%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP106^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n003.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 5.82s 2.02s
% Output : Refutation 5.82s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP106^2 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n003.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:25:55 EST 2026
% 0.17/0.35 % CPUTime :
% 0.17/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.31/0.47 ---- Original DTF file ---
% 0.31/0.47 thf(spec,logic,$$dhol).
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 % File : ITP106^2 : TPTP v9.2.1. Released v7.5.0.
% 0.31/0.47 % Domain : Interactive Theorem Proving
% 0.31/0.47 % Problem : Sledgehammer Localization problem prob_1149__8998378_1
% 0.31/0.47 % Version : Especial.
% 0.31/0.47 % English :
% 0.31/0.47
% 0.31/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.31/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.31/0.47 % Source : [Des21]
% 0.31/0.47 % Names : Localization/prob_1149__8998378_1 [Des21]
% 0.31/0.47
% 0.31/0.47 % Status : Theorem
% 0.31/0.47 % Rating : 0.67 v9.1.0, 0.33 v8.1.0, 0.50 v7.5.0
% 0.31/0.47 % Syntax : Number of formulae : 331 ( 80 unt; 55 typ; 0 def)
% 0.31/0.47 % Number of atoms : 757 ( 326 equ; 0 cnn)
% 0.31/0.47 % Maximal formula atoms : 17 ( 2 avg)
% 0.31/0.47 % Number of connectives : 8774 ( 18 ~; 1 |; 14 &;8315 @)
% 0.31/0.47 % ( 0 <=>; 426 =>; 0 <=; 0 <~>)
% 0.31/0.47 % Maximal formula depth : 28 ( 9 avg)
% 0.31/0.47 % Number of types : 6 ( 5 usr)
% 0.31/0.47 % Number of type conns : 68 ( 68 >; 0 *; 0 +; 0 <<)
% 0.31/0.47 % Number of symbols : 51 ( 50 usr; 8 con; 0-6 aty)
% 0.31/0.47 % Number of variables : 929 ( 10 ^; 851 !; 13 ?; 929 :)
% 0.31/0.47 % ( 55 !>; 0 ?*; 0 @-; 0 @+)
% 0.31/0.47 % SPC : TH1_THM_EQU_NAR
% 0.31/0.47
% 0.31/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:25:31.034
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 % Could-be-implicit typings (11)
% 0.31/0.47 thf(ty_t_Congruence_Opartial__object_Opartial__object__ext,type,
% 0.31/0.47 partia1265057091ct_ext: $tType > $tType > $tType ).
% 0.31/0.47
% 0.31/0.47 thf(ty_t_Congruence_Oeq__object_Oeq__object__ext,type,
% 0.31/0.47 eq_eq_object_ext: $tType > $tType > $tType ).
% 0.31/0.47
% 0.31/0.47 thf(ty_t_Group_Omonoid_Omonoid__ext,type,
% 0.31/0.47 monoid_ext: $tType > $tType > $tType ).
% 0.31/0.47
% 0.31/0.47 thf(ty_t_Ring_Oring_Oring__ext,type,
% 0.31/0.47 ring_ext: $tType > $tType > $tType ).
% 0.31/0.47
% 0.31/0.47 thf(ty_t_Product__Type_Ounit,type,
% 0.31/0.47 product_unit: $tType ).
% 0.31/0.47
% 0.31/0.47 thf(ty_t_Product__Type_Oprod,type,
% 0.31/0.47 product_prod: $tType > $tType > $tType ).
% 0.31/0.47
% 0.31/0.47 thf(ty_t_Set_Oset,type,
% 0.31/0.47 set: $tType > $tType ).
% 0.31/0.47
% 0.31/0.47 thf(ty_t_Nat_Onat,type,
% 0.31/0.47 nat: $tType ).
% 0.31/0.47
% 0.31/0.47 thf(ty_t_Int_Oint,type,
% 0.31/0.47 int: $tType ).
% 0.31/0.47
% 0.31/0.47 thf(ty_tf_b,type,
% 0.31/0.47 b: $tType ).
% 0.31/0.47
% 0.31/0.47 thf(ty_tf_a,type,
% 0.31/0.47 a: $tType ).
% 0.31/0.47
% 0.31/0.47 % Explicit typings (44)
% 0.31/0.47 thf(sy_cl_Groups_Ogroup__add,type,
% 0.31/0.47 group_add:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Rings_Osemiring__1,type,
% 0.31/0.47 semiring_1:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Oab__group__add,type,
% 0.31/0.47 ab_group_add:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.31/0.47 semiring_char_0:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.31/0.47 semigroup_add:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.31/0.47 comm_monoid_add:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.31/0.47 ab_semigroup_add:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.31/0.47 cancel_semigroup_add:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.31/0.47 cancel1352612707id_add:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.31/0.47 cancel146912293up_add:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.31/0.47 ordere779506340up_add:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Congruence_Opartial__object_Ocarrier,type,
% 0.31/0.47 partial_carrier:
% 0.31/0.47 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ Z ) > ( set @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Group_OUnits,type,
% 0.31/0.47 units:
% 0.31/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > ( set @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Group_Ocomm__group,type,
% 0.31/0.47 comm_group:
% 0.31/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Group_Ogroup,type,
% 0.31/0.47 group:
% 0.31/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Group_Omonoid_Omult,type,
% 0.31/0.47 mult:
% 0.31/0.47 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ Z ) ) > A > A > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Group_Omonoid_Oone,type,
% 0.31/0.47 one:
% 0.31/0.47 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ Z ) ) > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Group_Ounits__of,type,
% 0.31/0.47 units_of:
% 0.31/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ product_unit ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.31/0.47 minus_minus:
% 0.31/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.31/0.47 plus_plus:
% 0.31/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.31/0.47 uminus_uminus:
% 0.31/0.47 !>[A: $tType] : ( A > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Nat_OSuc,type,
% 0.31/0.47 suc: nat > nat ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 0.31/0.47 semiring_1_of_nat:
% 0.31/0.47 !>[A: $tType] : ( nat > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Product__Type_OPair,type,
% 0.31/0.47 product_Pair:
% 0.31/0.47 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.31/0.47 product_rec_prod:
% 0.31/0.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Ring_Oa__inv,type,
% 0.31/0.47 a_inv:
% 0.31/0.47 !>[A: $tType,M: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ M ) ) ) > A > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Ring_Oa__minus,type,
% 0.31/0.47 a_minus:
% 0.31/0.47 !>[A: $tType,M: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ M ) ) ) > A > A > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Ring_Oabelian__group,type,
% 0.31/0.47 abelian_group:
% 0.31/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Ring_Oadd__pow,type,
% 0.31/0.47 add_pow:
% 0.31/0.47 !>[A: $tType,C: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ C ) ) ) > B > A > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Ring_Ofield,type,
% 0.31/0.47 field:
% 0.31/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Ring_Oring_Oadd,type,
% 0.31/0.47 add:
% 0.31/0.47 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ) ) > A > A > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Ring_Oring_Omore,type,
% 0.31/0.47 more:
% 0.31/0.47 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ) ) > Z ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Ring_Oring_Ozero,type,
% 0.31/0.47 zero:
% 0.31/0.47 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ) ) > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Ring_Osemiring,type,
% 0.31/0.47 semiring:
% 0.31/0.47 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Set_OCollect,type,
% 0.31/0.47 collect:
% 0.31/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_member,type,
% 0.31/0.47 member:
% 0.31/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_R,type,
% 0.31/0.47 r: partia1265057091ct_ext @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_r,type,
% 0.31/0.47 r2: a ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_r_H,type,
% 0.31/0.47 r3: a ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_r_H_H,type,
% 0.31/0.47 r4: a ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_rel,type,
% 0.31/0.47 rel: partia1265057091ct_ext @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_s,type,
% 0.31/0.47 s: a ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_s_H,type,
% 0.31/0.47 s2: a ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_s_H_H,type,
% 0.31/0.47 s3: a ).
% 0.31/0.47
% 0.31/0.47 % Relevant facts (255)
% 0.31/0.47 thf(fact_0_right__add__eq,axiom,
% 0.31/0.47 ! [A2: a,B2: a,C2: a] :
% 0.31/0.47 ( ( A2 = B2 )
% 0.31/0.47 => ( ( add @ a @ b @ r @ C2 @ A2 )
% 0.31/0.47 = ( add @ a @ b @ r @ C2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % right_add_eq
% 0.31/0.47 thf(fact_1_assms_I3_J,axiom,
% 0.31/0.47 member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ r4 @ s3 ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ).
% 0.31/0.47
% 0.31/0.47 % assms(3)
% 0.31/0.47 thf(fact_2_assms_I2_J,axiom,
% 0.31/0.47 member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ r3 @ s2 ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ).
% 0.31/0.47
% 0.31/0.47 % assms(2)
% 0.31/0.47 thf(fact_3_assms_I1_J,axiom,
% 0.31/0.47 member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ r2 @ s ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ).
% 0.31/0.47
% 0.31/0.47 % assms(1)
% 0.31/0.47 thf(fact_4_closed__rel__add,axiom,
% 0.31/0.47 ! [R: a,S: a,R2: a,S2: a] :
% 0.31/0.47 ( ( 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 ) )
% 0.31/0.47 => ( ( 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 ) )
% 0.31/0.47 => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ S2 @ R ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ S @ R2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ S @ S2 ) ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % closed_rel_add
% 0.31/0.47 thf(fact_5__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_061_As_A_092_060otimes_062_A_Is_H_H_A_092_060otimes_062_As_H_J_A_092_060otimes_062_As_H_H_092_060close_062,axiom,
% 0.31/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) )
% 0.31/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ s2 ) ) @ s3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % \<open>s \<otimes> s'' \<otimes> (s' \<otimes> s'') = s \<otimes> (s'' \<otimes> s') \<otimes> s''\<close>
% 0.31/0.47 thf(fact_6_f9,axiom,
% 0.31/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) )
% 0.31/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ s3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % f9
% 0.31/0.47 thf(fact_7__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_A_061_As_H_H_A_092_060otimes_062_As_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_092_060close_062,axiom,
% 0.31/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r4 ) )
% 0.31/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ s ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % \<open>s \<otimes> s'' \<otimes> (r' \<otimes> r'') = s'' \<otimes> s \<otimes> (r' \<otimes> r'')\<close>
% 0.31/0.47 thf(fact_8__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_092_060close_062,axiom,
% 0.31/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ r3 ) @ r4 ) )
% 0.31/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s2 ) @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % \<open>s \<otimes> s'' \<otimes> (s' \<otimes> s'') \<otimes> (s \<otimes> r' \<otimes> r'') = s \<otimes> s' \<otimes> s'' \<otimes> (s \<otimes> s'' \<otimes> (r' \<otimes> r''))\<close>
% 0.31/0.47 thf(fact_9_f10,axiom,
% 0.31/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r2 ) @ r4 ) )
% 0.31/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s2 ) @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % f10
% 0.31/0.47 thf(fact_10_f5,axiom,
% 0.31/0.47 member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r4 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ).
% 0.31/0.47
% 0.31/0.47 % f5
% 0.31/0.47 thf(fact_11_f4,axiom,
% 0.31/0.47 member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r4 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ).
% 0.31/0.47
% 0.31/0.47 % f4
% 0.31/0.47 thf(fact_12__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_A_092_060otimes_062_Ar_H_H_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_A_092_060otimes_062_Ar_H_H_J_092_060close_062,axiom,
% 0.31/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ r3 ) ) @ r4 ) )
% 0.31/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r2 ) @ r4 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ r3 ) @ r4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % \<open>s \<otimes> s'' \<otimes> (s' \<otimes> s'') \<otimes> ((s' \<otimes> r \<oplus> s \<otimes> r') \<otimes> r'') = s \<otimes> s'' \<otimes> (s' \<otimes> s'') \<otimes> (s' \<otimes> r \<otimes> r'' \<oplus> s \<otimes> r' \<otimes> r'')\<close>
% 0.31/0.47 thf(fact_13__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_A_092_060otimes_062_Ar_H_H_J_A_092_060oplus_062_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_092_060close_062,axiom,
% 0.31/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ r3 ) ) @ r4 ) )
% 0.31/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s2 ) @ s3 ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r4 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r4 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % \<open>s \<otimes> s'' \<otimes> (s' \<otimes> s'') \<otimes> ((s' \<otimes> r \<oplus> s \<otimes> r') \<otimes> r'') = s \<otimes> s' \<otimes> s'' \<otimes> (s' \<otimes> s'' \<otimes> (r \<otimes> r'') \<oplus> s \<otimes> s'' \<otimes> (r' \<otimes> r''))\<close>
% 0.31/0.47 thf(fact_14_f7,axiom,
% 0.31/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ r3 ) ) @ r4 ) )
% 0.31/0.47 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r2 ) @ r4 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ r3 ) @ r4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % f7
% 0.31/0.47 thf(fact_15_f8,axiom,
% 0.31/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s2 ) @ s3 ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r4 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r4 ) ) ) )
% 0.31/0.47 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s2 ) @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r4 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s2 ) @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r4 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % f8
% 0.31/0.47 thf(fact_16_f12,axiom,
% 0.31/0.47 member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ r3 ) ) @ r4 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s2 ) @ s3 ) ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ).
% 0.31/0.47
% 0.31/0.47 % f12
% 0.31/0.47 thf(fact_17_l__distr,axiom,
% 0.31/0.47 ! [X: a,Y: a,Z2: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add @ a @ b @ r @ X @ Y ) @ Z2 )
% 0.31/0.47 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Z2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % l_distr
% 0.31/0.47 thf(fact_18_r__distr,axiom,
% 0.31/0.47 ! [X: a,Y: a,Z2: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Z2 @ ( add @ a @ b @ r @ X @ Y ) )
% 0.31/0.47 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Z2 @ X ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Z2 @ Y ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % r_distr
% 0.31/0.47 thf(fact_19__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_A_092_060otimes_062_Ar_H_H_J_A_092_060oplus_062_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_A_061_A_092_060zero_062_092_060close_062,axiom,
% 0.31/0.47 ( ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ r3 ) ) @ r4 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s2 ) @ s3 ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r4 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r4 ) ) ) ) )
% 0.31/0.47 = ( zero @ a @ b @ r ) ) ).
% 0.31/0.47
% 0.31/0.47 % \<open>s \<otimes> s'' \<otimes> (s' \<otimes> s'') \<otimes> ((s' \<otimes> r \<oplus> s \<otimes> r') \<otimes> r'') \<ominus> s \<otimes> s' \<otimes> s'' \<otimes> (s' \<otimes> s'' \<otimes> (r \<otimes> r'') \<oplus> s \<otimes> s'' \<otimes> (r' \<otimes> r'')) = \<zero>\<close>
% 0.31/0.47 thf(fact_20_prod_Oinject,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 0.31/0.47 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 0.31/0.47 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.31/0.47 = ( ( X1 = Y1 )
% 0.31/0.47 & ( X2 = Y2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod.inject
% 0.31/0.47 thf(fact_21_old_Oprod_Oinject,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.31/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.31/0.47 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.31/0.47 = ( ( A2 = A3 )
% 0.31/0.47 & ( B2 = B3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % old.prod.inject
% 0.31/0.47 thf(fact_22_local_Osemiring__axioms,axiom,
% 0.31/0.47 semiring @ a @ b @ r ).
% 0.31/0.47
% 0.31/0.47 % local.semiring_axioms
% 0.31/0.47 thf(fact_23_m__assoc,axiom,
% 0.31/0.47 ! [X: a,Y: a,Z2: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ Z2 )
% 0.31/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % m_assoc
% 0.31/0.47 thf(fact_24_m__comm,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y )
% 0.31/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % m_comm
% 0.31/0.47 thf(fact_25_m__lcomm,axiom,
% 0.31/0.47 ! [X: a,Y: a,Z2: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) )
% 0.31/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Z2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % m_lcomm
% 0.31/0.47 thf(fact_26_add_Om__assoc,axiom,
% 0.31/0.47 ! [X: a,Y: a,Z2: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add @ a @ b @ r @ ( add @ a @ b @ r @ X @ Y ) @ Z2 )
% 0.31/0.47 = ( add @ a @ b @ r @ X @ ( add @ a @ b @ r @ Y @ Z2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.m_assoc
% 0.31/0.47 thf(fact_27_add_Om__comm,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add @ a @ b @ r @ X @ Y )
% 0.31/0.47 = ( add @ a @ b @ r @ Y @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.m_comm
% 0.31/0.47 thf(fact_28_add_Om__lcomm,axiom,
% 0.31/0.47 ! [X: a,Y: a,Z2: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add @ a @ b @ r @ X @ ( add @ a @ b @ r @ Y @ Z2 ) )
% 0.31/0.47 = ( add @ a @ b @ r @ Y @ ( add @ a @ b @ r @ X @ Z2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.m_lcomm
% 0.31/0.47 thf(fact_29_local_Oright__minus__eq,axiom,
% 0.31/0.47 ! [A2: a,B2: a,C2: a] :
% 0.31/0.47 ( ( A2 = B2 )
% 0.31/0.47 => ( ( a_minus @ a @ b @ r @ C2 @ A2 )
% 0.31/0.47 = ( a_minus @ a @ b @ r @ C2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % local.right_minus_eq
% 0.31/0.47 thf(fact_30_local_Ominus__unique,axiom,
% 0.31/0.47 ! [Y: a,X: a,Y3: a] :
% 0.31/0.47 ( ( ( add @ a @ b @ r @ Y @ X )
% 0.31/0.47 = ( zero @ a @ b @ r ) )
% 0.31/0.47 => ( ( ( add @ a @ b @ r @ X @ Y3 )
% 0.31/0.47 = ( zero @ a @ b @ r ) )
% 0.31/0.47 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( Y = Y3 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % local.minus_unique
% 0.31/0.47 thf(fact_31_add_Or__inv__ex,axiom,
% 0.31/0.47 ! [X: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ? [X3: a] :
% 0.31/0.47 ( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 & ( ( add @ a @ b @ r @ X @ X3 )
% 0.31/0.47 = ( zero @ a @ b @ r ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.r_inv_ex
% 0.31/0.47 thf(fact_32_add_Oone__unique,axiom,
% 0.31/0.47 ! [U: a] :
% 0.31/0.47 ( ( member @ a @ U @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ! [X3: a] :
% 0.31/0.47 ( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add @ a @ b @ r @ U @ X3 )
% 0.31/0.47 = X3 ) )
% 0.31/0.47 => ( U
% 0.31/0.47 = ( zero @ a @ b @ r ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.one_unique
% 0.31/0.47 thf(fact_33_add_Ol__inv__ex,axiom,
% 0.31/0.47 ! [X: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ? [X3: a] :
% 0.31/0.47 ( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 & ( ( add @ a @ b @ r @ X3 @ X )
% 0.31/0.47 = ( zero @ a @ b @ r ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.l_inv_ex
% 0.31/0.47 thf(fact_34_add_Oinv__comm,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.47 ( ( ( add @ a @ b @ r @ X @ Y )
% 0.31/0.47 = ( zero @ a @ b @ r ) )
% 0.31/0.47 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add @ a @ b @ r @ Y @ X )
% 0.31/0.47 = ( zero @ a @ b @ r ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.inv_comm
% 0.31/0.47 thf(fact_35_right__inv__add,axiom,
% 0.31/0.47 ! [A2: a,B2: a,C2: a] :
% 0.31/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( a_minus @ a @ b @ r @ ( a_minus @ a @ b @ r @ C2 @ A2 ) @ B2 )
% 0.31/0.47 = ( a_minus @ a @ b @ r @ C2 @ ( add @ a @ b @ r @ A2 @ B2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % right_inv_add
% 0.31/0.47 thf(fact_36_four__elem__comm,axiom,
% 0.31/0.47 ! [A2: a,B2: a,C2: a,D: a] :
% 0.31/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ D @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( a_minus @ a @ b @ r @ ( add @ a @ b @ r @ ( a_minus @ a @ b @ r @ A2 @ C2 ) @ B2 ) @ D )
% 0.31/0.47 = ( a_minus @ a @ b @ r @ ( a_minus @ a @ b @ r @ ( add @ a @ b @ r @ A2 @ B2 ) @ C2 ) @ D ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % four_elem_comm
% 0.31/0.47 thf(fact_37_local_Oadd_Oright__cancel,axiom,
% 0.31/0.47 ! [X: a,Y: a,Z2: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( ( add @ a @ b @ r @ Y @ X )
% 0.31/0.47 = ( add @ a @ b @ r @ Z2 @ X ) )
% 0.31/0.47 = ( Y = Z2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % local.add.right_cancel
% 0.31/0.47 thf(fact_38_add_Om__closed,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( member @ a @ ( add @ a @ b @ r @ X @ Y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.m_closed
% 0.31/0.47 thf(fact_39_zero__closed,axiom,
% 0.31/0.47 member @ a @ ( zero @ a @ b @ r ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.31/0.47
% 0.31/0.47 % zero_closed
% 0.31/0.47 thf(fact_40_semiring__simprules_I3_J,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % semiring_simprules(3)
% 0.31/0.47 thf(fact_41_minus__closed,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( member @ a @ ( a_minus @ a @ b @ r @ X @ Y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minus_closed
% 0.31/0.47 thf(fact_42_r__zero,axiom,
% 0.31/0.47 ! [X: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add @ a @ b @ r @ X @ ( zero @ a @ b @ r ) )
% 0.31/0.47 = X ) ) ).
% 0.31/0.47
% 0.31/0.47 % r_zero
% 0.31/0.47 thf(fact_43_l__zero,axiom,
% 0.31/0.47 ! [X: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add @ a @ b @ r @ ( zero @ a @ b @ r ) @ X )
% 0.31/0.47 = X ) ) ).
% 0.31/0.47
% 0.31/0.47 % l_zero
% 0.31/0.47 thf(fact_44_add_Or__cancel__one_H,axiom,
% 0.31/0.47 ! [X: a,A2: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( X
% 0.31/0.47 = ( add @ a @ b @ r @ A2 @ X ) )
% 0.31/0.47 = ( A2
% 0.31/0.47 = ( zero @ a @ b @ r ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.r_cancel_one'
% 0.31/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.31/0.47 ! [A: $tType,A2: A,P: A > $o] :
% 0.31/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.31/0.47 = ( P @ A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % mem_Collect_eq
% 0.31/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.31/0.47 ! [A: $tType,A4: set @ A] :
% 0.31/0.47 ( ( collect @ A
% 0.31/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
% 0.31/0.47 = A4 ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_mem_eq
% 0.31/0.47 thf(fact_47_Collect__cong,axiom,
% 0.31/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/0.47 ( ! [X3: A] :
% 0.31/0.47 ( ( P @ X3 )
% 0.31/0.47 = ( Q @ X3 ) )
% 0.31/0.47 => ( ( collect @ A @ P )
% 0.31/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_cong
% 0.31/0.47 thf(fact_48_ext,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.31/0.47 ( ! [X3: A] :
% 0.31/0.47 ( ( F @ X3 )
% 0.31/0.47 = ( G @ X3 ) )
% 0.31/0.47 => ( F = G ) ) ).
% 0.31/0.47
% 0.31/0.47 % ext
% 0.31/0.47 thf(fact_49_add_Or__cancel__one,axiom,
% 0.31/0.47 ! [X: a,A2: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( ( add @ a @ b @ r @ A2 @ X )
% 0.31/0.47 = X )
% 0.31/0.47 = ( A2
% 0.31/0.47 = ( zero @ a @ b @ r ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.r_cancel_one
% 0.31/0.47 thf(fact_50_add_Ol__cancel__one_H,axiom,
% 0.31/0.47 ! [X: a,A2: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( X
% 0.31/0.47 = ( add @ a @ b @ r @ X @ A2 ) )
% 0.31/0.47 = ( A2
% 0.31/0.47 = ( zero @ a @ b @ r ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.l_cancel_one'
% 0.31/0.47 thf(fact_51_add_Ol__cancel__one,axiom,
% 0.31/0.47 ! [X: a,A2: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( ( add @ a @ b @ r @ X @ A2 )
% 0.31/0.47 = X )
% 0.31/0.47 = ( A2
% 0.31/0.47 = ( zero @ a @ b @ r ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.l_cancel_one
% 0.31/0.47 thf(fact_52_r__null,axiom,
% 0.31/0.47 ! [X: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( zero @ a @ b @ r ) )
% 0.31/0.47 = ( zero @ a @ b @ r ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % r_null
% 0.31/0.47 thf(fact_53_l__null,axiom,
% 0.31/0.47 ! [X: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( zero @ a @ b @ r ) @ X )
% 0.31/0.47 = ( zero @ a @ b @ r ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % l_null
% 0.31/0.47 thf(fact_54_old_Oprod_Oinducts,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.31/0.47 ( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
% 0.31/0.47 => ( P @ Prod ) ) ).
% 0.31/0.47
% 0.31/0.47 % old.prod.inducts
% 0.31/0.47 thf(fact_55_old_Oprod_Oexhaust,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.31/0.47 ~ ! [A5: A,B4: B] :
% 0.31/0.47 ( Y
% 0.31/0.47 != ( product_Pair @ A @ B @ A5 @ B4 ) ) ).
% 0.31/0.47
% 0.31/0.47 % old.prod.exhaust
% 0.31/0.47 thf(fact_56_prod__induct7,axiom,
% 0.31/0.47 ! [G2: $tType,F2: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
% 0.31/0.47 ( ! [A5: A,B4: B,C3: C,D3: D2,E2: E,F3: F2,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C3 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
% 0.31/0.47 => ( P @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_induct7
% 0.31/0.47 thf(fact_57_prod__induct6,axiom,
% 0.31/0.47 ! [F2: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) )] :
% 0.31/0.47 ( ! [A5: A,B4: B,C3: C,D3: D2,E2: E,F3: F2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D2 @ ( product_prod @ E @ F2 ) @ D3 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
% 0.31/0.47 => ( P @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_induct6
% 0.31/0.47 thf(fact_58_prod__induct5,axiom,
% 0.31/0.47 ! [E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 0.31/0.47 ( ! [A5: A,B4: B,C3: C,D3: D2,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C3 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) )
% 0.31/0.47 => ( P @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_induct5
% 0.31/0.47 thf(fact_59_prod__induct4,axiom,
% 0.31/0.47 ! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 0.31/0.47 ( ! [A5: A,B4: B,C3: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B4 @ ( product_Pair @ C @ D2 @ C3 @ D3 ) ) ) )
% 0.31/0.47 => ( P @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_induct4
% 0.31/0.47 thf(fact_60_prod__induct3,axiom,
% 0.31/0.47 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.31/0.47 ( ! [A5: A,B4: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C3 ) ) )
% 0.31/0.47 => ( P @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_induct3
% 0.31/0.47 thf(fact_61_prod__cases7,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F2: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
% 0.31/0.47 ~ ! [A5: A,B4: B,C3: C,D3: D2,E2: E,F3: F2,G3: G2] :
% 0.31/0.47 ( Y
% 0.31/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C3 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_cases7
% 0.31/0.47 thf(fact_62_prod__cases6,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) )] :
% 0.31/0.47 ~ ! [A5: A,B4: B,C3: C,D3: D2,E2: E,F3: F2] :
% 0.31/0.47 ( Y
% 0.31/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D2 @ ( product_prod @ E @ F2 ) @ D3 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_cases6
% 0.31/0.47 thf(fact_63_prod__cases5,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 0.31/0.47 ~ ! [A5: A,B4: B,C3: C,D3: D2,E2: E] :
% 0.31/0.47 ( Y
% 0.31/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C3 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_cases5
% 0.31/0.47 thf(fact_64_prod__cases4,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,C: $tType,D2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 0.31/0.47 ~ ! [A5: A,B4: B,C3: C,D3: D2] :
% 0.31/0.47 ( Y
% 0.31/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B4 @ ( product_Pair @ C @ D2 @ C3 @ D3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_cases4
% 0.31/0.47 thf(fact_65_prod__cases3,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.31/0.47 ~ ! [A5: A,B4: B,C3: C] :
% 0.31/0.47 ( Y
% 0.31/0.47 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_cases3
% 0.31/0.47 thf(fact_66_Pair__inject,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.31/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.31/0.47 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.31/0.47 => ~ ( ( A2 = A3 )
% 0.31/0.47 => ( B2 != B3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Pair_inject
% 0.31/0.47 thf(fact_67_prod__cases,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.31/0.47 ( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
% 0.31/0.47 => ( P @ P2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % prod_cases
% 0.31/0.47 thf(fact_68_surj__pair,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.31/0.47 ? [X3: A,Y4: B] :
% 0.31/0.47 ( P2
% 0.31/0.47 = ( product_Pair @ A @ B @ X3 @ Y4 ) ) ).
% 0.31/0.47
% 0.31/0.47 % surj_pair
% 0.31/0.47 thf(fact_69_f11,axiom,
% 0.31/0.47 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ r3 ) ) @ r4 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s2 ) @ s3 ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r4 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r4 ) ) ) ) ) )
% 0.31/0.47 = ( zero @ a @ b @ r ) ) ).
% 0.31/0.47
% 0.31/0.47 % f11
% 0.31/0.47 thf(fact_70_semiring_Osemiring__simprules_I11_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.31/0.47 ( ( semiring @ A @ B @ R3 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( add @ A @ B @ R3 @ X @ ( zero @ A @ B @ R3 ) )
% 0.31/0.47 = X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % semiring.semiring_simprules(11)
% 0.31/0.47 thf(fact_71_semiring_Osemiring__simprules_I6_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.31/0.47 ( ( semiring @ A @ B @ R3 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( add @ A @ B @ R3 @ ( zero @ A @ B @ R3 ) @ X )
% 0.31/0.47 = X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % semiring.semiring_simprules(6)
% 0.31/0.47 thf(fact_72_semiring_Ol__null,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.31/0.47 ( ( semiring @ A @ B @ R3 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( zero @ A @ B @ R3 ) @ X )
% 0.31/0.47 = ( zero @ A @ B @ R3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % semiring.l_null
% 0.31/0.47 thf(fact_73_semiring_Or__null,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.31/0.47 ( ( semiring @ A @ B @ R3 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ ( zero @ A @ B @ R3 ) )
% 0.31/0.47 = ( zero @ A @ B @ R3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % semiring.r_null
% 0.31/0.47 thf(fact_74_semiring_Or__distr,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 0.31/0.47 ( ( semiring @ A @ B @ R3 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ Z2 @ ( add @ A @ B @ R3 @ X @ Y ) )
% 0.31/0.47 = ( add @ A @ B @ R3 @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ Z2 @ X ) @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ Z2 @ Y ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % semiring.r_distr
% 0.31/0.47 thf(fact_75_semiring_Ol__distr,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 0.31/0.47 ( ( semiring @ A @ B @ R3 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( add @ A @ B @ R3 @ X @ Y ) @ Z2 )
% 0.31/0.47 = ( add @ A @ B @ R3 @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ Z2 ) @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ Y @ Z2 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % semiring.l_distr
% 0.31/0.47 thf(fact_76_add__pow__rdistr__int,axiom,
% 0.31/0.47 ! [A2: a,B2: a,K: int] :
% 0.31/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ ( add_pow @ a @ b @ int @ r @ K @ B2 ) )
% 0.31/0.47 = ( add_pow @ a @ b @ int @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_pow_rdistr_int
% 0.31/0.47 thf(fact_77_add__pow__ldistr__int,axiom,
% 0.31/0.47 ! [A2: a,B2: a,K: int] :
% 0.31/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add_pow @ a @ b @ int @ r @ K @ A2 ) @ B2 )
% 0.31/0.47 = ( add_pow @ a @ b @ int @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_pow_ldistr_int
% 0.31/0.47 thf(fact_78_inv__unique,axiom,
% 0.31/0.47 ! [Y: a,X: a,Y3: a] :
% 0.31/0.47 ( ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ X )
% 0.31/0.47 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/0.47 => ( ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y3 )
% 0.31/0.47 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( Y = Y3 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inv_unique
% 0.31/0.47 thf(fact_79_one__unique,axiom,
% 0.31/0.47 ! [U: a] :
% 0.31/0.47 ( ( member @ a @ U @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ! [X3: a] :
% 0.31/0.47 ( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ U @ X3 )
% 0.31/0.47 = X3 ) )
% 0.31/0.47 => ( U
% 0.31/0.47 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % one_unique
% 0.31/0.47 thf(fact_80_add_Oint__pow__distrib,axiom,
% 0.31/0.47 ! [X: a,Y: a,I: int] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add_pow @ a @ b @ int @ r @ I @ ( add @ a @ b @ r @ X @ Y ) )
% 0.31/0.47 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( add_pow @ a @ b @ int @ r @ I @ Y ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.int_pow_distrib
% 0.31/0.47 thf(fact_81_add_Oint__pow__mult__distrib,axiom,
% 0.31/0.47 ! [X: a,Y: a,I: int] :
% 0.31/0.47 ( ( ( add @ a @ b @ r @ X @ Y )
% 0.31/0.47 = ( add @ a @ b @ r @ Y @ X ) )
% 0.31/0.47 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add_pow @ a @ b @ int @ r @ I @ ( add @ a @ b @ r @ X @ Y ) )
% 0.31/0.47 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( add_pow @ a @ b @ int @ r @ I @ Y ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.int_pow_mult_distrib
% 0.31/0.47 thf(fact_82_semiring__simprules_I4_J,axiom,
% 0.31/0.47 member @ a @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 0.31/0.47
% 0.31/0.47 % semiring_simprules(4)
% 0.31/0.47 thf(fact_83_add_Oint__pow__closed,axiom,
% 0.31/0.47 ! [X: a,I: int] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( member @ a @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.int_pow_closed
% 0.31/0.47 thf(fact_84_add_Oint__pow__one,axiom,
% 0.31/0.47 ! [Z2: int] :
% 0.31/0.47 ( ( add_pow @ a @ b @ int @ r @ Z2 @ ( zero @ a @ b @ r ) )
% 0.31/0.47 = ( zero @ a @ b @ r ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.int_pow_one
% 0.31/0.47 thf(fact_85_l__one,axiom,
% 0.31/0.47 ! [X: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ X )
% 0.31/0.47 = X ) ) ).
% 0.31/0.47
% 0.31/0.47 % l_one
% 0.31/0.47 thf(fact_86_r__one,axiom,
% 0.31/0.47 ! [X: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/0.47 = X ) ) ).
% 0.31/0.47
% 0.31/0.47 % r_one
% 0.31/0.47 thf(fact_87_semiring_Osemiring__simprules_I4_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/0.47 ( ( semiring @ A @ B @ R3 )
% 0.31/0.47 => ( member @ A @ ( one @ A @ ( ring_ext @ A @ B ) @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % semiring.semiring_simprules(4)
% 0.31/0.47 thf(fact_88_semiring_Osemiring__simprules_I9_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.31/0.47 ( ( semiring @ A @ B @ R3 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( one @ A @ ( ring_ext @ A @ B ) @ R3 ) @ X )
% 0.31/0.47 = X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % semiring.semiring_simprules(9)
% 0.31/0.47 thf(fact_89_semiring_Osemiring__simprules_I3_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 0.31/0.47 ( ( semiring @ A @ B @ R3 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( member @ A @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % semiring.semiring_simprules(3)
% 0.31/0.47 thf(fact_90_semiring_Osemiring__simprules_I8_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 0.31/0.47 ( ( semiring @ A @ B @ R3 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ Y ) @ Z2 )
% 0.31/0.47 = ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ Y @ Z2 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % semiring.semiring_simprules(8)
% 0.31/0.47 thf(fact_91_semiring_Osemiring__simprules_I1_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 0.31/0.47 ( ( semiring @ A @ B @ R3 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( member @ A @ ( add @ A @ B @ R3 @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % semiring.semiring_simprules(1)
% 0.31/0.47 thf(fact_92_semiring_Osemiring__simprules_I5_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 0.31/0.47 ( ( semiring @ A @ B @ R3 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( add @ A @ B @ R3 @ ( add @ A @ B @ R3 @ X @ Y ) @ Z2 )
% 0.31/0.47 = ( add @ A @ B @ R3 @ X @ ( add @ A @ B @ R3 @ Y @ Z2 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % semiring.semiring_simprules(5)
% 0.31/0.47 thf(fact_93_semiring_Osemiring__simprules_I7_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 0.31/0.47 ( ( semiring @ A @ B @ R3 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( add @ A @ B @ R3 @ X @ Y )
% 0.31/0.47 = ( add @ A @ B @ R3 @ Y @ X ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % semiring.semiring_simprules(7)
% 0.31/0.47 thf(fact_94_semiring_Osemiring__simprules_I12_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 0.31/0.47 ( ( semiring @ A @ B @ R3 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( add @ A @ B @ R3 @ X @ ( add @ A @ B @ R3 @ Y @ Z2 ) )
% 0.31/0.47 = ( add @ A @ B @ R3 @ Y @ ( add @ A @ B @ R3 @ X @ Z2 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % semiring.semiring_simprules(12)
% 0.31/0.47 thf(fact_95_semiring_Osemiring__simprules_I2_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/0.47 ( ( semiring @ A @ B @ R3 )
% 0.31/0.47 => ( member @ A @ ( zero @ A @ B @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % semiring.semiring_simprules(2)
% 0.31/0.47 thf(fact_96_cring__fieldI2,axiom,
% 0.31/0.47 ( ( ( zero @ a @ b @ r )
% 0.31/0.47 != ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/0.47 => ( ! [A5: a] :
% 0.31/0.47 ( ( member @ a @ A5 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( A5
% 0.31/0.47 != ( zero @ a @ b @ r ) )
% 0.31/0.47 => ? [X5: a] :
% 0.31/0.47 ( ( member @ a @ X5 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 & ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A5 @ X5 )
% 0.31/0.47 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) )
% 0.31/0.47 => ( field @ a @ b @ r ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % cring_fieldI2
% 0.31/0.47 thf(fact_97_add_Oint__pow__mult,axiom,
% 0.31/0.47 ! [X: a,I: int,J: int] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add_pow @ a @ b @ int @ r @ ( plus_plus @ int @ I @ J ) @ X )
% 0.31/0.47 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( add_pow @ a @ b @ int @ r @ J @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.int_pow_mult
% 0.31/0.47 thf(fact_98_add__pow__ldistr,axiom,
% 0.31/0.47 ! [A2: a,B2: a,K: nat] :
% 0.31/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add_pow @ a @ b @ nat @ r @ K @ A2 ) @ B2 )
% 0.31/0.47 = ( add_pow @ a @ b @ nat @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_pow_ldistr
% 0.31/0.47 thf(fact_99_add__pow__rdistr,axiom,
% 0.31/0.47 ! [A2: a,B2: a,K: nat] :
% 0.31/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ ( add_pow @ a @ b @ nat @ r @ K @ B2 ) )
% 0.31/0.47 = ( add_pow @ a @ b @ nat @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_pow_rdistr
% 0.31/0.47 thf(fact_100_old_Oprod_Orec,axiom,
% 0.31/0.47 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.31/0.47 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.31/0.47 = ( F1 @ A2 @ B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % old.prod.rec
% 0.31/0.47 thf(fact_101_add_Ogroup__commutes__pow,axiom,
% 0.31/0.47 ! [X: a,Y: a,N: nat] :
% 0.31/0.47 ( ( ( add @ a @ b @ r @ X @ Y )
% 0.31/0.47 = ( add @ a @ b @ r @ Y @ X ) )
% 0.31/0.47 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add @ a @ b @ r @ ( add_pow @ a @ b @ nat @ r @ N @ X ) @ Y )
% 0.31/0.47 = ( add @ a @ b @ r @ Y @ ( add_pow @ a @ b @ nat @ r @ N @ X ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.group_commutes_pow
% 0.31/0.47 thf(fact_102_add_Opow__mult__distrib,axiom,
% 0.31/0.47 ! [X: a,Y: a,N: nat] :
% 0.31/0.47 ( ( ( add @ a @ b @ r @ X @ Y )
% 0.31/0.47 = ( add @ a @ b @ r @ Y @ X ) )
% 0.31/0.47 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add_pow @ a @ b @ nat @ r @ N @ ( add @ a @ b @ r @ X @ Y ) )
% 0.31/0.47 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ nat @ r @ N @ X ) @ ( add_pow @ a @ b @ nat @ r @ N @ Y ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.pow_mult_distrib
% 0.31/0.47 thf(fact_103_add_Onat__pow__distrib,axiom,
% 0.31/0.47 ! [X: a,Y: a,N: nat] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add_pow @ a @ b @ nat @ r @ N @ ( add @ a @ b @ r @ X @ Y ) )
% 0.31/0.47 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ nat @ r @ N @ X ) @ ( add_pow @ a @ b @ nat @ r @ N @ Y ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.nat_pow_distrib
% 0.31/0.47 thf(fact_104_add_Onat__pow__comm,axiom,
% 0.31/0.47 ! [X: a,N: nat,M2: nat] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add @ a @ b @ r @ ( add_pow @ a @ b @ nat @ r @ N @ X ) @ ( add_pow @ a @ b @ nat @ r @ M2 @ X ) )
% 0.31/0.47 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ nat @ r @ M2 @ X ) @ ( add_pow @ a @ b @ nat @ r @ N @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.nat_pow_comm
% 0.31/0.47 thf(fact_105_add_Onat__pow__closed,axiom,
% 0.31/0.47 ! [X: a,N: nat] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( member @ a @ ( add_pow @ a @ b @ nat @ r @ N @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.nat_pow_closed
% 0.31/0.47 thf(fact_106_add_Onat__pow__one,axiom,
% 0.31/0.47 ! [N: nat] :
% 0.31/0.47 ( ( add_pow @ a @ b @ nat @ r @ N @ ( zero @ a @ b @ r ) )
% 0.31/0.47 = ( zero @ a @ b @ r ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.nat_pow_one
% 0.31/0.47 thf(fact_107_one__not__zero,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/0.47 ( ( field @ A @ B @ R3 )
% 0.31/0.47 => ( ( one @ A @ ( ring_ext @ A @ B ) @ R3 )
% 0.31/0.47 != ( zero @ A @ B @ R3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % one_not_zero
% 0.31/0.47 thf(fact_108_integral,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A] :
% 0.31/0.47 ( ( field @ A @ B @ R3 )
% 0.31/0.47 => ( ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 )
% 0.31/0.47 = ( zero @ A @ B @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( A2
% 0.31/0.47 = ( zero @ A @ B @ R3 ) )
% 0.31/0.47 | ( B2
% 0.31/0.47 = ( zero @ A @ B @ R3 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % integral
% 0.31/0.47 thf(fact_109_semiring_Oadd__pow__ldistr,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A,K: nat] :
% 0.31/0.47 ( ( semiring @ A @ B @ R3 )
% 0.31/0.47 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( add_pow @ A @ B @ nat @ R3 @ K @ A2 ) @ B2 )
% 0.31/0.47 = ( add_pow @ A @ B @ nat @ R3 @ K @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % semiring.add_pow_ldistr
% 0.31/0.47 thf(fact_110_semiring_Oadd__pow__rdistr,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A,K: nat] :
% 0.31/0.47 ( ( semiring @ A @ B @ R3 )
% 0.31/0.47 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ ( add_pow @ A @ B @ nat @ R3 @ K @ B2 ) )
% 0.31/0.47 = ( add_pow @ A @ B @ nat @ R3 @ K @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % semiring.add_pow_rdistr
% 0.31/0.47 thf(fact_111_add_Onat__pow__mult,axiom,
% 0.31/0.47 ! [X: a,N: nat,M2: nat] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add @ a @ b @ r @ ( add_pow @ a @ b @ nat @ r @ N @ X ) @ ( add_pow @ a @ b @ nat @ r @ M2 @ X ) )
% 0.31/0.47 = ( add_pow @ a @ b @ nat @ r @ ( plus_plus @ nat @ N @ M2 ) @ X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.nat_pow_mult
% 0.31/0.47 thf(fact_112_add_Onat__pow__Suc2,axiom,
% 0.31/0.47 ! [X: a,N: nat] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add_pow @ a @ b @ nat @ r @ ( suc @ N ) @ X )
% 0.31/0.47 = ( add @ a @ b @ r @ X @ ( add_pow @ a @ b @ nat @ r @ N @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.nat_pow_Suc2
% 0.31/0.47 thf(fact_113_add_Opow__eq__div2,axiom,
% 0.31/0.47 ! [X: a,M2: nat,N: nat] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( ( add_pow @ a @ b @ nat @ r @ M2 @ X )
% 0.31/0.47 = ( add_pow @ a @ b @ nat @ r @ N @ X ) )
% 0.31/0.47 => ( ( add_pow @ a @ b @ nat @ r @ ( minus_minus @ nat @ M2 @ N ) @ X )
% 0.31/0.47 = ( zero @ a @ b @ r ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.pow_eq_div2
% 0.31/0.47 thf(fact_114_minus__to__eq,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.47 ( ( abelian_group @ a @ b @ r )
% 0.31/0.47 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( ( a_minus @ a @ b @ r @ X @ Y )
% 0.31/0.47 = ( zero @ a @ b @ r ) )
% 0.31/0.47 => ( X = Y ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minus_to_eq
% 0.31/0.47 thf(fact_115_ring_Oequality,axiom,
% 0.31/0.47 ! [Z: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ),R2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) )] :
% 0.31/0.47 ( ( ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ) @ R )
% 0.31/0.47 = ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ) @ R2 ) )
% 0.31/0.47 => ( ( ( mult @ A @ ( ring_ext @ A @ Z ) @ R )
% 0.31/0.47 = ( mult @ A @ ( ring_ext @ A @ Z ) @ R2 ) )
% 0.31/0.47 => ( ( ( one @ A @ ( ring_ext @ A @ Z ) @ R )
% 0.31/0.47 = ( one @ A @ ( ring_ext @ A @ Z ) @ R2 ) )
% 0.31/0.47 => ( ( ( zero @ A @ Z @ R )
% 0.31/0.47 = ( zero @ A @ Z @ R2 ) )
% 0.31/0.47 => ( ( ( add @ A @ Z @ R )
% 0.31/0.47 = ( add @ A @ Z @ R2 ) )
% 0.31/0.47 => ( ( ( more @ A @ Z @ R )
% 0.31/0.47 = ( more @ A @ Z @ R2 ) )
% 0.31/0.47 => ( R = R2 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ring.equality
% 0.31/0.47 thf(fact_116_is__abelian__group,axiom,
% 0.31/0.47 abelian_group @ a @ b @ r ).
% 0.31/0.47
% 0.31/0.47 % is_abelian_group
% 0.31/0.47 thf(fact_117_add_Onat__pow__Suc,axiom,
% 0.31/0.47 ! [N: nat,X: a] :
% 0.31/0.47 ( ( add_pow @ a @ b @ nat @ r @ ( suc @ N ) @ X )
% 0.31/0.47 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ nat @ r @ N @ X ) @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.nat_pow_Suc
% 0.31/0.47 thf(fact_118_abelian__groupE_I1_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 0.31/0.47 ( ( abelian_group @ A @ B @ R3 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( member @ A @ ( add @ A @ B @ R3 @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % abelian_groupE(1)
% 0.31/0.47 thf(fact_119_abelian__groupE_I3_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 0.31/0.47 ( ( abelian_group @ A @ B @ R3 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( add @ A @ B @ R3 @ ( add @ A @ B @ R3 @ X @ Y ) @ Z2 )
% 0.31/0.47 = ( add @ A @ B @ R3 @ X @ ( add @ A @ B @ R3 @ Y @ Z2 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % abelian_groupE(3)
% 0.31/0.47 thf(fact_120_abelian__groupE_I4_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 0.31/0.47 ( ( abelian_group @ A @ B @ R3 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( add @ A @ B @ R3 @ X @ Y )
% 0.31/0.47 = ( add @ A @ B @ R3 @ Y @ X ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % abelian_groupE(4)
% 0.31/0.47 thf(fact_121_abelian__groupE_I2_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/0.47 ( ( abelian_group @ A @ B @ R3 )
% 0.31/0.47 => ( member @ A @ ( zero @ A @ B @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % abelian_groupE(2)
% 0.31/0.47 thf(fact_122_abelian__group_Ominus__closed,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 0.31/0.47 ( ( abelian_group @ A @ B @ G4 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.47 => ( member @ A @ ( a_minus @ A @ B @ G4 @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % abelian_group.minus_closed
% 0.31/0.47 thf(fact_123_abelian__groupI,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 0.31/0.47 ( ! [X3: A] :
% 0.31/0.47 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ! [Y4: A] :
% 0.31/0.47 ( ( member @ A @ Y4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( member @ A @ ( add @ A @ B @ R3 @ X3 @ Y4 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) )
% 0.31/0.47 => ( ( member @ A @ ( zero @ A @ B @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ! [X3: A] :
% 0.31/0.47 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ! [Y4: A] :
% 0.31/0.47 ( ( member @ A @ Y4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ! [Z3: A] :
% 0.31/0.47 ( ( member @ A @ Z3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( add @ A @ B @ R3 @ ( add @ A @ B @ R3 @ X3 @ Y4 ) @ Z3 )
% 0.31/0.47 = ( add @ A @ B @ R3 @ X3 @ ( add @ A @ B @ R3 @ Y4 @ Z3 ) ) ) ) ) )
% 0.31/0.47 => ( ! [X3: A] :
% 0.31/0.47 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ! [Y4: A] :
% 0.31/0.47 ( ( member @ A @ Y4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( add @ A @ B @ R3 @ X3 @ Y4 )
% 0.31/0.47 = ( add @ A @ B @ R3 @ Y4 @ X3 ) ) ) )
% 0.31/0.47 => ( ! [X3: A] :
% 0.31/0.47 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( add @ A @ B @ R3 @ ( zero @ A @ B @ R3 ) @ X3 )
% 0.31/0.47 = X3 ) )
% 0.31/0.47 => ( ! [X3: A] :
% 0.31/0.47 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ? [Xa: A] :
% 0.31/0.47 ( ( member @ A @ Xa @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 & ( ( add @ A @ B @ R3 @ Xa @ X3 )
% 0.31/0.47 = ( zero @ A @ B @ R3 ) ) ) )
% 0.31/0.47 => ( abelian_group @ A @ B @ R3 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % abelian_groupI
% 0.31/0.47 thf(fact_124_abelian__groupE_I5_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.31/0.47 ( ( abelian_group @ A @ B @ R3 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ( ( add @ A @ B @ R3 @ ( zero @ A @ B @ R3 ) @ X )
% 0.31/0.47 = X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % abelian_groupE(5)
% 0.31/0.47 thf(fact_125_abelian__groupE_I6_J,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.31/0.47 ( ( abelian_group @ A @ B @ R3 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 => ? [X3: A] :
% 0.31/0.47 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 0.31/0.47 & ( ( add @ A @ B @ R3 @ X3 @ X )
% 0.31/0.47 = ( zero @ A @ B @ R3 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % abelian_groupE(6)
% 0.31/0.47 thf(fact_126_abelian__group_Ofour__elem__comm,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A,C2: A,D: A] :
% 0.31/0.47 ( ( abelian_group @ A @ B @ G4 )
% 0.31/0.47 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.47 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.47 => ( ( member @ A @ C2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.47 => ( ( member @ A @ D @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.47 => ( ( a_minus @ A @ B @ G4 @ ( add @ A @ B @ G4 @ ( a_minus @ A @ B @ G4 @ A2 @ C2 ) @ B2 ) @ D )
% 0.31/0.47 = ( a_minus @ A @ B @ G4 @ ( a_minus @ A @ B @ G4 @ ( add @ A @ B @ G4 @ A2 @ B2 ) @ C2 ) @ D ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % abelian_group.four_elem_comm
% 0.31/0.47 thf(fact_127_abelian__group_Oright__inv__add,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A,C2: A] :
% 0.31/0.47 ( ( abelian_group @ A @ B @ G4 )
% 0.31/0.47 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.47 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.47 => ( ( member @ A @ C2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.47 => ( ( a_minus @ A @ B @ G4 @ ( a_minus @ A @ B @ G4 @ C2 @ A2 ) @ B2 )
% 0.31/0.47 = ( a_minus @ A @ B @ G4 @ C2 @ ( add @ A @ B @ G4 @ A2 @ B2 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % abelian_group.right_inv_add
% 0.31/0.47 thf(fact_128_abelian__group_Ominus__to__eq,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 0.31/0.47 ( ( abelian_group @ A @ B @ G4 )
% 0.31/0.47 => ( ( abelian_group @ A @ B @ G4 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.47 => ( ( ( a_minus @ A @ B @ G4 @ X @ Y )
% 0.31/0.47 = ( zero @ A @ B @ G4 ) )
% 0.31/0.47 => ( X = Y ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % abelian_group.minus_to_eq
% 0.31/0.47 thf(fact_129_diff__diff__left,axiom,
% 0.31/0.47 ! [I: nat,J: nat,K: nat] :
% 0.31/0.47 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 0.31/0.47 = ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_diff_left
% 0.31/0.47 thf(fact_130_Suc__diff__diff,axiom,
% 0.31/0.47 ! [M2: nat,N: nat,K: nat] :
% 0.31/0.47 ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
% 0.31/0.47 = ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ N ) @ K ) ) ).
% 0.31/0.47
% 0.31/0.47 % Suc_diff_diff
% 0.31/0.47 thf(fact_131_diff__Suc__Suc,axiom,
% 0.31/0.47 ! [M2: nat,N: nat] :
% 0.31/0.47 ( ( minus_minus @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
% 0.31/0.47 = ( minus_minus @ nat @ M2 @ N ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_Suc_Suc
% 0.31/0.47 thf(fact_132_add__Suc__right,axiom,
% 0.31/0.47 ! [M2: nat,N: nat] :
% 0.31/0.47 ( ( plus_plus @ nat @ M2 @ ( suc @ N ) )
% 0.31/0.47 = ( suc @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_Suc_right
% 0.31/0.47 thf(fact_133_nat_Oinject,axiom,
% 0.31/0.47 ! [X2: nat,Y2: nat] :
% 0.31/0.47 ( ( ( suc @ X2 )
% 0.31/0.47 = ( suc @ Y2 ) )
% 0.31/0.47 = ( X2 = Y2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % nat.inject
% 0.31/0.47 thf(fact_134_old_Onat_Oinject,axiom,
% 0.31/0.47 ! [Nat: nat,Nat2: nat] :
% 0.31/0.47 ( ( ( suc @ Nat )
% 0.31/0.47 = ( suc @ Nat2 ) )
% 0.31/0.47 = ( Nat = Nat2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % old.nat.inject
% 0.31/0.47 thf(fact_135_Suc__inject,axiom,
% 0.31/0.47 ! [X: nat,Y: nat] :
% 0.31/0.47 ( ( ( suc @ X )
% 0.31/0.47 = ( suc @ Y ) )
% 0.31/0.47 => ( X = Y ) ) ).
% 0.31/0.47
% 0.31/0.47 % Suc_inject
% 0.31/0.47 thf(fact_136_n__not__Suc__n,axiom,
% 0.31/0.47 ! [N: nat] :
% 0.31/0.47 ( N
% 0.31/0.47 != ( suc @ N ) ) ).
% 0.31/0.47
% 0.31/0.47 % n_not_Suc_n
% 0.31/0.47 thf(fact_137_diff__commute,axiom,
% 0.31/0.47 ! [I: nat,J: nat,K: nat] :
% 0.31/0.47 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 0.31/0.47 = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_commute
% 0.31/0.47 thf(fact_138_add__Suc,axiom,
% 0.31/0.47 ! [M2: nat,N: nat] :
% 0.31/0.47 ( ( plus_plus @ nat @ ( suc @ M2 ) @ N )
% 0.31/0.47 = ( suc @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_Suc
% 0.31/0.47 thf(fact_139_nat__arith_Osuc1,axiom,
% 0.31/0.47 ! [A4: nat,K: nat,A2: nat] :
% 0.31/0.47 ( ( A4
% 0.31/0.47 = ( plus_plus @ nat @ K @ A2 ) )
% 0.31/0.47 => ( ( suc @ A4 )
% 0.31/0.47 = ( plus_plus @ nat @ K @ ( suc @ A2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % nat_arith.suc1
% 0.31/0.47 thf(fact_140_add__Suc__shift,axiom,
% 0.31/0.47 ! [M2: nat,N: nat] :
% 0.31/0.47 ( ( plus_plus @ nat @ ( suc @ M2 ) @ N )
% 0.31/0.47 = ( plus_plus @ nat @ M2 @ ( suc @ N ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_Suc_shift
% 0.31/0.47 thf(fact_141_zero__induct__lemma,axiom,
% 0.31/0.47 ! [P: nat > $o,K: nat,I: nat] :
% 0.31/0.47 ( ( P @ K )
% 0.31/0.47 => ( ! [N2: nat] :
% 0.31/0.47 ( ( P @ ( suc @ N2 ) )
% 0.31/0.47 => ( P @ N2 ) )
% 0.31/0.47 => ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_induct_lemma
% 0.31/0.47 thf(fact_142_diff__add__inverse2,axiom,
% 0.31/0.47 ! [M2: nat,N: nat] :
% 0.31/0.47 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ N )
% 0.31/0.47 = M2 ) ).
% 0.31/0.47
% 0.31/0.47 % diff_add_inverse2
% 0.31/0.47 thf(fact_143_diff__add__inverse,axiom,
% 0.31/0.47 ! [N: nat,M2: nat] :
% 0.31/0.47 ( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M2 ) @ N )
% 0.31/0.47 = M2 ) ).
% 0.31/0.47
% 0.31/0.47 % diff_add_inverse
% 0.31/0.47 thf(fact_144_diff__cancel2,axiom,
% 0.31/0.47 ! [M2: nat,K: nat,N: nat] :
% 0.31/0.47 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) )
% 0.31/0.47 = ( minus_minus @ nat @ M2 @ N ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_cancel2
% 0.31/0.47 thf(fact_145_Nat_Odiff__cancel,axiom,
% 0.31/0.47 ! [K: nat,M2: nat,N: nat] :
% 0.31/0.47 ( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
% 0.31/0.47 = ( minus_minus @ nat @ M2 @ N ) ) ).
% 0.31/0.47
% 0.31/0.47 % Nat.diff_cancel
% 0.31/0.47 thf(fact_146_add__diff__cancel__right_H,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( cancel146912293up_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 0.31/0.47 = A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_diff_cancel_right'
% 0.31/0.47 thf(fact_147_add__diff__cancel__right,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( cancel146912293up_add @ A )
% 0.31/0.47 => ! [A2: A,C2: A,B2: A] :
% 0.31/0.47 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.31/0.47 = ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_diff_cancel_right
% 0.31/0.47 thf(fact_148_add__diff__cancel__left_H,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( cancel146912293up_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 )
% 0.31/0.47 = B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_diff_cancel_left'
% 0.31/0.47 thf(fact_149_add__diff__cancel__left,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( cancel146912293up_add @ A )
% 0.31/0.47 => ! [C2: A,A2: A,B2: A] :
% 0.31/0.47 ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 0.31/0.47 = ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_diff_cancel_left
% 0.31/0.47 thf(fact_150_add__right__cancel,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( cancel_semigroup_add @ A )
% 0.31/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.31/0.47 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.31/0.47 = ( plus_plus @ A @ C2 @ A2 ) )
% 0.31/0.47 = ( B2 = C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_right_cancel
% 0.31/0.47 thf(fact_151_add__left__cancel,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( cancel_semigroup_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.47 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.31/0.47 = ( plus_plus @ A @ A2 @ C2 ) )
% 0.31/0.47 = ( B2 = C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_left_cancel
% 0.31/0.47 thf(fact_152_add__diff__cancel,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 0.31/0.47 = A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_diff_cancel
% 0.31/0.47 thf(fact_153_diff__add__cancel,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
% 0.31/0.47 = A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_add_cancel
% 0.31/0.47 thf(fact_154_add__right__imp__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( cancel_semigroup_add @ A )
% 0.31/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.31/0.47 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.31/0.47 = ( plus_plus @ A @ C2 @ A2 ) )
% 0.31/0.47 => ( B2 = C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_right_imp_eq
% 0.31/0.47 thf(fact_155_add__left__imp__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( cancel_semigroup_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.47 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.31/0.47 = ( plus_plus @ A @ A2 @ C2 ) )
% 0.31/0.47 => ( B2 = C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_left_imp_eq
% 0.31/0.47 thf(fact_156_add_Oleft__commute,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ab_semigroup_add @ A )
% 0.31/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.31/0.47 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
% 0.31/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.left_commute
% 0.31/0.47 thf(fact_157_add_Ocommute,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ab_semigroup_add @ A )
% 0.31/0.47 => ( ( plus_plus @ A )
% 0.31/0.47 = ( ^ [A6: A,B5: A] : ( plus_plus @ A @ B5 @ A6 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.commute
% 0.31/0.47 thf(fact_158_group__add__class_Oadd_Oright__cancel,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.31/0.47 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.31/0.47 = ( plus_plus @ A @ C2 @ A2 ) )
% 0.31/0.47 = ( B2 = C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % group_add_class.add.right_cancel
% 0.31/0.47 thf(fact_159_add_Oleft__cancel,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.47 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.31/0.47 = ( plus_plus @ A @ A2 @ C2 ) )
% 0.31/0.47 = ( B2 = C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.left_cancel
% 0.31/0.47 thf(fact_160_add_Oassoc,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semigroup_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 0.31/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.assoc
% 0.31/0.47 thf(fact_161_group__cancel_Oadd2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( comm_monoid_add @ A )
% 0.31/0.47 => ! [B6: A,K: A,B2: A,A2: A] :
% 0.31/0.47 ( ( B6
% 0.31/0.47 = ( plus_plus @ A @ K @ B2 ) )
% 0.31/0.47 => ( ( plus_plus @ A @ A2 @ B6 )
% 0.31/0.47 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % group_cancel.add2
% 0.31/0.47 thf(fact_162_group__cancel_Oadd1,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( comm_monoid_add @ A )
% 0.31/0.47 => ! [A4: A,K: A,A2: A,B2: A] :
% 0.31/0.47 ( ( A4
% 0.31/0.47 = ( plus_plus @ A @ K @ A2 ) )
% 0.31/0.47 => ( ( plus_plus @ A @ A4 @ B2 )
% 0.31/0.47 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % group_cancel.add1
% 0.31/0.47 thf(fact_163_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere779506340up_add @ A )
% 0.31/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.31/0.47 ( ( ( I = J )
% 0.31/0.47 & ( K = L ) )
% 0.31/0.47 => ( ( plus_plus @ A @ I @ K )
% 0.31/0.47 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_mono_thms_linordered_semiring(4)
% 0.31/0.47 thf(fact_164_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ab_semigroup_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 0.31/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ab_semigroup_add_class.add_ac(1)
% 0.31/0.47 thf(fact_165_diff__right__commute,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( cancel146912293up_add @ A )
% 0.31/0.47 => ! [A2: A,C2: A,B2: A] :
% 0.31/0.47 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 )
% 0.31/0.47 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_right_commute
% 0.31/0.47 thf(fact_166_diff__eq__diff__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A,C2: A,D: A] :
% 0.31/0.47 ( ( ( minus_minus @ A @ A2 @ B2 )
% 0.31/0.47 = ( minus_minus @ A @ C2 @ D ) )
% 0.31/0.47 => ( ( A2 = B2 )
% 0.31/0.47 = ( C2 = D ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_eq_diff_eq
% 0.31/0.47 thf(fact_167_group__cancel_Osub1,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ab_group_add @ A )
% 0.31/0.47 => ! [A4: A,K: A,A2: A,B2: A] :
% 0.31/0.47 ( ( A4
% 0.31/0.47 = ( plus_plus @ A @ K @ A2 ) )
% 0.31/0.47 => ( ( minus_minus @ A @ A4 @ B2 )
% 0.31/0.47 = ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % group_cancel.sub1
% 0.31/0.47 thf(fact_168_diff__eq__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.47 ( ( ( minus_minus @ A @ A2 @ B2 )
% 0.31/0.47 = C2 )
% 0.31/0.47 = ( A2
% 0.31/0.47 = ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_eq_eq
% 0.31/0.47 thf(fact_169_eq__diff__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [A2: A,C2: A,B2: A] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 = ( minus_minus @ A @ C2 @ B2 ) )
% 0.31/0.47 = ( ( plus_plus @ A @ A2 @ B2 )
% 0.31/0.47 = C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % eq_diff_eq
% 0.31/0.47 thf(fact_170_add__diff__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.47 ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
% 0.31/0.47 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_diff_eq
% 0.31/0.47 thf(fact_171_diff__diff__eq2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.47 ( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
% 0.31/0.47 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_diff_eq2
% 0.31/0.47 thf(fact_172_diff__add__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ab_group_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.47 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
% 0.31/0.47 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_add_eq
% 0.31/0.47 thf(fact_173_diff__add__eq__diff__diff__swap,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.47 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.31/0.47 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_add_eq_diff_diff_swap
% 0.31/0.47 thf(fact_174_diff__diff__add,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( cancel146912293up_add @ A )
% 0.31/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.31/0.47 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
% 0.31/0.47 = ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % diff_diff_add
% 0.31/0.47 thf(fact_175_add__implies__diff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( cancel1352612707id_add @ A )
% 0.31/0.47 => ! [C2: A,B2: A,A2: A] :
% 0.31/0.47 ( ( ( plus_plus @ A @ C2 @ B2 )
% 0.31/0.47 = A2 )
% 0.31/0.47 => ( C2
% 0.31/0.47 = ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_implies_diff
% 0.31/0.47 thf(fact_176_Units__l__inv__ex,axiom,
% 0.31/0.47 ! [X: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/0.47 => ? [X3: a] :
% 0.31/0.47 ( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 & ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X3 @ X )
% 0.31/0.47 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Units_l_inv_ex
% 0.31/0.47 thf(fact_177_Units__r__inv__ex,axiom,
% 0.31/0.47 ! [X: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/0.47 => ? [X3: a] :
% 0.31/0.47 ( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 & ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ X3 )
% 0.31/0.47 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Units_r_inv_ex
% 0.31/0.47 thf(fact_178_group__l__invI,axiom,
% 0.31/0.47 ( ! [X3: a] :
% 0.31/0.47 ( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ? [Xa: a] :
% 0.31/0.47 ( ( member @ a @ Xa @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 & ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Xa @ X3 )
% 0.31/0.47 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) )
% 0.31/0.47 => ( group @ a @ ( ring_ext @ a @ b ) @ r ) ) ).
% 0.31/0.47
% 0.31/0.47 % group_l_invI
% 0.31/0.47 thf(fact_179_Units__closed,axiom,
% 0.31/0.47 ! [X: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/0.47 => ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Units_closed
% 0.31/0.47 thf(fact_180_Units__inv__comm,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.47 ( ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y )
% 0.31/0.47 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ X )
% 0.31/0.47 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Units_inv_comm
% 0.31/0.47 thf(fact_181_Units__m__closed,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/0.47 => ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Units_m_closed
% 0.31/0.47 thf(fact_182_Units__one__closed,axiom,
% 0.31/0.47 member @ a @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) ).
% 0.31/0.47
% 0.31/0.47 % Units_one_closed
% 0.31/0.47 thf(fact_183_Units__l__cancel,axiom,
% 0.31/0.47 ! [X: a,Y: a,Z2: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y )
% 0.31/0.47 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Z2 ) )
% 0.31/0.47 = ( Y = Z2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Units_l_cancel
% 0.31/0.47 thf(fact_184_groupI,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/0.47 ( ! [X3: A] :
% 0.31/0.47 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ! [Y4: A] :
% 0.31/0.47 ( ( member @ A @ Y4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ( member @ A @ ( mult @ A @ B @ G4 @ X3 @ Y4 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) ) ) )
% 0.31/0.47 => ( ( member @ A @ ( one @ A @ B @ G4 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ( ! [X3: A] :
% 0.31/0.47 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ! [Y4: A] :
% 0.31/0.47 ( ( member @ A @ Y4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ! [Z3: A] :
% 0.31/0.47 ( ( member @ A @ Z3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ( ( mult @ A @ B @ G4 @ ( mult @ A @ B @ G4 @ X3 @ Y4 ) @ Z3 )
% 0.31/0.47 = ( mult @ A @ B @ G4 @ X3 @ ( mult @ A @ B @ G4 @ Y4 @ Z3 ) ) ) ) ) )
% 0.31/0.47 => ( ! [X3: A] :
% 0.31/0.47 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ( ( mult @ A @ B @ G4 @ ( one @ A @ B @ G4 ) @ X3 )
% 0.31/0.47 = X3 ) )
% 0.31/0.47 => ( ! [X3: A] :
% 0.31/0.47 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ? [Xa: A] :
% 0.31/0.47 ( ( member @ A @ Xa @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 & ( ( mult @ A @ B @ G4 @ Xa @ X3 )
% 0.31/0.47 = ( one @ A @ B @ G4 ) ) ) )
% 0.31/0.47 => ( group @ A @ B @ G4 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % groupI
% 0.31/0.47 thf(fact_185_group_Oinv__comm,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A] :
% 0.31/0.47 ( ( group @ A @ B @ G4 )
% 0.31/0.47 => ( ( ( mult @ A @ B @ G4 @ X @ Y )
% 0.31/0.47 = ( one @ A @ B @ G4 ) )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ( ( mult @ A @ B @ G4 @ Y @ X )
% 0.31/0.47 = ( one @ A @ B @ G4 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % group.inv_comm
% 0.31/0.47 thf(fact_186_group_Ol__inv__ex,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
% 0.31/0.47 ( ( group @ A @ B @ G4 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ? [X3: A] :
% 0.31/0.47 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 & ( ( mult @ A @ B @ G4 @ X3 @ X )
% 0.31/0.47 = ( one @ A @ B @ G4 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % group.l_inv_ex
% 0.31/0.47 thf(fact_187_Group_Ogroup_Oright__cancel,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A,Z2: A] :
% 0.31/0.47 ( ( group @ A @ B @ G4 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ( ( ( mult @ A @ B @ G4 @ Y @ X )
% 0.31/0.47 = ( mult @ A @ B @ G4 @ Z2 @ X ) )
% 0.31/0.47 = ( Y = Z2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Group.group.right_cancel
% 0.31/0.47 thf(fact_188_group_OUnits__eq,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/0.47 ( ( group @ A @ B @ G4 )
% 0.31/0.47 => ( ( units @ A @ B @ G4 )
% 0.31/0.47 = ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % group.Units_eq
% 0.31/0.47 thf(fact_189_group_Or__cancel__one_H,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,A2: A] :
% 0.31/0.47 ( ( group @ A @ B @ G4 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ( ( X
% 0.31/0.47 = ( mult @ A @ B @ G4 @ A2 @ X ) )
% 0.31/0.47 = ( A2
% 0.31/0.47 = ( one @ A @ B @ G4 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % group.r_cancel_one'
% 0.31/0.47 thf(fact_190_group_Ol__cancel__one_H,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,A2: A] :
% 0.31/0.47 ( ( group @ A @ B @ G4 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ( ( X
% 0.31/0.47 = ( mult @ A @ B @ G4 @ X @ A2 ) )
% 0.31/0.47 = ( A2
% 0.31/0.47 = ( one @ A @ B @ G4 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % group.l_cancel_one'
% 0.31/0.47 thf(fact_191_group_Or__cancel__one,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,A2: A] :
% 0.31/0.47 ( ( group @ A @ B @ G4 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ( ( ( mult @ A @ B @ G4 @ A2 @ X )
% 0.31/0.47 = X )
% 0.31/0.47 = ( A2
% 0.31/0.47 = ( one @ A @ B @ G4 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % group.r_cancel_one
% 0.31/0.47 thf(fact_192_group_Ol__cancel__one,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,A2: A] :
% 0.31/0.47 ( ( group @ A @ B @ G4 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ( ( ( mult @ A @ B @ G4 @ X @ A2 )
% 0.31/0.47 = X )
% 0.31/0.47 = ( A2
% 0.31/0.47 = ( one @ A @ B @ G4 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % group.l_cancel_one
% 0.31/0.47 thf(fact_193_group_Or__inv__ex,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
% 0.31/0.47 ( ( group @ A @ B @ G4 )
% 0.31/0.47 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 => ? [X3: A] :
% 0.31/0.47 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.47 & ( ( mult @ A @ B @ G4 @ X @ X3 )
% 0.31/0.47 = ( one @ A @ B @ G4 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % group.r_inv_ex
% 0.31/0.47 thf(fact_194_units__group,axiom,
% 0.31/0.47 group @ a @ product_unit @ ( units_of @ a @ ( ring_ext @ a @ b ) @ r ) ).
% 0.31/0.47
% 0.31/0.47 % units_group
% 0.31/0.47 thf(fact_195_inv__add,axiom,
% 0.31/0.47 ! [A2: a,B2: a] :
% 0.31/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( a_inv @ a @ b @ r @ ( add @ a @ b @ r @ A2 @ B2 ) )
% 0.31/0.47 = ( a_minus @ a @ b @ r @ ( a_inv @ a @ b @ r @ A2 ) @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % inv_add
% 0.31/0.47 thf(fact_196_r__neg2,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add @ a @ b @ r @ X @ ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ Y ) )
% 0.31/0.47 = Y ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % r_neg2
% 0.31/0.47 thf(fact_197_r__neg1,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ ( add @ a @ b @ r @ X @ Y ) )
% 0.31/0.47 = Y ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % r_neg1
% 0.31/0.47 thf(fact_198_local_Ominus__add,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( a_inv @ a @ b @ r @ ( add @ a @ b @ r @ X @ Y ) )
% 0.31/0.47 = ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ ( a_inv @ a @ b @ r @ Y ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % local.minus_add
% 0.31/0.47 thf(fact_199_add_Oinv__solve__right_H,axiom,
% 0.31/0.47 ! [A2: a,B2: a,C2: a] :
% 0.31/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( ( add @ a @ b @ r @ B2 @ ( a_inv @ a @ b @ r @ C2 ) )
% 0.31/0.47 = A2 )
% 0.31/0.47 = ( B2
% 0.31/0.47 = ( add @ a @ b @ r @ A2 @ C2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.inv_solve_right'
% 0.31/0.47 thf(fact_200_add_Oinv__solve__right,axiom,
% 0.31/0.47 ! [A2: a,B2: a,C2: a] :
% 0.31/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( A2
% 0.31/0.47 = ( add @ a @ b @ r @ B2 @ ( a_inv @ a @ b @ r @ C2 ) ) )
% 0.31/0.47 = ( B2
% 0.31/0.47 = ( add @ a @ b @ r @ A2 @ C2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.inv_solve_right
% 0.31/0.47 thf(fact_201_add_Oinv__solve__left_H,axiom,
% 0.31/0.47 ! [A2: a,B2: a,C2: a] :
% 0.31/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ B2 ) @ C2 )
% 0.31/0.47 = A2 )
% 0.31/0.47 = ( C2
% 0.31/0.47 = ( add @ a @ b @ r @ B2 @ A2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.inv_solve_left'
% 0.31/0.47 thf(fact_202_add_Oinv__solve__left,axiom,
% 0.31/0.47 ! [A2: a,B2: a,C2: a] :
% 0.31/0.47 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( A2
% 0.31/0.47 = ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ B2 ) @ C2 ) )
% 0.31/0.47 = ( C2
% 0.31/0.47 = ( add @ a @ b @ r @ B2 @ A2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.inv_solve_left
% 0.31/0.47 thf(fact_203_add_Oinv__mult__group,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( a_inv @ a @ b @ r @ ( add @ a @ b @ r @ X @ Y ) )
% 0.31/0.47 = ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ Y ) @ ( a_inv @ a @ b @ r @ X ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.inv_mult_group
% 0.31/0.47 thf(fact_204_a__transpose__inv,axiom,
% 0.31/0.47 ! [X: a,Y: a,Z2: a] :
% 0.31/0.47 ( ( ( add @ a @ b @ r @ X @ Y )
% 0.31/0.47 = Z2 )
% 0.31/0.47 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ Z2 )
% 0.31/0.47 = Y ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % a_transpose_inv
% 0.31/0.47 thf(fact_205_r__minus,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( a_inv @ a @ b @ r @ Y ) )
% 0.31/0.47 = ( a_inv @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % r_minus
% 0.31/0.47 thf(fact_206_l__minus,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( a_inv @ a @ b @ r @ X ) @ Y )
% 0.31/0.47 = ( a_inv @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % l_minus
% 0.31/0.47 thf(fact_207_add_Onat__pow__inv,axiom,
% 0.31/0.47 ! [X: a,I: nat] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add_pow @ a @ b @ nat @ r @ I @ ( a_inv @ a @ b @ r @ X ) )
% 0.31/0.47 = ( a_inv @ a @ b @ r @ ( add_pow @ a @ b @ nat @ r @ I @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.nat_pow_inv
% 0.31/0.47 thf(fact_208_add_Oint__pow__inv,axiom,
% 0.31/0.47 ! [X: a,I: int] :
% 0.31/0.47 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.47 => ( ( add_pow @ a @ b @ int @ r @ I @ ( a_inv @ a @ b @ r @ X ) )
% 0.31/0.47 = ( a_inv @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.int_pow_inv
% 0.31/0.47 thf(fact_209_minus__eq,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.47 ( ( a_minus @ a @ b @ r @ X @ Y )
% 0.31/0.47 = ( add @ a @ b @ r @ X @ ( a_inv @ a @ b @ r @ Y ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minus_eq
% 0.31/0.47 thf(fact_210_sum__zero__eq__neg,axiom,
% 0.31/0.47 ! [X: a,Y: a] :
% 0.31/0.48 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.48 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.48 => ( ( ( add @ a @ b @ r @ X @ Y )
% 0.31/0.48 = ( zero @ a @ b @ r ) )
% 0.31/0.48 => ( X
% 0.31/0.48 = ( a_inv @ a @ b @ r @ Y ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % sum_zero_eq_neg
% 0.31/0.48 thf(fact_211_r__neg,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.48 => ( ( add @ a @ b @ r @ X @ ( a_inv @ a @ b @ r @ X ) )
% 0.31/0.48 = ( zero @ a @ b @ r ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % r_neg
% 0.31/0.48 thf(fact_212_minus__equality,axiom,
% 0.31/0.48 ! [Y: a,X: a] :
% 0.31/0.48 ( ( ( add @ a @ b @ r @ Y @ X )
% 0.31/0.48 = ( zero @ a @ b @ r ) )
% 0.31/0.48 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.48 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.48 => ( ( a_inv @ a @ b @ r @ X )
% 0.31/0.48 = Y ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % minus_equality
% 0.31/0.48 thf(fact_213_l__neg,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.48 => ( ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ X )
% 0.31/0.48 = ( zero @ a @ b @ r ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % l_neg
% 0.31/0.48 thf(fact_214_add_Oint__pow__diff,axiom,
% 0.31/0.48 ! [X: a,N: int,M2: int] :
% 0.31/0.48 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.48 => ( ( add_pow @ a @ b @ int @ r @ ( minus_minus @ int @ N @ M2 ) @ X )
% 0.31/0.48 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ N @ X ) @ ( a_inv @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ M2 @ X ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % add.int_pow_diff
% 0.31/0.48 thf(fact_215_local_Ominus__minus,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.48 => ( ( a_inv @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) )
% 0.31/0.48 = X ) ) ).
% 0.31/0.48
% 0.31/0.48 % local.minus_minus
% 0.31/0.48 thf(fact_216_add_Oinv__closed,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.48 => ( member @ a @ ( a_inv @ a @ b @ r @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % add.inv_closed
% 0.31/0.48 thf(fact_217_local_Ominus__zero,axiom,
% 0.31/0.48 ( ( a_inv @ a @ b @ r @ ( zero @ a @ b @ r ) )
% 0.31/0.48 = ( zero @ a @ b @ r ) ) ).
% 0.31/0.48
% 0.31/0.48 % local.minus_zero
% 0.31/0.48 thf(fact_218_add_Oinv__eq__1__iff,axiom,
% 0.31/0.48 ! [X: a] :
% 0.31/0.48 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.48 => ( ( ( a_inv @ a @ b @ r @ X )
% 0.31/0.48 = ( zero @ a @ b @ r ) )
% 0.31/0.48 = ( X
% 0.31/0.48 = ( zero @ a @ b @ r ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % add.inv_eq_1_iff
% 0.31/0.48 thf(fact_219_Units__minus__one__closed,axiom,
% 0.31/0.48 member @ a @ ( a_inv @ a @ b @ r @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) ).
% 0.31/0.48
% 0.31/0.48 % Units_minus_one_closed
% 0.31/0.48 thf(fact_220_units__of__mult,axiom,
% 0.31/0.48 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/0.48 ( ( mult @ A @ product_unit @ ( units_of @ A @ B @ G4 ) )
% 0.31/0.48 = ( mult @ A @ B @ G4 ) ) ).
% 0.31/0.48
% 0.31/0.48 % units_of_mult
% 0.31/0.48 thf(fact_221_abelian__group_Oa__inv__closed,axiom,
% 0.31/0.48 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.31/0.48 ( ( abelian_group @ A @ B @ G4 )
% 0.31/0.48 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.48 => ( member @ A @ ( a_inv @ A @ B @ G4 @ X ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % abelian_group.a_inv_closed
% 0.31/0.48 thf(fact_222_abelian__group_Ominus__minus,axiom,
% 0.31/0.48 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.31/0.48 ( ( abelian_group @ A @ B @ G4 )
% 0.31/0.48 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.48 => ( ( a_inv @ A @ B @ G4 @ ( a_inv @ A @ B @ G4 @ X ) )
% 0.31/0.48 = X ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % abelian_group.minus_minus
% 0.31/0.48 thf(fact_223_a__minus__def,axiom,
% 0.31/0.48 ! [M: $tType,A: $tType] :
% 0.31/0.48 ( ( a_minus @ A @ M )
% 0.31/0.48 = ( ^ [R4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ M ) ),X4: A,Y5: A] : ( add @ A @ M @ R4 @ X4 @ ( a_inv @ A @ M @ R4 @ Y5 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % a_minus_def
% 0.31/0.48 thf(fact_224_units__of__one,axiom,
% 0.31/0.48 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/0.48 ( ( one @ A @ product_unit @ ( units_of @ A @ B @ G4 ) )
% 0.31/0.48 = ( one @ A @ B @ G4 ) ) ).
% 0.31/0.48
% 0.31/0.48 % units_of_one
% 0.31/0.48 thf(fact_225_abelian__group_Ominus__add,axiom,
% 0.31/0.48 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 0.31/0.48 ( ( abelian_group @ A @ B @ G4 )
% 0.31/0.48 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.48 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.48 => ( ( a_inv @ A @ B @ G4 @ ( add @ A @ B @ G4 @ X @ Y ) )
% 0.31/0.48 = ( add @ A @ B @ G4 @ ( a_inv @ A @ B @ G4 @ X ) @ ( a_inv @ A @ B @ G4 @ Y ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % abelian_group.minus_add
% 0.31/0.48 thf(fact_226_abelian__group_Or__neg2,axiom,
% 0.31/0.48 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 0.31/0.48 ( ( abelian_group @ A @ B @ G4 )
% 0.31/0.48 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.48 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.48 => ( ( add @ A @ B @ G4 @ X @ ( add @ A @ B @ G4 @ ( a_inv @ A @ B @ G4 @ X ) @ Y ) )
% 0.31/0.48 = Y ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % abelian_group.r_neg2
% 0.31/0.48 thf(fact_227_abelian__group_Or__neg1,axiom,
% 0.31/0.48 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 0.31/0.48 ( ( abelian_group @ A @ B @ G4 )
% 0.31/0.48 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.48 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.48 => ( ( add @ A @ B @ G4 @ ( a_inv @ A @ B @ G4 @ X ) @ ( add @ A @ B @ G4 @ X @ Y ) )
% 0.31/0.48 = Y ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % abelian_group.r_neg1
% 0.31/0.48 thf(fact_228_abelian__group_Ominus__eq,axiom,
% 0.31/0.48 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 0.31/0.48 ( ( abelian_group @ A @ B @ G4 )
% 0.31/0.48 => ( ( a_minus @ A @ B @ G4 @ X @ Y )
% 0.31/0.48 = ( add @ A @ B @ G4 @ X @ ( a_inv @ A @ B @ G4 @ Y ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % abelian_group.minus_eq
% 0.31/0.48 thf(fact_229_abelian__group_Ol__neg,axiom,
% 0.31/0.48 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.31/0.48 ( ( abelian_group @ A @ B @ G4 )
% 0.31/0.48 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.48 => ( ( add @ A @ B @ G4 @ ( a_inv @ A @ B @ G4 @ X ) @ X )
% 0.31/0.48 = ( zero @ A @ B @ G4 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % abelian_group.l_neg
% 0.31/0.48 thf(fact_230_abelian__group_Or__neg,axiom,
% 0.31/0.48 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 0.31/0.48 ( ( abelian_group @ A @ B @ G4 )
% 0.31/0.48 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.48 => ( ( add @ A @ B @ G4 @ X @ ( a_inv @ A @ B @ G4 @ X ) )
% 0.31/0.48 = ( zero @ A @ B @ G4 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % abelian_group.r_neg
% 0.31/0.48 thf(fact_231_abelian__group_Ominus__equality,axiom,
% 0.31/0.48 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),Y: A,X: A] :
% 0.31/0.48 ( ( abelian_group @ A @ B @ G4 )
% 0.31/0.48 => ( ( ( add @ A @ B @ G4 @ Y @ X )
% 0.31/0.48 = ( zero @ A @ B @ G4 ) )
% 0.31/0.48 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.48 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.48 => ( ( a_inv @ A @ B @ G4 @ X )
% 0.31/0.48 = Y ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % abelian_group.minus_equality
% 0.31/0.48 thf(fact_232_abelian__group_Oinv__add,axiom,
% 0.31/0.48 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A] :
% 0.31/0.48 ( ( abelian_group @ A @ B @ G4 )
% 0.31/0.48 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.48 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.48 => ( ( a_inv @ A @ B @ G4 @ ( add @ A @ B @ G4 @ A2 @ B2 ) )
% 0.31/0.48 = ( a_minus @ A @ B @ G4 @ ( a_inv @ A @ B @ G4 @ A2 ) @ B2 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % abelian_group.inv_add
% 0.31/0.48 thf(fact_233_abelian__group_Oa__transpose__inv,axiom,
% 0.31/0.48 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 0.31/0.48 ( ( abelian_group @ A @ B @ G4 )
% 0.31/0.48 => ( ( ( add @ A @ B @ G4 @ X @ Y )
% 0.31/0.48 = Z2 )
% 0.31/0.48 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.48 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.48 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 0.31/0.48 => ( ( add @ A @ B @ G4 @ ( a_inv @ A @ B @ G4 @ X ) @ Z2 )
% 0.31/0.48 = Y ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % abelian_group.a_transpose_inv
% 0.31/0.48 thf(fact_234_add_Oint__pow__neg,axiom,
% 0.31/0.48 ! [X: a,I: int] :
% 0.31/0.48 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.48 => ( ( add_pow @ a @ b @ int @ r @ ( uminus_uminus @ int @ I ) @ X )
% 0.31/0.48 = ( a_inv @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % add.int_pow_neg
% 0.31/0.48 thf(fact_235_add__minus__cancel,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( group_add @ A )
% 0.31/0.48 => ! [A2: A,B2: A] :
% 0.31/0.48 ( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) )
% 0.31/0.48 = B2 ) ) ).
% 0.31/0.48
% 0.31/0.48 % add_minus_cancel
% 0.31/0.48 thf(fact_236_minus__add__cancel,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( group_add @ A )
% 0.31/0.48 => ! [A2: A,B2: A] :
% 0.31/0.48 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) )
% 0.31/0.48 = B2 ) ) ).
% 0.31/0.48
% 0.31/0.48 % minus_add_cancel
% 0.31/0.48 thf(fact_237_minus__add__distrib,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ab_group_add @ A )
% 0.31/0.48 => ! [A2: A,B2: A] :
% 0.31/0.48 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
% 0.31/0.48 = ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % minus_add_distrib
% 0.31/0.48 thf(fact_238_minus__diff__eq,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( group_add @ A )
% 0.31/0.48 => ! [A2: A,B2: A] :
% 0.31/0.48 ( ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B2 ) )
% 0.31/0.48 = ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % minus_diff_eq
% 0.31/0.48 thf(fact_239_diff__minus__eq__add,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( group_add @ A )
% 0.31/0.48 => ! [A2: A,B2: A] :
% 0.31/0.48 ( ( minus_minus @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
% 0.31/0.48 = ( plus_plus @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % diff_minus_eq_add
% 0.31/0.48 thf(fact_240_uminus__add__conv__diff,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ab_group_add @ A )
% 0.31/0.48 => ! [A2: A,B2: A] :
% 0.31/0.48 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
% 0.31/0.48 = ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % uminus_add_conv_diff
% 0.31/0.48 thf(fact_241_group__cancel_Osub2,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ab_group_add @ A )
% 0.31/0.48 => ! [B6: A,K: A,B2: A,A2: A] :
% 0.31/0.48 ( ( B6
% 0.31/0.48 = ( plus_plus @ A @ K @ B2 ) )
% 0.31/0.48 => ( ( minus_minus @ A @ A2 @ B6 )
% 0.31/0.48 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % group_cancel.sub2
% 0.31/0.48 thf(fact_242_diff__conv__add__uminus,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( group_add @ A )
% 0.31/0.48 => ( ( minus_minus @ A )
% 0.31/0.48 = ( ^ [A6: A,B5: A] : ( plus_plus @ A @ A6 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % diff_conv_add_uminus
% 0.31/0.48 thf(fact_243_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ab_group_add @ A )
% 0.31/0.48 => ( ( minus_minus @ A )
% 0.31/0.48 = ( ^ [A6: A,B5: A] : ( plus_plus @ A @ A6 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.31/0.48 thf(fact_244_group__cancel_Oneg1,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ab_group_add @ A )
% 0.31/0.48 => ! [A4: A,K: A,A2: A] :
% 0.31/0.48 ( ( A4
% 0.31/0.48 = ( plus_plus @ A @ K @ A2 ) )
% 0.31/0.48 => ( ( uminus_uminus @ A @ A4 )
% 0.31/0.48 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % group_cancel.neg1
% 0.31/0.48 thf(fact_245_add_Oinverse__distrib__swap,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( group_add @ A )
% 0.31/0.48 => ! [A2: A,B2: A] :
% 0.31/0.48 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
% 0.31/0.48 = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % add.inverse_distrib_swap
% 0.31/0.48 thf(fact_246_minus__diff__commute,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( ab_group_add @ A )
% 0.31/0.48 => ! [B2: A,A2: A] :
% 0.31/0.48 ( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A2 )
% 0.31/0.48 = ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % minus_diff_commute
% 0.31/0.48 thf(fact_247_add_Oint__pow__neg__int,axiom,
% 0.31/0.48 ! [X: a,N: nat] :
% 0.31/0.48 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 0.31/0.48 => ( ( add_pow @ a @ b @ int @ r @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) @ X )
% 0.31/0.48 = ( a_inv @ a @ b @ r @ ( add_pow @ a @ b @ nat @ r @ N @ X ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % add.int_pow_neg_int
% 0.31/0.48 thf(fact_248_units__comm__group,axiom,
% 0.31/0.48 comm_group @ a @ product_unit @ ( units_of @ a @ ( ring_ext @ a @ b ) @ r ) ).
% 0.31/0.48
% 0.31/0.48 % units_comm_group
% 0.31/0.48 thf(fact_249_of__nat__eq__iff,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( semiring_char_0 @ A )
% 0.31/0.48 => ! [M2: nat,N: nat] :
% 0.31/0.48 ( ( ( semiring_1_of_nat @ A @ M2 )
% 0.31/0.48 = ( semiring_1_of_nat @ A @ N ) )
% 0.31/0.48 = ( M2 = N ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % of_nat_eq_iff
% 0.31/0.48 thf(fact_250_of__nat__add,axiom,
% 0.31/0.48 ! [A: $tType] :
% 0.31/0.48 ( ( semiring_1 @ A )
% 0.31/0.48 => ! [M2: nat,N: nat] :
% 0.31/0.48 ( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M2 @ N ) )
% 0.31/0.48 = ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % of_nat_add
% 0.31/0.48 thf(fact_251_comm__groupE_I4_J,axiom,
% 0.31/0.48 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A] :
% 0.31/0.48 ( ( comm_group @ A @ B @ G4 )
% 0.31/0.48 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.48 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.48 => ( ( mult @ A @ B @ G4 @ X @ Y )
% 0.31/0.48 = ( mult @ A @ B @ G4 @ Y @ X ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % comm_groupE(4)
% 0.31/0.48 thf(fact_252_comm__groupE_I3_J,axiom,
% 0.31/0.48 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A,Z2: A] :
% 0.31/0.48 ( ( comm_group @ A @ B @ G4 )
% 0.31/0.48 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.48 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.48 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.48 => ( ( mult @ A @ B @ G4 @ ( mult @ A @ B @ G4 @ X @ Y ) @ Z2 )
% 0.31/0.48 = ( mult @ A @ B @ G4 @ X @ ( mult @ A @ B @ G4 @ Y @ Z2 ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % comm_groupE(3)
% 0.31/0.48 thf(fact_253_comm__groupE_I1_J,axiom,
% 0.31/0.48 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A] :
% 0.31/0.48 ( ( comm_group @ A @ B @ G4 )
% 0.31/0.48 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.48 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 0.31/0.48 => ( member @ A @ ( mult @ A @ B @ G4 @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % comm_groupE(1)
% 0.31/0.48 thf(fact_254_comm__groupE_I2_J,axiom,
% 0.31/0.48 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 0.31/0.48 ( ( comm_group @ A @ B @ G4 )
% 0.31/0.48 => ( member @ A @ ( one @ A @ B @ G4 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 % comm_groupE(2)
% 0.31/0.48
% 0.31/0.48 % Type constructors (20)
% 0.31/0.48 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
% 0.31/0.48 ordere779506340up_add @ int ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.31/0.48 cancel146912293up_add @ int ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 0.31/0.48 cancel1352612707id_add @ int ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
% 0.31/0.48 cancel_semigroup_add @ int ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
% 0.31/0.48 ab_semigroup_add @ int ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
% 0.31/0.48 comm_monoid_add @ int ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
% 0.31/0.48 semigroup_add @ int ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 0.31/0.48 semiring_char_0 @ int ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
% 0.31/0.48 ab_group_add @ int ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 0.31/0.48 semiring_1 @ int ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 0.31/0.48 group_add @ int ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_1,axiom,
% 0.31/0.48 ordere779506340up_add @ nat ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_2,axiom,
% 0.31/0.48 cancel146912293up_add @ nat ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_3,axiom,
% 0.31/0.48 cancel1352612707id_add @ nat ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_4,axiom,
% 0.31/0.48 cancel_semigroup_add @ nat ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_5,axiom,
% 0.31/0.48 ab_semigroup_add @ nat ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_6,axiom,
% 0.31/0.48 comm_monoid_add @ nat ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Nat_Onat___Groups_Osemigroup__add_7,axiom,
% 0.31/0.48 semigroup_add @ nat ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_8,axiom,
% 0.31/0.48 semiring_char_0 @ nat ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Nat_Onat___Rings_Osemiring__1_9,axiom,
% 0.31/0.48 semiring_1 @ nat ).
% 0.31/0.48
% 0.31/0.48 % Conjectures (1)
% 0.31/0.48 thf(conj_0,conjecture,
% 0.31/0.48 member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r4 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r4 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ).
% 0.31/0.48
% 0.31/0.48 %------------------------------------------------------------------------------
% 0.31/0.48 ------------------------
% 2.90/1.55 ---- Embedded in THF ---
% 2.90/1.55 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.90/1.55 %%% Generated on Tue Feb 24 08:25:57 EST 2026
% 2.90/1.55 %%% using '$$dhol' embedding, version 1.3.0.
% 2.90/1.55 %%% Logic specification used:
% 2.90/1.55 %%% thf(spec, logic, $$dhol).
% 2.90/1.55
% 2.90/1.55 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.Fb6urRC3jx/DTF2DTF19374.p
% 2.90/1.55 thf(ty_t_Congruence_Opartial__object_Opartial__object__ext, type, partia1265057091ct_ext: ($tType > ($tType > $tType))).
% 2.90/1.55 thf(ty_t_Congruence_Oeq__object_Oeq__object__ext, type, eq_eq_object_ext: ($tType > ($tType > $tType))).
% 2.90/1.55 thf(ty_t_Group_Omonoid_Omonoid__ext, type, monoid_ext: ($tType > ($tType > $tType))).
% 2.90/1.55 thf(ty_t_Ring_Oring_Oring__ext, type, ring_ext: ($tType > ($tType > $tType))).
% 2.90/1.55 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 2.90/1.55 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.90/1.55 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.90/1.55 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.90/1.55 thf(ty_t_Int_Oint, type, int: $tType).
% 2.90/1.55 thf(ty_tf_b, type, b: $tType).
% 2.90/1.55 thf(ty_tf_a, type, a: $tType).
% 2.90/1.55 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.90/1.55 thf(sy_cl_Rings_Osemiring__1, type, semiring_1: (!> [A:$tType]: ($o))).
% 2.90/1.55 thf(sy_cl_Groups_Oab__group__add, type, ab_group_add: (!> [A:$tType]: ($o))).
% 2.90/1.55 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: ($o))).
% 2.90/1.55 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: ($o))).
% 2.90/1.55 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: ($o))).
% 2.90/1.55 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: ($o))).
% 2.90/1.55 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: ($o))).
% 2.90/1.55 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: ($o))).
% 2.90/1.55 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: ($o))).
% 2.90/1.55 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: ($o))).
% 2.90/1.55 thf(sy_c_Congruence_Opartial__object_Ocarrier, type, partial_carrier: (!> [A:$tType,Z:$tType]: ((((partia1265057091ct_ext @ A) @ Z) > (set @ A))))).
% 2.90/1.55 thf(sy_c_Group_OUnits, type, units: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)) > (set @ A))))).
% 2.90/1.55 thf(sy_c_Group_Ocomm__group, type, comm_group: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)) > $o)))).
% 2.90/1.55 thf(sy_c_Group_Ogroup, type, group: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)) > $o)))).
% 2.90/1.55 thf(sy_c_Group_Omonoid_Omult, type, mult: (!> [A:$tType,Z:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ Z)) > (A > (A > A)))))).
% 2.90/1.55 thf(sy_c_Group_Omonoid_Oone, type, one: (!> [A:$tType,Z:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ Z)) > A)))).
% 2.90/1.55 thf(sy_c_Group_Ounits__of, type, units_of: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)) > ((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ product_unit)))))).
% 2.90/1.55 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.90/1.55 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 2.90/1.55 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 2.90/1.55 thf(sy_c_Nat_OSuc, type, suc: (nat > nat)).
% 2.90/1.55 thf(sy_c_Nat_Osemiring__1__class_Oof__nat, type, semiring_1_of_nat: (!> [A:$tType]: ((nat > A)))).
% 2.90/1.55 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.90/1.55 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.90/1.55 thf(sy_c_Ring_Oa__inv, type, a_inv: (!> [A:$tType,M:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ M))) > (A > A))))).
% 2.90/1.55 thf(sy_c_Ring_Oa__minus, type, a_minus: (!> [A:$tType,M:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ M))) > (A > (A > A)))))).
% 2.90/1.55 thf(sy_c_Ring_Oabelian__group, type, abelian_group: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > $o)))).
% 2.90/1.55 thf(sy_c_Ring_Oadd__pow, type, add_pow: (!> [A:$tType,C:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ C))) > (B > (A > A)))))).
% 2.90/1.55 thf(sy_c_Ring_Ofield, type, field: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > $o)))).
% 2.90/1.55 thf(sy_c_Ring_Oring_Oadd, type, add: (!> [A:$tType,Z:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ Z))) > (A > (A > A)))))).
% 2.90/1.55 thf(sy_c_Ring_Oring_Omore, type, more: (!> [A:$tType,Z:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ Z))) > Z)))).
% 2.90/1.55 thf(sy_c_Ring_Oring_Ozero, type, zero: (!> [A:$tType,Z:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ Z))) > A)))).
% 2.90/1.55 thf(sy_c_Ring_Osemiring, type, semiring: (!> [A:$tType,B:$tType]: ((((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) > $o)))).
% 2.90/1.55 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.90/1.55 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.90/1.55 thf(sy_v_R, type, r: ((partia1265057091ct_ext @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b)))).
% 2.90/1.55 thf(sy_v_r, type, r2: a).
% 2.90/1.55 thf(sy_v_r_H, type, r3: a).
% 2.90/1.55 thf(sy_v_r_H_H, type, r4: a).
% 2.90/1.55 thf(sy_v_rel, type, rel: ((partia1265057091ct_ext @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit))).
% 2.90/1.55 thf(sy_v_s, type, s: a).
% 2.90/1.55 thf(sy_v_s_H, type, s2: a).
% 2.90/1.55 thf(sy_v_s_H_H, type, s3: a).
% 2.90/1.55 thf(fact_0_right__add__eq, axiom, (! [A2:a,B2:a,C2:a]: (((A2 = B2) => ((((((add @ a) @ b) @ r) @ C2) @ A2) = (((((add @ a) @ b) @ r) @ C2) @ B2)))))).
% 2.90/1.55 thf(fact_1_assms_I3_J, axiom, (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ r4) @ s3)) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel))).
% 2.90/1.55 thf(fact_2_assms_I2_J, axiom, (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ r3) @ s2)) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel))).
% 2.90/1.55 thf(fact_3_assms_I1_J, axiom, (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ r2) @ s)) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel))).
% 2.90/1.55 thf(fact_4_closed__rel__add, 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)) => (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ S2) @ R)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ S) @ R2))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ S) @ S2))) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel))))))).
% 2.90/1.55 thf(fact_5__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_061_As_A_092_060otimes_062_A_Is_H_H_A_092_060otimes_062_As_H_J_A_092_060otimes_062_As_H_H_092_060close_062, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ s2))) @ s3))).
% 2.90/1.55 thf(fact_6_f9, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ s3)))).
% 2.90/1.55 thf(fact_7__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_A_061_As_H_H_A_092_060otimes_062_As_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_092_060close_062, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r3) @ r4)) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s3) @ s)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r3) @ r4)))).
% 2.90/1.55 thf(fact_8__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_092_060close_062, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ r3)) @ r4)) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s2)) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r3) @ r4))))).
% 2.90/1.55 thf(fact_9_f10, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r2)) @ r4)) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s2)) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r2) @ r4))))).
% 2.90/1.55 thf(fact_10_f5, axiom, (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r3) @ r4)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3))) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel))).
% 2.90/1.55 thf(fact_11_f4, axiom, (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r2) @ r4)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3))) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel))).
% 2.90/1.55 thf(fact_12__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_A_092_060otimes_062_Ar_H_H_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_A_092_060otimes_062_Ar_H_H_J_092_060close_062, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ r3))) @ r4)) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3))) @ (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r2)) @ r4)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ r3)) @ r4))))).
% 2.90/1.55 thf(fact_13__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_A_092_060otimes_062_Ar_H_H_J_A_092_060oplus_062_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_092_060close_062, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ r3))) @ r4)) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s2)) @ s3)) @ (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r2) @ r4))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r3) @ r4)))))).
% 2.90/1.55 thf(fact_14_f7, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ r3))) @ r4)) = (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r2)) @ r4))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ r3)) @ r4))))).
% 2.90/1.55 thf(fact_15_f8, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s2)) @ s3)) @ (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r2) @ r4))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r3) @ r4)))) = (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s2)) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r2) @ r4)))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s2)) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r3) @ r4)))))).
% 2.90/1.55 thf(fact_16_f12, axiom, (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ r3))) @ r4)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s2)) @ s3))) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel))).
% 2.90/1.55 thf(fact_17_l__distr, axiom, (! [X:a,Y:a,Z2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((add @ a) @ b) @ r) @ X) @ Y)) @ Z2) = (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Z2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Y) @ Z2))))))))).
% 2.90/1.55 thf(fact_18_r__distr, axiom, (! [X:a,Y:a,Z2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Z2) @ (((((add @ a) @ b) @ r) @ X) @ Y)) = (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Z2) @ X)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Z2) @ Y))))))))).
% 2.90/1.55 thf(fact_19__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_A_092_060otimes_062_Ar_H_H_J_A_092_060oplus_062_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_A_061_A_092_060zero_062_092_060close_062, axiom, ((((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ r3))) @ r4))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s2)) @ s3)) @ (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r2) @ r4))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r3) @ r4))))) = (((zero @ a) @ b) @ r))).
% 2.90/1.55 thf(fact_20_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X2:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X2) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X2 = Y2)))))).
% 2.90/1.55 thf(fact_21_old_Oprod_Oinject, axiom, (! [A:$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)))))).
% 2.90/1.55 thf(fact_22_local_Osemiring__axioms, axiom, (((semiring @ a) @ b) @ r)).
% 2.90/1.55 thf(fact_23_m__assoc, axiom, (! [X:a,Y:a,Z2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Y)) @ Z2) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Y) @ Z2))))))))).
% 2.90/1.55 thf(fact_24_m__comm, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Y) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Y) @ X))))))).
% 2.90/1.55 thf(fact_25_m__lcomm, axiom, (! [X:a,Y:a,Z2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Y) @ Z2)) = (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ Y) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Z2))))))))).
% 2.90/1.55 thf(fact_26_add_Om__assoc, axiom, (! [X:a,Y:a,Z2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ (((((add @ a) @ b) @ r) @ X) @ Y)) @ Z2) = (((((add @ a) @ b) @ r) @ X) @ (((((add @ a) @ b) @ r) @ Y) @ Z2))))))))).
% 2.90/1.55 thf(fact_27_add_Om__comm, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ X) @ Y) = (((((add @ a) @ b) @ r) @ Y) @ X))))))).
% 2.90/1.55 thf(fact_28_add_Om__lcomm, axiom, (! [X:a,Y:a,Z2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ X) @ (((((add @ a) @ b) @ r) @ Y) @ Z2)) = (((((add @ a) @ b) @ r) @ Y) @ (((((add @ a) @ b) @ r) @ X) @ Z2))))))))).
% 2.90/1.55 thf(fact_29_local_Oright__minus__eq, axiom, (! [A2:a,B2:a,C2:a]: (((A2 = B2) => ((((((a_minus @ a) @ b) @ r) @ C2) @ A2) = (((((a_minus @ a) @ b) @ r) @ C2) @ B2)))))).
% 2.90/1.55 thf(fact_30_local_Ominus__unique, axiom, (! [Y:a,X:a,Y3:a]: ((((((((add @ a) @ b) @ r) @ Y) @ X) = (((zero @ a) @ b) @ r)) => (((((((add @ a) @ b) @ r) @ X) @ Y3) = (((zero @ a) @ b) @ r)) => ((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y3) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (Y = Y3))))))))).
% 2.90/1.55 thf(fact_31_add_Or__inv__ex, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (? [X3:a]: (((((member @ a) @ X3) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) & ((((((add @ a) @ b) @ r) @ X) @ X3) = (((zero @ a) @ b) @ r))))))))).
% 2.90/1.55 thf(fact_32_add_Oone__unique, axiom, (! [U:a]: (((((member @ a) @ U) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((! [X3:a]: (((((member @ a) @ X3) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ U) @ X3) = X3)))) => (U = (((zero @ a) @ b) @ r))))))).
% 2.90/1.55 thf(fact_33_add_Ol__inv__ex, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (? [X3:a]: (((((member @ a) @ X3) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) & ((((((add @ a) @ b) @ r) @ X3) @ X) = (((zero @ a) @ b) @ r))))))))).
% 2.90/1.55 thf(fact_34_add_Oinv__comm, axiom, (! [X:a,Y:a]: ((((((((add @ a) @ b) @ r) @ X) @ Y) = (((zero @ a) @ b) @ r)) => ((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ Y) @ X) = (((zero @ a) @ b) @ r)))))))).
% 2.90/1.55 thf(fact_35_right__inv__add, axiom, (! [A2:a,B2:a,C2:a]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ C2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((a_minus @ a) @ b) @ r) @ (((((a_minus @ a) @ b) @ r) @ C2) @ A2)) @ B2) = (((((a_minus @ a) @ b) @ r) @ C2) @ (((((add @ a) @ b) @ r) @ A2) @ B2))))))))).
% 2.90/1.55 thf(fact_36_four__elem__comm, axiom, (! [A2:a,B2:a,C2:a,D:a]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ C2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ D) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((a_minus @ a) @ b) @ r) @ (((((add @ a) @ b) @ r) @ (((((a_minus @ a) @ b) @ r) @ A2) @ C2)) @ B2)) @ D) = (((((a_minus @ a) @ b) @ r) @ (((((a_minus @ a) @ b) @ r) @ (((((add @ a) @ b) @ r) @ A2) @ B2)) @ C2)) @ D))))))))).
% 2.90/1.55 thf(fact_37_local_Oadd_Oright__cancel, axiom, (! [X:a,Y:a,Z2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add @ a) @ b) @ r) @ Y) @ X) = (((((add @ a) @ b) @ r) @ Z2) @ X)) = (Y = Z2)))))))).
% 2.90/1.55 thf(fact_38_add_Om__closed, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((member @ a) @ (((((add @ a) @ b) @ r) @ X) @ Y)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))))))).
% 2.90/1.55 thf(fact_39_zero__closed, axiom, (((member @ a) @ (((zero @ a) @ b) @ r)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))).
% 2.90/1.55 thf(fact_40_semiring__simprules_I3_J, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((member @ a) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Y)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))))))).
% 2.90/1.55 thf(fact_41_minus__closed, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((member @ a) @ (((((a_minus @ a) @ b) @ r) @ X) @ Y)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r))))))).
% 2.90/1.55 thf(fact_42_r__zero, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ X) @ (((zero @ a) @ b) @ r)) = X))))).
% 2.90/1.55 thf(fact_43_l__zero, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ (((zero @ a) @ b) @ r)) @ X) = X))))).
% 2.90/1.55 thf(fact_44_add_Or__cancel__one_H, axiom, (! [X:a,A2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((X = (((((add @ a) @ b) @ r) @ A2) @ X)) = (A2 = (((zero @ a) @ b) @ r)))))))).
% 2.90/1.55 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.90/1.55 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A4)))) = A4)))).
% 2.90/1.55 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.90/1.55 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.90/1.55 thf(fact_49_add_Or__cancel__one, axiom, (! [X:a,A2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add @ a) @ b) @ r) @ A2) @ X) = X) = (A2 = (((zero @ a) @ b) @ r)))))))).
% 2.90/1.55 thf(fact_50_add_Ol__cancel__one_H, axiom, (! [X:a,A2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((X = (((((add @ a) @ b) @ r) @ X) @ A2)) = (A2 = (((zero @ a) @ b) @ r)))))))).
% 2.90/1.55 thf(fact_51_add_Ol__cancel__one, axiom, (! [X:a,A2:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add @ a) @ b) @ r) @ X) @ A2) = X) = (A2 = (((zero @ a) @ b) @ r)))))))).
% 2.90/1.55 thf(fact_52_r__null, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ (((zero @ a) @ b) @ r)) = (((zero @ a) @ b) @ r)))))).
% 2.90/1.55 thf(fact_53_l__null, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((zero @ a) @ b) @ r)) @ X) = (((zero @ a) @ b) @ r)))))).
% 2.90/1.55 thf(fact_54_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A5:A,B4:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B4)))) => (P @ Prod))))).
% 2.90/1.55 thf(fact_55_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A5:A,B4:B]: ((Y != ((((product_Pair @ A) @ B) @ A5) @ B4))))))))).
% 2.90/1.55 thf(fact_56_prod__induct7, axiom, (! [G2:$tType,F2:$tType,E:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))))))]: (((! [A5:A,B4:B,C3:C,D3:D2,E2:E,F3:F2,G3:G2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))) @ C3) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))) @ D3) @ ((((product_Pair @ E) @ ((product_prod @ F2) @ G2)) @ E2) @ ((((product_Pair @ F2) @ G2) @ F3) @ G3))))))))) => (P @ X))))).
% 2.90/1.55 thf(fact_57_prod__induct6, axiom, (! [F2:$tType,E:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2)))))]: (((! [A5:A,B4:B,C3:C,D3:D2,E2:E,F3:F2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2))) @ C3) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ F2)) @ D3) @ ((((product_Pair @ E) @ F2) @ E2) @ F3)))))))) => (P @ X))))).
% 2.90/1.55 thf(fact_58_prod__induct5, axiom, (! [E:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))))]: (((! [A5:A,B4:B,C3:C,D3:D2,E2:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ E)) @ C3) @ ((((product_Pair @ D2) @ E) @ D3) @ E2))))))) => (P @ X))))).
% 2.90/1.55 thf(fact_59_prod__induct4, axiom, (! [D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2)))]: (((! [A5:A,B4:B,C3:C,D3:D2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D2)) @ B4) @ ((((product_Pair @ C) @ D2) @ C3) @ D3)))))) => (P @ X))))).
% 2.90/1.55 thf(fact_60_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A5:A,B4:B,C3:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B4) @ C3))))) => (P @ X))))).
% 2.90/1.55 thf(fact_61_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E:$tType,F2:$tType,G2:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))))))]: ((~ ((! [A5:A,B4:B,C3:C,D3:D2,E2:E,F3:F2,G3:G2]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))) @ C3) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))) @ D3) @ ((((product_Pair @ E) @ ((product_prod @ F2) @ G2)) @ E2) @ ((((product_Pair @ F2) @ G2) @ F3) @ G3)))))))))))))).
% 2.90/1.55 thf(fact_62_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E:$tType,F2:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2)))))]: ((~ ((! [A5:A,B4:B,C3:C,D3:D2,E2:E,F3:F2]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2))) @ C3) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ F2)) @ D3) @ ((((product_Pair @ E) @ F2) @ E2) @ F3))))))))))))).
% 2.90/1.55 thf(fact_63_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))))]: ((~ ((! [A5:A,B4:B,C3:C,D3:D2,E2:E]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ E)) @ C3) @ ((((product_Pair @ D2) @ E) @ D3) @ E2)))))))))))).
% 2.90/1.55 thf(fact_64_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2)))]: ((~ ((! [A5:A,B4:B,C3:C,D3:D2]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D2)) @ B4) @ ((((product_Pair @ C) @ D2) @ C3) @ D3))))))))))).
% 2.90/1.55 thf(fact_65_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A5:A,B4:B,C3:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B4) @ C3)))))))))).
% 2.90/1.55 thf(fact_66_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)))))))).
% 2.90/1.55 thf(fact_67_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A5:A,B4:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B4)))) => (P @ P2))))).
% 2.90/1.55 thf(fact_68_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X3:A,Y4:B]: ((P2 = ((((product_Pair @ A) @ B) @ X3) @ Y4))))))).
% 2.90/1.55 thf(fact_69_f11, axiom, ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((one @ a) @ ((ring_ext @ a) @ b)) @ r)) @ (((((a_minus @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ r2)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ r3))) @ r4))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s2)) @ s3)) @ (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r2) @ r4))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r3) @ r4)))))) = (((zero @ a) @ b) @ r))).
% 2.90/1.55 thf(fact_70_semiring_Osemiring__simprules_I11_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((semiring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((add @ A) @ B) @ R3) @ X) @ (((zero @ A) @ B) @ R3)) = X)))))).
% 2.90/1.55 thf(fact_71_semiring_Osemiring__simprules_I6_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((semiring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((add @ A) @ B) @ R3) @ (((zero @ A) @ B) @ R3)) @ X) = X)))))).
% 2.90/1.55 thf(fact_72_semiring_Ol__null, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((semiring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ (((zero @ A) @ B) @ R3)) @ X) = (((zero @ A) @ B) @ R3))))))).
% 2.90/1.55 thf(fact_73_semiring_Or__null, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((semiring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ X) @ (((zero @ A) @ B) @ R3)) = (((zero @ A) @ B) @ R3))))))).
% 2.90/1.55 thf(fact_74_semiring_Or__distr, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A,Z2:A]: (((((semiring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ Z2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ Z2) @ (((((add @ A) @ B) @ R3) @ X) @ Y)) = (((((add @ A) @ B) @ R3) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ Z2) @ X)) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ Z2) @ Y)))))))))).
% 2.90/1.55 thf(fact_75_semiring_Ol__distr, axiom, (! [B:$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) @ (((((add @ A) @ B) @ R3) @ X) @ Y)) @ Z2) = (((((add @ A) @ B) @ R3) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ X) @ Z2)) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ Y) @ Z2)))))))))).
% 2.90/1.55 thf(fact_76_add__pow__rdistr__int, axiom, (! [A2:a,B2:a,K:int]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ A2) @ ((((((add_pow @ a) @ b) @ int) @ r) @ K) @ B2)) = ((((((add_pow @ a) @ b) @ int) @ r) @ K) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ A2) @ B2)))))))).
% 2.90/1.55 thf(fact_77_add__pow__ldistr__int, axiom, (! [A2:a,B2:a,K:int]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ ((((((add_pow @ a) @ b) @ int) @ r) @ K) @ A2)) @ B2) = ((((((add_pow @ a) @ b) @ int) @ r) @ K) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ A2) @ B2)))))))).
% 2.90/1.55 thf(fact_78_inv__unique, axiom, (! [Y:a,X:a,Y3: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) @ Y3) = (((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) @ Y3) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (Y = Y3))))))))).
% 2.90/1.55 thf(fact_79_one__unique, axiom, (! [U:a]: (((((member @ a) @ U) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((! [X3:a]: (((((member @ a) @ X3) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ U) @ X3) = X3)))) => (U = (((one @ a) @ ((ring_ext @ a) @ b)) @ r))))))).
% 2.90/1.55 thf(fact_80_add_Oint__pow__distrib, axiom, (! [X:a,Y:a,I:int]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add_pow @ a) @ b) @ int) @ r) @ I) @ (((((add @ a) @ b) @ r) @ X) @ Y)) = (((((add @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ int) @ r) @ I) @ X)) @ ((((((add_pow @ a) @ b) @ int) @ r) @ I) @ Y)))))))).
% 2.90/1.55 thf(fact_81_add_Oint__pow__mult__distrib, axiom, (! [X:a,Y:a,I:int]: ((((((((add @ a) @ b) @ r) @ X) @ Y) = (((((add @ a) @ b) @ r) @ Y) @ X)) => ((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add_pow @ a) @ b) @ int) @ r) @ I) @ (((((add @ a) @ b) @ r) @ X) @ Y)) = (((((add @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ int) @ r) @ I) @ X)) @ ((((((add_pow @ a) @ b) @ int) @ r) @ I) @ Y))))))))).
% 2.90/1.55 thf(fact_82_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))).
% 2.90/1.55 thf(fact_83_add_Oint__pow__closed, axiom, (! [X:a,I:int]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((member @ a) @ ((((((add_pow @ a) @ b) @ int) @ r) @ I) @ X)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)))))).
% 2.90/1.55 thf(fact_84_add_Oint__pow__one, axiom, (! [Z2:int]: ((((((((add_pow @ a) @ b) @ int) @ r) @ Z2) @ (((zero @ a) @ b) @ r)) = (((zero @ a) @ b) @ r))))).
% 2.90/1.55 thf(fact_85_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))))).
% 2.90/1.55 thf(fact_86_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))))).
% 2.90/1.55 thf(fact_87_semiring_Osemiring__simprules_I4_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((semiring @ A) @ B) @ R3) => (((member @ A) @ (((one @ A) @ ((ring_ext @ A) @ B)) @ R3)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)))))).
% 2.90/1.55 thf(fact_88_semiring_Osemiring__simprules_I9_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((semiring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ (((one @ A) @ ((ring_ext @ A) @ B)) @ R3)) @ X) = X)))))).
% 2.90/1.55 thf(fact_89_semiring_Osemiring__simprules_I3_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A]: (((((semiring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => (((member @ A) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ X) @ Y)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)))))))).
% 2.90/1.55 thf(fact_90_semiring_Osemiring__simprules_I8_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A,Z2:A]: (((((semiring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ Z2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ X) @ Y)) @ Z2) = (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ X) @ (((((mult @ A) @ ((ring_ext @ A) @ B)) @ R3) @ Y) @ Z2)))))))))).
% 2.90/1.55 thf(fact_91_semiring_Osemiring__simprules_I1_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A]: (((((semiring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => (((member @ A) @ (((((add @ A) @ B) @ R3) @ X) @ Y)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)))))))).
% 2.90/1.55 thf(fact_92_semiring_Osemiring__simprules_I5_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A,Z2:A]: (((((semiring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ Z2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((add @ A) @ B) @ R3) @ (((((add @ A) @ B) @ R3) @ X) @ Y)) @ Z2) = (((((add @ A) @ B) @ R3) @ X) @ (((((add @ A) @ B) @ R3) @ Y) @ Z2)))))))))).
% 2.90/1.55 thf(fact_93_semiring_Osemiring__simprules_I7_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A]: (((((semiring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((add @ A) @ B) @ R3) @ X) @ Y) = (((((add @ A) @ B) @ R3) @ Y) @ X)))))))).
% 2.90/1.55 thf(fact_94_semiring_Osemiring__simprules_I12_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A,Z2:A]: (((((semiring @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((member @ A) @ Z2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((add @ A) @ B) @ R3) @ X) @ (((((add @ A) @ B) @ R3) @ Y) @ Z2)) = (((((add @ A) @ B) @ R3) @ Y) @ (((((add @ A) @ B) @ R3) @ X) @ Z2)))))))))).
% 2.90/1.55 thf(fact_95_semiring_Osemiring__simprules_I2_J, axiom, (! [B:$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)))))).
% 2.90/1.55 thf(fact_96_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)) => (? [X5:a]: (((((member @ a) @ X5) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) & ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ A5) @ X5) = (((one @ a) @ ((ring_ext @ a) @ b)) @ r))))))))) => (((field @ a) @ b) @ r)))).
% 2.90/1.55 thf(fact_97_add_Oint__pow__mult, axiom, (! [X:a,I:int,J:int]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add_pow @ a) @ b) @ int) @ r) @ (((plus_plus @ int) @ I) @ J)) @ X) = (((((add @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ int) @ r) @ I) @ X)) @ ((((((add_pow @ a) @ b) @ int) @ r) @ J) @ X))))))).
% 2.90/1.55 thf(fact_98_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)))))))).
% 2.90/1.55 thf(fact_99_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)))))))).
% 2.90/1.55 thf(fact_100_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))))).
% 2.90/1.55 thf(fact_101_add_Ogroup__commutes__pow, axiom, (! [X:a,Y:a,N:nat]: ((((((((add @ a) @ b) @ r) @ X) @ Y) = (((((add @ a) @ b) @ r) @ Y) @ X)) => ((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ nat) @ r) @ N) @ X)) @ Y) = (((((add @ a) @ b) @ r) @ Y) @ ((((((add_pow @ a) @ b) @ nat) @ r) @ N) @ X))))))))).
% 2.90/1.55 thf(fact_102_add_Opow__mult__distrib, axiom, (! [X:a,Y:a,N:nat]: ((((((((add @ a) @ b) @ r) @ X) @ Y) = (((((add @ a) @ b) @ r) @ Y) @ X)) => ((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add_pow @ a) @ b) @ nat) @ r) @ N) @ (((((add @ a) @ b) @ r) @ X) @ Y)) = (((((add @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ nat) @ r) @ N) @ X)) @ ((((((add_pow @ a) @ b) @ nat) @ r) @ N) @ Y))))))))).
% 2.90/1.55 thf(fact_103_add_Onat__pow__distrib, axiom, (! [X:a,Y:a,N:nat]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add_pow @ a) @ b) @ nat) @ r) @ N) @ (((((add @ a) @ b) @ r) @ X) @ Y)) = (((((add @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ nat) @ r) @ N) @ X)) @ ((((((add_pow @ a) @ b) @ nat) @ r) @ N) @ Y)))))))).
% 2.90/1.55 thf(fact_104_add_Onat__pow__comm, axiom, (! [X:a,N:nat,M2:nat]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ nat) @ r) @ N) @ X)) @ ((((((add_pow @ a) @ b) @ nat) @ r) @ M2) @ X)) = (((((add @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ nat) @ r) @ M2) @ X)) @ ((((((add_pow @ a) @ b) @ nat) @ r) @ N) @ X))))))).
% 2.90/1.55 thf(fact_105_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)))))).
% 2.90/1.55 thf(fact_106_add_Onat__pow__one, axiom, (! [N:nat]: ((((((((add_pow @ a) @ b) @ nat) @ r) @ N) @ (((zero @ a) @ b) @ r)) = (((zero @ a) @ b) @ r))))).
% 2.90/1.55 thf(fact_107_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)))))).
% 2.90/1.55 thf(fact_108_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)))))))))).
% 2.90/1.55 thf(fact_109_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))))))))).
% 2.90/1.55 thf(fact_110_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))))))))).
% 2.90/1.55 thf(fact_111_add_Onat__pow__mult, axiom, (! [X:a,N:nat,M2:nat]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ nat) @ r) @ N) @ X)) @ ((((((add_pow @ a) @ b) @ nat) @ r) @ M2) @ X)) = ((((((add_pow @ a) @ b) @ nat) @ r) @ (((plus_plus @ nat) @ N) @ M2)) @ X)))))).
% 2.90/1.55 thf(fact_112_add_Onat__pow__Suc2, axiom, (! [X:a,N:nat]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add_pow @ a) @ b) @ nat) @ r) @ (suc @ N)) @ X) = (((((add @ a) @ b) @ r) @ X) @ ((((((add_pow @ a) @ b) @ nat) @ r) @ N) @ X))))))).
% 2.90/1.55 thf(fact_113_add_Opow__eq__div2, axiom, (! [X:a,M2:nat,N:nat]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((((add_pow @ a) @ b) @ nat) @ r) @ M2) @ X) = ((((((add_pow @ a) @ b) @ nat) @ r) @ N) @ X)) => (((((((add_pow @ a) @ b) @ nat) @ r) @ (((minus_minus @ nat) @ M2) @ N)) @ X) = (((zero @ a) @ b) @ r))))))).
% 2.90/1.55 thf(fact_114_minus__to__eq, axiom, (! [X:a,Y:a]: (((((abelian_group @ a) @ b) @ r) => ((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((a_minus @ a) @ b) @ r) @ X) @ Y) = (((zero @ a) @ b) @ r)) => (X = Y)))))))).
% 2.90/1.55 thf(fact_115_ring_Oequality, axiom, (! [Z:$tType,A:$tType,R:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ Z))),R2:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ Z)))]: ((((((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ Z))) @ R) = (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ Z))) @ R2)) => (((((mult @ A) @ ((ring_ext @ A) @ Z)) @ R) = (((mult @ A) @ ((ring_ext @ A) @ Z)) @ R2)) => (((((one @ A) @ ((ring_ext @ A) @ Z)) @ R) = (((one @ A) @ ((ring_ext @ A) @ Z)) @ R2)) => (((((zero @ A) @ Z) @ R) = (((zero @ A) @ Z) @ R2)) => (((((add @ A) @ Z) @ R) = (((add @ A) @ Z) @ R2)) => (((((more @ A) @ Z) @ R) = (((more @ A) @ Z) @ R2)) => (R = R2)))))))))).
% 2.90/1.55 thf(fact_116_is__abelian__group, axiom, (((abelian_group @ a) @ b) @ r)).
% 2.90/1.55 thf(fact_117_add_Onat__pow__Suc, axiom, (! [N:nat,X:a]: ((((((((add_pow @ a) @ b) @ nat) @ r) @ (suc @ N)) @ X) = (((((add @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ nat) @ r) @ N) @ X)) @ X))))).
% 2.90/1.55 thf(fact_118_abelian__groupE_I1_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A]: (((((abelian_group @ 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) @ (((((add @ A) @ B) @ R3) @ X) @ Y)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)))))))).
% 2.90/1.55 thf(fact_119_abelian__groupE_I3_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A,Z2:A]: (((((abelian_group @ 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)) => ((((((add @ A) @ B) @ R3) @ (((((add @ A) @ B) @ R3) @ X) @ Y)) @ Z2) = (((((add @ A) @ B) @ R3) @ X) @ (((((add @ A) @ B) @ R3) @ Y) @ Z2)))))))))).
% 2.90/1.55 thf(fact_120_abelian__groupE_I4_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A]: (((((abelian_group @ 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)) => ((((((add @ A) @ B) @ R3) @ X) @ Y) = (((((add @ A) @ B) @ R3) @ Y) @ X)))))))).
% 2.90/1.55 thf(fact_121_abelian__groupE_I2_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((((abelian_group @ A) @ B) @ R3) => (((member @ A) @ (((zero @ A) @ B) @ R3)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)))))).
% 2.90/1.55 thf(fact_122_abelian__group_Ominus__closed, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A]: (((((abelian_group @ A) @ B) @ G4) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => (((member @ A) @ (((((a_minus @ A) @ B) @ G4) @ X) @ Y)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)))))))).
% 2.90/1.55 thf(fact_123_abelian__groupI, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B)))]: (((! [X3:A]: (((((member @ A) @ X3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => (! [Y4:A]: (((((member @ A) @ Y4) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => (((member @ A) @ (((((add @ A) @ B) @ R3) @ X3) @ Y4)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)))))))) => ((((member @ A) @ (((zero @ A) @ B) @ R3)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((! [X3:A]: (((((member @ A) @ X3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => (! [Y4:A]: (((((member @ A) @ Y4) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => (! [Z3:A]: (((((member @ A) @ Z3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((add @ A) @ B) @ R3) @ (((((add @ A) @ B) @ R3) @ X3) @ Y4)) @ Z3) = (((((add @ A) @ B) @ R3) @ X3) @ (((((add @ A) @ B) @ R3) @ Y4) @ Z3)))))))))))) => ((! [X3:A]: (((((member @ A) @ X3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => (! [Y4:A]: (((((member @ A) @ Y4) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((add @ A) @ B) @ R3) @ X3) @ Y4) = (((((add @ A) @ B) @ R3) @ Y4) @ X3)))))))) => ((! [X3:A]: (((((member @ A) @ X3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((add @ A) @ B) @ R3) @ (((zero @ A) @ B) @ R3)) @ X3) = X3)))) => ((! [X3:A]: (((((member @ A) @ X3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => (? [Xa:A]: (((((member @ A) @ Xa) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) & ((((((add @ A) @ B) @ R3) @ Xa) @ X3) = (((zero @ A) @ B) @ R3)))))))) => (((abelian_group @ A) @ B) @ R3)))))))))).
% 2.90/1.55 thf(fact_124_abelian__groupE_I5_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((abelian_group @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => ((((((add @ A) @ B) @ R3) @ (((zero @ A) @ B) @ R3)) @ X) = X)))))).
% 2.90/1.55 thf(fact_125_abelian__groupE_I6_J, axiom, (! [B:$tType,A:$tType,R3:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((abelian_group @ A) @ B) @ R3) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) => (? [X3:A]: (((((member @ A) @ X3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ R3)) & ((((((add @ A) @ B) @ R3) @ X3) @ X) = (((zero @ A) @ B) @ R3)))))))))).
% 2.90/1.55 thf(fact_126_abelian__group_Ofour__elem__comm, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),A2:A,B2:A,C2:A,D:A]: (((((abelian_group @ A) @ B) @ G4) => ((((member @ A) @ A2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((member @ A) @ B2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((member @ A) @ C2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((member @ A) @ D) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((((a_minus @ A) @ B) @ G4) @ (((((add @ A) @ B) @ G4) @ (((((a_minus @ A) @ B) @ G4) @ A2) @ C2)) @ B2)) @ D) = (((((a_minus @ A) @ B) @ G4) @ (((((a_minus @ A) @ B) @ G4) @ (((((add @ A) @ B) @ G4) @ A2) @ B2)) @ C2)) @ D)))))))))).
% 2.90/1.55 thf(fact_127_abelian__group_Oright__inv__add, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),A2:A,B2:A,C2:A]: (((((abelian_group @ A) @ B) @ G4) => ((((member @ A) @ A2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((member @ A) @ B2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((member @ A) @ C2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((((a_minus @ A) @ B) @ G4) @ (((((a_minus @ A) @ B) @ G4) @ C2) @ A2)) @ B2) = (((((a_minus @ A) @ B) @ G4) @ C2) @ (((((add @ A) @ B) @ G4) @ A2) @ B2)))))))))).
% 2.90/1.55 thf(fact_128_abelian__group_Ominus__to__eq, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A]: (((((abelian_group @ A) @ B) @ G4) => ((((abelian_group @ A) @ B) @ G4) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => (((((((a_minus @ A) @ B) @ G4) @ X) @ Y) = (((zero @ A) @ B) @ G4)) => (X = Y))))))))).
% 2.90/1.55 thf(fact_129_diff__diff__left, axiom, (! [I:nat,J:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ K) = (((minus_minus @ nat) @ I) @ (((plus_plus @ nat) @ J) @ K)))))).
% 2.90/1.55 thf(fact_130_Suc__diff__diff, axiom, (! [M2:nat,N:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ (suc @ M2)) @ N)) @ (suc @ K)) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ M2) @ N)) @ K))))).
% 2.90/1.55 thf(fact_131_diff__Suc__Suc, axiom, (! [M2:nat,N:nat]: (((((minus_minus @ nat) @ (suc @ M2)) @ (suc @ N)) = (((minus_minus @ nat) @ M2) @ N))))).
% 2.90/1.55 thf(fact_132_add__Suc__right, axiom, (! [M2:nat,N:nat]: (((((plus_plus @ nat) @ M2) @ (suc @ N)) = (suc @ (((plus_plus @ nat) @ M2) @ N)))))).
% 2.90/1.55 thf(fact_133_nat_Oinject, axiom, (! [X2:nat,Y2:nat]: ((((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))).
% 2.90/1.55 thf(fact_134_old_Onat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))).
% 2.90/1.55 thf(fact_135_Suc__inject, axiom, (! [X:nat,Y:nat]: ((((suc @ X) = (suc @ Y)) => (X = Y))))).
% 2.90/1.55 thf(fact_136_n__not__Suc__n, axiom, (! [N:nat]: ((N != (suc @ N))))).
% 2.90/1.55 thf(fact_137_diff__commute, axiom, (! [I:nat,J:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ K) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ K)) @ J))))).
% 2.90/1.55 thf(fact_138_add__Suc, axiom, (! [M2:nat,N:nat]: (((((plus_plus @ nat) @ (suc @ M2)) @ N) = (suc @ (((plus_plus @ nat) @ M2) @ N)))))).
% 2.90/1.55 thf(fact_139_nat__arith_Osuc1, axiom, (! [A4:nat,K:nat,A2:nat]: (((A4 = (((plus_plus @ nat) @ K) @ A2)) => ((suc @ A4) = (((plus_plus @ nat) @ K) @ (suc @ A2))))))).
% 2.90/1.55 thf(fact_140_add__Suc__shift, axiom, (! [M2:nat,N:nat]: (((((plus_plus @ nat) @ (suc @ M2)) @ N) = (((plus_plus @ nat) @ M2) @ (suc @ N)))))).
% 2.90/1.55 thf(fact_141_zero__induct__lemma, axiom, (! [P:(nat > $o),K:nat,I:nat]: (((P @ K) => ((! [N2:nat]: (((P @ (suc @ N2)) => (P @ N2)))) => (P @ (((minus_minus @ nat) @ K) @ I))))))).
% 2.90/1.55 thf(fact_142_diff__add__inverse2, axiom, (! [M2:nat,N:nat]: (((((minus_minus @ nat) @ (((plus_plus @ nat) @ M2) @ N)) @ N) = M2)))).
% 2.90/1.55 thf(fact_143_diff__add__inverse, axiom, (! [N:nat,M2:nat]: (((((minus_minus @ nat) @ (((plus_plus @ nat) @ N) @ M2)) @ N) = M2)))).
% 2.90/1.55 thf(fact_144_diff__cancel2, axiom, (! [M2:nat,K:nat,N:nat]: (((((minus_minus @ nat) @ (((plus_plus @ nat) @ M2) @ K)) @ (((plus_plus @ nat) @ N) @ K)) = (((minus_minus @ nat) @ M2) @ N))))).
% 2.90/1.55 thf(fact_145_Nat_Odiff__cancel, axiom, (! [K:nat,M2:nat,N:nat]: (((((minus_minus @ nat) @ (((plus_plus @ nat) @ K) @ M2)) @ (((plus_plus @ nat) @ K) @ N)) = (((minus_minus @ nat) @ M2) @ N))))).
% 2.90/1.55 thf(fact_146_add__diff__cancel__right_H, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = A2))))))).
% 2.90/1.55 thf(fact_147_add__diff__cancel__right, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,C2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ A2) @ B2)))))))).
% 2.90/1.55 thf(fact_148_add__diff__cancel__left_H, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ A2) = B2))))))).
% 2.90/1.55 thf(fact_149_add__diff__cancel__left, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [C2:A,A2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2)) = (((minus_minus @ A) @ A2) @ B2)))))))).
% 2.90/1.55 thf(fact_150_add__right__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A2:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C2) @ A2)) = (B2 = C2)))))))).
% 2.90/1.55 thf(fact_151_add__left__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C2)) = (B2 = C2)))))))).
% 2.90/1.55 thf(fact_152_add__diff__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = A2))))))).
% 2.90/1.55 thf(fact_153_diff__add__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ B2) = A2))))))).
% 2.90/1.55 thf(fact_154_add__right__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A2:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C2) @ A2)) => (B2 = C2)))))))).
% 2.90/1.55 thf(fact_155_add__left__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C2)) => (B2 = C2)))))))).
% 2.90/1.55 thf(fact_156_add_Oleft__commute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [B2:A,A2:A,C2:A]: (((((plus_plus @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C2)) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 2.90/1.55 thf(fact_157_add_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => ((plus_plus @ A) = (^ [A6:A,B5:A]: ((((plus_plus @ A) @ B5) @ A6)))))))).
% 2.90/1.55 thf(fact_158_group__add__class_Oadd_Oright__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [B2:A,A2:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C2) @ A2)) = (B2 = C2)))))))).
% 2.90/1.55 thf(fact_159_add_Oleft__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C2)) = (B2 = C2)))))))).
% 2.90/1.55 thf(fact_160_add_Oassoc, axiom, (! [A:$tType]: (((semigroup_add @ A) => (! [A2:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 2.90/1.55 thf(fact_161_group__cancel_Oadd2, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [B6:A,K:A,B2:A,A2:A]: (((B6 = (((plus_plus @ A) @ K) @ B2)) => ((((plus_plus @ A) @ A2) @ B6) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A2) @ B2)))))))))).
% 2.90/1.55 thf(fact_162_group__cancel_Oadd1, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A4:A,K:A,A2:A,B2:A]: (((A4 = (((plus_plus @ A) @ K) @ A2)) => ((((plus_plus @ A) @ A4) @ B2) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A2) @ B2)))))))))).
% 2.90/1.55 thf(fact_163_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (K = L)) => ((((plus_plus @ A) @ I) @ K) = (((plus_plus @ A) @ J) @ L))))))))).
% 2.90/1.55 thf(fact_164_ab__semigroup__add__class_Oadd__ac_I1_J, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [A2:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 2.90/1.55 thf(fact_165_diff__right__commute, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,C2:A,B2:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ B2) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2)))))))).
% 2.90/1.55 thf(fact_166_diff__eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A,D:A]: ((((((minus_minus @ A) @ A2) @ B2) = (((minus_minus @ A) @ C2) @ D)) => ((A2 = B2) = (C2 = D))))))))).
% 2.90/1.55 thf(fact_167_group__cancel_Osub1, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A4:A,K:A,A2:A,B2:A]: (((A4 = (((plus_plus @ A) @ K) @ A2)) => ((((minus_minus @ A) @ A4) @ B2) = (((plus_plus @ A) @ K) @ (((minus_minus @ A) @ A2) @ B2)))))))))).
% 2.90/1.55 thf(fact_168_diff__eq__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: ((((((minus_minus @ A) @ A2) @ B2) = C2) = (A2 = (((plus_plus @ A) @ C2) @ B2))))))))).
% 2.90/1.55 thf(fact_169_eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,C2:A,B2:A]: (((A2 = (((minus_minus @ A) @ C2) @ B2)) = ((((plus_plus @ A) @ A2) @ B2) = C2)))))))).
% 2.90/1.55 thf(fact_170_add__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((plus_plus @ A) @ A2) @ (((minus_minus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2)))))))).
% 2.90/1.55 thf(fact_171_diff__diff__eq2, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((minus_minus @ A) @ A2) @ (((minus_minus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ B2)))))))).
% 2.90/1.55 thf(fact_172_diff__add__eq, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2) = (((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ B2)))))))).
% 2.90/1.55 thf(fact_173_diff__add__eq__diff__diff__swap, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((minus_minus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ B2)))))))).
% 2.90/1.55 thf(fact_174_diff__diff__add, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,B2:A,C2:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2) = (((minus_minus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 2.90/1.55 thf(fact_175_add__implies__diff, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [C2:A,B2:A,A2:A]: ((((((plus_plus @ A) @ C2) @ B2) = A2) => (C2 = (((minus_minus @ A) @ A2) @ B2))))))))).
% 2.90/1.55 thf(fact_176_Units__l__inv__ex, axiom, (! [X:a]: (((((member @ a) @ X) @ (((units @ a) @ ((ring_ext @ a) @ b)) @ r)) => (? [X3:a]: (((((member @ a) @ X3) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) & ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X3) @ X) = (((one @ a) @ ((ring_ext @ a) @ b)) @ r))))))))).
% 2.90/1.55 thf(fact_177_Units__r__inv__ex, axiom, (! [X:a]: (((((member @ a) @ X) @ (((units @ a) @ ((ring_ext @ a) @ b)) @ r)) => (? [X3:a]: (((((member @ a) @ X3) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) & ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ X3) = (((one @ a) @ ((ring_ext @ a) @ b)) @ r))))))))).
% 2.90/1.55 thf(fact_178_group__l__invI, axiom, ((! [X3:a]: (((((member @ a) @ X3) @ (((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) @ X3) = (((one @ a) @ ((ring_ext @ a) @ b)) @ r)))))))) => (((group @ a) @ ((ring_ext @ a) @ b)) @ r))).
% 2.90/1.55 thf(fact_179_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)))))).
% 2.90/1.55 thf(fact_180_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)))))))).
% 2.90/1.55 thf(fact_181_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))))))).
% 2.90/1.55 thf(fact_182_Units__one__closed, axiom, (((member @ a) @ (((one @ a) @ ((ring_ext @ a) @ b)) @ r)) @ (((units @ a) @ ((ring_ext @ a) @ b)) @ r))).
% 2.90/1.55 thf(fact_183_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)))))))).
% 2.90/1.55 thf(fact_184_groupI, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((! [X3:A]: (((((member @ A) @ X3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => (! [Y4:A]: (((((member @ A) @ Y4) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => (((member @ A) @ (((((mult @ A) @ B) @ G4) @ X3) @ Y4)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)))))))) => ((((member @ A) @ (((one @ A) @ B) @ G4)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => ((! [X3:A]: (((((member @ A) @ X3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => (! [Y4:A]: (((((member @ A) @ Y4) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => (! [Z3:A]: (((((member @ A) @ Z3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => ((((((mult @ A) @ B) @ G4) @ (((((mult @ A) @ B) @ G4) @ X3) @ Y4)) @ Z3) = (((((mult @ A) @ B) @ G4) @ X3) @ (((((mult @ A) @ B) @ G4) @ Y4) @ Z3)))))))))))) => ((! [X3:A]: (((((member @ A) @ X3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => ((((((mult @ A) @ B) @ G4) @ (((one @ A) @ B) @ G4)) @ X3) = X3)))) => ((! [X3:A]: (((((member @ A) @ X3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => (? [Xa:A]: (((((member @ A) @ Xa) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) & ((((((mult @ A) @ B) @ G4) @ Xa) @ X3) = (((one @ A) @ B) @ G4)))))))) => (((group @ A) @ B) @ G4))))))))).
% 2.90/1.55 thf(fact_185_group_Oinv__comm, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A,Y:A]: (((((group @ A) @ B) @ G4) => (((((((mult @ A) @ B) @ G4) @ X) @ Y) = (((one @ A) @ B) @ G4)) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => ((((((mult @ A) @ B) @ G4) @ Y) @ X) = (((one @ A) @ B) @ G4))))))))).
% 2.90/1.55 thf(fact_186_group_Ol__inv__ex, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A]: (((((group @ A) @ B) @ G4) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => (? [X3:A]: (((((member @ A) @ X3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) & ((((((mult @ A) @ B) @ G4) @ X3) @ X) = (((one @ A) @ B) @ G4)))))))))).
% 2.90/1.55 thf(fact_187_Group_Ogroup_Oright__cancel, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A,Y:A,Z2:A]: (((((group @ A) @ B) @ G4) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => ((((member @ A) @ Z2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => (((((((mult @ A) @ B) @ G4) @ Y) @ X) = (((((mult @ A) @ B) @ G4) @ Z2) @ X)) = (Y = Z2))))))))).
% 2.90/1.55 thf(fact_188_group_OUnits__eq, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((((group @ A) @ B) @ G4) => ((((units @ A) @ B) @ G4) = (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)))))).
% 2.90/1.55 thf(fact_189_group_Or__cancel__one_H, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A,A2:A]: (((((group @ A) @ B) @ G4) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => ((((member @ A) @ A2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => ((X = (((((mult @ A) @ B) @ G4) @ A2) @ X)) = (A2 = (((one @ A) @ B) @ G4))))))))).
% 2.90/1.55 thf(fact_190_group_Ol__cancel__one_H, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A,A2:A]: (((((group @ A) @ B) @ G4) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => ((((member @ A) @ A2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => ((X = (((((mult @ A) @ B) @ G4) @ X) @ A2)) = (A2 = (((one @ A) @ B) @ G4))))))))).
% 2.90/1.55 thf(fact_191_group_Or__cancel__one, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A,A2:A]: (((((group @ A) @ B) @ G4) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => ((((member @ A) @ A2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => (((((((mult @ A) @ B) @ G4) @ A2) @ X) = X) = (A2 = (((one @ A) @ B) @ G4))))))))).
% 2.90/1.55 thf(fact_192_group_Ol__cancel__one, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A,A2:A]: (((((group @ A) @ B) @ G4) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => ((((member @ A) @ A2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => (((((((mult @ A) @ B) @ G4) @ X) @ A2) = X) = (A2 = (((one @ A) @ B) @ G4))))))))).
% 2.90/1.55 thf(fact_193_group_Or__inv__ex, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A]: (((((group @ A) @ B) @ G4) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => (? [X3:A]: (((((member @ A) @ X3) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) & ((((((mult @ A) @ B) @ G4) @ X) @ X3) = (((one @ A) @ B) @ G4)))))))))).
% 2.90/1.55 thf(fact_194_units__group, axiom, (((group @ a) @ product_unit) @ (((units_of @ a) @ ((ring_ext @ a) @ b)) @ r))).
% 2.90/1.55 thf(fact_195_inv__add, axiom, (! [A2:a,B2:a]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((a_inv @ a) @ b) @ r) @ (((((add @ a) @ b) @ r) @ A2) @ B2)) = (((((a_minus @ a) @ b) @ r) @ ((((a_inv @ a) @ b) @ r) @ A2)) @ B2))))))).
% 2.90/1.55 thf(fact_196_r__neg2, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ X) @ (((((add @ a) @ b) @ r) @ ((((a_inv @ a) @ b) @ r) @ X)) @ Y)) = Y)))))).
% 2.90/1.55 thf(fact_197_r__neg1, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ ((((a_inv @ a) @ b) @ r) @ X)) @ (((((add @ a) @ b) @ r) @ X) @ Y)) = Y)))))).
% 2.90/1.55 thf(fact_198_local_Ominus__add, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((a_inv @ a) @ b) @ r) @ (((((add @ a) @ b) @ r) @ X) @ Y)) = (((((add @ a) @ b) @ r) @ ((((a_inv @ a) @ b) @ r) @ X)) @ ((((a_inv @ a) @ b) @ r) @ Y)))))))).
% 2.90/1.55 thf(fact_199_add_Oinv__solve__right_H, axiom, (! [A2:a,B2:a,C2:a]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ C2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add @ a) @ b) @ r) @ B2) @ ((((a_inv @ a) @ b) @ r) @ C2)) = A2) = (B2 = (((((add @ a) @ b) @ r) @ A2) @ C2))))))))).
% 2.90/1.55 thf(fact_200_add_Oinv__solve__right, axiom, (! [A2:a,B2:a,C2:a]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ C2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((A2 = (((((add @ a) @ b) @ r) @ B2) @ ((((a_inv @ a) @ b) @ r) @ C2))) = (B2 = (((((add @ a) @ b) @ r) @ A2) @ C2))))))))).
% 2.90/1.55 thf(fact_201_add_Oinv__solve__left_H, axiom, (! [A2:a,B2:a,C2:a]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ C2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add @ a) @ b) @ r) @ ((((a_inv @ a) @ b) @ r) @ B2)) @ C2) = A2) = (C2 = (((((add @ a) @ b) @ r) @ B2) @ A2))))))))).
% 2.90/1.55 thf(fact_202_add_Oinv__solve__left, axiom, (! [A2:a,B2:a,C2:a]: (((((member @ a) @ A2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ B2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ C2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((A2 = (((((add @ a) @ b) @ r) @ ((((a_inv @ a) @ b) @ r) @ B2)) @ C2)) = (C2 = (((((add @ a) @ b) @ r) @ B2) @ A2))))))))).
% 2.90/1.55 thf(fact_203_add_Oinv__mult__group, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((a_inv @ a) @ b) @ r) @ (((((add @ a) @ b) @ r) @ X) @ Y)) = (((((add @ a) @ b) @ r) @ ((((a_inv @ a) @ b) @ r) @ Y)) @ ((((a_inv @ a) @ b) @ r) @ X)))))))).
% 2.90/1.55 thf(fact_204_a__transpose__inv, axiom, (! [X:a,Y:a,Z2:a]: ((((((((add @ a) @ b) @ r) @ X) @ Y) = Z2) => ((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Z2) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ ((((a_inv @ a) @ b) @ r) @ X)) @ Z2) = Y)))))))).
% 2.90/1.55 thf(fact_205_r__minus, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ ((((a_inv @ a) @ b) @ r) @ Y)) = ((((a_inv @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Y)))))))).
% 2.90/1.55 thf(fact_206_l__minus, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ ((((a_inv @ a) @ b) @ r) @ X)) @ Y) = ((((a_inv @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ X) @ Y)))))))).
% 2.90/1.55 thf(fact_207_add_Onat__pow__inv, axiom, (! [X:a,I:nat]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add_pow @ a) @ b) @ nat) @ r) @ I) @ ((((a_inv @ a) @ b) @ r) @ X)) = ((((a_inv @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ nat) @ r) @ I) @ X))))))).
% 2.90/1.55 thf(fact_208_add_Oint__pow__inv, axiom, (! [X:a,I:int]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add_pow @ a) @ b) @ int) @ r) @ I) @ ((((a_inv @ a) @ b) @ r) @ X)) = ((((a_inv @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ int) @ r) @ I) @ X))))))).
% 2.90/1.55 thf(fact_209_minus__eq, axiom, (! [X:a,Y:a]: (((((((a_minus @ a) @ b) @ r) @ X) @ Y) = (((((add @ a) @ b) @ r) @ X) @ ((((a_inv @ a) @ b) @ r) @ Y)))))).
% 2.90/1.55 thf(fact_210_sum__zero__eq__neg, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add @ a) @ b) @ r) @ X) @ Y) = (((zero @ a) @ b) @ r)) => (X = ((((a_inv @ a) @ b) @ r) @ Y)))))))).
% 2.90/1.55 thf(fact_211_r__neg, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ X) @ ((((a_inv @ a) @ b) @ r) @ X)) = (((zero @ a) @ b) @ r)))))).
% 2.90/1.55 thf(fact_212_minus__equality, axiom, (! [Y:a,X:a]: ((((((((add @ a) @ b) @ r) @ Y) @ X) = (((zero @ a) @ b) @ r)) => ((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((member @ a) @ Y) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((a_inv @ a) @ b) @ r) @ X) = Y))))))).
% 2.90/1.55 thf(fact_213_l__neg, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((add @ a) @ b) @ r) @ ((((a_inv @ a) @ b) @ r) @ X)) @ X) = (((zero @ a) @ b) @ r)))))).
% 2.90/1.55 thf(fact_214_add_Oint__pow__diff, axiom, (! [X:a,N:int,M2:int]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add_pow @ a) @ b) @ int) @ r) @ (((minus_minus @ int) @ N) @ M2)) @ X) = (((((add @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ int) @ r) @ N) @ X)) @ ((((a_inv @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ int) @ r) @ M2) @ X)))))))).
% 2.90/1.55 thf(fact_215_local_Ominus__minus, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((a_inv @ a) @ b) @ r) @ ((((a_inv @ a) @ b) @ r) @ X)) = X))))).
% 2.90/1.55 thf(fact_216_add_Oinv__closed, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((member @ a) @ ((((a_inv @ a) @ b) @ r) @ X)) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)))))).
% 2.90/1.55 thf(fact_217_local_Ominus__zero, axiom, (((((a_inv @ a) @ b) @ r) @ (((zero @ a) @ b) @ r)) = (((zero @ a) @ b) @ r))).
% 2.90/1.55 thf(fact_218_add_Oinv__eq__1__iff, axiom, (! [X:a]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => ((((((a_inv @ a) @ b) @ r) @ X) = (((zero @ a) @ b) @ r)) = (X = (((zero @ a) @ b) @ r))))))).
% 2.90/1.55 thf(fact_219_Units__minus__one__closed, axiom, (((member @ a) @ ((((a_inv @ a) @ b) @ r) @ (((one @ a) @ ((ring_ext @ a) @ b)) @ r))) @ (((units @ a) @ ((ring_ext @ a) @ b)) @ r))).
% 2.90/1.55 thf(fact_220_units__of__mult, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((((mult @ A) @ product_unit) @ (((units_of @ A) @ B) @ G4)) = (((mult @ A) @ B) @ G4))))).
% 2.90/1.55 thf(fact_221_abelian__group_Oa__inv__closed, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((abelian_group @ A) @ B) @ G4) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => (((member @ A) @ ((((a_inv @ A) @ B) @ G4) @ X)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4))))))).
% 2.90/1.55 thf(fact_222_abelian__group_Ominus__minus, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((abelian_group @ A) @ B) @ G4) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => (((((a_inv @ A) @ B) @ G4) @ ((((a_inv @ A) @ B) @ G4) @ X)) = X)))))).
% 2.90/1.55 thf(fact_223_a__minus__def, axiom, (! [M:$tType,A:$tType]: ((((a_minus @ A) @ M) = (^ [R4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ M))),X4:A,Y5:A]: ((((((add @ A) @ M) @ R4) @ X4) @ ((((a_inv @ A) @ M) @ R4) @ Y5)))))))).
% 2.90/1.55 thf(fact_224_units__of__one, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((((one @ A) @ product_unit) @ (((units_of @ A) @ B) @ G4)) = (((one @ A) @ B) @ G4))))).
% 2.90/1.55 thf(fact_225_abelian__group_Ominus__add, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A]: (((((abelian_group @ A) @ B) @ G4) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => (((((a_inv @ A) @ B) @ G4) @ (((((add @ A) @ B) @ G4) @ X) @ Y)) = (((((add @ A) @ B) @ G4) @ ((((a_inv @ A) @ B) @ G4) @ X)) @ ((((a_inv @ A) @ B) @ G4) @ Y))))))))).
% 2.90/1.55 thf(fact_226_abelian__group_Or__neg2, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A]: (((((abelian_group @ A) @ B) @ G4) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((((add @ A) @ B) @ G4) @ X) @ (((((add @ A) @ B) @ G4) @ ((((a_inv @ A) @ B) @ G4) @ X)) @ Y)) = Y))))))).
% 2.90/1.55 thf(fact_227_abelian__group_Or__neg1, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A]: (((((abelian_group @ A) @ B) @ G4) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((((add @ A) @ B) @ G4) @ ((((a_inv @ A) @ B) @ G4) @ X)) @ (((((add @ A) @ B) @ G4) @ X) @ Y)) = Y))))))).
% 2.90/1.55 thf(fact_228_abelian__group_Ominus__eq, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A]: (((((abelian_group @ A) @ B) @ G4) => ((((((a_minus @ A) @ B) @ G4) @ X) @ Y) = (((((add @ A) @ B) @ G4) @ X) @ ((((a_inv @ A) @ B) @ G4) @ Y))))))).
% 2.90/1.55 thf(fact_229_abelian__group_Ol__neg, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((abelian_group @ A) @ B) @ G4) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((((add @ A) @ B) @ G4) @ ((((a_inv @ A) @ B) @ G4) @ X)) @ X) = (((zero @ A) @ B) @ G4))))))).
% 2.90/1.55 thf(fact_230_abelian__group_Or__neg, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A]: (((((abelian_group @ A) @ B) @ G4) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((((add @ A) @ B) @ G4) @ X) @ ((((a_inv @ A) @ B) @ G4) @ X)) = (((zero @ A) @ B) @ G4))))))).
% 2.90/1.55 thf(fact_231_abelian__group_Ominus__equality, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),Y:A,X:A]: (((((abelian_group @ A) @ B) @ G4) => (((((((add @ A) @ B) @ G4) @ Y) @ X) = (((zero @ A) @ B) @ G4)) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => (((((a_inv @ A) @ B) @ G4) @ X) = Y)))))))).
% 2.90/1.55 thf(fact_232_abelian__group_Oinv__add, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),A2:A,B2:A]: (((((abelian_group @ A) @ B) @ G4) => ((((member @ A) @ A2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((member @ A) @ B2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => (((((a_inv @ A) @ B) @ G4) @ (((((add @ A) @ B) @ G4) @ A2) @ B2)) = (((((a_minus @ A) @ B) @ G4) @ ((((a_inv @ A) @ B) @ G4) @ A2)) @ B2)))))))).
% 2.90/1.55 thf(fact_233_abelian__group_Oa__transpose__inv, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))),X:A,Y:A,Z2:A]: (((((abelian_group @ A) @ B) @ G4) => (((((((add @ A) @ B) @ G4) @ X) @ Y) = Z2) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((member @ A) @ Z2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ ((ring_ext @ A) @ B))) @ G4)) => ((((((add @ A) @ B) @ G4) @ ((((a_inv @ A) @ B) @ G4) @ X)) @ Z2) = Y))))))))).
% 2.90/1.55 thf(fact_234_add_Oint__pow__neg, axiom, (! [X:a,I:int]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add_pow @ a) @ b) @ int) @ r) @ ((uminus_uminus @ int) @ I)) @ X) = ((((a_inv @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ int) @ r) @ I) @ X))))))).
% 2.90/1.55 thf(fact_235_add__minus__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ B2)) = B2))))))).
% 2.90/1.55 thf(fact_236_minus__add__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ (((plus_plus @ A) @ A2) @ B2)) = B2))))))).
% 2.90/1.55 thf(fact_237_minus__add__distrib, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,B2:A]: ((((uminus_uminus @ A) @ (((plus_plus @ A) @ A2) @ B2)) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ ((uminus_uminus @ A) @ B2))))))))).
% 2.90/1.55 thf(fact_238_minus__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: ((((uminus_uminus @ A) @ (((minus_minus @ A) @ A2) @ B2)) = (((minus_minus @ A) @ B2) @ A2)))))))).
% 2.90/1.55 thf(fact_239_diff__minus__eq__add, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ A2) @ ((uminus_uminus @ A) @ B2)) = (((plus_plus @ A) @ A2) @ B2)))))))).
% 2.90/1.55 thf(fact_240_uminus__add__conv__diff, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ B2) = (((minus_minus @ A) @ B2) @ A2)))))))).
% 2.90/1.55 thf(fact_241_group__cancel_Osub2, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [B6:A,K:A,B2:A,A2:A]: (((B6 = (((plus_plus @ A) @ K) @ B2)) => ((((minus_minus @ A) @ A2) @ B6) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ K)) @ (((minus_minus @ A) @ A2) @ B2)))))))))).
% 2.90/1.55 thf(fact_242_diff__conv__add__uminus, axiom, (! [A:$tType]: (((group_add @ A) => ((minus_minus @ A) = (^ [A6:A,B5:A]: ((((plus_plus @ A) @ A6) @ ((uminus_uminus @ A) @ B5))))))))).
% 2.90/1.55 thf(fact_243_ab__group__add__class_Oab__diff__conv__add__uminus, axiom, (! [A:$tType]: (((ab_group_add @ A) => ((minus_minus @ A) = (^ [A6:A,B5:A]: ((((plus_plus @ A) @ A6) @ ((uminus_uminus @ A) @ B5))))))))).
% 2.90/1.55 thf(fact_244_group__cancel_Oneg1, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A4:A,K:A,A2:A]: (((A4 = (((plus_plus @ A) @ K) @ A2)) => (((uminus_uminus @ A) @ A4) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ K)) @ ((uminus_uminus @ A) @ A2)))))))))).
% 2.90/1.55 thf(fact_245_add_Oinverse__distrib__swap, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: ((((uminus_uminus @ A) @ (((plus_plus @ A) @ A2) @ B2)) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2))))))))).
% 2.90/1.55 thf(fact_246_minus__diff__commute, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [B2:A,A2:A]: (((((minus_minus @ A) @ ((uminus_uminus @ A) @ B2)) @ A2) = (((minus_minus @ A) @ ((uminus_uminus @ A) @ A2)) @ B2)))))))).
% 2.90/1.55 thf(fact_247_add_Oint__pow__neg__int, axiom, (! [X:a,N:nat]: (((((member @ a) @ X) @ (((partial_carrier @ a) @ ((monoid_ext @ a) @ ((ring_ext @ a) @ b))) @ r)) => (((((((add_pow @ a) @ b) @ int) @ r) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N))) @ X) = ((((a_inv @ a) @ b) @ r) @ ((((((add_pow @ a) @ b) @ nat) @ r) @ N) @ X))))))).
% 2.90/1.55 thf(fact_248_units__comm__group, axiom, (((comm_group @ a) @ product_unit) @ (((units_of @ a) @ ((ring_ext @ a) @ b)) @ r))).
% 2.90/1.55 thf(fact_249_of__nat__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M2:nat,N:nat]: (((((semiring_1_of_nat @ A) @ M2) = ((semiring_1_of_nat @ A) @ N)) = (M2 = N)))))))).
% 2.90/1.55 thf(fact_250_of__nat__add, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [M2:nat,N:nat]: ((((semiring_1_of_nat @ A) @ (((plus_plus @ nat) @ M2) @ N)) = (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ M2)) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 2.90/1.55 thf(fact_251_comm__groupE_I4_J, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A,Y:A]: (((((comm_group @ A) @ B) @ G4) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => ((((((mult @ A) @ B) @ G4) @ X) @ Y) = (((((mult @ A) @ B) @ G4) @ Y) @ X)))))))).
% 2.90/1.55 thf(fact_252_comm__groupE_I3_J, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A,Y:A,Z2:A]: (((((comm_group @ A) @ B) @ G4) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => ((((member @ A) @ Z2) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => ((((((mult @ A) @ B) @ G4) @ (((((mult @ A) @ B) @ G4) @ X) @ Y)) @ Z2) = (((((mult @ A) @ B) @ G4) @ X) @ (((((mult @ A) @ B) @ G4) @ Y) @ Z2)))))))))).
% 2.90/1.55 thf(fact_253_comm__groupE_I1_J, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B)),X:A,Y:A]: (((((comm_group @ A) @ B) @ G4) => ((((member @ A) @ X) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => ((((member @ A) @ Y) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)) => (((member @ A) @ (((((mult @ A) @ B) @ G4) @ X) @ Y)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)))))))).
% 2.90/1.55 thf(fact_254_comm__groupE_I2_J, axiom, (! [B:$tType,A:$tType,G4:((partia1265057091ct_ext @ A) @ ((monoid_ext @ A) @ B))]: (((((comm_group @ A) @ B) @ G4) => (((member @ A) @ (((one @ A) @ B) @ G4)) @ (((partial_carrier @ A) @ ((monoid_ext @ A) @ B)) @ G4)))))).
% 2.90/1.55 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add, axiom, (ordere779506340up_add @ int)).
% 2.90/1.55 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add, axiom, (cancel146912293up_add @ int)).
% 2.90/1.55 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add, axiom, (cancel1352612707id_add @ int)).
% 2.90/1.55 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add, axiom, (cancel_semigroup_add @ int)).
% 2.90/1.55 thf(tcon_Int_Oint___Groups_Oab__semigroup__add, axiom, (ab_semigroup_add @ int)).
% 2.90/1.55 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add, axiom, (comm_monoid_add @ int)).
% 2.90/1.55 thf(tcon_Int_Oint___Groups_Osemigroup__add, axiom, (semigroup_add @ int)).
% 2.90/1.55 thf(tcon_Int_Oint___Nat_Osemiring__char__0, axiom, (semiring_char_0 @ int)).
% 2.90/1.55 thf(tcon_Int_Oint___Groups_Oab__group__add, axiom, (ab_group_add @ int)).
% 2.90/1.55 thf(tcon_Int_Oint___Rings_Osemiring__1, axiom, (semiring_1 @ int)).
% 2.90/1.55 thf(tcon_Int_Oint___Groups_Ogroup__add, axiom, (group_add @ int)).
% 2.90/1.55 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_1, axiom, (ordere779506340up_add @ nat)).
% 2.90/1.55 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_2, axiom, (cancel146912293up_add @ nat)).
% 2.90/1.55 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_3, axiom, (cancel1352612707id_add @ nat)).
% 2.90/1.55 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_4, axiom, (cancel_semigroup_add @ nat)).
% 2.90/1.55 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_5, axiom, (ab_semigroup_add @ nat)).
% 2.90/1.55 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_6, axiom, (comm_monoid_add @ nat)).
% 2.90/1.55 thf(tcon_Nat_Onat___Groups_Osemigroup__add_7, axiom, (semigroup_add @ nat)).
% 2.90/1.55 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_8, axiom, (semiring_char_0 @ nat)).
% 2.90/1.55 thf(tcon_Nat_Onat___Rings_Osemiring__1_9, axiom, (semiring_1 @ nat)).
% 2.90/1.55 thf(conj_0, conjecture, (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ (((((add @ a) @ b) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r2) @ r4))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ r3) @ r4)))) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s) @ s3)) @ (((((mult @ a) @ ((ring_ext @ a) @ b)) @ r) @ s2) @ s3)))) @ (((partial_carrier @ ((product_prod @ a) @ a)) @ ((eq_eq_object_ext @ ((product_prod @ a) @ a)) @ product_unit)) @ rel))).
% 2.90/1.55 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.Fb6urRC3jx/DTF2DTF19374.p
% 2.90/1.55 ------------------------
% 2.90/1.58 ---- Cleaned THF ---
% 2.90/1.58 thf(ty_t_Congruence_Opartial__object_Opartial__object__ext,type,
% 2.90/1.58 partia1265057091ct_ext: $tType > $tType > $tType ).
% 2.90/1.58
% 2.90/1.58 thf(ty_t_Congruence_Oeq__object_Oeq__object__ext,type,
% 2.90/1.58 eq_eq_object_ext: $tType > $tType > $tType ).
% 2.90/1.58
% 2.90/1.58 thf(ty_t_Group_Omonoid_Omonoid__ext,type,
% 2.90/1.58 monoid_ext: $tType > $tType > $tType ).
% 2.90/1.58
% 2.90/1.58 thf(ty_t_Ring_Oring_Oring__ext,type,
% 2.90/1.58 ring_ext: $tType > $tType > $tType ).
% 2.90/1.58
% 2.90/1.58 thf(ty_t_Product__Type_Ounit,type,
% 2.90/1.58 product_unit: $tType ).
% 2.90/1.58
% 2.90/1.58 thf(ty_t_Product__Type_Oprod,type,
% 2.90/1.58 product_prod: $tType > $tType > $tType ).
% 2.90/1.58
% 2.90/1.58 thf(ty_t_Set_Oset,type,
% 2.90/1.58 set: $tType > $tType ).
% 2.90/1.58
% 2.90/1.58 thf(ty_t_Nat_Onat,type,
% 2.90/1.58 nat: $tType ).
% 2.90/1.58
% 2.90/1.58 thf(ty_t_Int_Oint,type,
% 2.90/1.58 int: $tType ).
% 2.90/1.58
% 2.90/1.58 thf(ty_tf_b,type,
% 2.90/1.58 b: $tType ).
% 2.90/1.58
% 2.90/1.58 thf(ty_tf_a,type,
% 2.90/1.58 a: $tType ).
% 2.90/1.58
% 2.90/1.58 thf(sy_cl_Groups_Ogroup__add,type,
% 2.90/1.58 group_add:
% 2.90/1.58 !>[A: $tType] : $o ).
% 2.90/1.58
% 2.90/1.58 thf(sy_cl_Rings_Osemiring__1,type,
% 2.90/1.58 semiring_1:
% 2.90/1.58 !>[A: $tType] : $o ).
% 2.90/1.58
% 2.90/1.58 thf(sy_cl_Groups_Oab__group__add,type,
% 2.90/1.58 ab_group_add:
% 2.90/1.58 !>[A: $tType] : $o ).
% 2.90/1.58
% 2.90/1.58 thf(sy_cl_Nat_Osemiring__char__0,type,
% 2.90/1.58 semiring_char_0:
% 2.90/1.58 !>[A: $tType] : $o ).
% 2.90/1.58
% 2.90/1.58 thf(sy_cl_Groups_Osemigroup__add,type,
% 2.90/1.58 semigroup_add:
% 2.90/1.58 !>[A: $tType] : $o ).
% 2.90/1.58
% 2.90/1.58 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 2.90/1.58 comm_monoid_add:
% 2.90/1.58 !>[A: $tType] : $o ).
% 2.90/1.58
% 2.90/1.58 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 2.90/1.58 ab_semigroup_add:
% 2.90/1.58 !>[A: $tType] : $o ).
% 2.90/1.58
% 2.90/1.58 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 2.90/1.58 cancel_semigroup_add:
% 2.90/1.58 !>[A: $tType] : $o ).
% 2.90/1.58
% 2.90/1.58 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 2.90/1.58 cancel1352612707id_add:
% 2.90/1.58 !>[A: $tType] : $o ).
% 2.90/1.58
% 2.90/1.58 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.90/1.58 cancel146912293up_add:
% 2.90/1.58 !>[A: $tType] : $o ).
% 2.90/1.58
% 2.90/1.58 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 2.90/1.58 ordere779506340up_add:
% 2.90/1.58 !>[A: $tType] : $o ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Congruence_Opartial__object_Ocarrier,type,
% 2.90/1.58 partial_carrier:
% 2.90/1.58 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ Z ) > ( set @ A ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Group_OUnits,type,
% 2.90/1.58 units:
% 2.90/1.58 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > ( set @ A ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Group_Ocomm__group,type,
% 2.90/1.58 comm_group:
% 2.90/1.58 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > $o ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Group_Ogroup,type,
% 2.90/1.58 group:
% 2.90/1.58 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > $o ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Group_Omonoid_Omult,type,
% 2.90/1.58 mult:
% 2.90/1.58 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ Z ) ) > A > A > A ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Group_Omonoid_Oone,type,
% 2.90/1.58 one:
% 2.90/1.58 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ Z ) ) > A ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Group_Ounits__of,type,
% 2.90/1.58 units_of:
% 2.90/1.58 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ product_unit ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.90/1.58 minus_minus:
% 2.90/1.58 !>[A: $tType] : ( A > A > A ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 2.90/1.58 plus_plus:
% 2.90/1.58 !>[A: $tType] : ( A > A > A ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 2.90/1.58 uminus_uminus:
% 2.90/1.58 !>[A: $tType] : ( A > A ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Nat_OSuc,type,
% 2.90/1.58 suc: nat > nat ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 2.90/1.58 semiring_1_of_nat:
% 2.90/1.58 !>[A: $tType] : ( nat > A ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Product__Type_OPair,type,
% 2.90/1.58 product_Pair:
% 2.90/1.58 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.90/1.58 product_rec_prod:
% 2.90/1.58 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Ring_Oa__inv,type,
% 2.90/1.58 a_inv:
% 2.90/1.58 !>[A: $tType,M: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ M ) ) ) > A > A ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Ring_Oa__minus,type,
% 2.90/1.58 a_minus:
% 2.90/1.58 !>[A: $tType,M: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ M ) ) ) > A > A > A ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Ring_Oabelian__group,type,
% 2.90/1.58 abelian_group:
% 2.90/1.58 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Ring_Oadd__pow,type,
% 2.90/1.58 add_pow:
% 2.90/1.58 !>[A: $tType,C: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ C ) ) ) > B > A > A ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Ring_Ofield,type,
% 2.90/1.58 field:
% 2.90/1.58 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Ring_Oring_Oadd,type,
% 2.90/1.58 add:
% 2.90/1.58 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ) ) > A > A > A ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Ring_Oring_Omore,type,
% 2.90/1.58 more:
% 2.90/1.58 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ) ) > Z ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Ring_Oring_Ozero,type,
% 2.90/1.58 zero:
% 2.90/1.58 !>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ) ) > A ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Ring_Osemiring,type,
% 2.90/1.58 semiring:
% 2.90/1.58 !>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_Set_OCollect,type,
% 2.90/1.58 collect:
% 2.90/1.58 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_c_member,type,
% 2.90/1.58 member:
% 2.90/1.58 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_v_R,type,
% 2.90/1.58 r: partia1265057091ct_ext @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_v_r,type,
% 2.90/1.58 r2: a ).
% 2.90/1.58
% 2.90/1.58 thf(sy_v_r_H,type,
% 2.90/1.58 r3: a ).
% 2.90/1.58
% 2.90/1.58 thf(sy_v_r_H_H,type,
% 2.90/1.58 r4: a ).
% 2.90/1.58
% 2.90/1.58 thf(sy_v_rel,type,
% 2.90/1.58 rel: partia1265057091ct_ext @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) ).
% 2.90/1.58
% 2.90/1.58 thf(sy_v_s,type,
% 2.90/1.58 s: a ).
% 2.90/1.58
% 2.90/1.58 thf(sy_v_s_H,type,
% 2.90/1.58 s2: a ).
% 2.90/1.58
% 2.90/1.58 thf(sy_v_s_H_H,type,
% 2.90/1.58 s3: a ).
% 2.90/1.58
% 2.90/1.58 thf(fact_0_right__add__eq,axiom,
% 2.90/1.58 ! [A2: a,B2: a,C2: a] :
% 2.90/1.58 ( ( A2 = B2 )
% 2.90/1.58 => ( ( add @ a @ b @ r @ C2 @ A2 )
% 2.90/1.58 = ( add @ a @ b @ r @ C2 @ B2 ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_1_assms_I3_J,axiom,
% 2.90/1.58 member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ r4 @ s3 ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_2_assms_I2_J,axiom,
% 2.90/1.58 member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ r3 @ s2 ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_3_assms_I1_J,axiom,
% 2.90/1.58 member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ r2 @ s ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_4_closed__rel__add,axiom,
% 2.90/1.58 ! [R: a,S: a,R2: a,S2: a] :
% 2.90/1.58 ( ( 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 ) )
% 2.90/1.58 => ( ( 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 ) )
% 2.90/1.58 => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ S2 @ R ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ S @ R2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ S @ S2 ) ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_5__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_061_As_A_092_060otimes_062_A_Is_H_H_A_092_060otimes_062_As_H_J_A_092_060otimes_062_As_H_H_092_060close_062,axiom,
% 2.90/1.58 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) )
% 2.90/1.58 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ s2 ) ) @ s3 ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_6_f9,axiom,
% 2.90/1.58 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) )
% 2.90/1.58 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ s3 ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_7__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_A_061_As_H_H_A_092_060otimes_062_As_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_092_060close_062,axiom,
% 2.90/1.58 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r4 ) )
% 2.90/1.58 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ s ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r4 ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_8__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_092_060close_062,axiom,
% 2.90/1.58 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ r3 ) @ r4 ) )
% 2.90/1.58 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s2 ) @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r4 ) ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_9_f10,axiom,
% 2.90/1.58 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r2 ) @ r4 ) )
% 2.90/1.58 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s2 ) @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r4 ) ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_10_f5,axiom,
% 2.90/1.58 member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r4 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_11_f4,axiom,
% 2.90/1.58 member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r4 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_12__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_A_092_060otimes_062_Ar_H_H_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_A_092_060otimes_062_Ar_H_H_J_092_060close_062,axiom,
% 2.90/1.58 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ r3 ) ) @ r4 ) )
% 2.90/1.58 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r2 ) @ r4 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ r3 ) @ r4 ) ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_13__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_A_092_060otimes_062_Ar_H_H_J_A_092_060oplus_062_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_092_060close_062,axiom,
% 2.90/1.58 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ r3 ) ) @ r4 ) )
% 2.90/1.58 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s2 ) @ s3 ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r4 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r4 ) ) ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_14_f7,axiom,
% 2.90/1.58 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ r3 ) ) @ r4 ) )
% 2.90/1.58 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r2 ) @ r4 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ r3 ) @ r4 ) ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_15_f8,axiom,
% 2.90/1.58 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s2 ) @ s3 ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r4 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r4 ) ) ) )
% 2.90/1.58 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s2 ) @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r4 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s2 ) @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r4 ) ) ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_16_f12,axiom,
% 2.90/1.58 member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ r3 ) ) @ r4 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s2 ) @ s3 ) ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_17_l__distr,axiom,
% 2.90/1.58 ! [X: a,Y: a,Z2: a] :
% 2.90/1.58 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add @ a @ b @ r @ X @ Y ) @ Z2 )
% 2.90/1.58 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Z2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) ) ) ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_18_r__distr,axiom,
% 2.90/1.58 ! [X: a,Y: a,Z2: a] :
% 2.90/1.58 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Z2 @ ( add @ a @ b @ r @ X @ Y ) )
% 2.90/1.58 = ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Z2 @ X ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Z2 @ Y ) ) ) ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_19__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_A_092_060otimes_062_Ar_H_H_J_A_092_060oplus_062_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_A_061_A_092_060zero_062_092_060close_062,axiom,
% 2.90/1.58 ( ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ r3 ) ) @ r4 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s2 ) @ s3 ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r4 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r4 ) ) ) ) )
% 2.90/1.58 = ( zero @ a @ b @ r ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_20_prod_Oinject,axiom,
% 2.90/1.58 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 2.90/1.58 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 2.90/1.58 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 2.90/1.58 = ( ( X1 = Y1 )
% 2.90/1.58 & ( X2 = Y2 ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_21_old_Oprod_Oinject,axiom,
% 2.90/1.58 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 2.90/1.58 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.90/1.58 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.90/1.58 = ( ( A2 = A3 )
% 2.90/1.58 & ( B2 = B3 ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_22_local_Osemiring__axioms,axiom,
% 2.90/1.58 semiring @ a @ b @ r ).
% 2.90/1.58
% 2.90/1.58 thf(fact_23_m__assoc,axiom,
% 2.90/1.58 ! [X: a,Y: a,Z2: a] :
% 2.90/1.58 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ Z2 )
% 2.90/1.58 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) ) ) ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_24_m__comm,axiom,
% 2.90/1.58 ! [X: a,Y: a] :
% 2.90/1.58 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y )
% 2.90/1.58 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ X ) ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_25_m__lcomm,axiom,
% 2.90/1.58 ! [X: a,Y: a,Z2: a] :
% 2.90/1.58 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) )
% 2.90/1.58 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Z2 ) ) ) ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_26_add_Om__assoc,axiom,
% 2.90/1.58 ! [X: a,Y: a,Z2: a] :
% 2.90/1.58 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( add @ a @ b @ r @ ( add @ a @ b @ r @ X @ Y ) @ Z2 )
% 2.90/1.58 = ( add @ a @ b @ r @ X @ ( add @ a @ b @ r @ Y @ Z2 ) ) ) ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_27_add_Om__comm,axiom,
% 2.90/1.58 ! [X: a,Y: a] :
% 2.90/1.58 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( add @ a @ b @ r @ X @ Y )
% 2.90/1.58 = ( add @ a @ b @ r @ Y @ X ) ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_28_add_Om__lcomm,axiom,
% 2.90/1.58 ! [X: a,Y: a,Z2: a] :
% 2.90/1.58 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( add @ a @ b @ r @ X @ ( add @ a @ b @ r @ Y @ Z2 ) )
% 2.90/1.58 = ( add @ a @ b @ r @ Y @ ( add @ a @ b @ r @ X @ Z2 ) ) ) ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_29_local_Oright__minus__eq,axiom,
% 2.90/1.58 ! [A2: a,B2: a,C2: a] :
% 2.90/1.58 ( ( A2 = B2 )
% 2.90/1.58 => ( ( a_minus @ a @ b @ r @ C2 @ A2 )
% 2.90/1.58 = ( a_minus @ a @ b @ r @ C2 @ B2 ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_30_local_Ominus__unique,axiom,
% 2.90/1.58 ! [Y: a,X: a,Y3: a] :
% 2.90/1.58 ( ( ( add @ a @ b @ r @ Y @ X )
% 2.90/1.58 = ( zero @ a @ b @ r ) )
% 2.90/1.58 => ( ( ( add @ a @ b @ r @ X @ Y3 )
% 2.90/1.58 = ( zero @ a @ b @ r ) )
% 2.90/1.58 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( member @ a @ Y3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( Y = Y3 ) ) ) ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_31_add_Or__inv__ex,axiom,
% 2.90/1.58 ! [X: a] :
% 2.90/1.58 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ? [X3: a] :
% 2.90/1.58 ( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 & ( ( add @ a @ b @ r @ X @ X3 )
% 2.90/1.58 = ( zero @ a @ b @ r ) ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_32_add_Oone__unique,axiom,
% 2.90/1.58 ! [U: a] :
% 2.90/1.58 ( ( member @ a @ U @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ! [X3: a] :
% 2.90/1.58 ( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ( ( add @ a @ b @ r @ U @ X3 )
% 2.90/1.58 = X3 ) )
% 2.90/1.58 => ( U
% 2.90/1.58 = ( zero @ a @ b @ r ) ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_33_add_Ol__inv__ex,axiom,
% 2.90/1.58 ! [X: a] :
% 2.90/1.58 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 => ? [X3: a] :
% 2.90/1.58 ( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.58 & ( ( add @ a @ b @ r @ X3 @ X )
% 2.90/1.58 = ( zero @ a @ b @ r ) ) ) ) ).
% 2.90/1.58
% 2.90/1.58 thf(fact_34_add_Oinv__comm,axiom,
% 2.90/1.58 ! [X: a,Y: a] :
% 2.90/1.58 ( ( ( add @ a @ b @ r @ X @ Y )
% 2.90/1.58 = ( zero @ a @ b @ r ) )
% 2.90/1.58 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add @ a @ b @ r @ Y @ X )
% 2.90/1.59 = ( zero @ a @ b @ r ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_35_right__inv__add,axiom,
% 2.90/1.59 ! [A2: a,B2: a,C2: a] :
% 2.90/1.59 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( a_minus @ a @ b @ r @ ( a_minus @ a @ b @ r @ C2 @ A2 ) @ B2 )
% 2.90/1.59 = ( a_minus @ a @ b @ r @ C2 @ ( add @ a @ b @ r @ A2 @ B2 ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_36_four__elem__comm,axiom,
% 2.90/1.59 ! [A2: a,B2: a,C2: a,D: a] :
% 2.90/1.59 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ D @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( a_minus @ a @ b @ r @ ( add @ a @ b @ r @ ( a_minus @ a @ b @ r @ A2 @ C2 ) @ B2 ) @ D )
% 2.90/1.59 = ( a_minus @ a @ b @ r @ ( a_minus @ a @ b @ r @ ( add @ a @ b @ r @ A2 @ B2 ) @ C2 ) @ D ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_37_local_Oadd_Oright__cancel,axiom,
% 2.90/1.59 ! [X: a,Y: a,Z2: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( ( add @ a @ b @ r @ Y @ X )
% 2.90/1.59 = ( add @ a @ b @ r @ Z2 @ X ) )
% 2.90/1.59 = ( Y = Z2 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_38_add_Om__closed,axiom,
% 2.90/1.59 ! [X: a,Y: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( member @ a @ ( add @ a @ b @ r @ X @ Y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_39_zero__closed,axiom,
% 2.90/1.59 member @ a @ ( zero @ a @ b @ r ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_40_semiring__simprules_I3_J,axiom,
% 2.90/1.59 ! [X: a,Y: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_41_minus__closed,axiom,
% 2.90/1.59 ! [X: a,Y: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( member @ a @ ( a_minus @ a @ b @ r @ X @ Y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_42_r__zero,axiom,
% 2.90/1.59 ! [X: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add @ a @ b @ r @ X @ ( zero @ a @ b @ r ) )
% 2.90/1.59 = X ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_43_l__zero,axiom,
% 2.90/1.59 ! [X: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add @ a @ b @ r @ ( zero @ a @ b @ r ) @ X )
% 2.90/1.59 = X ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_44_add_Or__cancel__one_H,axiom,
% 2.90/1.59 ! [X: a,A2: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( X
% 2.90/1.59 = ( add @ a @ b @ r @ A2 @ X ) )
% 2.90/1.59 = ( A2
% 2.90/1.59 = ( zero @ a @ b @ r ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_45_mem__Collect__eq,axiom,
% 2.90/1.59 ! [A: $tType,A2: A,P: A > $o] :
% 2.90/1.59 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.90/1.59 = ( P @ A2 ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_46_Collect__mem__eq,axiom,
% 2.90/1.59 ! [A: $tType,A4: set @ A] :
% 2.90/1.59 ( ( collect @ A
% 2.90/1.59 @ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
% 2.90/1.59 = A4 ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_47_Collect__cong,axiom,
% 2.90/1.59 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.90/1.59 ( ! [X3: A] :
% 2.90/1.59 ( ( P @ X3 )
% 2.90/1.59 = ( Q @ X3 ) )
% 2.90/1.59 => ( ( collect @ A @ P )
% 2.90/1.59 = ( collect @ A @ Q ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_48_ext,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.90/1.59 ( ! [X3: A] :
% 2.90/1.59 ( ( F @ X3 )
% 2.90/1.59 = ( G @ X3 ) )
% 2.90/1.59 => ( F = G ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_49_add_Or__cancel__one,axiom,
% 2.90/1.59 ! [X: a,A2: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( ( add @ a @ b @ r @ A2 @ X )
% 2.90/1.59 = X )
% 2.90/1.59 = ( A2
% 2.90/1.59 = ( zero @ a @ b @ r ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_50_add_Ol__cancel__one_H,axiom,
% 2.90/1.59 ! [X: a,A2: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( X
% 2.90/1.59 = ( add @ a @ b @ r @ X @ A2 ) )
% 2.90/1.59 = ( A2
% 2.90/1.59 = ( zero @ a @ b @ r ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_51_add_Ol__cancel__one,axiom,
% 2.90/1.59 ! [X: a,A2: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( ( add @ a @ b @ r @ X @ A2 )
% 2.90/1.59 = X )
% 2.90/1.59 = ( A2
% 2.90/1.59 = ( zero @ a @ b @ r ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_52_r__null,axiom,
% 2.90/1.59 ! [X: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( zero @ a @ b @ r ) )
% 2.90/1.59 = ( zero @ a @ b @ r ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_53_l__null,axiom,
% 2.90/1.59 ! [X: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( zero @ a @ b @ r ) @ X )
% 2.90/1.59 = ( zero @ a @ b @ r ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_54_old_Oprod_Oinducts,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.90/1.59 ( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
% 2.90/1.59 => ( P @ Prod ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_55_old_Oprod_Oexhaust,axiom,
% 2.90/1.59 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 2.90/1.59 ~ ! [A5: A,B4: B] :
% 2.90/1.59 ( Y
% 2.90/1.59 != ( product_Pair @ A @ B @ A5 @ B4 ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_56_prod__induct7,axiom,
% 2.90/1.59 ! [G2: $tType,F2: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
% 2.90/1.59 ( ! [A5: A,B4: B,C3: C,D3: D2,E2: E,F3: F2,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C3 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
% 2.90/1.59 => ( P @ X ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_57_prod__induct6,axiom,
% 2.90/1.59 ! [F2: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) )] :
% 2.90/1.59 ( ! [A5: A,B4: B,C3: C,D3: D2,E2: E,F3: F2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D2 @ ( product_prod @ E @ F2 ) @ D3 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
% 2.90/1.59 => ( P @ X ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_58_prod__induct5,axiom,
% 2.90/1.59 ! [E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 2.90/1.59 ( ! [A5: A,B4: B,C3: C,D3: D2,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C3 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) )
% 2.90/1.59 => ( P @ X ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_59_prod__induct4,axiom,
% 2.90/1.59 ! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 2.90/1.59 ( ! [A5: A,B4: B,C3: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B4 @ ( product_Pair @ C @ D2 @ C3 @ D3 ) ) ) )
% 2.90/1.59 => ( P @ X ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_60_prod__induct3,axiom,
% 2.90/1.59 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.90/1.59 ( ! [A5: A,B4: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C3 ) ) )
% 2.90/1.59 => ( P @ X ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_61_prod__cases7,axiom,
% 2.90/1.59 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F2: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
% 2.90/1.59 ~ ! [A5: A,B4: B,C3: C,D3: D2,E2: E,F3: F2,G3: G2] :
% 2.90/1.59 ( Y
% 2.90/1.59 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C3 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_62_prod__cases6,axiom,
% 2.90/1.59 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) )] :
% 2.90/1.59 ~ ! [A5: A,B4: B,C3: C,D3: D2,E2: E,F3: F2] :
% 2.90/1.59 ( Y
% 2.90/1.59 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D2 @ ( product_prod @ E @ F2 ) @ D3 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_63_prod__cases5,axiom,
% 2.90/1.59 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 2.90/1.59 ~ ! [A5: A,B4: B,C3: C,D3: D2,E2: E] :
% 2.90/1.59 ( Y
% 2.90/1.59 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C3 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_64_prod__cases4,axiom,
% 2.90/1.59 ! [A: $tType,B: $tType,C: $tType,D2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 2.90/1.59 ~ ! [A5: A,B4: B,C3: C,D3: D2] :
% 2.90/1.59 ( Y
% 2.90/1.59 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B4 @ ( product_Pair @ C @ D2 @ C3 @ D3 ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_65_prod__cases3,axiom,
% 2.90/1.59 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.90/1.59 ~ ! [A5: A,B4: B,C3: C] :
% 2.90/1.59 ( Y
% 2.90/1.59 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C3 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_66_Pair__inject,axiom,
% 2.90/1.59 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 2.90/1.59 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.90/1.59 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.90/1.59 => ~ ( ( A2 = A3 )
% 2.90/1.59 => ( B2 != B3 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_67_prod__cases,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 2.90/1.59 ( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
% 2.90/1.59 => ( P @ P2 ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_68_surj__pair,axiom,
% 2.90/1.59 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 2.90/1.59 ? [X3: A,Y4: B] :
% 2.90/1.59 ( P2
% 2.90/1.59 = ( product_Pair @ A @ B @ X3 @ Y4 ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_69_f11,axiom,
% 2.90/1.59 ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ r3 ) ) @ r4 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s2 ) @ s3 ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r4 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r4 ) ) ) ) ) )
% 2.90/1.59 = ( zero @ a @ b @ r ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_70_semiring_Osemiring__simprules_I11_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 2.90/1.59 ( ( semiring @ A @ B @ R3 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( add @ A @ B @ R3 @ X @ ( zero @ A @ B @ R3 ) )
% 2.90/1.59 = X ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_71_semiring_Osemiring__simprules_I6_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 2.90/1.59 ( ( semiring @ A @ B @ R3 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( add @ A @ B @ R3 @ ( zero @ A @ B @ R3 ) @ X )
% 2.90/1.59 = X ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_72_semiring_Ol__null,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 2.90/1.59 ( ( semiring @ A @ B @ R3 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( zero @ A @ B @ R3 ) @ X )
% 2.90/1.59 = ( zero @ A @ B @ R3 ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_73_semiring_Or__null,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 2.90/1.59 ( ( semiring @ A @ B @ R3 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ ( zero @ A @ B @ R3 ) )
% 2.90/1.59 = ( zero @ A @ B @ R3 ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_74_semiring_Or__distr,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 2.90/1.59 ( ( semiring @ A @ B @ R3 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ Z2 @ ( add @ A @ B @ R3 @ X @ Y ) )
% 2.90/1.59 = ( add @ A @ B @ R3 @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ Z2 @ X ) @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ Z2 @ Y ) ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_75_semiring_Ol__distr,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 2.90/1.59 ( ( semiring @ A @ B @ R3 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( add @ A @ B @ R3 @ X @ Y ) @ Z2 )
% 2.90/1.59 = ( add @ A @ B @ R3 @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ Z2 ) @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ Y @ Z2 ) ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_76_add__pow__rdistr__int,axiom,
% 2.90/1.59 ! [A2: a,B2: a,K: int] :
% 2.90/1.59 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ ( add_pow @ a @ b @ int @ r @ K @ B2 ) )
% 2.90/1.59 = ( add_pow @ a @ b @ int @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_77_add__pow__ldistr__int,axiom,
% 2.90/1.59 ! [A2: a,B2: a,K: int] :
% 2.90/1.59 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add_pow @ a @ b @ int @ r @ K @ A2 ) @ B2 )
% 2.90/1.59 = ( add_pow @ a @ b @ int @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_78_inv__unique,axiom,
% 2.90/1.59 ! [Y: a,X: a,Y3: a] :
% 2.90/1.59 ( ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ X )
% 2.90/1.59 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 2.90/1.59 => ( ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y3 )
% 2.90/1.59 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( Y = Y3 ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_79_one__unique,axiom,
% 2.90/1.59 ! [U: a] :
% 2.90/1.59 ( ( member @ a @ U @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ! [X3: a] :
% 2.90/1.59 ( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ U @ X3 )
% 2.90/1.59 = X3 ) )
% 2.90/1.59 => ( U
% 2.90/1.59 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_80_add_Oint__pow__distrib,axiom,
% 2.90/1.59 ! [X: a,Y: a,I: int] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add_pow @ a @ b @ int @ r @ I @ ( add @ a @ b @ r @ X @ Y ) )
% 2.90/1.59 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( add_pow @ a @ b @ int @ r @ I @ Y ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_81_add_Oint__pow__mult__distrib,axiom,
% 2.90/1.59 ! [X: a,Y: a,I: int] :
% 2.90/1.59 ( ( ( add @ a @ b @ r @ X @ Y )
% 2.90/1.59 = ( add @ a @ b @ r @ Y @ X ) )
% 2.90/1.59 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add_pow @ a @ b @ int @ r @ I @ ( add @ a @ b @ r @ X @ Y ) )
% 2.90/1.59 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( add_pow @ a @ b @ int @ r @ I @ Y ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_82_semiring__simprules_I4_J,axiom,
% 2.90/1.59 member @ a @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_83_add_Oint__pow__closed,axiom,
% 2.90/1.59 ! [X: a,I: int] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( member @ a @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_84_add_Oint__pow__one,axiom,
% 2.90/1.59 ! [Z2: int] :
% 2.90/1.59 ( ( add_pow @ a @ b @ int @ r @ Z2 @ ( zero @ a @ b @ r ) )
% 2.90/1.59 = ( zero @ a @ b @ r ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_85_l__one,axiom,
% 2.90/1.59 ! [X: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ X )
% 2.90/1.59 = X ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_86_r__one,axiom,
% 2.90/1.59 ! [X: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 2.90/1.59 = X ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_87_semiring_Osemiring__simprules_I4_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 2.90/1.59 ( ( semiring @ A @ B @ R3 )
% 2.90/1.59 => ( member @ A @ ( one @ A @ ( ring_ext @ A @ B ) @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_88_semiring_Osemiring__simprules_I9_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 2.90/1.59 ( ( semiring @ A @ B @ R3 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( one @ A @ ( ring_ext @ A @ B ) @ R3 ) @ X )
% 2.90/1.59 = X ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_89_semiring_Osemiring__simprules_I3_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 2.90/1.59 ( ( semiring @ A @ B @ R3 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( member @ A @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_90_semiring_Osemiring__simprules_I8_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 2.90/1.59 ( ( semiring @ A @ B @ R3 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ Y ) @ Z2 )
% 2.90/1.59 = ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ Y @ Z2 ) ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_91_semiring_Osemiring__simprules_I1_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 2.90/1.59 ( ( semiring @ A @ B @ R3 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( member @ A @ ( add @ A @ B @ R3 @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_92_semiring_Osemiring__simprules_I5_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 2.90/1.59 ( ( semiring @ A @ B @ R3 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( add @ A @ B @ R3 @ ( add @ A @ B @ R3 @ X @ Y ) @ Z2 )
% 2.90/1.59 = ( add @ A @ B @ R3 @ X @ ( add @ A @ B @ R3 @ Y @ Z2 ) ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_93_semiring_Osemiring__simprules_I7_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 2.90/1.59 ( ( semiring @ A @ B @ R3 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( add @ A @ B @ R3 @ X @ Y )
% 2.90/1.59 = ( add @ A @ B @ R3 @ Y @ X ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_94_semiring_Osemiring__simprules_I12_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 2.90/1.59 ( ( semiring @ A @ B @ R3 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( add @ A @ B @ R3 @ X @ ( add @ A @ B @ R3 @ Y @ Z2 ) )
% 2.90/1.59 = ( add @ A @ B @ R3 @ Y @ ( add @ A @ B @ R3 @ X @ Z2 ) ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_95_semiring_Osemiring__simprules_I2_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 2.90/1.59 ( ( semiring @ A @ B @ R3 )
% 2.90/1.59 => ( member @ A @ ( zero @ A @ B @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_96_cring__fieldI2,axiom,
% 2.90/1.59 ( ( ( zero @ a @ b @ r )
% 2.90/1.59 != ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 2.90/1.59 => ( ! [A5: a] :
% 2.90/1.59 ( ( member @ a @ A5 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( A5
% 2.90/1.59 != ( zero @ a @ b @ r ) )
% 2.90/1.59 => ? [X5: a] :
% 2.90/1.59 ( ( member @ a @ X5 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 & ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A5 @ X5 )
% 2.90/1.59 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) )
% 2.90/1.59 => ( field @ a @ b @ r ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_97_add_Oint__pow__mult,axiom,
% 2.90/1.59 ! [X: a,I: int,J: int] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add_pow @ a @ b @ int @ r @ ( plus_plus @ int @ I @ J ) @ X )
% 2.90/1.59 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( add_pow @ a @ b @ int @ r @ J @ X ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_98_add__pow__ldistr,axiom,
% 2.90/1.59 ! [A2: a,B2: a,K: nat] :
% 2.90/1.59 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add_pow @ a @ b @ nat @ r @ K @ A2 ) @ B2 )
% 2.90/1.59 = ( add_pow @ a @ b @ nat @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_99_add__pow__rdistr,axiom,
% 2.90/1.59 ! [A2: a,B2: a,K: nat] :
% 2.90/1.59 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ ( add_pow @ a @ b @ nat @ r @ K @ B2 ) )
% 2.90/1.59 = ( add_pow @ a @ b @ nat @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_100_old_Oprod_Orec,axiom,
% 2.90/1.59 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 2.90/1.59 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.90/1.59 = ( F1 @ A2 @ B2 ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_101_add_Ogroup__commutes__pow,axiom,
% 2.90/1.59 ! [X: a,Y: a,N: nat] :
% 2.90/1.59 ( ( ( add @ a @ b @ r @ X @ Y )
% 2.90/1.59 = ( add @ a @ b @ r @ Y @ X ) )
% 2.90/1.59 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add @ a @ b @ r @ ( add_pow @ a @ b @ nat @ r @ N @ X ) @ Y )
% 2.90/1.59 = ( add @ a @ b @ r @ Y @ ( add_pow @ a @ b @ nat @ r @ N @ X ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_102_add_Opow__mult__distrib,axiom,
% 2.90/1.59 ! [X: a,Y: a,N: nat] :
% 2.90/1.59 ( ( ( add @ a @ b @ r @ X @ Y )
% 2.90/1.59 = ( add @ a @ b @ r @ Y @ X ) )
% 2.90/1.59 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add_pow @ a @ b @ nat @ r @ N @ ( add @ a @ b @ r @ X @ Y ) )
% 2.90/1.59 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ nat @ r @ N @ X ) @ ( add_pow @ a @ b @ nat @ r @ N @ Y ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_103_add_Onat__pow__distrib,axiom,
% 2.90/1.59 ! [X: a,Y: a,N: nat] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add_pow @ a @ b @ nat @ r @ N @ ( add @ a @ b @ r @ X @ Y ) )
% 2.90/1.59 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ nat @ r @ N @ X ) @ ( add_pow @ a @ b @ nat @ r @ N @ Y ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_104_add_Onat__pow__comm,axiom,
% 2.90/1.59 ! [X: a,N: nat,M2: nat] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add @ a @ b @ r @ ( add_pow @ a @ b @ nat @ r @ N @ X ) @ ( add_pow @ a @ b @ nat @ r @ M2 @ X ) )
% 2.90/1.59 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ nat @ r @ M2 @ X ) @ ( add_pow @ a @ b @ nat @ r @ N @ X ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_105_add_Onat__pow__closed,axiom,
% 2.90/1.59 ! [X: a,N: nat] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( member @ a @ ( add_pow @ a @ b @ nat @ r @ N @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_106_add_Onat__pow__one,axiom,
% 2.90/1.59 ! [N: nat] :
% 2.90/1.59 ( ( add_pow @ a @ b @ nat @ r @ N @ ( zero @ a @ b @ r ) )
% 2.90/1.59 = ( zero @ a @ b @ r ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_107_one__not__zero,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 2.90/1.59 ( ( field @ A @ B @ R3 )
% 2.90/1.59 => ( ( one @ A @ ( ring_ext @ A @ B ) @ R3 )
% 2.90/1.59 != ( zero @ A @ B @ R3 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_108_integral,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A] :
% 2.90/1.59 ( ( field @ A @ B @ R3 )
% 2.90/1.59 => ( ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 )
% 2.90/1.59 = ( zero @ A @ B @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( A2
% 2.90/1.59 = ( zero @ A @ B @ R3 ) )
% 2.90/1.59 | ( B2
% 2.90/1.59 = ( zero @ A @ B @ R3 ) ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_109_semiring_Oadd__pow__ldistr,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A,K: nat] :
% 2.90/1.59 ( ( semiring @ A @ B @ R3 )
% 2.90/1.59 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( add_pow @ A @ B @ nat @ R3 @ K @ A2 ) @ B2 )
% 2.90/1.59 = ( add_pow @ A @ B @ nat @ R3 @ K @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_110_semiring_Oadd__pow__rdistr,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A,K: nat] :
% 2.90/1.59 ( ( semiring @ A @ B @ R3 )
% 2.90/1.59 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ ( add_pow @ A @ B @ nat @ R3 @ K @ B2 ) )
% 2.90/1.59 = ( add_pow @ A @ B @ nat @ R3 @ K @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_111_add_Onat__pow__mult,axiom,
% 2.90/1.59 ! [X: a,N: nat,M2: nat] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add @ a @ b @ r @ ( add_pow @ a @ b @ nat @ r @ N @ X ) @ ( add_pow @ a @ b @ nat @ r @ M2 @ X ) )
% 2.90/1.59 = ( add_pow @ a @ b @ nat @ r @ ( plus_plus @ nat @ N @ M2 ) @ X ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_112_add_Onat__pow__Suc2,axiom,
% 2.90/1.59 ! [X: a,N: nat] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add_pow @ a @ b @ nat @ r @ ( suc @ N ) @ X )
% 2.90/1.59 = ( add @ a @ b @ r @ X @ ( add_pow @ a @ b @ nat @ r @ N @ X ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_113_add_Opow__eq__div2,axiom,
% 2.90/1.59 ! [X: a,M2: nat,N: nat] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( ( add_pow @ a @ b @ nat @ r @ M2 @ X )
% 2.90/1.59 = ( add_pow @ a @ b @ nat @ r @ N @ X ) )
% 2.90/1.59 => ( ( add_pow @ a @ b @ nat @ r @ ( minus_minus @ nat @ M2 @ N ) @ X )
% 2.90/1.59 = ( zero @ a @ b @ r ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_114_minus__to__eq,axiom,
% 2.90/1.59 ! [X: a,Y: a] :
% 2.90/1.59 ( ( abelian_group @ a @ b @ r )
% 2.90/1.59 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( ( a_minus @ a @ b @ r @ X @ Y )
% 2.90/1.59 = ( zero @ a @ b @ r ) )
% 2.90/1.59 => ( X = Y ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_115_ring_Oequality,axiom,
% 2.90/1.59 ! [Z: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ),R2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) )] :
% 2.90/1.59 ( ( ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ) @ R )
% 2.90/1.59 = ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ) @ R2 ) )
% 2.90/1.59 => ( ( ( mult @ A @ ( ring_ext @ A @ Z ) @ R )
% 2.90/1.59 = ( mult @ A @ ( ring_ext @ A @ Z ) @ R2 ) )
% 2.90/1.59 => ( ( ( one @ A @ ( ring_ext @ A @ Z ) @ R )
% 2.90/1.59 = ( one @ A @ ( ring_ext @ A @ Z ) @ R2 ) )
% 2.90/1.59 => ( ( ( zero @ A @ Z @ R )
% 2.90/1.59 = ( zero @ A @ Z @ R2 ) )
% 2.90/1.59 => ( ( ( add @ A @ Z @ R )
% 2.90/1.59 = ( add @ A @ Z @ R2 ) )
% 2.90/1.59 => ( ( ( more @ A @ Z @ R )
% 2.90/1.59 = ( more @ A @ Z @ R2 ) )
% 2.90/1.59 => ( R = R2 ) ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_116_is__abelian__group,axiom,
% 2.90/1.59 abelian_group @ a @ b @ r ).
% 2.90/1.59
% 2.90/1.59 thf(fact_117_add_Onat__pow__Suc,axiom,
% 2.90/1.59 ! [N: nat,X: a] :
% 2.90/1.59 ( ( add_pow @ a @ b @ nat @ r @ ( suc @ N ) @ X )
% 2.90/1.59 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ nat @ r @ N @ X ) @ X ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_118_abelian__groupE_I1_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ R3 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( member @ A @ ( add @ A @ B @ R3 @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_119_abelian__groupE_I3_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ R3 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( add @ A @ B @ R3 @ ( add @ A @ B @ R3 @ X @ Y ) @ Z2 )
% 2.90/1.59 = ( add @ A @ B @ R3 @ X @ ( add @ A @ B @ R3 @ Y @ Z2 ) ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_120_abelian__groupE_I4_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ R3 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( add @ A @ B @ R3 @ X @ Y )
% 2.90/1.59 = ( add @ A @ B @ R3 @ Y @ X ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_121_abelian__groupE_I2_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ R3 )
% 2.90/1.59 => ( member @ A @ ( zero @ A @ B @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_122_abelian__group_Ominus__closed,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( member @ A @ ( a_minus @ A @ B @ G4 @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_123_abelian__groupI,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
% 2.90/1.59 ( ! [X3: A] :
% 2.90/1.59 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ! [Y4: A] :
% 2.90/1.59 ( ( member @ A @ Y4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( member @ A @ ( add @ A @ B @ R3 @ X3 @ Y4 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) )
% 2.90/1.59 => ( ( member @ A @ ( zero @ A @ B @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ! [X3: A] :
% 2.90/1.59 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ! [Y4: A] :
% 2.90/1.59 ( ( member @ A @ Y4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ! [Z3: A] :
% 2.90/1.59 ( ( member @ A @ Z3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( add @ A @ B @ R3 @ ( add @ A @ B @ R3 @ X3 @ Y4 ) @ Z3 )
% 2.90/1.59 = ( add @ A @ B @ R3 @ X3 @ ( add @ A @ B @ R3 @ Y4 @ Z3 ) ) ) ) ) )
% 2.90/1.59 => ( ! [X3: A] :
% 2.90/1.59 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ! [Y4: A] :
% 2.90/1.59 ( ( member @ A @ Y4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( add @ A @ B @ R3 @ X3 @ Y4 )
% 2.90/1.59 = ( add @ A @ B @ R3 @ Y4 @ X3 ) ) ) )
% 2.90/1.59 => ( ! [X3: A] :
% 2.90/1.59 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( add @ A @ B @ R3 @ ( zero @ A @ B @ R3 ) @ X3 )
% 2.90/1.59 = X3 ) )
% 2.90/1.59 => ( ! [X3: A] :
% 2.90/1.59 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ? [Xa: A] :
% 2.90/1.59 ( ( member @ A @ Xa @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 & ( ( add @ A @ B @ R3 @ Xa @ X3 )
% 2.90/1.59 = ( zero @ A @ B @ R3 ) ) ) )
% 2.90/1.59 => ( abelian_group @ A @ B @ R3 ) ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_124_abelian__groupE_I5_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ R3 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ( ( add @ A @ B @ R3 @ ( zero @ A @ B @ R3 ) @ X )
% 2.90/1.59 = X ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_125_abelian__groupE_I6_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ R3 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 => ? [X3: A] :
% 2.90/1.59 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
% 2.90/1.59 & ( ( add @ A @ B @ R3 @ X3 @ X )
% 2.90/1.59 = ( zero @ A @ B @ R3 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_126_abelian__group_Ofour__elem__comm,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A,C2: A,D: A] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ C2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ D @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( a_minus @ A @ B @ G4 @ ( add @ A @ B @ G4 @ ( a_minus @ A @ B @ G4 @ A2 @ C2 ) @ B2 ) @ D )
% 2.90/1.59 = ( a_minus @ A @ B @ G4 @ ( a_minus @ A @ B @ G4 @ ( add @ A @ B @ G4 @ A2 @ B2 ) @ C2 ) @ D ) ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_127_abelian__group_Oright__inv__add,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A,C2: A] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ C2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( a_minus @ A @ B @ G4 @ ( a_minus @ A @ B @ G4 @ C2 @ A2 ) @ B2 )
% 2.90/1.59 = ( a_minus @ A @ B @ G4 @ C2 @ ( add @ A @ B @ G4 @ A2 @ B2 ) ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_128_abelian__group_Ominus__to__eq,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ G4 )
% 2.90/1.59 => ( ( abelian_group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( ( a_minus @ A @ B @ G4 @ X @ Y )
% 2.90/1.59 = ( zero @ A @ B @ G4 ) )
% 2.90/1.59 => ( X = Y ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_129_diff__diff__left,axiom,
% 2.90/1.59 ! [I: nat,J: nat,K: nat] :
% 2.90/1.59 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 2.90/1.59 = ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_130_Suc__diff__diff,axiom,
% 2.90/1.59 ! [M2: nat,N: nat,K: nat] :
% 2.90/1.59 ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
% 2.90/1.59 = ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ N ) @ K ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_131_diff__Suc__Suc,axiom,
% 2.90/1.59 ! [M2: nat,N: nat] :
% 2.90/1.59 ( ( minus_minus @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
% 2.90/1.59 = ( minus_minus @ nat @ M2 @ N ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_132_add__Suc__right,axiom,
% 2.90/1.59 ! [M2: nat,N: nat] :
% 2.90/1.59 ( ( plus_plus @ nat @ M2 @ ( suc @ N ) )
% 2.90/1.59 = ( suc @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_133_nat_Oinject,axiom,
% 2.90/1.59 ! [X2: nat,Y2: nat] :
% 2.90/1.59 ( ( ( suc @ X2 )
% 2.90/1.59 = ( suc @ Y2 ) )
% 2.90/1.59 = ( X2 = Y2 ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_134_old_Onat_Oinject,axiom,
% 2.90/1.59 ! [Nat: nat,Nat2: nat] :
% 2.90/1.59 ( ( ( suc @ Nat )
% 2.90/1.59 = ( suc @ Nat2 ) )
% 2.90/1.59 = ( Nat = Nat2 ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_135_Suc__inject,axiom,
% 2.90/1.59 ! [X: nat,Y: nat] :
% 2.90/1.59 ( ( ( suc @ X )
% 2.90/1.59 = ( suc @ Y ) )
% 2.90/1.59 => ( X = Y ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_136_n__not__Suc__n,axiom,
% 2.90/1.59 ! [N: nat] :
% 2.90/1.59 ( N
% 2.90/1.59 != ( suc @ N ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_137_diff__commute,axiom,
% 2.90/1.59 ! [I: nat,J: nat,K: nat] :
% 2.90/1.59 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 2.90/1.59 = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_138_add__Suc,axiom,
% 2.90/1.59 ! [M2: nat,N: nat] :
% 2.90/1.59 ( ( plus_plus @ nat @ ( suc @ M2 ) @ N )
% 2.90/1.59 = ( suc @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_139_nat__arith_Osuc1,axiom,
% 2.90/1.59 ! [A4: nat,K: nat,A2: nat] :
% 2.90/1.59 ( ( A4
% 2.90/1.59 = ( plus_plus @ nat @ K @ A2 ) )
% 2.90/1.59 => ( ( suc @ A4 )
% 2.90/1.59 = ( plus_plus @ nat @ K @ ( suc @ A2 ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_140_add__Suc__shift,axiom,
% 2.90/1.59 ! [M2: nat,N: nat] :
% 2.90/1.59 ( ( plus_plus @ nat @ ( suc @ M2 ) @ N )
% 2.90/1.59 = ( plus_plus @ nat @ M2 @ ( suc @ N ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_141_zero__induct__lemma,axiom,
% 2.90/1.59 ! [P: nat > $o,K: nat,I: nat] :
% 2.90/1.59 ( ( P @ K )
% 2.90/1.59 => ( ! [N2: nat] :
% 2.90/1.59 ( ( P @ ( suc @ N2 ) )
% 2.90/1.59 => ( P @ N2 ) )
% 2.90/1.59 => ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_142_diff__add__inverse2,axiom,
% 2.90/1.59 ! [M2: nat,N: nat] :
% 2.90/1.59 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ N )
% 2.90/1.59 = M2 ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_143_diff__add__inverse,axiom,
% 2.90/1.59 ! [N: nat,M2: nat] :
% 2.90/1.59 ( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M2 ) @ N )
% 2.90/1.59 = M2 ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_144_diff__cancel2,axiom,
% 2.90/1.59 ! [M2: nat,K: nat,N: nat] :
% 2.90/1.59 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) )
% 2.90/1.59 = ( minus_minus @ nat @ M2 @ N ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_145_Nat_Odiff__cancel,axiom,
% 2.90/1.59 ! [K: nat,M2: nat,N: nat] :
% 2.90/1.59 ( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
% 2.90/1.59 = ( minus_minus @ nat @ M2 @ N ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_146_add__diff__cancel__right_H,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( cancel146912293up_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A] :
% 2.90/1.59 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 2.90/1.59 = A2 ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_147_add__diff__cancel__right,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( cancel146912293up_add @ A )
% 2.90/1.59 => ! [A2: A,C2: A,B2: A] :
% 2.90/1.59 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 2.90/1.59 = ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_148_add__diff__cancel__left_H,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( cancel146912293up_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A] :
% 2.90/1.59 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 )
% 2.90/1.59 = B2 ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_149_add__diff__cancel__left,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( cancel146912293up_add @ A )
% 2.90/1.59 => ! [C2: A,A2: A,B2: A] :
% 2.90/1.59 ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 2.90/1.59 = ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_150_add__right__cancel,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( cancel_semigroup_add @ A )
% 2.90/1.59 => ! [B2: A,A2: A,C2: A] :
% 2.90/1.59 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.90/1.59 = ( plus_plus @ A @ C2 @ A2 ) )
% 2.90/1.59 = ( B2 = C2 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_151_add__left__cancel,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( cancel_semigroup_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.90/1.59 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.90/1.59 = ( plus_plus @ A @ A2 @ C2 ) )
% 2.90/1.59 = ( B2 = C2 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_152_add__diff__cancel,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( group_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A] :
% 2.90/1.59 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 2.90/1.59 = A2 ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_153_diff__add__cancel,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( group_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A] :
% 2.90/1.59 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
% 2.90/1.59 = A2 ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_154_add__right__imp__eq,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( cancel_semigroup_add @ A )
% 2.90/1.59 => ! [B2: A,A2: A,C2: A] :
% 2.90/1.59 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.90/1.59 = ( plus_plus @ A @ C2 @ A2 ) )
% 2.90/1.59 => ( B2 = C2 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_155_add__left__imp__eq,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( cancel_semigroup_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.90/1.59 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.90/1.59 = ( plus_plus @ A @ A2 @ C2 ) )
% 2.90/1.59 => ( B2 = C2 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_156_add_Oleft__commute,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( ab_semigroup_add @ A )
% 2.90/1.59 => ! [B2: A,A2: A,C2: A] :
% 2.90/1.59 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
% 2.90/1.59 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_157_add_Ocommute,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( ab_semigroup_add @ A )
% 2.90/1.59 => ( ( plus_plus @ A )
% 2.90/1.59 = ( ^ [A6: A,B5: A] : ( plus_plus @ A @ B5 @ A6 ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_158_group__add__class_Oadd_Oright__cancel,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( group_add @ A )
% 2.90/1.59 => ! [B2: A,A2: A,C2: A] :
% 2.90/1.59 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.90/1.59 = ( plus_plus @ A @ C2 @ A2 ) )
% 2.90/1.59 = ( B2 = C2 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_159_add_Oleft__cancel,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( group_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.90/1.59 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.90/1.59 = ( plus_plus @ A @ A2 @ C2 ) )
% 2.90/1.59 = ( B2 = C2 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_160_add_Oassoc,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( semigroup_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.90/1.59 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 2.90/1.59 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_161_group__cancel_Oadd2,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( comm_monoid_add @ A )
% 2.90/1.59 => ! [B6: A,K: A,B2: A,A2: A] :
% 2.90/1.59 ( ( B6
% 2.90/1.59 = ( plus_plus @ A @ K @ B2 ) )
% 2.90/1.59 => ( ( plus_plus @ A @ A2 @ B6 )
% 2.90/1.59 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_162_group__cancel_Oadd1,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( comm_monoid_add @ A )
% 2.90/1.59 => ! [A4: A,K: A,A2: A,B2: A] :
% 2.90/1.59 ( ( A4
% 2.90/1.59 = ( plus_plus @ A @ K @ A2 ) )
% 2.90/1.59 => ( ( plus_plus @ A @ A4 @ B2 )
% 2.90/1.59 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_163_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( ordere779506340up_add @ A )
% 2.90/1.59 => ! [I: A,J: A,K: A,L: A] :
% 2.90/1.59 ( ( ( I = J )
% 2.90/1.59 & ( K = L ) )
% 2.90/1.59 => ( ( plus_plus @ A @ I @ K )
% 2.90/1.59 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_164_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( ab_semigroup_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.90/1.59 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 2.90/1.59 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_165_diff__right__commute,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( cancel146912293up_add @ A )
% 2.90/1.59 => ! [A2: A,C2: A,B2: A] :
% 2.90/1.59 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 )
% 2.90/1.59 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_166_diff__eq__diff__eq,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( group_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A,C2: A,D: A] :
% 2.90/1.59 ( ( ( minus_minus @ A @ A2 @ B2 )
% 2.90/1.59 = ( minus_minus @ A @ C2 @ D ) )
% 2.90/1.59 => ( ( A2 = B2 )
% 2.90/1.59 = ( C2 = D ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_167_group__cancel_Osub1,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( ab_group_add @ A )
% 2.90/1.59 => ! [A4: A,K: A,A2: A,B2: A] :
% 2.90/1.59 ( ( A4
% 2.90/1.59 = ( plus_plus @ A @ K @ A2 ) )
% 2.90/1.59 => ( ( minus_minus @ A @ A4 @ B2 )
% 2.90/1.59 = ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_168_diff__eq__eq,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( group_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.90/1.59 ( ( ( minus_minus @ A @ A2 @ B2 )
% 2.90/1.59 = C2 )
% 2.90/1.59 = ( A2
% 2.90/1.59 = ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_169_eq__diff__eq,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( group_add @ A )
% 2.90/1.59 => ! [A2: A,C2: A,B2: A] :
% 2.90/1.59 ( ( A2
% 2.90/1.59 = ( minus_minus @ A @ C2 @ B2 ) )
% 2.90/1.59 = ( ( plus_plus @ A @ A2 @ B2 )
% 2.90/1.59 = C2 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_170_add__diff__eq,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( group_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.90/1.59 ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
% 2.90/1.59 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_171_diff__diff__eq2,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( group_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.90/1.59 ( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
% 2.90/1.59 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_172_diff__add__eq,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( ab_group_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.90/1.59 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
% 2.90/1.59 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_173_diff__add__eq__diff__diff__swap,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( group_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.90/1.59 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
% 2.90/1.59 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_174_diff__diff__add,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( cancel146912293up_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A,C2: A] :
% 2.90/1.59 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
% 2.90/1.59 = ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_175_add__implies__diff,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( cancel1352612707id_add @ A )
% 2.90/1.59 => ! [C2: A,B2: A,A2: A] :
% 2.90/1.59 ( ( ( plus_plus @ A @ C2 @ B2 )
% 2.90/1.59 = A2 )
% 2.90/1.59 => ( C2
% 2.90/1.59 = ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_176_Units__l__inv__ex,axiom,
% 2.90/1.59 ! [X: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 2.90/1.59 => ? [X3: a] :
% 2.90/1.59 ( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 & ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X3 @ X )
% 2.90/1.59 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_177_Units__r__inv__ex,axiom,
% 2.90/1.59 ! [X: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 2.90/1.59 => ? [X3: a] :
% 2.90/1.59 ( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 & ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ X3 )
% 2.90/1.59 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_178_group__l__invI,axiom,
% 2.90/1.59 ( ! [X3: a] :
% 2.90/1.59 ( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ? [Xa: a] :
% 2.90/1.59 ( ( member @ a @ Xa @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 & ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Xa @ X3 )
% 2.90/1.59 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) )
% 2.90/1.59 => ( group @ a @ ( ring_ext @ a @ b ) @ r ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_179_Units__closed,axiom,
% 2.90/1.59 ! [X: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 2.90/1.59 => ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_180_Units__inv__comm,axiom,
% 2.90/1.59 ! [X: a,Y: a] :
% 2.90/1.59 ( ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y )
% 2.90/1.59 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 2.90/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ X )
% 2.90/1.59 = ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_181_Units__m__closed,axiom,
% 2.90/1.59 ! [X: a,Y: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 2.90/1.59 => ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_182_Units__one__closed,axiom,
% 2.90/1.59 member @ a @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_183_Units__l__cancel,axiom,
% 2.90/1.59 ! [X: a,Y: a,Z2: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y )
% 2.90/1.59 = ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Z2 ) )
% 2.90/1.59 = ( Y = Z2 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_184_groupI,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 2.90/1.59 ( ! [X3: A] :
% 2.90/1.59 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ! [Y4: A] :
% 2.90/1.59 ( ( member @ A @ Y4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( member @ A @ ( mult @ A @ B @ G4 @ X3 @ Y4 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) ) ) )
% 2.90/1.59 => ( ( member @ A @ ( one @ A @ B @ G4 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ! [X3: A] :
% 2.90/1.59 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ! [Y4: A] :
% 2.90/1.59 ( ( member @ A @ Y4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ! [Z3: A] :
% 2.90/1.59 ( ( member @ A @ Z3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( mult @ A @ B @ G4 @ ( mult @ A @ B @ G4 @ X3 @ Y4 ) @ Z3 )
% 2.90/1.59 = ( mult @ A @ B @ G4 @ X3 @ ( mult @ A @ B @ G4 @ Y4 @ Z3 ) ) ) ) ) )
% 2.90/1.59 => ( ! [X3: A] :
% 2.90/1.59 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( mult @ A @ B @ G4 @ ( one @ A @ B @ G4 ) @ X3 )
% 2.90/1.59 = X3 ) )
% 2.90/1.59 => ( ! [X3: A] :
% 2.90/1.59 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ? [Xa: A] :
% 2.90/1.59 ( ( member @ A @ Xa @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 & ( ( mult @ A @ B @ G4 @ Xa @ X3 )
% 2.90/1.59 = ( one @ A @ B @ G4 ) ) ) )
% 2.90/1.59 => ( group @ A @ B @ G4 ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_185_group_Oinv__comm,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A] :
% 2.90/1.59 ( ( group @ A @ B @ G4 )
% 2.90/1.59 => ( ( ( mult @ A @ B @ G4 @ X @ Y )
% 2.90/1.59 = ( one @ A @ B @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( mult @ A @ B @ G4 @ Y @ X )
% 2.90/1.59 = ( one @ A @ B @ G4 ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_186_group_Ol__inv__ex,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
% 2.90/1.59 ( ( group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ? [X3: A] :
% 2.90/1.59 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 & ( ( mult @ A @ B @ G4 @ X3 @ X )
% 2.90/1.59 = ( one @ A @ B @ G4 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_187_Group_Ogroup_Oright__cancel,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A,Z2: A] :
% 2.90/1.59 ( ( group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( ( mult @ A @ B @ G4 @ Y @ X )
% 2.90/1.59 = ( mult @ A @ B @ G4 @ Z2 @ X ) )
% 2.90/1.59 = ( Y = Z2 ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_188_group_OUnits__eq,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 2.90/1.59 ( ( group @ A @ B @ G4 )
% 2.90/1.59 => ( ( units @ A @ B @ G4 )
% 2.90/1.59 = ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_189_group_Or__cancel__one_H,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,A2: A] :
% 2.90/1.59 ( ( group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( X
% 2.90/1.59 = ( mult @ A @ B @ G4 @ A2 @ X ) )
% 2.90/1.59 = ( A2
% 2.90/1.59 = ( one @ A @ B @ G4 ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_190_group_Ol__cancel__one_H,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,A2: A] :
% 2.90/1.59 ( ( group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( X
% 2.90/1.59 = ( mult @ A @ B @ G4 @ X @ A2 ) )
% 2.90/1.59 = ( A2
% 2.90/1.59 = ( one @ A @ B @ G4 ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_191_group_Or__cancel__one,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,A2: A] :
% 2.90/1.59 ( ( group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( ( mult @ A @ B @ G4 @ A2 @ X )
% 2.90/1.59 = X )
% 2.90/1.59 = ( A2
% 2.90/1.59 = ( one @ A @ B @ G4 ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_192_group_Ol__cancel__one,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,A2: A] :
% 2.90/1.59 ( ( group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( ( mult @ A @ B @ G4 @ X @ A2 )
% 2.90/1.59 = X )
% 2.90/1.59 = ( A2
% 2.90/1.59 = ( one @ A @ B @ G4 ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_193_group_Or__inv__ex,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
% 2.90/1.59 ( ( group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ? [X3: A] :
% 2.90/1.59 ( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 & ( ( mult @ A @ B @ G4 @ X @ X3 )
% 2.90/1.59 = ( one @ A @ B @ G4 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_194_units__group,axiom,
% 2.90/1.59 group @ a @ product_unit @ ( units_of @ a @ ( ring_ext @ a @ b ) @ r ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_195_inv__add,axiom,
% 2.90/1.59 ! [A2: a,B2: a] :
% 2.90/1.59 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( a_inv @ a @ b @ r @ ( add @ a @ b @ r @ A2 @ B2 ) )
% 2.90/1.59 = ( a_minus @ a @ b @ r @ ( a_inv @ a @ b @ r @ A2 ) @ B2 ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_196_r__neg2,axiom,
% 2.90/1.59 ! [X: a,Y: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add @ a @ b @ r @ X @ ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ Y ) )
% 2.90/1.59 = Y ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_197_r__neg1,axiom,
% 2.90/1.59 ! [X: a,Y: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ ( add @ a @ b @ r @ X @ Y ) )
% 2.90/1.59 = Y ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_198_local_Ominus__add,axiom,
% 2.90/1.59 ! [X: a,Y: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( a_inv @ a @ b @ r @ ( add @ a @ b @ r @ X @ Y ) )
% 2.90/1.59 = ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ ( a_inv @ a @ b @ r @ Y ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_199_add_Oinv__solve__right_H,axiom,
% 2.90/1.59 ! [A2: a,B2: a,C2: a] :
% 2.90/1.59 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( ( add @ a @ b @ r @ B2 @ ( a_inv @ a @ b @ r @ C2 ) )
% 2.90/1.59 = A2 )
% 2.90/1.59 = ( B2
% 2.90/1.59 = ( add @ a @ b @ r @ A2 @ C2 ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_200_add_Oinv__solve__right,axiom,
% 2.90/1.59 ! [A2: a,B2: a,C2: a] :
% 2.90/1.59 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( A2
% 2.90/1.59 = ( add @ a @ b @ r @ B2 @ ( a_inv @ a @ b @ r @ C2 ) ) )
% 2.90/1.59 = ( B2
% 2.90/1.59 = ( add @ a @ b @ r @ A2 @ C2 ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_201_add_Oinv__solve__left_H,axiom,
% 2.90/1.59 ! [A2: a,B2: a,C2: a] :
% 2.90/1.59 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ B2 ) @ C2 )
% 2.90/1.59 = A2 )
% 2.90/1.59 = ( C2
% 2.90/1.59 = ( add @ a @ b @ r @ B2 @ A2 ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_202_add_Oinv__solve__left,axiom,
% 2.90/1.59 ! [A2: a,B2: a,C2: a] :
% 2.90/1.59 ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( A2
% 2.90/1.59 = ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ B2 ) @ C2 ) )
% 2.90/1.59 = ( C2
% 2.90/1.59 = ( add @ a @ b @ r @ B2 @ A2 ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_203_add_Oinv__mult__group,axiom,
% 2.90/1.59 ! [X: a,Y: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( a_inv @ a @ b @ r @ ( add @ a @ b @ r @ X @ Y ) )
% 2.90/1.59 = ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ Y ) @ ( a_inv @ a @ b @ r @ X ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_204_a__transpose__inv,axiom,
% 2.90/1.59 ! [X: a,Y: a,Z2: a] :
% 2.90/1.59 ( ( ( add @ a @ b @ r @ X @ Y )
% 2.90/1.59 = Z2 )
% 2.90/1.59 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ Z2 )
% 2.90/1.59 = Y ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_205_r__minus,axiom,
% 2.90/1.59 ! [X: a,Y: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( a_inv @ a @ b @ r @ Y ) )
% 2.90/1.59 = ( a_inv @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_206_l__minus,axiom,
% 2.90/1.59 ! [X: a,Y: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( a_inv @ a @ b @ r @ X ) @ Y )
% 2.90/1.59 = ( a_inv @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_207_add_Onat__pow__inv,axiom,
% 2.90/1.59 ! [X: a,I: nat] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add_pow @ a @ b @ nat @ r @ I @ ( a_inv @ a @ b @ r @ X ) )
% 2.90/1.59 = ( a_inv @ a @ b @ r @ ( add_pow @ a @ b @ nat @ r @ I @ X ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_208_add_Oint__pow__inv,axiom,
% 2.90/1.59 ! [X: a,I: int] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add_pow @ a @ b @ int @ r @ I @ ( a_inv @ a @ b @ r @ X ) )
% 2.90/1.59 = ( a_inv @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_209_minus__eq,axiom,
% 2.90/1.59 ! [X: a,Y: a] :
% 2.90/1.59 ( ( a_minus @ a @ b @ r @ X @ Y )
% 2.90/1.59 = ( add @ a @ b @ r @ X @ ( a_inv @ a @ b @ r @ Y ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_210_sum__zero__eq__neg,axiom,
% 2.90/1.59 ! [X: a,Y: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( ( add @ a @ b @ r @ X @ Y )
% 2.90/1.59 = ( zero @ a @ b @ r ) )
% 2.90/1.59 => ( X
% 2.90/1.59 = ( a_inv @ a @ b @ r @ Y ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_211_r__neg,axiom,
% 2.90/1.59 ! [X: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add @ a @ b @ r @ X @ ( a_inv @ a @ b @ r @ X ) )
% 2.90/1.59 = ( zero @ a @ b @ r ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_212_minus__equality,axiom,
% 2.90/1.59 ! [Y: a,X: a] :
% 2.90/1.59 ( ( ( add @ a @ b @ r @ Y @ X )
% 2.90/1.59 = ( zero @ a @ b @ r ) )
% 2.90/1.59 => ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( a_inv @ a @ b @ r @ X )
% 2.90/1.59 = Y ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_213_l__neg,axiom,
% 2.90/1.59 ! [X: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ X )
% 2.90/1.59 = ( zero @ a @ b @ r ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_214_add_Oint__pow__diff,axiom,
% 2.90/1.59 ! [X: a,N: int,M2: int] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add_pow @ a @ b @ int @ r @ ( minus_minus @ int @ N @ M2 ) @ X )
% 2.90/1.59 = ( add @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ N @ X ) @ ( a_inv @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ M2 @ X ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_215_local_Ominus__minus,axiom,
% 2.90/1.59 ! [X: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( a_inv @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) )
% 2.90/1.59 = X ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_216_add_Oinv__closed,axiom,
% 2.90/1.59 ! [X: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( member @ a @ ( a_inv @ a @ b @ r @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_217_local_Ominus__zero,axiom,
% 2.90/1.59 ( ( a_inv @ a @ b @ r @ ( zero @ a @ b @ r ) )
% 2.90/1.59 = ( zero @ a @ b @ r ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_218_add_Oinv__eq__1__iff,axiom,
% 2.90/1.59 ! [X: a] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( ( a_inv @ a @ b @ r @ X )
% 2.90/1.59 = ( zero @ a @ b @ r ) )
% 2.90/1.59 = ( X
% 2.90/1.59 = ( zero @ a @ b @ r ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_219_Units__minus__one__closed,axiom,
% 2.90/1.59 member @ a @ ( a_inv @ a @ b @ r @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_220_units__of__mult,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 2.90/1.59 ( ( mult @ A @ product_unit @ ( units_of @ A @ B @ G4 ) )
% 2.90/1.59 = ( mult @ A @ B @ G4 ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_221_abelian__group_Oa__inv__closed,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( member @ A @ ( a_inv @ A @ B @ G4 @ X ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_222_abelian__group_Ominus__minus,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( a_inv @ A @ B @ G4 @ ( a_inv @ A @ B @ G4 @ X ) )
% 2.90/1.59 = X ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_223_a__minus__def,axiom,
% 2.90/1.59 ! [M: $tType,A: $tType] :
% 2.90/1.59 ( ( a_minus @ A @ M )
% 2.90/1.59 = ( ^ [R4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ M ) ),X4: A,Y5: A] : ( add @ A @ M @ R4 @ X4 @ ( a_inv @ A @ M @ R4 @ Y5 ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_224_units__of__one,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 2.90/1.59 ( ( one @ A @ product_unit @ ( units_of @ A @ B @ G4 ) )
% 2.90/1.59 = ( one @ A @ B @ G4 ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_225_abelian__group_Ominus__add,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( a_inv @ A @ B @ G4 @ ( add @ A @ B @ G4 @ X @ Y ) )
% 2.90/1.59 = ( add @ A @ B @ G4 @ ( a_inv @ A @ B @ G4 @ X ) @ ( a_inv @ A @ B @ G4 @ Y ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_226_abelian__group_Or__neg2,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( add @ A @ B @ G4 @ X @ ( add @ A @ B @ G4 @ ( a_inv @ A @ B @ G4 @ X ) @ Y ) )
% 2.90/1.59 = Y ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_227_abelian__group_Or__neg1,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( add @ A @ B @ G4 @ ( a_inv @ A @ B @ G4 @ X ) @ ( add @ A @ B @ G4 @ X @ Y ) )
% 2.90/1.59 = Y ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_228_abelian__group_Ominus__eq,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ G4 )
% 2.90/1.59 => ( ( a_minus @ A @ B @ G4 @ X @ Y )
% 2.90/1.59 = ( add @ A @ B @ G4 @ X @ ( a_inv @ A @ B @ G4 @ Y ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_229_abelian__group_Ol__neg,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( add @ A @ B @ G4 @ ( a_inv @ A @ B @ G4 @ X ) @ X )
% 2.90/1.59 = ( zero @ A @ B @ G4 ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_230_abelian__group_Or__neg,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( add @ A @ B @ G4 @ X @ ( a_inv @ A @ B @ G4 @ X ) )
% 2.90/1.59 = ( zero @ A @ B @ G4 ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_231_abelian__group_Ominus__equality,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),Y: A,X: A] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ G4 )
% 2.90/1.59 => ( ( ( add @ A @ B @ G4 @ Y @ X )
% 2.90/1.59 = ( zero @ A @ B @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( a_inv @ A @ B @ G4 @ X )
% 2.90/1.59 = Y ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_232_abelian__group_Oinv__add,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( a_inv @ A @ B @ G4 @ ( add @ A @ B @ G4 @ A2 @ B2 ) )
% 2.90/1.59 = ( a_minus @ A @ B @ G4 @ ( a_inv @ A @ B @ G4 @ A2 ) @ B2 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_233_abelian__group_Oa__transpose__inv,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
% 2.90/1.59 ( ( abelian_group @ A @ B @ G4 )
% 2.90/1.59 => ( ( ( add @ A @ B @ G4 @ X @ Y )
% 2.90/1.59 = Z2 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G4 ) )
% 2.90/1.59 => ( ( add @ A @ B @ G4 @ ( a_inv @ A @ B @ G4 @ X ) @ Z2 )
% 2.90/1.59 = Y ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_234_add_Oint__pow__neg,axiom,
% 2.90/1.59 ! [X: a,I: int] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add_pow @ a @ b @ int @ r @ ( uminus_uminus @ int @ I ) @ X )
% 2.90/1.59 = ( a_inv @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_235_add__minus__cancel,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( group_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A] :
% 2.90/1.59 ( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) )
% 2.90/1.59 = B2 ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_236_minus__add__cancel,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( group_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A] :
% 2.90/1.59 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) )
% 2.90/1.59 = B2 ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_237_minus__add__distrib,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( ab_group_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A] :
% 2.90/1.59 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
% 2.90/1.59 = ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_238_minus__diff__eq,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( group_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A] :
% 2.90/1.59 ( ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B2 ) )
% 2.90/1.59 = ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_239_diff__minus__eq__add,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( group_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A] :
% 2.90/1.59 ( ( minus_minus @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
% 2.90/1.59 = ( plus_plus @ A @ A2 @ B2 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_240_uminus__add__conv__diff,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( ab_group_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A] :
% 2.90/1.59 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
% 2.90/1.59 = ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_241_group__cancel_Osub2,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( ab_group_add @ A )
% 2.90/1.59 => ! [B6: A,K: A,B2: A,A2: A] :
% 2.90/1.59 ( ( B6
% 2.90/1.59 = ( plus_plus @ A @ K @ B2 ) )
% 2.90/1.59 => ( ( minus_minus @ A @ A2 @ B6 )
% 2.90/1.59 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_242_diff__conv__add__uminus,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( group_add @ A )
% 2.90/1.59 => ( ( minus_minus @ A )
% 2.90/1.59 = ( ^ [A6: A,B5: A] : ( plus_plus @ A @ A6 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_243_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( ab_group_add @ A )
% 2.90/1.59 => ( ( minus_minus @ A )
% 2.90/1.59 = ( ^ [A6: A,B5: A] : ( plus_plus @ A @ A6 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_244_group__cancel_Oneg1,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( ab_group_add @ A )
% 2.90/1.59 => ! [A4: A,K: A,A2: A] :
% 2.90/1.59 ( ( A4
% 2.90/1.59 = ( plus_plus @ A @ K @ A2 ) )
% 2.90/1.59 => ( ( uminus_uminus @ A @ A4 )
% 2.90/1.59 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_245_add_Oinverse__distrib__swap,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( group_add @ A )
% 2.90/1.59 => ! [A2: A,B2: A] :
% 2.90/1.59 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
% 2.90/1.59 = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_246_minus__diff__commute,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( ab_group_add @ A )
% 2.90/1.59 => ! [B2: A,A2: A] :
% 2.90/1.59 ( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A2 )
% 2.90/1.59 = ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_247_add_Oint__pow__neg__int,axiom,
% 2.90/1.59 ! [X: a,N: nat] :
% 2.90/1.59 ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
% 2.90/1.59 => ( ( add_pow @ a @ b @ int @ r @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) @ X )
% 2.90/1.59 = ( a_inv @ a @ b @ r @ ( add_pow @ a @ b @ nat @ r @ N @ X ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_248_units__comm__group,axiom,
% 2.90/1.59 comm_group @ a @ product_unit @ ( units_of @ a @ ( ring_ext @ a @ b ) @ r ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_249_of__nat__eq__iff,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( semiring_char_0 @ A )
% 2.90/1.59 => ! [M2: nat,N: nat] :
% 2.90/1.59 ( ( ( semiring_1_of_nat @ A @ M2 )
% 2.90/1.59 = ( semiring_1_of_nat @ A @ N ) )
% 2.90/1.59 = ( M2 = N ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_250_of__nat__add,axiom,
% 2.90/1.59 ! [A: $tType] :
% 2.90/1.59 ( ( semiring_1 @ A )
% 2.90/1.59 => ! [M2: nat,N: nat] :
% 2.90/1.59 ( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M2 @ N ) )
% 2.90/1.59 = ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_251_comm__groupE_I4_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A] :
% 2.90/1.59 ( ( comm_group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( mult @ A @ B @ G4 @ X @ Y )
% 2.90/1.59 = ( mult @ A @ B @ G4 @ Y @ X ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_252_comm__groupE_I3_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A,Z2: A] :
% 2.90/1.59 ( ( comm_group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( mult @ A @ B @ G4 @ ( mult @ A @ B @ G4 @ X @ Y ) @ Z2 )
% 2.90/1.59 = ( mult @ A @ B @ G4 @ X @ ( mult @ A @ B @ G4 @ Y @ Z2 ) ) ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_253_comm__groupE_I1_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A] :
% 2.90/1.59 ( ( comm_group @ A @ B @ G4 )
% 2.90/1.59 => ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) )
% 2.90/1.59 => ( member @ A @ ( mult @ A @ B @ G4 @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) ) ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(fact_254_comm__groupE_I2_J,axiom,
% 2.90/1.59 ! [B: $tType,A: $tType,G4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
% 2.90/1.59 ( ( comm_group @ A @ B @ G4 )
% 2.90/1.59 => ( member @ A @ ( one @ A @ B @ G4 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G4 ) ) ) ).
% 2.90/1.59
% 2.90/1.59 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
% 2.90/1.59 ordere779506340up_add @ int ).
% 2.90/1.59
% 2.90/1.59 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
% 2.90/1.59 cancel146912293up_add @ int ).
% 2.90/1.59
% 2.90/1.59 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 2.90/1.59 cancel1352612707id_add @ int ).
% 2.90/1.59
% 2.90/1.59 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
% 2.90/1.59 cancel_semigroup_add @ int ).
% 2.90/1.59
% 2.90/1.59 thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
% 2.90/1.59 ab_semigroup_add @ int ).
% 2.90/1.59
% 2.90/1.59 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
% 2.90/1.59 comm_monoid_add @ int ).
% 2.90/1.59
% 2.90/1.59 thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
% 2.90/1.59 semigroup_add @ int ).
% 2.90/1.59
% 2.90/1.59 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 2.90/1.59 semiring_char_0 @ int ).
% 2.90/1.59
% 2.90/1.59 thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
% 2.90/1.59 ab_group_add @ int ).
% 2.90/1.59
% 2.90/1.59 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 2.90/1.59 semiring_1 @ int ).
% 2.90/1.59
% 2.90/1.59 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 2.90/1.59 group_add @ int ).
% 2.90/1.59
% 2.90/1.59 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_1,axiom,
% 2.90/1.59 ordere779506340up_add @ nat ).
% 2.90/1.59
% 2.90/1.59 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_2,axiom,
% 2.90/1.59 cancel146912293up_add @ nat ).
% 2.90/1.59
% 2.90/1.59 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_3,axiom,
% 2.90/1.59 cancel1352612707id_add @ nat ).
% 2.90/1.59
% 2.90/1.59 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_4,axiom,
% 2.90/1.59 cancel_semigroup_add @ nat ).
% 2.90/1.59
% 2.90/1.59 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_5,axiom,
% 2.90/1.59 ab_semigroup_add @ nat ).
% 2.90/1.59
% 2.90/1.59 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_6,axiom,
% 2.90/1.59 comm_monoid_add @ nat ).
% 2.90/1.59
% 2.90/1.59 thf(tcon_Nat_Onat___Groups_Osemigroup__add_7,axiom,
% 2.90/1.59 semigroup_add @ nat ).
% 2.90/1.59
% 2.90/1.59 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_8,axiom,
% 2.90/1.59 semiring_char_0 @ nat ).
% 2.90/1.59
% 2.90/1.59 thf(tcon_Nat_Onat___Rings_Osemiring__1_9,axiom,
% 2.90/1.59 semiring_1 @ nat ).
% 2.90/1.59
% 2.90/1.59 thf(conj_0,conjecture,
% 2.90/1.59 member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r4 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r4 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ).
% 2.90/1.59 ------------------------
% 3.05/1.62 % (19704)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_19374 for (2999ds/4Mi)
% 3.05/1.62 % (19707)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_19374 for (2999ds/2Mi)
% 3.05/1.62 % (19706)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_19374 for (2999ds/2Mi)
% 3.05/1.62 % (19704)Instruction limit reached!
% 3.05/1.62 % (19704)------------------------------
% 3.05/1.62 % (19704)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.62 % (19704)Termination reason: Unknown
% 3.05/1.62 % (19704)Termination phase: shuffling
% 3.05/1.62
% 3.05/1.62 % (19704)Memory used [KB]: 1407
% 3.05/1.62 % (19704)Time elapsed: 0.003 s
% 3.05/1.62 % (19704)Instructions burned: 6 (million)
% 3.05/1.62 % (19704)------------------------------
% 3.05/1.62 % (19704)------------------------------
% 3.05/1.62 % (19706)Instruction limit reached!
% 3.05/1.62 % (19706)------------------------------
% 3.05/1.62 % (19706)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.62 % (19706)Termination reason: Unknown
% 3.05/1.62 % (19706)Termination phase: shuffling
% 3.05/1.62
% 3.05/1.62 % (19706)Memory used [KB]: 1279
% 3.05/1.62 % (19706)Time elapsed: 0.002 s
% 3.05/1.62 % (19706)Instructions burned: 2 (million)
% 3.05/1.62 % (19706)------------------------------
% 3.05/1.62 % (19706)------------------------------
% 3.05/1.62 % (19707)Instruction limit reached!
% 3.05/1.62 % (19707)------------------------------
% 3.05/1.62 % (19707)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.62 % (19707)Termination reason: Unknown
% 3.05/1.62 % (19707)Termination phase: shuffling
% 3.05/1.62
% 3.05/1.62 % (19707)Memory used [KB]: 1407
% 3.05/1.62 % (19707)Time elapsed: 0.002 s
% 3.05/1.62 % (19707)Instructions burned: 3 (million)
% 3.05/1.62 % (19707)------------------------------
% 3.05/1.62 % (19707)------------------------------
% 3.05/1.63 % (19703)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_19374 for (2999ds/183Mi)
% 3.05/1.63 % (19705)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_19374 for (2999ds/27Mi)
% 3.05/1.63 % (19708)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_19374 for (2999ds/275Mi)
% 3.05/1.63 % (19709)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_19374 for (2999ds/18Mi)
% 3.05/1.63 % (19710)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_19374 for (2999ds/3Mi)
% 3.05/1.63 % (19711)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_19374 for (2999ds/37Mi)
% 3.05/1.63 % (19712)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_19374 for (2999ds/15Mi)
% 3.05/1.63 % (19710)Instruction limit reached!
% 3.05/1.63 % (19710)------------------------------
% 3.05/1.63 % (19710)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.63 % (19710)Termination reason: Unknown
% 3.05/1.63 % (19710)Termination phase: shuffling
% 3.05/1.63
% 3.05/1.63 % (19710)Memory used [KB]: 1407
% 3.05/1.63 % (19710)Time elapsed: 0.003 s
% 3.05/1.63 % (19710)Instructions burned: 5 (million)
% 3.05/1.63 % (19710)------------------------------
% 3.05/1.63 % (19710)------------------------------
% 3.05/1.64 % (19712)Instruction limit reached!
% 3.05/1.64 % (19712)------------------------------
% 3.05/1.64 % (19712)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.64 % (19712)Termination reason: Unknown
% 3.05/1.64 % (19712)Termination phase: shuffling
% 3.05/1.64
% 3.05/1.64 % (19712)Memory used [KB]: 1535
% 3.05/1.64 % (19712)Time elapsed: 0.007 s
% 3.05/1.64 % (19712)Instructions burned: 17 (million)
% 3.05/1.64 % (19712)------------------------------
% 3.05/1.64 % (19712)------------------------------
% 3.05/1.64 % (19709)Instruction limit reached!
% 3.05/1.64 % (19709)------------------------------
% 3.05/1.64 % (19709)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.64 % (19709)Termination reason: Unknown
% 3.05/1.64 % (19709)Termination phase: shuffling
% 3.05/1.64
% 3.05/1.64 % (19709)Memory used [KB]: 1535
% 3.05/1.64 % (19709)Time elapsed: 0.014 s
% 3.05/1.64 % (19709)Instructions burned: 19 (million)
% 3.05/1.64 % (19709)------------------------------
% 3.05/1.64 % (19709)------------------------------
% 3.05/1.64 % (19713)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_19374 for (2999ds/3Mi)
% 3.05/1.64 % (19711)Instruction limit reached!
% 3.05/1.64 % (19711)------------------------------
% 3.05/1.64 % (19711)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.64 % (19711)Termination reason: Unknown
% 3.05/1.64 % (19711)Termination phase: shuffling
% 3.05/1.64
% 3.05/1.64 % (19711)Memory used [KB]: 2046
% 3.05/1.64 % (19711)Time elapsed: 0.013 s
% 3.05/1.64 % (19711)Instructions burned: 38 (million)
% 3.05/1.64 % (19711)------------------------------
% 3.05/1.64 % (19711)------------------------------
% 3.05/1.64 % (19713)Instruction limit reached!
% 3.05/1.64 % (19713)------------------------------
% 3.05/1.64 % (19713)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.64 % (19713)Termination reason: Unknown
% 3.05/1.64 % (19713)Termination phase: shuffling
% 3.05/1.64
% 3.05/1.64 % (19713)Memory used [KB]: 1407
% 3.05/1.64 % (19713)Time elapsed: 0.002 s
% 3.05/1.64 % (19713)Instructions burned: 3 (million)
% 3.05/1.64 % (19713)------------------------------
% 3.05/1.64 % (19713)------------------------------
% 3.05/1.65 % (19705)Instruction limit reached!
% 3.05/1.65 % (19705)------------------------------
% 3.05/1.65 % (19705)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.65 % (19705)Termination reason: Unknown
% 3.05/1.65 % (19705)Termination phase: shuffling
% 3.05/1.65
% 3.05/1.65 % (19705)Memory used [KB]: 1663
% 3.05/1.65 % (19705)Time elapsed: 0.019 s
% 3.05/1.65 % (19705)Instructions burned: 27 (million)
% 3.05/1.65 % (19705)------------------------------
% 3.05/1.65 % (19705)------------------------------
% 3.05/1.65 % (19714)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_19374 for (2999ds/1041Mi)
% 3.05/1.65 % (19717)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_19374 for (2999ds/3Mi)
% 3.05/1.65 % (19716)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_19374 for (2999ds/16Mi)
% 3.05/1.66 % (19717)Instruction limit reached!
% 3.05/1.66 % (19717)------------------------------
% 3.05/1.66 % (19717)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.66 % (19717)Termination reason: Unknown
% 3.05/1.66 % (19717)Termination phase: shuffling
% 3.05/1.66
% 3.05/1.66 % (19717)Memory used [KB]: 1407
% 3.05/1.66 % (19717)Time elapsed: 0.003 s
% 3.05/1.66 % (19717)Instructions burned: 5 (million)
% 3.05/1.66 % (19717)------------------------------
% 3.05/1.66 % (19717)------------------------------
% 3.05/1.66 % (19715)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_19374 for (2999ds/7Mi)
% 3.05/1.66 % (19716)Instruction limit reached!
% 3.05/1.66 % (19716)------------------------------
% 3.05/1.66 % (19716)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.66 % (19716)Termination reason: Unknown
% 3.05/1.66 % (19716)Termination phase: shuffling
% 3.05/1.66
% 3.05/1.66 % (19716)Memory used [KB]: 1535
% 3.05/1.66 % (19716)Time elapsed: 0.008 s
% 3.05/1.66 % (19716)Instructions burned: 18 (million)
% 3.05/1.66 % (19716)------------------------------
% 3.05/1.66 % (19716)------------------------------
% 3.05/1.66 % (19715)Instruction limit reached!
% 3.05/1.66 % (19715)------------------------------
% 3.05/1.66 % (19715)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.66 % (19715)Termination reason: Unknown
% 3.05/1.66 % (19715)Termination phase: shuffling
% 3.05/1.66
% 3.05/1.66 % (19715)Memory used [KB]: 1407
% 3.05/1.66 % (19715)Time elapsed: 0.006 s
% 3.05/1.66 % (19715)Instructions burned: 7 (million)
% 3.05/1.66 % (19715)------------------------------
% 3.05/1.66 % (19715)------------------------------
% 3.05/1.66 % (19718)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_19374 for (2999ds/3Mi)
% 3.05/1.67 % (19719)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_19374 for (2999ds/7Mi)
% 3.05/1.67 % (19718)Instruction limit reached!
% 3.05/1.67 % (19718)------------------------------
% 3.05/1.67 % (19718)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.67 % (19718)Termination reason: Unknown
% 3.05/1.67 % (19718)Termination phase: shuffling
% 3.05/1.67
% 3.05/1.67 % (19718)Memory used [KB]: 1407
% 3.05/1.67 % (19718)Time elapsed: 0.004 s
% 3.05/1.67 % (19718)Instructions burned: 4 (million)
% 3.05/1.67 % (19718)------------------------------
% 3.05/1.67 % (19718)------------------------------
% 3.05/1.67 % (19719)Instruction limit reached!
% 3.05/1.67 % (19719)------------------------------
% 3.05/1.67 % (19719)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.67 % (19719)Termination reason: Unknown
% 3.05/1.67 % (19719)Termination phase: shuffling
% 3.05/1.67
% 3.05/1.67 % (19719)Memory used [KB]: 1407
% 3.05/1.67 % (19719)Time elapsed: 0.003 s
% 3.05/1.67 % (19719)Instructions burned: 7 (million)
% 3.05/1.67 % (19719)------------------------------
% 3.05/1.67 % (19719)------------------------------
% 3.05/1.67 % (19720)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_19374 for (2999ds/3Mi)
% 3.05/1.67 % (19720)Instruction limit reached!
% 3.05/1.67 % (19720)------------------------------
% 3.05/1.67 % (19720)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.67 % (19720)Termination reason: Unknown
% 3.05/1.67 % (19720)Termination phase: shuffling
% 3.05/1.67
% 3.05/1.67 % (19720)Memory used [KB]: 1407
% 3.05/1.67 % (19720)Time elapsed: 0.003 s
% 3.05/1.67 % (19720)Instructions burned: 4 (million)
% 3.05/1.67 % (19720)------------------------------
% 3.05/1.67 % (19720)------------------------------
% 3.05/1.68 % (19723)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_19374 for (2999ds/710Mi)
% 3.05/1.68 % (19722)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_19374 for (2999ds/18Mi)
% 3.05/1.68 % (19721)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_19374 for (2999ds/4Mi)
% 3.05/1.68 % (19724)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_19374 for (2999ds/6Mi)
% 3.05/1.68 % (19721)Instruction limit reached!
% 3.05/1.68 % (19721)------------------------------
% 3.05/1.68 % (19721)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.68 % (19721)Termination reason: Unknown
% 3.05/1.68 % (19721)Termination phase: shuffling
% 3.05/1.68
% 3.05/1.68 % (19721)Memory used [KB]: 1407
% 3.05/1.68 % (19721)Time elapsed: 0.005 s
% 3.05/1.68 % (19721)Instructions burned: 5 (million)
% 3.05/1.68 % (19721)------------------------------
% 3.05/1.68 % (19721)------------------------------
% 3.05/1.68 % (19724)Instruction limit reached!
% 3.05/1.68 % (19724)------------------------------
% 3.05/1.68 % (19724)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.68 % (19724)Termination reason: Unknown
% 3.05/1.68 % (19724)Termination phase: shuffling
% 3.05/1.68
% 3.05/1.68 % (19724)Memory used [KB]: 1407
% 3.05/1.68 % (19724)Time elapsed: 0.004 s
% 3.05/1.68 % (19724)Instructions burned: 6 (million)
% 3.05/1.68 % (19724)------------------------------
% 3.05/1.68 % (19724)------------------------------
% 3.05/1.69 % (19722)Instruction limit reached!
% 3.05/1.69 % (19722)------------------------------
% 3.05/1.69 % (19722)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.69 % (19722)Termination reason: Unknown
% 3.05/1.69 % (19722)Termination phase: shuffling
% 3.05/1.69
% 3.05/1.69 % (19722)Memory used [KB]: 1535
% 3.05/1.69 % (19722)Time elapsed: 0.008 s
% 3.05/1.69 % (19722)Instructions burned: 19 (million)
% 3.05/1.69 % (19722)------------------------------
% 3.05/1.69 % (19722)------------------------------
% 3.05/1.69 % (19708)Refutation not found, incomplete strategy
% 3.05/1.69 % (19708)------------------------------
% 3.05/1.69 % (19708)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.69 % (19708)Termination reason: Refutation not found, incomplete strategy
% 3.05/1.69
% 3.05/1.69
% 3.05/1.69 % (19708)Memory used [KB]: 6908
% 3.05/1.69 % (19708)Time elapsed: 0.064 s
% 3.05/1.69 % (19708)Instructions burned: 152 (million)
% 3.05/1.69 % (19708)------------------------------
% 3.05/1.69 % (19708)------------------------------
% 3.05/1.69 % (19726)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_19374 for (2999ds/21Mi)
% 3.05/1.70 % (19727)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_19374 for (2999ds/5Mi)
% 3.05/1.70 % (19727)Instruction limit reached!
% 3.05/1.70 % (19727)------------------------------
% 3.05/1.70 % (19727)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.70 % (19727)Termination reason: Unknown
% 3.05/1.70 % (19727)Termination phase: shuffling
% 3.05/1.70
% 3.05/1.70 % (19727)Memory used [KB]: 1407
% 3.05/1.70 % (19727)Time elapsed: 0.004 s
% 3.05/1.70 % (19727)Instructions burned: 7 (million)
% 3.05/1.70 % (19727)------------------------------
% 3.05/1.70 % (19727)------------------------------
% 3.05/1.70 % (19725)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_19374 for (2999ds/902Mi)
% 3.05/1.70 % (19728)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_19374 for (2999ds/6Mi)
% 3.05/1.70 % (19726)Instruction limit reached!
% 3.05/1.70 % (19726)------------------------------
% 3.05/1.70 % (19726)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.70 % (19726)Termination reason: Unknown
% 3.05/1.70 % (19726)Termination phase: shuffling
% 3.05/1.70
% 3.05/1.70 % (19726)Memory used [KB]: 1535
% 3.05/1.70 % (19726)Time elapsed: 0.009 s
% 3.05/1.70 % (19726)Instructions burned: 22 (million)
% 3.05/1.70 % (19726)------------------------------
% 3.05/1.70 % (19726)------------------------------
% 3.05/1.70 % (19728)Instruction limit reached!
% 3.05/1.70 % (19728)------------------------------
% 3.05/1.70 % (19728)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.70 % (19728)Termination reason: Unknown
% 3.05/1.70 % (19728)Termination phase: shuffling
% 3.05/1.70
% 3.05/1.70 % (19728)Memory used [KB]: 1407
% 3.05/1.70 % (19728)Time elapsed: 0.003 s
% 3.05/1.70 % (19728)Instructions burned: 7 (million)
% 3.05/1.70 % (19728)------------------------------
% 3.05/1.70 % (19728)------------------------------
% 3.05/1.71 % (19729)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_19374 for (2999ds/377Mi)
% 3.05/1.71 % (19730)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_19374 for (2998ds/779Mi)
% 3.05/1.71 % (19731)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_19374 for (2998ds/19Mi)
% 3.05/1.72 % (19731)Instruction limit reached!
% 3.05/1.72 % (19731)------------------------------
% 3.05/1.72 % (19731)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.72 % (19731)Termination reason: Unknown
% 3.05/1.72 % (19731)Termination phase: shuffling
% 3.05/1.72
% 3.05/1.72 % (19731)Memory used [KB]: 1663
% 3.05/1.72 % (19731)Time elapsed: 0.006 s
% 3.05/1.72 % (19731)Instructions burned: 19 (million)
% 3.05/1.72 % (19731)------------------------------
% 3.05/1.72 % (19731)------------------------------
% 3.05/1.72 % (19703)Instruction limit reached!
% 3.05/1.72 % (19703)------------------------------
% 3.05/1.72 % (19703)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.72 % (19703)Termination reason: Unknown
% 3.05/1.72 % (19703)Termination phase: Saturation
% 3.05/1.72
% 3.05/1.72 % (19703)Memory used [KB]: 6908
% 3.05/1.72 % (19703)Time elapsed: 0.092 s
% 3.05/1.72 % (19703)Instructions burned: 183 (million)
% 3.05/1.72 % (19703)------------------------------
% 3.05/1.72 % (19703)------------------------------
% 3.05/1.73 % (19732)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_19374 for (2998ds/879Mi)
% 3.05/1.73 % (19733)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_19374 for (2998ds/17Mi)
% 3.05/1.74 % (19733)Instruction limit reached!
% 3.05/1.74 % (19733)------------------------------
% 3.05/1.74 % (19733)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.74 % (19733)Termination reason: Unknown
% 3.05/1.74 % (19733)Termination phase: shuffling
% 3.05/1.74
% 3.05/1.74 % (19733)Memory used [KB]: 1535
% 3.05/1.74 % (19733)Time elapsed: 0.008 s
% 3.05/1.74 % (19733)Instructions burned: 18 (million)
% 3.05/1.74 % (19733)------------------------------
% 3.05/1.74 % (19733)------------------------------
% 3.05/1.75 % (19734)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_19374 for (2998ds/3Mi)
% 3.05/1.75 % (19734)Instruction limit reached!
% 3.05/1.75 % (19734)------------------------------
% 3.05/1.75 % (19734)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.75 % (19734)Termination reason: Unknown
% 3.05/1.75 % (19734)Termination phase: shuffling
% 3.05/1.75
% 3.05/1.75 % (19734)Memory used [KB]: 1279
% 3.05/1.75 % (19734)Time elapsed: 0.003 s
% 3.05/1.75 % (19734)Instructions burned: 4 (million)
% 3.05/1.75 % (19734)------------------------------
% 3.05/1.75 % (19734)------------------------------
% 3.05/1.76 % (19735)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_19374 for (2998ds/30Mi)
% 3.05/1.77 % (19735)Instruction limit reached!
% 3.05/1.77 % (19735)------------------------------
% 3.05/1.77 % (19735)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.77 % (19735)Termination reason: Unknown
% 3.05/1.77 % (19735)Termination phase: shuffling
% 3.05/1.77
% 3.05/1.77 % (19735)Memory used [KB]: 1663
% 3.05/1.77 % (19735)Time elapsed: 0.012 s
% 3.05/1.77 % (19735)Instructions burned: 30 (million)
% 3.05/1.77 % (19735)------------------------------
% 3.05/1.77 % (19735)------------------------------
% 3.05/1.78 % (19736)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_19374 for (2998ds/127Mi)
% 3.05/1.82 % (19736)Instruction limit reached!
% 3.05/1.82 % (19736)------------------------------
% 3.05/1.82 % (19736)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.82 % (19736)Termination reason: Unknown
% 3.05/1.82 % (19736)Termination phase: Property scanning
% 3.05/1.82
% 3.05/1.82 % (19736)Memory used [KB]: 2174
% 3.05/1.82 % (19736)Time elapsed: 0.041 s
% 3.05/1.82 % (19736)Instructions burned: 129 (million)
% 3.05/1.82 % (19736)------------------------------
% 3.05/1.82 % (19736)------------------------------
% 3.05/1.83 % (19737)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_19374 for (2997ds/100Mi)
% 3.05/1.83 % (19729)Instruction limit reached!
% 3.05/1.83 % (19729)------------------------------
% 3.05/1.83 % (19729)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.83 % (19729)Termination reason: Unknown
% 3.05/1.83 % (19729)Termination phase: Saturation
% 3.05/1.83
% 3.05/1.83 % (19729)Memory used [KB]: 7931
% 3.05/1.83 % (19729)Time elapsed: 0.126 s
% 3.05/1.83 % (19729)Instructions burned: 379 (million)
% 3.05/1.83 % (19729)------------------------------
% 3.05/1.83 % (19729)------------------------------
% 3.05/1.84 % (19738)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_19374 for (2997ds/3Mi)
% 3.05/1.84 % (19738)Instruction limit reached!
% 3.05/1.84 % (19738)------------------------------
% 3.05/1.84 % (19738)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.84 % (19738)Termination reason: Unknown
% 3.05/1.84 % (19738)Termination phase: shuffling
% 3.05/1.84
% 3.05/1.84 % (19738)Memory used [KB]: 1407
% 3.05/1.84 % (19738)Time elapsed: 0.003 s
% 3.05/1.84 % (19738)Instructions burned: 4 (million)
% 3.05/1.84 % (19738)------------------------------
% 3.05/1.84 % (19738)------------------------------
% 3.05/1.85 % (19739)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_19374 for (2997ds/20Mi)
% 3.05/1.86 % (19737)Instruction limit reached!
% 3.05/1.86 % (19737)------------------------------
% 3.05/1.86 % (19737)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.86 % (19737)Termination reason: Unknown
% 3.05/1.86 % (19737)Termination phase: Preprocessing 3
% 3.05/1.86
% 3.05/1.86 % (19737)Memory used [KB]: 2302
% 3.05/1.86 % (19737)Time elapsed: 0.030 s
% 3.05/1.86 % (19737)Instructions burned: 100 (million)
% 3.05/1.86 % (19737)------------------------------
% 3.05/1.86 % (19737)------------------------------
% 3.05/1.86 % (19739)Instruction limit reached!
% 3.05/1.86 % (19739)------------------------------
% 3.05/1.86 % (19739)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.86 % (19739)Termination reason: Unknown
% 3.05/1.86 % (19739)Termination phase: shuffling
% 3.05/1.86
% 3.05/1.86 % (19739)Memory used [KB]: 1535
% 3.05/1.86 % (19739)Time elapsed: 0.009 s
% 3.05/1.86 % (19739)Instructions burned: 21 (million)
% 3.05/1.86 % (19739)------------------------------
% 3.05/1.86 % (19739)------------------------------
% 3.05/1.87 % (19740)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_19374 for (2997ds/86Mi)
% 3.05/1.87 % (19741)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_19374 for (2997ds/107Mi)
% 3.05/1.89 % (19740)Instruction limit reached!
% 3.05/1.89 % (19740)------------------------------
% 3.05/1.89 % (19740)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.89 % (19740)Termination reason: Unknown
% 3.05/1.89 % (19740)Termination phase: Property scanning
% 3.05/1.89
% 3.05/1.89 % (19740)Memory used [KB]: 2174
% 3.05/1.89 % (19740)Time elapsed: 0.029 s
% 3.05/1.89 % (19740)Instructions burned: 87 (million)
% 3.05/1.89 % (19740)------------------------------
% 3.05/1.89 % (19740)------------------------------
% 3.05/1.89 % (19723)Instruction limit reached!
% 3.05/1.89 % (19723)------------------------------
% 3.05/1.89 % (19723)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.89 % (19723)Termination reason: Unknown
% 3.05/1.89 % (19723)Termination phase: Saturation
% 3.05/1.89
% 3.05/1.89 % (19723)Memory used [KB]: 8315
% 3.05/1.89 % (19723)Time elapsed: 0.218 s
% 3.05/1.89 % (19723)Instructions burned: 713 (million)
% 3.05/1.89 % (19723)------------------------------
% 3.05/1.89 % (19723)------------------------------
% 3.05/1.90 % (19742)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_19374 for (2996ds/39Mi)
% 3.05/1.90 % (19743)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_19374 for (2996ds/448Mi)
% 3.05/1.91 % (19741)Instruction limit reached!
% 3.05/1.91 % (19741)------------------------------
% 3.05/1.91 % (19741)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.91 % (19741)Termination reason: Unknown
% 3.05/1.91 % (19741)Termination phase: Property scanning
% 3.05/1.91
% 3.05/1.91 % (19741)Memory used [KB]: 2174
% 3.05/1.91 % (19741)Time elapsed: 0.035 s
% 3.05/1.91 % (19741)Instructions burned: 108 (million)
% 3.05/1.91 % (19741)------------------------------
% 3.05/1.91 % (19741)------------------------------
% 3.05/1.91 % (19742)Instruction limit reached!
% 3.05/1.91 % (19742)------------------------------
% 3.05/1.91 % (19742)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.91 % (19742)Termination reason: Unknown
% 3.05/1.91 % (19742)Termination phase: shuffling
% 3.05/1.91
% 3.05/1.91 % (19742)Memory used [KB]: 2046
% 3.05/1.91 % (19742)Time elapsed: 0.013 s
% 3.05/1.91 % (19742)Instructions burned: 39 (million)
% 3.05/1.91 % (19742)------------------------------
% 3.05/1.91 % (19742)------------------------------
% 3.05/1.92 % (19744)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_19374 for (2996ds/46Mi)
% 3.05/1.92 % (19745)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_19374 for (2996ds/98Mi)
% 3.05/1.93 % (19725)Refutation not found, incomplete strategy
% 3.05/1.93 % (19725)------------------------------
% 3.05/1.93 % (19725)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.93 % (19725)Termination reason: Refutation not found, incomplete strategy
% 3.05/1.93
% 3.05/1.93
% 3.05/1.93 % (19725)Memory used [KB]: 7803
% 3.05/1.93 % (19725)Time elapsed: 0.227 s
% 3.05/1.93 % (19725)Instructions burned: 682 (million)
% 3.05/1.93 % (19725)------------------------------
% 3.05/1.93 % (19725)------------------------------
% 3.05/1.93 % (19744)Instruction limit reached!
% 3.05/1.93 % (19744)------------------------------
% 3.05/1.93 % (19744)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.93 % (19744)Termination reason: Unknown
% 3.05/1.93 % (19744)Termination phase: Property scanning
% 3.05/1.93
% 3.05/1.93 % (19744)Memory used [KB]: 1791
% 3.05/1.93 % (19744)Time elapsed: 0.018 s
% 3.05/1.93 % (19744)Instructions burned: 49 (million)
% 3.05/1.93 % (19744)------------------------------
% 3.05/1.93 % (19744)------------------------------
% 3.05/1.94 % (19746)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_19374 for (2996ds/507Mi)
% 3.05/1.94 % (19747)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_19374 for (2996ds/149Mi)
% 3.05/1.94 % (19730)Instruction limit reached!
% 3.05/1.94 % (19730)------------------------------
% 3.05/1.94 % (19730)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.94 % (19730)Termination reason: Unknown
% 3.05/1.94 % (19730)Termination phase: Saturation
% 3.05/1.94
% 3.05/1.94 % (19730)Memory used [KB]: 8059
% 3.05/1.94 % (19730)Time elapsed: 0.232 s
% 3.05/1.94 % (19730)Instructions burned: 781 (million)
% 3.05/1.94 % (19730)------------------------------
% 3.05/1.94 % (19730)------------------------------
% 3.05/1.95 % (19748)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_19374 for (2996ds/39Mi)
% 3.05/1.95 % (19745)Instruction limit reached!
% 3.05/1.95 % (19745)------------------------------
% 3.05/1.95 % (19745)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.95 % (19745)Termination reason: Unknown
% 3.05/1.95 % (19745)Termination phase: Preprocessing 3
% 3.05/1.95
% 3.05/1.95 % (19745)Memory used [KB]: 2302
% 3.05/1.95 % (19745)Time elapsed: 0.030 s
% 3.05/1.95 % (19745)Instructions burned: 99 (million)
% 3.05/1.95 % (19745)------------------------------
% 3.05/1.95 % (19745)------------------------------
% 3.05/1.96 % (19714)Instruction limit reached!
% 3.05/1.96 % (19714)------------------------------
% 3.05/1.96 % (19714)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.96 % (19714)Termination reason: Unknown
% 3.05/1.96 % (19714)Termination phase: Saturation
% 3.05/1.96
% 3.05/1.96 % (19714)Memory used [KB]: 9210
% 3.05/1.96 % (19714)Time elapsed: 0.311 s
% 3.05/1.96 % (19714)Instructions burned: 1042 (million)
% 3.05/1.96 % (19714)------------------------------
% 3.05/1.96 % (19714)------------------------------
% 3.05/1.96 % (19749)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_19374 for (2996ds/6Mi)
% 3.05/1.96 % (19749)Instruction limit reached!
% 3.05/1.96 % (19749)------------------------------
% 3.05/1.96 % (19749)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.96 % (19749)Termination reason: Unknown
% 3.05/1.96 % (19749)Termination phase: shuffling
% 3.05/1.96
% 3.05/1.96 % (19749)Memory used [KB]: 1407
% 3.05/1.96 % (19749)Time elapsed: 0.004 s
% 3.05/1.96 % (19749)Instructions burned: 8 (million)
% 3.05/1.96 % (19749)------------------------------
% 3.05/1.96 % (19749)------------------------------
% 3.05/1.96 % (19748)Instruction limit reached!
% 3.05/1.96 % (19748)------------------------------
% 3.05/1.96 % (19748)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.96 % (19748)Termination reason: Unknown
% 3.05/1.96 % (19748)Termination phase: Property scanning
% 3.05/1.96
% 3.05/1.97 % (19748)Memory used [KB]: 1791
% 3.05/1.97 % (19748)Time elapsed: 0.014 s
% 3.05/1.97 % (19748)Instructions burned: 39 (million)
% 3.05/1.97 % (19748)------------------------------
% 3.05/1.97 % (19748)------------------------------
% 3.05/1.97 % (19750)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_19374 for (2996ds/1087Mi)
% 3.05/1.97 % (19752)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_19374 for (2996ds/44Mi)
% 3.05/1.98 % (19751)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_19374 for (2996ds/47Mi)
% 3.05/1.99 % (19752)Instruction limit reached!
% 3.05/1.99 % (19752)------------------------------
% 3.05/1.99 % (19752)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.99 % (19752)Termination reason: Unknown
% 3.05/1.99 % (19752)Termination phase: shuffling
% 3.05/1.99
% 3.05/1.99 % (19752)Memory used [KB]: 2174
% 3.05/1.99 % (19752)Time elapsed: 0.013 s
% 3.05/1.99 % (19752)Instructions burned: 46 (million)
% 3.05/1.99 % (19752)------------------------------
% 3.05/1.99 % (19752)------------------------------
% 3.05/1.99 % (19732)Instruction limit reached!
% 3.05/1.99 % (19732)------------------------------
% 3.05/1.99 % (19732)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.99 % (19732)Termination reason: Unknown
% 3.05/1.99 % (19732)Termination phase: Saturation
% 3.05/1.99
% 3.05/1.99 % (19732)Memory used [KB]: 7675
% 3.05/1.99 % (19732)Time elapsed: 0.261 s
% 3.05/1.99 % (19732)Instructions burned: 879 (million)
% 3.05/1.99 % (19732)------------------------------
% 3.05/1.99 % (19732)------------------------------
% 3.05/1.99 % (19747)Instruction limit reached!
% 3.05/1.99 % (19747)------------------------------
% 3.05/1.99 % (19747)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.99 % (19747)Termination reason: Unknown
% 3.05/1.99 % (19747)Termination phase: Property scanning
% 3.05/1.99
% 3.05/1.99 % (19747)Memory used [KB]: 2174
% 3.05/1.99 % (19747)Time elapsed: 0.048 s
% 3.05/1.99 % (19747)Instructions burned: 151 (million)
% 3.05/1.99 % (19747)------------------------------
% 3.05/1.99 % (19747)------------------------------
% 3.05/2.00 % (19753)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_19374 for (2996ds/432Mi)
% 3.05/2.00 % (19754)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_19374 for (2996ds/62Mi)
% 3.05/2.00 % (19755)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_19374 for (2996ds/31Mi)
% 3.05/2.01 % (19751)Instruction limit reached!
% 3.05/2.01 % (19751)------------------------------
% 3.05/2.01 % (19751)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/2.01 % (19751)Termination reason: Unknown
% 3.05/2.01 % (19751)Termination phase: Property scanning
% 3.05/2.01
% 3.05/2.01 % (19751)Memory used [KB]: 1791
% 3.05/2.01 % (19751)Time elapsed: 0.030 s
% 3.05/2.01 % (19751)Instructions burned: 47 (million)
% 3.05/2.01 % (19751)------------------------------
% 3.05/2.01 % (19751)------------------------------
% 3.05/2.01 % (19755)Instruction limit reached!
% 3.05/2.01 % (19755)------------------------------
% 3.05/2.01 % (19755)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/2.01 % (19755)Termination reason: Unknown
% 3.05/2.01 % (19755)Termination phase: shuffling
% 3.05/2.01
% 3.05/2.01 % (19755)Memory used [KB]: 1663
% 3.05/2.01 % (19755)Time elapsed: 0.013 s
% 3.05/2.01 % (19755)Instructions burned: 33 (million)
% 3.05/2.01 % (19755)------------------------------
% 3.05/2.01 % (19755)------------------------------
% 5.82/2.01 % (19754)Instruction limit reached!
% 5.82/2.01 % (19754)------------------------------
% 5.82/2.01 % (19754)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.82/2.01 % (19754)Termination reason: Unknown
% 5.82/2.01 % (19754)Termination phase: Property scanning
% 5.82/2.01
% 5.82/2.01 % (19754)Memory used [KB]: 2174
% 5.82/2.01 % (19754)Time elapsed: 0.017 s
% 5.82/2.01 % (19754)Instructions burned: 62 (million)
% 5.82/2.01 % (19754)------------------------------
% 5.82/2.01 % (19754)------------------------------
% 5.82/2.02 % (19746)First to succeed.
% 5.82/2.02 % (19746)Refutation found. Thanks to Tanya!
% 5.82/2.02 % SZS status Theorem for DTF2THF_19374
% 5.82/2.02 % SZS output start Proof for DTF2THF_19374
% 5.82/2.02 thf(type_def_5, type, partia1265057091ct_ext: ($tType * $tType) > $tType).
% 5.82/2.02 thf(type_def_6, type, set: ($tType) > $tType).
% 5.82/2.02 thf(type_def_8, type, monoid_ext: ($tType * $tType) > $tType).
% 5.82/2.02 thf(type_def_9, type, product_unit: $tType).
% 5.82/2.02 thf(type_def_10, type, nat: $tType).
% 5.82/2.02 thf(type_def_11, type, product_prod: ($tType * $tType) > $tType).
% 5.82/2.02 thf(type_def_12, type, ring_ext: ($tType * $tType) > $tType).
% 5.82/2.02 thf(type_def_13, type, a: $tType).
% 5.82/2.02 thf(type_def_14, type, b: $tType).
% 5.82/2.02 thf(type_def_15, type, eq_eq_object_ext: ($tType * $tType) > $tType).
% 5.82/2.02 thf(type_def_16, type, int: $tType).
% 5.82/2.02 thf(func_def_0, type, partia1265057091ct_ext: ($tType * $tType) > $tType).
% 5.82/2.02 thf(func_def_1, type, eq_eq_object_ext: ($tType * $tType) > $tType).
% 5.82/2.02 thf(func_def_2, type, monoid_ext: ($tType * $tType) > $tType).
% 5.82/2.02 thf(func_def_3, type, ring_ext: ($tType * $tType) > $tType).
% 5.82/2.02 thf(func_def_4, type, product_unit: $tType).
% 5.82/2.02 thf(func_def_5, type, product_prod: ($tType * $tType) > $tType).
% 5.82/2.02 thf(func_def_6, type, set: ($tType) > $tType).
% 5.82/2.02 thf(func_def_7, type, nat: $tType).
% 5.82/2.02 thf(func_def_8, type, int: $tType).
% 5.82/2.02 thf(func_def_9, type, b: $tType).
% 5.82/2.02 thf(func_def_10, type, a: $tType).
% 5.82/2.02 thf(func_def_11, type, group_add: !>[X0: $tType]:($o)).
% 5.82/2.02 thf(func_def_12, type, semiring_1: !>[X0: $tType]:($o)).
% 5.82/2.02 thf(func_def_13, type, ab_group_add: !>[X0: $tType]:($o)).
% 5.82/2.02 thf(func_def_14, type, semiring_char_0: !>[X0: $tType]:($o)).
% 5.82/2.02 thf(func_def_15, type, semigroup_add: !>[X0: $tType]:($o)).
% 5.82/2.02 thf(func_def_16, type, comm_monoid_add: !>[X0: $tType]:($o)).
% 5.82/2.02 thf(func_def_17, type, ab_semigroup_add: !>[X0: $tType]:($o)).
% 5.82/2.02 thf(func_def_18, type, cancel_semigroup_add: !>[X0: $tType]:($o)).
% 5.82/2.02 thf(func_def_19, type, cancel1352612707id_add: !>[X0: $tType]:($o)).
% 5.82/2.02 thf(func_def_20, type, cancel146912293up_add: !>[X0: $tType]:($o)).
% 5.82/2.02 thf(func_def_21, type, ordere779506340up_add: !>[X0: $tType]:($o)).
% 5.82/2.02 thf(func_def_22, type, partial_carrier: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ X1 > set @ X0)).
% 5.82/2.02 thf(func_def_23, type, units: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ X1 > set @ X0)).
% 5.82/2.02 thf(func_def_24, type, comm_group: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ X1 > $o)).
% 5.82/2.02 thf(func_def_25, type, group: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ X1 > $o)).
% 5.82/2.02 thf(func_def_26, type, mult: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ X1 > X0 > X0 > X0)).
% 5.82/2.02 thf(func_def_27, type, one: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ X1 > X0)).
% 5.82/2.02 thf(func_def_28, type, units_of: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ X1 > partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ product_unit)).
% 5.82/2.02 thf(func_def_29, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 5.82/2.02 thf(func_def_30, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 5.82/2.02 thf(func_def_31, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 5.82/2.02 thf(func_def_32, type, suc: nat > nat).
% 5.82/2.02 thf(func_def_33, type, semiring_1_of_nat: !>[X0: $tType]:(nat > X0)).
% 5.82/2.02 thf(func_def_34, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 5.82/2.02 thf(func_def_35, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 5.82/2.02 thf(func_def_36, type, a_inv: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > X0 > X0)).
% 5.82/2.02 thf(func_def_37, type, a_minus: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > X0 > X0 > X0)).
% 5.82/2.02 thf(func_def_38, type, abelian_group: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > $o)).
% 5.82/2.02 thf(func_def_39, type, add_pow: !>[X0: $tType, X1: $tType, X2: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > X2 > X0 > X0)).
% 5.82/2.02 thf(func_def_40, type, field: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > $o)).
% 5.82/2.02 thf(func_def_41, type, add: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > X0 > X0 > X0)).
% 5.82/2.02 thf(func_def_42, type, more: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > X1)).
% 5.82/2.02 thf(func_def_43, type, zero: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > X0)).
% 5.82/2.02 thf(func_def_44, type, semiring: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X0 @ monoid_ext @ X0 @ ring_ext @ X0 @ X1 > $o)).
% 5.82/2.02 thf(func_def_45, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 5.82/2.02 thf(func_def_46, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 5.82/2.02 thf(func_def_47, type, r: partia1265057091ct_ext @ a @ monoid_ext @ a @ ring_ext @ a @ b).
% 5.82/2.02 thf(func_def_48, type, r2: a).
% 5.82/2.02 thf(func_def_49, type, r3: a).
% 5.82/2.02 thf(func_def_50, type, r4: a).
% 5.82/2.02 thf(func_def_51, type, rel: partia1265057091ct_ext @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit).
% 5.82/2.02 thf(func_def_52, type, s: a).
% 5.82/2.02 thf(func_def_53, type, s2: a).
% 5.82/2.02 thf(func_def_54, type, s3: a).
% 5.82/2.02 thf(func_def_64, type, sK0: !>[X0: $tType]:((X0 > $o) > X0)).
% 5.82/2.02 thf(func_def_65, type, sK1: !>[X0: $tType]:((X0 > $o) > X0)).
% 5.82/2.02 thf(func_def_71, type, sP2: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > $o)).
% 5.82/2.02 thf(func_def_72, type, sP3: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > $o)).
% 5.82/2.02 thf(func_def_73, type, sP4: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ ring_ext @ X1 @ X0 > $o)).
% 5.82/2.02 thf(func_def_74, type, sP5: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ ring_ext @ X1 @ X0 > $o)).
% 5.82/2.02 thf(func_def_75, type, sP6: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ ring_ext @ X1 @ X0 > $o)).
% 5.82/2.02 thf(func_def_76, type, sK7: !>[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)).
% 5.82/2.02 thf(func_def_77, type, sK8: !>[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)).
% 5.82/2.02 thf(func_def_78, type, sK9: !>[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)).
% 5.82/2.02 thf(func_def_79, type, sK10: !>[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)).
% 5.82/2.02 thf(func_def_80, type, sK11: !>[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)).
% 5.82/2.02 thf(func_def_81, type, sK12: !>[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)).
% 5.82/2.02 thf(func_def_82, type, sK13: !>[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)).
% 5.82/2.02 thf(func_def_83, type, sK14: !>[X0: $tType, X1: $tType]:(X1 > partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 5.82/2.02 thf(func_def_84, type, sK15: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 5.82/2.02 thf(func_def_85, type, sK16: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 5.82/2.02 thf(func_def_86, type, sK17: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 5.82/2.02 thf(func_def_87, type, sK18: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 5.82/2.02 thf(func_def_88, type, sK19: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 5.82/2.02 thf(func_def_89, type, sK20: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 5.82/2.02 thf(func_def_90, type, sK21: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 5.82/2.02 thf(func_def_91, type, sK22: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 5.82/2.02 thf(func_def_92, type, sK23: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 5.82/2.02 thf(func_def_93, type, sK24: !>[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)).
% 5.82/2.02 thf(func_def_94, type, sK25: !>[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)).
% 5.82/2.02 thf(func_def_95, type, sK26: !>[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)).
% 5.82/2.02 thf(func_def_96, type, sK27: !>[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)).
% 5.82/2.02 thf(func_def_97, type, sK28: !>[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)).
% 5.82/2.02 thf(func_def_98, type, sK29: !>[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)).
% 5.82/2.02 thf(func_def_99, type, sK30: a > a).
% 5.82/2.02 thf(func_def_100, type, sK31: !>[X0: $tType, X1: $tType]:(X1 > partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 5.82/2.02 thf(func_def_101, type, sK32: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 5.82/2.02 thf(func_def_102, type, sK33: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 5.82/2.02 thf(func_def_103, type, sK34: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 5.82/2.02 thf(func_def_104, type, sK35: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 5.82/2.02 thf(func_def_105, type, sK36: !>[X0: $tType, X1: $tType]:(X1 > partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 5.82/2.02 thf(func_def_106, type, sK37: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 5.82/2.02 thf(func_def_107, type, sK38: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 5.82/2.02 thf(func_def_108, type, sK39: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 5.82/2.02 thf(func_def_109, type, sK40: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 5.82/2.02 thf(func_def_110, type, sK41: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 5.82/2.02 thf(func_def_111, type, sK42: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 5.82/2.02 thf(func_def_112, type, sK43: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ X0 > X1)).
% 5.82/2.02 thf(func_def_113, type, sK44: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 5.82/2.02 thf(func_def_114, type, sK45: !>[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)).
% 5.82/2.02 thf(func_def_115, type, sK46: !>[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)).
% 5.82/2.02 thf(func_def_116, type, sK47: !>[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)).
% 5.82/2.02 thf(func_def_117, type, sK48: !>[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)).
% 5.82/2.02 thf(func_def_118, type, sK49: !>[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)).
% 5.82/2.02 thf(func_def_119, type, sK50: (nat > $o) > nat).
% 5.82/2.02 thf(func_def_120, type, sK51: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 5.82/2.02 thf(func_def_121, type, sK52: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 5.82/2.02 thf(func_def_122, type, sK53: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 5.82/2.02 thf(func_def_123, type, sK54: !>[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)).
% 5.82/2.02 thf(func_def_124, type, sK55: !>[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)).
% 5.82/2.02 thf(func_def_125, type, sK56: !>[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)).
% 5.82/2.02 thf(func_def_126, type, sK57: !>[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)).
% 5.82/2.02 thf(func_def_127, type, sK58: !>[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)).
% 5.82/2.02 thf(func_def_128, type, sK59: !>[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)).
% 5.82/2.02 thf(func_def_129, type, sK60: a > a).
% 5.82/2.02 thf(func_def_130, type, sK61: a > a).
% 5.82/2.02 thf(func_def_131, type, sK62: a > a).
% 5.82/2.02 thf(func_def_132, type, sK63: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 5.82/2.02 thf(func_def_133, type, sK64: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 5.82/2.02 thf(func_def_134, type, sK65: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 5.82/2.02 thf(func_def_135, type, sK66: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 5.82/2.02 thf(func_def_136, type, sK67: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 5.82/2.02 thf(func_def_137, type, sK68: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 5.82/2.02 thf(func_def_138, type, sK69: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 5.82/2.02 thf(func_def_139, type, sK70: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 5.82/2.02 thf(func_def_140, type, sK71: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 5.82/2.02 thf(func_def_141, type, sK72: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 5.82/2.02 thf(func_def_142, type, sK73: !>[X0: $tType, X1: $tType]:(partia1265057091ct_ext @ X1 @ monoid_ext @ X1 @ ring_ext @ X1 @ X0 > X1)).
% 5.82/2.02 thf(func_def_143, type, sK74: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 5.82/2.02 thf(func_def_144, type, sK75: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 5.82/2.02 thf(func_def_145, type, sK76: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 5.82/2.02 thf(func_def_146, type, sK77: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 5.82/2.02 thf(func_def_147, 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) > X6)).
% 5.82/2.02 thf(func_def_148, 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) > X5)).
% 5.82/2.02 thf(func_def_149, 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) > X4)).
% 5.82/2.02 thf(func_def_150, type, sK81: !>[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)).
% 5.82/2.02 thf(func_def_151, type, sK82: !>[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)).
% 5.82/2.02 thf(func_def_152, type, sK83: !>[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)).
% 5.82/2.02 thf(func_def_153, type, sK84: !>[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)).
% 5.82/2.02 thf(func_def_154, type, sK85: a).
% 5.82/2.02 thf(func_def_155, type, sK86: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 5.82/2.02 thf(func_def_156, type, sK87: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 5.82/2.02 thf(func_def_157, type, sK88: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 5.82/2.02 thf(func_def_158, type, sK89: a > a).
% 5.82/2.02 thf(func_def_159, type, sK90: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 5.82/2.02 thf(func_def_160, type, sK91: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 5.82/2.02 thf(func_def_161, type, sK92: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 5.82/2.02 thf(func_def_162, type, sK93: a > a).
% 5.82/2.02 thf(func_def_163, type, sK94: a).
% 5.82/2.02 thf(func_def_165, type, ph96: !>[X0: $tType]:(X0)).
% 5.82/2.02 thf(f1860,plain,(
% 5.82/2.02 $false),
% 5.82/2.02 inference(subsumption_resolution,[status(thm)],[f1859,f1520])).
% 5.82/2.02 thf(f1520,plain,(
% 5.82/2.02 ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (mult @ a @ ring_ext @ a @ b @ r @ r3 @ r4) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3)) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel)) = $true)),
% 5.82/2.02 inference(cnf_transformation,[status(thm)],[f314])).
% 5.82/2.02 thf(f314,plain,(
% 5.82/2.02 ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (mult @ a @ ring_ext @ a @ b @ r @ r3 @ r4) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3)) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel)) = $true)),
% 5.82/2.02 inference(fool_elimination,[status(thm)],[f313])).
% 5.82/2.02 thf(f313,plain,(
% 5.82/2.02 (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (mult @ a @ ring_ext @ a @ b @ r @ r3 @ r4) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3)) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel))),
% 5.82/2.02 inference(rectify,[status(thm)],[f11])).
% 5.82/2.02 thf(f11,axiom,(
% 5.82/2.02 (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (mult @ a @ ring_ext @ a @ b @ r @ r3 @ r4) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3)) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel))),
% 5.82/2.02 file('/export/starexec/sandbox/tmp/tmp.Fb6urRC3jx/DTF2THF_19374.p',fact_10_f5)).
% 5.82/2.02 thf(f1859,plain,(
% 5.82/2.02 ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (mult @ a @ ring_ext @ a @ b @ r @ r3 @ r4) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3)) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel)) != $true)),
% 5.82/2.02 inference(subsumption_resolution,[status(thm)],[f1652,f1464])).
% 5.82/2.02 thf(f1464,plain,(
% 5.82/2.02 ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (mult @ a @ ring_ext @ a @ b @ r @ r2 @ r4) @ (mult @ a @ ring_ext @ a @ b @ r @ s @ s3)) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel)) = $true)),
% 5.82/2.02 inference(cnf_transformation,[status(thm)],[f367])).
% 5.82/2.02 thf(f367,plain,(
% 5.82/2.02 ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (mult @ a @ ring_ext @ a @ b @ r @ r2 @ r4) @ (mult @ a @ ring_ext @ a @ b @ r @ s @ s3)) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel)) = $true)),
% 5.82/2.02 inference(fool_elimination,[status(thm)],[f366])).
% 5.82/2.02 thf(f366,plain,(
% 5.82/2.02 (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (mult @ a @ ring_ext @ a @ b @ r @ r2 @ r4) @ (mult @ a @ ring_ext @ a @ b @ r @ s @ s3)) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel))),
% 5.82/2.02 inference(rectify,[status(thm)],[f12])).
% 5.82/2.02 thf(f12,axiom,(
% 5.82/2.02 (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (mult @ a @ ring_ext @ a @ b @ r @ r2 @ r4) @ (mult @ a @ ring_ext @ a @ b @ r @ s @ s3)) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel))),
% 5.82/2.02 file('/export/starexec/sandbox/tmp/tmp.Fb6urRC3jx/DTF2THF_19374.p',fact_11_f4)).
% 5.82/2.02 thf(f1652,plain,(
% 5.82/2.02 ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (mult @ a @ ring_ext @ a @ b @ r @ r2 @ r4) @ (mult @ a @ ring_ext @ a @ b @ r @ s @ s3)) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel)) != $true) | ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (mult @ a @ ring_ext @ a @ b @ r @ r3 @ r4) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3)) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel)) != $true)),
% 5.82/2.02 inference(trivial_inequality_removal,[status(thm)],[f1651])).
% 5.82/2.02 thf(f1651,plain,(
% 5.82/2.02 ($true != $true) | ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (mult @ a @ ring_ext @ a @ b @ r @ r2 @ r4) @ (mult @ a @ ring_ext @ a @ b @ r @ s @ s3)) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel)) != $true) | ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (mult @ a @ ring_ext @ a @ b @ r @ r3 @ r4) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3)) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel)) != $true)),
% 5.82/2.02 inference(superposition,[status(thm)],[f1515,f1449])).
% 5.82/2.02 thf(f1449,plain,(
% 5.82/2.02 ( ! [X2 : a,X3 : a,X0 : a,X1 : a] : (($true = (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ X3 @ X0) @ (mult @ a @ ring_ext @ a @ b @ r @ X1 @ X2)) @ (mult @ a @ ring_ext @ a @ b @ r @ X1 @ X3)) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel))) | ($true != (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ X0 @ X1) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel))) | ($true != (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ X2 @ X3) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel)))) )),
% 5.82/2.02 inference(cnf_transformation,[status(thm)],[f1052])).
% 5.82/2.02 thf(f1052,plain,(
% 5.82/2.02 ! [X0 : a,X1 : a,X2 : a,X3 : a] : (($true != (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ X2 @ X3) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel))) | ($true = (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ X3 @ X0) @ (mult @ a @ ring_ext @ a @ b @ r @ X1 @ X2)) @ (mult @ a @ ring_ext @ a @ b @ r @ X1 @ X3)) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel))) | ($true != (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ X0 @ X1) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel))))),
% 5.82/2.02 inference(flattening,[status(thm)],[f1051])).
% 5.82/2.02 thf(f1051,plain,(
% 5.82/2.02 ! [X0 : a,X1 : a,X2 : a,X3 : a] : ((($true = (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ X3 @ X0) @ (mult @ a @ ring_ext @ a @ b @ r @ X1 @ X2)) @ (mult @ a @ ring_ext @ a @ b @ r @ X1 @ X3)) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel))) | ($true != (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ X2 @ X3) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel)))) | ($true != (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ X0 @ X1) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel))))),
% 5.82/2.02 inference(ennf_transformation,[status(thm)],[f649])).
% 5.82/2.02 thf(f649,plain,(
% 5.82/2.02 ! [X0 : a,X1 : a,X2 : a,X3 : a] : (($true = (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ X0 @ X1) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel))) => (($true = (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ X2 @ X3) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel))) => ($true = (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ X3 @ X0) @ (mult @ a @ ring_ext @ a @ b @ r @ X1 @ X2)) @ (mult @ a @ ring_ext @ a @ b @ r @ X1 @ X3)) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel)))))),
% 5.82/2.02 inference(fool_elimination,[status(thm)],[f648])).
% 5.82/2.02 thf(f648,plain,(
% 5.82/2.02 ! [X0 : a,X1 : a,X2 : a,X3 : a] : ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ X0 @ X1) @ (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 @ X2 @ X3) @ (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 @ (add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ X3 @ X0) @ (mult @ a @ ring_ext @ a @ b @ r @ X1 @ X2)) @ (mult @ a @ ring_ext @ a @ b @ r @ X1 @ X3)) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel))))),
% 5.82/2.02 inference(rectify,[status(thm)],[f5])).
% 5.82/2.02 thf(f5,axiom,(
% 5.82/2.02 ! [X9 : a,X10 : a,X11 : a,X12 : a] : ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ X9 @ X10) @ (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 @ X11 @ X12) @ (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 @ (add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ X12 @ X9) @ (mult @ a @ ring_ext @ a @ b @ r @ X10 @ X11)) @ (mult @ a @ ring_ext @ a @ b @ r @ X10 @ X12)) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel))))),
% 5.82/2.02 file('/export/starexec/sandbox/tmp/tmp.Fb6urRC3jx/DTF2THF_19374.p',fact_4_closed__rel__add)).
% 5.82/2.02 thf(f1515,plain,(
% 5.82/2.02 ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ r2 @ r4)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ r3 @ r4))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3))) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel)) != $true)),
% 5.82/2.02 inference(cnf_transformation,[status(thm)],[f836])).
% 5.82/2.02 thf(f836,plain,(
% 5.82/2.02 ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ r2 @ r4)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ r3 @ r4))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3))) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel)) != $true)),
% 5.82/2.02 inference(flattening,[status(thm)],[f403])).
% 5.82/2.02 thf(f403,plain,(
% 5.82/2.02 ~((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ r2 @ r4)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ r3 @ r4))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3))) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel)) = $true)),
% 5.82/2.02 inference(fool_elimination,[status(thm)],[f402])).
% 5.82/2.02 thf(f402,plain,(
% 5.82/2.02 ~(member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ r2 @ r4)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ r3 @ r4))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3))) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel))),
% 5.82/2.02 inference(rectify,[status(thm)],[f277])).
% 5.82/2.02 thf(f277,negated_conjecture,(
% 5.82/2.02 ~(member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ r2 @ r4)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ r3 @ r4))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3))) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel))),
% 5.82/2.02 inference(negated_conjecture,[status(cth)],[f276])).
% 5.82/2.02 thf(f276,conjecture,(
% 5.82/2.02 (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (add @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ r2 @ r4)) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ r3 @ r4))) @ (mult @ a @ ring_ext @ a @ b @ r @ (mult @ a @ ring_ext @ a @ b @ r @ s @ s3) @ (mult @ a @ ring_ext @ a @ b @ r @ s2 @ s3))) @ (partial_carrier @ product_prod @ a @ a @ eq_eq_object_ext @ product_prod @ a @ a @ product_unit @ rel))),
% 5.82/2.02 file('/export/starexec/sandbox/tmp/tmp.Fb6urRC3jx/DTF2THF_19374.p',conj_0)).
% 5.82/2.02 % SZS output end Proof for DTF2THF_19374
% 5.82/2.02 % (19746)------------------------------
% 5.82/2.02 % (19746)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.82/2.02 % (19746)Termination reason: Refutation
% 5.82/2.02
% 5.82/2.02 % (19746)Memory used [KB]: 7675
% 5.82/2.02 % (19746)Time elapsed: 0.082 s
% 5.82/2.02 % (19746)Instructions burned: 254 (million)
% 5.82/2.02 % (19746)------------------------------
% 5.82/2.02 % (19746)------------------------------
% 5.82/2.02 % (19702)Success in time 0.416 s
%------------------------------------------------------------------------------