0.28/0.48 ( ( additive_subgroup @ a @ b @ H @ r )
% 0.28/0.48 => ( ( additive_subgroup @ a @ b @ K2 @ r )
% 0.28/0.48 => ( additive_subgroup @ a @ b @ ( set_add @ a @ b @ r @ H @ K2 ) @ r ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % add_additive_subgroups
% 0.28/0.48 thf(fact_180_a__rcos__assoc__lcos,axiom,
% 0.28/0.48 ! [H: set @ a,K2: set @ a,X: a] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.48 => ( ( ord_less_eq @ ( set @ a ) @ K2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.48 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.48 => ( ( set_add @ a @ b @ r @ ( a_r_coset @ a @ b @ r @ H @ X ) @ K2 )
% 0.28/0.48 = ( set_add @ a @ b @ r @ H @ ( a_l_coset @ a @ b @ r @ X @ K2 ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % a_rcos_assoc_lcos
% 0.28/0.48 thf(fact_181_a__r__coset__subset__G,axiom,
% 0.28/0.48 ! [H: set @ a,X: a] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.48 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.48 => ( ord_less_eq @ ( set @ a ) @ ( a_r_coset @ a @ b @ r @ H @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % a_r_coset_subset_G
% 0.28/0.48 thf(fact_182_a__setmult__rcos__assoc,axiom,
% 0.28/0.48 ! [H: set @ a,K2: set @ a,X: a] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.48 => ( ( ord_less_eq @ ( set @ a ) @ K2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.48 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.48 => ( ( set_add @ a @ b @ r @ H @ ( a_r_coset @ a @ b @ r @ K2 @ X ) )
% 0.28/0.48 = ( a_r_coset @ a @ b @ r @ ( set_add @ a @ b @ r @ H @ K2 ) @ X ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % a_setmult_rcos_assoc
% 0.28/0.48 thf(fact_183_units__of__units,axiom,
% 0.28/0.48 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.28/0.48 ( ( units @ A @ product_unit @ ( units_of @ A @ B @ G2 ) )
% 0.28/0.48 = ( units @ A @ B @ G2 ) ) ).
% 0.28/0.48
% 0.28/0.48 % units_of_units
% 0.28/0.48 thf(fact_184_a__coset__add__zero,axiom,
% 0.28/0.48 ! [M: set @ a] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ a ) @ M @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.48 => ( ( a_r_coset @ a @ b @ r @ M @ ( zero @ a @ b @ r ) )
% 0.28/0.48 = M ) ) ).
% 0.28/0.48
% 0.28/0.48 % a_coset_add_zero
% 0.28/0.48 thf(fact_185_units__of__carrier,axiom,
% 0.28/0.48 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.28/0.48 ( ( partial_carrier @ A @ ( monoid_ext @ A @ product_unit ) @ ( units_of @ A @ B @ G2 ) )
% 0.28/0.48 = ( units @ A @ B @ G2 ) ) ).
% 0.28/0.48
% 0.28/0.48 % units_of_carrier
% 0.28/0.48 thf(fact_186_monoid_Ounits__group,axiom,
% 0.28/0.48 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.28/0.48 ( ( monoid @ A @ B @ G2 )
% 0.28/0.48 => ( group @ A @ product_unit @ ( units_of @ A @ B @ G2 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % monoid.units_group
% 0.28/0.48 thf(fact_187_units__of__mult,axiom,
% 0.28/0.48 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.28/0.48 ( ( mult @ A @ product_unit @ ( units_of @ A @ B @ G2 ) )
% 0.28/0.48 = ( mult @ A @ B @ G2 ) ) ).
% 0.28/0.48
% 0.28/0.48 % units_of_mult
% 0.28/0.48 thf(fact_188_units__of__one,axiom,
% 0.28/0.48 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.28/0.48 ( ( one @ A @ product_unit @ ( units_of @ A @ B @ G2 ) )
% 0.28/0.48 = ( one @ A @ B @ G2 ) ) ).
% 0.28/0.48
% 0.28/0.48 % units_of_one
% 0.28/0.48 thf(fact_189_a__rcosetsI,axiom,
% 0.28/0.48 ! [H: set @ a,X: a] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.48 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.28/0.48 => ( member @ ( set @ a ) @ ( a_r_coset @ a @ b @ r @ H @ X ) @ ( a_RCOSETS @ a @ b @ r @ H ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % a_rcosetsI
% 0.28/0.48 thf(fact_190_one__zeroI,axiom,
% 0.28/0.48 ( ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 0.28/0.48 = ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) )
% 0.28/0.48 => ( ( one @ a @ ( ring_ext @ a @ b ) @ r )
% 0.28/0.48 = ( zero @ a @ b @ r ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % one_zeroI
% 0.28/0.48 thf(fact_191_one__zeroD,axiom,
% 0.28/0.48 ( ( ( one @ a @ ( ring_ext @ a @ b ) @ r )
% 0.28/0.48 = ( zero @ a @ b @ r ) )
% 0.28/0.48 => ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 0.28/0.48 = ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % one_zeroD
% 0.28/0.48 thf(fact_192_insert__absorb2,axiom,
% 0.28/0.48 ! [A: $tType,X: A,A4: set @ A] :
% 0.28/0.48 ( ( insert @ A @ X @ ( insert @ A @ X @ A4 ) )
% 0.28/0.48 = ( insert @ A @ X @ A4 ) ) ).
% 0.28/0.48
% 0.28/0.48 % insert_absorb2
% 0.28/0.48 thf(fact_193_insert__iff,axiom,
% 0.28/0.48 ! [A: $tType,A2: A,B2: A,A4: set @ A] :
% 0.28/0.48 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
% 0.28/0.48 = ( ( A2 = B2 )
% 0.28/0.48 | ( member @ A @ A2 @ A4 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % insert_iff
% 0.28/0.48 thf(fact_194_insertCI,axiom,
% 0.28/0.48 ! [A: $tType,A2: A,B5: set @ A,B2: A] :
% 0.28/0.48 ( ( ~ ( member @ A @ A2 @ B5 )
% 0.28/0.48 => ( A2 = B2 ) )
% 0.28/0.48 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % insertCI
% 0.28/0.48 thf(fact_195_carrier__one__not__zero,axiom,
% 0.28/0.48 ( ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 0.28/0.48 != ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) )
% 0.28/0.48 = ( ( one @ a @ ( ring_ext @ a @ b ) @ r )
% 0.28/0.48 != ( zero @ a @ b @ r ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % carrier_one_not_zero
% 0.28/0.48 thf(fact_196_carrier__one__zero,axiom,
% 0.28/0.48 ( ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 0.28/0.48 = ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) )
% 0.28/0.48 = ( ( one @ a @ ( ring_ext @ a @ b ) @ r )
% 0.28/0.48 = ( zero @ a @ b @ r ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % carrier_one_zero
% 0.28/0.48 thf(fact_197_singletonI,axiom,
% 0.28/0.48 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % singletonI
% 0.28/0.48 thf(fact_198_insert__subset,axiom,
% 0.28/0.48 ! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B5 )
% 0.28/0.48 = ( ( member @ A @ X @ B5 )
% 0.28/0.48 & ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % insert_subset
% 0.28/0.48 thf(fact_199_singleton__insert__inj__eq_H,axiom,
% 0.28/0.48 ! [A: $tType,A2: A,A4: set @ A,B2: A] :
% 0.28/0.48 ( ( ( insert @ A @ A2 @ A4 )
% 0.28/0.48 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.48 = ( ( A2 = B2 )
% 0.28/0.48 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % singleton_insert_inj_eq'
% 0.28/0.48 thf(fact_200_singleton__insert__inj__eq,axiom,
% 0.28/0.48 ! [A: $tType,B2: A,A2: A,A4: set @ A] :
% 0.28/0.48 ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
% 0.28/0.48 = ( insert @ A @ A2 @ A4 ) )
% 0.28/0.48 = ( ( A2 = B2 )
% 0.28/0.48 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % singleton_insert_inj_eq
% 0.28/0.48 thf(fact_201_subset__singleton__iff,axiom,
% 0.28/0.48 ! [A: $tType,X5: set @ A,A2: A] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.48 = ( ( X5
% 0.28/0.48 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.48 | ( X5
% 0.28/0.48 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_singleton_iff
% 0.28/0.48 thf(fact_202_subset__singletonD,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,X: A] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.48 => ( ( A4
% 0.28/0.48 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.48 | ( A4
% 0.28/0.48 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_singletonD
% 0.28/0.48 thf(fact_203_mk__disjoint__insert,axiom,
% 0.28/0.48 ! [A: $tType,A2: A,A4: set @ A] :
% 0.28/0.48 ( ( member @ A @ A2 @ A4 )
% 0.28/0.48 => ? [B7: set @ A] :
% 0.28/0.48 ( ( A4
% 0.28/0.48 = ( insert @ A @ A2 @ B7 ) )
% 0.28/0.48 & ~ ( member @ A @ A2 @ B7 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % mk_disjoint_insert
% 0.28/0.48 thf(fact_204_insert__commute,axiom,
% 0.28/0.48 ! [A: $tType,X: A,Y: A,A4: set @ A] :
% 0.28/0.48 ( ( insert @ A @ X @ ( insert @ A @ Y @ A4 ) )
% 0.28/0.48 = ( insert @ A @ Y @ ( insert @ A @ X @ A4 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % insert_commute
% 0.28/0.48 thf(fact_205_insert__eq__iff,axiom,
% 0.28/0.48 ! [A: $tType,A2: A,A4: set @ A,B2: A,B5: set @ A] :
% 0.28/0.48 ( ~ ( member @ A @ A2 @ A4 )
% 0.28/0.48 => ( ~ ( member @ A @ B2 @ B5 )
% 0.28/0.48 => ( ( ( insert @ A @ A2 @ A4 )
% 0.28/0.48 = ( insert @ A @ B2 @ B5 ) )
% 0.28/0.48 = ( ( ( A2 = B2 )
% 0.28/0.48 => ( A4 = B5 ) )
% 0.28/0.48 & ( ( A2 != B2 )
% 0.28/0.48 => ? [C5: set @ A] :
% 0.28/0.48 ( ( A4
% 0.28/0.48 = ( insert @ A @ B2 @ C5 ) )
% 0.28/0.48 & ~ ( member @ A @ B2 @ C5 )
% 0.28/0.48 & ( B5
% 0.28/0.48 = ( insert @ A @ A2 @ C5 ) )
% 0.28/0.48 & ~ ( member @ A @ A2 @ C5 ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % insert_eq_iff
% 0.28/0.48 thf(fact_206_insert__absorb,axiom,
% 0.28/0.48 ! [A: $tType,A2: A,A4: set @ A] :
% 0.28/0.48 ( ( member @ A @ A2 @ A4 )
% 0.28/0.48 => ( ( insert @ A @ A2 @ A4 )
% 0.28/0.48 = A4 ) ) ).
% 0.28/0.48
% 0.28/0.48 % insert_absorb
% 0.28/0.48 thf(fact_207_insert__ident,axiom,
% 0.28/0.48 ! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
% 0.28/0.48 ( ~ ( member @ A @ X @ A4 )
% 0.28/0.48 => ( ~ ( member @ A @ X @ B5 )
% 0.28/0.48 => ( ( ( insert @ A @ X @ A4 )
% 0.28/0.48 = ( insert @ A @ X @ B5 ) )
% 0.28/0.48 = ( A4 = B5 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % insert_ident
% 0.28/0.48 thf(fact_208_Set_Oset__insert,axiom,
% 0.28/0.48 ! [A: $tType,X: A,A4: set @ A] :
% 0.28/0.48 ( ( member @ A @ X @ A4 )
% 0.28/0.48 => ~ ! [B7: set @ A] :
% 0.28/0.48 ( ( A4
% 0.28/0.48 = ( insert @ A @ X @ B7 ) )
% 0.28/0.48 => ( member @ A @ X @ B7 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Set.set_insert
% 0.28/0.48 thf(fact_209_insertI2,axiom,
% 0.28/0.48 ! [A: $tType,A2: A,B5: set @ A,B2: A] :
% 0.28/0.48 ( ( member @ A @ A2 @ B5 )
% 0.28/0.48 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % insertI2
% 0.28/0.48 thf(fact_210_insertI1,axiom,
% 0.28/0.48 ! [A: $tType,A2: A,B5: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B5 ) ) ).
% 0.28/0.48
% 0.28/0.48 % insertI1
% 0.28/0.48 thf(fact_211_insertE,axiom,
% 0.28/0.48 ! [A: $tType,A2: A,B2: A,A4: set @ A] :
% 0.28/0.48 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
% 0.28/0.48 => ( ( A2 != B2 )
% 0.28/0.48 => ( member @ A @ A2 @ A4 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % insertE
% 0.28/0.48 thf(fact_212_singleton__inject,axiom,
% 0.28/0.48 ! [A: $tType,A2: A,B2: A] :
% 0.28/0.48 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.28/0.48 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.48 => ( A2 = B2 ) ) ).
% 0.28/0.48
% 0.28/0.48 % singleton_inject
% 0.28/0.48 thf(fact_213_insert__not__empty,axiom,
% 0.28/0.48 ! [A: $tType,A2: A,A4: set @ A] :
% 0.28/0.48 ( ( insert @ A @ A2 @ A4 )
% 0.28/0.48 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % insert_not_empty
% 0.28/0.48 thf(fact_214_doubleton__eq__iff,axiom,
% 0.28/0.48 ! [A: $tType,A2: A,B2: A,C3: A,D3: A] :
% 0.28/0.48 ( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.48 = ( insert @ A @ C3 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.28/0.48 = ( ( ( A2 = C3 )
% 0.28/0.48 & ( B2 = D3 ) )
% 0.28/0.48 | ( ( A2 = D3 )
% 0.28/0.48 & ( B2 = C3 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % doubleton_eq_iff
% 0.28/0.48 thf(fact_215_singleton__iff,axiom,
% 0.28/0.48 ! [A: $tType,B2: A,A2: A] :
% 0.28/0.48 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.48 = ( B2 = A2 ) ) ).
% 0.28/0.48
% 0.28/0.48 % singleton_iff
% 0.28/0.48 thf(fact_216_singletonD,axiom,
% 0.28/0.48 ! [A: $tType,B2: A,A2: A] :
% 0.28/0.48 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.48 => ( B2 = A2 ) ) ).
% 0.28/0.48
% 0.28/0.48 % singletonD
% 0.28/0.48 thf(fact_217_insert__mono,axiom,
% 0.28/0.48 ! [A: $tType,C4: set @ A,D4: set @ A,A2: A] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ A ) @ C4 @ D4 )
% 0.28/0.48 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C4 ) @ ( insert @ A @ A2 @ D4 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % insert_mono
% 0.28/0.48 thf(fact_218_subset__insert,axiom,
% 0.28/0.48 ! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
% 0.28/0.48 ( ~ ( member @ A @ X @ A4 )
% 0.28/0.48 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B5 ) )
% 0.28/0.48 = ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_insert
% 0.28/0.48 thf(fact_219_subset__insertI,axiom,
% 0.28/0.48 ! [A: $tType,B5: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert @ A @ A2 @ B5 ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_insertI
% 0.28/0.48 thf(fact_220_subset__insertI2,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,B5: set @ A,B2: A] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.28/0.48 => ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_insertI2
% 0.28/0.48 thf(fact_221_semiring_Ocarrier__one__not__zero,axiom,
% 0.28/0.48 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.28/0.48 ( ( semiring @ A @ B @ R3 )
% 0.28/0.48 => ( ( ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 )
% 0.28/0.48 != ( insert @ A @ ( zero @ A @ B @ R3 ) @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.48 = ( ( one @ A @ ( ring_ext @ A @ B ) @ R3 )
% 0.28/0.48 != ( zero @ A @ B @ R3 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % semiring.carrier_one_not_zero
% 0.28/0.48 thf(fact_222_semiring_Ocarrier__one__zero,axiom,
% 0.28/0.48 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.28/0.48 ( ( semiring @ A @ B @ R3 )
% 0.28/0.48 => ( ( ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 )
% 0.28/0.48 = ( insert @ A @ ( zero @ A @ B @ R3 ) @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.48 = ( ( one @ A @ ( ring_ext @ A @ B ) @ R3 )
% 0.28/0.48 = ( zero @ A @ B @ R3 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % semiring.carrier_one_zero
% 0.28/0.48 thf(fact_223_semiring_Oone__zeroI,axiom,
% 0.28/0.48 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.28/0.48 ( ( semiring @ A @ B @ R3 )
% 0.28/0.48 => ( ( ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 )
% 0.28/0.48 = ( insert @ A @ ( zero @ A @ B @ R3 ) @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.48 => ( ( one @ A @ ( ring_ext @ A @ B ) @ R3 )
% 0.28/0.48 = ( zero @ A @ B @ R3 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % semiring.one_zeroI
% 0.28/0.48 thf(fact_224_semiring_Oone__zeroD,axiom,
% 0.28/0.48 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.28/0.48 ( ( semiring @ A @ B @ R3 )
% 0.28/0.48 => ( ( ( one @ A @ ( ring_ext @ A @ B ) @ R3 )
% 0.28/0.48 = ( zero @ A @ B @ R3 ) )
% 0.28/0.48 => ( ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 )
% 0.28/0.48 = ( insert @ A @ ( zero @ A @ B @ R3 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % semiring.one_zeroD
% 0.28/0.48 thf(fact_225_field__intro2,axiom,
% 0.28/0.48 ( ( ( zero @ a @ b @ r )
% 0.28/0.48 != ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.28/0.48 => ( ! [X2: a] :
% 0.28/0.48 ( ( member @ a @ X2 @ ( minus_minus @ ( set @ a ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) @ ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) ) )
% 0.28/0.48 => ( member @ a @ X2 @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) ) )
% 0.28/0.48 => ( field @ a @ b @ r ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % field_intro2
% 0.28/0.48 thf(fact_226_cring__fieldI,axiom,
% 0.28/0.48 ( ( ( units @ a @ ( ring_ext @ a @ b ) @ r )
% 0.28/0.48 = ( minus_minus @ ( set @ a ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) @ ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) ) )
% 0.28/0.48 => ( field @ a @ b @ r ) ) ).
% 0.28/0.48
% 0.28/0.48 % cring_fieldI
% 0.28/0.48 thf(fact_227_DiffI,axiom,
% 0.28/0.48 ! [A: $tType,C3: A,A4: set @ A,B5: set @ A] :
% 0.28/0.48 ( ( member @ A @ C3 @ A4 )
% 0.28/0.48 => ( ~ ( member @ A @ C3 @ B5 )
% 0.28/0.48 => ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % DiffI
% 0.28/0.48 thf(fact_228_Diff__iff,axiom,
% 0.28/0.48 ! [A: $tType,C3: A,A4: set @ A,B5: set @ A] :
% 0.28/0.48 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) )
% 0.28/0.48 = ( ( member @ A @ C3 @ A4 )
% 0.28/0.48 & ~ ( member @ A @ C3 @ B5 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Diff_iff
% 0.28/0.48 thf(fact_229_Diff__idemp,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.28/0.48 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) @ B5 )
% 0.28/0.48 = ( minus_minus @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.28/0.48
% 0.28/0.48 % Diff_idemp
% 0.28/0.48 thf(fact_230_Diff__empty,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A] :
% 0.28/0.48 ( ( minus_minus @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 0.28/0.48 = A4 ) ).
% 0.28/0.48
% 0.28/0.48 % Diff_empty
% 0.28/0.48 thf(fact_231_empty__Diff,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A] :
% 0.28/0.48 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 )
% 0.28/0.48 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % empty_Diff
% 0.28/0.48 thf(fact_232_Diff__cancel,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A] :
% 0.28/0.48 ( ( minus_minus @ ( set @ A ) @ A4 @ A4 )
% 0.28/0.48 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Diff_cancel
% 0.28/0.48 thf(fact_233_insert__Diff1,axiom,
% 0.28/0.48 ! [A: $tType,X: A,B5: set @ A,A4: set @ A] :
% 0.28/0.48 ( ( member @ A @ X @ B5 )
% 0.28/0.48 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B5 )
% 0.28/0.48 = ( minus_minus @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % insert_Diff1
% 0.28/0.48 thf(fact_234_Diff__insert0,axiom,
% 0.28/0.48 ! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
% 0.28/0.48 ( ~ ( member @ A @ X @ A4 )
% 0.28/0.48 => ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B5 ) )
% 0.28/0.48 = ( minus_minus @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Diff_insert0
% 0.28/0.48 thf(fact_235_Diff__eq__empty__iff,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.28/0.48 ( ( ( minus_minus @ ( set @ A ) @ A4 @ B5 )
% 0.28/0.48 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.48 = ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.28/0.48
% 0.28/0.48 % Diff_eq_empty_iff
% 0.28/0.48 thf(fact_236_insert__Diff__single,axiom,
% 0.28/0.48 ! [A: $tType,A2: A,A4: set @ A] :
% 0.28/0.48 ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.28/0.48 = ( insert @ A @ A2 @ A4 ) ) ).
% 0.28/0.48
% 0.28/0.48 % insert_Diff_single
% 0.28/0.48 thf(fact_237_insert__Diff__if,axiom,
% 0.28/0.48 ! [A: $tType,X: A,B5: set @ A,A4: set @ A] :
% 0.28/0.48 ( ( ( member @ A @ X @ B5 )
% 0.28/0.48 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B5 )
% 0.28/0.48 = ( minus_minus @ ( set @ A ) @ A4 @ B5 ) ) )
% 0.28/0.48 & ( ~ ( member @ A @ X @ B5 )
% 0.28/0.48 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B5 )
% 0.28/0.48 = ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % insert_Diff_if
% 0.28/0.48 thf(fact_238_Diff__insert__absorb,axiom,
% 0.28/0.48 ! [A: $tType,X: A,A4: set @ A] :
% 0.28/0.48 ( ~ ( member @ A @ X @ A4 )
% 0.28/0.48 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.48 = A4 ) ) ).
% 0.28/0.48
% 0.28/0.48 % Diff_insert_absorb
% 0.28/0.48 thf(fact_239_Diff__insert2,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,A2: A,B5: set @ A] :
% 0.28/0.48 ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B5 ) )
% 0.28/0.48 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) ) ).
% 0.28/0.48
% 0.28/0.48 % Diff_insert2
% 0.28/0.48 thf(fact_240_insert__Diff,axiom,
% 0.28/0.48 ! [A: $tType,A2: A,A4: set @ A] :
% 0.28/0.48 ( ( member @ A @ A2 @ A4 )
% 0.28/0.48 => ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.28/0.48 = A4 ) ) ).
% 0.28/0.48
% 0.28/0.48 % insert_Diff
% 0.28/0.48 thf(fact_241_Diff__insert,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,A2: A,B5: set @ A] :
% 0.28/0.48 ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B5 ) )
% 0.28/0.48 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Diff_insert
% 0.28/0.48 thf(fact_242_subset__Diff__insert,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,B5: set @ A,X: A,C4: set @ A] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B5 @ ( insert @ A @ X @ C4 ) ) )
% 0.28/0.48 = ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B5 @ C4 ) )
% 0.28/0.48 & ~ ( member @ A @ X @ A4 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_Diff_insert
% 0.28/0.48 thf(fact_243_double__diff,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,B5: set @ A,C4: set @ A] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.28/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
% 0.28/0.48 => ( ( minus_minus @ ( set @ A ) @ B5 @ ( minus_minus @ ( set @ A ) @ C4 @ A4 ) )
% 0.28/0.48 = A4 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % double_diff
% 0.28/0.48 thf(fact_244_Diff__subset,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) @ A4 ) ).
% 0.28/0.48
% 0.28/0.48 % Diff_subset
% 0.28/0.48 thf(fact_245_Diff__mono,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,C4: set @ A,D4: set @ A,B5: set @ A] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ C4 )
% 0.28/0.48 => ( ( ord_less_eq @ ( set @ A ) @ D4 @ B5 )
% 0.28/0.48 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) @ ( minus_minus @ ( set @ A ) @ C4 @ D4 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Diff_mono
% 0.28/0.48 thf(fact_246_DiffE,axiom,
% 0.28/0.48 ! [A: $tType,C3: A,A4: set @ A,B5: set @ A] :
% 0.28/0.48 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) )
% 0.28/0.48 => ~ ( ( member @ A @ C3 @ A4 )
% 0.28/0.48 => ( member @ A @ C3 @ B5 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % DiffE
% 0.28/0.48 thf(fact_247_DiffD1,axiom,
% 0.28/0.48 ! [A: $tType,C3: A,A4: set @ A,B5: set @ A] :
% 0.28/0.48 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) )
% 0.28/0.48 => ( member @ A @ C3 @ A4 ) ) ).
% 0.28/0.48
% 0.28/0.48 % DiffD1
% 0.28/0.48 thf(fact_248_DiffD2,axiom,
% 0.28/0.48 ! [A: $tType,C3: A,A4: set @ A,B5: set @ A] :
% 0.28/0.48 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) )
% 0.28/0.48 => ~ ( member @ A @ C3 @ B5 ) ) ).
% 0.28/0.48
% 0.28/0.48 % DiffD2
% 0.28/0.48 thf(fact_249_subset__insert__iff,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,X: A,B5: set @ A] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B5 ) )
% 0.28/0.48 = ( ( ( member @ A @ X @ A4 )
% 0.28/0.48 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) )
% 0.28/0.48 & ( ~ ( member @ A @ X @ A4 )
% 0.28/0.48 => ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_insert_iff
% 0.28/0.48 thf(fact_250_Diff__single__insert,axiom,
% 0.28/0.48 ! [A: $tType,A4: set @ A,X: A,B5: set @ A] :
% 0.28/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 )
% 0.28/0.48 => ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B5 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Diff_single_insert
% 0.28/0.48 thf(fact_251_field_Ofield__Units,axiom,
% 0.28/0.48 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.28/0.48 ( ( field @ A @ B @ R3 )
% 0.28/0.48 => ( ( units @ A @ ( ring_ext @ A @ B ) @ R3 )
% 0.28/0.48 = ( minus_minus @ ( set @ A ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) @ ( insert @ A @ ( zero @ A @ B @ R3 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % field.field_Units
% 0.28/0.48 thf(fact_252_rcosets__subset__PowG,axiom,
% 0.28/0.48 ! [H: set @ a] :
% 0.28/0.48 ( ( additive_subgroup @ a @ b @ H @ r )
% 0.28/0.48 => ( ord_less_eq @ ( set @ ( set @ a ) ) @ ( a_RCOSETS @ a @ b @ r @ H ) @ ( pow @ a @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % rcosets_subset_PowG
% 0.28/0.48 thf(fact_253_abelian__monoid_Oset__add__closed,axiom,
% 0.28/0.48 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A4: set @ A,B5: set @ A] :
% 0.28/0.48 ( ( abelian_monoid @ A @ B @ G2 )
% 0.28/0.48 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G2 ) )
% 0.28/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G2 ) )
% 0.28/0.48 => ( ord_less_eq @ ( set @ A ) @ ( set_add @ A @ B @ G2 @ A4 @ B5 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G2 ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % abelian_monoid.set_add_closed
% 0.28/0.48 thf(fact_254_Pow__singleton__iff,axiom,
% 0.28/0.48 ! [A: $tType,X5: set @ A,Y6: set @ A] :
% 0.28/0.48 ( ( ( pow @ A @ X5 )
% 0.28/0.48 = ( insert @ ( set @ A ) @ Y6 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
% 0.28/0.48 = ( ( X5
% 0.28/0.48 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.48 & ( Y6
% 0.28/0.48 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Pow_singleton_iff
% 0.28/0.48 thf(fact_255_Pow__empty,axiom,
% 0.28/0.48 ! [A: $tType] :
% 0.28/0.48 ( ( pow @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.28/0.48 = ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Pow_empty
% 0.28/0.48
% 0.28/0.48 % Conjectures (1)
% 0.28/0.48 thf(conj_0,conjecture,
% 0.28/0.48 ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) )
% 0.28/0.48 = ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 %------------------------------------------------------------------------------
% 0.28/0.48 ------------------------
% 0.31/1.50 ---- Embedded in THF ---
% 0.31/1.50 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.31/1.50 %%% Generated on Tue Feb 24 08:25:48 EST 2026
% 0.31/1.50 %%% using '$$dhol' embedding, version 1.3.0.
% 0.31/1.50 %%% Logic specification used:
% 0.31/1.50 %%% thf(spec, logic, $$dhol).
% 0.31/1.50
% 0.31/1.50 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.ZsShBFnucq/DTF2DTF27309.p
% 0.31/1.50 thf(ty_t_Congruence_Opartial__object_Opartial__object__ext, type, partia1265057091ct_ext: ($tType > ($tType > $tType))).
% 0.31/1.50 thf(ty_t_Congruence_Oeq__object_Oeq__object__ext, type, eq_eq_object_ext: ($tType > ($tType > $tType))).
% 0.31/1.50 thf(ty_t_Group_Omonoid_Omonoid__ext, type, monoid_ext: ($tType > ($tType > $tType))).
% 0.31/1.50 thf(ty_t_Ring_Oring_Oring__ext, type, ring_ext: ($tType > ($tType > $tType))).
% 0.31/1.50 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 0.31/1.50 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 0.31/1.50 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 0.31/1.50 thf(ty_t_Nat_Onat, type, nat: $tType).
% 0.31/1.50 thf(ty_t_Int_Oint, type, int: $tType).
% 0.31/1.50 thf(ty_tf_b, type, b: $tType).
% 0.31/1.50 thf(ty_tf_a, type, a: $tType).
% 0.31/1.50 thf(sy_c_AbelCoset_OA__RCOSETS, type, a_RCOSETS: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > ((set @ A) > (set @ (set @ A))))))).
% 0.31/1.50 thf(sy_c_AbelCoset_Oa__l__coset, type, a_l_coset: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > (A > ((set @ A) > (set @ A))))))).
% 0.31/1.50 thf(sy_c_AbelCoset_Oa__r__coset, type, a_r_coset: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > ((set @ A) > (A > (set @ A))))))).
% 0.31/1.50 thf(sy_c_AbelCoset_Oadditive__subgroup, type, additive_subgroup: (!> [A:$tType,B:$tType]: (((set @ A) > (((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > $o))))).
% 0.31/1.50 thf(sy_c_AbelCoset_Oset__add, type, set_add: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > ((set @ A) > ((set @ A) > (set @ A))))))).
% 0.31/1.50 thf(sy_c_Congruence_Oequivalence, type, equivalence: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((eq_eq_object_ext @ A) @ B)) > $o)))).
% 0.31/1.50 thf(sy_c_Congruence_Opartial__object_Ocarrier, type, partial_carrier: (!> [A:$tType,Z:$tType]: ((((partia1265057091ct_ext @ A) @ Z) > (set @ A))))).
% 0.31/1.50 thf(sy_c_Group_OUnits, type, units: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)) > (set @ A))))).
% 0.31/1.50 thf(sy_c_Group_Ogroup, type, group: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)) > $o)))).
% 0.31/1.50 thf(sy_c_Group_Omonoid, type, monoid: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)) > $o)))).
% 0.31/1.50 thf(sy_c_Group_Omonoid_Omult, type, mult: (!> [A:$tType,Z:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ Z)) > (A > (A > A)))))).
% 0.31/1.50 thf(sy_c_Group_Omonoid_Oone, type, one: (!> [A:$tType,Z:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ Z)) > A)))).
% 0.31/1.50 thf(sy_c_Group_Ounits__of, type, units_of: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)) > ((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ product_unit)))))).
% 0.31/1.50 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 0.31/1.50 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Oeq__class__of__rng__of__frac, type, locali1641774487f_frac: (!> [A:$tType,B:$tType,C:$tType]: ((((partia1265057091ct_ext @ ((product_prod @ A) @ B)) @ ((eq_eq_object_ext @ ((product_prod @ A) @ B)) @ C)) > (A > (B > (set @ ((product_prod @ A) @ B)))))))).
% 0.31/1.50 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Oeq__obj__rng__of__frac, type, locali1648044335f_frac: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > ((set @ A) > $o))))).
% 0.31/1.50 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Oeq__obj__rng__of__frac_Orec__monoid__rng__of__frac, type, locali1768110497f_frac: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > ((set @ A) > ((partia1265057091ct_ext @ (set @ ((product_prod @ A) @ A))) @ ((monoid_ext @ (set @ ((product_prod @ A) @ A))) @ product_unit))))))).
% 0.31/1.50 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Omult__submonoid__of__crng, type, locali7914339f_crng: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > ((set @ A) > $o))))).
% 0.31/1.50 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Omult__submonoid__of__rng, type, locali1402578320of_rng: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > ((set @ A) > $o))))).
% 0.31/1.50 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Oset__eq__class__of__rng__of__frac, type, locali990921413f_frac: (!> [A:$tType,B:$tType,C:$tType]: ((((partia1265057091ct_ext @ ((product_prod @ A) @ B)) @ ((eq_eq_object_ext @ ((product_prod @ A) @ B)) @ C)) > (set @ (set @ ((product_prod @ A) @ B))))))).
% 0.31/1.50 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Osubmonoid, type, locali1441642228monoid: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)) > ((set @ A) > $o))))).
% 0.31/1.50 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 0.31/1.50 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 0.31/1.50 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 0.31/1.50 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 0.31/1.50 thf(sy_c_Ring_Oabelian__monoid, type, abelian_monoid: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > $o)))).
% 0.31/1.50 thf(sy_c_Ring_Oadd__pow, type, add_pow: (!> [A:$tType,C:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ C))) > (B > (A > A)))))).
% 0.31/1.50 thf(sy_c_Ring_Ofield, type, field: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > $o)))).
% 0.31/1.50 thf(sy_c_Ring_Oring, type, ring: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > $o)))).
% 0.31/1.50 thf(sy_c_Ring_Oring_Ozero, type, zero: (!> [A:$tType,Z:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ Z))) > A)))).
% 0.31/1.50 thf(sy_c_Ring_Osemiring, type, semiring: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > $o)))).
% 0.31/1.50 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 0.31/1.50 thf(sy_c_Set_OPow, type, pow: (!> [A:$tType]: (((set @ A) > (set @ (set @ A)))))).
% 0.31/1.50 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 0.31/1.50 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 0.31/1.50 thf(sy_v_R, type, r: ((partia1265057091ct_ext @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b)))).
% 0.31/1.50 thf(sy_v_S, type, s: (set @ a)).
% 0.31/1.50 thf(sy_v_r, type, r2: a).
% 0.31/1.50 thf(sy_v_r_H, type, r3: a).
% 0.31/1.50 thf(sy_v_rel, type, rel: ((partia1265057091ct_ext @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit))).
% 0.31/1.50 thf(sy_v_s, type, s2: a).
% 0.31/1.50 thf(sy_v_s_H, type, s3: a).
% 0.31/1.50 thf(fact_0_f3, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r2) @ r3) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r3) @ r2))).
% 0.31/1.50 thf(fact_1_f4, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ s2))).
% 0.31/1.50 thf(fact_2_eq__obj__rng__of__frac__axioms, axiom, ((((locali1648044335f_frac @ a) @ b) @ r) @ s)).
% 0.31/1.50 thf(fact_3_f1, axiom, ((((((mult @ (set @ ((product_prod @ a) @ a))) @ product_unit) @ ((((locali1768110497f_frac @ a) @ b) @ r) @ s)) @ ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ r2) @ s2)) @ ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ r3) @ s3)) = ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r2) @ r3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)))).
% 0.31/1.50 thf(fact_4_f2, axiom, ((((((mult @ (set @ ((product_prod @ a) @ a))) @ product_unit) @ ((((locali1768110497f_frac @ a) @ b) @ r) @ s)) @ ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ r3) @ s3)) @ ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ r2) @ s2)) = ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r3) @ r2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ s2)))).
% 0.31/1.50 thf(fact_5_mult__submonoid__of__crng__axioms, axiom, ((((locali7914339f_crng @ a) @ b) @ r) @ s)).
% 0.31/1.50 thf(fact_6_mult__submonoid__of__rng__axioms, axiom, ((((locali1402578320of_rng @ a) @ b) @ r) @ s)).
% 0.31/1.50 thf(fact_7_eq__obj__rng__of__frac_Orec__monoid__rng__of__frac_Ocong, axiom, (! [B:$tType,A:$tType]: ((((locali1768110497f_frac @ A) @ B) = ((locali1768110497f_frac @ A) @ B))))).
% 0.31/1.50 thf(fact_8_submonoid__axioms, axiom, ((((locali1441642228monoid @ a) @ ((ring_ext @ a) @ b)) @ r) @ s)).
% 0.31/1.50 thf(fact_9_m__closed, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ s) => ((((member @ a) @ Y) @ s) => (((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Y)) @ s)))))).
% 0.31/1.50 thf(fact_10_monoid__rng__of__frac, axiom, (((monoid @ (set @ ((product_prod @ a) @ a))) @ product_unit) @ ((((locali1768110497f_frac @ a) @ b) @ r) @ s))).
% 0.31/1.50 thf(fact_11_local_Osemiring__axioms, axiom, (((semiring @ a) @ b) @ r)).
% 0.31/1.50 thf(fact_12_one__closed, axiom, (((member @ a) @ (((one @ a) @ ((ring_ext @ a) @ b)) @ r)) @ s)).
% 0.31/1.50 thf(fact_13_equiv__obj__rng__of__frac, axiom, (((equivalence @ ((product_prod @ a) @ a)) @ product_unit) @ rel)).
% 0.31/1.50 thf(fact_14_assms_I2_J, axiom, (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ r3) @ s3)) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel))).
% 0.31/1.50 thf(fact_15_assms_I1_J, axiom, (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ r2) @ s2)) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel))).
% 0.31/1.50 thf(fact_16_assoc__mult__rng__of__frac, axiom, (! [R:a,S:a,T2:a,U:a,V:a,W:a]: (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ R) @ S)) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel)) => ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ T2) @ U)) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel)) => ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ V) @ W)) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel)) => ((((((mult @ (set @ ((product_prod @ a) @ a))) @ product_unit) @ ((((locali1768110497f_frac @ a) @ b) @ r) @ s)) @ (((((mult @ (set @ ((product_prod @ a) @ a))) @ product_unit) @ ((((locali1768110497f_frac @ a) @ b) @ r) @ s)) @ ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ R) @ S)) @ ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ T2) @ U))) @ ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ V) @ W)) = (((((mult @ (set @ ((product_prod @ a) @ a))) @ product_unit) @ ((((locali1768110497f_frac @ a) @ b) @ r) @ s)) @ ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ R) @ S)) @ (((((mult @ (set @ ((product_prod @ a) @ a))) @ product_unit) @ ((((locali1768110497f_frac @ a) @ b) @ r) @ s)) @ ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ T2) @ U)) @ ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ V) @ W)))))))))).
% 0.31/1.50 thf(fact_17_abelian__monoid__axioms, axiom, (((abelian_monoid @ a) @ b) @ r)).
% 0.31/1.50 thf(fact_18_mult__rng__of__frac__fundamental__lemma, axiom, (! [R:a,S:a,R2:a,S2:a]: (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ R) @ S)) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel)) => ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ R2) @ S2)) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel)) => ((((((mult @ (set @ ((product_prod @ a) @ a))) @ product_unit) @ ((((locali1768110497f_frac @ a) @ b) @ r) @ s)) @ ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ R) @ S)) @ ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ R2) @ S2)) = ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ R) @ R2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ S) @ S2)))))))).
% 0.31/1.50 thf(fact_19_local_Oring__axioms, axiom, (((ring @ a) @ b) @ r)).
% 0.31/1.50 thf(fact_20_closed__mult__rng__of__frac, axiom, (! [R:a,S:a,T2:a,U:a]: (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ R) @ S)) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel)) => ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ T2) @ U)) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel)) => (((member @ (set @ ((product_prod @ a) @ a))) @ (((((mult @ (set @ ((product_prod @ a) @ a))) @ product_unit) @ ((((locali1768110497f_frac @ a) @ b) @ r) @ s)) @ ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ R) @ S)) @ ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ T2) @ U))) @ ((((locali990921413f_frac @ a) @ a) @ product_unit) @ rel))))))).
% 0.31/1.50 thf(fact_21_submonoid_Oaxioms_I1_J, axiom, (! [B:$tType,A:$tType,M:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),S3:(set @ A)]: ((((((locali1441642228monoid @ A) @ B) @ M) @ S3) => (((monoid @ A) @ B) @ M))))).
% 0.31/1.50 thf(fact_22_eq__obj__rng__of__frac_Oaxioms_I2_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),S3:(set @ A)]: ((((((locali1648044335f_frac @ A) @ B) @ R3) @ S3) => ((((locali7914339f_crng @ A) @ B) @ R3) @ S3))))).
% 0.31/1.50 thf(fact_23_mult__submonoid__of__rng_Oaxioms_I2_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),S3:(set @ A)]: ((((((locali1402578320of_rng @ A) @ B) @ R3) @ S3) => ((((locali1441642228monoid @ A) @ ((ring_ext @ A) @ B)) @ R3) @ S3))))).
% 0.31/1.50 thf(fact_24_mult__submonoid__of__rng_Oaxioms_I1_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),S3:(set @ A)]: ((((((locali1402578320of_rng @ A) @ B) @ R3) @ S3) => (((ring @ A) @ B) @ R3))))).
% 0.31/1.50 thf(fact_25_mult__submonoid__of__crng_Oaxioms_I2_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),S3:(set @ A)]: ((((((locali7914339f_crng @ A) @ B) @ R3) @ S3) => ((((locali1402578320of_rng @ A) @ B) @ R3) @ S3))))).
% 0.31/1.50 thf(fact_26_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Oone__closed, axiom, (! [B:$tType,A:$tType,M:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),S3:(set @ A)]: ((((((locali1441642228monoid @ A) @ B) @ M) @ S3) => (((member @ A) @ (((one @ A) @ B) @ M)) @ S3))))).
% 0.31/1.50 thf(fact_27_mult__submonoid__of__rng__def, axiom, (! [B:$tType,A:$tType]: ((((locali1402578320of_rng @ A) @ B) = (^ [R4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),S4:(set @ A)]: (((((ring @ A) @ B) @ R4) & ((((locali1441642228monoid @ A) @ ((ring_ext @ A) @ B)) @ R4) @ S4)))))))).
% 0.31/1.50 thf(fact_28_mult__submonoid__of__rng_Ointro, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),S3:(set @ A)]: (((((ring @ A) @ B) @ R3) => (((((locali1441642228monoid @ A) @ ((ring_ext @ A) @ B)) @ R3) @ S3) => ((((locali1402578320of_rng @ A) @ B) @ R3) @ S3)))))).
% 0.31/1.50 thf(fact_29_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Ois__submonoid, axiom, (! [B:$tType,A:$tType,M:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),S3:(set @ A)]: ((((((locali1441642228monoid @ A) @ B) @ M) @ S3) => ((((locali1441642228monoid @ A) @ B) @ M) @ S3))))).
% 0.31/1.50 thf(fact_30_eq__obj__rng__of__frac_Omonoid__rng__of__frac, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),S3:(set @ A)]: ((((((locali1648044335f_frac @ A) @ B) @ R3) @ S3) => (((monoid @ (set @ ((product_prod @ A) @ A))) @ product_unit) @ ((((locali1768110497f_frac @ A) @ B) @ R3) @ S3)))))).
% 0.31/1.50 thf(fact_31_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Om__closed, axiom, (! [B:$tType,A:$tType,M:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),S3:(set @ A),X:A,Y:A]: ((((((locali1441642228monoid @ A) @ B) @ M) @ S3) => ((((member @ A) @ X) @ S3) => ((((member @ A) @ Y) @ S3) => (((member @ A) @ (((((mult @ A) @ B) @ M) @ X) @ Y)) @ S3))))))).
% 0.31/1.50 thf(fact_32_right__unit__mult__rng__of__frac, axiom, (! [R:a,S:a]: (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ R) @ S)) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel)) => ((((((mult @ (set @ ((product_prod @ a) @ a))) @ product_unit) @ ((((locali1768110497f_frac @ a) @ b) @ r) @ s)) @ ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ R) @ S)) @ (((one @ (set @ ((product_prod @ a) @ a))) @ product_unit) @ ((((locali1768110497f_frac @ a) @ b) @ r) @ s))) = ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ R) @ S)))))).
% 0.31/1.50 thf(fact_33_left__unit__mult__rng__of__frac, axiom, (! [R:a,S:a]: (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ R) @ S)) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel)) => ((((((mult @ (set @ ((product_prod @ a) @ a))) @ product_unit) @ ((((locali1768110497f_frac @ a) @ b) @ r) @ s)) @ (((one @ (set @ ((product_prod @ a) @ a))) @ product_unit) @ ((((locali1768110497f_frac @ a) @ b) @ r) @ s))) @ ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ R) @ S)) = ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ R) @ S)))))).
% 0.31/1.50 thf(fact_34_zero__in__mult__submonoid, axiom, (! [R:a,S:a,R2:a,S2:a]: (((((member @ a) @ (((zero @ a) @ b) @ r)) @ s) => ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ R) @ S)) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel)) => ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ R2) @ S2)) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel)) => (((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ R) @ S) = ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ R2) @ S2)))))))).
% 0.31/1.50 thf(fact_35_non__empty__class, axiom, (! [R:a,S:a]: (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ R) @ S)) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel)) => (((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ R) @ S) != (bot_bot @ (set @ ((product_prod @ a) @ a)))))))).
% 0.31/1.50 thf(fact_36_one__unique, axiom, (! [U:a]: (((((member @ a) @ U) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((! [X2:a]: (((((member @ a) @ X2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ U) @ X2) = X2)))) => (U = (((one @ a) @ ((ring_ext @ a) @ b)) @ r))))))).
% 0.31/1.50 thf(fact_37_inv__unique, axiom, (! [Y:a,X:a,Y2:a]: ((((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Y) @ X) = (((one @ a) @ ((ring_ext @ a) @ b)) @ r)) => (((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Y2) = (((one @ a) @ ((ring_ext @ a) @ b)) @ r)) => ((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (Y = Y2))))))))).
% 0.31/1.50 thf(fact_38_Units__inv__comm, axiom, (! [X:a,Y:a]: ((((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Y) = (((one @ a) @ ((ring_ext @ a) @ b)) @ r)) => ((((member @ a) @ X) @ (((units @ a) @ ((ring_ext @ a) @ b)) @ r)) => ((((member @ a) @ Y) @ (((units @ a) @ ((ring_ext @ a) @ b)) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Y) @ X) = (((one @ a) @ ((ring_ext @ a) @ b)) @ r)))))))).
% 0.31/1.50 thf(fact_39_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y22:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y22)) = ((X1 = Y1) & (X22 = Y22)))))).
% 0.31/1.50 thf(fact_40_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A3:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A3) @ B3)) = ((A2 = A3) & (B2 = B3)))))).
% 0.31/1.50 thf(fact_41_semiring_Oaxioms_I1_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((semiring @ A) @ B) @ R3) => (((abelian_monoid @ A) @ B) @ R3))))).
% 0.31/1.50 thf(fact_42_monoid__axioms, axiom, (((monoid @ a) @ ((ring_ext @ a) @ b)) @ r)).
% 0.31/1.50 thf(fact_43_m__assoc, axiom, (! [X:a,Y:a,Z2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Y)) @ Z2) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Y) @ Z2))))))))).
% 0.31/1.50 thf(fact_44_m__comm, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Y) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Y) @ X))))))).
% 0.31/1.50 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 0.31/1.50 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A4)))) = A4)))).
% 0.31/1.50 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) = (Q @ X2)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 0.31/1.50 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X2:A]: (((F @ X2) = (G @ X2)))) => (F = G))))).
% 0.31/1.50 thf(fact_49_m__lcomm, axiom, (! [X:a,Y:a,Z2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Y) @ Z2)) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Y) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Z2))))))))).
% 0.31/1.50 thf(fact_50_Units__closed, axiom, (! [X:a]: (((((member @ a) @ X) @ (((units @ a) @ ((ring_ext @ a) @ b)) @ r)) => (((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)))))).
% 0.31/1.50 thf(fact_51_Units__l__inv__ex, axiom, (! [X:a]: (((((member @ a) @ X) @ (((units @ a) @ ((ring_ext @ a) @ b)) @ r)) => (? [X2:a]: (((((member @ a) @ X2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) & ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X2) @ X) = (((one @ a) @ ((ring_ext @ a) @ b)) @ r))))))))).
% 0.31/1.50 thf(fact_52_Units__r__inv__ex, axiom, (! [X:a]: (((((member @ a) @ X) @ (((units @ a) @ ((ring_ext @ a) @ b)) @ r)) => (? [X2:a]: (((((member @ a) @ X2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) & ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ X2) = (((one @ a) @ ((ring_ext @ a) @ b)) @ r))))))))).
% 0.31/1.50 thf(fact_53_zero__closed, axiom, (((member @ a) @ (((zero @ a) @ b) @ r)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 0.31/1.50 thf(fact_54_semiring__simprules_I3_J, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Y)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))))))).
% 0.31/1.50 thf(fact_55_semiring__simprules_I4_J, axiom, (((member @ a) @ (((one @ a) @ ((ring_ext @ a) @ b)) @ r)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 0.31/1.50 thf(fact_56_Units__m__closed, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((units @ a) @ ((ring_ext @ a) @ b)) @ r)) => ((((member @ a) @ Y) @ (((units @ a) @ ((ring_ext @ a) @ b)) @ r)) => (((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Y)) @ (((units @ a) @ ((ring_ext @ a) @ b)) @ r))))))).
% 0.31/1.50 thf(fact_57_Units__one__closed, axiom, (((member @ a) @ (((one @ a) @ ((ring_ext @ a) @ b)) @ r)) @ (((units @ a) @ ((ring_ext @ a) @ b)) @ r))).
% 0.31/1.50 thf(fact_58_l__null, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((zero @ a) @ b) @ r)) @ X) = (((zero @ a) @ b) @ r)))))).
% 0.31/1.50 thf(fact_59_r__null, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ (((zero @ a) @ b) @ r)) = (((zero @ a) @ b) @ r)))))).
% 0.31/1.50 thf(fact_60_l__one, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((one @ a) @ ((ring_ext @ a) @ b)) @ r)) @ X) = X))))).
% 0.31/1.50 thf(fact_61_r__one, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ (((one @ a) @ ((ring_ext @ a) @ b)) @ r)) = X))))).
% 0.31/1.50 thf(fact_62_Units__l__cancel, axiom, (! [X:a,Y:a,Z2:a]: (((((member @ a) @ X) @ (((units @ a) @ ((ring_ext @ a) @ b)) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Y) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Z2)) = (Y = Z2)))))))).
% 0.31/1.50 thf(fact_63_abelian__monoidE_I2_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((abelian_monoid @ A) @ B) @ R3) => (((member @ A) @ (((zero @ A) @ B) @ R3)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)))))).
% 0.31/1.50 thf(fact_64_ring_Oring__simprules_I2_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((ring @ A) @ B) @ R3) => (((member @ A) @ (((zero @ A) @ B) @ R3)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)))))).
% 0.31/1.50 thf(fact_65_semiring_Osemiring__simprules_I2_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((semiring @ A) @ B) @ R3) => (((member @ A) @ (((zero @ A) @ B) @ R3)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)))))).
% 0.31/1.50 thf(fact_66_abelian__monoid_Ozero__closed, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((abelian_monoid @ A) @ B) @ G2) => (((member @ A) @ (((zero @ A) @ B) @ G2)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G2)))))).
% 0.31/1.50 thf(fact_67_ring_Ois__monoid, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((ring @ A) @ B) @ R3) => (((monoid @ A) @ ((ring_ext @ A) @ B)) @ R3))))).
% 0.31/1.50 thf(fact_68_semiring_Oaxioms_I2_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((semiring @ A) @ B) @ R3) => (((monoid @ A) @ ((ring_ext @ A) @ B)) @ R3))))).
% 0.31/1.50 thf(fact_69_ring_Oring__simprules_I24_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((ring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ (((zero @ A) @ B) @ R3)) @ X) = (((zero @ A) @ B) @ R3))))))).
% 0.31/1.50 thf(fact_70_ring_Oring__simprules_I25_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((ring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ X) @ (((zero @ A) @ B) @ R3)) = (((zero @ A) @ B) @ R3))))))).
% 0.31/1.50 thf(fact_71_semiring_Or__null, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((semiring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ X) @ (((zero @ A) @ B) @ R3)) = (((zero @ A) @ B) @ R3))))))).
% 0.31/1.50 thf(fact_72_semiring_Ol__null, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((semiring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ (((zero @ A) @ B) @ R3)) @ X) = (((zero @ A) @ B) @ R3))))))).
% 0.31/1.50 thf(fact_73_ring_Oring__simprules_I5_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A]: (((((ring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => (((member @ A) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ X) @ Y)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)))))))).
% 0.31/1.50 thf(fact_74_ring_Oring__simprules_I11_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A,Z2:A]: (((((ring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ Z2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ X) @ Y)) @ Z2) = (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ X) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ Y) @ Z2)))))))))).
% 0.31/1.50 thf(fact_75_semiring_Osemiring__simprules_I3_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A]: (((((semiring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => (((member @ A) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ X) @ Y)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)))))))).
% 0.31/1.50 thf(fact_76_semiring_Osemiring__simprules_I8_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A,Z2:A]: (((((semiring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ Z2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ X) @ Y)) @ Z2) = (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ X) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ Y) @ Z2)))))))))).
% 0.31/1.50 thf(fact_77_ring_Oring__simprules_I6_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((ring @ A) @ B) @ R3) => (((member @ A) @ (((one @ A) @ ((ring_ext @ A) @ B)) @ R3)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)))))).
% 0.31/1.50 thf(fact_78_semiring_Osemiring__simprules_I4_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((semiring @ A) @ B) @ R3) => (((member @ A) @ (((one @ A) @ ((ring_ext @ A) @ B)) @ R3)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)))))).
% 0.31/1.50 thf(fact_79_ring_Oring__simprules_I12_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((ring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ (((one @ A) @ ((ring_ext @ A) @ B)) @ R3)) @ X) = X)))))).
% 0.31/1.50 thf(fact_80_semiring_Osemiring__simprules_I9_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((semiring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ (((one @ A) @ ((ring_ext @ A) @ B)) @ R3)) @ X) = X)))))).
% 0.31/1.50 thf(fact_81_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A5:A,B4:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B4)))) => (P @ Prod))))).
% 0.31/1.50 thf(fact_82_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A5:A,B4:B]: ((Y != ((((product_Pair @ A) @ B) @ A5) @ B4))))))))).
% 0.31/1.50 thf(fact_83_prod__induct7, axiom, (! [G3:$tType,F2:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3))))))]: (((! [A5:A,B4:B,C2:C,D2:D,E2:E,F3:F2,G4:G3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F2) @ G3)) @ E2) @ ((((product_Pair @ F2) @ G3) @ F3) @ G4))))))))) => (P @ X))))).
% 0.31/1.50 thf(fact_84_prod__induct6, axiom, (! [F2:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2)))))]: (((! [A5:A,B4:B,C2:C,D2:D,E2:E,F3:F2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F2)) @ D2) @ ((((product_Pair @ E) @ F2) @ E2) @ F3)))))))) => (P @ X))))).
% 0.31/1.50 thf(fact_85_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A5:A,B4:B,C2:C,D2:D,E2:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P @ X))))).
% 0.31/1.50 thf(fact_86_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A5:A,B4:B,C2:C,D2:D]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C2) @ D2)))))) => (P @ X))))).
% 0.31/1.50 thf(fact_87_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A5:A,B4:B,C2:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B4) @ C2))))) => (P @ X))))).
% 0.31/1.50 thf(fact_88_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F2:$tType,G3:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3))))))]: ((~ ((! [A5:A,B4:B,C2:C,D2:D,E2:E,F3:F2,G4:G3]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G3))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F2) @ G3)) @ E2) @ ((((product_Pair @ F2) @ G3) @ F3) @ G4)))))))))))))).
% 0.31/1.50 thf(fact_89_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F2:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2)))))]: ((~ ((! [A5:A,B4:B,C2:C,D2:D,E2:E,F3:F2]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F2)) @ D2) @ ((((product_Pair @ E) @ F2) @ E2) @ F3))))))))))))).
% 0.31/1.50 thf(fact_90_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A5:A,B4:B,C2:C,D2:D,E2:E]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 0.31/1.50 thf(fact_91_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A5:A,B4:B,C2:C,D2:D]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C2) @ D2))))))))))).
% 0.31/1.50 thf(fact_92_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A5:A,B4:B,C2:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B4) @ C2)))))))))).
% 0.31/1.50 thf(fact_93_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A3:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A3) @ B3)) => (~ (((A2 = A3) => (B2 != B3)))))))).
% 0.31/1.50 thf(fact_94_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A5:A,B4:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B4)))) => (P @ P2))))).
% 0.31/1.50 thf(fact_95_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X2:A,Y3:B]: ((P2 = ((((product_Pair @ A) @ B) @ X2) @ Y3))))))).
% 0.31/1.50 thf(fact_96_ring_Ois__ring, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((ring @ A) @ B) @ R3) => (((ring @ A) @ B) @ R3))))).
% 0.31/1.50 thf(fact_97_cring__fieldI2, axiom, (((((zero @ a) @ b) @ r) != (((one @ a) @ ((ring_ext @ a) @ b)) @ r)) => ((! [A5:a]: (((((member @ a) @ A5) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((A5 != (((zero @ a) @ b) @ r)) => (? [X4:a]: (((((member @ a) @ X4) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) & ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ A5) @ X4) = (((one @ a) @ ((ring_ext @ a) @ b)) @ r))))))))) => (((field @ a) @ b) @ r)))).
% 0.31/1.50 thf(fact_98_monoid_Or__one, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A]: (((((monoid @ A) @ B) @ G2) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((((mult @ A) @ B) @ G2) @ X) @ (((one @ A) @ B) @ G2)) = X)))))).
% 0.31/1.50 thf(fact_99_monoid_Ol__one, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A]: (((((monoid @ A) @ B) @ G2) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((((mult @ A) @ B) @ G2) @ (((one @ A) @ B) @ G2)) @ X) = X)))))).
% 0.31/1.50 thf(fact_100_monoid_Oone__closed, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((((monoid @ A) @ B) @ G2) => (((member @ A) @ (((one @ A) @ B) @ G2)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)))))).
% 0.31/1.50 thf(fact_101_monoid_OUnits__r__inv__ex, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A]: (((((monoid @ A) @ B) @ G2) => ((((member @ A) @ X) @ (((units @ A) @ B) @ G2)) => (? [X2:A]: (((((member @ A) @ X2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) & ((((((mult @ A) @ B) @ G2) @ X) @ X2) = (((one @ A) @ B) @ G2)))))))))).
% 0.31/1.50 thf(fact_102_monoid_OUnits__l__inv__ex, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A]: (((((monoid @ A) @ B) @ G2) => ((((member @ A) @ X) @ (((units @ A) @ B) @ G2)) => (? [X2:A]: (((((member @ A) @ X2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) & ((((((mult @ A) @ B) @ G2) @ X2) @ X) = (((one @ A) @ B) @ G2)))))))))).
% 0.31/1.50 thf(fact_103_monoid_OUnits__inv__comm, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A,Y:A]: (((((monoid @ A) @ B) @ G2) => (((((((mult @ A) @ B) @ G2) @ X) @ Y) = (((one @ A) @ B) @ G2)) => ((((member @ A) @ X) @ (((units @ A) @ B) @ G2)) => ((((member @ A) @ Y) @ (((units @ A) @ B) @ G2)) => ((((((mult @ A) @ B) @ G2) @ Y) @ X) = (((one @ A) @ B) @ G2))))))))).
% 0.31/1.50 thf(fact_104_add__pow__ldistr__int, axiom, (! [A2:a,B2:a,K:int]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ ((((((add_pow @ a) @ b) @ int) @ r) @ K) @ A2)) @ B2) = ((((((add_pow @ a) @ b) @ int) @ r) @ K) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ A2) @ B2)))))))).
% 0.31/1.50 thf(fact_105_add__pow__rdistr__int, axiom, (! [A2:a,B2:a,K:int]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ A2) @ ((((((add_pow @ a) @ b) @ int) @ r) @ K) @ B2)) = ((((((add_pow @ a) @ b) @ int) @ r) @ K) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ A2) @ B2)))))))).
% 0.31/1.50 thf(fact_106_add_Oint__pow__closed, axiom, (! [X:a,I:int]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((member @ a) @ ((((((add_pow @ a) @ b) @ int) @ r) @ I) @ X)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)))))).
% 0.31/1.50 thf(fact_107_add_Oint__pow__one, axiom, (! [Z2:int]: ((((((((add_pow @ a) @ b) @ int) @ r) @ Z2) @ (((zero @ a) @ b) @ r)) = (((zero @ a) @ b) @ r))))).
% 0.31/1.50 thf(fact_108_field_Ois__ring, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((field @ A) @ B) @ R3) => (((ring @ A) @ B) @ R3))))).
% 0.31/1.50 thf(fact_109_one__not__zero, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((field @ A) @ B) @ R3) => ((((one @ A) @ ((ring_ext @ A) @ B)) @ R3) != (((zero @ A) @ B) @ R3)))))).
% 0.31/1.50 thf(fact_110_integral, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),A2:A,B2:A]: (((((field @ A) @ B) @ R3) => (((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ A2) @ B2) = (((zero @ A) @ B) @ R3)) => ((((member @ A) @ A2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ B2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((A2 = (((zero @ A) @ B) @ R3)) | (B2 = (((zero @ A) @ B) @ R3)))))))))).
% 0.31/1.50 thf(fact_111_ring_Oadd__pow__ldistr__int, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),A2:A,B2:A,K:int]: (((((ring @ A) @ B) @ R3) => ((((member @ A) @ A2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ B2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ ((((((add_pow @ A) @ B) @ int) @ R3) @ K) @ A2)) @ B2) = ((((((add_pow @ A) @ B) @ int) @ R3) @ K) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ A2) @ B2))))))))).
% 0.31/1.50 thf(fact_112_ring_Oadd__pow__rdistr__int, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),A2:A,B2:A,K:int]: (((((ring @ A) @ B) @ R3) => ((((member @ A) @ A2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ B2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ A2) @ ((((((add_pow @ A) @ B) @ int) @ R3) @ K) @ B2)) = ((((((add_pow @ A) @ B) @ int) @ R3) @ K) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ A2) @ B2))))))))).
% 0.31/1.50 thf(fact_113_monoid_Ocarrier__not__empty, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((((monoid @ A) @ B) @ G2) => ((((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2) != (bot_bot @ (set @ A))))))).
% 0.31/1.50 thf(fact_114_monoid_Om__assoc, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A,Y:A,Z2:A]: (((((monoid @ A) @ B) @ G2) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((member @ A) @ Z2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((((mult @ A) @ B) @ G2) @ (((((mult @ A) @ B) @ G2) @ X) @ Y)) @ Z2) = (((((mult @ A) @ B) @ G2) @ X) @ (((((mult @ A) @ B) @ G2) @ Y) @ Z2)))))))))).
% 0.31/1.50 thf(fact_115_monoid_Om__closed, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A,Y:A]: (((((monoid @ A) @ B) @ G2) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => (((member @ A) @ (((((mult @ A) @ B) @ G2) @ X) @ Y)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)))))))).
% 0.31/1.50 thf(fact_116_monoid_OUnits__closed, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A]: (((((monoid @ A) @ B) @ G2) => ((((member @ A) @ X) @ (((units @ A) @ B) @ G2)) => (((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2))))))).
% 0.31/1.50 thf(fact_117_monoid_OUnits__m__closed, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A,Y:A]: (((((monoid @ A) @ B) @ G2) => ((((member @ A) @ X) @ (((units @ A) @ B) @ G2)) => ((((member @ A) @ Y) @ (((units @ A) @ B) @ G2)) => (((member @ A) @ (((((mult @ A) @ B) @ G2) @ X) @ Y)) @ (((units @ A) @ B) @ G2)))))))).
% 0.31/1.50 thf(fact_118_monoid_OUnits__one__closed, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((((monoid @ A) @ B) @ G2) => (((member @ A) @ (((one @ A) @ B) @ G2)) @ (((units @ A) @ B) @ G2)))))).
% 0.31/1.50 thf(fact_119_monoidI, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((! [X2:A,Y3:A]: (((((member @ A) @ X2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((member @ A) @ Y3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => (((member @ A) @ (((((mult @ A) @ B) @ G2) @ X2) @ Y3)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)))))) => ((((member @ A) @ (((one @ A) @ B) @ G2)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((! [X2:A,Y3:A,Z3:A]: (((((member @ A) @ X2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((member @ A) @ Y3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((member @ A) @ Z3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((((mult @ A) @ B) @ G2) @ (((((mult @ A) @ B) @ G2) @ X2) @ Y3)) @ Z3) = (((((mult @ A) @ B) @ G2) @ X2) @ (((((mult @ A) @ B) @ G2) @ Y3) @ Z3)))))))) => ((! [X2:A]: (((((member @ A) @ X2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((((mult @ A) @ B) @ G2) @ (((one @ A) @ B) @ G2)) @ X2) = X2)))) => ((! [X2:A]: (((((member @ A) @ X2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((((mult @ A) @ B) @ G2) @ X2) @ (((one @ A) @ B) @ G2)) = X2)))) => (((monoid @ A) @ B) @ G2))))))))).
% 0.31/1.50 thf(fact_120_Group_Omonoid__def, axiom, (! [B:$tType,A:$tType]: ((((monoid @ A) @ B) = (^ [G5:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((! [X3:A,Y4:A]: (((((member @ A) @ X3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G5)) => ((((member @ A) @ Y4) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G5)) => (((member @ A) @ (((((mult @ A) @ B) @ G5) @ X3) @ Y4)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G5)))))) & ((! [X3:A,Y4:A,Z4:A]: (((((member @ A) @ X3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G5)) => ((((member @ A) @ Y4) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G5)) => ((((member @ A) @ Z4) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G5)) => ((((((mult @ A) @ B) @ G5) @ (((((mult @ A) @ B) @ G5) @ X3) @ Y4)) @ Z4) = (((((mult @ A) @ B) @ G5) @ X3) @ (((((mult @ A) @ B) @ G5) @ Y4) @ Z4)))))))) & ((((member @ A) @ (((one @ A) @ B) @ G5)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G5)) & ((! [X3:A]: (((((member @ A) @ X3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G5)) => ((((((mult @ A) @ B) @ G5) @ (((one @ A) @ B) @ G5)) @ X3) = X3)))) & (! [X3:A]: (((((member @ A) @ X3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G5)) => ((((((mult @ A) @ B) @ G5) @ X3) @ (((one @ A) @ B) @ G5)) = X3)))))))))))))).
% 0.31/1.50 thf(fact_121_Group_Omonoid_Ointro, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((! [X2:A,Y3:A]: (((((member @ A) @ X2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((member @ A) @ Y3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => (((member @ A) @ (((((mult @ A) @ B) @ G2) @ X2) @ Y3)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)))))) => ((! [X2:A,Y3:A,Z3:A]: (((((member @ A) @ X2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((member @ A) @ Y3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((member @ A) @ Z3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((((mult @ A) @ B) @ G2) @ (((((mult @ A) @ B) @ G2) @ X2) @ Y3)) @ Z3) = (((((mult @ A) @ B) @ G2) @ X2) @ (((((mult @ A) @ B) @ G2) @ Y3) @ Z3)))))))) => ((((member @ A) @ (((one @ A) @ B) @ G2)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((! [X2:A]: (((((member @ A) @ X2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((((mult @ A) @ B) @ G2) @ (((one @ A) @ B) @ G2)) @ X2) = X2)))) => ((! [X2:A]: (((((member @ A) @ X2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((((mult @ A) @ B) @ G2) @ X2) @ (((one @ A) @ B) @ G2)) = X2)))) => (((monoid @ A) @ B) @ G2))))))))).
% 0.31/1.50 thf(fact_122_monoid_Oinv__unique, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),Y:A,X:A,Y2:A]: (((((monoid @ A) @ B) @ G2) => (((((((mult @ A) @ B) @ G2) @ Y) @ X) = (((one @ A) @ B) @ G2)) => (((((((mult @ A) @ B) @ G2) @ X) @ Y2) = (((one @ A) @ B) @ G2)) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((member @ A) @ Y2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => (Y = Y2)))))))))).
% 0.31/1.50 thf(fact_123_monoid_Oone__unique, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),U:A]: (((((monoid @ A) @ B) @ G2) => ((((member @ A) @ U) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((! [X2:A]: (((((member @ A) @ X2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((((mult @ A) @ B) @ G2) @ U) @ X2) = X2)))) => (U = (((one @ A) @ B) @ G2)))))))).
% 0.31/1.50 thf(fact_124_monoid_OUnits__l__cancel, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A,Y:A,Z2:A]: (((((monoid @ A) @ B) @ G2) => ((((member @ A) @ X) @ (((units @ A) @ B) @ G2)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((member @ A) @ Z2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => (((((((mult @ A) @ B) @ G2) @ X) @ Y) = (((((mult @ A) @ B) @ G2) @ X) @ Z2)) = (Y = Z2))))))))).
% 0.31/1.50 thf(fact_125_subset, axiom, (((ord_less_eq @ (set @ a)) @ s) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 0.31/1.50 thf(fact_126_add__pow__rdistr, axiom, (! [A2:a,B2:a,K:nat]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ A2) @ ((((((add_pow @ a) @ b) @ nat) @ r) @ K) @ B2)) = ((((((add_pow @ a) @ b) @ nat) @ r) @ K) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ A2) @ B2)))))))).
% 0.31/1.50 thf(fact_127_add__pow__ldistr, axiom, (! [A2:a,B2:a,K:nat]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ ((((((add_pow @ a) @ b) @ nat) @ r) @ K) @ A2)) @ B2) = ((((((add_pow @ a) @ b) @ nat) @ r) @ K) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ A2) @ B2)))))))).
% 0.31/1.50 thf(fact_128_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))).
% 0.31/1.50 thf(fact_129_group__l__invI, axiom, ((! [X2:a]: (((((member @ a) @ X2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (? [Xa:a]: (((((member @ a) @ Xa) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) & ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Xa) @ X2) = (((one @ a) @ ((ring_ext @ a) @ b)) @ r)))))))) => (((group @ a) @ ((ring_ext @ a) @ b)) @ r))).
% 0.31/1.50 thf(fact_130_carrier__not__empty, axiom, ((((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r) != (bot_bot @ (set @ a)))).
% 0.31/1.50 thf(fact_131_add_Onat__pow__closed, axiom, (! [X:a,N:nat]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((member @ a) @ ((((((add_pow @ a) @ b) @ nat) @ r) @ N) @ X)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)))))).
% 0.31/1.50 thf(fact_132_add_Onat__pow__one, axiom, (! [N:nat]: ((((((((add_pow @ a) @ b) @ nat) @ r) @ N) @ (((zero @ a) @ b) @ r)) = (((zero @ a) @ b) @ r))))).
% 0.31/1.50 thf(fact_133_group_Ois__group, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((((group @ A) @ B) @ G2) => (((group @ A) @ B) @ G2))))).
% 0.31/1.50 thf(fact_134_group_OUnits, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((((group @ A) @ B) @ G2) => (((ord_less_eq @ (set @ A)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) @ (((units @ A) @ B) @ G2)))))).
% 0.31/1.50 thf(fact_135_group_Ois__monoid, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((((group @ A) @ B) @ G2) => (((monoid @ A) @ B) @ G2))))).
% 0.31/1.50 thf(fact_136_Group_Ogroup_Oright__cancel, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A,Y:A,Z2:A]: (((((group @ A) @ B) @ G2) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((member @ A) @ Z2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => (((((((mult @ A) @ B) @ G2) @ Y) @ X) = (((((mult @ A) @ B) @ G2) @ Z2) @ X)) = (Y = Z2))))))))).
% 0.31/1.50 thf(fact_137_group_OUnits__eq, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((((group @ A) @ B) @ G2) => ((((units @ A) @ B) @ G2) = (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)))))).
% 0.31/1.50 thf(fact_138_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Osubset, axiom, (! [B:$tType,A:$tType,M:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),S3:(set @ A)]: ((((((locali1441642228monoid @ A) @ B) @ M) @ S3) => (((ord_less_eq @ (set @ A)) @ S3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ M)))))).
% 0.31/1.50 thf(fact_139_group_Or__cancel__one_H, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A,A2:A]: (((((group @ A) @ B) @ G2) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((member @ A) @ A2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((X = (((((mult @ A) @ B) @ G2) @ A2) @ X)) = (A2 = (((one @ A) @ B) @ G2))))))))).
% 0.31/1.50 thf(fact_140_group_Ol__cancel__one_H, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A,A2:A]: (((((group @ A) @ B) @ G2) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((member @ A) @ A2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((X = (((((mult @ A) @ B) @ G2) @ X) @ A2)) = (A2 = (((one @ A) @ B) @ G2))))))))).
% 0.31/1.50 thf(fact_141_group_Or__cancel__one, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A,A2:A]: (((((group @ A) @ B) @ G2) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((member @ A) @ A2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => (((((((mult @ A) @ B) @ G2) @ A2) @ X) = X) = (A2 = (((one @ A) @ B) @ G2))))))))).
% 0.31/1.50 thf(fact_142_group_Ol__cancel__one, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A,A2:A]: (((((group @ A) @ B) @ G2) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((member @ A) @ A2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => (((((((mult @ A) @ B) @ G2) @ X) @ A2) = X) = (A2 = (((one @ A) @ B) @ G2))))))))).
% 0.31/1.50 thf(fact_143_group_Or__inv__ex, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A]: (((((group @ A) @ B) @ G2) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => (? [X2:A]: (((((member @ A) @ X2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) & ((((((mult @ A) @ B) @ G2) @ X) @ X2) = (((one @ A) @ B) @ G2)))))))))).
% 0.31/1.50 thf(fact_144_group_Ol__inv__ex, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A]: (((((group @ A) @ B) @ G2) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => (? [X2:A]: (((((member @ A) @ X2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) & ((((((mult @ A) @ B) @ G2) @ X2) @ X) = (((one @ A) @ B) @ G2)))))))))).
% 0.31/1.50 thf(fact_145_group_Oinv__comm, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A,Y:A]: (((((group @ A) @ B) @ G2) => (((((((mult @ A) @ B) @ G2) @ X) @ Y) = (((one @ A) @ B) @ G2)) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((((mult @ A) @ B) @ G2) @ Y) @ X) = (((one @ A) @ B) @ G2))))))))).
% 0.31/1.50 thf(fact_146_groupI, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((! [X2:A]: (((((member @ A) @ X2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => (! [Y3:A]: (((((member @ A) @ Y3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => (((member @ A) @ (((((mult @ A) @ B) @ G2) @ X2) @ Y3)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)))))))) => ((((member @ A) @ (((one @ A) @ B) @ G2)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((! [X2:A]: (((((member @ A) @ X2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => (! [Y3:A]: (((((member @ A) @ Y3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => (! [Z3:A]: (((((member @ A) @ Z3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((((mult @ A) @ B) @ G2) @ (((((mult @ A) @ B) @ G2) @ X2) @ Y3)) @ Z3) = (((((mult @ A) @ B) @ G2) @ X2) @ (((((mult @ A) @ B) @ G2) @ Y3) @ Z3)))))))))))) => ((! [X2:A]: (((((member @ A) @ X2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => ((((((mult @ A) @ B) @ G2) @ (((one @ A) @ B) @ G2)) @ X2) = X2)))) => ((! [X2:A]: (((((member @ A) @ X2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => (? [Xa:A]: (((((member @ A) @ Xa) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) & ((((((mult @ A) @ B) @ G2) @ Xa) @ X2) = (((one @ A) @ B) @ G2)))))))) => (((group @ A) @ B) @ G2))))))))).
% 0.31/1.50 thf(fact_147_monoid_Ogroup__l__invI, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((((monoid @ A) @ B) @ G2) => ((! [X2:A]: (((((member @ A) @ X2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) => (? [Xa:A]: (((((member @ A) @ Xa) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G2)) & ((((((mult @ A) @ B) @ G2) @ Xa) @ X2) = (((one @ A) @ B) @ G2)))))))) => (((group @ A) @ B) @ G2)))))).
% 0.31/1.50 thf(fact_148_semiring_Oadd__pow__rdistr, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),A2:A,B2:A,K:nat]: (((((semiring @ A) @ B) @ R3) => ((((member @ A) @ A2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ B2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ A2) @ ((((((add_pow @ A) @ B) @ nat) @ R3) @ K) @ B2)) = ((((((add_pow @ A) @ B) @ nat) @ R3) @ K) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ A2) @ B2))))))))).
% 0.31/1.50 thf(fact_149_semiring_Oadd__pow__ldistr, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),A2:A,B2:A,K:nat]: (((((semiring @ A) @ B) @ R3) => ((((member @ A) @ A2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ B2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ ((((((add_pow @ A) @ B) @ nat) @ R3) @ K) @ A2)) @ B2) = ((((((add_pow @ A) @ B) @ nat) @ R3) @ K) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ A2) @ B2))))))))).
% 0.31/1.50 thf(fact_150_a__lcos__mult__one, axiom, (! [M:(set @ a)]: (((((ord_less_eq @ (set @ a)) @ M) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((a_l_coset @ a) @ b) @ r) @ (((zero @ a) @ b) @ r)) @ M) = M))))).
% 0.31/1.50 thf(fact_151_set__add__closed, axiom, (! [A4:(set @ a),B5:(set @ a)]: (((((ord_less_eq @ (set @ a)) @ A4) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((ord_less_eq @ (set @ a)) @ B5) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((ord_less_eq @ (set @ a)) @ (((((set_add @ a) @ b) @ r) @ A4) @ B5)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))))))).
% 0.31/1.50 thf(fact_152_setadd__subset__G, axiom, (! [H:(set @ a),K2:(set @ a)]: (((((ord_less_eq @ (set @ a)) @ H) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((ord_less_eq @ (set @ a)) @ K2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((ord_less_eq @ (set @ a)) @ (((((set_add @ a) @ b) @ r) @ H) @ K2)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))))))).
% 0.31/1.50 thf(fact_153_subset__empty, axiom, (! [A:$tType,A4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ (bot_bot @ (set @ A))) = (A4 = (bot_bot @ (set @ A))))))).
% 0.31/1.50 thf(fact_154_empty__iff, axiom, (! [A:$tType,C3:A]: ((~ ((((member @ A) @ C3) @ (bot_bot @ (set @ A)))))))).
% 0.31/1.50 thf(fact_155_all__not__in__conv, axiom, (! [A:$tType,A4:(set @ A)]: (((! [X3:A]: ((~ ((((member @ A) @ X3) @ A4))))) = (A4 = (bot_bot @ (set @ A))))))).
% 0.31/1.50 thf(fact_156_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X3:A]: ((~ ((P @ X3))))))))).
% 0.31/1.50 thf(fact_157_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X3:A]: ((~ ((P @ X3))))))))).
% 0.31/1.50 thf(fact_158_subsetI, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: (((! [X2:A]: (((((member @ A) @ X2) @ A4) => (((member @ A) @ X2) @ B5)))) => (((ord_less_eq @ (set @ A)) @ A4) @ B5))))).
% 0.31/1.50 thf(fact_159_subset__antisym, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ A4) => (A4 = B5)))))).
% 0.31/1.50 thf(fact_160_a__l__coset__subset__G, axiom, (! [H:(set @ a),X:a]: (((((ord_less_eq @ (set @ a)) @ H) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((ord_less_eq @ (set @ a)) @ (((((a_l_coset @ a) @ b) @ r) @ X) @ H)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))))))).
% 0.31/1.50 thf(fact_161_empty__subsetI, axiom, (! [A:$tType,A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A4)))).
% 0.31/1.50 thf(fact_162_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 0.31/1.50 thf(fact_163_equals0D, axiom, (! [A:$tType,A4:(set @ A),A2:A]: (((A4 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A4))))))).
% 0.31/1.50 thf(fact_164_equals0I, axiom, (! [A:$tType,A4:(set @ A)]: (((! [Y3:A]: ((~ ((((member @ A) @ Y3) @ A4))))) => (A4 = (bot_bot @ (set @ A))))))).
% 0.31/1.50 thf(fact_165_ex__in__conv, axiom, (! [A:$tType,A4:(set @ A)]: (((? [X3:A]: ((((member @ A) @ X3) @ A4))) = (A4 != (bot_bot @ (set @ A))))))).
% 0.31/1.50 thf(fact_166_in__mono, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B5) => ((((member @ A) @ X) @ A4) => (((member @ A) @ X) @ B5)))))).
% 0.31/1.50 thf(fact_167_subsetD, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B5) => ((((member @ A) @ C3) @ A4) => (((member @ A) @ C3) @ B5)))))).
% 0.31/1.50 thf(fact_168_equalityE, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: (((A4 = B5) => (~ (((((ord_less_eq @ (set @ A)) @ A4) @ B5) => (~ ((((ord_less_eq @ (set @ A)) @ B5) @ A4)))))))))).
% 0.31/1.50 thf(fact_169_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((member @ A) @ X3) @ B6)))))))))).
% 0.31/1.50 thf(fact_170_equalityD1, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: (((A4 = B5) => (((ord_less_eq @ (set @ A)) @ A4) @ B5))))).
% 0.31/1.50 thf(fact_171_equalityD2, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: (((A4 = B5) => (((ord_less_eq @ (set @ A)) @ B5) @ A4))))).
% 0.31/1.50 thf(fact_172_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((! [T3:A]: (((((member @ A) @ T3) @ A6) => (((member @ A) @ T3) @ B6)))))))))).
% 0.31/1.50 thf(fact_173_subset__refl, axiom, (! [A:$tType,A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A4) @ A4)))).
% 0.31/1.50 thf(fact_174_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) => (Q @ X2)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 0.31/1.50 thf(fact_175_subset__trans, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ C4) => (((ord_less_eq @ (set @ A)) @ A4) @ C4)))))).
% 0.31/1.50 thf(fact_176_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y5:(set @ A),Z5:(set @ A)]: ((Y5 = Z5))) = (^ [A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) & (((ord_less_eq @ (set @ A)) @ B6) @ A6)))))))).
% 0.31/1.50 thf(fact_177_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X3:A]: (((P @ X3) => (Q @ X3)))))))).
% 0.31/1.50 thf(fact_178_units__group, axiom, (((group @ a) @ product_unit) @ (((units_of @ a) @ ((ring_ext @ a) @ b)) @ r))).
% 0.31/1.50 thf(fact_179_add__additive__subgroups, axiom, (! [H:(set @ a),K2:(set @ a)]: ((((((additive_subgroup @ a) @ b) @ H) @ r) => (((((additive_subgroup @ a) @ b) @ K2) @ r) => ((((additive_subgroup @ a) @ b) @ (((((set_add @ a) @ b) @ r) @ H) @ K2)) @ r)))))).
% 0.31/1.50 thf(fact_180_a__rcos__assoc__lcos, axiom, (! [H:(set @ a),K2:(set @ a),X:a]: (((((ord_less_eq @ (set @ a)) @ H) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((ord_less_eq @ (set @ a)) @ K2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((set_add @ a) @ b) @ r) @ (((((a_r_coset @ a) @ b) @ r) @ H) @ X)) @ K2) = (((((set_add @ a) @ b) @ r) @ H) @ (((((a_l_coset @ a) @ b) @ r) @ X) @ K2))))))))).
% 0.31/1.50 thf(fact_181_a__r__coset__subset__G, axiom, (! [H:(set @ a),X:a]: (((((ord_less_eq @ (set @ a)) @ H) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((ord_less_eq @ (set @ a)) @ (((((a_r_coset @ a) @ b) @ r) @ H) @ X)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))))))).
% 0.31/1.50 thf(fact_182_a__setmult__rcos__assoc, axiom, (! [H:(set @ a),K2:(set @ a),X:a]: (((((ord_less_eq @ (set @ a)) @ H) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((ord_less_eq @ (set @ a)) @ K2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((set_add @ a) @ b) @ r) @ H) @ (((((a_r_coset @ a) @ b) @ r) @ K2) @ X)) = (((((a_r_coset @ a) @ b) @ r) @ (((((set_add @ a) @ b) @ r) @ H) @ K2)) @ X)))))))).
% 0.31/1.50 thf(fact_183_units__of__units, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((((units @ A) @ product_unit) @ (((units_of @ A) @ B) @ G2)) = (((units @ A) @ B) @ G2))))).
% 0.31/1.50 thf(fact_184_a__coset__add__zero, axiom, (! [M:(set @ a)]: (((((ord_less_eq @ (set @ a)) @ M) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((a_r_coset @ a) @ b) @ r) @ M) @ (((zero @ a) @ b) @ r)) = M))))).
% 0.31/1.50 thf(fact_185_units__of__carrier, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((((partial_carrier @ A) @ ((monoid_ext @ A) @ product_unit)) @ (((units_of @ A) @ B) @ G2)) = (((units @ A) @ B) @ G2))))).
% 0.31/1.50 thf(fact_186_monoid_Ounits__group, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((((monoid @ A) @ B) @ G2) => (((group @ A) @ product_unit) @ (((units_of @ A) @ B) @ G2)))))).
% 0.31/1.50 thf(fact_187_units__of__mult, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((((mult @ A) @ product_unit) @ (((units_of @ A) @ B) @ G2)) = (((mult @ A) @ B) @ G2))))).
% 0.31/1.50 thf(fact_188_units__of__one, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((((one @ A) @ product_unit) @ (((units_of @ A) @ B) @ G2)) = (((one @ A) @ B) @ G2))))).
% 0.31/1.50 thf(fact_189_a__rcosetsI, axiom, (! [H:(set @ a),X:a]: (((((ord_less_eq @ (set @ a)) @ H) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((member @ (set @ a)) @ (((((a_r_coset @ a) @ b) @ r) @ H) @ X)) @ ((((a_RCOSETS @ a) @ b) @ r) @ H))))))).
% 0.31/1.50 thf(fact_190_one__zeroI, axiom, (((((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r) = (((insert @ a) @ (((zero @ a) @ b) @ r)) @ (bot_bot @ (set @ a)))) => ((((one @ a) @ ((ring_ext @ a) @ b)) @ r) = (((zero @ a) @ b) @ r)))).
% 0.31/1.50 thf(fact_191_one__zeroD, axiom, (((((one @ a) @ ((ring_ext @ a) @ b)) @ r) = (((zero @ a) @ b) @ r)) => ((((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r) = (((insert @ a) @ (((zero @ a) @ b) @ r)) @ (bot_bot @ (set @ a)))))).
% 0.31/1.50 thf(fact_192_insert__absorb2, axiom, (! [A:$tType,X:A,A4:(set @ A)]: (((((insert @ A) @ X) @ (((insert @ A) @ X) @ A4)) = (((insert @ A) @ X) @ A4))))).
% 0.31/1.50 thf(fact_193_insert__iff, axiom, (! [A:$tType,A2:A,B2:A,A4:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B2) @ A4)) = ((A2 = B2) | (((member @ A) @ A2) @ A4)))))).
% 0.31/1.50 thf(fact_194_insertCI, axiom, (! [A:$tType,A2:A,B5:(set @ A),B2:A]: ((((~ ((((member @ A) @ A2) @ B5))) => (A2 = B2)) => (((member @ A) @ A2) @ (((insert @ A) @ B2) @ B5)))))).
% 0.31/1.50 thf(fact_195_carrier__one__not__zero, axiom, (((((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r) != (((insert @ a) @ (((zero @ a) @ b) @ r)) @ (bot_bot @ (set @ a)))) = ((((one @ a) @ ((ring_ext @ a) @ b)) @ r) != (((zero @ a) @ b) @ r)))).
% 0.31/1.50 thf(fact_196_carrier__one__zero, axiom, (((((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r) = (((insert @ a) @ (((zero @ a) @ b) @ r)) @ (bot_bot @ (set @ a)))) = ((((one @ a) @ ((ring_ext @ a) @ b)) @ r) = (((zero @ a) @ b) @ r)))).
% 0.31/1.50 thf(fact_197_singletonI, axiom, (! [A:$tType,A2:A]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 0.31/1.50 thf(fact_198_insert__subset, axiom, (! [A:$tType,X:A,A4:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X) @ A4)) @ B5) = ((((member @ A) @ X) @ B5) & (((ord_less_eq @ (set @ A)) @ A4) @ B5)))))).
% 0.31/1.50 thf(fact_199_singleton__insert__inj__eq_H, axiom, (! [A:$tType,A2:A,A4:(set @ A),B2:A]: ((((((insert @ A) @ A2) @ A4) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = ((A2 = B2) & (((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))))))).
% 0.31/1.50 thf(fact_200_singleton__insert__inj__eq, axiom, (! [A:$tType,B2:A,A2:A,A4:(set @ A)]: ((((((insert @ A) @ B2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ A2) @ A4)) = ((A2 = B2) & (((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))))))).
% 0.31/1.50 thf(fact_201_subset__singleton__iff, axiom, (! [A:$tType,X5:(set @ A),A2:A]: (((((ord_less_eq @ (set @ A)) @ X5) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = ((X5 = (bot_bot @ (set @ A))) | (X5 = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))))).
% 0.31/1.50 thf(fact_202_subset__singletonD, axiom, (! [A:$tType,A4:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) => ((A4 = (bot_bot @ (set @ A))) | (A4 = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))).
% 0.31/1.50 thf(fact_203_mk__disjoint__insert, axiom, (! [A:$tType,A2:A,A4:(set @ A)]: (((((member @ A) @ A2) @ A4) => (? [B7:(set @ A)]: (((A4 = (((insert @ A) @ A2) @ B7)) & (~ ((((member @ A) @ A2) @ B7)))))))))).
% 0.31/1.50 thf(fact_204_insert__commute, axiom, (! [A:$tType,X:A,Y:A,A4:(set @ A)]: (((((insert @ A) @ X) @ (((insert @ A) @ Y) @ A4)) = (((insert @ A) @ Y) @ (((insert @ A) @ X) @ A4)))))).
% 0.31/1.50 thf(fact_205_insert__eq__iff, axiom, (! [A:$tType,A2:A,A4:(set @ A),B2:A,B5:(set @ A)]: (((~ ((((member @ A) @ A2) @ A4))) => ((~ ((((member @ A) @ B2) @ B5))) => (((((insert @ A) @ A2) @ A4) = (((insert @ A) @ B2) @ B5)) = (((A2 = B2) => (A4 = B5)) & ((A2 != B2) => (? [C5:(set @ A)]: (((A4 = (((insert @ A) @ B2) @ C5)) & ((~ ((((member @ A) @ B2) @ C5))) & ((B5 = (((insert @ A) @ A2) @ C5)) & (~ ((((member @ A) @ A2) @ C5)))))))))))))))).
% 0.31/1.50 thf(fact_206_insert__absorb, axiom, (! [A:$tType,A2:A,A4:(set @ A)]: (((((member @ A) @ A2) @ A4) => ((((insert @ A) @ A2) @ A4) = A4))))).
% 0.31/1.50 thf(fact_207_insert__ident, axiom, (! [A:$tType,X:A,A4:(set @ A),B5:(set @ A)]: (((~ ((((member @ A) @ X) @ A4))) => ((~ ((((member @ A) @ X) @ B5))) => (((((insert @ A) @ X) @ A4) = (((insert @ A) @ X) @ B5)) = (A4 = B5))))))).
% 0.31/1.50 thf(fact_208_Set_Oset__insert, axiom, (! [A:$tType,X:A,A4:(set @ A)]: (((((member @ A) @ X) @ A4) => (~ ((! [B7:(set @ A)]: (((A4 = (((insert @ A) @ X) @ B7)) => (((member @ A) @ X) @ B7)))))))))).
% 0.31/1.50 thf(fact_209_insertI2, axiom, (! [A:$tType,A2:A,B5:(set @ A),B2:A]: (((((member @ A) @ A2) @ B5) => (((member @ A) @ A2) @ (((insert @ A) @ B2) @ B5)))))).
% 0.31/1.50 thf(fact_210_insertI1, axiom, (! [A:$tType,A2:A,B5:(set @ A)]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ B5))))).
% 0.31/1.50 thf(fact_211_insertE, axiom, (! [A:$tType,A2:A,B2:A,A4:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B2) @ A4)) => ((A2 != B2) => (((member @ A) @ A2) @ A4)))))).
% 0.31/1.50 thf(fact_212_singleton__inject, axiom, (! [A:$tType,A2:A,B2:A]: ((((((insert @ A) @ A2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) => (A2 = B2))))).
% 0.31/1.50 thf(fact_213_insert__not__empty, axiom, (! [A:$tType,A2:A,A4:(set @ A)]: (((((insert @ A) @ A2) @ A4) != (bot_bot @ (set @ A)))))).
% 0.31/1.50 thf(fact_214_doubleton__eq__iff, axiom, (! [A:$tType,A2:A,B2:A,C3:A,D3:A]: ((((((insert @ A) @ A2) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C3) @ (((insert @ A) @ D3) @ (bot_bot @ (set @ A))))) = (((A2 = C3) & (B2 = D3)) | ((A2 = D3) & (B2 = C3))))))).
% 0.31/1.50 thf(fact_215_singleton__iff, axiom, (! [A:$tType,B2:A,A2:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = (B2 = A2))))).
% 0.31/1.50 thf(fact_216_singletonD, axiom, (! [A:$tType,B2:A,A2:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) => (B2 = A2))))).
% 0.31/1.50 thf(fact_217_insert__mono, axiom, (! [A:$tType,C4:(set @ A),D4:(set @ A),A2:A]: (((((ord_less_eq @ (set @ A)) @ C4) @ D4) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ A2) @ C4)) @ (((insert @ A) @ A2) @ D4)))))).
% 0.31/1.50 thf(fact_218_subset__insert, axiom, (! [A:$tType,X:A,A4:(set @ A),B5:(set @ A)]: (((~ ((((member @ A) @ X) @ A4))) => ((((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ X) @ B5)) = (((ord_less_eq @ (set @ A)) @ A4) @ B5)))))).
% 0.31/1.50 thf(fact_219_subset__insertI, axiom, (! [A:$tType,B5:(set @ A),A2:A]: ((((ord_less_eq @ (set @ A)) @ B5) @ (((insert @ A) @ A2) @ B5))))).
% 0.31/1.50 thf(fact_220_subset__insertI2, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),B2:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B5) => (((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ B2) @ B5)))))).
% 0.31/1.50 thf(fact_221_semiring_Ocarrier__one__not__zero, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((semiring @ A) @ B) @ R3) => (((((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3) != (((insert @ A) @ (((zero @ A) @ B) @ R3)) @ (bot_bot @ (set @ A)))) = ((((one @ A) @ ((ring_ext @ A) @ B)) @ R3) != (((zero @ A) @ B) @ R3))))))).
% 0.31/1.50 thf(fact_222_semiring_Ocarrier__one__zero, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((semiring @ A) @ B) @ R3) => (((((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3) = (((insert @ A) @ (((zero @ A) @ B) @ R3)) @ (bot_bot @ (set @ A)))) = ((((one @ A) @ ((ring_ext @ A) @ B)) @ R3) = (((zero @ A) @ B) @ R3))))))).
% 0.31/1.50 thf(fact_223_semiring_Oone__zeroI, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((semiring @ A) @ B) @ R3) => (((((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3) = (((insert @ A) @ (((zero @ A) @ B) @ R3)) @ (bot_bot @ (set @ A)))) => ((((one @ A) @ ((ring_ext @ A) @ B)) @ R3) = (((zero @ A) @ B) @ R3))))))).
% 0.31/1.50 thf(fact_224_semiring_Oone__zeroD, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((semiring @ A) @ B) @ R3) => (((((one @ A) @ ((ring_ext @ A) @ B)) @ R3) = (((zero @ A) @ B) @ R3)) => ((((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3) = (((insert @ A) @ (((zero @ A) @ B) @ R3)) @ (bot_bot @ (set @ A))))))))).
% 0.31/1.50 thf(fact_225_field__intro2, axiom, (((((zero @ a) @ b) @ r) != (((one @ a) @ ((ring_ext @ a) @ b)) @ r)) => ((! [X2:a]: (((((member @ a) @ X2) @ (((minus_minus @ (set @ a)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) @ (((insert @ a) @ (((zero @ a) @ b) @ r)) @ (bot_bot @ (set @ a))))) => (((member @ a) @ X2) @ (((units @ a) @ ((ring_ext @ a) @ b)) @ r))))) => (((field @ a) @ b) @ r)))).
% 0.31/1.50 thf(fact_226_cring__fieldI, axiom, (((((units @ a) @ ((ring_ext @ a) @ b)) @ r) = (((minus_minus @ (set @ a)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) @ (((insert @ a) @ (((zero @ a) @ b) @ r)) @ (bot_bot @ (set @ a))))) => (((field @ a) @ b) @ r))).
% 0.31/1.50 thf(fact_227_DiffI, axiom, (! [A:$tType,C3:A,A4:(set @ A),B5:(set @ A)]: (((((member @ A) @ C3) @ A4) => ((~ ((((member @ A) @ C3) @ B5))) => (((member @ A) @ C3) @ (((minus_minus @ (set @ A)) @ A4) @ B5))))))).
% 0.31/1.50 thf(fact_228_Diff__iff, axiom, (! [A:$tType,C3:A,A4:(set @ A),B5:(set @ A)]: (((((member @ A) @ C3) @ (((minus_minus @ (set @ A)) @ A4) @ B5)) = ((((member @ A) @ C3) @ A4) & (~ ((((member @ A) @ C3) @ B5)))))))).
% 0.31/1.50 thf(fact_229_Diff__idemp, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A4) @ B5)) @ B5) = (((minus_minus @ (set @ A)) @ A4) @ B5))))).
% 0.31/1.50 thf(fact_230_Diff__empty, axiom, (! [A:$tType,A4:(set @ A)]: (((((minus_minus @ (set @ A)) @ A4) @ (bot_bot @ (set @ A))) = A4)))).
% 0.31/1.50 thf(fact_231_empty__Diff, axiom, (! [A:$tType,A4:(set @ A)]: (((((minus_minus @ (set @ A)) @ (bot_bot @ (set @ A))) @ A4) = (bot_bot @ (set @ A)))))).
% 0.31/1.50 thf(fact_232_Diff__cancel, axiom, (! [A:$tType,A4:(set @ A)]: (((((minus_minus @ (set @ A)) @ A4) @ A4) = (bot_bot @ (set @ A)))))).
% 0.31/1.50 thf(fact_233_insert__Diff1, axiom, (! [A:$tType,X:A,B5:(set @ A),A4:(set @ A)]: (((((member @ A) @ X) @ B5) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X) @ A4)) @ B5) = (((minus_minus @ (set @ A)) @ A4) @ B5)))))).
% 0.31/1.50 thf(fact_234_Diff__insert0, axiom, (! [A:$tType,X:A,A4:(set @ A),B5:(set @ A)]: (((~ ((((member @ A) @ X) @ A4))) => ((((minus_minus @ (set @ A)) @ A4) @ (((insert @ A) @ X) @ B5)) = (((minus_minus @ (set @ A)) @ A4) @ B5)))))).
% 0.31/1.50 thf(fact_235_Diff__eq__empty__iff, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: ((((((minus_minus @ (set @ A)) @ A4) @ B5) = (bot_bot @ (set @ A))) = (((ord_less_eq @ (set @ A)) @ A4) @ B5))))).
% 0.31/1.50 thf(fact_236_insert__Diff__single, axiom, (! [A:$tType,A2:A,A4:(set @ A)]: (((((insert @ A) @ A2) @ (((minus_minus @ (set @ A)) @ A4) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = (((insert @ A) @ A2) @ A4))))).
% 0.31/1.50 thf(fact_237_insert__Diff__if, axiom, (! [A:$tType,X:A,B5:(set @ A),A4:(set @ A)]: ((((((member @ A) @ X) @ B5) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X) @ A4)) @ B5) = (((minus_minus @ (set @ A)) @ A4) @ B5))) & ((~ ((((member @ A) @ X) @ B5))) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X) @ A4)) @ B5) = (((insert @ A) @ X) @ (((minus_minus @ (set @ A)) @ A4) @ B5)))))))).
% 0.31/1.50 thf(fact_238_Diff__insert__absorb, axiom, (! [A:$tType,X:A,A4:(set @ A)]: (((~ ((((member @ A) @ X) @ A4))) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X) @ A4)) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = A4))))).
% 0.31/1.50 thf(fact_239_Diff__insert2, axiom, (! [A:$tType,A4:(set @ A),A2:A,B5:(set @ A)]: (((((minus_minus @ (set @ A)) @ A4) @ (((insert @ A) @ A2) @ B5)) = (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A4) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) @ B5))))).
% 0.31/1.50 thf(fact_240_insert__Diff, axiom, (! [A:$tType,A2:A,A4:(set @ A)]: (((((member @ A) @ A2) @ A4) => ((((insert @ A) @ A2) @ (((minus_minus @ (set @ A)) @ A4) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = A4))))).
% 0.31/1.50 thf(fact_241_Diff__insert, axiom, (! [A:$tType,A4:(set @ A),A2:A,B5:(set @ A)]: (((((minus_minus @ (set @ A)) @ A4) @ (((insert @ A) @ A2) @ B5)) = (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A4) @ B5)) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 0.31/1.50 thf(fact_242_subset__Diff__insert, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),X:A,C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ (((minus_minus @ (set @ A)) @ B5) @ (((insert @ A) @ X) @ C4))) = ((((ord_less_eq @ (set @ A)) @ A4) @ (((minus_minus @ (set @ A)) @ B5) @ C4)) & (~ ((((member @ A) @ X) @ A4)))))))).
% 0.31/1.50 thf(fact_243_double__diff, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ C4) => ((((minus_minus @ (set @ A)) @ B5) @ (((minus_minus @ (set @ A)) @ C4) @ A4)) = A4)))))).
% 0.31/1.50 thf(fact_244_Diff__subset, axiom, (! [A:$tType,A4:(set @ A),B5:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A4) @ B5)) @ A4)))).
% 0.31/1.50 thf(fact_245_Diff__mono, axiom, (! [A:$tType,A4:(set @ A),C4:(set @ A),D4:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ C4) => ((((ord_less_eq @ (set @ A)) @ D4) @ B5) => (((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A4) @ B5)) @ (((minus_minus @ (set @ A)) @ C4) @ D4))))))).
% 0.31/1.50 thf(fact_246_DiffE, axiom, (! [A:$tType,C3:A,A4:(set @ A),B5:(set @ A)]: (((((member @ A) @ C3) @ (((minus_minus @ (set @ A)) @ A4) @ B5)) => (~ (((((member @ A) @ C3) @ A4) => (((member @ A) @ C3) @ B5)))))))).
% 0.31/1.50 thf(fact_247_DiffD1, axiom, (! [A:$tType,C3:A,A4:(set @ A),B5:(set @ A)]: (((((member @ A) @ C3) @ (((minus_minus @ (set @ A)) @ A4) @ B5)) => (((member @ A) @ C3) @ A4))))).
% 0.31/1.50 thf(fact_248_DiffD2, axiom, (! [A:$tType,C3:A,A4:(set @ A),B5:(set @ A)]: (((((member @ A) @ C3) @ (((minus_minus @ (set @ A)) @ A4) @ B5)) => (~ ((((member @ A) @ C3) @ B5))))))).
% 0.31/1.50 thf(fact_249_subset__insert__iff, axiom, (! [A:$tType,A4:(set @ A),X:A,B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ X) @ B5)) = (((((member @ A) @ X) @ A4) => (((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A4) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) @ B5)) & ((~ ((((member @ A) @ X) @ A4))) => (((ord_less_eq @ (set @ A)) @ A4) @ B5))))))).
% 0.31/1.50 thf(fact_250_Diff__single__insert, axiom, (! [A:$tType,A4:(set @ A),X:A,B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A4) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) @ B5) => (((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ X) @ B5)))))).
% 0.31/1.50 thf(fact_251_field_Ofield__Units, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((field @ A) @ B) @ R3) => ((((units @ A) @ ((ring_ext @ A) @ B)) @ R3) = (((minus_minus @ (set @ A)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) @ (((insert @ A) @ (((zero @ A) @ B) @ R3)) @ (bot_bot @ (set @ A))))))))).
% 0.31/1.50 thf(fact_252_rcosets__subset__PowG, axiom, (! [H:(set @ a)]: ((((((additive_subgroup @ a) @ b) @ H) @ r) => (((ord_less_eq @ (set @ (set @ a))) @ ((((a_RCOSETS @ a) @ b) @ r) @ H)) @ ((pow @ a) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))))))).
% 0.31/1.50 thf(fact_253_abelian__monoid_Oset__add__closed, axiom, (! [B:$tType,A:$tType,G2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),A4:(set @ A),B5:(set @ A)]: (((((abelian_monoid @ A) @ B) @ G2) => ((((ord_less_eq @ (set @ A)) @ A4) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G2)) => ((((ord_less_eq @ (set @ A)) @ B5) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G2)) => (((ord_less_eq @ (set @ A)) @ (((((set_add @ A) @ B) @ G2) @ A4) @ B5)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G2)))))))).
% 0.31/1.50 thf(fact_254_Pow__singleton__iff, axiom, (! [A:$tType,X5:(set @ A),Y6:(set @ A)]: (((((pow @ A) @ X5) = (((insert @ (set @ A)) @ Y6) @ (bot_bot @ (set @ (set @ A))))) = ((X5 = (bot_bot @ (set @ A))) & (Y6 = (bot_bot @ (set @ A)))))))).
% 0.31/1.50 thf(fact_255_Pow__empty, axiom, (! [A:$tType]: ((((pow @ A) @ (bot_bot @ (set @ A))) = (((insert @ (set @ A)) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ (set @ A)))))))).
% 0.31/1.50 thf(conj_0, conjecture, ((((((mult @ (set @ ((product_prod @ a) @ a))) @ product_unit) @ ((((locali1768110497f_frac @ a) @ b) @ r) @ s)) @ ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ r2) @ s2)) @ ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ r3) @ s3)) = (((((mult @ (set @ ((product_prod @ a) @ a))) @ product_unit) @ ((((locali1768110497f_frac @ a) @ b) @ r) @ s)) @ ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ r3) @ s3)) @ ((((((locali1641774487f_frac @ a) @ a) @ product_unit) @ rel) @ r2) @ s2)))).
% 0.31/1.50 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.ZsShBFnucq/DTF2DTF27309.p
% 0.31/1.50 ------------------------
% 0.31/1.56 ---- Cleaned THF ---
% 0.31/1.56 thf(ty_t_Congruence_Opartial__object_Opartial__object__ext,type,
% 0.31/1.56 partia1265057091ct_ext: $tType > $tType > $tType ).
% 0.31/1.56
% 0.31/1.56 thf(ty_t_Congruence_Oeq__object_Oeq__object__ext,type,
% 0.31/1.56 eq_eq_object_ext: $tType > $tType > $tType ).
% 0.31/1.56
% 0.31/1.56 thf(ty_t_Group_Omonoid_Omonoid__ext,type,
% 0.31/1.56 monoid_ext: $tType > $tType > $tType ).
% 0.31/1.56
% 0.31/1.56 thf(ty_t_Ring_Oring_Oring__ext,type,
% 0.31/1.56 ring_ext: $tType > $tType > $tType ).
% 0.31/1.56
% 0.31/1.56 thf(ty_t_Product__Type_Ounit,type,
% 0.31/1.56 product_unit: $tType ).
% 0.31/1.56
% 0.31/1.56 thf(ty_t_Product__Type_Oprod,type,
% 0.31/1.56 product_prod: $tType > $tType > $tType ).
% 0.31/1.56
% 0.31/1.56 thf(ty_t_Set_Oset,type,
% 0.31/1.56 set: $tType > $tType ).
% 0.31/1.56
% 0.31/1.56 thf(ty_t_Nat_Onat,type,
% 0.31/1.56 nat: $tType ).
% 0.31/1.56
% 0.31/1.56 thf(ty_t_Int_Oint,type,
% 0.31/1.56 int: $tType ).
% 0.31/1.56
% 0.31/1.56 thf(ty_tf_b,type,
% 0.31/1.56 b: $tType ).
% 0.31/1.56
% 0.31/1.56 thf(ty_tf_a,type,
% 0.31/1.56 a: $tType ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_AbelCoset_OA__RCOSETS,type,
% 0.31/1.56 a_RCOSETS:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_AbelCoset_Oa__l__coset,type,
% 0.31/1.56 a_l_coset:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > A > ( set @ A ) > ( set @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_AbelCoset_Oa__r__coset,type,
% 0.31/1.56 a_r_coset:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > A > ( set @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_AbelCoset_Oadditive__subgroup,type,
% 0.31/1.56 additive_subgroup:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_AbelCoset_Oset__add,type,
% 0.31/1.56 set_add:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > ( set @ A ) > ( set @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Congruence_Oequivalence,type,
% 0.31/1.56 equivalence:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( eq_eq_object_ext @ A @ B ) ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Congruence_Opartial__object_Ocarrier,type,
% 0.31/1.56 partial_carrier:
% 0.31/1.56 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ Z ) > ( set @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Group_OUnits,type,
% 0.31/1.56 units:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > ( set @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Group_Ogroup,type,
% 0.31/1.56 group:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Group_Omonoid,type,
% 0.31/1.56 monoid:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Group_Omonoid_Omult,type,
% 0.31/1.56 mult:
% 0.31/1.56 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ Z ) ) > A > A > A ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Group_Omonoid_Oone,type,
% 0.31/1.56 one:
% 0.31/1.56 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ Z ) ) > A ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Group_Ounits__of,type,
% 0.31/1.56 units_of:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ product_unit ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.31/1.56 minus_minus:
% 0.31/1.56 !>[A: $tType] : ( A > A > A ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Oeq__class__of__rng__of__frac,type,
% 0.31/1.56 locali1641774487f_frac:
% 0.31/1.56 !>[A: $tType,B: $tType,C: $tType] : ( ( partia1265057091ct_ext @ ( product_prod @ A @ B ) @ ( eq_eq_object_ext @ ( product_prod @ A @ B ) @ C ) ) > A > B > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Oeq__obj__rng__of__frac,type,
% 0.31/1.56 locali1648044335f_frac:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Oeq__obj__rng__of__frac_Orec__monoid__rng__of__frac,type,
% 0.31/1.56 locali1768110497f_frac:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > ( partia1265057091ct_ext @ ( set @ ( product_prod @ A @ A ) ) @ ( monoid_ext @ ( set @ ( product_prod @ A @ A ) ) @ product_unit ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Omult__submonoid__of__crng,type,
% 0.31/1.56 locali7914339f_crng:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Omult__submonoid__of__rng,type,
% 0.31/1.56 locali1402578320of_rng:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Oset__eq__class__of__rng__of__frac,type,
% 0.31/1.56 locali990921413f_frac:
% 0.31/1.56 !>[A: $tType,B: $tType,C: $tType] : ( ( partia1265057091ct_ext @ ( product_prod @ A @ B ) @ ( eq_eq_object_ext @ ( product_prod @ A @ B ) @ C ) ) > ( set @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Localization__Mirabelle__ojffvtlkod_Osubmonoid,type,
% 0.31/1.56 locali1441642228monoid:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > ( set @ A ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.31/1.56 bot_bot:
% 0.31/1.56 !>[A: $tType] : A ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.31/1.56 ord_less_eq:
% 0.31/1.56 !>[A: $tType] : ( A > A > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Product__Type_OPair,type,
% 0.31/1.56 product_Pair:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.31/1.56 product_rec_prod:
% 0.31/1.56 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Ring_Oabelian__monoid,type,
% 0.31/1.56 abelian_monoid:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Ring_Oadd__pow,type,
% 0.31/1.56 add_pow:
% 0.31/1.56 !>[A: $tType,C: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ C ) ) ) > B > A > A ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Ring_Ofield,type,
% 0.31/1.56 field:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Ring_Oring,type,
% 0.31/1.56 ring:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Ring_Oring_Ozero,type,
% 0.31/1.56 zero:
% 0.31/1.56 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ) ) > A ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Ring_Osemiring,type,
% 0.31/1.56 semiring:
% 0.31/1.56 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Set_OCollect,type,
% 0.31/1.56 collect:
% 0.31/1.56 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Set_OPow,type,
% 0.31/1.56 pow:
% 0.31/1.56 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_Set_Oinsert,type,
% 0.31/1.56 insert:
% 0.31/1.56 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_c_member,type,
% 0.31/1.56 member:
% 0.31/1.56 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_v_R,type,
% 0.31/1.56 r: partia1265057091ct_ext @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_v_S,type,
% 0.31/1.56 s: set @ a ).
% 0.31/1.56
% 0.31/1.56 thf(sy_v_r,type,
% 0.31/1.56 r2: a ).
% 0.31/1.56
% 0.31/1.56 thf(sy_v_r_H,type,
% 0.31/1.56 r3: a ).
% 0.31/1.56
% 0.31/1.56 thf(sy_v_rel,type,
% 0.31/1.56 rel: partia1265057091ct_ext @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) ).
% 0.31/1.56
% 0.31/1.56 thf(sy_v_s,type,
% 0.31/1.56 s2: a ).
% 0.31/1.56
% 0.31/1.56 thf(sy_v_s_H,type,
% 0.31/1.56 s3: a ).
% 0.31/1.56
% 0.31/1.56 thf(fact_0_f3,axiom,
% 0.31/1.56 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r3 )
% 0.31/1.56 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r2 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_1_f4,axiom,
% 0.31/1.56 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 )
% 0.31/1.56 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ s2 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_2_eq__obj__rng__of__frac__axioms,axiom,
% 0.31/1.56 locali1648044335f_frac @ a @ b @ r @ s ).
% 0.31/1.56
% 0.31/1.56 thf(fact_3_f1,axiom,
% 0.31/1.56 ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) )
% 0.31/1.56 = ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_4_f2,axiom,
% 0.31/1.56 ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) )
% 0.31/1.56 = ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ s2 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_5_mult__submonoid__of__crng__axioms,axiom,
% 0.31/1.56 locali7914339f_crng @ a @ b @ r @ s ).
% 0.31/1.56
% 0.31/1.56 thf(fact_6_mult__submonoid__of__rng__axioms,axiom,
% 0.31/1.56 locali1402578320of_rng @ a @ b @ r @ s ).
% 0.31/1.56
% 0.31/1.56 thf(fact_7_eq__obj__rng__of__frac_Orec__monoid__rng__of__frac_Ocong,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType] :
% 0.31/1.56 ( ( locali1768110497f_frac @ A @ B )
% 0.31/1.56 = ( locali1768110497f_frac @ A @ B ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_8_submonoid__axioms,axiom,
% 0.31/1.56 locali1441642228monoid @ a @ ( ring_ext @ a @ b ) @ r @ s ).
% 0.31/1.56
% 0.31/1.56 thf(fact_9_m__closed,axiom,
% 0.31/1.56 ! [X: a,Y: a] :
% 0.31/1.56 ( ( member @ a @ X @ s )
% 0.31/1.56 => ( ( member @ a @ Y @ s )
% 0.31/1.56 => ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ s ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_10_monoid__rng__of__frac,axiom,
% 0.31/1.56 monoid @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_11_local_Osemiring__axioms,axiom,
% 0.31/1.56 semiring @ a @ b @ r ).
% 0.31/1.56
% 0.31/1.56 thf(fact_12_one__closed,axiom,
% 0.31/1.56 member @ a @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ s ).
% 0.31/1.56
% 0.31/1.56 thf(fact_13_equiv__obj__rng__of__frac,axiom,
% 0.31/1.56 equivalence @ ( product_prod @ a @ a ) @ product_unit @ rel ).
% 0.31/1.56
% 0.31/1.56 thf(fact_14_assms_I2_J,axiom,
% 0.31/1.56 member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ r3 @ s3 ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_15_assms_I1_J,axiom,
% 0.31/1.56 member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ r2 @ s2 ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_16_assoc__mult__rng__of__frac,axiom,
% 0.31/1.56 ! [R: a,S: a,T2: a,U: a,V: a,W: a] :
% 0.31/1.56 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.31/1.56 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ T2 @ U ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.31/1.56 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ V @ W ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.31/1.56 => ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ T2 @ U ) ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ V @ W ) )
% 0.31/1.56 = ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) @ ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ T2 @ U ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ V @ W ) ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_17_abelian__monoid__axioms,axiom,
% 0.31/1.56 abelian_monoid @ a @ b @ r ).
% 0.31/1.56
% 0.31/1.56 thf(fact_18_mult__rng__of__frac__fundamental__lemma,axiom,
% 0.31/1.56 ! [R: a,S: a,R2: a,S2: a] :
% 0.31/1.56 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.31/1.56 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R2 @ S2 ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.31/1.56 => ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R2 @ S2 ) )
% 0.31/1.56 = ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ R @ R2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ S @ S2 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_19_local_Oring__axioms,axiom,
% 0.31/1.56 ring @ a @ b @ r ).
% 0.31/1.56
% 0.31/1.56 thf(fact_20_closed__mult__rng__of__frac,axiom,
% 0.31/1.56 ! [R: a,S: a,T2: a,U: a] :
% 0.31/1.56 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.31/1.56 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ T2 @ U ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.31/1.56 => ( member @ ( set @ ( product_prod @ a @ a ) ) @ ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ T2 @ U ) ) @ ( locali990921413f_frac @ a @ a @ product_unit @ rel ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_21_submonoid_Oaxioms_I1_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,M: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S3: set @ A] :
% 0.31/1.56 ( ( locali1441642228monoid @ A @ B @ M @ S3 )
% 0.31/1.56 => ( monoid @ A @ B @ M ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_22_eq__obj__rng__of__frac_Oaxioms_I2_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
% 0.31/1.56 ( ( locali1648044335f_frac @ A @ B @ R3 @ S3 )
% 0.31/1.56 => ( locali7914339f_crng @ A @ B @ R3 @ S3 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_23_mult__submonoid__of__rng_Oaxioms_I2_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
% 0.31/1.56 ( ( locali1402578320of_rng @ A @ B @ R3 @ S3 )
% 0.31/1.56 => ( locali1441642228monoid @ A @ ( ring_ext @ A @ B ) @ R3 @ S3 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_24_mult__submonoid__of__rng_Oaxioms_I1_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
% 0.31/1.56 ( ( locali1402578320of_rng @ A @ B @ R3 @ S3 )
% 0.31/1.56 => ( ring @ A @ B @ R3 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_25_mult__submonoid__of__crng_Oaxioms_I2_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
% 0.31/1.56 ( ( locali7914339f_crng @ A @ B @ R3 @ S3 )
% 0.31/1.56 => ( locali1402578320of_rng @ A @ B @ R3 @ S3 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_26_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Oone__closed,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,M: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S3: set @ A] :
% 0.31/1.56 ( ( locali1441642228monoid @ A @ B @ M @ S3 )
% 0.31/1.56 => ( member @ A @ ( one @ A @ B @ M ) @ S3 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_27_mult__submonoid__of__rng__def,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType] :
% 0.31/1.56 ( ( locali1402578320of_rng @ A @ B )
% 0.31/1.56 = ( ^ [R4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S4: set @ A] :
% 0.31/1.56 ( ( ring @ A @ B @ R4 )
% 0.31/1.56 & ( locali1441642228monoid @ A @ ( ring_ext @ A @ B ) @ R4 @ S4 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_28_mult__submonoid__of__rng_Ointro,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
% 0.31/1.56 ( ( ring @ A @ B @ R3 )
% 0.31/1.56 => ( ( locali1441642228monoid @ A @ ( ring_ext @ A @ B ) @ R3 @ S3 )
% 0.31/1.56 => ( locali1402578320of_rng @ A @ B @ R3 @ S3 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_29_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Ois__submonoid,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,M: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S3: set @ A] :
% 0.31/1.56 ( ( locali1441642228monoid @ A @ B @ M @ S3 )
% 0.31/1.56 => ( locali1441642228monoid @ A @ B @ M @ S3 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_30_eq__obj__rng__of__frac_Omonoid__rng__of__frac,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
% 0.31/1.56 ( ( locali1648044335f_frac @ A @ B @ R3 @ S3 )
% 0.31/1.56 => ( monoid @ ( set @ ( product_prod @ A @ A ) ) @ product_unit @ ( locali1768110497f_frac @ A @ B @ R3 @ S3 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_31_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Om__closed,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,M: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S3: set @ A,X: A,Y: A] :
% 0.31/1.56 ( ( locali1441642228monoid @ A @ B @ M @ S3 )
% 0.31/1.56 => ( ( member @ A @ X @ S3 )
% 0.31/1.56 => ( ( member @ A @ Y @ S3 )
% 0.31/1.56 => ( member @ A @ ( mult @ A @ B @ M @ X @ Y ) @ S3 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_32_right__unit__mult__rng__of__frac,axiom,
% 0.31/1.56 ! [R: a,S: a] :
% 0.31/1.56 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.31/1.56 => ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) @ ( one @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) ) )
% 0.31/1.56 = ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_33_left__unit__mult__rng__of__frac,axiom,
% 0.31/1.56 ! [R: a,S: a] :
% 0.31/1.56 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.31/1.56 => ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( one @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) )
% 0.31/1.56 = ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_34_zero__in__mult__submonoid,axiom,
% 0.31/1.56 ! [R: a,S: a,R2: a,S2: a] :
% 0.31/1.56 ( ( member @ a @ ( zero @ a @ b @ r ) @ s )
% 0.31/1.56 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.31/1.56 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R2 @ S2 ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.31/1.56 => ( ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S )
% 0.31/1.56 = ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R2 @ S2 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_35_non__empty__class,axiom,
% 0.31/1.56 ! [R: a,S: a] :
% 0.31/1.56 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
% 0.31/1.56 => ( ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S )
% 0.31/1.56 != ( bot_bot @ ( set @ ( product_prod @ a @ a ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_36_one__unique,axiom,
% 0.31/1.56 ! [U: a] :
% 0.31/1.56 ( ( member @ a @ U @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( ! [X2: a] :
% 0.31/1.56 ( ( member @ a @ X2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ U @ X2 )
% 0.31/1.56 = X2 ) )
% 0.31/1.56 => ( U
% 0.31/1.56 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_37_inv__unique,axiom,
% 0.31/1.56 ! [Y: a,X: a,Y2: a] :
% 0.31/1.56 ( ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ X )
% 0.31/1.56 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/1.56 => ( ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y2 )
% 0.31/1.56 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/1.56 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( ( member @ a @ Y2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( Y = Y2 ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_38_Units__inv__comm,axiom,
% 0.31/1.56 ! [X: a,Y: a] :
% 0.31/1.56 ( ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y )
% 0.31/1.56 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/1.56 => ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/1.56 => ( ( member @ a @ Y @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/1.56 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ X )
% 0.31/1.56 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_39_prod_Oinject,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 0.31/1.56 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.31/1.56 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 0.31/1.56 = ( ( X1 = Y1 )
% 0.31/1.56 & ( X22 = Y22 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_40_old_Oprod_Oinject,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.31/1.56 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.31/1.56 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.31/1.56 = ( ( A2 = A3 )
% 0.31/1.56 & ( B2 = B3 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_41_semiring_Oaxioms_I1_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/1.56 ( ( semiring @ A @ B @ R3 )
% 0.31/1.56 => ( abelian_monoid @ A @ B @ R3 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_42_monoid__axioms,axiom,
% 0.31/1.56 monoid @ a @ ( ring_ext @ a @ b ) @ r ).
% 0.31/1.56
% 0.31/1.56 thf(fact_43_m__assoc,axiom,
% 0.31/1.56 ! [X: a,Y: a,Z2: a] :
% 0.31/1.56 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ Z2 )
% 0.31/1.56 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_44_m__comm,axiom,
% 0.31/1.56 ! [X: a,Y: a] :
% 0.31/1.56 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y )
% 0.31/1.56 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ X ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_45_mem__Collect__eq,axiom,
% 0.31/1.56 ! [A: $tType,A2: A,P: A > $o] :
% 0.31/1.56 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.31/1.56 = ( P @ A2 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_46_Collect__mem__eq,axiom,
% 0.31/1.56 ! [A: $tType,A4: set @ A] :
% 0.31/1.56 ( ( collect @ A
% 0.31/1.56 @ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
% 0.31/1.56 = A4 ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_47_Collect__cong,axiom,
% 0.31/1.56 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/1.56 ( ! [X2: A] :
% 0.31/1.56 ( ( P @ X2 )
% 0.31/1.56 = ( Q @ X2 ) )
% 0.31/1.56 => ( ( collect @ A @ P )
% 0.31/1.56 = ( collect @ A @ Q ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_48_ext,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.31/1.56 ( ! [X2: A] :
% 0.31/1.56 ( ( F @ X2 )
% 0.31/1.56 = ( G @ X2 ) )
% 0.31/1.56 => ( F = G ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_49_m__lcomm,axiom,
% 0.31/1.56 ! [X: a,Y: a,Z2: a] :
% 0.31/1.56 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) )
% 0.31/1.56 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Z2 ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_50_Units__closed,axiom,
% 0.31/1.56 ! [X: a] :
% 0.31/1.56 ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/1.56 => ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_51_Units__l__inv__ex,axiom,
% 0.31/1.56 ! [X: a] :
% 0.31/1.56 ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/1.56 => ? [X2: a] :
% 0.31/1.56 ( ( member @ a @ X2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 & ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X2 @ X )
% 0.31/1.56 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_52_Units__r__inv__ex,axiom,
% 0.31/1.56 ! [X: a] :
% 0.31/1.56 ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/1.56 => ? [X2: a] :
% 0.31/1.56 ( ( member @ a @ X2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 & ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ X2 )
% 0.31/1.56 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_53_zero__closed,axiom,
% 0.31/1.56 member @ a @ ( zero @ a @ b @ r ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_54_semiring__simprules_I3_J,axiom,
% 0.31/1.56 ! [X: a,Y: a] :
% 0.31/1.56 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_55_semiring__simprules_I4_J,axiom,
% 0.31/1.56 member @ a @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_56_Units__m__closed,axiom,
% 0.31/1.56 ! [X: a,Y: a] :
% 0.31/1.56 ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/1.56 => ( ( member @ a @ Y @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/1.56 => ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_57_Units__one__closed,axiom,
% 0.31/1.56 member @ a @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_58_l__null,axiom,
% 0.31/1.56 ! [X: a] :
% 0.31/1.56 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( zero @ a @ b @ r ) @ X )
% 0.31/1.56 = ( zero @ a @ b @ r ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_59_r__null,axiom,
% 0.31/1.56 ! [X: a] :
% 0.31/1.56 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( zero @ a @ b @ r ) )
% 0.31/1.56 = ( zero @ a @ b @ r ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_60_l__one,axiom,
% 0.31/1.56 ! [X: a] :
% 0.31/1.56 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ X )
% 0.31/1.56 = X ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_61_r__one,axiom,
% 0.31/1.56 ! [X: a] :
% 0.31/1.56 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/1.56 = X ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_62_Units__l__cancel,axiom,
% 0.31/1.56 ! [X: a,Y: a,Z2: a] :
% 0.31/1.56 ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/1.56 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y )
% 0.31/1.56 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Z2 ) )
% 0.31/1.56 = ( Y = Z2 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_63_abelian__monoidE_I2_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/1.56 ( ( abelian_monoid @ A @ B @ R3 )
% 0.31/1.56 => ( member @ A @ ( zero @ A @ B @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_64_ring_Oring__simprules_I2_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/1.56 ( ( ring @ A @ B @ R3 )
% 0.31/1.56 => ( member @ A @ ( zero @ A @ B @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_65_semiring_Osemiring__simprules_I2_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/1.56 ( ( semiring @ A @ B @ R3 )
% 0.31/1.56 => ( member @ A @ ( zero @ A @ B @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_66_abelian__monoid_Ozero__closed,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/1.56 ( ( abelian_monoid @ A @ B @ G2 )
% 0.31/1.56 => ( member @ A @ ( zero @ A @ B @ G2 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G2 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_67_ring_Ois__monoid,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/1.56 ( ( ring @ A @ B @ R3 )
% 0.31/1.56 => ( monoid @ A @ ( ring_ext @ A @ B ) @ R3 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_68_semiring_Oaxioms_I2_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/1.56 ( ( semiring @ A @ B @ R3 )
% 0.31/1.56 => ( monoid @ A @ ( ring_ext @ A @ B ) @ R3 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_69_ring_Oring__simprules_I24_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.31/1.56 ( ( ring @ A @ B @ R3 )
% 0.31/1.56 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.56 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( zero @ A @ B @ R3 ) @ X )
% 0.31/1.56 = ( zero @ A @ B @ R3 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_70_ring_Oring__simprules_I25_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.31/1.56 ( ( ring @ A @ B @ R3 )
% 0.31/1.56 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.56 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ ( zero @ A @ B @ R3 ) )
% 0.31/1.56 = ( zero @ A @ B @ R3 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_71_semiring_Or__null,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.31/1.56 ( ( semiring @ A @ B @ R3 )
% 0.31/1.56 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.56 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ ( zero @ A @ B @ R3 ) )
% 0.31/1.56 = ( zero @ A @ B @ R3 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_72_semiring_Ol__null,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.31/1.56 ( ( semiring @ A @ B @ R3 )
% 0.31/1.56 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.56 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( zero @ A @ B @ R3 ) @ X )
% 0.31/1.56 = ( zero @ A @ B @ R3 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_73_ring_Oring__simprules_I5_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 0.31/1.56 ( ( ring @ A @ B @ R3 )
% 0.31/1.56 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.56 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.56 => ( member @ A @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_74_ring_Oring__simprules_I11_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 0.31/1.56 ( ( ring @ A @ B @ R3 )
% 0.31/1.56 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.56 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.56 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.56 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ Y ) @ Z2 )
% 0.31/1.56 = ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ Y @ Z2 ) ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_75_semiring_Osemiring__simprules_I3_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 0.31/1.56 ( ( semiring @ A @ B @ R3 )
% 0.31/1.56 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.56 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.56 => ( member @ A @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_76_semiring_Osemiring__simprules_I8_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 0.31/1.56 ( ( semiring @ A @ B @ R3 )
% 0.31/1.56 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.56 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.56 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.56 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ Y ) @ Z2 )
% 0.31/1.56 = ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ Y @ Z2 ) ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_77_ring_Oring__simprules_I6_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/1.56 ( ( ring @ A @ B @ R3 )
% 0.31/1.56 => ( member @ A @ ( one @ A @ ( ring_ext @ A @ B ) @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_78_semiring_Osemiring__simprules_I4_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/1.56 ( ( semiring @ A @ B @ R3 )
% 0.31/1.56 => ( member @ A @ ( one @ A @ ( ring_ext @ A @ B ) @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_79_ring_Oring__simprules_I12_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.31/1.56 ( ( ring @ A @ B @ R3 )
% 0.31/1.56 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.56 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( one @ A @ ( ring_ext @ A @ B ) @ R3 ) @ X )
% 0.31/1.56 = X ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_80_semiring_Osemiring__simprules_I9_J,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.31/1.56 ( ( semiring @ A @ B @ R3 )
% 0.31/1.56 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.56 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( one @ A @ ( ring_ext @ A @ B ) @ R3 ) @ X )
% 0.31/1.56 = X ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_81_old_Oprod_Oinducts,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.31/1.56 ( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
% 0.31/1.56 => ( P @ Prod ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_82_old_Oprod_Oexhaust,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.31/1.56 ~ ! [A5: A,B4: B] :
% 0.31/1.56 ( Y
% 0.31/1.56 != ( product_Pair @ A @ B @ A5 @ B4 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_83_prod__induct7,axiom,
% 0.31/1.56 ! [G3: $tType,F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) )] :
% 0.31/1.56 ( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G3 ) @ E2 @ ( product_Pair @ F2 @ G3 @ F3 @ G4 ) ) ) ) ) ) )
% 0.31/1.56 => ( P @ X ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_84_prod__induct6,axiom,
% 0.31/1.56 ! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
% 0.31/1.56 ( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
% 0.31/1.56 => ( P @ X ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_85_prod__induct5,axiom,
% 0.31/1.56 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.31/1.56 ( ! [A5: A,B4: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.31/1.56 => ( P @ X ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_86_prod__induct4,axiom,
% 0.31/1.56 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.31/1.56 ( ! [A5: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 0.31/1.56 => ( P @ X ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_87_prod__induct3,axiom,
% 0.31/1.56 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.31/1.56 ( ! [A5: A,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
% 0.31/1.56 => ( P @ X ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_88_prod__cases7,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) )] :
% 0.31/1.56 ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2,G4: G3] :
% 0.31/1.56 ( Y
% 0.31/1.56 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G3 ) @ E2 @ ( product_Pair @ F2 @ G3 @ F3 @ G4 ) ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_89_prod__cases6,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
% 0.31/1.56 ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2] :
% 0.31/1.56 ( Y
% 0.31/1.56 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_90_prod__cases5,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.31/1.56 ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E] :
% 0.31/1.56 ( Y
% 0.31/1.56 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_91_prod__cases4,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.31/1.56 ~ ! [A5: A,B4: B,C2: C,D2: D] :
% 0.31/1.56 ( Y
% 0.31/1.56 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_92_prod__cases3,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.31/1.56 ~ ! [A5: A,B4: B,C2: C] :
% 0.31/1.56 ( Y
% 0.31/1.56 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_93_Pair__inject,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.31/1.56 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.31/1.56 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.31/1.56 => ~ ( ( A2 = A3 )
% 0.31/1.56 => ( B2 != B3 ) ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_94_prod__cases,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.31/1.56 ( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
% 0.31/1.56 => ( P @ P2 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_95_surj__pair,axiom,
% 0.31/1.56 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.31/1.56 ? [X2: A,Y3: B] :
% 0.31/1.56 ( P2
% 0.31/1.56 = ( product_Pair @ A @ B @ X2 @ Y3 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_96_ring_Ois__ring,axiom,
% 0.31/1.56 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/1.56 ( ( ring @ A @ B @ R3 )
% 0.31/1.56 => ( ring @ A @ B @ R3 ) ) ).
% 0.31/1.56
% 0.31/1.56 thf(fact_97_cring__fieldI2,axiom,
% 0.31/1.56 ( ( ( zero @ a @ b @ r )
% 0.31/1.56 != ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/1.56 => ( ! [A5: a] :
% 0.31/1.56 ( ( member @ a @ A5 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.56 => ( ( A5
% 0.31/1.56 != ( zero @ a @ b @ r ) )
% 0.31/1.56 => ? [X4: a] :
% 0.31/1.56 ( ( member @ a @ X4 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 & ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A5 @ X4 )
% 0.31/1.57 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) )
% 0.31/1.57 => ( field @ a @ b @ r ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_98_monoid_Or__one,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
% 0.31/1.57 ( ( monoid @ A @ B @ G2 )
% 0.31/1.57 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( mult @ A @ B @ G2 @ X @ ( one @ A @ B @ G2 ) )
% 0.31/1.57 = X ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_99_monoid_Ol__one,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
% 0.31/1.57 ( ( monoid @ A @ B @ G2 )
% 0.31/1.57 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( mult @ A @ B @ G2 @ ( one @ A @ B @ G2 ) @ X )
% 0.31/1.57 = X ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_100_monoid_Oone__closed,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/1.57 ( ( monoid @ A @ B @ G2 )
% 0.31/1.57 => ( member @ A @ ( one @ A @ B @ G2 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_101_monoid_OUnits__r__inv__ex,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
% 0.31/1.57 ( ( monoid @ A @ B @ G2 )
% 0.31/1.57 => ( ( member @ A @ X @ ( units @ A @ B @ G2 ) )
% 0.31/1.57 => ? [X2: A] :
% 0.31/1.57 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 & ( ( mult @ A @ B @ G2 @ X @ X2 )
% 0.31/1.57 = ( one @ A @ B @ G2 ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_102_monoid_OUnits__l__inv__ex,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
% 0.31/1.57 ( ( monoid @ A @ B @ G2 )
% 0.31/1.57 => ( ( member @ A @ X @ ( units @ A @ B @ G2 ) )
% 0.31/1.57 => ? [X2: A] :
% 0.31/1.57 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 & ( ( mult @ A @ B @ G2 @ X2 @ X )
% 0.31/1.57 = ( one @ A @ B @ G2 ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_103_monoid_OUnits__inv__comm,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A] :
% 0.31/1.57 ( ( monoid @ A @ B @ G2 )
% 0.31/1.57 => ( ( ( mult @ A @ B @ G2 @ X @ Y )
% 0.31/1.57 = ( one @ A @ B @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ X @ ( units @ A @ B @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ Y @ ( units @ A @ B @ G2 ) )
% 0.31/1.57 => ( ( mult @ A @ B @ G2 @ Y @ X )
% 0.31/1.57 = ( one @ A @ B @ G2 ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_104_add__pow__ldistr__int,axiom,
% 0.31/1.57 ! [A2: a,B2: a,K: int] :
% 0.31/1.57 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add_pow @ a @ b @ int @ r @ K @ A2 ) @ B2 )
% 0.31/1.57 = ( add_pow @ a @ b @ int @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_105_add__pow__rdistr__int,axiom,
% 0.31/1.57 ! [A2: a,B2: a,K: int] :
% 0.31/1.57 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ ( add_pow @ a @ b @ int @ r @ K @ B2 ) )
% 0.31/1.57 = ( add_pow @ a @ b @ int @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_106_add_Oint__pow__closed,axiom,
% 0.31/1.57 ! [X: a,I: int] :
% 0.31/1.57 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( member @ a @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_107_add_Oint__pow__one,axiom,
% 0.31/1.57 ! [Z2: int] :
% 0.31/1.57 ( ( add_pow @ a @ b @ int @ r @ Z2 @ ( zero @ a @ b @ r ) )
% 0.31/1.57 = ( zero @ a @ b @ r ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_108_field_Ois__ring,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/1.57 ( ( field @ A @ B @ R3 )
% 0.31/1.57 => ( ring @ A @ B @ R3 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_109_one__not__zero,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/1.57 ( ( field @ A @ B @ R3 )
% 0.31/1.57 => ( ( one @ A @ ( ring_ext @ A @ B ) @ R3 )
% 0.31/1.57 != ( zero @ A @ B @ R3 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_110_integral,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A] :
% 0.31/1.57 ( ( field @ A @ B @ R3 )
% 0.31/1.57 => ( ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 )
% 0.31/1.57 = ( zero @ A @ B @ R3 ) )
% 0.31/1.57 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.57 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.57 => ( ( A2
% 0.31/1.57 = ( zero @ A @ B @ R3 ) )
% 0.31/1.57 | ( B2
% 0.31/1.57 = ( zero @ A @ B @ R3 ) ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_111_ring_Oadd__pow__ldistr__int,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A,K: int] :
% 0.31/1.57 ( ( ring @ A @ B @ R3 )
% 0.31/1.57 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.57 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.57 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( add_pow @ A @ B @ int @ R3 @ K @ A2 ) @ B2 )
% 0.31/1.57 = ( add_pow @ A @ B @ int @ R3 @ K @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_112_ring_Oadd__pow__rdistr__int,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A,K: int] :
% 0.31/1.57 ( ( ring @ A @ B @ R3 )
% 0.31/1.57 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.57 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.57 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ ( add_pow @ A @ B @ int @ R3 @ K @ B2 ) )
% 0.31/1.57 = ( add_pow @ A @ B @ int @ R3 @ K @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_113_monoid_Ocarrier__not__empty,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/1.57 ( ( monoid @ A @ B @ G2 )
% 0.31/1.57 => ( ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 )
% 0.31/1.57 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_114_monoid_Om__assoc,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A,Z2: A] :
% 0.31/1.57 ( ( monoid @ A @ B @ G2 )
% 0.31/1.57 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( mult @ A @ B @ G2 @ ( mult @ A @ B @ G2 @ X @ Y ) @ Z2 )
% 0.31/1.57 = ( mult @ A @ B @ G2 @ X @ ( mult @ A @ B @ G2 @ Y @ Z2 ) ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_115_monoid_Om__closed,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A] :
% 0.31/1.57 ( ( monoid @ A @ B @ G2 )
% 0.31/1.57 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( member @ A @ ( mult @ A @ B @ G2 @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_116_monoid_OUnits__closed,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
% 0.31/1.57 ( ( monoid @ A @ B @ G2 )
% 0.31/1.57 => ( ( member @ A @ X @ ( units @ A @ B @ G2 ) )
% 0.31/1.57 => ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_117_monoid_OUnits__m__closed,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A] :
% 0.31/1.57 ( ( monoid @ A @ B @ G2 )
% 0.31/1.57 => ( ( member @ A @ X @ ( units @ A @ B @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ Y @ ( units @ A @ B @ G2 ) )
% 0.31/1.57 => ( member @ A @ ( mult @ A @ B @ G2 @ X @ Y ) @ ( units @ A @ B @ G2 ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_118_monoid_OUnits__one__closed,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/1.57 ( ( monoid @ A @ B @ G2 )
% 0.31/1.57 => ( member @ A @ ( one @ A @ B @ G2 ) @ ( units @ A @ B @ G2 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_119_monoidI,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/1.57 ( ! [X2: A,Y3: A] :
% 0.31/1.57 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ Y3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( member @ A @ ( mult @ A @ B @ G2 @ X2 @ Y3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) )
% 0.31/1.57 => ( ( member @ A @ ( one @ A @ B @ G2 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ! [X2: A,Y3: A,Z3: A] :
% 0.31/1.57 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ Y3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ Z3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( mult @ A @ B @ G2 @ ( mult @ A @ B @ G2 @ X2 @ Y3 ) @ Z3 )
% 0.31/1.57 = ( mult @ A @ B @ G2 @ X2 @ ( mult @ A @ B @ G2 @ Y3 @ Z3 ) ) ) ) ) )
% 0.31/1.57 => ( ! [X2: A] :
% 0.31/1.57 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( mult @ A @ B @ G2 @ ( one @ A @ B @ G2 ) @ X2 )
% 0.31/1.57 = X2 ) )
% 0.31/1.57 => ( ! [X2: A] :
% 0.31/1.57 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( mult @ A @ B @ G2 @ X2 @ ( one @ A @ B @ G2 ) )
% 0.31/1.57 = X2 ) )
% 0.31/1.57 => ( monoid @ A @ B @ G2 ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_120_Group_Omonoid__def,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType] :
% 0.31/1.57 ( ( monoid @ A @ B )
% 0.31/1.57 = ( ^ [G5: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/1.57 ( ! [X3: A,Y4: A] :
% 0.31/1.57 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
% 0.31/1.57 => ( ( member @ A @ Y4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
% 0.31/1.57 => ( member @ A @ ( mult @ A @ B @ G5 @ X3 @ Y4 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) ) ) )
% 0.31/1.57 & ! [X3: A,Y4: A,Z4: A] :
% 0.31/1.57 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
% 0.31/1.57 => ( ( member @ A @ Y4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
% 0.31/1.57 => ( ( member @ A @ Z4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
% 0.31/1.57 => ( ( mult @ A @ B @ G5 @ ( mult @ A @ B @ G5 @ X3 @ Y4 ) @ Z4 )
% 0.31/1.57 = ( mult @ A @ B @ G5 @ X3 @ ( mult @ A @ B @ G5 @ Y4 @ Z4 ) ) ) ) ) )
% 0.31/1.57 & ( member @ A @ ( one @ A @ B @ G5 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
% 0.31/1.57 & ! [X3: A] :
% 0.31/1.57 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
% 0.31/1.57 => ( ( mult @ A @ B @ G5 @ ( one @ A @ B @ G5 ) @ X3 )
% 0.31/1.57 = X3 ) )
% 0.31/1.57 & ! [X3: A] :
% 0.31/1.57 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
% 0.31/1.57 => ( ( mult @ A @ B @ G5 @ X3 @ ( one @ A @ B @ G5 ) )
% 0.31/1.57 = X3 ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_121_Group_Omonoid_Ointro,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/1.57 ( ! [X2: A,Y3: A] :
% 0.31/1.57 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ Y3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( member @ A @ ( mult @ A @ B @ G2 @ X2 @ Y3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) )
% 0.31/1.57 => ( ! [X2: A,Y3: A,Z3: A] :
% 0.31/1.57 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ Y3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ Z3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( mult @ A @ B @ G2 @ ( mult @ A @ B @ G2 @ X2 @ Y3 ) @ Z3 )
% 0.31/1.57 = ( mult @ A @ B @ G2 @ X2 @ ( mult @ A @ B @ G2 @ Y3 @ Z3 ) ) ) ) ) )
% 0.31/1.57 => ( ( member @ A @ ( one @ A @ B @ G2 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ! [X2: A] :
% 0.31/1.57 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( mult @ A @ B @ G2 @ ( one @ A @ B @ G2 ) @ X2 )
% 0.31/1.57 = X2 ) )
% 0.31/1.57 => ( ! [X2: A] :
% 0.31/1.57 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( mult @ A @ B @ G2 @ X2 @ ( one @ A @ B @ G2 ) )
% 0.31/1.57 = X2 ) )
% 0.31/1.57 => ( monoid @ A @ B @ G2 ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_122_monoid_Oinv__unique,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),Y: A,X: A,Y2: A] :
% 0.31/1.57 ( ( monoid @ A @ B @ G2 )
% 0.31/1.57 => ( ( ( mult @ A @ B @ G2 @ Y @ X )
% 0.31/1.57 = ( one @ A @ B @ G2 ) )
% 0.31/1.57 => ( ( ( mult @ A @ B @ G2 @ X @ Y2 )
% 0.31/1.57 = ( one @ A @ B @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ Y2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( Y = Y2 ) ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_123_monoid_Oone__unique,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),U: A] :
% 0.31/1.57 ( ( monoid @ A @ B @ G2 )
% 0.31/1.57 => ( ( member @ A @ U @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ! [X2: A] :
% 0.31/1.57 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( mult @ A @ B @ G2 @ U @ X2 )
% 0.31/1.57 = X2 ) )
% 0.31/1.57 => ( U
% 0.31/1.57 = ( one @ A @ B @ G2 ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_124_monoid_OUnits__l__cancel,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A,Z2: A] :
% 0.31/1.57 ( ( monoid @ A @ B @ G2 )
% 0.31/1.57 => ( ( member @ A @ X @ ( units @ A @ B @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( ( mult @ A @ B @ G2 @ X @ Y )
% 0.31/1.57 = ( mult @ A @ B @ G2 @ X @ Z2 ) )
% 0.31/1.57 = ( Y = Z2 ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_125_subset,axiom,
% 0.31/1.57 ord_less_eq @ ( set @ a ) @ s @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_126_add__pow__rdistr,axiom,
% 0.31/1.57 ! [A2: a,B2: a,K: nat] :
% 0.31/1.57 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ ( add_pow @ a @ b @ nat @ r @ K @ B2 ) )
% 0.31/1.57 = ( add_pow @ a @ b @ nat @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_127_add__pow__ldistr,axiom,
% 0.31/1.57 ! [A2: a,B2: a,K: nat] :
% 0.31/1.57 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add_pow @ a @ b @ nat @ r @ K @ A2 ) @ B2 )
% 0.31/1.57 = ( add_pow @ a @ b @ nat @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_128_old_Oprod_Orec,axiom,
% 0.31/1.57 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.31/1.57 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.31/1.57 = ( F1 @ A2 @ B2 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_129_group__l__invI,axiom,
% 0.31/1.57 ( ! [X2: a] :
% 0.31/1.57 ( ( member @ a @ X2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ? [Xa: a] :
% 0.31/1.57 ( ( member @ a @ Xa @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 & ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Xa @ X2 )
% 0.31/1.57 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) )
% 0.31/1.57 => ( group @ a @ ( ring_ext @ a @ b ) @ r ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_130_carrier__not__empty,axiom,
% 0.31/1.57 ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 0.31/1.57 != ( bot_bot @ ( set @ a ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_131_add_Onat__pow__closed,axiom,
% 0.31/1.57 ! [X: a,N: nat] :
% 0.31/1.57 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( member @ a @ ( add_pow @ a @ b @ nat @ r @ N @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_132_add_Onat__pow__one,axiom,
% 0.31/1.57 ! [N: nat] :
% 0.31/1.57 ( ( add_pow @ a @ b @ nat @ r @ N @ ( zero @ a @ b @ r ) )
% 0.31/1.57 = ( zero @ a @ b @ r ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_133_group_Ois__group,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/1.57 ( ( group @ A @ B @ G2 )
% 0.31/1.57 => ( group @ A @ B @ G2 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_134_group_OUnits,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/1.57 ( ( group @ A @ B @ G2 )
% 0.31/1.57 => ( ord_less_eq @ ( set @ A ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) @ ( units @ A @ B @ G2 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_135_group_Ois__monoid,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/1.57 ( ( group @ A @ B @ G2 )
% 0.31/1.57 => ( monoid @ A @ B @ G2 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_136_Group_Ogroup_Oright__cancel,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A,Z2: A] :
% 0.31/1.57 ( ( group @ A @ B @ G2 )
% 0.31/1.57 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( ( mult @ A @ B @ G2 @ Y @ X )
% 0.31/1.57 = ( mult @ A @ B @ G2 @ Z2 @ X ) )
% 0.31/1.57 = ( Y = Z2 ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_137_group_OUnits__eq,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/1.57 ( ( group @ A @ B @ G2 )
% 0.31/1.57 => ( ( units @ A @ B @ G2 )
% 0.31/1.57 = ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_138_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Osubset,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,M: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S3: set @ A] :
% 0.31/1.57 ( ( locali1441642228monoid @ A @ B @ M @ S3 )
% 0.31/1.57 => ( ord_less_eq @ ( set @ A ) @ S3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ M ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_139_group_Or__cancel__one_H,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,A2: A] :
% 0.31/1.57 ( ( group @ A @ B @ G2 )
% 0.31/1.57 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( X
% 0.31/1.57 = ( mult @ A @ B @ G2 @ A2 @ X ) )
% 0.31/1.57 = ( A2
% 0.31/1.57 = ( one @ A @ B @ G2 ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_140_group_Ol__cancel__one_H,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,A2: A] :
% 0.31/1.57 ( ( group @ A @ B @ G2 )
% 0.31/1.57 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( X
% 0.31/1.57 = ( mult @ A @ B @ G2 @ X @ A2 ) )
% 0.31/1.57 = ( A2
% 0.31/1.57 = ( one @ A @ B @ G2 ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_141_group_Or__cancel__one,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,A2: A] :
% 0.31/1.57 ( ( group @ A @ B @ G2 )
% 0.31/1.57 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( ( mult @ A @ B @ G2 @ A2 @ X )
% 0.31/1.57 = X )
% 0.31/1.57 = ( A2
% 0.31/1.57 = ( one @ A @ B @ G2 ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_142_group_Ol__cancel__one,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,A2: A] :
% 0.31/1.57 ( ( group @ A @ B @ G2 )
% 0.31/1.57 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( ( mult @ A @ B @ G2 @ X @ A2 )
% 0.31/1.57 = X )
% 0.31/1.57 = ( A2
% 0.31/1.57 = ( one @ A @ B @ G2 ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_143_group_Or__inv__ex,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
% 0.31/1.57 ( ( group @ A @ B @ G2 )
% 0.31/1.57 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ? [X2: A] :
% 0.31/1.57 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 & ( ( mult @ A @ B @ G2 @ X @ X2 )
% 0.31/1.57 = ( one @ A @ B @ G2 ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_144_group_Ol__inv__ex,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
% 0.31/1.57 ( ( group @ A @ B @ G2 )
% 0.31/1.57 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ? [X2: A] :
% 0.31/1.57 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 & ( ( mult @ A @ B @ G2 @ X2 @ X )
% 0.31/1.57 = ( one @ A @ B @ G2 ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_145_group_Oinv__comm,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A] :
% 0.31/1.57 ( ( group @ A @ B @ G2 )
% 0.31/1.57 => ( ( ( mult @ A @ B @ G2 @ X @ Y )
% 0.31/1.57 = ( one @ A @ B @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( mult @ A @ B @ G2 @ Y @ X )
% 0.31/1.57 = ( one @ A @ B @ G2 ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_146_groupI,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/1.57 ( ! [X2: A] :
% 0.31/1.57 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ! [Y3: A] :
% 0.31/1.57 ( ( member @ A @ Y3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( member @ A @ ( mult @ A @ B @ G2 @ X2 @ Y3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) )
% 0.31/1.57 => ( ( member @ A @ ( one @ A @ B @ G2 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ! [X2: A] :
% 0.31/1.57 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ! [Y3: A] :
% 0.31/1.57 ( ( member @ A @ Y3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ! [Z3: A] :
% 0.31/1.57 ( ( member @ A @ Z3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( mult @ A @ B @ G2 @ ( mult @ A @ B @ G2 @ X2 @ Y3 ) @ Z3 )
% 0.31/1.57 = ( mult @ A @ B @ G2 @ X2 @ ( mult @ A @ B @ G2 @ Y3 @ Z3 ) ) ) ) ) )
% 0.31/1.57 => ( ! [X2: A] :
% 0.31/1.57 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ( ( mult @ A @ B @ G2 @ ( one @ A @ B @ G2 ) @ X2 )
% 0.31/1.57 = X2 ) )
% 0.31/1.57 => ( ! [X2: A] :
% 0.31/1.57 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ? [Xa: A] :
% 0.31/1.57 ( ( member @ A @ Xa @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 & ( ( mult @ A @ B @ G2 @ Xa @ X2 )
% 0.31/1.57 = ( one @ A @ B @ G2 ) ) ) )
% 0.31/1.57 => ( group @ A @ B @ G2 ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_147_monoid_Ogroup__l__invI,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/1.57 ( ( monoid @ A @ B @ G2 )
% 0.31/1.57 => ( ! [X2: A] :
% 0.31/1.57 ( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 => ? [Xa: A] :
% 0.31/1.57 ( ( member @ A @ Xa @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
% 0.31/1.57 & ( ( mult @ A @ B @ G2 @ Xa @ X2 )
% 0.31/1.57 = ( one @ A @ B @ G2 ) ) ) )
% 0.31/1.57 => ( group @ A @ B @ G2 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_148_semiring_Oadd__pow__rdistr,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A,K: nat] :
% 0.31/1.57 ( ( semiring @ A @ B @ R3 )
% 0.31/1.57 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.57 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.57 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ ( add_pow @ A @ B @ nat @ R3 @ K @ B2 ) )
% 0.31/1.57 = ( add_pow @ A @ B @ nat @ R3 @ K @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_149_semiring_Oadd__pow__ldistr,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A,K: nat] :
% 0.31/1.57 ( ( semiring @ A @ B @ R3 )
% 0.31/1.57 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.57 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/1.57 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( add_pow @ A @ B @ nat @ R3 @ K @ A2 ) @ B2 )
% 0.31/1.57 = ( add_pow @ A @ B @ nat @ R3 @ K @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_150_a__lcos__mult__one,axiom,
% 0.31/1.57 ! [M: set @ a] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ a ) @ M @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( a_l_coset @ a @ b @ r @ ( zero @ a @ b @ r ) @ M )
% 0.31/1.57 = M ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_151_set__add__closed,axiom,
% 0.31/1.57 ! [A4: set @ a,B5: set @ a] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ a ) @ A4 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( ord_less_eq @ ( set @ a ) @ B5 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ord_less_eq @ ( set @ a ) @ ( set_add @ a @ b @ r @ A4 @ B5 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_152_setadd__subset__G,axiom,
% 0.31/1.57 ! [H: set @ a,K2: set @ a] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( ord_less_eq @ ( set @ a ) @ K2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ord_less_eq @ ( set @ a ) @ ( set_add @ a @ b @ r @ H @ K2 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_153_subset__empty,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 0.31/1.57 = ( A4
% 0.31/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_154_empty__iff,axiom,
% 0.31/1.57 ! [A: $tType,C3: A] :
% 0.31/1.57 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_155_all__not__in__conv,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A] :
% 0.31/1.57 ( ( ! [X3: A] :
% 0.31/1.57 ~ ( member @ A @ X3 @ A4 ) )
% 0.31/1.57 = ( A4
% 0.31/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_156_Collect__empty__eq,axiom,
% 0.31/1.57 ! [A: $tType,P: A > $o] :
% 0.31/1.57 ( ( ( collect @ A @ P )
% 0.31/1.57 = ( bot_bot @ ( set @ A ) ) )
% 0.31/1.57 = ( ! [X3: A] :
% 0.31/1.57 ~ ( P @ X3 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_157_empty__Collect__eq,axiom,
% 0.31/1.57 ! [A: $tType,P: A > $o] :
% 0.31/1.57 ( ( ( bot_bot @ ( set @ A ) )
% 0.31/1.57 = ( collect @ A @ P ) )
% 0.31/1.57 = ( ! [X3: A] :
% 0.31/1.57 ~ ( P @ X3 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_158_subsetI,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.31/1.57 ( ! [X2: A] :
% 0.31/1.57 ( ( member @ A @ X2 @ A4 )
% 0.31/1.57 => ( member @ A @ X2 @ B5 ) )
% 0.31/1.57 => ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_159_subset__antisym,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.31/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ A4 )
% 0.31/1.57 => ( A4 = B5 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_160_a__l__coset__subset__G,axiom,
% 0.31/1.57 ! [H: set @ a,X: a] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ord_less_eq @ ( set @ a ) @ ( a_l_coset @ a @ b @ r @ X @ H ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_161_empty__subsetI,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_162_emptyE,axiom,
% 0.31/1.57 ! [A: $tType,A2: A] :
% 0.31/1.57 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_163_equals0D,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,A2: A] :
% 0.31/1.57 ( ( A4
% 0.31/1.57 = ( bot_bot @ ( set @ A ) ) )
% 0.31/1.57 => ~ ( member @ A @ A2 @ A4 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_164_equals0I,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A] :
% 0.31/1.57 ( ! [Y3: A] :
% 0.31/1.57 ~ ( member @ A @ Y3 @ A4 )
% 0.31/1.57 => ( A4
% 0.31/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_165_ex__in__conv,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A] :
% 0.31/1.57 ( ( ? [X3: A] : ( member @ A @ X3 @ A4 ) )
% 0.31/1.57 = ( A4
% 0.31/1.57 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_166_in__mono,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,B5: set @ A,X: A] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.31/1.57 => ( ( member @ A @ X @ A4 )
% 0.31/1.57 => ( member @ A @ X @ B5 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_167_subsetD,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,B5: set @ A,C3: A] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.31/1.57 => ( ( member @ A @ C3 @ A4 )
% 0.31/1.57 => ( member @ A @ C3 @ B5 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_168_equalityE,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.31/1.57 ( ( A4 = B5 )
% 0.31/1.57 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.31/1.57 => ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_169_subset__eq,axiom,
% 0.31/1.57 ! [A: $tType] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ A ) )
% 0.31/1.57 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.31/1.57 ! [X3: A] :
% 0.31/1.57 ( ( member @ A @ X3 @ A6 )
% 0.31/1.57 => ( member @ A @ X3 @ B6 ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_170_equalityD1,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.31/1.57 ( ( A4 = B5 )
% 0.31/1.57 => ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_171_equalityD2,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.31/1.57 ( ( A4 = B5 )
% 0.31/1.57 => ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_172_subset__iff,axiom,
% 0.31/1.57 ! [A: $tType] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ A ) )
% 0.31/1.57 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.31/1.57 ! [T3: A] :
% 0.31/1.57 ( ( member @ A @ T3 @ A6 )
% 0.31/1.57 => ( member @ A @ T3 @ B6 ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_173_subset__refl,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_174_Collect__mono,axiom,
% 0.31/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/1.57 ( ! [X2: A] :
% 0.31/1.57 ( ( P @ X2 )
% 0.31/1.57 => ( Q @ X2 ) )
% 0.31/1.57 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_175_subset__trans,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,B5: set @ A,C4: set @ A] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.31/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
% 0.31/1.57 => ( ord_less_eq @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_176_set__eq__subset,axiom,
% 0.31/1.57 ! [A: $tType] :
% 0.31/1.57 ( ( ^ [Y5: set @ A,Z5: set @ A] : ( Y5 = Z5 ) )
% 0.31/1.57 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.31/1.57 & ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_177_Collect__mono__iff,axiom,
% 0.31/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.31/1.57 = ( ! [X3: A] :
% 0.31/1.57 ( ( P @ X3 )
% 0.31/1.57 => ( Q @ X3 ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_178_units__group,axiom,
% 0.31/1.57 group @ a @ product_unit @ ( units_of @ a @ ( ring_ext @ a @ b ) @ r ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_179_add__additive__subgroups,axiom,
% 0.31/1.57 ! [H: set @ a,K2: set @ a] :
% 0.31/1.57 ( ( additive_subgroup @ a @ b @ H @ r )
% 0.31/1.57 => ( ( additive_subgroup @ a @ b @ K2 @ r )
% 0.31/1.57 => ( additive_subgroup @ a @ b @ ( set_add @ a @ b @ r @ H @ K2 ) @ r ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_180_a__rcos__assoc__lcos,axiom,
% 0.31/1.57 ! [H: set @ a,K2: set @ a,X: a] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( ord_less_eq @ ( set @ a ) @ K2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( set_add @ a @ b @ r @ ( a_r_coset @ a @ b @ r @ H @ X ) @ K2 )
% 0.31/1.57 = ( set_add @ a @ b @ r @ H @ ( a_l_coset @ a @ b @ r @ X @ K2 ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_181_a__r__coset__subset__G,axiom,
% 0.31/1.57 ! [H: set @ a,X: a] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ord_less_eq @ ( set @ a ) @ ( a_r_coset @ a @ b @ r @ H @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_182_a__setmult__rcos__assoc,axiom,
% 0.31/1.57 ! [H: set @ a,K2: set @ a,X: a] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( ord_less_eq @ ( set @ a ) @ K2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( set_add @ a @ b @ r @ H @ ( a_r_coset @ a @ b @ r @ K2 @ X ) )
% 0.31/1.57 = ( a_r_coset @ a @ b @ r @ ( set_add @ a @ b @ r @ H @ K2 ) @ X ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_183_units__of__units,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/1.57 ( ( units @ A @ product_unit @ ( units_of @ A @ B @ G2 ) )
% 0.31/1.57 = ( units @ A @ B @ G2 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_184_a__coset__add__zero,axiom,
% 0.31/1.57 ! [M: set @ a] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ a ) @ M @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( a_r_coset @ a @ b @ r @ M @ ( zero @ a @ b @ r ) )
% 0.31/1.57 = M ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_185_units__of__carrier,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/1.57 ( ( partial_carrier @ A @ ( monoid_ext @ A @ product_unit ) @ ( units_of @ A @ B @ G2 ) )
% 0.31/1.57 = ( units @ A @ B @ G2 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_186_monoid_Ounits__group,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/1.57 ( ( monoid @ A @ B @ G2 )
% 0.31/1.57 => ( group @ A @ product_unit @ ( units_of @ A @ B @ G2 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_187_units__of__mult,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/1.57 ( ( mult @ A @ product_unit @ ( units_of @ A @ B @ G2 ) )
% 0.31/1.57 = ( mult @ A @ B @ G2 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_188_units__of__one,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/1.57 ( ( one @ A @ product_unit @ ( units_of @ A @ B @ G2 ) )
% 0.31/1.57 = ( one @ A @ B @ G2 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_189_a__rcosetsI,axiom,
% 0.31/1.57 ! [H: set @ a,X: a] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/1.57 => ( member @ ( set @ a ) @ ( a_r_coset @ a @ b @ r @ H @ X ) @ ( a_RCOSETS @ a @ b @ r @ H ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_190_one__zeroI,axiom,
% 0.31/1.57 ( ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 0.31/1.57 = ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) )
% 0.31/1.57 => ( ( one @ a @ ( ring_ext @ a @ b ) @ r )
% 0.31/1.57 = ( zero @ a @ b @ r ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_191_one__zeroD,axiom,
% 0.31/1.57 ( ( ( one @ a @ ( ring_ext @ a @ b ) @ r )
% 0.31/1.57 = ( zero @ a @ b @ r ) )
% 0.31/1.57 => ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 0.31/1.57 = ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_192_insert__absorb2,axiom,
% 0.31/1.57 ! [A: $tType,X: A,A4: set @ A] :
% 0.31/1.57 ( ( insert @ A @ X @ ( insert @ A @ X @ A4 ) )
% 0.31/1.57 = ( insert @ A @ X @ A4 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_193_insert__iff,axiom,
% 0.31/1.57 ! [A: $tType,A2: A,B2: A,A4: set @ A] :
% 0.31/1.57 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
% 0.31/1.57 = ( ( A2 = B2 )
% 0.31/1.57 | ( member @ A @ A2 @ A4 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_194_insertCI,axiom,
% 0.31/1.57 ! [A: $tType,A2: A,B5: set @ A,B2: A] :
% 0.31/1.57 ( ( ~ ( member @ A @ A2 @ B5 )
% 0.31/1.57 => ( A2 = B2 ) )
% 0.31/1.57 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_195_carrier__one__not__zero,axiom,
% 0.31/1.57 ( ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 0.31/1.57 != ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) )
% 0.31/1.57 = ( ( one @ a @ ( ring_ext @ a @ b ) @ r )
% 0.31/1.57 != ( zero @ a @ b @ r ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_196_carrier__one__zero,axiom,
% 0.31/1.57 ( ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
% 0.31/1.57 = ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) )
% 0.31/1.57 = ( ( one @ a @ ( ring_ext @ a @ b ) @ r )
% 0.31/1.57 = ( zero @ a @ b @ r ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_197_singletonI,axiom,
% 0.31/1.57 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_198_insert__subset,axiom,
% 0.31/1.57 ! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B5 )
% 0.31/1.57 = ( ( member @ A @ X @ B5 )
% 0.31/1.57 & ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_199_singleton__insert__inj__eq_H,axiom,
% 0.31/1.57 ! [A: $tType,A2: A,A4: set @ A,B2: A] :
% 0.31/1.57 ( ( ( insert @ A @ A2 @ A4 )
% 0.31/1.57 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.31/1.57 = ( ( A2 = B2 )
% 0.31/1.57 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_200_singleton__insert__inj__eq,axiom,
% 0.31/1.57 ! [A: $tType,B2: A,A2: A,A4: set @ A] :
% 0.31/1.57 ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
% 0.31/1.57 = ( insert @ A @ A2 @ A4 ) )
% 0.31/1.57 = ( ( A2 = B2 )
% 0.31/1.57 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_201_subset__singleton__iff,axiom,
% 0.31/1.57 ! [A: $tType,X5: set @ A,A2: A] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.31/1.57 = ( ( X5
% 0.31/1.57 = ( bot_bot @ ( set @ A ) ) )
% 0.31/1.57 | ( X5
% 0.31/1.57 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_202_subset__singletonD,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,X: A] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.31/1.57 => ( ( A4
% 0.31/1.57 = ( bot_bot @ ( set @ A ) ) )
% 0.31/1.57 | ( A4
% 0.31/1.57 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_203_mk__disjoint__insert,axiom,
% 0.31/1.57 ! [A: $tType,A2: A,A4: set @ A] :
% 0.31/1.57 ( ( member @ A @ A2 @ A4 )
% 0.31/1.57 => ? [B7: set @ A] :
% 0.31/1.57 ( ( A4
% 0.31/1.57 = ( insert @ A @ A2 @ B7 ) )
% 0.31/1.57 & ~ ( member @ A @ A2 @ B7 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_204_insert__commute,axiom,
% 0.31/1.57 ! [A: $tType,X: A,Y: A,A4: set @ A] :
% 0.31/1.57 ( ( insert @ A @ X @ ( insert @ A @ Y @ A4 ) )
% 0.31/1.57 = ( insert @ A @ Y @ ( insert @ A @ X @ A4 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_205_insert__eq__iff,axiom,
% 0.31/1.57 ! [A: $tType,A2: A,A4: set @ A,B2: A,B5: set @ A] :
% 0.31/1.57 ( ~ ( member @ A @ A2 @ A4 )
% 0.31/1.57 => ( ~ ( member @ A @ B2 @ B5 )
% 0.31/1.57 => ( ( ( insert @ A @ A2 @ A4 )
% 0.31/1.57 = ( insert @ A @ B2 @ B5 ) )
% 0.31/1.57 = ( ( ( A2 = B2 )
% 0.31/1.57 => ( A4 = B5 ) )
% 0.31/1.57 & ( ( A2 != B2 )
% 0.31/1.57 => ? [C5: set @ A] :
% 0.31/1.57 ( ( A4
% 0.31/1.57 = ( insert @ A @ B2 @ C5 ) )
% 0.31/1.57 & ~ ( member @ A @ B2 @ C5 )
% 0.31/1.57 & ( B5
% 0.31/1.57 = ( insert @ A @ A2 @ C5 ) )
% 0.31/1.57 & ~ ( member @ A @ A2 @ C5 ) ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_206_insert__absorb,axiom,
% 0.31/1.57 ! [A: $tType,A2: A,A4: set @ A] :
% 0.31/1.57 ( ( member @ A @ A2 @ A4 )
% 0.31/1.57 => ( ( insert @ A @ A2 @ A4 )
% 0.31/1.57 = A4 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_207_insert__ident,axiom,
% 0.31/1.57 ! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
% 0.31/1.57 ( ~ ( member @ A @ X @ A4 )
% 0.31/1.57 => ( ~ ( member @ A @ X @ B5 )
% 0.31/1.57 => ( ( ( insert @ A @ X @ A4 )
% 0.31/1.57 = ( insert @ A @ X @ B5 ) )
% 0.31/1.57 = ( A4 = B5 ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_208_Set_Oset__insert,axiom,
% 0.31/1.57 ! [A: $tType,X: A,A4: set @ A] :
% 0.31/1.57 ( ( member @ A @ X @ A4 )
% 0.31/1.57 => ~ ! [B7: set @ A] :
% 0.31/1.57 ( ( A4
% 0.31/1.57 = ( insert @ A @ X @ B7 ) )
% 0.31/1.57 => ( member @ A @ X @ B7 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_209_insertI2,axiom,
% 0.31/1.57 ! [A: $tType,A2: A,B5: set @ A,B2: A] :
% 0.31/1.57 ( ( member @ A @ A2 @ B5 )
% 0.31/1.57 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_210_insertI1,axiom,
% 0.31/1.57 ! [A: $tType,A2: A,B5: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B5 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_211_insertE,axiom,
% 0.31/1.57 ! [A: $tType,A2: A,B2: A,A4: set @ A] :
% 0.31/1.57 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
% 0.31/1.57 => ( ( A2 != B2 )
% 0.31/1.57 => ( member @ A @ A2 @ A4 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_212_singleton__inject,axiom,
% 0.31/1.57 ! [A: $tType,A2: A,B2: A] :
% 0.31/1.57 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.31/1.57 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.31/1.57 => ( A2 = B2 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_213_insert__not__empty,axiom,
% 0.31/1.57 ! [A: $tType,A2: A,A4: set @ A] :
% 0.31/1.57 ( ( insert @ A @ A2 @ A4 )
% 0.31/1.57 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_214_doubleton__eq__iff,axiom,
% 0.31/1.57 ! [A: $tType,A2: A,B2: A,C3: A,D3: A] :
% 0.31/1.57 ( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.31/1.57 = ( insert @ A @ C3 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.31/1.57 = ( ( ( A2 = C3 )
% 0.31/1.57 & ( B2 = D3 ) )
% 0.31/1.57 | ( ( A2 = D3 )
% 0.31/1.57 & ( B2 = C3 ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_215_singleton__iff,axiom,
% 0.31/1.57 ! [A: $tType,B2: A,A2: A] :
% 0.31/1.57 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.31/1.57 = ( B2 = A2 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_216_singletonD,axiom,
% 0.31/1.57 ! [A: $tType,B2: A,A2: A] :
% 0.31/1.57 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.31/1.57 => ( B2 = A2 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_217_insert__mono,axiom,
% 0.31/1.57 ! [A: $tType,C4: set @ A,D4: set @ A,A2: A] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ A ) @ C4 @ D4 )
% 0.31/1.57 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C4 ) @ ( insert @ A @ A2 @ D4 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_218_subset__insert,axiom,
% 0.31/1.57 ! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
% 0.31/1.57 ( ~ ( member @ A @ X @ A4 )
% 0.31/1.57 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B5 ) )
% 0.31/1.57 = ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_219_subset__insertI,axiom,
% 0.31/1.57 ! [A: $tType,B5: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert @ A @ A2 @ B5 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_220_subset__insertI2,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,B5: set @ A,B2: A] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.31/1.57 => ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_221_semiring_Ocarrier__one__not__zero,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/1.57 ( ( semiring @ A @ B @ R3 )
% 0.31/1.57 => ( ( ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 )
% 0.31/1.57 != ( insert @ A @ ( zero @ A @ B @ R3 ) @ ( bot_bot @ ( set @ A ) ) ) )
% 0.31/1.57 = ( ( one @ A @ ( ring_ext @ A @ B ) @ R3 )
% 0.31/1.57 != ( zero @ A @ B @ R3 ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_222_semiring_Ocarrier__one__zero,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/1.57 ( ( semiring @ A @ B @ R3 )
% 0.31/1.57 => ( ( ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 )
% 0.31/1.57 = ( insert @ A @ ( zero @ A @ B @ R3 ) @ ( bot_bot @ ( set @ A ) ) ) )
% 0.31/1.57 = ( ( one @ A @ ( ring_ext @ A @ B ) @ R3 )
% 0.31/1.57 = ( zero @ A @ B @ R3 ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_223_semiring_Oone__zeroI,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/1.57 ( ( semiring @ A @ B @ R3 )
% 0.31/1.57 => ( ( ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 )
% 0.31/1.57 = ( insert @ A @ ( zero @ A @ B @ R3 ) @ ( bot_bot @ ( set @ A ) ) ) )
% 0.31/1.57 => ( ( one @ A @ ( ring_ext @ A @ B ) @ R3 )
% 0.31/1.57 = ( zero @ A @ B @ R3 ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_224_semiring_Oone__zeroD,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/1.57 ( ( semiring @ A @ B @ R3 )
% 0.31/1.57 => ( ( ( one @ A @ ( ring_ext @ A @ B ) @ R3 )
% 0.31/1.57 = ( zero @ A @ B @ R3 ) )
% 0.31/1.57 => ( ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 )
% 0.31/1.57 = ( insert @ A @ ( zero @ A @ B @ R3 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_225_field__intro2,axiom,
% 0.31/1.57 ( ( ( zero @ a @ b @ r )
% 0.31/1.57 != ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/1.57 => ( ! [X2: a] :
% 0.31/1.57 ( ( member @ a @ X2 @ ( minus_minus @ ( set @ a ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) @ ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) ) )
% 0.31/1.57 => ( member @ a @ X2 @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) ) )
% 0.31/1.57 => ( field @ a @ b @ r ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_226_cring__fieldI,axiom,
% 0.31/1.57 ( ( ( units @ a @ ( ring_ext @ a @ b ) @ r )
% 0.31/1.57 = ( minus_minus @ ( set @ a ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) @ ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) ) )
% 0.31/1.57 => ( field @ a @ b @ r ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_227_DiffI,axiom,
% 0.31/1.57 ! [A: $tType,C3: A,A4: set @ A,B5: set @ A] :
% 0.31/1.57 ( ( member @ A @ C3 @ A4 )
% 0.31/1.57 => ( ~ ( member @ A @ C3 @ B5 )
% 0.31/1.57 => ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_228_Diff__iff,axiom,
% 0.31/1.57 ! [A: $tType,C3: A,A4: set @ A,B5: set @ A] :
% 0.31/1.57 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) )
% 0.31/1.57 = ( ( member @ A @ C3 @ A4 )
% 0.31/1.57 & ~ ( member @ A @ C3 @ B5 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_229_Diff__idemp,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.31/1.57 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) @ B5 )
% 0.31/1.57 = ( minus_minus @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_230_Diff__empty,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A] :
% 0.31/1.57 ( ( minus_minus @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 0.31/1.57 = A4 ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_231_empty__Diff,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A] :
% 0.31/1.57 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 )
% 0.31/1.57 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_232_Diff__cancel,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A] :
% 0.31/1.57 ( ( minus_minus @ ( set @ A ) @ A4 @ A4 )
% 0.31/1.57 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_233_insert__Diff1,axiom,
% 0.31/1.57 ! [A: $tType,X: A,B5: set @ A,A4: set @ A] :
% 0.31/1.57 ( ( member @ A @ X @ B5 )
% 0.31/1.57 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B5 )
% 0.31/1.57 = ( minus_minus @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_234_Diff__insert0,axiom,
% 0.31/1.57 ! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
% 0.31/1.57 ( ~ ( member @ A @ X @ A4 )
% 0.31/1.57 => ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B5 ) )
% 0.31/1.57 = ( minus_minus @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_235_Diff__eq__empty__iff,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,B5: set @ A] :
% 0.31/1.57 ( ( ( minus_minus @ ( set @ A ) @ A4 @ B5 )
% 0.31/1.57 = ( bot_bot @ ( set @ A ) ) )
% 0.31/1.57 = ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_236_insert__Diff__single,axiom,
% 0.31/1.57 ! [A: $tType,A2: A,A4: set @ A] :
% 0.31/1.57 ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.31/1.57 = ( insert @ A @ A2 @ A4 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_237_insert__Diff__if,axiom,
% 0.31/1.57 ! [A: $tType,X: A,B5: set @ A,A4: set @ A] :
% 0.31/1.57 ( ( ( member @ A @ X @ B5 )
% 0.31/1.57 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B5 )
% 0.31/1.57 = ( minus_minus @ ( set @ A ) @ A4 @ B5 ) ) )
% 0.31/1.57 & ( ~ ( member @ A @ X @ B5 )
% 0.31/1.57 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B5 )
% 0.31/1.57 = ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_238_Diff__insert__absorb,axiom,
% 0.31/1.57 ! [A: $tType,X: A,A4: set @ A] :
% 0.31/1.57 ( ~ ( member @ A @ X @ A4 )
% 0.31/1.57 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.31/1.57 = A4 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_239_Diff__insert2,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,A2: A,B5: set @ A] :
% 0.31/1.57 ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B5 ) )
% 0.31/1.57 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_240_insert__Diff,axiom,
% 0.31/1.57 ! [A: $tType,A2: A,A4: set @ A] :
% 0.31/1.57 ( ( member @ A @ A2 @ A4 )
% 0.31/1.57 => ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.31/1.57 = A4 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_241_Diff__insert,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,A2: A,B5: set @ A] :
% 0.31/1.57 ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B5 ) )
% 0.31/1.57 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_242_subset__Diff__insert,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,B5: set @ A,X: A,C4: set @ A] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B5 @ ( insert @ A @ X @ C4 ) ) )
% 0.31/1.57 = ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B5 @ C4 ) )
% 0.31/1.57 & ~ ( member @ A @ X @ A4 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_243_double__diff,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,B5: set @ A,C4: set @ A] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
% 0.31/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
% 0.31/1.57 => ( ( minus_minus @ ( set @ A ) @ B5 @ ( minus_minus @ ( set @ A ) @ C4 @ A4 ) )
% 0.31/1.57 = A4 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_244_Diff__subset,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) @ A4 ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_245_Diff__mono,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,C4: set @ A,D4: set @ A,B5: set @ A] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ A ) @ A4 @ C4 )
% 0.31/1.57 => ( ( ord_less_eq @ ( set @ A ) @ D4 @ B5 )
% 0.31/1.57 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) @ ( minus_minus @ ( set @ A ) @ C4 @ D4 ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_246_DiffE,axiom,
% 0.31/1.57 ! [A: $tType,C3: A,A4: set @ A,B5: set @ A] :
% 0.31/1.57 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) )
% 0.31/1.57 => ~ ( ( member @ A @ C3 @ A4 )
% 0.31/1.57 => ( member @ A @ C3 @ B5 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_247_DiffD1,axiom,
% 0.31/1.57 ! [A: $tType,C3: A,A4: set @ A,B5: set @ A] :
% 0.31/1.57 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) )
% 0.31/1.57 => ( member @ A @ C3 @ A4 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_248_DiffD2,axiom,
% 0.31/1.57 ! [A: $tType,C3: A,A4: set @ A,B5: set @ A] :
% 0.31/1.57 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) )
% 0.31/1.57 => ~ ( member @ A @ C3 @ B5 ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_249_subset__insert__iff,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,X: A,B5: set @ A] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B5 ) )
% 0.31/1.57 = ( ( ( member @ A @ X @ A4 )
% 0.31/1.57 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) )
% 0.31/1.57 & ( ~ ( member @ A @ X @ A4 )
% 0.31/1.57 => ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_250_Diff__single__insert,axiom,
% 0.31/1.57 ! [A: $tType,A4: set @ A,X: A,B5: set @ A] :
% 0.31/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 )
% 0.31/1.57 => ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B5 ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_251_field_Ofield__Units,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/1.57 ( ( field @ A @ B @ R3 )
% 0.31/1.57 => ( ( units @ A @ ( ring_ext @ A @ B ) @ R3 )
% 0.31/1.57 = ( minus_minus @ ( set @ A ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) @ ( insert @ A @ ( zero @ A @ B @ R3 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_252_rcosets__subset__PowG,axiom,
% 0.31/1.57 ! [H: set @ a] :
% 0.31/1.57 ( ( additive_subgroup @ a @ b @ H @ r )
% 0.31/1.57 => ( ord_less_eq @ ( set @ ( set @ a ) ) @ ( a_RCOSETS @ a @ b @ r @ H ) @ ( pow @ a @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_253_abelian__monoid_Oset__add__closed,axiom,
% 0.31/1.57 ! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A4: set @ A,B5: set @ A] :
% 0.31/1.57 ( ( abelian_monoid @ A @ B @ G2 )
% 0.31/1.57 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G2 ) )
% 0.31/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G2 ) )
% 0.31/1.57 => ( ord_less_eq @ ( set @ A ) @ ( set_add @ A @ B @ G2 @ A4 @ B5 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G2 ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_254_Pow__singleton__iff,axiom,
% 0.31/1.57 ! [A: $tType,X5: set @ A,Y6: set @ A] :
% 0.31/1.57 ( ( ( pow @ A @ X5 )
% 0.31/1.57 = ( insert @ ( set @ A ) @ Y6 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
% 0.31/1.57 = ( ( X5
% 0.31/1.57 = ( bot_bot @ ( set @ A ) ) )
% 0.31/1.57 & ( Y6
% 0.31/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(fact_255_Pow__empty,axiom,
% 0.31/1.57 ! [A: $tType] :
% 0.31/1.57 ( ( pow @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.31/1.57 = ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% 0.31/1.57
% 0.31/1.57 thf(conj_0,conjecture,
% 0.31/1.57 ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) )
% 0.31/1.57 = ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) ) ) ).
% 0.31/1.57 ------------------------
% 0.31/1.59 % (27586)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_27309 for (2999ds/275Mi)
% 0.31/1.59 % (27582)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_27309 for (2999ds/4Mi)
% 0.31/1.59 % (27587)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_27309 for (2999ds/18Mi)
% 0.31/1.59 % (27582)Instruction limit reached!
% 0.31/1.59 % (27582)------------------------------
% 0.31/1.59 % (27582)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.59 % (27582)Termination reason: Unknown
% 0.31/1.59 % (27582)Termination phase: shuffling
% 0.31/1.59
% 0.31/1.59 % (27582)Memory used [KB]: 1407
% 0.31/1.59 % (27582)Time elapsed: 0.003 s
% 0.31/1.59 % (27582)Instructions burned: 6 (million)
% 0.31/1.59 % (27582)------------------------------
% 0.31/1.59 % (27582)------------------------------
% 2.97/1.60 % (27587)Instruction limit reached!
% 2.97/1.60 % (27587)------------------------------
% 2.97/1.60 % (27587)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.60 % (27587)Termination reason: Unknown
% 2.97/1.60 % (27587)Termination phase: shuffling
% 2.97/1.60
% 2.97/1.60 % (27587)Memory used [KB]: 1535
% 2.97/1.60 % (27587)Time elapsed: 0.008 s
% 2.97/1.60 % (27587)Instructions burned: 19 (million)
% 2.97/1.60 % (27587)------------------------------
% 2.97/1.60 % (27587)------------------------------
% 2.97/1.60 % (27583)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_27309 for (2999ds/27Mi)
% 2.97/1.60 % (27584)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_27309 for (2999ds/2Mi)
% 2.97/1.60 % (27585)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_27309 for (2999ds/2Mi)
% 2.97/1.60 % (27581)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_27309 for (2999ds/183Mi)
% 2.97/1.60 % (27584)Instruction limit reached!
% 2.97/1.60 % (27584)------------------------------
% 2.97/1.60 % (27584)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.60 % (27584)Termination reason: Unknown
% 2.97/1.60 % (27584)Termination phase: shuffling
% 2.97/1.60
% 2.97/1.60 % (27584)Memory used [KB]: 1279
% 2.97/1.60 % (27584)Time elapsed: 0.004 s
% 2.97/1.60 % (27584)Instructions burned: 3 (million)
% 2.97/1.60 % (27584)------------------------------
% 2.97/1.60 % (27584)------------------------------
% 2.97/1.60 % (27585)Instruction limit reached!
% 2.97/1.60 % (27585)------------------------------
% 2.97/1.60 % (27585)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.60 % (27585)Termination reason: Unknown
% 2.97/1.60 % (27585)Termination phase: shuffling
% 2.97/1.60
% 2.97/1.60 % (27585)Memory used [KB]: 1407
% 2.97/1.60 % (27585)Time elapsed: 0.004 s
% 2.97/1.60 % (27585)Instructions burned: 3 (million)
% 2.97/1.60 % (27585)------------------------------
% 2.97/1.60 % (27585)------------------------------
% 2.97/1.60 % (27588)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_27309 for (2999ds/3Mi)
% 2.97/1.61 % (27588)Instruction limit reached!
% 2.97/1.61 % (27588)------------------------------
% 2.97/1.61 % (27588)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.61 % (27588)Termination reason: Unknown
% 2.97/1.61 % (27588)Termination phase: shuffling
% 2.97/1.61
% 2.97/1.61 % (27588)Memory used [KB]: 1407
% 2.97/1.61 % (27588)Time elapsed: 0.002 s
% 2.97/1.61 % (27588)Instructions burned: 3 (million)
% 2.97/1.61 % (27588)------------------------------
% 2.97/1.61 % (27588)------------------------------
% 2.97/1.61 % (27589)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_27309 for (2999ds/37Mi)
% 2.97/1.62 % (27592)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_27309 for (2999ds/1041Mi)
% 2.97/1.62 % (27583)Instruction limit reached!
% 2.97/1.62 % (27583)------------------------------
% 2.97/1.62 % (27583)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.62 % (27583)Termination reason: Unknown
% 2.97/1.62 % (27583)Termination phase: shuffling
% 2.97/1.62
% 2.97/1.62 % (27583)Memory used [KB]: 1663
% 2.97/1.62 % (27583)Time elapsed: 0.020 s
% 2.97/1.62 % (27583)Instructions burned: 28 (million)
% 2.97/1.62 % (27583)------------------------------
% 2.97/1.62 % (27583)------------------------------
% 2.97/1.62 % (27590)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_27309 for (2999ds/15Mi)
% 2.97/1.62 % (27591)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_27309 for (2999ds/3Mi)
% 2.97/1.62 % (27591)Instruction limit reached!
% 2.97/1.62 % (27591)------------------------------
% 2.97/1.62 % (27591)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.62 % (27591)Termination reason: Unknown
% 2.97/1.62 % (27591)Termination phase: shuffling
% 2.97/1.62
% 2.97/1.62 % (27591)Memory used [KB]: 1279
% 2.97/1.62 % (27591)Time elapsed: 0.004 s
% 2.97/1.62 % (27591)Instructions burned: 3 (million)
% 2.97/1.62 % (27591)------------------------------
% 2.97/1.62 % (27591)------------------------------
% 2.97/1.62 % (27589)Instruction limit reached!
% 2.97/1.62 % (27589)------------------------------
% 2.97/1.62 % (27589)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.62 % (27589)Termination reason: Unknown
% 2.97/1.62 % (27589)Termination phase: shuffling
% 2.97/1.62
% 2.97/1.62 % (27589)Memory used [KB]: 2046
% 2.97/1.62 % (27589)Time elapsed: 0.013 s
% 2.97/1.62 % (27589)Instructions burned: 40 (million)
% 2.97/1.62 % (27589)------------------------------
% 2.97/1.62 % (27589)------------------------------
% 2.97/1.62 % (27586)First to succeed.
% 2.97/1.63 % (27586)Refutation found. Thanks to Tanya!
% 2.97/1.63 % SZS status Theorem for DTF2THF_27309
% 2.97/1.63 % SZS output start Proof for DTF2THF_27309
% 2.97/1.63 thf(type_def_5, type, ring_ext: ($tType * $tType) > $tType).
% 2.97/1.63 thf(type_def_6, type, monoid_ext: ($tType * $tType) > $tType).
% 2.97/1.63 thf(type_def_7, type, partia1265057091ct_ext: ($tType * $tType) > $tType).
% 2.97/1.63 thf(type_def_8, type, set: ($tType) > $tType).
% 2.97/1.63 thf(type_def_10, type, eq_eq_object_ext: ($tType * $tType) > $tType).
% 2.97/1.63 thf(type_def_11, type, product_unit: $tType).
% 2.97/1.63 thf(type_def_12, type, product_prod: ($tType * $tType) > $tType).
% 2.97/1.63 thf(type_def_13, type, a: $tType).
% 2.97/1.63 thf(type_def_14, type, b: $tType).
% 2.97/1.63 thf(type_def_15, type, int: $tType).
% 2.97/1.63 thf(type_def_16, type, nat: $tType).
% 2.97/1.63 thf(func_def_0, type, partia1265057091ct_ext: ($tType * $tType) > $tType).
% 2.97/1.63 thf(func_def_1, type, eq_eq_object_ext: ($tType * $tType) > $tType).
% 2.97/1.63 thf(func_def_2, type, monoid_ext: ($tType * $tType) > $tType).
% 2.97/1.63 thf(func_def_3, type, ring_ext: ($tType * $tType) > $tType).
% 2.97/1.63 thf(func_def_4, type, product_unit: $tType).
% 2.97/1.63 thf(func_def_5, type, product_prod: ($tType * $tType) > $tType).
% 2.97/1.63 thf(func_def_6, type, set: ($tType) > $tType).
% 2.97/1.63 thf(func_def_7, type, nat: $tType).
% 2.97/1.63 thf(func_def_8, type, int: $tType).
% 2.97/1.63 thf(func_def_9, type, b: $tType).
% 2.97/1.63 thf(func_def_10, type, a: $tType).
% 2.97/1.63 thf(func_def_11, type, a_RCOSETS: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X0 > set @ set @ X0)).
% 2.97/1.63 thf(func_def_12, type, a_l_coset: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > X0 > set @ X0 > set @ X0)).
% 2.97/1.63 thf(func_def_13, type, a_r_coset: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X0 > X0 > set @ X0)).
% 2.97/1.63 thf(func_def_14, type, additive_subgroup: !>[X0: $tType, X1: $tType]:(set @ X0 > partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > $o)).
% 2.97/1.63 thf(func_def_15, type, set_add: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X0 > set @ X0 > set @ X0)).
% 2.97/1.63 thf(func_def_16, type, equivalence: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ eq_eq_object_ext @ X0 @ X1 > $o)).
% 2.97/1.63 thf(func_def_17, type, partial_carrier: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ X1 > set @ X0)).
% 2.97/1.63 thf(func_def_18, type, units: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ X1 > set @ X0)).
% 2.97/1.63 thf(func_def_19, type, group: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ X1 > $o)).
% 2.97/1.63 thf(func_def_20, type, monoid: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ X1 > $o)).
% 2.97/1.63 thf(func_def_21, type, mult: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ X1 > X0 > X0 > X0)).
% 2.97/1.63 thf(func_def_22, type, one: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ X1 > X0)).
% 2.97/1.63 thf(func_def_23, type, units_of: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ X1 > partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ product_unit)).
% 2.97/1.63 thf(func_def_24, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.97/1.63 thf(func_def_25, type, locali1641774487f_frac: !>[X0: $tType, X1: $tType, X2: $tType]:(partia1265057091ct_ext @ product_prod @ X0 @ X1 @ eq_eq_object_ext @ product_prod @ X0 @ X1 @ X2 > X0 > X1 > set @ product_prod @ X0 @ X1)).
% 2.97/1.63 thf(func_def_26, type, locali1648044335f_frac: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X0 > $o)).
% 2.97/1.63 thf(func_def_27, type, locali1768110497f_frac: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X0 > partia1265057091ct_ext @ set @ product_prod @ X0 @ X0 @ monoid_ext @ set @ product_prod @ X0 @ X0 @ product_unit)).
% 2.97/1.63 thf(func_def_28, type, locali7914339f_crng: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X0 > $o)).
% 2.97/1.63 thf(func_def_29, type, locali1402578320of_rng: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X0 > $o)).
% 2.97/1.63 thf(func_def_30, type, locali990921413f_frac: !>[X0: $tType, X1: $tType, X2: $tType]:(partia1265057091ct_ext @ product_prod @ X0 @ X1 @ eq_eq_object_ext @ product_prod @ X0 @ X1 @ X2 > set @ set @ product_prod @ X0 @ X1)).
% 2.97/1.63 thf(func_def_31, type, locali1441642228monoid: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ X1 > set @ X0 > $o)).
% 2.97/1.63 thf(func_def_32, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.97/1.63 thf(func_def_33, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.97/1.63 thf(func_def_34, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 2.97/1.63 thf(func_def_35, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.97/1.63 thf(func_def_36, type, abelian_monoid: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > $o)).
% 2.97/1.63 thf(func_def_37, type, add_pow: !>[X0: $tType, X1: $tType, X2: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > X2 > X0 > X0)).
% 2.97/1.63 thf(func_def_38, type, field: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > $o)).
% 2.97/1.63 thf(func_def_39, type, ring: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > $o)).
% 2.97/1.63 thf(func_def_40, type, zero: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > X0)).
% 2.97/1.63 thf(func_def_41, type, semiring: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > $o)).
% 2.97/1.63 thf(func_def_42, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.97/1.63 thf(func_def_43, type, pow: !>[X0: $tType]:(set @ X0 > set @ set @ X0)).
% 2.97/1.63 thf(func_def_44, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 2.97/1.63 thf(func_def_45, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.97/1.63 thf(func_def_46, type, r: partia1265057091ct_ext @ a @ monoid_ext @ a @ ring_ext @ a @ b).
% 2.97/1.63 thf(func_def_47, type, s: set @ a).
% 2.97/1.63 thf(func_def_48, type, r2: a).
% 2.97/1.63 thf(func_def_49, type, r3: a).
% 2.97/1.63 thf(func_def_50, type, rel: partia1265057091ct_ext @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit).
% 2.97/1.63 thf(func_def_51, type, s2: a).
% 2.97/1.63 thf(func_def_52, type, s3: a).
% 2.97/1.63 thf(func_def_65, type, sP0: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > $o)).
% 2.97/1.63 thf(func_def_66, type, sP1: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > $o)).
% 2.97/1.63 thf(func_def_67, type, sP2: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > $o)).
% 2.97/1.63 thf(func_def_68, type, sP3: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > $o)).
% 2.97/1.63 thf(func_def_69, type, sK4: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 2.97/1.63 thf(func_def_70, type, sK5: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 2.97/1.63 thf(func_def_71, type, sK6: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 2.97/1.63 thf(func_def_72, type, sK7: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 2.97/1.63 thf(func_def_73, type, sK8: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.97/1.63 thf(func_def_74, type, sK9: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.97/1.63 thf(func_def_75, type, sK10: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.97/1.63 thf(func_def_76, type, sK11: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.97/1.63 thf(func_def_77, type, sK12: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.97/1.63 thf(func_def_78, type, sK13: !>[X0: $tType]:(set @ X0 > X0)).
% 2.97/1.63 thf(func_def_79, type, sK14: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.97/1.63 thf(func_def_80, type, sK15: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.97/1.63 thf(func_def_81, type, sK16: !>[X0: $tType]:(set @ X0 > X0)).
% 2.97/1.63 thf(func_def_82, type, sK17: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.97/1.63 thf(func_def_83, type, sK18: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.97/1.63 thf(func_def_84, type, sK19: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.97/1.63 thf(func_def_85, type, sK20: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.97/1.63 thf(func_def_86, type, sK21: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.97/1.63 thf(func_def_87, type, sK22: a > a).
% 2.97/1.63 thf(func_def_88, type, sK23: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.97/1.63 thf(func_def_89, type, sK24: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.97/1.63 thf(func_def_90, type, sK25: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 2.97/1.63 thf(func_def_91, type, sK26: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 2.97/1.63 thf(func_def_92, type, sK27: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 2.97/1.63 thf(func_def_93, type, sK28: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 2.97/1.63 thf(func_def_94, type, sK29: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 2.97/1.63 thf(func_def_95, type, sK30: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 2.97/1.63 thf(func_def_96, type, sK31: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 2.97/1.63 thf(func_def_97, type, sK32: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.97/1.63 thf(func_def_98, type, sK33: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.97/1.63 thf(func_def_99, type, sK34: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.97/1.63 thf(func_def_100, type, sK35: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.97/1.63 thf(func_def_101, type, sK36: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X0)).
% 2.97/1.63 thf(func_def_102, type, sK37: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X1)).
% 2.97/1.63 thf(func_def_103, type, sK38: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X2)).
% 2.97/1.63 thf(func_def_104, type, sK39: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X3)).
% 2.97/1.63 thf(func_def_105, type, sK40: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X4)).
% 2.97/1.63 thf(func_def_106, type, sK41: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X5)).
% 2.97/1.63 thf(func_def_107, type, sK42: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X6)).
% 2.97/1.63 thf(func_def_108, type, sK43: !>[X0: $tType]:(set @ X0 > X0)).
% 2.97/1.63 thf(func_def_109, type, sK44: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.97/1.63 thf(func_def_110, type, sK45: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.97/1.63 thf(func_def_111, type, sK46: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 2.97/1.63 thf(func_def_112, type, sK47: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 2.97/1.63 thf(func_def_113, type, sK48: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 2.97/1.63 thf(func_def_114, type, sK49: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 2.97/1.63 thf(func_def_115, type, sK50: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 2.97/1.63 thf(func_def_116, type, sK51: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 2.97/1.63 thf(func_def_117, type, sK52: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 2.97/1.63 thf(func_def_118, type, sK53: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X0)).
% 2.97/1.63 thf(func_def_119, type, sK54: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X1)).
% 2.97/1.63 thf(func_def_120, type, sK55: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X2)).
% 2.97/1.63 thf(func_def_121, type, sK56: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X3)).
% 2.97/1.63 thf(func_def_122, type, sK57: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X4)).
% 2.97/1.63 thf(func_def_123, type, sK58: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X5)).
% 2.97/1.63 thf(func_def_124, type, sK59: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 2.97/1.63 thf(func_def_125, type, sK60: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 2.97/1.63 thf(func_def_126, type, sK61: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 2.97/1.63 thf(func_def_127, type, sK62: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 2.97/1.63 thf(func_def_128, type, sK63: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 2.97/1.63 thf(func_def_129, type, sK64: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 2.97/1.63 thf(func_def_130, type, sK65: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 2.97/1.63 thf(func_def_131, type, sK66: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 2.97/1.63 thf(func_def_132, type, sK67: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 2.97/1.63 thf(func_def_133, type, sK68: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.97/1.63 thf(func_def_134, type, sK69: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.97/1.63 thf(func_def_135, type, sK70: !>[X0: $tType, X1: $tType]:(X1 > partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 2.97/1.63 thf(func_def_136, type, sK71: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.97/1.63 thf(func_def_137, type, sK72: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.97/1.63 thf(func_def_138, type, sK73: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.97/1.63 thf(func_def_139, type, sK74: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X6)).
% 2.97/1.63 thf(func_def_140, type, sK75: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 2.97/1.63 thf(func_def_141, type, sK76: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.97/1.63 thf(func_def_142, type, sK77: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.97/1.63 thf(func_def_143, type, sK78: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.97/1.63 thf(func_def_144, type, sK79: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.97/1.63 thf(func_def_145, type, sK80: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.97/1.63 thf(func_def_146, type, sK81: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.97/1.63 thf(func_def_147, type, sK82: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.97/1.63 thf(func_def_149, type, ph84: !>[X0: $tType]:(X0)).
% 2.97/1.63 thf(f1059,plain,(
% 2.97/1.63 $false),
% 2.97/1.63 inference(subsumption_resolution,[status(thm)],[f1058,f988])).
% 2.97/1.63 thf(f988,plain,(
% 2.97/1.63 ((mult @ set @ product_prod @ a @ a @ product_unit @ (locali1768110497f_frac @ a @ b @ r @ s) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3)) = (locali1641774487f_frac @ a @ a @ product_unit @ rel @ (mult @ a @ ring_ext @ a @ b @ r @ r2 @ r3) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3)))),
% 2.97/1.63 inference(cnf_transformation,[status(thm)],[f4])).
% 2.97/1.63 thf(f4,axiom,(
% 2.97/1.63 ((mult @ set @ product_prod @ a @ a @ product_unit @ (locali1768110497f_frac @ a @ b @ r @ s) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3)) = (locali1641774487f_frac @ a @ a @ product_unit @ rel @ (mult @ a @ ring_ext @ a @ b @ r @ r2 @ r3) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3)))),
% 2.97/1.63 file('/export/starexec/sandbox2/tmp/tmp.ZsShBFnucq/DTF2THF_27309.p',fact_3_f1)).
% 2.97/1.63 thf(f1058,plain,(
% 2.97/1.63 ((mult @ set @ product_prod @ a @ a @ product_unit @ (locali1768110497f_frac @ a @ b @ r @ s) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3)) != (locali1641774487f_frac @ a @ a @ product_unit @ rel @ (mult @ a @ ring_ext @ a @ b @ r @ r2 @ r3) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3)))),
% 2.97/1.63 inference(forward_demodulation,[status(thm)],[f1057,f938])).
% 2.97/1.63 thf(f938,plain,(
% 2.97/1.63 ((mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) = (mult @ a @ ring_ext @ a @ b @ r @ s3 @ s2))),
% 2.97/1.63 inference(cnf_transformation,[status(thm)],[f2])).
% 2.97/1.63 thf(f2,axiom,(
% 2.97/1.63 ((mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) = (mult @ a @ ring_ext @ a @ b @ r @ s3 @ s2))),
% 2.97/1.63 file('/export/starexec/sandbox2/tmp/tmp.ZsShBFnucq/DTF2THF_27309.p',fact_1_f4)).
% 2.97/1.63 thf(f1057,plain,(
% 2.97/1.63 ((mult @ set @ product_prod @ a @ a @ product_unit @ (locali1768110497f_frac @ a @ b @ r @ s) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3)) != (locali1641774487f_frac @ a @ a @ product_unit @ rel @ (mult @ a @ ring_ext @ a @ b @ r @ r2 @ r3) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ s2)))),
% 2.97/1.63 inference(forward_demodulation,[status(thm)],[f1056,f950])).
% 2.97/1.63 thf(f950,plain,(
% 2.97/1.63 ((mult @ a @ ring_ext @ a @ b @ r @ r2 @ r3) = (mult @ a @ ring_ext @ a @ b @ r @ r3 @ r2))),
% 2.97/1.63 inference(cnf_transformation,[status(thm)],[f1])).
% 2.97/1.63 thf(f1,axiom,(
% 2.97/1.63 ((mult @ a @ ring_ext @ a @ b @ r @ r2 @ r3) = (mult @ a @ ring_ext @ a @ b @ r @ r3 @ r2))),
% 2.97/1.63 file('/export/starexec/sandbox2/tmp/tmp.ZsShBFnucq/DTF2THF_27309.p',fact_0_f3)).
% 2.97/1.63 thf(f1056,plain,(
% 2.97/1.63 ((mult @ set @ product_prod @ a @ a @ product_unit @ (locali1768110497f_frac @ a @ b @ r @ s) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3)) != (locali1641774487f_frac @ a @ a @ product_unit @ rel @ (mult @ a @ ring_ext @ a @ b @ r @ r3 @ r2) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ s2)))),
% 2.97/1.63 inference(forward_demodulation,[status(thm)],[f981,f1013])).
% 2.97/1.63 thf(f1013,plain,(
% 2.97/1.63 ((mult @ set @ product_prod @ a @ a @ product_unit @ (locali1768110497f_frac @ a @ b @ r @ s) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2)) = (locali1641774487f_frac @ a @ a @ product_unit @ rel @ (mult @ a @ ring_ext @ a @ b @ r @ r3 @ r2) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ s2)))),
% 2.97/1.63 inference(cnf_transformation,[status(thm)],[f5])).
% 2.97/1.63 thf(f5,axiom,(
% 2.97/1.63 ((mult @ set @ product_prod @ a @ a @ product_unit @ (locali1768110497f_frac @ a @ b @ r @ s) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2)) = (locali1641774487f_frac @ a @ a @ product_unit @ rel @ (mult @ a @ ring_ext @ a @ b @ r @ r3 @ r2) @ (mult @ a @ ring_ext @ a @ b @ r @ s3 @ s2)))),
% 2.97/1.63 file('/export/starexec/sandbox2/tmp/tmp.ZsShBFnucq/DTF2THF_27309.p',fact_4_f2)).
% 2.97/1.63 thf(f981,plain,(
% 2.97/1.63 ((mult @ set @ product_prod @ a @ a @ product_unit @ (locali1768110497f_frac @ a @ b @ r @ s) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3)) != (mult @ set @ product_prod @ a @ a @ product_unit @ (locali1768110497f_frac @ a @ b @ r @ s) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2)))),
% 2.97/1.63 inference(cnf_transformation,[status(thm)],[f694])).
% 2.97/1.63 thf(f694,plain,(
% 2.97/1.63 ((mult @ set @ product_prod @ a @ a @ product_unit @ (locali1768110497f_frac @ a @ b @ r @ s) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3)) != (mult @ set @ product_prod @ a @ a @ product_unit @ (locali1768110497f_frac @ a @ b @ r @ s) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2)))),
% 2.97/1.63 inference(flattening,[status(thm)],[f258])).
% 2.97/1.63 thf(f258,negated_conjecture,(
% 2.97/1.63 ~((mult @ set @ product_prod @ a @ a @ product_unit @ (locali1768110497f_frac @ a @ b @ r @ s) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3)) = (mult @ set @ product_prod @ a @ a @ product_unit @ (locali1768110497f_frac @ a @ b @ r @ s) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2)))),
% 2.97/1.63 inference(negated_conjecture,[status(cth)],[f257])).
% 2.97/1.63 thf(f257,conjecture,(
% 2.97/1.63 ((mult @ set @ product_prod @ a @ a @ product_unit @ (locali1768110497f_frac @ a @ b @ r @ s) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3)) = (mult @ set @ product_prod @ a @ a @ product_unit @ (locali1768110497f_frac @ a @ b @ r @ s) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3) @ (locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2)))),
% 2.97/1.63 file('/export/starexec/sandbox2/tmp/tmp.ZsShBFnucq/DTF2THF_27309.p',conj_0)).
% 2.97/1.63 % SZS output end Proof for DTF2THF_27309
% 2.97/1.63 % (27586)------------------------------
% 2.97/1.63 % (27586)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.63 % (27586)Termination reason: Refutation
% 2.97/1.63
% 2.97/1.63 % (27586)Memory used [KB]: 6908
% 2.97/1.63 % (27586)Time elapsed: 0.034 s
% 2.97/1.63 % (27586)Instructions burned: 103 (million)
% 2.97/1.63 % (27586)------------------------------
% 2.97/1.63 % (27586)------------------------------
% 2.97/1.63 % (27580)Success in time 0.054 s
%------------------------------------------------------------------------------