TSTP Solution File: ITP106^1 by cvc5---1.0.5
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cvc5---1.0.5
% Problem : ITP106^1 : TPTP v8.1.2. Released v7.5.0.
% Transfm : none
% Format : tptp
% Command : do_cvc5 %s %d
% Computer : n016.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 : Thu Aug 31 03:18:17 EDT 2023
% Result : Theorem 0.48s 0.86s
% Output : Proof 0.48s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.17 % Problem : ITP106^1 : TPTP v8.1.2. Released v7.5.0.
% 0.17/0.18 % Command : do_cvc5 %s %d
% 0.17/0.39 % Computer : n016.cluster.edu
% 0.17/0.39 % Model : x86_64 x86_64
% 0.17/0.39 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.39 % Memory : 8042.1875MB
% 0.17/0.39 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.39 % CPULimit : 300
% 0.17/0.39 % WCLimit : 300
% 0.17/0.39 % DateTime : Sun Aug 27 16:59:44 EDT 2023
% 0.17/0.39 % CPUTime :
% 0.24/0.57 %----Proving TH0
% 0.24/0.57 %------------------------------------------------------------------------------
% 0.24/0.57 % File : ITP106^1 : TPTP v8.1.2. Released v7.5.0.
% 0.24/0.57 % Domain : Interactive Theorem Proving
% 0.24/0.57 % Problem : Sledgehammer Localization problem prob_1149__8998378_1
% 0.24/0.57 % Version : Especial.
% 0.24/0.57 % English :
% 0.24/0.57
% 0.24/0.57 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.24/0.57 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.24/0.57 % Source : [Des21]
% 0.24/0.57 % Names : Localization/prob_1149__8998378_1 [Des21]
% 0.24/0.57
% 0.24/0.57 % Status : Theorem
% 0.24/0.57 % Rating : 0.31 v8.1.0, 0.36 v7.5.0
% 0.24/0.57 % Syntax : Number of formulae : 339 ( 106 unt; 56 typ; 0 def)
% 0.24/0.57 % Number of atoms : 801 ( 382 equ; 0 cnn)
% 0.24/0.57 % Maximal formula atoms : 17 ( 2 avg)
% 0.24/0.57 % Number of connectives : 3859 ( 8 ~; 1 |; 18 &;3401 @)
% 0.24/0.57 % ( 0 <=>; 431 =>; 0 <=; 0 <~>)
% 0.24/0.57 % Maximal formula depth : 16 ( 6 avg)
% 0.24/0.57 % Number of types : 11 ( 10 usr)
% 0.24/0.57 % Number of type conns : 69 ( 69 >; 0 *; 0 +; 0 <<)
% 0.24/0.57 % Number of symbols : 47 ( 46 usr; 8 con; 0-3 aty)
% 0.24/0.57 % Number of variables : 682 ( 13 ^; 653 !; 16 ?; 682 :)
% 0.24/0.57 % SPC : TH0_THM_EQU_NAR
% 0.24/0.57
% 0.24/0.57 % Comments : This file was generated by Sledgehammer 2021-02-23 15:40:36.834
% 0.24/0.57 %------------------------------------------------------------------------------
% 0.24/0.57 % Could-be-implicit typings (10)
% 0.24/0.57 thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Congruence__Oeq____object__Oeq____object____ext_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Ounit_J_J,type,
% 0.24/0.57 partia1993116613t_unit: $tType ).
% 0.24/0.57
% 0.24/0.57 thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_Itf__a_Mt__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J_J,type,
% 0.24/0.57 partia1833973666xt_a_b: $tType ).
% 0.24/0.57
% 0.24/0.57 thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_Itf__a_Mt__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Product____Type__Ounit_J_J,type,
% 0.24/0.57 partia96731725t_unit: $tType ).
% 0.24/0.57
% 0.24/0.57 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.24/0.57 set_Product_prod_a_a: $tType ).
% 0.24/0.57
% 0.24/0.57 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.24/0.57 product_prod_a_a: $tType ).
% 0.24/0.57
% 0.24/0.57 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.24/0.57 set_a: $tType ).
% 0.24/0.57
% 0.24/0.57 thf(ty_n_t__Nat__Onat,type,
% 0.24/0.57 nat: $tType ).
% 0.24/0.57
% 0.24/0.57 thf(ty_n_t__Int__Oint,type,
% 0.24/0.57 int: $tType ).
% 0.24/0.57
% 0.24/0.57 thf(ty_n_tf__b,type,
% 0.24/0.57 b: $tType ).
% 0.24/0.57
% 0.24/0.57 thf(ty_n_tf__a,type,
% 0.24/0.57 a: $tType ).
% 0.24/0.57
% 0.24/0.57 % Explicit typings (46)
% 0.24/0.57 thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Congruence__Oeq____object__Oeq____object____ext_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Ounit_J,type,
% 0.24/0.57 partia206007992t_unit: partia1993116613t_unit > set_Product_prod_a_a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Product____Type__Ounit_J,type,
% 0.24/0.57 partia1955795460t_unit: partia96731725t_unit > set_a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J,type,
% 0.24/0.57 partia1066395285xt_a_b: partia1833973666xt_a_b > set_a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Group_OUnits_001tf__a_001t__Product____Type__Ounit,type,
% 0.24/0.57 units_a_Product_unit: partia96731725t_unit > set_a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Group_OUnits_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 0.24/0.57 units_a_ring_ext_a_b: partia1833973666xt_a_b > set_a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Group_Ocomm__group_001tf__a_001t__Product____Type__Ounit,type,
% 0.24/0.57 comm_g1684316527t_unit: partia96731725t_unit > $o ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Group_Ocomm__group_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 0.24/0.57 comm_g791708116xt_a_b: partia1833973666xt_a_b > $o ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Group_Ogroup_001tf__a_001t__Product____Type__Ounit,type,
% 0.24/0.57 group_a_Product_unit: partia96731725t_unit > $o ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Group_Ogroup_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 0.24/0.57 group_a_ring_ext_a_b: partia1833973666xt_a_b > $o ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Group_Omonoid_Omult_001tf__a_001t__Product____Type__Ounit,type,
% 0.24/0.57 mult_a_Product_unit: partia96731725t_unit > a > a > a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Group_Omonoid_Omult_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 0.24/0.57 mult_a_ring_ext_a_b: partia1833973666xt_a_b > a > a > a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Group_Omonoid_Oone_001tf__a_001t__Product____Type__Ounit,type,
% 0.24/0.57 one_a_Product_unit: partia96731725t_unit > a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Group_Omonoid_Oone_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 0.24/0.57 one_a_ring_ext_a_b: partia1833973666xt_a_b > a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Group_Ounits__of_001tf__a_001t__Product____Type__Ounit,type,
% 0.24/0.57 units_873712258t_unit: partia96731725t_unit > partia96731725t_unit ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Group_Ounits__of_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 0.24/0.57 units_1411277569xt_a_b: partia1833973666xt_a_b > partia96731725t_unit ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
% 0.24/0.57 minus_minus_int: int > int > int ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 0.24/0.57 minus_minus_nat: nat > nat > nat ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
% 0.24/0.57 plus_plus_int: int > int > int ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 0.24/0.57 plus_plus_nat: nat > nat > nat ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
% 0.24/0.57 uminus_uminus_int: int > int ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Nat_OSuc,type,
% 0.24/0.57 suc: nat > nat ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 0.24/0.57 semiri2019852685at_int: nat > int ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 0.24/0.57 semiri1382578993at_nat: nat > nat ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
% 0.24/0.57 product_Pair_a_a: a > a > product_prod_a_a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Ring_Oa__inv_001tf__a_001tf__b,type,
% 0.24/0.57 a_inv_a_b: partia1833973666xt_a_b > a > a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Ring_Oa__minus_001tf__a_001tf__b,type,
% 0.24/0.57 a_minus_a_b: partia1833973666xt_a_b > a > a > a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Ring_Oabelian__group_001tf__a_001tf__b,type,
% 0.24/0.57 abelian_group_a_b: partia1833973666xt_a_b > $o ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Int__Oint,type,
% 0.24/0.57 add_pow_a_b_int: partia1833973666xt_a_b > int > a > a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Nat__Onat,type,
% 0.24/0.57 add_pow_a_b_nat: partia1833973666xt_a_b > nat > a > a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Ring_Ofield_001tf__a_001tf__b,type,
% 0.24/0.57 field_a_b: partia1833973666xt_a_b > $o ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
% 0.24/0.57 add_a_b: partia1833973666xt_a_b > a > a > a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Ring_Oring_Omore_001tf__a_001tf__b,type,
% 0.24/0.57 more_a_b: partia1833973666xt_a_b > b ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
% 0.24/0.57 zero_a_b: partia1833973666xt_a_b > a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
% 0.24/0.57 semiring_a_b: partia1833973666xt_a_b > $o ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.24/0.57 collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.24/0.57 collect_a: ( a > $o ) > set_a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.24/0.57 member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
% 0.24/0.57
% 0.24/0.57 thf(sy_c_member_001tf__a,type,
% 0.24/0.57 member_a: a > set_a > $o ).
% 0.24/0.57
% 0.24/0.57 thf(sy_v_R,type,
% 0.24/0.57 r: partia1833973666xt_a_b ).
% 0.24/0.57
% 0.24/0.57 thf(sy_v_r,type,
% 0.24/0.57 r2: a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_v_r_H,type,
% 0.24/0.57 r3: a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_v_r_H_H,type,
% 0.24/0.57 r4: a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_v_rel,type,
% 0.24/0.57 rel: partia1993116613t_unit ).
% 0.24/0.57
% 0.24/0.57 thf(sy_v_s,type,
% 0.24/0.57 s: a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_v_s_H,type,
% 0.24/0.57 s2: a ).
% 0.24/0.57
% 0.24/0.57 thf(sy_v_s_H_H,type,
% 0.24/0.57 s3: a ).
% 0.24/0.57
% 0.24/0.57 % Relevant facts (282)
% 0.24/0.57 thf(fact_0_right__add__eq,axiom,
% 0.24/0.57 ! [A: a,B: a,C: a] :
% 0.24/0.57 ( ( A = B )
% 0.24/0.57 => ( ( add_a_b @ r @ C @ A )
% 0.24/0.57 = ( add_a_b @ r @ C @ B ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % right_add_eq
% 0.24/0.57 thf(fact_1_assms_I3_J,axiom,
% 0.24/0.57 member449909584od_a_a @ ( product_Pair_a_a @ r4 @ s3 ) @ ( partia206007992t_unit @ rel ) ).
% 0.24/0.57
% 0.24/0.57 % assms(3)
% 0.24/0.57 thf(fact_2_assms_I2_J,axiom,
% 0.24/0.57 member449909584od_a_a @ ( product_Pair_a_a @ r3 @ s2 ) @ ( partia206007992t_unit @ rel ) ).
% 0.24/0.57
% 0.24/0.57 % assms(2)
% 0.24/0.57 thf(fact_3_assms_I1_J,axiom,
% 0.24/0.57 member449909584od_a_a @ ( product_Pair_a_a @ r2 @ s ) @ ( partia206007992t_unit @ rel ) ).
% 0.24/0.57
% 0.24/0.57 % assms(1)
% 0.24/0.57 thf(fact_4_closed__rel__add,axiom,
% 0.24/0.57 ! [R: a,S: a,R2: a,S2: a] :
% 0.24/0.57 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
% 0.24/0.57 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S2 ) @ ( partia206007992t_unit @ rel ) )
% 0.24/0.57 => ( member449909584od_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 ) ) @ ( partia206007992t_unit @ rel ) ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % closed_rel_add
% 0.24/0.57 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,
% 0.24/0.57 ( ( 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 ) )
% 0.24/0.57 = ( 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 ) ) ).
% 0.24/0.57
% 0.24/0.57 % \<open>s \<otimes> s'' \<otimes> (s' \<otimes> s'') = s \<otimes> (s'' \<otimes> s') \<otimes> s''\<close>
% 0.24/0.57 thf(fact_6_f9,axiom,
% 0.24/0.57 ( ( 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 ) )
% 0.24/0.57 = ( 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 ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % f9
% 0.24/0.57 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,
% 0.24/0.57 ( ( 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 ) )
% 0.24/0.57 = ( 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 ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % \<open>s \<otimes> s'' \<otimes> (r' \<otimes> r'') = s'' \<otimes> s \<otimes> (r' \<otimes> r'')\<close>
% 0.24/0.57 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,
% 0.24/0.57 ( ( 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 ) )
% 0.24/0.57 = ( 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 ) ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % \<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>
% 0.24/0.57 thf(fact_9_f10,axiom,
% 0.24/0.57 ( ( 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 ) )
% 0.24/0.57 = ( 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 ) ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % f10
% 0.24/0.57 thf(fact_10_f5,axiom,
% 0.24/0.57 member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( partia206007992t_unit @ rel ) ).
% 0.24/0.57
% 0.24/0.57 % f5
% 0.24/0.57 thf(fact_11_f4,axiom,
% 0.24/0.57 member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) ) @ ( partia206007992t_unit @ rel ) ).
% 0.24/0.57
% 0.24/0.57 % f4
% 0.24/0.57 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,
% 0.24/0.57 ( ( 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 ) )
% 0.24/0.57 = ( 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 ) ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % \<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>
% 0.24/0.57 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,
% 0.24/0.57 ( ( 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 ) )
% 0.24/0.57 = ( 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 ) ) ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % \<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>
% 0.24/0.57 thf(fact_14_f7,axiom,
% 0.24/0.57 ( ( 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 ) )
% 0.24/0.57 = ( 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 ) ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % f7
% 0.24/0.57 thf(fact_15_f8,axiom,
% 0.24/0.57 ( ( 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 ) ) ) )
% 0.24/0.57 = ( 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 ) ) ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % f8
% 0.24/0.57 thf(fact_16_f12,axiom,
% 0.24/0.57 member449909584od_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 ) ) @ ( partia206007992t_unit @ rel ) ).
% 0.24/0.57
% 0.24/0.57 % f12
% 0.24/0.57 thf(fact_17_l__distr,axiom,
% 0.24/0.57 ! [X: a,Y: a,Z: a] :
% 0.24/0.57 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
% 0.24/0.57 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % l_distr
% 0.24/0.57 thf(fact_18_r__distr,axiom,
% 0.24/0.57 ! [X: a,Y: a,Z: a] :
% 0.24/0.57 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X @ Y ) )
% 0.24/0.57 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % r_distr
% 0.24/0.57 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,
% 0.24/0.57 ( ( 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 ) ) ) ) )
% 0.24/0.57 = ( zero_a_b @ r ) ) ).
% 0.24/0.57
% 0.24/0.57 % \<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>
% 0.24/0.57 thf(fact_20_prod_Oinject,axiom,
% 0.24/0.57 ! [X1: a,X2: a,Y1: a,Y2: a] :
% 0.24/0.57 ( ( ( product_Pair_a_a @ X1 @ X2 )
% 0.24/0.57 = ( product_Pair_a_a @ Y1 @ Y2 ) )
% 0.24/0.57 = ( ( X1 = Y1 )
% 0.24/0.57 & ( X2 = Y2 ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % prod.inject
% 0.24/0.57 thf(fact_21_old_Oprod_Oinject,axiom,
% 0.24/0.57 ! [A: a,B: a,A2: a,B2: a] :
% 0.24/0.57 ( ( ( product_Pair_a_a @ A @ B )
% 0.24/0.57 = ( product_Pair_a_a @ A2 @ B2 ) )
% 0.24/0.57 = ( ( A = A2 )
% 0.24/0.57 & ( B = B2 ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % old.prod.inject
% 0.24/0.57 thf(fact_22_local_Osemiring__axioms,axiom,
% 0.24/0.57 semiring_a_b @ r ).
% 0.24/0.57
% 0.24/0.57 % local.semiring_axioms
% 0.24/0.57 thf(fact_23_m__assoc,axiom,
% 0.24/0.57 ! [X: a,Y: a,Z: a] :
% 0.24/0.57 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z )
% 0.24/0.57 = ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % m_assoc
% 0.24/0.57 thf(fact_24_m__comm,axiom,
% 0.24/0.57 ! [X: a,Y: a] :
% 0.24/0.57 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
% 0.24/0.57 = ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % m_comm
% 0.24/0.57 thf(fact_25_m__lcomm,axiom,
% 0.24/0.57 ! [X: a,Y: a,Z: a] :
% 0.24/0.57 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
% 0.24/0.57 = ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % m_lcomm
% 0.24/0.57 thf(fact_26_add_Om__assoc,axiom,
% 0.24/0.57 ! [X: a,Y: a,Z: a] :
% 0.24/0.57 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
% 0.24/0.57 = ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % add.m_assoc
% 0.24/0.57 thf(fact_27_add_Om__comm,axiom,
% 0.24/0.57 ! [X: a,Y: a] :
% 0.24/0.57 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( add_a_b @ r @ X @ Y )
% 0.24/0.57 = ( add_a_b @ r @ Y @ X ) ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % add.m_comm
% 0.24/0.57 thf(fact_28_add_Om__lcomm,axiom,
% 0.24/0.57 ! [X: a,Y: a,Z: a] :
% 0.24/0.57 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) )
% 0.24/0.57 = ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % add.m_lcomm
% 0.24/0.57 thf(fact_29_local_Oright__minus__eq,axiom,
% 0.24/0.57 ! [A: a,B: a,C: a] :
% 0.24/0.57 ( ( A = B )
% 0.24/0.57 => ( ( a_minus_a_b @ r @ C @ A )
% 0.24/0.57 = ( a_minus_a_b @ r @ C @ B ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % local.right_minus_eq
% 0.24/0.57 thf(fact_30_local_Ominus__unique,axiom,
% 0.24/0.57 ! [Y: a,X: a,Y3: a] :
% 0.24/0.57 ( ( ( add_a_b @ r @ Y @ X )
% 0.24/0.57 = ( zero_a_b @ r ) )
% 0.24/0.57 => ( ( ( add_a_b @ r @ X @ Y3 )
% 0.24/0.57 = ( zero_a_b @ r ) )
% 0.24/0.57 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( Y = Y3 ) ) ) ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % local.minus_unique
% 0.24/0.57 thf(fact_31_add_Or__inv__ex,axiom,
% 0.24/0.57 ! [X: a] :
% 0.24/0.57 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ? [X3: a] :
% 0.24/0.57 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 & ( ( add_a_b @ r @ X @ X3 )
% 0.24/0.57 = ( zero_a_b @ r ) ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % add.r_inv_ex
% 0.24/0.57 thf(fact_32_add_Oone__unique,axiom,
% 0.24/0.57 ! [U: a] :
% 0.24/0.57 ( ( member_a @ U @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ! [X3: a] :
% 0.24/0.57 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( add_a_b @ r @ U @ X3 )
% 0.24/0.57 = X3 ) )
% 0.24/0.57 => ( U
% 0.24/0.57 = ( zero_a_b @ r ) ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % add.one_unique
% 0.24/0.57 thf(fact_33_add_Ol__inv__ex,axiom,
% 0.24/0.57 ! [X: a] :
% 0.24/0.57 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ? [X3: a] :
% 0.24/0.57 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 & ( ( add_a_b @ r @ X3 @ X )
% 0.24/0.57 = ( zero_a_b @ r ) ) ) ) ).
% 0.24/0.57
% 0.24/0.57 % add.l_inv_ex
% 0.24/0.57 thf(fact_34_add_Oinv__comm,axiom,
% 0.24/0.57 ! [X: a,Y: a] :
% 0.24/0.57 ( ( ( add_a_b @ r @ X @ Y )
% 0.24/0.57 = ( zero_a_b @ r ) )
% 0.24/0.57 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.57 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_a_b @ r @ Y @ X )
% 0.24/0.58 = ( zero_a_b @ r ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.inv_comm
% 0.24/0.58 thf(fact_35_right__inv__add,axiom,
% 0.24/0.58 ! [A: a,B: a,C: a] :
% 0.24/0.58 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( a_minus_a_b @ r @ ( a_minus_a_b @ r @ C @ A ) @ B )
% 0.24/0.58 = ( a_minus_a_b @ r @ C @ ( add_a_b @ r @ A @ B ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % right_inv_add
% 0.24/0.58 thf(fact_36_four__elem__comm,axiom,
% 0.24/0.58 ! [A: a,B: a,C: a,D: a] :
% 0.24/0.58 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ D @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( a_minus_a_b @ r @ ( add_a_b @ r @ ( a_minus_a_b @ r @ A @ C ) @ B ) @ D )
% 0.24/0.58 = ( a_minus_a_b @ r @ ( a_minus_a_b @ r @ ( add_a_b @ r @ A @ B ) @ C ) @ D ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % four_elem_comm
% 0.24/0.58 thf(fact_37_local_Oadd_Oright__cancel,axiom,
% 0.24/0.58 ! [X: a,Y: a,Z: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( ( add_a_b @ r @ Y @ X )
% 0.24/0.58 = ( add_a_b @ r @ Z @ X ) )
% 0.24/0.58 = ( Y = Z ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % local.add.right_cancel
% 0.24/0.58 thf(fact_38_add_Om__closed,axiom,
% 0.24/0.58 ! [X: a,Y: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( member_a @ ( add_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.m_closed
% 0.24/0.58 thf(fact_39_zero__closed,axiom,
% 0.24/0.58 member_a @ ( zero_a_b @ r ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.24/0.58
% 0.24/0.58 % zero_closed
% 0.24/0.58 thf(fact_40_semiring__simprules_I3_J,axiom,
% 0.24/0.58 ! [X: a,Y: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % semiring_simprules(3)
% 0.24/0.58 thf(fact_41_minus__closed,axiom,
% 0.24/0.58 ! [X: a,Y: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( member_a @ ( a_minus_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % minus_closed
% 0.24/0.58 thf(fact_42_r__zero,axiom,
% 0.24/0.58 ! [X: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_a_b @ r @ X @ ( zero_a_b @ r ) )
% 0.24/0.58 = X ) ) ).
% 0.24/0.58
% 0.24/0.58 % r_zero
% 0.24/0.58 thf(fact_43_l__zero,axiom,
% 0.24/0.58 ! [X: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X )
% 0.24/0.58 = X ) ) ).
% 0.24/0.58
% 0.24/0.58 % l_zero
% 0.24/0.58 thf(fact_44_add_Or__cancel__one_H,axiom,
% 0.24/0.58 ! [X: a,A: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( X
% 0.24/0.58 = ( add_a_b @ r @ A @ X ) )
% 0.24/0.58 = ( A
% 0.24/0.58 = ( zero_a_b @ r ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.r_cancel_one'
% 0.24/0.58 thf(fact_45_mem__Collect__eq,axiom,
% 0.24/0.58 ! [A: product_prod_a_a,P: product_prod_a_a > $o] :
% 0.24/0.58 ( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P ) )
% 0.24/0.58 = ( P @ A ) ) ).
% 0.24/0.58
% 0.24/0.58 % mem_Collect_eq
% 0.24/0.58 thf(fact_46_mem__Collect__eq,axiom,
% 0.24/0.58 ! [A: a,P: a > $o] :
% 0.24/0.58 ( ( member_a @ A @ ( collect_a @ P ) )
% 0.24/0.58 = ( P @ A ) ) ).
% 0.24/0.58
% 0.24/0.58 % mem_Collect_eq
% 0.24/0.58 thf(fact_47_Collect__mem__eq,axiom,
% 0.24/0.58 ! [A3: set_Product_prod_a_a] :
% 0.24/0.58 ( ( collec645855634od_a_a
% 0.24/0.58 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ A3 ) )
% 0.24/0.58 = A3 ) ).
% 0.24/0.58
% 0.24/0.58 % Collect_mem_eq
% 0.24/0.58 thf(fact_48_Collect__mem__eq,axiom,
% 0.24/0.58 ! [A3: set_a] :
% 0.24/0.58 ( ( collect_a
% 0.24/0.58 @ ^ [X4: a] : ( member_a @ X4 @ A3 ) )
% 0.24/0.58 = A3 ) ).
% 0.24/0.58
% 0.24/0.58 % Collect_mem_eq
% 0.24/0.58 thf(fact_49_Collect__cong,axiom,
% 0.24/0.58 ! [P: a > $o,Q: a > $o] :
% 0.24/0.58 ( ! [X3: a] :
% 0.24/0.58 ( ( P @ X3 )
% 0.24/0.58 = ( Q @ X3 ) )
% 0.24/0.58 => ( ( collect_a @ P )
% 0.24/0.58 = ( collect_a @ Q ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % Collect_cong
% 0.24/0.58 thf(fact_50_Collect__cong,axiom,
% 0.24/0.58 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 0.24/0.58 ( ! [X3: product_prod_a_a] :
% 0.24/0.58 ( ( P @ X3 )
% 0.24/0.58 = ( Q @ X3 ) )
% 0.24/0.58 => ( ( collec645855634od_a_a @ P )
% 0.24/0.58 = ( collec645855634od_a_a @ Q ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % Collect_cong
% 0.24/0.58 thf(fact_51_add_Or__cancel__one,axiom,
% 0.24/0.58 ! [X: a,A: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( ( add_a_b @ r @ A @ X )
% 0.24/0.58 = X )
% 0.24/0.58 = ( A
% 0.24/0.58 = ( zero_a_b @ r ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.r_cancel_one
% 0.24/0.58 thf(fact_52_add_Ol__cancel__one_H,axiom,
% 0.24/0.58 ! [X: a,A: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( X
% 0.24/0.58 = ( add_a_b @ r @ X @ A ) )
% 0.24/0.58 = ( A
% 0.24/0.58 = ( zero_a_b @ r ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.l_cancel_one'
% 0.24/0.58 thf(fact_53_add_Ol__cancel__one,axiom,
% 0.24/0.58 ! [X: a,A: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( ( add_a_b @ r @ X @ A )
% 0.24/0.58 = X )
% 0.24/0.58 = ( A
% 0.24/0.58 = ( zero_a_b @ r ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.l_cancel_one
% 0.24/0.58 thf(fact_54_r__null,axiom,
% 0.24/0.58 ! [X: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
% 0.24/0.58 = ( zero_a_b @ r ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % r_null
% 0.24/0.58 thf(fact_55_l__null,axiom,
% 0.24/0.58 ! [X: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
% 0.24/0.58 = ( zero_a_b @ r ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % l_null
% 0.24/0.58 thf(fact_56_old_Oprod_Oinducts,axiom,
% 0.24/0.58 ! [P: product_prod_a_a > $o,Prod: product_prod_a_a] :
% 0.24/0.58 ( ! [A4: a,B3: a] : ( P @ ( product_Pair_a_a @ A4 @ B3 ) )
% 0.24/0.58 => ( P @ Prod ) ) ).
% 0.24/0.58
% 0.24/0.58 % old.prod.inducts
% 0.24/0.58 thf(fact_57_old_Oprod_Oexhaust,axiom,
% 0.24/0.58 ! [Y: product_prod_a_a] :
% 0.24/0.58 ~ ! [A4: a,B3: a] :
% 0.24/0.58 ( Y
% 0.24/0.58 != ( product_Pair_a_a @ A4 @ B3 ) ) ).
% 0.24/0.58
% 0.24/0.58 % old.prod.exhaust
% 0.24/0.58 thf(fact_58_Pair__inject,axiom,
% 0.24/0.58 ! [A: a,B: a,A2: a,B2: a] :
% 0.24/0.58 ( ( ( product_Pair_a_a @ A @ B )
% 0.24/0.58 = ( product_Pair_a_a @ A2 @ B2 ) )
% 0.24/0.58 => ~ ( ( A = A2 )
% 0.24/0.58 => ( B != B2 ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % Pair_inject
% 0.24/0.58 thf(fact_59_prod__cases,axiom,
% 0.24/0.58 ! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
% 0.24/0.58 ( ! [A4: a,B3: a] : ( P @ ( product_Pair_a_a @ A4 @ B3 ) )
% 0.24/0.58 => ( P @ P2 ) ) ).
% 0.24/0.58
% 0.24/0.58 % prod_cases
% 0.24/0.58 thf(fact_60_surj__pair,axiom,
% 0.24/0.58 ! [P2: product_prod_a_a] :
% 0.24/0.58 ? [X3: a,Y4: a] :
% 0.24/0.58 ( P2
% 0.24/0.58 = ( product_Pair_a_a @ X3 @ Y4 ) ) ).
% 0.24/0.58
% 0.24/0.58 % surj_pair
% 0.24/0.58 thf(fact_61_f11,axiom,
% 0.24/0.58 ( ( 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 ) ) ) ) ) )
% 0.24/0.58 = ( zero_a_b @ r ) ) ).
% 0.24/0.58
% 0.24/0.58 % f11
% 0.24/0.58 thf(fact_62_semiring_Osemiring__simprules_I11_J,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,X: a] :
% 0.24/0.58 ( ( semiring_a_b @ R3 )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( add_a_b @ R3 @ X @ ( zero_a_b @ R3 ) )
% 0.24/0.58 = X ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % semiring.semiring_simprules(11)
% 0.24/0.58 thf(fact_63_semiring_Osemiring__simprules_I6_J,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,X: a] :
% 0.24/0.58 ( ( semiring_a_b @ R3 )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( add_a_b @ R3 @ ( zero_a_b @ R3 ) @ X )
% 0.24/0.58 = X ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % semiring.semiring_simprules(6)
% 0.24/0.58 thf(fact_64_semiring_Ol__null,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,X: a] :
% 0.24/0.58 ( ( semiring_a_b @ R3 )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ R3 @ ( zero_a_b @ R3 ) @ X )
% 0.24/0.58 = ( zero_a_b @ R3 ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % semiring.l_null
% 0.24/0.58 thf(fact_65_semiring_Or__null,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,X: a] :
% 0.24/0.58 ( ( semiring_a_b @ R3 )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ R3 @ X @ ( zero_a_b @ R3 ) )
% 0.24/0.58 = ( zero_a_b @ R3 ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % semiring.r_null
% 0.24/0.58 thf(fact_66_semiring_Or__distr,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.24/0.58 ( ( semiring_a_b @ R3 )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ R3 @ Z @ ( add_a_b @ R3 @ X @ Y ) )
% 0.24/0.58 = ( add_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R3 @ Z @ Y ) ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % semiring.r_distr
% 0.24/0.58 thf(fact_67_semiring_Ol__distr,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.24/0.58 ( ( semiring_a_b @ R3 )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ R3 @ ( add_a_b @ R3 @ X @ Y ) @ Z )
% 0.24/0.58 = ( add_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % semiring.l_distr
% 0.24/0.58 thf(fact_68_add__pow__rdistr__int,axiom,
% 0.24/0.58 ! [A: a,B: a,K: int] :
% 0.24/0.58 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_int @ r @ K @ B ) )
% 0.24/0.58 = ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_pow_rdistr_int
% 0.24/0.58 thf(fact_69_add__pow__ldistr__int,axiom,
% 0.24/0.58 ! [A: a,B: a,K: int] :
% 0.24/0.58 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_int @ r @ K @ A ) @ B )
% 0.24/0.58 = ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_pow_ldistr_int
% 0.24/0.58 thf(fact_70_inv__unique,axiom,
% 0.24/0.58 ! [Y: a,X: a,Y3: a] :
% 0.24/0.58 ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
% 0.24/0.58 = ( one_a_ring_ext_a_b @ r ) )
% 0.24/0.58 => ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y3 )
% 0.24/0.58 = ( one_a_ring_ext_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( Y = Y3 ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % inv_unique
% 0.24/0.58 thf(fact_71_one__unique,axiom,
% 0.24/0.58 ! [U: a] :
% 0.24/0.58 ( ( member_a @ U @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ! [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ r @ U @ X3 )
% 0.24/0.58 = X3 ) )
% 0.24/0.58 => ( U
% 0.24/0.58 = ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % one_unique
% 0.24/0.58 thf(fact_72_add_Oint__pow__distrib,axiom,
% 0.24/0.58 ! [X: a,Y: a,I: int] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
% 0.24/0.58 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.int_pow_distrib
% 0.24/0.58 thf(fact_73_add_Oint__pow__mult__distrib,axiom,
% 0.24/0.58 ! [X: a,Y: a,I: int] :
% 0.24/0.58 ( ( ( add_a_b @ r @ X @ Y )
% 0.24/0.58 = ( add_a_b @ r @ Y @ X ) )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
% 0.24/0.58 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.int_pow_mult_distrib
% 0.24/0.58 thf(fact_74_semiring__simprules_I4_J,axiom,
% 0.24/0.58 member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.24/0.58
% 0.24/0.58 % semiring_simprules(4)
% 0.24/0.58 thf(fact_75_add_Oint__pow__closed,axiom,
% 0.24/0.58 ! [X: a,I: int] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( member_a @ ( add_pow_a_b_int @ r @ I @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.int_pow_closed
% 0.24/0.58 thf(fact_76_add_Oint__pow__one,axiom,
% 0.24/0.58 ! [Z: int] :
% 0.24/0.58 ( ( add_pow_a_b_int @ r @ Z @ ( zero_a_b @ r ) )
% 0.24/0.58 = ( zero_a_b @ r ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.int_pow_one
% 0.24/0.58 thf(fact_77_l__one,axiom,
% 0.24/0.58 ! [X: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X )
% 0.24/0.58 = X ) ) ).
% 0.24/0.58
% 0.24/0.58 % l_one
% 0.24/0.58 thf(fact_78_r__one,axiom,
% 0.24/0.58 ! [X: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( one_a_ring_ext_a_b @ r ) )
% 0.24/0.58 = X ) ) ).
% 0.24/0.58
% 0.24/0.58 % r_one
% 0.24/0.58 thf(fact_79_semiring_Osemiring__simprules_I4_J,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b] :
% 0.24/0.58 ( ( semiring_a_b @ R3 )
% 0.24/0.58 => ( member_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % semiring.semiring_simprules(4)
% 0.24/0.58 thf(fact_80_semiring_Osemiring__simprules_I9_J,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,X: a] :
% 0.24/0.58 ( ( semiring_a_b @ R3 )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ R3 @ ( one_a_ring_ext_a_b @ R3 ) @ X )
% 0.24/0.58 = X ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % semiring.semiring_simprules(9)
% 0.24/0.58 thf(fact_81_semiring_Osemiring__simprules_I3_J,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,X: a,Y: a] :
% 0.24/0.58 ( ( semiring_a_b @ R3 )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( member_a @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % semiring.semiring_simprules(3)
% 0.24/0.58 thf(fact_82_semiring_Osemiring__simprules_I8_J,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.24/0.58 ( ( semiring_a_b @ R3 )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) @ Z )
% 0.24/0.58 = ( mult_a_ring_ext_a_b @ R3 @ X @ ( mult_a_ring_ext_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % semiring.semiring_simprules(8)
% 0.24/0.58 thf(fact_83_semiring_Osemiring__simprules_I1_J,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,X: a,Y: a] :
% 0.24/0.58 ( ( semiring_a_b @ R3 )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( member_a @ ( add_a_b @ R3 @ X @ Y ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % semiring.semiring_simprules(1)
% 0.24/0.58 thf(fact_84_semiring_Osemiring__simprules_I5_J,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.24/0.58 ( ( semiring_a_b @ R3 )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( add_a_b @ R3 @ ( add_a_b @ R3 @ X @ Y ) @ Z )
% 0.24/0.58 = ( add_a_b @ R3 @ X @ ( add_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % semiring.semiring_simprules(5)
% 0.24/0.58 thf(fact_85_semiring_Osemiring__simprules_I7_J,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,X: a,Y: a] :
% 0.24/0.58 ( ( semiring_a_b @ R3 )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( add_a_b @ R3 @ X @ Y )
% 0.24/0.58 = ( add_a_b @ R3 @ Y @ X ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % semiring.semiring_simprules(7)
% 0.24/0.58 thf(fact_86_semiring_Osemiring__simprules_I12_J,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.24/0.58 ( ( semiring_a_b @ R3 )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( add_a_b @ R3 @ X @ ( add_a_b @ R3 @ Y @ Z ) )
% 0.24/0.58 = ( add_a_b @ R3 @ Y @ ( add_a_b @ R3 @ X @ Z ) ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % semiring.semiring_simprules(12)
% 0.24/0.58 thf(fact_87_semiring_Osemiring__simprules_I2_J,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b] :
% 0.24/0.58 ( ( semiring_a_b @ R3 )
% 0.24/0.58 => ( member_a @ ( zero_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % semiring.semiring_simprules(2)
% 0.24/0.58 thf(fact_88_cring__fieldI2,axiom,
% 0.24/0.58 ( ( ( zero_a_b @ r )
% 0.24/0.58 != ( one_a_ring_ext_a_b @ r ) )
% 0.24/0.58 => ( ! [A4: a] :
% 0.24/0.58 ( ( member_a @ A4 @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( A4
% 0.24/0.58 != ( zero_a_b @ r ) )
% 0.24/0.58 => ? [X5: a] :
% 0.24/0.58 ( ( member_a @ X5 @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 & ( ( mult_a_ring_ext_a_b @ r @ A4 @ X5 )
% 0.24/0.58 = ( one_a_ring_ext_a_b @ r ) ) ) ) )
% 0.24/0.58 => ( field_a_b @ r ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % cring_fieldI2
% 0.24/0.58 thf(fact_89_add_Oint__pow__mult,axiom,
% 0.24/0.58 ! [X: a,I: int,J: int] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_pow_a_b_int @ r @ ( plus_plus_int @ I @ J ) @ X )
% 0.24/0.58 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ J @ X ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.int_pow_mult
% 0.24/0.58 thf(fact_90_add__pow__ldistr,axiom,
% 0.24/0.58 ! [A: a,B: a,K: nat] :
% 0.24/0.58 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_nat @ r @ K @ A ) @ B )
% 0.24/0.58 = ( add_pow_a_b_nat @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_pow_ldistr
% 0.24/0.58 thf(fact_91_add__pow__rdistr,axiom,
% 0.24/0.58 ! [A: a,B: a,K: nat] :
% 0.24/0.58 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_nat @ r @ K @ B ) )
% 0.24/0.58 = ( add_pow_a_b_nat @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_pow_rdistr
% 0.24/0.58 thf(fact_92_add_Ogroup__commutes__pow,axiom,
% 0.24/0.58 ! [X: a,Y: a,N: nat] :
% 0.24/0.58 ( ( ( add_a_b @ r @ X @ Y )
% 0.24/0.58 = ( add_a_b @ r @ Y @ X ) )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ Y )
% 0.24/0.58 = ( add_a_b @ r @ Y @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.group_commutes_pow
% 0.24/0.58 thf(fact_93_add_Opow__mult__distrib,axiom,
% 0.24/0.58 ! [X: a,Y: a,N: nat] :
% 0.24/0.58 ( ( ( add_a_b @ r @ X @ Y )
% 0.24/0.58 = ( add_a_b @ r @ Y @ X ) )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_pow_a_b_nat @ r @ N @ ( add_a_b @ r @ X @ Y ) )
% 0.24/0.58 = ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ N @ Y ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.pow_mult_distrib
% 0.24/0.58 thf(fact_94_add_Onat__pow__distrib,axiom,
% 0.24/0.58 ! [X: a,Y: a,N: nat] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_pow_a_b_nat @ r @ N @ ( add_a_b @ r @ X @ Y ) )
% 0.24/0.58 = ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ N @ Y ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.nat_pow_distrib
% 0.24/0.58 thf(fact_95_add_Onat__pow__comm,axiom,
% 0.24/0.58 ! [X: a,N: nat,M: nat] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ M @ X ) )
% 0.24/0.58 = ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ M @ X ) @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.nat_pow_comm
% 0.24/0.58 thf(fact_96_add_Onat__pow__closed,axiom,
% 0.24/0.58 ! [X: a,N: nat] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( member_a @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.nat_pow_closed
% 0.24/0.58 thf(fact_97_add_Onat__pow__one,axiom,
% 0.24/0.58 ! [N: nat] :
% 0.24/0.58 ( ( add_pow_a_b_nat @ r @ N @ ( zero_a_b @ r ) )
% 0.24/0.58 = ( zero_a_b @ r ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.nat_pow_one
% 0.24/0.58 thf(fact_98_one__not__zero,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b] :
% 0.24/0.58 ( ( field_a_b @ R3 )
% 0.24/0.58 => ( ( one_a_ring_ext_a_b @ R3 )
% 0.24/0.58 != ( zero_a_b @ R3 ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % one_not_zero
% 0.24/0.58 thf(fact_99_integral,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,A: a,B: a] :
% 0.24/0.58 ( ( field_a_b @ R3 )
% 0.24/0.58 => ( ( ( mult_a_ring_ext_a_b @ R3 @ A @ B )
% 0.24/0.58 = ( zero_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( A
% 0.24/0.58 = ( zero_a_b @ R3 ) )
% 0.24/0.58 | ( B
% 0.24/0.58 = ( zero_a_b @ R3 ) ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % integral
% 0.24/0.58 thf(fact_100_semiring_Oadd__pow__ldistr,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,A: a,B: a,K: nat] :
% 0.24/0.58 ( ( semiring_a_b @ R3 )
% 0.24/0.58 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ R3 @ ( add_pow_a_b_nat @ R3 @ K @ A ) @ B )
% 0.24/0.58 = ( add_pow_a_b_nat @ R3 @ K @ ( mult_a_ring_ext_a_b @ R3 @ A @ B ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % semiring.add_pow_ldistr
% 0.24/0.58 thf(fact_101_semiring_Oadd__pow__rdistr,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,A: a,B: a,K: nat] :
% 0.24/0.58 ( ( semiring_a_b @ R3 )
% 0.24/0.58 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ R3 @ A @ ( add_pow_a_b_nat @ R3 @ K @ B ) )
% 0.24/0.58 = ( add_pow_a_b_nat @ R3 @ K @ ( mult_a_ring_ext_a_b @ R3 @ A @ B ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % semiring.add_pow_rdistr
% 0.24/0.58 thf(fact_102_add_Onat__pow__mult,axiom,
% 0.24/0.58 ! [X: a,N: nat,M: nat] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ M @ X ) )
% 0.24/0.58 = ( add_pow_a_b_nat @ r @ ( plus_plus_nat @ N @ M ) @ X ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.nat_pow_mult
% 0.24/0.58 thf(fact_103_add_Onat__pow__Suc2,axiom,
% 0.24/0.58 ! [X: a,N: nat] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_pow_a_b_nat @ r @ ( suc @ N ) @ X )
% 0.24/0.58 = ( add_a_b @ r @ X @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.nat_pow_Suc2
% 0.24/0.58 thf(fact_104_add_Opow__eq__div2,axiom,
% 0.24/0.58 ! [X: a,M: nat,N: nat] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( ( add_pow_a_b_nat @ r @ M @ X )
% 0.24/0.58 = ( add_pow_a_b_nat @ r @ N @ X ) )
% 0.24/0.58 => ( ( add_pow_a_b_nat @ r @ ( minus_minus_nat @ M @ N ) @ X )
% 0.24/0.58 = ( zero_a_b @ r ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.pow_eq_div2
% 0.24/0.58 thf(fact_105_minus__to__eq,axiom,
% 0.24/0.58 ! [X: a,Y: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ r )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( ( a_minus_a_b @ r @ X @ Y )
% 0.24/0.58 = ( zero_a_b @ r ) )
% 0.24/0.58 => ( X = Y ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % minus_to_eq
% 0.24/0.58 thf(fact_106_ring_Oequality,axiom,
% 0.24/0.58 ! [R: partia1833973666xt_a_b,R2: partia1833973666xt_a_b] :
% 0.24/0.58 ( ( ( partia1066395285xt_a_b @ R )
% 0.24/0.58 = ( partia1066395285xt_a_b @ R2 ) )
% 0.24/0.58 => ( ( ( mult_a_ring_ext_a_b @ R )
% 0.24/0.58 = ( mult_a_ring_ext_a_b @ R2 ) )
% 0.24/0.58 => ( ( ( one_a_ring_ext_a_b @ R )
% 0.24/0.58 = ( one_a_ring_ext_a_b @ R2 ) )
% 0.24/0.58 => ( ( ( zero_a_b @ R )
% 0.24/0.58 = ( zero_a_b @ R2 ) )
% 0.24/0.58 => ( ( ( add_a_b @ R )
% 0.24/0.58 = ( add_a_b @ R2 ) )
% 0.24/0.58 => ( ( ( more_a_b @ R )
% 0.24/0.58 = ( more_a_b @ R2 ) )
% 0.24/0.58 => ( R = R2 ) ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % ring.equality
% 0.24/0.58 thf(fact_107_is__abelian__group,axiom,
% 0.24/0.58 abelian_group_a_b @ r ).
% 0.24/0.58
% 0.24/0.58 % is_abelian_group
% 0.24/0.58 thf(fact_108_add_Onat__pow__Suc,axiom,
% 0.24/0.58 ! [N: nat,X: a] :
% 0.24/0.58 ( ( add_pow_a_b_nat @ r @ ( suc @ N ) @ X )
% 0.24/0.58 = ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ X ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.nat_pow_Suc
% 0.24/0.58 thf(fact_109_abelian__groupE_I1_J,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,X: a,Y: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ R3 )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( member_a @ ( add_a_b @ R3 @ X @ Y ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_groupE(1)
% 0.24/0.58 thf(fact_110_abelian__groupE_I3_J,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ R3 )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( add_a_b @ R3 @ ( add_a_b @ R3 @ X @ Y ) @ Z )
% 0.24/0.58 = ( add_a_b @ R3 @ X @ ( add_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_groupE(3)
% 0.24/0.58 thf(fact_111_abelian__groupE_I4_J,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,X: a,Y: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ R3 )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( add_a_b @ R3 @ X @ Y )
% 0.24/0.58 = ( add_a_b @ R3 @ Y @ X ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_groupE(4)
% 0.24/0.58 thf(fact_112_abelian__groupE_I2_J,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b] :
% 0.24/0.58 ( ( abelian_group_a_b @ R3 )
% 0.24/0.58 => ( member_a @ ( zero_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_groupE(2)
% 0.24/0.58 thf(fact_113_abelian__group_Ominus__closed,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( member_a @ ( a_minus_a_b @ G @ X @ Y ) @ ( partia1066395285xt_a_b @ G ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_group.minus_closed
% 0.24/0.58 thf(fact_114_abelian__groupI,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b] :
% 0.24/0.58 ( ! [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ! [Y4: a] :
% 0.24/0.58 ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( member_a @ ( add_a_b @ R3 @ X3 @ Y4 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) )
% 0.24/0.58 => ( ( member_a @ ( zero_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ! [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ! [Y4: a] :
% 0.24/0.58 ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ! [Z2: a] :
% 0.24/0.58 ( ( member_a @ Z2 @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( add_a_b @ R3 @ ( add_a_b @ R3 @ X3 @ Y4 ) @ Z2 )
% 0.24/0.58 = ( add_a_b @ R3 @ X3 @ ( add_a_b @ R3 @ Y4 @ Z2 ) ) ) ) ) )
% 0.24/0.58 => ( ! [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ! [Y4: a] :
% 0.24/0.58 ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( add_a_b @ R3 @ X3 @ Y4 )
% 0.24/0.58 = ( add_a_b @ R3 @ Y4 @ X3 ) ) ) )
% 0.24/0.58 => ( ! [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( add_a_b @ R3 @ ( zero_a_b @ R3 ) @ X3 )
% 0.24/0.58 = X3 ) )
% 0.24/0.58 => ( ! [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ? [Xa: a] :
% 0.24/0.58 ( ( member_a @ Xa @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 & ( ( add_a_b @ R3 @ Xa @ X3 )
% 0.24/0.58 = ( zero_a_b @ R3 ) ) ) )
% 0.24/0.58 => ( abelian_group_a_b @ R3 ) ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_groupI
% 0.24/0.58 thf(fact_115_abelian__groupE_I5_J,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,X: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ R3 )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ( ( add_a_b @ R3 @ ( zero_a_b @ R3 ) @ X )
% 0.24/0.58 = X ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_groupE(5)
% 0.24/0.58 thf(fact_116_abelian__groupE_I6_J,axiom,
% 0.24/0.58 ! [R3: partia1833973666xt_a_b,X: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ R3 )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 => ? [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ R3 ) )
% 0.24/0.58 & ( ( add_a_b @ R3 @ X3 @ X )
% 0.24/0.58 = ( zero_a_b @ R3 ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_groupE(6)
% 0.24/0.58 thf(fact_117_abelian__group_Ofour__elem__comm,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,A: a,B: a,C: a,D: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ G )
% 0.24/0.58 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ D @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( a_minus_a_b @ G @ ( add_a_b @ G @ ( a_minus_a_b @ G @ A @ C ) @ B ) @ D )
% 0.24/0.58 = ( a_minus_a_b @ G @ ( a_minus_a_b @ G @ ( add_a_b @ G @ A @ B ) @ C ) @ D ) ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_group.four_elem_comm
% 0.24/0.58 thf(fact_118_abelian__group_Oright__inv__add,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,A: a,B: a,C: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ G )
% 0.24/0.58 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( a_minus_a_b @ G @ ( a_minus_a_b @ G @ C @ A ) @ B )
% 0.24/0.58 = ( a_minus_a_b @ G @ C @ ( add_a_b @ G @ A @ B ) ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_group.right_inv_add
% 0.24/0.58 thf(fact_119_abelian__group_Ominus__to__eq,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ G )
% 0.24/0.58 => ( ( abelian_group_a_b @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( ( a_minus_a_b @ G @ X @ Y )
% 0.24/0.58 = ( zero_a_b @ G ) )
% 0.24/0.58 => ( X = Y ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_group.minus_to_eq
% 0.24/0.58 thf(fact_120_diff__diff__left,axiom,
% 0.24/0.58 ! [I: nat,J: nat,K: nat] :
% 0.24/0.58 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.24/0.58 = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % diff_diff_left
% 0.24/0.58 thf(fact_121_Suc__diff__diff,axiom,
% 0.24/0.58 ! [M: nat,N: nat,K: nat] :
% 0.24/0.58 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 0.24/0.58 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 0.24/0.58
% 0.24/0.58 % Suc_diff_diff
% 0.24/0.58 thf(fact_122_diff__Suc__Suc,axiom,
% 0.24/0.58 ! [M: nat,N: nat] :
% 0.24/0.58 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.24/0.58 = ( minus_minus_nat @ M @ N ) ) ).
% 0.24/0.58
% 0.24/0.58 % diff_Suc_Suc
% 0.24/0.58 thf(fact_123_add__Suc__right,axiom,
% 0.24/0.58 ! [M: nat,N: nat] :
% 0.24/0.58 ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 0.24/0.58 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_Suc_right
% 0.24/0.58 thf(fact_124_nat_Oinject,axiom,
% 0.24/0.58 ! [X2: nat,Y2: nat] :
% 0.24/0.58 ( ( ( suc @ X2 )
% 0.24/0.58 = ( suc @ Y2 ) )
% 0.24/0.58 = ( X2 = Y2 ) ) ).
% 0.24/0.58
% 0.24/0.58 % nat.inject
% 0.24/0.58 thf(fact_125_old_Onat_Oinject,axiom,
% 0.24/0.58 ! [Nat: nat,Nat2: nat] :
% 0.24/0.58 ( ( ( suc @ Nat )
% 0.24/0.58 = ( suc @ Nat2 ) )
% 0.24/0.58 = ( Nat = Nat2 ) ) ).
% 0.24/0.58
% 0.24/0.58 % old.nat.inject
% 0.24/0.58 thf(fact_126_Suc__inject,axiom,
% 0.24/0.58 ! [X: nat,Y: nat] :
% 0.24/0.58 ( ( ( suc @ X )
% 0.24/0.58 = ( suc @ Y ) )
% 0.24/0.58 => ( X = Y ) ) ).
% 0.24/0.58
% 0.24/0.58 % Suc_inject
% 0.24/0.58 thf(fact_127_n__not__Suc__n,axiom,
% 0.24/0.58 ! [N: nat] :
% 0.24/0.58 ( N
% 0.24/0.58 != ( suc @ N ) ) ).
% 0.24/0.58
% 0.24/0.58 % n_not_Suc_n
% 0.24/0.58 thf(fact_128_diff__commute,axiom,
% 0.24/0.58 ! [I: nat,J: nat,K: nat] :
% 0.24/0.58 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.24/0.58 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 0.24/0.58
% 0.24/0.58 % diff_commute
% 0.24/0.58 thf(fact_129_add__Suc,axiom,
% 0.24/0.58 ! [M: nat,N: nat] :
% 0.24/0.58 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.24/0.58 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_Suc
% 0.24/0.58 thf(fact_130_nat__arith_Osuc1,axiom,
% 0.24/0.58 ! [A3: nat,K: nat,A: nat] :
% 0.24/0.58 ( ( A3
% 0.24/0.58 = ( plus_plus_nat @ K @ A ) )
% 0.24/0.58 => ( ( suc @ A3 )
% 0.24/0.58 = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % nat_arith.suc1
% 0.24/0.58 thf(fact_131_add__Suc__shift,axiom,
% 0.24/0.58 ! [M: nat,N: nat] :
% 0.24/0.58 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.24/0.58 = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_Suc_shift
% 0.24/0.58 thf(fact_132_zero__induct__lemma,axiom,
% 0.24/0.58 ! [P: nat > $o,K: nat,I: nat] :
% 0.24/0.58 ( ( P @ K )
% 0.24/0.58 => ( ! [N2: nat] :
% 0.24/0.58 ( ( P @ ( suc @ N2 ) )
% 0.24/0.58 => ( P @ N2 ) )
% 0.24/0.58 => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % zero_induct_lemma
% 0.24/0.58 thf(fact_133_diff__add__inverse2,axiom,
% 0.24/0.58 ! [M: nat,N: nat] :
% 0.24/0.58 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 0.24/0.58 = M ) ).
% 0.24/0.58
% 0.24/0.58 % diff_add_inverse2
% 0.24/0.58 thf(fact_134_diff__add__inverse,axiom,
% 0.24/0.58 ! [N: nat,M: nat] :
% 0.24/0.58 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 0.24/0.58 = M ) ).
% 0.24/0.58
% 0.24/0.58 % diff_add_inverse
% 0.24/0.58 thf(fact_135_diff__cancel2,axiom,
% 0.24/0.58 ! [M: nat,K: nat,N: nat] :
% 0.24/0.58 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 0.24/0.58 = ( minus_minus_nat @ M @ N ) ) ).
% 0.24/0.58
% 0.24/0.58 % diff_cancel2
% 0.24/0.58 thf(fact_136_Nat_Odiff__cancel,axiom,
% 0.24/0.58 ! [K: nat,M: nat,N: nat] :
% 0.24/0.58 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.24/0.58 = ( minus_minus_nat @ M @ N ) ) ).
% 0.24/0.58
% 0.24/0.58 % Nat.diff_cancel
% 0.24/0.58 thf(fact_137_add__diff__cancel__right_H,axiom,
% 0.24/0.58 ! [A: nat,B: nat] :
% 0.24/0.58 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.24/0.58 = A ) ).
% 0.24/0.58
% 0.24/0.58 % add_diff_cancel_right'
% 0.24/0.58 thf(fact_138_add__diff__cancel__right_H,axiom,
% 0.24/0.58 ! [A: int,B: int] :
% 0.24/0.58 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.24/0.58 = A ) ).
% 0.24/0.58
% 0.24/0.58 % add_diff_cancel_right'
% 0.24/0.58 thf(fact_139_add__diff__cancel__right,axiom,
% 0.24/0.58 ! [A: nat,C: nat,B: nat] :
% 0.24/0.58 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.24/0.58 = ( minus_minus_nat @ A @ B ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_diff_cancel_right
% 0.24/0.58 thf(fact_140_add__diff__cancel__right,axiom,
% 0.24/0.58 ! [A: int,C: int,B: int] :
% 0.24/0.58 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.24/0.58 = ( minus_minus_int @ A @ B ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_diff_cancel_right
% 0.24/0.58 thf(fact_141_add__diff__cancel__left_H,axiom,
% 0.24/0.58 ! [A: nat,B: nat] :
% 0.24/0.58 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 0.24/0.58 = B ) ).
% 0.24/0.58
% 0.24/0.58 % add_diff_cancel_left'
% 0.24/0.58 thf(fact_142_add__diff__cancel__left_H,axiom,
% 0.24/0.58 ! [A: int,B: int] :
% 0.24/0.58 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 0.24/0.58 = B ) ).
% 0.24/0.58
% 0.24/0.58 % add_diff_cancel_left'
% 0.24/0.58 thf(fact_143_add__diff__cancel__left,axiom,
% 0.24/0.58 ! [C: nat,A: nat,B: nat] :
% 0.24/0.58 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.24/0.58 = ( minus_minus_nat @ A @ B ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_diff_cancel_left
% 0.24/0.58 thf(fact_144_add__diff__cancel__left,axiom,
% 0.24/0.58 ! [C: int,A: int,B: int] :
% 0.24/0.58 ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.24/0.58 = ( minus_minus_int @ A @ B ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_diff_cancel_left
% 0.24/0.58 thf(fact_145_add__right__cancel,axiom,
% 0.24/0.58 ! [B: int,A: int,C: int] :
% 0.24/0.58 ( ( ( plus_plus_int @ B @ A )
% 0.24/0.58 = ( plus_plus_int @ C @ A ) )
% 0.24/0.58 = ( B = C ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_right_cancel
% 0.24/0.58 thf(fact_146_add__right__cancel,axiom,
% 0.24/0.58 ! [B: nat,A: nat,C: nat] :
% 0.24/0.58 ( ( ( plus_plus_nat @ B @ A )
% 0.24/0.58 = ( plus_plus_nat @ C @ A ) )
% 0.24/0.58 = ( B = C ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_right_cancel
% 0.24/0.58 thf(fact_147_add__left__cancel,axiom,
% 0.24/0.58 ! [A: int,B: int,C: int] :
% 0.24/0.58 ( ( ( plus_plus_int @ A @ B )
% 0.24/0.58 = ( plus_plus_int @ A @ C ) )
% 0.24/0.58 = ( B = C ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_left_cancel
% 0.24/0.58 thf(fact_148_add__left__cancel,axiom,
% 0.24/0.58 ! [A: nat,B: nat,C: nat] :
% 0.24/0.58 ( ( ( plus_plus_nat @ A @ B )
% 0.24/0.58 = ( plus_plus_nat @ A @ C ) )
% 0.24/0.58 = ( B = C ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_left_cancel
% 0.24/0.58 thf(fact_149_add__diff__cancel,axiom,
% 0.24/0.58 ! [A: int,B: int] :
% 0.24/0.58 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.24/0.58 = A ) ).
% 0.24/0.58
% 0.24/0.58 % add_diff_cancel
% 0.24/0.58 thf(fact_150_diff__add__cancel,axiom,
% 0.24/0.58 ! [A: int,B: int] :
% 0.24/0.58 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.24/0.58 = A ) ).
% 0.24/0.58
% 0.24/0.58 % diff_add_cancel
% 0.24/0.58 thf(fact_151_add__right__imp__eq,axiom,
% 0.24/0.58 ! [B: int,A: int,C: int] :
% 0.24/0.58 ( ( ( plus_plus_int @ B @ A )
% 0.24/0.58 = ( plus_plus_int @ C @ A ) )
% 0.24/0.58 => ( B = C ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_right_imp_eq
% 0.24/0.58 thf(fact_152_add__right__imp__eq,axiom,
% 0.24/0.58 ! [B: nat,A: nat,C: nat] :
% 0.24/0.58 ( ( ( plus_plus_nat @ B @ A )
% 0.24/0.58 = ( plus_plus_nat @ C @ A ) )
% 0.24/0.58 => ( B = C ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_right_imp_eq
% 0.24/0.58 thf(fact_153_add__left__imp__eq,axiom,
% 0.24/0.58 ! [A: int,B: int,C: int] :
% 0.24/0.58 ( ( ( plus_plus_int @ A @ B )
% 0.24/0.58 = ( plus_plus_int @ A @ C ) )
% 0.24/0.58 => ( B = C ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_left_imp_eq
% 0.24/0.58 thf(fact_154_add__left__imp__eq,axiom,
% 0.24/0.58 ! [A: nat,B: nat,C: nat] :
% 0.24/0.58 ( ( ( plus_plus_nat @ A @ B )
% 0.24/0.58 = ( plus_plus_nat @ A @ C ) )
% 0.24/0.58 => ( B = C ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_left_imp_eq
% 0.24/0.58 thf(fact_155_add_Oleft__commute,axiom,
% 0.24/0.58 ! [B: int,A: int,C: int] :
% 0.24/0.58 ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 0.24/0.58 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.left_commute
% 0.24/0.58 thf(fact_156_add_Oleft__commute,axiom,
% 0.24/0.58 ! [B: nat,A: nat,C: nat] :
% 0.24/0.58 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 0.24/0.58 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.left_commute
% 0.24/0.58 thf(fact_157_add_Ocommute,axiom,
% 0.24/0.58 ( plus_plus_int
% 0.24/0.58 = ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.commute
% 0.24/0.58 thf(fact_158_add_Ocommute,axiom,
% 0.24/0.58 ( plus_plus_nat
% 0.24/0.58 = ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.commute
% 0.24/0.58 thf(fact_159_group__add__class_Oadd_Oright__cancel,axiom,
% 0.24/0.58 ! [B: int,A: int,C: int] :
% 0.24/0.58 ( ( ( plus_plus_int @ B @ A )
% 0.24/0.58 = ( plus_plus_int @ C @ A ) )
% 0.24/0.58 = ( B = C ) ) ).
% 0.24/0.58
% 0.24/0.58 % group_add_class.add.right_cancel
% 0.24/0.58 thf(fact_160_add_Oleft__cancel,axiom,
% 0.24/0.58 ! [A: int,B: int,C: int] :
% 0.24/0.58 ( ( ( plus_plus_int @ A @ B )
% 0.24/0.58 = ( plus_plus_int @ A @ C ) )
% 0.24/0.58 = ( B = C ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.left_cancel
% 0.24/0.58 thf(fact_161_add_Oassoc,axiom,
% 0.24/0.58 ! [A: int,B: int,C: int] :
% 0.24/0.58 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.24/0.58 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.assoc
% 0.24/0.58 thf(fact_162_add_Oassoc,axiom,
% 0.24/0.58 ! [A: nat,B: nat,C: nat] :
% 0.24/0.58 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.24/0.58 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.assoc
% 0.24/0.58 thf(fact_163_group__cancel_Oadd2,axiom,
% 0.24/0.58 ! [B5: int,K: int,B: int,A: int] :
% 0.24/0.58 ( ( B5
% 0.24/0.58 = ( plus_plus_int @ K @ B ) )
% 0.24/0.58 => ( ( plus_plus_int @ A @ B5 )
% 0.24/0.58 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group_cancel.add2
% 0.24/0.58 thf(fact_164_group__cancel_Oadd2,axiom,
% 0.24/0.58 ! [B5: nat,K: nat,B: nat,A: nat] :
% 0.24/0.58 ( ( B5
% 0.24/0.58 = ( plus_plus_nat @ K @ B ) )
% 0.24/0.58 => ( ( plus_plus_nat @ A @ B5 )
% 0.24/0.58 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group_cancel.add2
% 0.24/0.58 thf(fact_165_group__cancel_Oadd1,axiom,
% 0.24/0.58 ! [A3: int,K: int,A: int,B: int] :
% 0.24/0.58 ( ( A3
% 0.24/0.58 = ( plus_plus_int @ K @ A ) )
% 0.24/0.58 => ( ( plus_plus_int @ A3 @ B )
% 0.24/0.58 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group_cancel.add1
% 0.24/0.58 thf(fact_166_group__cancel_Oadd1,axiom,
% 0.24/0.58 ! [A3: nat,K: nat,A: nat,B: nat] :
% 0.24/0.58 ( ( A3
% 0.24/0.58 = ( plus_plus_nat @ K @ A ) )
% 0.24/0.58 => ( ( plus_plus_nat @ A3 @ B )
% 0.24/0.58 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group_cancel.add1
% 0.24/0.58 thf(fact_167_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.24/0.58 ! [I: int,J: int,K: int,L: int] :
% 0.24/0.58 ( ( ( I = J )
% 0.24/0.58 & ( K = L ) )
% 0.24/0.58 => ( ( plus_plus_int @ I @ K )
% 0.24/0.58 = ( plus_plus_int @ J @ L ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_mono_thms_linordered_semiring(4)
% 0.24/0.58 thf(fact_168_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.24/0.58 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.24/0.58 ( ( ( I = J )
% 0.24/0.58 & ( K = L ) )
% 0.24/0.58 => ( ( plus_plus_nat @ I @ K )
% 0.24/0.58 = ( plus_plus_nat @ J @ L ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_mono_thms_linordered_semiring(4)
% 0.24/0.58 thf(fact_169_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.24/0.58 ! [A: int,B: int,C: int] :
% 0.24/0.58 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.24/0.58 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % ab_semigroup_add_class.add_ac(1)
% 0.24/0.58 thf(fact_170_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.24/0.58 ! [A: nat,B: nat,C: nat] :
% 0.24/0.58 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.24/0.58 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % ab_semigroup_add_class.add_ac(1)
% 0.24/0.58 thf(fact_171_diff__right__commute,axiom,
% 0.24/0.58 ! [A: nat,C: nat,B: nat] :
% 0.24/0.58 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.24/0.58 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.24/0.58
% 0.24/0.58 % diff_right_commute
% 0.24/0.58 thf(fact_172_diff__right__commute,axiom,
% 0.24/0.58 ! [A: int,C: int,B: int] :
% 0.24/0.58 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 0.24/0.58 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.24/0.58
% 0.24/0.58 % diff_right_commute
% 0.24/0.58 thf(fact_173_diff__eq__diff__eq,axiom,
% 0.24/0.58 ! [A: int,B: int,C: int,D: int] :
% 0.24/0.58 ( ( ( minus_minus_int @ A @ B )
% 0.24/0.58 = ( minus_minus_int @ C @ D ) )
% 0.24/0.58 => ( ( A = B )
% 0.24/0.58 = ( C = D ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % diff_eq_diff_eq
% 0.24/0.58 thf(fact_174_group__cancel_Osub1,axiom,
% 0.24/0.58 ! [A3: int,K: int,A: int,B: int] :
% 0.24/0.58 ( ( A3
% 0.24/0.58 = ( plus_plus_int @ K @ A ) )
% 0.24/0.58 => ( ( minus_minus_int @ A3 @ B )
% 0.24/0.58 = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group_cancel.sub1
% 0.24/0.58 thf(fact_175_diff__eq__eq,axiom,
% 0.24/0.58 ! [A: int,B: int,C: int] :
% 0.24/0.58 ( ( ( minus_minus_int @ A @ B )
% 0.24/0.58 = C )
% 0.24/0.58 = ( A
% 0.24/0.58 = ( plus_plus_int @ C @ B ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % diff_eq_eq
% 0.24/0.58 thf(fact_176_eq__diff__eq,axiom,
% 0.24/0.58 ! [A: int,C: int,B: int] :
% 0.24/0.58 ( ( A
% 0.24/0.58 = ( minus_minus_int @ C @ B ) )
% 0.24/0.58 = ( ( plus_plus_int @ A @ B )
% 0.24/0.58 = C ) ) ).
% 0.24/0.58
% 0.24/0.58 % eq_diff_eq
% 0.24/0.58 thf(fact_177_add__diff__eq,axiom,
% 0.24/0.58 ! [A: int,B: int,C: int] :
% 0.24/0.58 ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.24/0.58 = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_diff_eq
% 0.24/0.58 thf(fact_178_diff__diff__eq2,axiom,
% 0.24/0.58 ! [A: int,B: int,C: int] :
% 0.24/0.58 ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.24/0.58 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.24/0.58
% 0.24/0.58 % diff_diff_eq2
% 0.24/0.58 thf(fact_179_diff__add__eq,axiom,
% 0.24/0.58 ! [A: int,B: int,C: int] :
% 0.24/0.58 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.24/0.58 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.24/0.58
% 0.24/0.58 % diff_add_eq
% 0.24/0.58 thf(fact_180_diff__add__eq__diff__diff__swap,axiom,
% 0.24/0.58 ! [A: int,B: int,C: int] :
% 0.24/0.58 ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.24/0.58 = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 0.24/0.58
% 0.24/0.58 % diff_add_eq_diff_diff_swap
% 0.24/0.58 thf(fact_181_diff__diff__add,axiom,
% 0.24/0.58 ! [A: nat,B: nat,C: nat] :
% 0.24/0.58 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 0.24/0.58 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % diff_diff_add
% 0.24/0.58 thf(fact_182_diff__diff__add,axiom,
% 0.24/0.58 ! [A: int,B: int,C: int] :
% 0.24/0.58 ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.24/0.58 = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % diff_diff_add
% 0.24/0.58 thf(fact_183_add__implies__diff,axiom,
% 0.24/0.58 ! [C: nat,B: nat,A: nat] :
% 0.24/0.58 ( ( ( plus_plus_nat @ C @ B )
% 0.24/0.58 = A )
% 0.24/0.58 => ( C
% 0.24/0.58 = ( minus_minus_nat @ A @ B ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_implies_diff
% 0.24/0.58 thf(fact_184_add__implies__diff,axiom,
% 0.24/0.58 ! [C: int,B: int,A: int] :
% 0.24/0.58 ( ( ( plus_plus_int @ C @ B )
% 0.24/0.58 = A )
% 0.24/0.58 => ( C
% 0.24/0.58 = ( minus_minus_int @ A @ B ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add_implies_diff
% 0.24/0.58 thf(fact_185_Units__l__inv__ex,axiom,
% 0.24/0.58 ! [X: a] :
% 0.24/0.58 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 0.24/0.58 => ? [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 & ( ( mult_a_ring_ext_a_b @ r @ X3 @ X )
% 0.24/0.58 = ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % Units_l_inv_ex
% 0.24/0.58 thf(fact_186_Units__r__inv__ex,axiom,
% 0.24/0.58 ! [X: a] :
% 0.24/0.58 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 0.24/0.58 => ? [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 & ( ( mult_a_ring_ext_a_b @ r @ X @ X3 )
% 0.24/0.58 = ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % Units_r_inv_ex
% 0.24/0.58 thf(fact_187_group__l__invI,axiom,
% 0.24/0.58 ( ! [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ? [Xa: a] :
% 0.24/0.58 ( ( member_a @ Xa @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 & ( ( mult_a_ring_ext_a_b @ r @ Xa @ X3 )
% 0.24/0.58 = ( one_a_ring_ext_a_b @ r ) ) ) )
% 0.24/0.58 => ( group_a_ring_ext_a_b @ r ) ) ).
% 0.24/0.58
% 0.24/0.58 % group_l_invI
% 0.24/0.58 thf(fact_188_Units__closed,axiom,
% 0.24/0.58 ! [X: a] :
% 0.24/0.58 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 0.24/0.58 => ( member_a @ X @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % Units_closed
% 0.24/0.58 thf(fact_189_Units__inv__comm,axiom,
% 0.24/0.58 ! [X: a,Y: a] :
% 0.24/0.58 ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
% 0.24/0.58 = ( one_a_ring_ext_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
% 0.24/0.58 = ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % Units_inv_comm
% 0.24/0.58 thf(fact_190_Units__m__closed,axiom,
% 0.24/0.58 ! [X: a,Y: a] :
% 0.24/0.58 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
% 0.24/0.58 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % Units_m_closed
% 0.24/0.58 thf(fact_191_Units__one__closed,axiom,
% 0.24/0.58 member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) ).
% 0.24/0.58
% 0.24/0.58 % Units_one_closed
% 0.24/0.58 thf(fact_192_Units__l__cancel,axiom,
% 0.24/0.58 ! [X: a,Y: a,Z: a] :
% 0.24/0.58 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
% 0.24/0.58 = ( mult_a_ring_ext_a_b @ r @ X @ Z ) )
% 0.24/0.58 = ( Y = Z ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % Units_l_cancel
% 0.24/0.58 thf(fact_193_groupI,axiom,
% 0.24/0.58 ! [G: partia96731725t_unit] :
% 0.24/0.58 ( ! [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ! [Y4: a] :
% 0.24/0.58 ( ( member_a @ Y4 @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( member_a @ ( mult_a_Product_unit @ G @ X3 @ Y4 ) @ ( partia1955795460t_unit @ G ) ) ) )
% 0.24/0.58 => ( ( member_a @ ( one_a_Product_unit @ G ) @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ! [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ! [Y4: a] :
% 0.24/0.58 ( ( member_a @ Y4 @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ! [Z2: a] :
% 0.24/0.58 ( ( member_a @ Z2 @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( mult_a_Product_unit @ G @ ( mult_a_Product_unit @ G @ X3 @ Y4 ) @ Z2 )
% 0.24/0.58 = ( mult_a_Product_unit @ G @ X3 @ ( mult_a_Product_unit @ G @ Y4 @ Z2 ) ) ) ) ) )
% 0.24/0.58 => ( ! [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( mult_a_Product_unit @ G @ ( one_a_Product_unit @ G ) @ X3 )
% 0.24/0.58 = X3 ) )
% 0.24/0.58 => ( ! [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ? [Xa: a] :
% 0.24/0.58 ( ( member_a @ Xa @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 & ( ( mult_a_Product_unit @ G @ Xa @ X3 )
% 0.24/0.58 = ( one_a_Product_unit @ G ) ) ) )
% 0.24/0.58 => ( group_a_Product_unit @ G ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % groupI
% 0.24/0.58 thf(fact_194_groupI,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b] :
% 0.24/0.58 ( ! [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ! [Y4: a] :
% 0.24/0.58 ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X3 @ Y4 ) @ ( partia1066395285xt_a_b @ G ) ) ) )
% 0.24/0.58 => ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ! [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ! [Y4: a] :
% 0.24/0.58 ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ! [Z2: a] :
% 0.24/0.58 ( ( member_a @ Z2 @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X3 @ Y4 ) @ Z2 )
% 0.24/0.58 = ( mult_a_ring_ext_a_b @ G @ X3 @ ( mult_a_ring_ext_a_b @ G @ Y4 @ Z2 ) ) ) ) ) )
% 0.24/0.58 => ( ! [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X3 )
% 0.24/0.58 = X3 ) )
% 0.24/0.58 => ( ! [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ? [Xa: a] :
% 0.24/0.58 ( ( member_a @ Xa @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 & ( ( mult_a_ring_ext_a_b @ G @ Xa @ X3 )
% 0.24/0.58 = ( one_a_ring_ext_a_b @ G ) ) ) )
% 0.24/0.58 => ( group_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % groupI
% 0.24/0.58 thf(fact_195_group_Oinv__comm,axiom,
% 0.24/0.58 ! [G: partia96731725t_unit,X: a,Y: a] :
% 0.24/0.58 ( ( group_a_Product_unit @ G )
% 0.24/0.58 => ( ( ( mult_a_Product_unit @ G @ X @ Y )
% 0.24/0.58 = ( one_a_Product_unit @ G ) )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( mult_a_Product_unit @ G @ Y @ X )
% 0.24/0.58 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group.inv_comm
% 0.24/0.58 thf(fact_196_group_Oinv__comm,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.24/0.58 ( ( group_a_ring_ext_a_b @ G )
% 0.24/0.58 => ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
% 0.24/0.58 = ( one_a_ring_ext_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
% 0.24/0.58 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group.inv_comm
% 0.24/0.58 thf(fact_197_group_Ol__inv__ex,axiom,
% 0.24/0.58 ! [G: partia96731725t_unit,X: a] :
% 0.24/0.58 ( ( group_a_Product_unit @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ? [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 & ( ( mult_a_Product_unit @ G @ X3 @ X )
% 0.24/0.58 = ( one_a_Product_unit @ G ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group.l_inv_ex
% 0.24/0.58 thf(fact_198_group_Ol__inv__ex,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a] :
% 0.24/0.58 ( ( group_a_ring_ext_a_b @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ? [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 & ( ( mult_a_ring_ext_a_b @ G @ X3 @ X )
% 0.24/0.58 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group.l_inv_ex
% 0.24/0.58 thf(fact_199_Group_Ogroup_Oright__cancel,axiom,
% 0.24/0.58 ! [G: partia96731725t_unit,X: a,Y: a,Z: a] :
% 0.24/0.58 ( ( group_a_Product_unit @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( member_a @ Z @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( ( mult_a_Product_unit @ G @ Y @ X )
% 0.24/0.58 = ( mult_a_Product_unit @ G @ Z @ X ) )
% 0.24/0.58 = ( Y = Z ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % Group.group.right_cancel
% 0.24/0.58 thf(fact_200_Group_Ogroup_Oright__cancel,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.24/0.58 ( ( group_a_ring_ext_a_b @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
% 0.24/0.58 = ( mult_a_ring_ext_a_b @ G @ Z @ X ) )
% 0.24/0.58 = ( Y = Z ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % Group.group.right_cancel
% 0.24/0.58 thf(fact_201_group_OUnits__eq,axiom,
% 0.24/0.58 ! [G: partia96731725t_unit] :
% 0.24/0.58 ( ( group_a_Product_unit @ G )
% 0.24/0.58 => ( ( units_a_Product_unit @ G )
% 0.24/0.58 = ( partia1955795460t_unit @ G ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group.Units_eq
% 0.24/0.58 thf(fact_202_group_OUnits__eq,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b] :
% 0.24/0.58 ( ( group_a_ring_ext_a_b @ G )
% 0.24/0.58 => ( ( units_a_ring_ext_a_b @ G )
% 0.24/0.58 = ( partia1066395285xt_a_b @ G ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group.Units_eq
% 0.24/0.58 thf(fact_203_group_Or__cancel__one_H,axiom,
% 0.24/0.58 ! [G: partia96731725t_unit,X: a,A: a] :
% 0.24/0.58 ( ( group_a_Product_unit @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( X
% 0.24/0.58 = ( mult_a_Product_unit @ G @ A @ X ) )
% 0.24/0.58 = ( A
% 0.24/0.58 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group.r_cancel_one'
% 0.24/0.58 thf(fact_204_group_Or__cancel__one_H,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a,A: a] :
% 0.24/0.58 ( ( group_a_ring_ext_a_b @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( X
% 0.24/0.58 = ( mult_a_ring_ext_a_b @ G @ A @ X ) )
% 0.24/0.58 = ( A
% 0.24/0.58 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group.r_cancel_one'
% 0.24/0.58 thf(fact_205_group_Ol__cancel__one_H,axiom,
% 0.24/0.58 ! [G: partia96731725t_unit,X: a,A: a] :
% 0.24/0.58 ( ( group_a_Product_unit @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( X
% 0.24/0.58 = ( mult_a_Product_unit @ G @ X @ A ) )
% 0.24/0.58 = ( A
% 0.24/0.58 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group.l_cancel_one'
% 0.24/0.58 thf(fact_206_group_Ol__cancel__one_H,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a,A: a] :
% 0.24/0.58 ( ( group_a_ring_ext_a_b @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( X
% 0.24/0.58 = ( mult_a_ring_ext_a_b @ G @ X @ A ) )
% 0.24/0.58 = ( A
% 0.24/0.58 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group.l_cancel_one'
% 0.24/0.58 thf(fact_207_group_Or__cancel__one,axiom,
% 0.24/0.58 ! [G: partia96731725t_unit,X: a,A: a] :
% 0.24/0.58 ( ( group_a_Product_unit @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( ( mult_a_Product_unit @ G @ A @ X )
% 0.24/0.58 = X )
% 0.24/0.58 = ( A
% 0.24/0.58 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group.r_cancel_one
% 0.24/0.58 thf(fact_208_group_Or__cancel__one,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a,A: a] :
% 0.24/0.58 ( ( group_a_ring_ext_a_b @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( ( mult_a_ring_ext_a_b @ G @ A @ X )
% 0.24/0.58 = X )
% 0.24/0.58 = ( A
% 0.24/0.58 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group.r_cancel_one
% 0.24/0.58 thf(fact_209_group_Ol__cancel__one,axiom,
% 0.24/0.58 ! [G: partia96731725t_unit,X: a,A: a] :
% 0.24/0.58 ( ( group_a_Product_unit @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( ( mult_a_Product_unit @ G @ X @ A )
% 0.24/0.58 = X )
% 0.24/0.58 = ( A
% 0.24/0.58 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group.l_cancel_one
% 0.24/0.58 thf(fact_210_group_Ol__cancel__one,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a,A: a] :
% 0.24/0.58 ( ( group_a_ring_ext_a_b @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( ( mult_a_ring_ext_a_b @ G @ X @ A )
% 0.24/0.58 = X )
% 0.24/0.58 = ( A
% 0.24/0.58 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group.l_cancel_one
% 0.24/0.58 thf(fact_211_group_Or__inv__ex,axiom,
% 0.24/0.58 ! [G: partia96731725t_unit,X: a] :
% 0.24/0.58 ( ( group_a_Product_unit @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ? [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 & ( ( mult_a_Product_unit @ G @ X @ X3 )
% 0.24/0.58 = ( one_a_Product_unit @ G ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group.r_inv_ex
% 0.24/0.58 thf(fact_212_group_Or__inv__ex,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a] :
% 0.24/0.58 ( ( group_a_ring_ext_a_b @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ? [X3: a] :
% 0.24/0.58 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 & ( ( mult_a_ring_ext_a_b @ G @ X @ X3 )
% 0.24/0.58 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group.r_inv_ex
% 0.24/0.58 thf(fact_213_units__group,axiom,
% 0.24/0.58 group_a_Product_unit @ ( units_1411277569xt_a_b @ r ) ).
% 0.24/0.58
% 0.24/0.58 % units_group
% 0.24/0.58 thf(fact_214_inv__add,axiom,
% 0.24/0.58 ! [A: a,B: a] :
% 0.24/0.58 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ A @ B ) )
% 0.24/0.58 = ( a_minus_a_b @ r @ ( a_inv_a_b @ r @ A ) @ B ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % inv_add
% 0.24/0.58 thf(fact_215_r__neg2,axiom,
% 0.24/0.58 ! [X: a,Y: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_a_b @ r @ X @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y ) )
% 0.24/0.58 = Y ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % r_neg2
% 0.24/0.58 thf(fact_216_r__neg1,axiom,
% 0.24/0.58 ! [X: a,Y: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( add_a_b @ r @ X @ Y ) )
% 0.24/0.58 = Y ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % r_neg1
% 0.24/0.58 thf(fact_217_local_Ominus__add,axiom,
% 0.24/0.58 ! [X: a,Y: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
% 0.24/0.58 = ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % local.minus_add
% 0.24/0.58 thf(fact_218_add_Oinv__solve__right_H,axiom,
% 0.24/0.58 ! [A: a,B: a,C: a] :
% 0.24/0.58 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
% 0.24/0.58 = A )
% 0.24/0.58 = ( B
% 0.24/0.58 = ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.inv_solve_right'
% 0.24/0.58 thf(fact_219_add_Oinv__solve__right,axiom,
% 0.24/0.58 ! [A: a,B: a,C: a] :
% 0.24/0.58 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( A
% 0.24/0.58 = ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
% 0.24/0.58 = ( B
% 0.24/0.58 = ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.inv_solve_right
% 0.24/0.58 thf(fact_220_add_Oinv__solve__left_H,axiom,
% 0.24/0.58 ! [A: a,B: a,C: a] :
% 0.24/0.58 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
% 0.24/0.58 = A )
% 0.24/0.58 = ( C
% 0.24/0.58 = ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.inv_solve_left'
% 0.24/0.58 thf(fact_221_add_Oinv__solve__left,axiom,
% 0.24/0.58 ! [A: a,B: a,C: a] :
% 0.24/0.58 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( A
% 0.24/0.58 = ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
% 0.24/0.58 = ( C
% 0.24/0.58 = ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.inv_solve_left
% 0.24/0.58 thf(fact_222_add_Oinv__mult__group,axiom,
% 0.24/0.58 ! [X: a,Y: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
% 0.24/0.58 = ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.inv_mult_group
% 0.24/0.58 thf(fact_223_a__transpose__inv,axiom,
% 0.24/0.58 ! [X: a,Y: a,Z: a] :
% 0.24/0.58 ( ( ( add_a_b @ r @ X @ Y )
% 0.24/0.58 = Z )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Z )
% 0.24/0.58 = Y ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % a_transpose_inv
% 0.24/0.58 thf(fact_224_r__minus,axiom,
% 0.24/0.58 ! [X: a,Y: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) )
% 0.24/0.58 = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % r_minus
% 0.24/0.58 thf(fact_225_l__minus,axiom,
% 0.24/0.58 ! [X: a,Y: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y )
% 0.24/0.58 = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % l_minus
% 0.24/0.58 thf(fact_226_add_Onat__pow__inv,axiom,
% 0.24/0.58 ! [X: a,I: nat] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_pow_a_b_nat @ r @ I @ ( a_inv_a_b @ r @ X ) )
% 0.24/0.58 = ( a_inv_a_b @ r @ ( add_pow_a_b_nat @ r @ I @ X ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.nat_pow_inv
% 0.24/0.58 thf(fact_227_add_Oint__pow__inv,axiom,
% 0.24/0.58 ! [X: a,I: int] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_pow_a_b_int @ r @ I @ ( a_inv_a_b @ r @ X ) )
% 0.24/0.58 = ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.int_pow_inv
% 0.24/0.58 thf(fact_228_minus__eq,axiom,
% 0.24/0.58 ! [X: a,Y: a] :
% 0.24/0.58 ( ( a_minus_a_b @ r @ X @ Y )
% 0.24/0.58 = ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % minus_eq
% 0.24/0.58 thf(fact_229_sum__zero__eq__neg,axiom,
% 0.24/0.58 ! [X: a,Y: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( ( add_a_b @ r @ X @ Y )
% 0.24/0.58 = ( zero_a_b @ r ) )
% 0.24/0.58 => ( X
% 0.24/0.58 = ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % sum_zero_eq_neg
% 0.24/0.58 thf(fact_230_r__neg,axiom,
% 0.24/0.58 ! [X: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ X ) )
% 0.24/0.58 = ( zero_a_b @ r ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % r_neg
% 0.24/0.58 thf(fact_231_minus__equality,axiom,
% 0.24/0.58 ! [Y: a,X: a] :
% 0.24/0.58 ( ( ( add_a_b @ r @ Y @ X )
% 0.24/0.58 = ( zero_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( a_inv_a_b @ r @ X )
% 0.24/0.58 = Y ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % minus_equality
% 0.24/0.58 thf(fact_232_l__neg,axiom,
% 0.24/0.58 ! [X: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ X )
% 0.24/0.58 = ( zero_a_b @ r ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % l_neg
% 0.24/0.58 thf(fact_233_add_Oint__pow__diff,axiom,
% 0.24/0.58 ! [X: a,N: int,M: int] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_pow_a_b_int @ r @ ( minus_minus_int @ N @ M ) @ X )
% 0.24/0.58 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ N @ X ) @ ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ M @ X ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.int_pow_diff
% 0.24/0.58 thf(fact_234_local_Ominus__minus,axiom,
% 0.24/0.58 ! [X: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X ) )
% 0.24/0.58 = X ) ) ).
% 0.24/0.58
% 0.24/0.58 % local.minus_minus
% 0.24/0.58 thf(fact_235_add_Oinv__closed,axiom,
% 0.24/0.58 ! [X: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( member_a @ ( a_inv_a_b @ r @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.inv_closed
% 0.24/0.58 thf(fact_236_local_Ominus__zero,axiom,
% 0.24/0.58 ( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
% 0.24/0.58 = ( zero_a_b @ r ) ) ).
% 0.24/0.58
% 0.24/0.58 % local.minus_zero
% 0.24/0.58 thf(fact_237_add_Oinv__eq__1__iff,axiom,
% 0.24/0.58 ! [X: a] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( ( a_inv_a_b @ r @ X )
% 0.24/0.58 = ( zero_a_b @ r ) )
% 0.24/0.58 = ( X
% 0.24/0.58 = ( zero_a_b @ r ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.inv_eq_1_iff
% 0.24/0.58 thf(fact_238_Units__minus__one__closed,axiom,
% 0.24/0.58 member_a @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) @ ( units_a_ring_ext_a_b @ r ) ).
% 0.24/0.58
% 0.24/0.58 % Units_minus_one_closed
% 0.24/0.58 thf(fact_239_units__of__mult,axiom,
% 0.24/0.58 ! [G: partia96731725t_unit] :
% 0.24/0.58 ( ( mult_a_Product_unit @ ( units_873712258t_unit @ G ) )
% 0.24/0.58 = ( mult_a_Product_unit @ G ) ) ).
% 0.24/0.58
% 0.24/0.58 % units_of_mult
% 0.24/0.58 thf(fact_240_units__of__mult,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b] :
% 0.24/0.58 ( ( mult_a_Product_unit @ ( units_1411277569xt_a_b @ G ) )
% 0.24/0.58 = ( mult_a_ring_ext_a_b @ G ) ) ).
% 0.24/0.58
% 0.24/0.58 % units_of_mult
% 0.24/0.58 thf(fact_241_abelian__group_Oa__inv__closed,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( member_a @ ( a_inv_a_b @ G @ X ) @ ( partia1066395285xt_a_b @ G ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_group.a_inv_closed
% 0.24/0.58 thf(fact_242_abelian__group_Ominus__minus,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( a_inv_a_b @ G @ ( a_inv_a_b @ G @ X ) )
% 0.24/0.58 = X ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_group.minus_minus
% 0.24/0.58 thf(fact_243_a__minus__def,axiom,
% 0.24/0.58 ( a_minus_a_b
% 0.24/0.58 = ( ^ [R4: partia1833973666xt_a_b,X4: a,Y5: a] : ( add_a_b @ R4 @ X4 @ ( a_inv_a_b @ R4 @ Y5 ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % a_minus_def
% 0.24/0.58 thf(fact_244_units__of__one,axiom,
% 0.24/0.58 ! [G: partia96731725t_unit] :
% 0.24/0.58 ( ( one_a_Product_unit @ ( units_873712258t_unit @ G ) )
% 0.24/0.58 = ( one_a_Product_unit @ G ) ) ).
% 0.24/0.58
% 0.24/0.58 % units_of_one
% 0.24/0.58 thf(fact_245_units__of__one,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b] :
% 0.24/0.58 ( ( one_a_Product_unit @ ( units_1411277569xt_a_b @ G ) )
% 0.24/0.58 = ( one_a_ring_ext_a_b @ G ) ) ).
% 0.24/0.58
% 0.24/0.58 % units_of_one
% 0.24/0.58 thf(fact_246_abelian__group_Ominus__add,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( a_inv_a_b @ G @ ( add_a_b @ G @ X @ Y ) )
% 0.24/0.58 = ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ ( a_inv_a_b @ G @ Y ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_group.minus_add
% 0.24/0.58 thf(fact_247_abelian__group_Or__neg2,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( add_a_b @ G @ X @ ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ Y ) )
% 0.24/0.58 = Y ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_group.r_neg2
% 0.24/0.58 thf(fact_248_abelian__group_Or__neg1,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ ( add_a_b @ G @ X @ Y ) )
% 0.24/0.58 = Y ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_group.r_neg1
% 0.24/0.58 thf(fact_249_abelian__group_Ominus__eq,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ G )
% 0.24/0.58 => ( ( a_minus_a_b @ G @ X @ Y )
% 0.24/0.58 = ( add_a_b @ G @ X @ ( a_inv_a_b @ G @ Y ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_group.minus_eq
% 0.24/0.58 thf(fact_250_abelian__group_Ol__neg,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ X )
% 0.24/0.58 = ( zero_a_b @ G ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_group.l_neg
% 0.24/0.58 thf(fact_251_abelian__group_Or__neg,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( add_a_b @ G @ X @ ( a_inv_a_b @ G @ X ) )
% 0.24/0.58 = ( zero_a_b @ G ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_group.r_neg
% 0.24/0.58 thf(fact_252_abelian__group_Ominus__equality,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,Y: a,X: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ G )
% 0.24/0.58 => ( ( ( add_a_b @ G @ Y @ X )
% 0.24/0.58 = ( zero_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( a_inv_a_b @ G @ X )
% 0.24/0.58 = Y ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_group.minus_equality
% 0.24/0.58 thf(fact_253_abelian__group_Oinv__add,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,A: a,B: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ G )
% 0.24/0.58 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( a_inv_a_b @ G @ ( add_a_b @ G @ A @ B ) )
% 0.24/0.58 = ( a_minus_a_b @ G @ ( a_inv_a_b @ G @ A ) @ B ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_group.inv_add
% 0.24/0.58 thf(fact_254_abelian__group_Oa__transpose__inv,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.24/0.58 ( ( abelian_group_a_b @ G )
% 0.24/0.58 => ( ( ( add_a_b @ G @ X @ Y )
% 0.24/0.58 = Z )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ Z )
% 0.24/0.58 = Y ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % abelian_group.a_transpose_inv
% 0.24/0.58 thf(fact_255_add_Oint__pow__neg,axiom,
% 0.24/0.58 ! [X: a,I: int] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_pow_a_b_int @ r @ ( uminus_uminus_int @ I ) @ X )
% 0.24/0.58 = ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.int_pow_neg
% 0.24/0.58 thf(fact_256_add__minus__cancel,axiom,
% 0.24/0.58 ! [A: int,B: int] :
% 0.24/0.58 ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 0.24/0.58 = B ) ).
% 0.24/0.58
% 0.24/0.58 % add_minus_cancel
% 0.24/0.58 thf(fact_257_minus__add__cancel,axiom,
% 0.24/0.58 ! [A: int,B: int] :
% 0.24/0.58 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 0.24/0.58 = B ) ).
% 0.24/0.58
% 0.24/0.58 % minus_add_cancel
% 0.24/0.58 thf(fact_258_minus__add__distrib,axiom,
% 0.24/0.58 ! [A: int,B: int] :
% 0.24/0.58 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.24/0.58 = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % minus_add_distrib
% 0.24/0.58 thf(fact_259_minus__diff__eq,axiom,
% 0.24/0.58 ! [A: int,B: int] :
% 0.24/0.58 ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 0.24/0.58 = ( minus_minus_int @ B @ A ) ) ).
% 0.24/0.58
% 0.24/0.58 % minus_diff_eq
% 0.24/0.58 thf(fact_260_diff__minus__eq__add,axiom,
% 0.24/0.58 ! [A: int,B: int] :
% 0.24/0.58 ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 0.24/0.58 = ( plus_plus_int @ A @ B ) ) ).
% 0.24/0.58
% 0.24/0.58 % diff_minus_eq_add
% 0.24/0.58 thf(fact_261_uminus__add__conv__diff,axiom,
% 0.24/0.58 ! [A: int,B: int] :
% 0.24/0.58 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 0.24/0.58 = ( minus_minus_int @ B @ A ) ) ).
% 0.24/0.58
% 0.24/0.58 % uminus_add_conv_diff
% 0.24/0.58 thf(fact_262_group__cancel_Osub2,axiom,
% 0.24/0.58 ! [B5: int,K: int,B: int,A: int] :
% 0.24/0.58 ( ( B5
% 0.24/0.58 = ( plus_plus_int @ K @ B ) )
% 0.24/0.58 => ( ( minus_minus_int @ A @ B5 )
% 0.24/0.58 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group_cancel.sub2
% 0.24/0.58 thf(fact_263_diff__conv__add__uminus,axiom,
% 0.24/0.58 ( minus_minus_int
% 0.24/0.58 = ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % diff_conv_add_uminus
% 0.24/0.58 thf(fact_264_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.24/0.58 ( minus_minus_int
% 0.24/0.58 = ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.24/0.58 thf(fact_265_group__cancel_Oneg1,axiom,
% 0.24/0.58 ! [A3: int,K: int,A: int] :
% 0.24/0.58 ( ( A3
% 0.24/0.58 = ( plus_plus_int @ K @ A ) )
% 0.24/0.58 => ( ( uminus_uminus_int @ A3 )
% 0.24/0.58 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % group_cancel.neg1
% 0.24/0.58 thf(fact_266_add_Oinverse__distrib__swap,axiom,
% 0.24/0.58 ! [A: int,B: int] :
% 0.24/0.58 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.24/0.58 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.inverse_distrib_swap
% 0.24/0.58 thf(fact_267_minus__diff__commute,axiom,
% 0.24/0.58 ! [B: int,A: int] :
% 0.24/0.58 ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 0.24/0.58 = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.24/0.58
% 0.24/0.58 % minus_diff_commute
% 0.24/0.58 thf(fact_268_add_Oint__pow__neg__int,axiom,
% 0.24/0.58 ! [X: a,N: nat] :
% 0.24/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.24/0.58 => ( ( add_pow_a_b_int @ r @ ( uminus_uminus_int @ ( semiri2019852685at_int @ N ) ) @ X )
% 0.24/0.58 = ( a_inv_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % add.int_pow_neg_int
% 0.24/0.58 thf(fact_269_units__comm__group,axiom,
% 0.24/0.58 comm_g1684316527t_unit @ ( units_1411277569xt_a_b @ r ) ).
% 0.24/0.58
% 0.24/0.58 % units_comm_group
% 0.24/0.58 thf(fact_270_of__nat__eq__iff,axiom,
% 0.24/0.58 ! [M: nat,N: nat] :
% 0.24/0.58 ( ( ( semiri1382578993at_nat @ M )
% 0.24/0.58 = ( semiri1382578993at_nat @ N ) )
% 0.24/0.58 = ( M = N ) ) ).
% 0.24/0.58
% 0.24/0.58 % of_nat_eq_iff
% 0.24/0.58 thf(fact_271_of__nat__eq__iff,axiom,
% 0.24/0.58 ! [M: nat,N: nat] :
% 0.24/0.58 ( ( ( semiri2019852685at_int @ M )
% 0.24/0.58 = ( semiri2019852685at_int @ N ) )
% 0.24/0.58 = ( M = N ) ) ).
% 0.24/0.58
% 0.24/0.58 % of_nat_eq_iff
% 0.24/0.58 thf(fact_272_of__nat__add,axiom,
% 0.24/0.58 ! [M: nat,N: nat] :
% 0.24/0.58 ( ( semiri1382578993at_nat @ ( plus_plus_nat @ M @ N ) )
% 0.24/0.58 = ( plus_plus_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % of_nat_add
% 0.24/0.58 thf(fact_273_of__nat__add,axiom,
% 0.24/0.58 ! [M: nat,N: nat] :
% 0.24/0.58 ( ( semiri2019852685at_int @ ( plus_plus_nat @ M @ N ) )
% 0.24/0.58 = ( plus_plus_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % of_nat_add
% 0.24/0.58 thf(fact_274_comm__groupE_I4_J,axiom,
% 0.24/0.58 ! [G: partia96731725t_unit,X: a,Y: a] :
% 0.24/0.58 ( ( comm_g1684316527t_unit @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( mult_a_Product_unit @ G @ X @ Y )
% 0.24/0.58 = ( mult_a_Product_unit @ G @ Y @ X ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % comm_groupE(4)
% 0.24/0.58 thf(fact_275_comm__groupE_I4_J,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.24/0.58 ( ( comm_g791708116xt_a_b @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
% 0.24/0.58 = ( mult_a_ring_ext_a_b @ G @ Y @ X ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % comm_groupE(4)
% 0.24/0.58 thf(fact_276_comm__groupE_I3_J,axiom,
% 0.24/0.58 ! [G: partia96731725t_unit,X: a,Y: a,Z: a] :
% 0.24/0.58 ( ( comm_g1684316527t_unit @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( member_a @ Z @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( mult_a_Product_unit @ G @ ( mult_a_Product_unit @ G @ X @ Y ) @ Z )
% 0.24/0.58 = ( mult_a_Product_unit @ G @ X @ ( mult_a_Product_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % comm_groupE(3)
% 0.24/0.58 thf(fact_277_comm__groupE_I3_J,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.24/0.58 ( ( comm_g791708116xt_a_b @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ Z )
% 0.24/0.58 = ( mult_a_ring_ext_a_b @ G @ X @ ( mult_a_ring_ext_a_b @ G @ Y @ Z ) ) ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % comm_groupE(3)
% 0.24/0.58 thf(fact_278_comm__groupE_I1_J,axiom,
% 0.24/0.58 ! [G: partia96731725t_unit,X: a,Y: a] :
% 0.24/0.58 ( ( comm_g1684316527t_unit @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
% 0.24/0.58 => ( member_a @ ( mult_a_Product_unit @ G @ X @ Y ) @ ( partia1955795460t_unit @ G ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % comm_groupE(1)
% 0.24/0.58 thf(fact_279_comm__groupE_I1_J,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.24/0.58 ( ( comm_g791708116xt_a_b @ G )
% 0.24/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.24/0.58 => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ ( partia1066395285xt_a_b @ G ) ) ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % comm_groupE(1)
% 0.24/0.58 thf(fact_280_comm__groupE_I2_J,axiom,
% 0.24/0.58 ! [G: partia96731725t_unit] :
% 0.24/0.58 ( ( comm_g1684316527t_unit @ G )
% 0.24/0.58 => ( member_a @ ( one_a_Product_unit @ G ) @ ( partia1955795460t_unit @ G ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % comm_groupE(2)
% 0.24/0.58 thf(fact_281_comm__groupE_I2_J,axiom,
% 0.24/0.58 ! [G: partia1833973666xt_a_b] :
% 0.24/0.58 ( ( comm_g791708116xt_a_b @ G )
% 0.24/0.58 => ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) ) ) ).
% 0.24/0.58
% 0.24/0.58 % comm_groupE(2)
% 0.24/0.58
% 0.24/0.58 % Conjectures (1)
% 0.24/0.58 thf(conj_0,conjecture,
% 0.24/0.58 member449909584od_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 ) ) ) @ ( partia206007992t_unit @ rel ) ).
% 0.24/0.66
% 0.24/0.66 %------------------------------------------------------------------------------
% 0.24/0.66 ------- convert to smt2 : /export/starexec/sandbox2/tmp/tmp.PPC8jj8PT6/cvc5---1.0.5_6716.p...
% 0.24/0.66 (declare-sort $$unsorted 0)
% 0.24/0.66 (declare-sort tptp.partia1993116613t_unit 0)
% 0.24/0.66 (declare-sort tptp.partia1833973666xt_a_b 0)
% 0.24/0.66 (declare-sort tptp.partia96731725t_unit 0)
% 0.24/0.66 (declare-sort tptp.set_Product_prod_a_a 0)
% 0.24/0.66 (declare-sort tptp.product_prod_a_a 0)
% 0.24/0.66 (declare-sort tptp.set_a 0)
% 0.24/0.66 (declare-sort tptp.nat 0)
% 0.24/0.66 (declare-sort tptp.int 0)
% 0.24/0.66 (declare-sort tptp.b 0)
% 0.24/0.66 (declare-sort tptp.a 0)
% 0.24/0.66 (declare-fun tptp.partia206007992t_unit (tptp.partia1993116613t_unit) tptp.set_Product_prod_a_a)
% 0.24/0.66 (declare-fun tptp.partia1955795460t_unit (tptp.partia96731725t_unit) tptp.set_a)
% 0.24/0.66 (declare-fun tptp.partia1066395285xt_a_b (tptp.partia1833973666xt_a_b) tptp.set_a)
% 0.24/0.66 (declare-fun tptp.units_a_Product_unit (tptp.partia96731725t_unit) tptp.set_a)
% 0.24/0.66 (declare-fun tptp.units_a_ring_ext_a_b (tptp.partia1833973666xt_a_b) tptp.set_a)
% 0.24/0.66 (declare-fun tptp.comm_g1684316527t_unit (tptp.partia96731725t_unit) Bool)
% 0.24/0.66 (declare-fun tptp.comm_g791708116xt_a_b (tptp.partia1833973666xt_a_b) Bool)
% 0.24/0.66 (declare-fun tptp.group_a_Product_unit (tptp.partia96731725t_unit) Bool)
% 0.24/0.66 (declare-fun tptp.group_a_ring_ext_a_b (tptp.partia1833973666xt_a_b) Bool)
% 0.24/0.66 (declare-fun tptp.mult_a_Product_unit (tptp.partia96731725t_unit tptp.a tptp.a) tptp.a)
% 0.24/0.66 (declare-fun tptp.mult_a_ring_ext_a_b (tptp.partia1833973666xt_a_b tptp.a tptp.a) tptp.a)
% 0.24/0.66 (declare-fun tptp.one_a_Product_unit (tptp.partia96731725t_unit) tptp.a)
% 0.24/0.66 (declare-fun tptp.one_a_ring_ext_a_b (tptp.partia1833973666xt_a_b) tptp.a)
% 0.24/0.66 (declare-fun tptp.units_873712258t_unit (tptp.partia96731725t_unit) tptp.partia96731725t_unit)
% 0.24/0.66 (declare-fun tptp.units_1411277569xt_a_b (tptp.partia1833973666xt_a_b) tptp.partia96731725t_unit)
% 0.24/0.66 (declare-fun tptp.minus_minus_int (tptp.int tptp.int) tptp.int)
% 0.24/0.66 (declare-fun tptp.minus_minus_nat (tptp.nat tptp.nat) tptp.nat)
% 0.24/0.66 (declare-fun tptp.plus_plus_int (tptp.int tptp.int) tptp.int)
% 0.24/0.66 (declare-fun tptp.plus_plus_nat (tptp.nat tptp.nat) tptp.nat)
% 0.24/0.66 (declare-fun tptp.uminus_uminus_int (tptp.int) tptp.int)
% 0.24/0.66 (declare-fun tptp.suc (tptp.nat) tptp.nat)
% 0.24/0.66 (declare-fun tptp.semiri2019852685at_int (tptp.nat) tptp.int)
% 0.24/0.66 (declare-fun tptp.semiri1382578993at_nat (tptp.nat) tptp.nat)
% 0.24/0.66 (declare-fun tptp.product_Pair_a_a (tptp.a tptp.a) tptp.product_prod_a_a)
% 0.24/0.66 (declare-fun tptp.a_inv_a_b (tptp.partia1833973666xt_a_b tptp.a) tptp.a)
% 0.24/0.66 (declare-fun tptp.a_minus_a_b (tptp.partia1833973666xt_a_b tptp.a tptp.a) tptp.a)
% 0.24/0.66 (declare-fun tptp.abelian_group_a_b (tptp.partia1833973666xt_a_b) Bool)
% 0.24/0.66 (declare-fun tptp.add_pow_a_b_int (tptp.partia1833973666xt_a_b tptp.int tptp.a) tptp.a)
% 0.24/0.66 (declare-fun tptp.add_pow_a_b_nat (tptp.partia1833973666xt_a_b tptp.nat tptp.a) tptp.a)
% 0.24/0.66 (declare-fun tptp.field_a_b (tptp.partia1833973666xt_a_b) Bool)
% 0.24/0.66 (declare-fun tptp.add_a_b (tptp.partia1833973666xt_a_b tptp.a tptp.a) tptp.a)
% 0.24/0.66 (declare-fun tptp.more_a_b (tptp.partia1833973666xt_a_b) tptp.b)
% 0.24/0.66 (declare-fun tptp.zero_a_b (tptp.partia1833973666xt_a_b) tptp.a)
% 0.24/0.66 (declare-fun tptp.semiring_a_b (tptp.partia1833973666xt_a_b) Bool)
% 0.24/0.66 (declare-fun tptp.collec645855634od_a_a ((-> tptp.product_prod_a_a Bool)) tptp.set_Product_prod_a_a)
% 0.24/0.66 (declare-fun tptp.collect_a ((-> tptp.a Bool)) tptp.set_a)
% 0.24/0.66 (declare-fun tptp.member449909584od_a_a (tptp.product_prod_a_a tptp.set_Product_prod_a_a) Bool)
% 0.24/0.66 (declare-fun tptp.member_a (tptp.a tptp.set_a) Bool)
% 0.24/0.66 (declare-fun tptp.r () tptp.partia1833973666xt_a_b)
% 0.24/0.66 (declare-fun tptp.r2 () tptp.a)
% 0.24/0.66 (declare-fun tptp.r3 () tptp.a)
% 0.24/0.66 (declare-fun tptp.r4 () tptp.a)
% 0.24/0.66 (declare-fun tptp.rel () tptp.partia1993116613t_unit)
% 0.24/0.66 (declare-fun tptp.s () tptp.a)
% 0.24/0.66 (declare-fun tptp.s2 () tptp.a)
% 0.24/0.66 (declare-fun tptp.s3 () tptp.a)
% 0.24/0.66 (assert (forall ((A tptp.a) (B tptp.a) (C tptp.a)) (let ((_let_1 (@ (@ tptp.add_a_b tptp.r) C))) (=> (= A B) (= (@ _let_1 A) (@ _let_1 B))))))
% 0.24/0.66 (assert (@ (@ tptp.member449909584od_a_a (@ (@ tptp.product_Pair_a_a tptp.r4) tptp.s3)) (@ tptp.partia206007992t_unit tptp.rel)))
% 0.24/0.66 (assert (@ (@ tptp.member449909584od_a_a (@ (@ tptp.product_Pair_a_a tptp.r3) tptp.s2)) (@ tptp.partia206007992t_unit tptp.rel)))
% 0.24/0.66 (assert (@ (@ tptp.member449909584od_a_a (@ (@ tptp.product_Pair_a_a tptp.r2) tptp.s)) (@ tptp.partia206007992t_unit tptp.rel)))
% 0.24/0.66 (assert (forall ((R tptp.a) (S tptp.a) (R2 tptp.a) (S2 tptp.a)) (let ((_let_1 (@ tptp.partia206007992t_unit tptp.rel))) (let ((_let_2 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_3 (@ _let_2 S))) (=> (@ (@ tptp.member449909584od_a_a (@ (@ tptp.product_Pair_a_a R) S)) _let_1) (=> (@ (@ tptp.member449909584od_a_a (@ (@ tptp.product_Pair_a_a R2) S2)) _let_1) (@ (@ tptp.member449909584od_a_a (@ (@ tptp.product_Pair_a_a (@ (@ (@ tptp.add_a_b tptp.r) (@ (@ _let_2 S2) R)) (@ _let_3 R2))) (@ _let_3 S2))) _let_1))))))))
% 0.24/0.66 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 tptp.s))) (= (@ (@ _let_1 (@ _let_2 tptp.s3)) (@ (@ _let_1 tptp.s2) tptp.s3)) (@ (@ _let_1 (@ _let_2 (@ (@ _let_1 tptp.s3) tptp.s2))) tptp.s3)))))
% 0.24/0.66 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 tptp.s))) (= (@ (@ _let_1 (@ _let_2 tptp.s3)) (@ (@ _let_1 tptp.s2) tptp.s3)) (@ (@ _let_1 (@ _let_2 tptp.s2)) (@ (@ _let_1 tptp.s3) tptp.s3))))))
% 0.24/0.66 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ (@ _let_1 tptp.r3) tptp.r4))) (= (@ (@ _let_1 (@ (@ _let_1 tptp.s) tptp.s3)) _let_2) (@ (@ _let_1 (@ (@ _let_1 tptp.s3) tptp.s)) _let_2)))))
% 0.24/0.66 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 tptp.s))) (let ((_let_3 (@ _let_1 (@ _let_2 tptp.s3)))) (= (@ (@ _let_1 (@ _let_3 (@ (@ _let_1 tptp.s2) tptp.s3))) (@ (@ _let_1 (@ _let_2 tptp.r3)) tptp.r4)) (@ (@ _let_1 (@ (@ _let_1 (@ _let_2 tptp.s2)) tptp.s3)) (@ _let_3 (@ (@ _let_1 tptp.r3) tptp.r4))))))))
% 0.24/0.66 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 tptp.s2))) (let ((_let_3 (@ _let_2 tptp.s3))) (let ((_let_4 (@ _let_1 tptp.s))) (= (@ (@ _let_1 (@ (@ _let_1 (@ _let_4 tptp.s3)) _let_3)) (@ (@ _let_1 (@ _let_2 tptp.r2)) tptp.r4)) (@ (@ _let_1 (@ (@ _let_1 (@ _let_4 tptp.s2)) tptp.s3)) (@ (@ _let_1 _let_3) (@ (@ _let_1 tptp.r2) tptp.r4)))))))))
% 0.24/0.66 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (@ (@ tptp.member449909584od_a_a (@ (@ tptp.product_Pair_a_a (@ (@ _let_1 tptp.r3) tptp.r4)) (@ (@ _let_1 tptp.s2) tptp.s3))) (@ tptp.partia206007992t_unit tptp.rel))))
% 0.24/0.66 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (@ (@ tptp.member449909584od_a_a (@ (@ tptp.product_Pair_a_a (@ (@ _let_1 tptp.r2) tptp.r4)) (@ (@ _let_1 tptp.s) tptp.s3))) (@ tptp.partia206007992t_unit tptp.rel))))
% 0.24/0.66 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 tptp.s))) (let ((_let_3 (@ _let_2 tptp.r3))) (let ((_let_4 (@ _let_1 tptp.s2))) (let ((_let_5 (@ _let_4 tptp.r2))) (let ((_let_6 (@ tptp.add_a_b tptp.r))) (let ((_let_7 (@ _let_1 (@ (@ _let_1 (@ _let_2 tptp.s3)) (@ _let_4 tptp.s3))))) (= (@ _let_7 (@ (@ _let_1 (@ (@ _let_6 _let_5) _let_3)) tptp.r4)) (@ _let_7 (@ (@ _let_6 (@ (@ _let_1 _let_5) tptp.r4)) (@ (@ _let_1 _let_3) tptp.r4))))))))))))
% 0.24/0.66 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 tptp.s))) (let ((_let_3 (@ _let_1 (@ _let_2 tptp.s3)))) (let ((_let_4 (@ _let_1 tptp.s2))) (let ((_let_5 (@ _let_4 tptp.s3))) (let ((_let_6 (@ tptp.add_a_b tptp.r))) (= (@ (@ _let_1 (@ _let_3 _let_5)) (@ (@ _let_1 (@ (@ _let_6 (@ _let_4 tptp.r2)) (@ _let_2 tptp.r3))) tptp.r4)) (@ (@ _let_1 (@ (@ _let_1 (@ _let_2 tptp.s2)) tptp.s3)) (@ (@ _let_6 (@ (@ _let_1 _let_5) (@ (@ _let_1 tptp.r2) tptp.r4))) (@ _let_3 (@ (@ _let_1 tptp.r3) tptp.r4))))))))))))
% 0.24/0.66 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 tptp.s))) (let ((_let_3 (@ _let_2 tptp.r3))) (let ((_let_4 (@ _let_1 tptp.s2))) (let ((_let_5 (@ _let_1 (@ (@ _let_1 (@ _let_2 tptp.s3)) (@ _let_4 tptp.s3))))) (let ((_let_6 (@ _let_4 tptp.r2))) (let ((_let_7 (@ tptp.add_a_b tptp.r))) (= (@ _let_5 (@ (@ _let_1 (@ (@ _let_7 _let_6) _let_3)) tptp.r4)) (@ (@ _let_7 (@ _let_5 (@ (@ _let_1 _let_6) tptp.r4))) (@ _let_5 (@ (@ _let_1 _let_3) tptp.r4))))))))))))
% 0.24/0.66 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 tptp.s))) (let ((_let_3 (@ (@ _let_1 (@ _let_2 tptp.s3)) (@ (@ _let_1 tptp.r3) tptp.r4)))) (let ((_let_4 (@ _let_1 (@ (@ _let_1 (@ _let_2 tptp.s2)) tptp.s3)))) (let ((_let_5 (@ (@ _let_1 (@ (@ _let_1 tptp.s2) tptp.s3)) (@ (@ _let_1 tptp.r2) tptp.r4)))) (let ((_let_6 (@ tptp.add_a_b tptp.r))) (= (@ _let_4 (@ (@ _let_6 _let_5) _let_3)) (@ (@ _let_6 (@ _let_4 _let_5)) (@ _let_4 _let_3))))))))))
% 0.24/0.66 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 tptp.s))) (@ (@ tptp.member449909584od_a_a (@ (@ tptp.product_Pair_a_a (@ (@ _let_1 (@ (@ (@ tptp.add_a_b tptp.r) (@ (@ _let_1 tptp.s2) tptp.r2)) (@ _let_2 tptp.r3))) tptp.r4)) (@ (@ _let_1 (@ _let_2 tptp.s2)) tptp.s3))) (@ tptp.partia206007992t_unit tptp.rel)))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ tptp.add_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ (@ _let_1 (@ (@ _let_2 X) Y)) Z) (@ (@ _let_2 (@ (@ _let_1 X) Z)) (@ (@ _let_1 Y) Z)))))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) Z))) (let ((_let_2 (@ tptp.add_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ _let_1 (@ (@ _let_2 X) Y)) (@ (@ _let_2 (@ _let_1 X)) (@ _let_1 Y)))))))))))
% 0.24/0.66 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 tptp.s))) (let ((_let_3 (@ _let_1 (@ _let_2 tptp.s3)))) (let ((_let_4 (@ _let_1 tptp.s2))) (let ((_let_5 (@ _let_4 tptp.s3))) (let ((_let_6 (@ tptp.add_a_b tptp.r))) (= (@ (@ (@ tptp.a_minus_a_b tptp.r) (@ (@ _let_1 (@ _let_3 _let_5)) (@ (@ _let_1 (@ (@ _let_6 (@ _let_4 tptp.r2)) (@ _let_2 tptp.r3))) tptp.r4))) (@ (@ _let_1 (@ (@ _let_1 (@ _let_2 tptp.s2)) tptp.s3)) (@ (@ _let_6 (@ (@ _let_1 _let_5) (@ (@ _let_1 tptp.r2) tptp.r4))) (@ _let_3 (@ (@ _let_1 tptp.r3) tptp.r4))))) (@ tptp.zero_a_b tptp.r)))))))))
% 0.24/0.66 (assert (forall ((X1 tptp.a) (X2 tptp.a) (Y1 tptp.a) (Y2 tptp.a)) (= (= (@ (@ tptp.product_Pair_a_a X1) X2) (@ (@ tptp.product_Pair_a_a Y1) Y2)) (and (= X1 Y1) (= X2 Y2)))))
% 0.24/0.66 (assert (forall ((A tptp.a) (B tptp.a) (A2 tptp.a) (B2 tptp.a)) (= (= (@ (@ tptp.product_Pair_a_a A) B) (@ (@ tptp.product_Pair_a_a A2) B2)) (and (= A A2) (= B B2)))))
% 0.24/0.66 (assert (@ tptp.semiring_a_b tptp.r))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 X))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ (@ _let_1 (@ _let_2 Y)) Z) (@ _let_2 (@ (@ _let_1 Y) Z)))))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (= (@ (@ _let_1 X) Y) (@ (@ _let_1 Y) X))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 X))) (let ((_let_3 (@ _let_1 Y))) (let ((_let_4 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_4) (=> (@ (@ tptp.member_a Y) _let_4) (=> (@ (@ tptp.member_a Z) _let_4) (= (@ _let_2 (@ _let_3 Z)) (@ _let_3 (@ _let_2 Z))))))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ _let_1 X))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ (@ _let_1 (@ _let_2 Y)) Z) (@ _let_2 (@ (@ _let_1 Y) Z)))))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (= (@ (@ _let_1 X) Y) (@ (@ _let_1 Y) X))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ _let_1 X))) (let ((_let_3 (@ _let_1 Y))) (let ((_let_4 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_4) (=> (@ (@ tptp.member_a Y) _let_4) (=> (@ (@ tptp.member_a Z) _let_4) (= (@ _let_2 (@ _let_3 Z)) (@ _let_3 (@ _let_2 Z))))))))))))
% 0.24/0.66 (assert (forall ((A tptp.a) (B tptp.a) (C tptp.a)) (let ((_let_1 (@ (@ tptp.a_minus_a_b tptp.r) C))) (=> (= A B) (= (@ _let_1 A) (@ _let_1 B))))))
% 0.24/0.66 (assert (forall ((Y tptp.a) (X tptp.a) (Y3 tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (let ((_let_2 (@ tptp.zero_a_b tptp.r))) (let ((_let_3 (@ tptp.add_a_b tptp.r))) (=> (= (@ (@ _let_3 Y) X) _let_2) (=> (= (@ (@ _let_3 X) Y3) _let_2) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (=> (@ (@ tptp.member_a Y3) _let_1) (= Y Y3)))))))))))
% 0.24/0.66 (assert (forall ((X tptp.a)) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (exists ((X3 tptp.a)) (and (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.add_a_b tptp.r) X) X3) (@ tptp.zero_a_b tptp.r)))))))
% 0.24/0.66 (assert (forall ((U tptp.a)) (=> (@ (@ tptp.member_a U) (@ tptp.partia1066395285xt_a_b tptp.r)) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.add_a_b tptp.r) U) X3) X3))) (= U (@ tptp.zero_a_b tptp.r))))))
% 0.24/0.66 (assert (forall ((X tptp.a)) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (exists ((X3 tptp.a)) (and (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.add_a_b tptp.r) X3) X) (@ tptp.zero_a_b tptp.r)))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.zero_a_b tptp.r))) (let ((_let_2 (@ tptp.add_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (= (@ (@ _let_2 X) Y) _let_1) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ (@ _let_2 Y) X) _let_1)))))))))
% 0.24/0.66 (assert (forall ((A tptp.a) (B tptp.a) (C tptp.a)) (let ((_let_1 (@ tptp.a_minus_a_b tptp.r))) (let ((_let_2 (@ _let_1 C))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_3) (=> (@ (@ tptp.member_a B) _let_3) (=> (@ (@ tptp.member_a C) _let_3) (= (@ (@ _let_1 (@ _let_2 A)) B) (@ _let_2 (@ (@ (@ tptp.add_a_b tptp.r) A) B)))))))))))
% 0.24/0.66 (assert (forall ((A tptp.a) (B tptp.a) (C tptp.a) (D tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.a_minus_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_3) (=> (@ (@ tptp.member_a B) _let_3) (=> (@ (@ tptp.member_a C) _let_3) (=> (@ (@ tptp.member_a D) _let_3) (= (@ (@ _let_2 (@ (@ _let_1 (@ (@ _let_2 A) C)) B)) D) (@ (@ _let_2 (@ (@ _let_2 (@ (@ _let_1 A) B)) C)) D)))))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (=> (@ (@ tptp.member_a Z) _let_2) (= (= (@ (@ _let_1 Y) X) (@ (@ _let_1 Z) X)) (= Y Z)))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.add_a_b tptp.r) X) Y)) _let_1))))))
% 0.24/0.66 (assert (@ (@ tptp.member_a (@ tptp.zero_a_b tptp.r)) (@ tptp.partia1066395285xt_a_b tptp.r)))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) X) Y)) _let_1))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.a_minus_a_b tptp.r) X) Y)) _let_1))))))
% 0.24/0.66 (assert (forall ((X tptp.a)) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.add_a_b tptp.r) X) (@ tptp.zero_a_b tptp.r)) X))))
% 0.24/0.66 (assert (forall ((X tptp.a)) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.add_a_b tptp.r) (@ tptp.zero_a_b tptp.r)) X) X))))
% 0.24/0.66 (assert (forall ((X tptp.a) (A tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a A) _let_1) (= (= X (@ (@ (@ tptp.add_a_b tptp.r) A) X)) (= A (@ tptp.zero_a_b tptp.r))))))))
% 0.24/0.66 (assert (forall ((A tptp.product_prod_a_a) (P (-> tptp.product_prod_a_a Bool))) (= (@ (@ tptp.member449909584od_a_a A) (@ tptp.collec645855634od_a_a P)) (@ P A))))
% 0.24/0.66 (assert (forall ((A tptp.a) (P (-> tptp.a Bool))) (= (@ (@ tptp.member_a A) (@ tptp.collect_a P)) (@ P A))))
% 0.24/0.66 (assert (forall ((A3 tptp.set_Product_prod_a_a)) (= (@ tptp.collec645855634od_a_a (lambda ((X4 tptp.product_prod_a_a)) (@ (@ tptp.member449909584od_a_a X4) A3))) A3)))
% 0.24/0.66 (assert (forall ((A3 tptp.set_a)) (= (@ tptp.collect_a (lambda ((X4 tptp.a)) (@ (@ tptp.member_a X4) A3))) A3)))
% 0.24/0.66 (assert (forall ((P (-> tptp.a Bool)) (Q (-> tptp.a Bool))) (=> (forall ((X3 tptp.a)) (= (@ P X3) (@ Q X3))) (= (@ tptp.collect_a P) (@ tptp.collect_a Q)))))
% 0.24/0.66 (assert (forall ((P (-> tptp.product_prod_a_a Bool)) (Q (-> tptp.product_prod_a_a Bool))) (=> (forall ((X3 tptp.product_prod_a_a)) (= (@ P X3) (@ Q X3))) (= (@ tptp.collec645855634od_a_a P) (@ tptp.collec645855634od_a_a Q)))))
% 0.24/0.66 (assert (forall ((X tptp.a) (A tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a A) _let_1) (= (= (@ (@ (@ tptp.add_a_b tptp.r) A) X) X) (= A (@ tptp.zero_a_b tptp.r))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (A tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a A) _let_1) (= (= X (@ (@ (@ tptp.add_a_b tptp.r) X) A)) (= A (@ tptp.zero_a_b tptp.r))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (A tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a A) _let_1) (= (= (@ (@ (@ tptp.add_a_b tptp.r) X) A) X) (= A (@ tptp.zero_a_b tptp.r))))))))
% 0.24/0.66 (assert (forall ((X tptp.a)) (let ((_let_1 (@ tptp.zero_a_b tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) X) _let_1) _let_1)))))
% 0.24/0.66 (assert (forall ((X tptp.a)) (let ((_let_1 (@ tptp.zero_a_b tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) _let_1) X) _let_1)))))
% 0.24/0.66 (assert (forall ((P (-> tptp.product_prod_a_a Bool)) (Prod tptp.product_prod_a_a)) (=> (forall ((A4 tptp.a) (B3 tptp.a)) (@ P (@ (@ tptp.product_Pair_a_a A4) B3))) (@ P Prod))))
% 0.24/0.66 (assert (forall ((Y tptp.product_prod_a_a)) (not (forall ((A4 tptp.a) (B3 tptp.a)) (not (= Y (@ (@ tptp.product_Pair_a_a A4) B3)))))))
% 0.24/0.66 (assert (forall ((A tptp.a) (B tptp.a) (A2 tptp.a) (B2 tptp.a)) (=> (= (@ (@ tptp.product_Pair_a_a A) B) (@ (@ tptp.product_Pair_a_a A2) B2)) (not (=> (= A A2) (not (= B B2)))))))
% 0.24/0.66 (assert (forall ((P (-> tptp.product_prod_a_a Bool)) (P2 tptp.product_prod_a_a)) (=> (forall ((A4 tptp.a) (B3 tptp.a)) (@ P (@ (@ tptp.product_Pair_a_a A4) B3))) (@ P P2))))
% 0.24/0.66 (assert (forall ((P2 tptp.product_prod_a_a)) (exists ((X3 tptp.a) (Y4 tptp.a)) (= P2 (@ (@ tptp.product_Pair_a_a X3) Y4)))))
% 0.24/0.66 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 tptp.s))) (let ((_let_3 (@ _let_1 (@ _let_2 tptp.s3)))) (let ((_let_4 (@ _let_1 tptp.s2))) (let ((_let_5 (@ _let_4 tptp.s3))) (let ((_let_6 (@ tptp.add_a_b tptp.r))) (= (@ (@ _let_1 (@ tptp.one_a_ring_ext_a_b tptp.r)) (@ (@ (@ tptp.a_minus_a_b tptp.r) (@ (@ _let_1 (@ _let_3 _let_5)) (@ (@ _let_1 (@ (@ _let_6 (@ _let_4 tptp.r2)) (@ _let_2 tptp.r3))) tptp.r4))) (@ (@ _let_1 (@ (@ _let_1 (@ _let_2 tptp.s2)) tptp.s3)) (@ (@ _let_6 (@ (@ _let_1 _let_5) (@ (@ _let_1 tptp.r2) tptp.r4))) (@ _let_3 (@ (@ _let_1 tptp.r3) tptp.r4)))))) (@ tptp.zero_a_b tptp.r)))))))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a)) (=> (@ tptp.semiring_a_b R3) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b R3)) (= (@ (@ (@ tptp.add_a_b R3) X) (@ tptp.zero_a_b R3)) X)))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a)) (=> (@ tptp.semiring_a_b R3) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b R3)) (= (@ (@ (@ tptp.add_a_b R3) (@ tptp.zero_a_b R3)) X) X)))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a)) (let ((_let_1 (@ tptp.zero_a_b R3))) (=> (@ tptp.semiring_a_b R3) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b R3)) (= (@ (@ (@ tptp.mult_a_ring_ext_a_b R3) _let_1) X) _let_1))))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a)) (let ((_let_1 (@ tptp.zero_a_b R3))) (=> (@ tptp.semiring_a_b R3) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b R3)) (= (@ (@ (@ tptp.mult_a_ring_ext_a_b R3) X) _let_1) _let_1))))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ (@ tptp.mult_a_ring_ext_a_b R3) Z))) (let ((_let_2 (@ tptp.add_a_b R3))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b R3))) (=> (@ tptp.semiring_a_b R3) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ _let_1 (@ (@ _let_2 X) Y)) (@ (@ _let_2 (@ _let_1 X)) (@ _let_1 Y))))))))))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b R3))) (let ((_let_2 (@ tptp.add_a_b R3))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b R3))) (=> (@ tptp.semiring_a_b R3) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ (@ _let_1 (@ (@ _let_2 X) Y)) Z) (@ (@ _let_2 (@ (@ _let_1 X) Z)) (@ (@ _let_1 Y) Z))))))))))))
% 0.24/0.66 (assert (forall ((A tptp.a) (B tptp.a) (K tptp.int)) (let ((_let_1 (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) A))) (let ((_let_2 (@ (@ tptp.add_pow_a_b_int tptp.r) K))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_3) (=> (@ (@ tptp.member_a B) _let_3) (= (@ _let_1 (@ _let_2 B)) (@ _let_2 (@ _let_1 B))))))))))
% 0.24/0.66 (assert (forall ((A tptp.a) (B tptp.a) (K tptp.int)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ (@ tptp.add_pow_a_b_int tptp.r) K))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_3) (=> (@ (@ tptp.member_a B) _let_3) (= (@ (@ _let_1 (@ _let_2 A)) B) (@ _let_2 (@ (@ _let_1 A) B))))))))))
% 0.24/0.66 (assert (forall ((Y tptp.a) (X tptp.a) (Y3 tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (let ((_let_2 (@ tptp.one_a_ring_ext_a_b tptp.r))) (let ((_let_3 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (=> (= (@ (@ _let_3 Y) X) _let_2) (=> (= (@ (@ _let_3 X) Y3) _let_2) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (=> (@ (@ tptp.member_a Y3) _let_1) (= Y Y3)))))))))))
% 0.24/0.66 (assert (forall ((U tptp.a)) (=> (@ (@ tptp.member_a U) (@ tptp.partia1066395285xt_a_b tptp.r)) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) U) X3) X3))) (= U (@ tptp.one_a_ring_ext_a_b tptp.r))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a) (I tptp.int)) (let ((_let_1 (@ (@ tptp.add_pow_a_b_int tptp.r) I))) (let ((_let_2 (@ tptp.add_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ _let_1 (@ (@ _let_2 X) Y)) (@ (@ _let_2 (@ _let_1 X)) (@ _let_1 Y))))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a) (I tptp.int)) (let ((_let_1 (@ (@ tptp.add_pow_a_b_int tptp.r) I))) (let ((_let_2 (@ tptp.add_a_b tptp.r))) (let ((_let_3 (@ (@ _let_2 X) Y))) (let ((_let_4 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (= _let_3 (@ (@ _let_2 Y) X)) (=> (@ (@ tptp.member_a X) _let_4) (=> (@ (@ tptp.member_a Y) _let_4) (= (@ _let_1 _let_3) (@ (@ _let_2 (@ _let_1 X)) (@ _let_1 Y))))))))))))
% 0.24/0.66 (assert (@ (@ tptp.member_a (@ tptp.one_a_ring_ext_a_b tptp.r)) (@ tptp.partia1066395285xt_a_b tptp.r)))
% 0.24/0.66 (assert (forall ((X tptp.a) (I tptp.int)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.add_pow_a_b_int tptp.r) I) X)) _let_1)))))
% 0.24/0.66 (assert (forall ((Z tptp.int)) (let ((_let_1 (@ tptp.zero_a_b tptp.r))) (= (@ (@ (@ tptp.add_pow_a_b_int tptp.r) Z) _let_1) _let_1))))
% 0.24/0.66 (assert (forall ((X tptp.a)) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) (@ tptp.one_a_ring_ext_a_b tptp.r)) X) X))))
% 0.24/0.66 (assert (forall ((X tptp.a)) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) X) (@ tptp.one_a_ring_ext_a_b tptp.r)) X))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b)) (=> (@ tptp.semiring_a_b R3) (@ (@ tptp.member_a (@ tptp.one_a_ring_ext_a_b R3)) (@ tptp.partia1066395285xt_a_b R3)))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a)) (=> (@ tptp.semiring_a_b R3) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b R3)) (= (@ (@ (@ tptp.mult_a_ring_ext_a_b R3) (@ tptp.one_a_ring_ext_a_b R3)) X) X)))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b R3))) (=> (@ tptp.semiring_a_b R3) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_ring_ext_a_b R3) X) Y)) _let_1)))))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b R3))) (let ((_let_2 (@ _let_1 X))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b R3))) (=> (@ tptp.semiring_a_b R3) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ (@ _let_1 (@ _let_2 Y)) Z) (@ _let_2 (@ (@ _let_1 Y) Z))))))))))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b R3))) (=> (@ tptp.semiring_a_b R3) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.add_a_b R3) X) Y)) _let_1)))))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.add_a_b R3))) (let ((_let_2 (@ _let_1 X))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b R3))) (=> (@ tptp.semiring_a_b R3) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ (@ _let_1 (@ _let_2 Y)) Z) (@ _let_2 (@ (@ _let_1 Y) Z))))))))))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.add_a_b R3))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b R3))) (=> (@ tptp.semiring_a_b R3) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (= (@ (@ _let_1 X) Y) (@ (@ _let_1 Y) X)))))))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.add_a_b R3))) (let ((_let_2 (@ _let_1 X))) (let ((_let_3 (@ _let_1 Y))) (let ((_let_4 (@ tptp.partia1066395285xt_a_b R3))) (=> (@ tptp.semiring_a_b R3) (=> (@ (@ tptp.member_a X) _let_4) (=> (@ (@ tptp.member_a Y) _let_4) (=> (@ (@ tptp.member_a Z) _let_4) (= (@ _let_2 (@ _let_3 Z)) (@ _let_3 (@ _let_2 Z)))))))))))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b)) (=> (@ tptp.semiring_a_b R3) (@ (@ tptp.member_a (@ tptp.zero_a_b R3)) (@ tptp.partia1066395285xt_a_b R3)))))
% 0.24/0.66 (assert (=> (not (= (@ tptp.zero_a_b tptp.r) (@ tptp.one_a_ring_ext_a_b tptp.r))) (=> (forall ((A4 tptp.a)) (=> (@ (@ tptp.member_a A4) (@ tptp.partia1066395285xt_a_b tptp.r)) (=> (not (= A4 (@ tptp.zero_a_b tptp.r))) (exists ((X5 tptp.a)) (and (@ (@ tptp.member_a X5) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) A4) X5) (@ tptp.one_a_ring_ext_a_b tptp.r))))))) (@ tptp.field_a_b tptp.r))))
% 0.24/0.66 (assert (forall ((X tptp.a) (I tptp.int) (J tptp.int)) (let ((_let_1 (@ tptp.add_pow_a_b_int tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ _let_1 (@ (@ tptp.plus_plus_int I) J)) X) (@ (@ (@ tptp.add_a_b tptp.r) (@ (@ _let_1 I) X)) (@ (@ _let_1 J) X)))))))
% 0.24/0.66 (assert (forall ((A tptp.a) (B tptp.a) (K tptp.nat)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ (@ tptp.add_pow_a_b_nat tptp.r) K))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_3) (=> (@ (@ tptp.member_a B) _let_3) (= (@ (@ _let_1 (@ _let_2 A)) B) (@ _let_2 (@ (@ _let_1 A) B))))))))))
% 0.24/0.66 (assert (forall ((A tptp.a) (B tptp.a) (K tptp.nat)) (let ((_let_1 (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) A))) (let ((_let_2 (@ (@ tptp.add_pow_a_b_nat tptp.r) K))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_3) (=> (@ (@ tptp.member_a B) _let_3) (= (@ _let_1 (@ _let_2 B)) (@ _let_2 (@ _let_1 B))))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a) (N tptp.nat)) (let ((_let_1 (@ (@ (@ tptp.add_pow_a_b_nat tptp.r) N) X))) (let ((_let_2 (@ tptp.add_a_b tptp.r))) (let ((_let_3 (@ _let_2 Y))) (let ((_let_4 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (= (@ (@ _let_2 X) Y) (@ _let_3 X)) (=> (@ (@ tptp.member_a X) _let_4) (=> (@ (@ tptp.member_a Y) _let_4) (= (@ (@ _let_2 _let_1) Y) (@ _let_3 _let_1)))))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a) (N tptp.nat)) (let ((_let_1 (@ (@ tptp.add_pow_a_b_nat tptp.r) N))) (let ((_let_2 (@ tptp.add_a_b tptp.r))) (let ((_let_3 (@ (@ _let_2 X) Y))) (let ((_let_4 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (= _let_3 (@ (@ _let_2 Y) X)) (=> (@ (@ tptp.member_a X) _let_4) (=> (@ (@ tptp.member_a Y) _let_4) (= (@ _let_1 _let_3) (@ (@ _let_2 (@ _let_1 X)) (@ _let_1 Y))))))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a) (N tptp.nat)) (let ((_let_1 (@ (@ tptp.add_pow_a_b_nat tptp.r) N))) (let ((_let_2 (@ tptp.add_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ _let_1 (@ (@ _let_2 X) Y)) (@ (@ _let_2 (@ _let_1 X)) (@ _let_1 Y))))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (N tptp.nat) (M tptp.nat)) (let ((_let_1 (@ tptp.add_pow_a_b_nat tptp.r))) (let ((_let_2 (@ (@ _let_1 N) X))) (let ((_let_3 (@ (@ _let_1 M) X))) (let ((_let_4 (@ tptp.add_a_b tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ _let_4 _let_2) _let_3) (@ (@ _let_4 _let_3) _let_2)))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (N tptp.nat)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.add_pow_a_b_nat tptp.r) N) X)) _let_1)))))
% 0.24/0.66 (assert (forall ((N tptp.nat)) (let ((_let_1 (@ tptp.zero_a_b tptp.r))) (= (@ (@ (@ tptp.add_pow_a_b_nat tptp.r) N) _let_1) _let_1))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b)) (=> (@ tptp.field_a_b R3) (not (= (@ tptp.one_a_ring_ext_a_b R3) (@ tptp.zero_a_b R3))))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (A tptp.a) (B tptp.a)) (let ((_let_1 (@ tptp.zero_a_b R3))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b R3))) (=> (@ tptp.field_a_b R3) (=> (= (@ (@ (@ tptp.mult_a_ring_ext_a_b R3) A) B) _let_1) (=> (@ (@ tptp.member_a A) _let_2) (=> (@ (@ tptp.member_a B) _let_2) (or (= A _let_1) (= B _let_1))))))))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (A tptp.a) (B tptp.a) (K tptp.nat)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b R3))) (let ((_let_2 (@ (@ tptp.add_pow_a_b_nat R3) K))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b R3))) (=> (@ tptp.semiring_a_b R3) (=> (@ (@ tptp.member_a A) _let_3) (=> (@ (@ tptp.member_a B) _let_3) (= (@ (@ _let_1 (@ _let_2 A)) B) (@ _let_2 (@ (@ _let_1 A) B)))))))))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (A tptp.a) (B tptp.a) (K tptp.nat)) (let ((_let_1 (@ (@ tptp.mult_a_ring_ext_a_b R3) A))) (let ((_let_2 (@ (@ tptp.add_pow_a_b_nat R3) K))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b R3))) (=> (@ tptp.semiring_a_b R3) (=> (@ (@ tptp.member_a A) _let_3) (=> (@ (@ tptp.member_a B) _let_3) (= (@ _let_1 (@ _let_2 B)) (@ _let_2 (@ _let_1 B)))))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (N tptp.nat) (M tptp.nat)) (let ((_let_1 (@ tptp.add_pow_a_b_nat tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.add_a_b tptp.r) (@ (@ _let_1 N) X)) (@ (@ _let_1 M) X)) (@ (@ _let_1 (@ (@ tptp.plus_plus_nat N) M)) X))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (N tptp.nat)) (let ((_let_1 (@ tptp.add_pow_a_b_nat tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ _let_1 (@ tptp.suc N)) X) (@ (@ (@ tptp.add_a_b tptp.r) X) (@ (@ _let_1 N) X)))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (M tptp.nat) (N tptp.nat)) (let ((_let_1 (@ tptp.add_pow_a_b_nat tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (=> (= (@ (@ _let_1 M) X) (@ (@ _let_1 N) X)) (= (@ (@ _let_1 (@ (@ tptp.minus_minus_nat M) N)) X) (@ tptp.zero_a_b tptp.r)))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ tptp.abelian_group_a_b tptp.r) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (=> (= (@ (@ (@ tptp.a_minus_a_b tptp.r) X) Y) (@ tptp.zero_a_b tptp.r)) (= X Y))))))))
% 0.24/0.66 (assert (forall ((R tptp.partia1833973666xt_a_b) (R2 tptp.partia1833973666xt_a_b)) (=> (= (@ tptp.partia1066395285xt_a_b R) (@ tptp.partia1066395285xt_a_b R2)) (=> (= (@ tptp.mult_a_ring_ext_a_b R) (@ tptp.mult_a_ring_ext_a_b R2)) (=> (= (@ tptp.one_a_ring_ext_a_b R) (@ tptp.one_a_ring_ext_a_b R2)) (=> (= (@ tptp.zero_a_b R) (@ tptp.zero_a_b R2)) (=> (= (@ tptp.add_a_b R) (@ tptp.add_a_b R2)) (=> (= (@ tptp.more_a_b R) (@ tptp.more_a_b R2)) (= R R2)))))))))
% 0.24/0.66 (assert (@ tptp.abelian_group_a_b tptp.r))
% 0.24/0.66 (assert (forall ((N tptp.nat) (X tptp.a)) (let ((_let_1 (@ tptp.add_pow_a_b_nat tptp.r))) (= (@ (@ _let_1 (@ tptp.suc N)) X) (@ (@ (@ tptp.add_a_b tptp.r) (@ (@ _let_1 N) X)) X)))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b R3))) (=> (@ tptp.abelian_group_a_b R3) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.add_a_b R3) X) Y)) _let_1)))))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.add_a_b R3))) (let ((_let_2 (@ _let_1 X))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b R3))) (=> (@ tptp.abelian_group_a_b R3) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ (@ _let_1 (@ _let_2 Y)) Z) (@ _let_2 (@ (@ _let_1 Y) Z))))))))))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.add_a_b R3))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b R3))) (=> (@ tptp.abelian_group_a_b R3) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (= (@ (@ _let_1 X) Y) (@ (@ _let_1 Y) X)))))))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b)) (=> (@ tptp.abelian_group_a_b R3) (@ (@ tptp.member_a (@ tptp.zero_a_b R3)) (@ tptp.partia1066395285xt_a_b R3)))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.a_minus_a_b G) X) Y)) _let_1)))))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b)) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b R3)) (forall ((Y4 tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b R3))) (=> (@ (@ tptp.member_a Y4) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.add_a_b R3) X3) Y4)) _let_1)))))) (=> (@ (@ tptp.member_a (@ tptp.zero_a_b R3)) (@ tptp.partia1066395285xt_a_b R3)) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b R3)) (forall ((Y4 tptp.a)) (=> (@ (@ tptp.member_a Y4) (@ tptp.partia1066395285xt_a_b R3)) (forall ((Z2 tptp.a)) (let ((_let_1 (@ tptp.add_a_b R3))) (let ((_let_2 (@ _let_1 X3))) (=> (@ (@ tptp.member_a Z2) (@ tptp.partia1066395285xt_a_b R3)) (= (@ (@ _let_1 (@ _let_2 Y4)) Z2) (@ _let_2 (@ (@ _let_1 Y4) Z2))))))))))) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b R3)) (forall ((Y4 tptp.a)) (let ((_let_1 (@ tptp.add_a_b R3))) (=> (@ (@ tptp.member_a Y4) (@ tptp.partia1066395285xt_a_b R3)) (= (@ (@ _let_1 X3) Y4) (@ (@ _let_1 Y4) X3))))))) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b R3)) (= (@ (@ (@ tptp.add_a_b R3) (@ tptp.zero_a_b R3)) X3) X3))) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b R3)) (exists ((Xa tptp.a)) (and (@ (@ tptp.member_a Xa) (@ tptp.partia1066395285xt_a_b R3)) (= (@ (@ (@ tptp.add_a_b R3) Xa) X3) (@ tptp.zero_a_b R3)))))) (@ tptp.abelian_group_a_b R3)))))))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a)) (=> (@ tptp.abelian_group_a_b R3) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b R3)) (= (@ (@ (@ tptp.add_a_b R3) (@ tptp.zero_a_b R3)) X) X)))))
% 0.24/0.66 (assert (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a)) (=> (@ tptp.abelian_group_a_b R3) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b R3)) (exists ((X3 tptp.a)) (and (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b R3)) (= (@ (@ (@ tptp.add_a_b R3) X3) X) (@ tptp.zero_a_b R3))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (A tptp.a) (B tptp.a) (C tptp.a) (D tptp.a)) (let ((_let_1 (@ tptp.add_a_b G))) (let ((_let_2 (@ tptp.a_minus_a_b G))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a A) _let_3) (=> (@ (@ tptp.member_a B) _let_3) (=> (@ (@ tptp.member_a C) _let_3) (=> (@ (@ tptp.member_a D) _let_3) (= (@ (@ _let_2 (@ (@ _let_1 (@ (@ _let_2 A) C)) B)) D) (@ (@ _let_2 (@ (@ _let_2 (@ (@ _let_1 A) B)) C)) D))))))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (A tptp.a) (B tptp.a) (C tptp.a)) (let ((_let_1 (@ tptp.a_minus_a_b G))) (let ((_let_2 (@ _let_1 C))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a A) _let_3) (=> (@ (@ tptp.member_a B) _let_3) (=> (@ (@ tptp.member_a C) _let_3) (= (@ (@ _let_1 (@ _let_2 A)) B) (@ _let_2 (@ (@ (@ tptp.add_a_b G) A) B))))))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b G))) (let ((_let_2 (@ tptp.abelian_group_a_b G))) (=> _let_2 (=> _let_2 (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (=> (= (@ (@ (@ tptp.a_minus_a_b G) X) Y) (@ tptp.zero_a_b G)) (= X Y))))))))))
% 0.24/0.66 (assert (forall ((I tptp.nat) (J tptp.nat) (K tptp.nat)) (let ((_let_1 (@ tptp.minus_minus_nat I))) (= (@ (@ tptp.minus_minus_nat (@ _let_1 J)) K) (@ _let_1 (@ (@ tptp.plus_plus_nat J) K))))))
% 0.24/0.66 (assert (forall ((M tptp.nat) (N tptp.nat) (K tptp.nat)) (= (@ (@ tptp.minus_minus_nat (@ (@ tptp.minus_minus_nat (@ tptp.suc M)) N)) (@ tptp.suc K)) (@ (@ tptp.minus_minus_nat (@ (@ tptp.minus_minus_nat M) N)) K))))
% 0.24/0.66 (assert (forall ((M tptp.nat) (N tptp.nat)) (= (@ (@ tptp.minus_minus_nat (@ tptp.suc M)) (@ tptp.suc N)) (@ (@ tptp.minus_minus_nat M) N))))
% 0.24/0.66 (assert (forall ((M tptp.nat) (N tptp.nat)) (let ((_let_1 (@ tptp.plus_plus_nat M))) (= (@ _let_1 (@ tptp.suc N)) (@ tptp.suc (@ _let_1 N))))))
% 0.24/0.66 (assert (forall ((X2 tptp.nat) (Y2 tptp.nat)) (= (= (@ tptp.suc X2) (@ tptp.suc Y2)) (= X2 Y2))))
% 0.24/0.66 (assert (forall ((Nat tptp.nat) (Nat2 tptp.nat)) (= (= (@ tptp.suc Nat) (@ tptp.suc Nat2)) (= Nat Nat2))))
% 0.24/0.66 (assert (forall ((X tptp.nat) (Y tptp.nat)) (=> (= (@ tptp.suc X) (@ tptp.suc Y)) (= X Y))))
% 0.24/0.66 (assert (forall ((N tptp.nat)) (not (= N (@ tptp.suc N)))))
% 0.24/0.66 (assert (forall ((I tptp.nat) (J tptp.nat) (K tptp.nat)) (let ((_let_1 (@ tptp.minus_minus_nat I))) (= (@ (@ tptp.minus_minus_nat (@ _let_1 J)) K) (@ (@ tptp.minus_minus_nat (@ _let_1 K)) J)))))
% 0.24/0.66 (assert (forall ((M tptp.nat) (N tptp.nat)) (= (@ (@ tptp.plus_plus_nat (@ tptp.suc M)) N) (@ tptp.suc (@ (@ tptp.plus_plus_nat M) N)))))
% 0.24/0.66 (assert (forall ((A3 tptp.nat) (K tptp.nat) (A tptp.nat)) (let ((_let_1 (@ tptp.plus_plus_nat K))) (=> (= A3 (@ _let_1 A)) (= (@ tptp.suc A3) (@ _let_1 (@ tptp.suc A)))))))
% 0.24/0.66 (assert (forall ((M tptp.nat) (N tptp.nat)) (= (@ (@ tptp.plus_plus_nat (@ tptp.suc M)) N) (@ (@ tptp.plus_plus_nat M) (@ tptp.suc N)))))
% 0.24/0.66 (assert (forall ((P (-> tptp.nat Bool)) (K tptp.nat) (I tptp.nat)) (=> (@ P K) (=> (forall ((N2 tptp.nat)) (=> (@ P (@ tptp.suc N2)) (@ P N2))) (@ P (@ (@ tptp.minus_minus_nat K) I))))))
% 0.24/0.66 (assert (forall ((M tptp.nat) (N tptp.nat)) (= (@ (@ tptp.minus_minus_nat (@ (@ tptp.plus_plus_nat M) N)) N) M)))
% 0.24/0.66 (assert (forall ((N tptp.nat) (M tptp.nat)) (= (@ (@ tptp.minus_minus_nat (@ (@ tptp.plus_plus_nat N) M)) N) M)))
% 0.24/0.66 (assert (forall ((M tptp.nat) (K tptp.nat) (N tptp.nat)) (= (@ (@ tptp.minus_minus_nat (@ (@ tptp.plus_plus_nat M) K)) (@ (@ tptp.plus_plus_nat N) K)) (@ (@ tptp.minus_minus_nat M) N))))
% 0.24/0.66 (assert (forall ((K tptp.nat) (M tptp.nat) (N tptp.nat)) (let ((_let_1 (@ tptp.plus_plus_nat K))) (= (@ (@ tptp.minus_minus_nat (@ _let_1 M)) (@ _let_1 N)) (@ (@ tptp.minus_minus_nat M) N)))))
% 0.24/0.66 (assert (forall ((A tptp.nat) (B tptp.nat)) (= (@ (@ tptp.minus_minus_nat (@ (@ tptp.plus_plus_nat A) B)) B) A)))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) B)) B) A)))
% 0.24/0.66 (assert (forall ((A tptp.nat) (C tptp.nat) (B tptp.nat)) (= (@ (@ tptp.minus_minus_nat (@ (@ tptp.plus_plus_nat A) C)) (@ (@ tptp.plus_plus_nat B) C)) (@ (@ tptp.minus_minus_nat A) B))))
% 0.24/0.66 (assert (forall ((A tptp.int) (C tptp.int) (B tptp.int)) (= (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) C)) (@ (@ tptp.plus_plus_int B) C)) (@ (@ tptp.minus_minus_int A) B))))
% 0.24/0.66 (assert (forall ((A tptp.nat) (B tptp.nat)) (= (@ (@ tptp.minus_minus_nat (@ (@ tptp.plus_plus_nat A) B)) A) B)))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) B)) A) B)))
% 0.24/0.66 (assert (forall ((C tptp.nat) (A tptp.nat) (B tptp.nat)) (let ((_let_1 (@ tptp.plus_plus_nat C))) (= (@ (@ tptp.minus_minus_nat (@ _let_1 A)) (@ _let_1 B)) (@ (@ tptp.minus_minus_nat A) B)))))
% 0.24/0.66 (assert (forall ((C tptp.int) (A tptp.int) (B tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int C))) (= (@ (@ tptp.minus_minus_int (@ _let_1 A)) (@ _let_1 B)) (@ (@ tptp.minus_minus_int A) B)))))
% 0.24/0.66 (assert (forall ((B tptp.int) (A tptp.int) (C tptp.int)) (= (= (@ (@ tptp.plus_plus_int B) A) (@ (@ tptp.plus_plus_int C) A)) (= B C))))
% 0.24/0.66 (assert (forall ((B tptp.nat) (A tptp.nat) (C tptp.nat)) (= (= (@ (@ tptp.plus_plus_nat B) A) (@ (@ tptp.plus_plus_nat C) A)) (= B C))))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (= (= (@ _let_1 B) (@ _let_1 C)) (= B C)))))
% 0.24/0.66 (assert (forall ((A tptp.nat) (B tptp.nat) (C tptp.nat)) (let ((_let_1 (@ tptp.plus_plus_nat A))) (= (= (@ _let_1 B) (@ _let_1 C)) (= B C)))))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) B)) B) A)))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.plus_plus_int (@ (@ tptp.minus_minus_int A) B)) B) A)))
% 0.24/0.66 (assert (forall ((B tptp.int) (A tptp.int) (C tptp.int)) (=> (= (@ (@ tptp.plus_plus_int B) A) (@ (@ tptp.plus_plus_int C) A)) (= B C))))
% 0.24/0.66 (assert (forall ((B tptp.nat) (A tptp.nat) (C tptp.nat)) (=> (= (@ (@ tptp.plus_plus_nat B) A) (@ (@ tptp.plus_plus_nat C) A)) (= B C))))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (=> (= (@ _let_1 B) (@ _let_1 C)) (= B C)))))
% 0.24/0.66 (assert (forall ((A tptp.nat) (B tptp.nat) (C tptp.nat)) (let ((_let_1 (@ tptp.plus_plus_nat A))) (=> (= (@ _let_1 B) (@ _let_1 C)) (= B C)))))
% 0.24/0.66 (assert (forall ((B tptp.int) (A tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int B))) (let ((_let_2 (@ tptp.plus_plus_int A))) (= (@ _let_1 (@ _let_2 C)) (@ _let_2 (@ _let_1 C)))))))
% 0.24/0.66 (assert (forall ((B tptp.nat) (A tptp.nat) (C tptp.nat)) (let ((_let_1 (@ tptp.plus_plus_nat B))) (let ((_let_2 (@ tptp.plus_plus_nat A))) (= (@ _let_1 (@ _let_2 C)) (@ _let_2 (@ _let_1 C)))))))
% 0.24/0.66 (assert (= tptp.plus_plus_int (lambda ((A5 tptp.int) (B4 tptp.int)) (@ (@ tptp.plus_plus_int B4) A5))))
% 0.24/0.66 (assert (= tptp.plus_plus_nat (lambda ((A5 tptp.nat) (B4 tptp.nat)) (@ (@ tptp.plus_plus_nat B4) A5))))
% 0.24/0.66 (assert (forall ((B tptp.int) (A tptp.int) (C tptp.int)) (= (= (@ (@ tptp.plus_plus_int B) A) (@ (@ tptp.plus_plus_int C) A)) (= B C))))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (= (= (@ _let_1 B) (@ _let_1 C)) (= B C)))))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (= (@ (@ tptp.plus_plus_int (@ _let_1 B)) C) (@ _let_1 (@ (@ tptp.plus_plus_int B) C))))))
% 0.24/0.66 (assert (forall ((A tptp.nat) (B tptp.nat) (C tptp.nat)) (let ((_let_1 (@ tptp.plus_plus_nat A))) (= (@ (@ tptp.plus_plus_nat (@ _let_1 B)) C) (@ _let_1 (@ (@ tptp.plus_plus_nat B) C))))))
% 0.24/0.66 (assert (forall ((B5 tptp.int) (K tptp.int) (B tptp.int) (A tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (let ((_let_2 (@ tptp.plus_plus_int K))) (=> (= B5 (@ _let_2 B)) (= (@ _let_1 B5) (@ _let_2 (@ _let_1 B))))))))
% 0.24/0.66 (assert (forall ((B5 tptp.nat) (K tptp.nat) (B tptp.nat) (A tptp.nat)) (let ((_let_1 (@ tptp.plus_plus_nat A))) (let ((_let_2 (@ tptp.plus_plus_nat K))) (=> (= B5 (@ _let_2 B)) (= (@ _let_1 B5) (@ _let_2 (@ _let_1 B))))))))
% 0.24/0.66 (assert (forall ((A3 tptp.int) (K tptp.int) (A tptp.int) (B tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int K))) (=> (= A3 (@ _let_1 A)) (= (@ (@ tptp.plus_plus_int A3) B) (@ _let_1 (@ (@ tptp.plus_plus_int A) B)))))))
% 0.24/0.66 (assert (forall ((A3 tptp.nat) (K tptp.nat) (A tptp.nat) (B tptp.nat)) (let ((_let_1 (@ tptp.plus_plus_nat K))) (=> (= A3 (@ _let_1 A)) (= (@ (@ tptp.plus_plus_nat A3) B) (@ _let_1 (@ (@ tptp.plus_plus_nat A) B)))))))
% 0.24/0.66 (assert (forall ((I tptp.int) (J tptp.int) (K tptp.int) (L tptp.int)) (=> (and (= I J) (= K L)) (= (@ (@ tptp.plus_plus_int I) K) (@ (@ tptp.plus_plus_int J) L)))))
% 0.24/0.66 (assert (forall ((I tptp.nat) (J tptp.nat) (K tptp.nat) (L tptp.nat)) (=> (and (= I J) (= K L)) (= (@ (@ tptp.plus_plus_nat I) K) (@ (@ tptp.plus_plus_nat J) L)))))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (= (@ (@ tptp.plus_plus_int (@ _let_1 B)) C) (@ _let_1 (@ (@ tptp.plus_plus_int B) C))))))
% 0.24/0.66 (assert (forall ((A tptp.nat) (B tptp.nat) (C tptp.nat)) (let ((_let_1 (@ tptp.plus_plus_nat A))) (= (@ (@ tptp.plus_plus_nat (@ _let_1 B)) C) (@ _let_1 (@ (@ tptp.plus_plus_nat B) C))))))
% 0.24/0.66 (assert (forall ((A tptp.nat) (C tptp.nat) (B tptp.nat)) (let ((_let_1 (@ tptp.minus_minus_nat A))) (= (@ (@ tptp.minus_minus_nat (@ _let_1 C)) B) (@ (@ tptp.minus_minus_nat (@ _let_1 B)) C)))))
% 0.24/0.66 (assert (forall ((A tptp.int) (C tptp.int) (B tptp.int)) (let ((_let_1 (@ tptp.minus_minus_int A))) (= (@ (@ tptp.minus_minus_int (@ _let_1 C)) B) (@ (@ tptp.minus_minus_int (@ _let_1 B)) C)))))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int) (D tptp.int)) (=> (= (@ (@ tptp.minus_minus_int A) B) (@ (@ tptp.minus_minus_int C) D)) (= (= A B) (= C D)))))
% 0.24/0.66 (assert (forall ((A3 tptp.int) (K tptp.int) (A tptp.int) (B tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int K))) (=> (= A3 (@ _let_1 A)) (= (@ (@ tptp.minus_minus_int A3) B) (@ _let_1 (@ (@ tptp.minus_minus_int A) B)))))))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (= (= (@ (@ tptp.minus_minus_int A) B) C) (= A (@ (@ tptp.plus_plus_int C) B)))))
% 0.24/0.66 (assert (forall ((A tptp.int) (C tptp.int) (B tptp.int)) (= (= A (@ (@ tptp.minus_minus_int C) B)) (= (@ (@ tptp.plus_plus_int A) B) C))))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (= (@ _let_1 (@ (@ tptp.minus_minus_int B) C)) (@ (@ tptp.minus_minus_int (@ _let_1 B)) C)))))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (= (@ (@ tptp.minus_minus_int A) (@ (@ tptp.minus_minus_int B) C)) (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) C)) B))))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (= (@ (@ tptp.plus_plus_int (@ (@ tptp.minus_minus_int A) B)) C) (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) C)) B))))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.minus_minus_int A))) (= (@ _let_1 (@ (@ tptp.plus_plus_int B) C)) (@ (@ tptp.minus_minus_int (@ _let_1 C)) B)))))
% 0.24/0.66 (assert (forall ((A tptp.nat) (B tptp.nat) (C tptp.nat)) (let ((_let_1 (@ tptp.minus_minus_nat A))) (= (@ (@ tptp.minus_minus_nat (@ _let_1 B)) C) (@ _let_1 (@ (@ tptp.plus_plus_nat B) C))))))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.minus_minus_int A))) (= (@ (@ tptp.minus_minus_int (@ _let_1 B)) C) (@ _let_1 (@ (@ tptp.plus_plus_int B) C))))))
% 0.24/0.66 (assert (forall ((C tptp.nat) (B tptp.nat) (A tptp.nat)) (=> (= (@ (@ tptp.plus_plus_nat C) B) A) (= C (@ (@ tptp.minus_minus_nat A) B)))))
% 0.24/0.66 (assert (forall ((C tptp.int) (B tptp.int) (A tptp.int)) (=> (= (@ (@ tptp.plus_plus_int C) B) A) (= C (@ (@ tptp.minus_minus_int A) B)))))
% 0.24/0.66 (assert (forall ((X tptp.a)) (=> (@ (@ tptp.member_a X) (@ tptp.units_a_ring_ext_a_b tptp.r)) (exists ((X3 tptp.a)) (and (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) X3) X) (@ tptp.one_a_ring_ext_a_b tptp.r)))))))
% 0.24/0.66 (assert (forall ((X tptp.a)) (=> (@ (@ tptp.member_a X) (@ tptp.units_a_ring_ext_a_b tptp.r)) (exists ((X3 tptp.a)) (and (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) X) X3) (@ tptp.one_a_ring_ext_a_b tptp.r)))))))
% 0.24/0.66 (assert (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b tptp.r)) (exists ((Xa tptp.a)) (and (@ (@ tptp.member_a Xa) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) Xa) X3) (@ tptp.one_a_ring_ext_a_b tptp.r)))))) (@ tptp.group_a_ring_ext_a_b tptp.r)))
% 0.24/0.66 (assert (forall ((X tptp.a)) (let ((_let_1 (@ tptp.member_a X))) (=> (@ _let_1 (@ tptp.units_a_ring_ext_a_b tptp.r)) (@ _let_1 (@ tptp.partia1066395285xt_a_b tptp.r))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.one_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_3 (@ tptp.units_a_ring_ext_a_b tptp.r))) (=> (= (@ (@ _let_2 X) Y) _let_1) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ (@ _let_2 Y) X) _let_1)))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.units_a_ring_ext_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) X) Y)) _let_1))))))
% 0.24/0.66 (assert (@ (@ tptp.member_a (@ tptp.one_a_ring_ext_a_b tptp.r)) (@ tptp.units_a_ring_ext_a_b tptp.r)))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) X))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.units_a_ring_ext_a_b tptp.r)) (=> (@ (@ tptp.member_a Y) _let_2) (=> (@ (@ tptp.member_a Z) _let_2) (= (= (@ _let_1 Y) (@ _let_1 Z)) (= Y Z)))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia96731725t_unit)) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1955795460t_unit G)) (forall ((Y4 tptp.a)) (let ((_let_1 (@ tptp.partia1955795460t_unit G))) (=> (@ (@ tptp.member_a Y4) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_Product_unit G) X3) Y4)) _let_1)))))) (=> (@ (@ tptp.member_a (@ tptp.one_a_Product_unit G)) (@ tptp.partia1955795460t_unit G)) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1955795460t_unit G)) (forall ((Y4 tptp.a)) (=> (@ (@ tptp.member_a Y4) (@ tptp.partia1955795460t_unit G)) (forall ((Z2 tptp.a)) (let ((_let_1 (@ tptp.mult_a_Product_unit G))) (let ((_let_2 (@ _let_1 X3))) (=> (@ (@ tptp.member_a Z2) (@ tptp.partia1955795460t_unit G)) (= (@ (@ _let_1 (@ _let_2 Y4)) Z2) (@ _let_2 (@ (@ _let_1 Y4) Z2))))))))))) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1955795460t_unit G)) (= (@ (@ (@ tptp.mult_a_Product_unit G) (@ tptp.one_a_Product_unit G)) X3) X3))) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1955795460t_unit G)) (exists ((Xa tptp.a)) (and (@ (@ tptp.member_a Xa) (@ tptp.partia1955795460t_unit G)) (= (@ (@ (@ tptp.mult_a_Product_unit G) Xa) X3) (@ tptp.one_a_Product_unit G)))))) (@ tptp.group_a_Product_unit G))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b)) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b G)) (forall ((Y4 tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b G))) (=> (@ (@ tptp.member_a Y4) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_ring_ext_a_b G) X3) Y4)) _let_1)))))) (=> (@ (@ tptp.member_a (@ tptp.one_a_ring_ext_a_b G)) (@ tptp.partia1066395285xt_a_b G)) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b G)) (forall ((Y4 tptp.a)) (=> (@ (@ tptp.member_a Y4) (@ tptp.partia1066395285xt_a_b G)) (forall ((Z2 tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b G))) (let ((_let_2 (@ _let_1 X3))) (=> (@ (@ tptp.member_a Z2) (@ tptp.partia1066395285xt_a_b G)) (= (@ (@ _let_1 (@ _let_2 Y4)) Z2) (@ _let_2 (@ (@ _let_1 Y4) Z2))))))))))) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b G)) (= (@ (@ (@ tptp.mult_a_ring_ext_a_b G) (@ tptp.one_a_ring_ext_a_b G)) X3) X3))) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b G)) (exists ((Xa tptp.a)) (and (@ (@ tptp.member_a Xa) (@ tptp.partia1066395285xt_a_b G)) (= (@ (@ (@ tptp.mult_a_ring_ext_a_b G) Xa) X3) (@ tptp.one_a_ring_ext_a_b G)))))) (@ tptp.group_a_ring_ext_a_b G))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia96731725t_unit) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.one_a_Product_unit G))) (let ((_let_2 (@ tptp.mult_a_Product_unit G))) (let ((_let_3 (@ tptp.partia1955795460t_unit G))) (=> (@ tptp.group_a_Product_unit G) (=> (= (@ (@ _let_2 X) Y) _let_1) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ (@ _let_2 Y) X) _let_1))))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.one_a_ring_ext_a_b G))) (let ((_let_2 (@ tptp.mult_a_ring_ext_a_b G))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.group_a_ring_ext_a_b G) (=> (= (@ (@ _let_2 X) Y) _let_1) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ (@ _let_2 Y) X) _let_1))))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia96731725t_unit) (X tptp.a)) (=> (@ tptp.group_a_Product_unit G) (=> (@ (@ tptp.member_a X) (@ tptp.partia1955795460t_unit G)) (exists ((X3 tptp.a)) (and (@ (@ tptp.member_a X3) (@ tptp.partia1955795460t_unit G)) (= (@ (@ (@ tptp.mult_a_Product_unit G) X3) X) (@ tptp.one_a_Product_unit G))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a)) (=> (@ tptp.group_a_ring_ext_a_b G) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b G)) (exists ((X3 tptp.a)) (and (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b G)) (= (@ (@ (@ tptp.mult_a_ring_ext_a_b G) X3) X) (@ tptp.one_a_ring_ext_a_b G))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia96731725t_unit) (X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.mult_a_Product_unit G))) (let ((_let_2 (@ tptp.partia1955795460t_unit G))) (=> (@ tptp.group_a_Product_unit G) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (=> (@ (@ tptp.member_a Z) _let_2) (= (= (@ (@ _let_1 Y) X) (@ (@ _let_1 Z) X)) (= Y Z))))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b G))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.group_a_ring_ext_a_b G) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (=> (@ (@ tptp.member_a Z) _let_2) (= (= (@ (@ _let_1 Y) X) (@ (@ _let_1 Z) X)) (= Y Z))))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia96731725t_unit)) (=> (@ tptp.group_a_Product_unit G) (= (@ tptp.units_a_Product_unit G) (@ tptp.partia1955795460t_unit G)))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b)) (=> (@ tptp.group_a_ring_ext_a_b G) (= (@ tptp.units_a_ring_ext_a_b G) (@ tptp.partia1066395285xt_a_b G)))))
% 0.24/0.66 (assert (forall ((G tptp.partia96731725t_unit) (X tptp.a) (A tptp.a)) (let ((_let_1 (@ tptp.partia1955795460t_unit G))) (=> (@ tptp.group_a_Product_unit G) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a A) _let_1) (= (= X (@ (@ (@ tptp.mult_a_Product_unit G) A) X)) (= A (@ tptp.one_a_Product_unit G)))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (A tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.group_a_ring_ext_a_b G) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a A) _let_1) (= (= X (@ (@ (@ tptp.mult_a_ring_ext_a_b G) A) X)) (= A (@ tptp.one_a_ring_ext_a_b G)))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia96731725t_unit) (X tptp.a) (A tptp.a)) (let ((_let_1 (@ tptp.partia1955795460t_unit G))) (=> (@ tptp.group_a_Product_unit G) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a A) _let_1) (= (= X (@ (@ (@ tptp.mult_a_Product_unit G) X) A)) (= A (@ tptp.one_a_Product_unit G)))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (A tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.group_a_ring_ext_a_b G) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a A) _let_1) (= (= X (@ (@ (@ tptp.mult_a_ring_ext_a_b G) X) A)) (= A (@ tptp.one_a_ring_ext_a_b G)))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia96731725t_unit) (X tptp.a) (A tptp.a)) (let ((_let_1 (@ tptp.partia1955795460t_unit G))) (=> (@ tptp.group_a_Product_unit G) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a A) _let_1) (= (= (@ (@ (@ tptp.mult_a_Product_unit G) A) X) X) (= A (@ tptp.one_a_Product_unit G)))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (A tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.group_a_ring_ext_a_b G) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a A) _let_1) (= (= (@ (@ (@ tptp.mult_a_ring_ext_a_b G) A) X) X) (= A (@ tptp.one_a_ring_ext_a_b G)))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia96731725t_unit) (X tptp.a) (A tptp.a)) (let ((_let_1 (@ tptp.partia1955795460t_unit G))) (=> (@ tptp.group_a_Product_unit G) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a A) _let_1) (= (= (@ (@ (@ tptp.mult_a_Product_unit G) X) A) X) (= A (@ tptp.one_a_Product_unit G)))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (A tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.group_a_ring_ext_a_b G) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a A) _let_1) (= (= (@ (@ (@ tptp.mult_a_ring_ext_a_b G) X) A) X) (= A (@ tptp.one_a_ring_ext_a_b G)))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia96731725t_unit) (X tptp.a)) (=> (@ tptp.group_a_Product_unit G) (=> (@ (@ tptp.member_a X) (@ tptp.partia1955795460t_unit G)) (exists ((X3 tptp.a)) (and (@ (@ tptp.member_a X3) (@ tptp.partia1955795460t_unit G)) (= (@ (@ (@ tptp.mult_a_Product_unit G) X) X3) (@ tptp.one_a_Product_unit G))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a)) (=> (@ tptp.group_a_ring_ext_a_b G) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b G)) (exists ((X3 tptp.a)) (and (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b G)) (= (@ (@ (@ tptp.mult_a_ring_ext_a_b G) X) X3) (@ tptp.one_a_ring_ext_a_b G))))))))
% 0.24/0.66 (assert (@ tptp.group_a_Product_unit (@ tptp.units_1411277569xt_a_b tptp.r)))
% 0.24/0.66 (assert (forall ((A tptp.a) (B tptp.a)) (let ((_let_1 (@ tptp.a_inv_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_2) (=> (@ (@ tptp.member_a B) _let_2) (= (@ _let_1 (@ (@ (@ tptp.add_a_b tptp.r) A) B)) (@ (@ (@ tptp.a_minus_a_b tptp.r) (@ _let_1 A)) B))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (= (@ (@ _let_1 X) (@ (@ _let_1 (@ (@ tptp.a_inv_a_b tptp.r) X)) Y)) Y)))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (= (@ (@ _let_1 (@ (@ tptp.a_inv_a_b tptp.r) X)) (@ (@ _let_1 X) Y)) Y)))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.a_inv_a_b tptp.r))) (let ((_let_2 (@ tptp.add_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ _let_1 (@ (@ _let_2 X) Y)) (@ (@ _let_2 (@ _let_1 X)) (@ _let_1 Y))))))))))
% 0.24/0.66 (assert (forall ((A tptp.a) (B tptp.a) (C tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_2) (=> (@ (@ tptp.member_a B) _let_2) (=> (@ (@ tptp.member_a C) _let_2) (= (= (@ (@ _let_1 B) (@ (@ tptp.a_inv_a_b tptp.r) C)) A) (= B (@ (@ _let_1 A) C))))))))))
% 0.24/0.66 (assert (forall ((A tptp.a) (B tptp.a) (C tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_2) (=> (@ (@ tptp.member_a B) _let_2) (=> (@ (@ tptp.member_a C) _let_2) (= (= A (@ (@ _let_1 B) (@ (@ tptp.a_inv_a_b tptp.r) C))) (= B (@ (@ _let_1 A) C))))))))))
% 0.24/0.66 (assert (forall ((A tptp.a) (B tptp.a) (C tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_2) (=> (@ (@ tptp.member_a B) _let_2) (=> (@ (@ tptp.member_a C) _let_2) (= (= (@ (@ _let_1 (@ (@ tptp.a_inv_a_b tptp.r) B)) C) A) (= C (@ (@ _let_1 B) A))))))))))
% 0.24/0.66 (assert (forall ((A tptp.a) (B tptp.a) (C tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_2) (=> (@ (@ tptp.member_a B) _let_2) (=> (@ (@ tptp.member_a C) _let_2) (= (= A (@ (@ _let_1 (@ (@ tptp.a_inv_a_b tptp.r) B)) C)) (= C (@ (@ _let_1 B) A))))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.a_inv_a_b tptp.r))) (let ((_let_2 (@ tptp.add_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ _let_1 (@ (@ _let_2 X) Y)) (@ (@ _let_2 (@ _let_1 Y)) (@ _let_1 X))))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (= (@ (@ _let_1 X) Y) Z) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (=> (@ (@ tptp.member_a Z) _let_2) (= (@ (@ _let_1 (@ (@ tptp.a_inv_a_b tptp.r) X)) Z) Y)))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) X))) (let ((_let_2 (@ tptp.a_inv_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ _let_1 (@ _let_2 Y)) (@ _let_2 (@ _let_1 Y))))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ tptp.a_inv_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ (@ _let_1 (@ _let_2 X)) Y) (@ _let_2 (@ (@ _let_1 X) Y))))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (I tptp.nat)) (let ((_let_1 (@ (@ tptp.add_pow_a_b_nat tptp.r) I))) (let ((_let_2 (@ tptp.a_inv_a_b tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ _let_1 (@ _let_2 X)) (@ _let_2 (@ _let_1 X))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (I tptp.int)) (let ((_let_1 (@ (@ tptp.add_pow_a_b_int tptp.r) I))) (let ((_let_2 (@ tptp.a_inv_a_b tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ _let_1 (@ _let_2 X)) (@ _let_2 (@ _let_1 X))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a)) (= (@ (@ (@ tptp.a_minus_a_b tptp.r) X) Y) (@ (@ (@ tptp.add_a_b tptp.r) X) (@ (@ tptp.a_inv_a_b tptp.r) Y)))))
% 0.24/0.66 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (=> (= (@ (@ (@ tptp.add_a_b tptp.r) X) Y) (@ tptp.zero_a_b tptp.r)) (= X (@ (@ tptp.a_inv_a_b tptp.r) Y))))))))
% 0.24/0.66 (assert (forall ((X tptp.a)) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.add_a_b tptp.r) X) (@ (@ tptp.a_inv_a_b tptp.r) X)) (@ tptp.zero_a_b tptp.r)))))
% 0.24/0.66 (assert (forall ((Y tptp.a) (X tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (= (@ (@ (@ tptp.add_a_b tptp.r) Y) X) (@ tptp.zero_a_b tptp.r)) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (= (@ (@ tptp.a_inv_a_b tptp.r) X) Y)))))))
% 0.24/0.66 (assert (forall ((X tptp.a)) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.add_a_b tptp.r) (@ (@ tptp.a_inv_a_b tptp.r) X)) X) (@ tptp.zero_a_b tptp.r)))))
% 0.24/0.66 (assert (forall ((X tptp.a) (N tptp.int) (M tptp.int)) (let ((_let_1 (@ tptp.add_pow_a_b_int tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ _let_1 (@ (@ tptp.minus_minus_int N) M)) X) (@ (@ (@ tptp.add_a_b tptp.r) (@ (@ _let_1 N) X)) (@ (@ tptp.a_inv_a_b tptp.r) (@ (@ _let_1 M) X))))))))
% 0.24/0.66 (assert (forall ((X tptp.a)) (let ((_let_1 (@ tptp.a_inv_a_b tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ _let_1 (@ _let_1 X)) X)))))
% 0.24/0.66 (assert (forall ((X tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (@ (@ tptp.member_a (@ (@ tptp.a_inv_a_b tptp.r) X)) _let_1)))))
% 0.24/0.66 (assert (let ((_let_1 (@ tptp.zero_a_b tptp.r))) (= (@ (@ tptp.a_inv_a_b tptp.r) _let_1) _let_1)))
% 0.24/0.66 (assert (forall ((X tptp.a)) (let ((_let_1 (@ tptp.zero_a_b tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (= (@ (@ tptp.a_inv_a_b tptp.r) X) _let_1) (= X _let_1))))))
% 0.24/0.66 (assert (@ (@ tptp.member_a (@ (@ tptp.a_inv_a_b tptp.r) (@ tptp.one_a_ring_ext_a_b tptp.r))) (@ tptp.units_a_ring_ext_a_b tptp.r)))
% 0.24/0.66 (assert (forall ((G tptp.partia96731725t_unit)) (= (@ tptp.mult_a_Product_unit (@ tptp.units_873712258t_unit G)) (@ tptp.mult_a_Product_unit G))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b)) (= (@ tptp.mult_a_Product_unit (@ tptp.units_1411277569xt_a_b G)) (@ tptp.mult_a_ring_ext_a_b G))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a X) _let_1) (@ (@ tptp.member_a (@ (@ tptp.a_inv_a_b G) X)) _let_1))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a)) (let ((_let_1 (@ tptp.a_inv_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b G)) (= (@ _let_1 (@ _let_1 X)) X))))))
% 0.24/0.66 (assert (= tptp.a_minus_a_b (lambda ((R4 tptp.partia1833973666xt_a_b) (X4 tptp.a) (Y5 tptp.a)) (@ (@ (@ tptp.add_a_b R4) X4) (@ (@ tptp.a_inv_a_b R4) Y5)))))
% 0.24/0.66 (assert (forall ((G tptp.partia96731725t_unit)) (= (@ tptp.one_a_Product_unit (@ tptp.units_873712258t_unit G)) (@ tptp.one_a_Product_unit G))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b)) (= (@ tptp.one_a_Product_unit (@ tptp.units_1411277569xt_a_b G)) (@ tptp.one_a_ring_ext_a_b G))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.a_inv_a_b G))) (let ((_let_2 (@ tptp.add_a_b G))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ _let_1 (@ (@ _let_2 X) Y)) (@ (@ _let_2 (@ _let_1 X)) (@ _let_1 Y)))))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.add_a_b G))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (= (@ (@ _let_1 X) (@ (@ _let_1 (@ (@ tptp.a_inv_a_b G) X)) Y)) Y))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.add_a_b G))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (= (@ (@ _let_1 (@ (@ tptp.a_inv_a_b G) X)) (@ (@ _let_1 X) Y)) Y))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (=> (@ tptp.abelian_group_a_b G) (= (@ (@ (@ tptp.a_minus_a_b G) X) Y) (@ (@ (@ tptp.add_a_b G) X) (@ (@ tptp.a_inv_a_b G) Y))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a)) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b G)) (= (@ (@ (@ tptp.add_a_b G) (@ (@ tptp.a_inv_a_b G) X)) X) (@ tptp.zero_a_b G))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a)) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b G)) (= (@ (@ (@ tptp.add_a_b G) X) (@ (@ tptp.a_inv_a_b G) X)) (@ tptp.zero_a_b G))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (Y tptp.a) (X tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (= (@ (@ (@ tptp.add_a_b G) Y) X) (@ tptp.zero_a_b G)) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (= (@ (@ tptp.a_inv_a_b G) X) Y))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (A tptp.a) (B tptp.a)) (let ((_let_1 (@ tptp.a_inv_a_b G))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a A) _let_2) (=> (@ (@ tptp.member_a B) _let_2) (= (@ _let_1 (@ (@ (@ tptp.add_a_b G) A) B)) (@ (@ (@ tptp.a_minus_a_b G) (@ _let_1 A)) B)))))))))
% 0.24/0.66 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.add_a_b G))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (= (@ (@ _let_1 X) Y) Z) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (=> (@ (@ tptp.member_a Z) _let_2) (= (@ (@ _let_1 (@ (@ tptp.a_inv_a_b G) X)) Z) Y))))))))))
% 0.24/0.66 (assert (forall ((X tptp.a) (I tptp.int)) (let ((_let_1 (@ tptp.add_pow_a_b_int tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ _let_1 (@ tptp.uminus_uminus_int I)) X) (@ (@ tptp.a_inv_a_b tptp.r) (@ (@ _let_1 I) X)))))))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.plus_plus_int A) (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int A)) B)) B)))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int A)) (@ (@ tptp.plus_plus_int A) B)) B)))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ tptp.uminus_uminus_int (@ (@ tptp.plus_plus_int A) B)) (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int A)) (@ tptp.uminus_uminus_int B)))))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ tptp.uminus_uminus_int (@ (@ tptp.minus_minus_int A) B)) (@ (@ tptp.minus_minus_int B) A))))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.minus_minus_int A) (@ tptp.uminus_uminus_int B)) (@ (@ tptp.plus_plus_int A) B))))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int A)) B) (@ (@ tptp.minus_minus_int B) A))))
% 0.24/0.66 (assert (forall ((B5 tptp.int) (K tptp.int) (B tptp.int) (A tptp.int)) (let ((_let_1 (@ tptp.minus_minus_int A))) (=> (= B5 (@ (@ tptp.plus_plus_int K) B)) (= (@ _let_1 B5) (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int K)) (@ _let_1 B)))))))
% 0.24/0.66 (assert (= tptp.minus_minus_int (lambda ((A5 tptp.int) (B4 tptp.int)) (@ (@ tptp.plus_plus_int A5) (@ tptp.uminus_uminus_int B4)))))
% 0.24/0.66 (assert (= tptp.minus_minus_int (lambda ((A5 tptp.int) (B4 tptp.int)) (@ (@ tptp.plus_plus_int A5) (@ tptp.uminus_uminus_int B4)))))
% 0.24/0.66 (assert (forall ((A3 tptp.int) (K tptp.int) (A tptp.int)) (=> (= A3 (@ (@ tptp.plus_plus_int K) A)) (= (@ tptp.uminus_uminus_int A3) (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int K)) (@ tptp.uminus_uminus_int A))))))
% 0.24/0.66 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ tptp.uminus_uminus_int (@ (@ tptp.plus_plus_int A) B)) (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int B)) (@ tptp.uminus_uminus_int A)))))
% 0.24/0.66 (assert (forall ((B tptp.int) (A tptp.int)) (= (@ (@ tptp.minus_minus_int (@ tptp.uminus_uminus_int B)) A) (@ (@ tptp.minus_minus_int (@ tptp.uminus_uminus_int A)) B))))
% 0.24/0.66 (assert (forall ((X tptp.a) (N tptp.nat)) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.add_pow_a_b_int tptp.r) (@ tptp.uminus_uminus_int (@ tptp.semiri2019852685at_int N))) X) (@ (@ tptp.a_inv_a_b tptp.r) (@ (@ (@ tptp.add_pow_a_b_nat tptp.r) N) X))))))
% 0.24/0.66 (assert (@ tptp.comm_g1684316527t_unit (@ tptp.units_1411277569xt_a_b tptp.r)))
% 0.24/0.66 (assert (forall ((M tptp.nat) (N tptp.nat)) (= (= (@ tptp.semiri1382578993at_nat M) (@ tptp.semiri1382578993at_nat N)) (= M N))))
% 0.48/0.86 (assert (forall ((M tptp.nat) (N tptp.nat)) (= (= (@ tptp.semiri2019852685at_int M) (@ tptp.semiri2019852685at_int N)) (= M N))))
% 0.48/0.86 (assert (forall ((M tptp.nat) (N tptp.nat)) (= (@ tptp.semiri1382578993at_nat (@ (@ tptp.plus_plus_nat M) N)) (@ (@ tptp.plus_plus_nat (@ tptp.semiri1382578993at_nat M)) (@ tptp.semiri1382578993at_nat N)))))
% 0.48/0.86 (assert (forall ((M tptp.nat) (N tptp.nat)) (= (@ tptp.semiri2019852685at_int (@ (@ tptp.plus_plus_nat M) N)) (@ (@ tptp.plus_plus_int (@ tptp.semiri2019852685at_int M)) (@ tptp.semiri2019852685at_int N)))))
% 0.48/0.86 (assert (forall ((G tptp.partia96731725t_unit) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.mult_a_Product_unit G))) (let ((_let_2 (@ tptp.partia1955795460t_unit G))) (=> (@ tptp.comm_g1684316527t_unit G) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (= (@ (@ _let_1 X) Y) (@ (@ _let_1 Y) X)))))))))
% 0.48/0.86 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b G))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.comm_g791708116xt_a_b G) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (= (@ (@ _let_1 X) Y) (@ (@ _let_1 Y) X)))))))))
% 0.48/0.86 (assert (forall ((G tptp.partia96731725t_unit) (X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.mult_a_Product_unit G))) (let ((_let_2 (@ _let_1 X))) (let ((_let_3 (@ tptp.partia1955795460t_unit G))) (=> (@ tptp.comm_g1684316527t_unit G) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ (@ _let_1 (@ _let_2 Y)) Z) (@ _let_2 (@ (@ _let_1 Y) Z))))))))))))
% 0.48/0.86 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b G))) (let ((_let_2 (@ _let_1 X))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.comm_g791708116xt_a_b G) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ (@ _let_1 (@ _let_2 Y)) Z) (@ _let_2 (@ (@ _let_1 Y) Z))))))))))))
% 0.48/0.86 (assert (forall ((G tptp.partia96731725t_unit) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1955795460t_unit G))) (=> (@ tptp.comm_g1684316527t_unit G) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_Product_unit G) X) Y)) _let_1)))))))
% 0.48/0.86 (assert (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.comm_g791708116xt_a_b G) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_ring_ext_a_b G) X) Y)) _let_1)))))))
% 0.48/0.86 (assert (forall ((G tptp.partia96731725t_unit)) (=> (@ tptp.comm_g1684316527t_unit G) (@ (@ tptp.member_a (@ tptp.one_a_Product_unit G)) (@ tptp.partia1955795460t_unit G)))))
% 0.48/0.86 (assert (forall ((G tptp.partia1833973666xt_a_b)) (=> (@ tptp.comm_g791708116xt_a_b G) (@ (@ tptp.member_a (@ tptp.one_a_ring_ext_a_b G)) (@ tptp.partia1066395285xt_a_b G)))))
% 0.48/0.86 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ (@ _let_1 tptp.s2) tptp.s3))) (let ((_let_3 (@ _let_1 (@ (@ _let_1 tptp.s) tptp.s3)))) (not (@ (@ tptp.member449909584od_a_a (@ (@ tptp.product_Pair_a_a (@ (@ (@ tptp.add_a_b tptp.r) (@ (@ _let_1 _let_2) (@ (@ _let_1 tptp.r2) tptp.r4))) (@ _let_3 (@ (@ _let_1 tptp.r3) tptp.r4)))) (@ _let_3 _let_2))) (@ tptp.partia206007992t_unit tptp.rel)))))))
% 0.48/0.86 (set-info :filename cvc5---1.0.5_6716)
% 0.48/0.86 (check-sat-assuming ( true ))
% 0.48/0.86 ------- get file name : TPTP file name is ITP106^1
% 0.48/0.86 ------- cvc5-thf : /export/starexec/sandbox2/solver/bin/cvc5---1.0.5_6716.smt2...
% 0.48/0.86 --- Run --ho-elim --full-saturate-quant at 10...
% 0.48/0.86 % SZS status Theorem for ITP106^1
% 0.48/0.86 % SZS output start Proof for ITP106^1
% 0.48/0.86 (
% 0.48/0.86 (let ((_let_1 (@ tptp.partia206007992t_unit tptp.rel))) (let ((_let_2 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_3 (@ _let_2 tptp.s2))) (let ((_let_4 (@ _let_3 tptp.s3))) (let ((_let_5 (@ _let_2 tptp.s))) (let ((_let_6 (@ _let_5 tptp.s3))) (let ((_let_7 (@ _let_2 _let_6))) (let ((_let_8 (@ _let_7 _let_4))) (let ((_let_9 (@ (@ _let_2 tptp.r3) tptp.r4))) (let ((_let_10 (@ _let_7 _let_9))) (let ((_let_11 (@ (@ _let_2 tptp.r2) tptp.r4))) (let ((_let_12 (@ (@ _let_2 _let_4) _let_11))) (let ((_let_13 (@ tptp.add_a_b tptp.r))) (let ((_let_14 (@ (@ _let_13 _let_12) _let_10))) (let ((_let_15 (not (@ (@ tptp.member449909584od_a_a (@ (@ tptp.product_Pair_a_a _let_14) _let_8)) _let_1)))) (let ((_let_16 (@ tptp.units_1411277569xt_a_b tptp.r))) (let ((_let_17 (@ tptp.units_a_ring_ext_a_b tptp.r))) (let ((_let_18 (@ tptp.one_a_ring_ext_a_b tptp.r))) (let ((_let_19 (@ tptp.a_inv_a_b tptp.r))) (let ((_let_20 (@ tptp.zero_a_b tptp.r))) (let ((_let_21 (@ tptp.member_a _let_18))) (let ((_let_22 (@ tptp.partia1066395285xt_a_b tptp.r))) (let ((_let_23 (@ _let_2 (@ _let_5 tptp.s2)))) (let ((_let_24 (@ _let_23 tptp.s3))) (let ((_let_25 (@ _let_2 _let_24))) (let ((_let_26 (@ _let_25 _let_14))) (let ((_let_27 (@ _let_5 tptp.r3))) (let ((_let_28 (@ _let_3 tptp.r2))) (let ((_let_29 (@ (@ _let_2 (@ (@ _let_13 _let_28) _let_27)) tptp.r4))) (let ((_let_30 (@ _let_2 _let_8))) (let ((_let_31 (@ _let_30 _let_29))) (let ((_let_32 (@ (@ (@ tptp.a_minus_a_b tptp.r) _let_31) _let_26))) (let ((_let_33 (@ _let_25 _let_10))) (let ((_let_34 (@ _let_25 _let_12))) (let ((_let_35 (@ (@ _let_2 _let_27) tptp.r4))) (let ((_let_36 (@ _let_30 _let_35))) (let ((_let_37 (@ (@ _let_2 _let_28) tptp.r4))) (let ((_let_38 (@ _let_30 _let_37))) (let ((_let_39 (@ (@ tptp.member449909584od_a_a (@ (@ tptp.product_Pair_a_a _let_11) _let_6)) _let_1))) (let ((_let_40 (@ (@ tptp.member449909584od_a_a (@ (@ tptp.product_Pair_a_a _let_9) _let_4)) _let_1))) (let ((_let_41 (@ _let_2 tptp.s3))) (let ((_let_42 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tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ _let_1 (@ tptp.suc N)) X) (@ (@ (@ tptp.add_a_b tptp.r) X) (@ (@ _let_1 N) X)))))) (forall ((X tptp.a) (M tptp.nat) (N tptp.nat)) (let ((_let_1 (@ tptp.add_pow_a_b_nat tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (=> (= (@ (@ _let_1 M) X) (@ (@ _let_1 N) X)) (= (@ (@ _let_1 (@ (@ tptp.minus_minus_nat M) N)) X) (@ tptp.zero_a_b tptp.r)))))) (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ tptp.abelian_group_a_b tptp.r) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (=> (= (@ (@ (@ tptp.a_minus_a_b tptp.r) X) Y) (@ tptp.zero_a_b tptp.r)) (= X Y))))))) (forall ((R tptp.partia1833973666xt_a_b) (R2 tptp.partia1833973666xt_a_b)) (=> (= (@ tptp.partia1066395285xt_a_b R) (@ tptp.partia1066395285xt_a_b R2)) (=> (= (@ tptp.mult_a_ring_ext_a_b R) (@ tptp.mult_a_ring_ext_a_b R2)) (=> (= (@ tptp.one_a_ring_ext_a_b R) (@ tptp.one_a_ring_ext_a_b R2)) (=> (= (@ tptp.zero_a_b R) (@ tptp.zero_a_b R2)) (=> (= (@ tptp.add_a_b R) (@ tptp.add_a_b R2)) (=> (= (@ tptp.more_a_b R) (@ tptp.more_a_b R2)) (= R R2)))))))) (@ tptp.abelian_group_a_b tptp.r) (forall ((N tptp.nat) (X tptp.a)) (let ((_let_1 (@ tptp.add_pow_a_b_nat tptp.r))) (= (@ (@ _let_1 (@ tptp.suc N)) X) (@ (@ (@ tptp.add_a_b tptp.r) (@ (@ _let_1 N) X)) X)))) (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b R3))) (=> (@ tptp.abelian_group_a_b R3) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.add_a_b R3) X) Y)) _let_1)))))) (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.add_a_b R3))) (let ((_let_2 (@ _let_1 X))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b R3))) (=> (@ tptp.abelian_group_a_b R3) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ 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(let ((_let_1 (@ tptp.partia1066395285xt_a_b R3))) (=> (@ (@ tptp.member_a Y4) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.add_a_b R3) X3) Y4)) _let_1)))))) (=> (@ (@ tptp.member_a (@ tptp.zero_a_b R3)) (@ tptp.partia1066395285xt_a_b R3)) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b R3)) (forall ((Y4 tptp.a)) (=> (@ (@ tptp.member_a Y4) (@ tptp.partia1066395285xt_a_b R3)) (forall ((Z2 tptp.a)) (let ((_let_1 (@ tptp.add_a_b R3))) (let ((_let_2 (@ _let_1 X3))) (=> (@ (@ tptp.member_a Z2) (@ tptp.partia1066395285xt_a_b R3)) (= (@ (@ _let_1 (@ _let_2 Y4)) Z2) (@ _let_2 (@ (@ _let_1 Y4) Z2))))))))))) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b R3)) (forall ((Y4 tptp.a)) (let ((_let_1 (@ tptp.add_a_b R3))) (=> (@ (@ tptp.member_a Y4) (@ tptp.partia1066395285xt_a_b R3)) (= (@ (@ _let_1 X3) Y4) (@ (@ _let_1 Y4) X3))))))) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b R3)) (= (@ (@ (@ tptp.add_a_b R3) (@ tptp.zero_a_b R3)) X3) X3))) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b R3)) (exists ((Xa tptp.a)) (and (@ (@ tptp.member_a Xa) (@ tptp.partia1066395285xt_a_b R3)) (= (@ (@ (@ tptp.add_a_b R3) Xa) X3) (@ tptp.zero_a_b R3)))))) (@ tptp.abelian_group_a_b R3)))))))) (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a)) (=> (@ tptp.abelian_group_a_b R3) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b R3)) (= (@ (@ (@ tptp.add_a_b R3) (@ tptp.zero_a_b R3)) X) X)))) (forall ((R3 tptp.partia1833973666xt_a_b) (X tptp.a)) (=> (@ tptp.abelian_group_a_b R3) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b R3)) (exists ((X3 tptp.a)) (and (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b R3)) (= (@ (@ (@ tptp.add_a_b R3) X3) X) (@ tptp.zero_a_b R3))))))) (forall ((G tptp.partia1833973666xt_a_b) (A tptp.a) (B tptp.a) (C tptp.a) (D tptp.a)) (let ((_let_1 (@ tptp.add_a_b G))) (let ((_let_2 (@ tptp.a_minus_a_b G))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a A) _let_3) (=> (@ (@ tptp.member_a B) _let_3) (=> (@ (@ tptp.member_a C) _let_3) (=> (@ (@ tptp.member_a D) _let_3) (= (@ (@ _let_2 (@ (@ _let_1 (@ (@ _let_2 A) C)) B)) D) (@ (@ _let_2 (@ (@ _let_2 (@ (@ _let_1 A) B)) C)) D))))))))))) (forall ((G tptp.partia1833973666xt_a_b) (A tptp.a) (B tptp.a) (C tptp.a)) (let ((_let_1 (@ tptp.a_minus_a_b G))) (let ((_let_2 (@ _let_1 C))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a A) _let_3) (=> (@ (@ tptp.member_a B) _let_3) (=> (@ (@ tptp.member_a C) _let_3) (= (@ (@ _let_1 (@ _let_2 A)) B) (@ _let_2 (@ (@ (@ tptp.add_a_b G) A) B))))))))))) (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b G))) (let ((_let_2 (@ tptp.abelian_group_a_b G))) (=> _let_2 (=> _let_2 (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (=> (= (@ (@ (@ tptp.a_minus_a_b G) X) Y) (@ tptp.zero_a_b G)) (= X Y))))))))) (forall ((I tptp.nat) (J tptp.nat) (K tptp.nat)) (let ((_let_1 (@ tptp.minus_minus_nat I))) (= (@ (@ tptp.minus_minus_nat (@ _let_1 J)) K) (@ _let_1 (@ (@ tptp.plus_plus_nat J) K))))) (forall ((M tptp.nat) (N tptp.nat) (K tptp.nat)) (= (@ (@ tptp.minus_minus_nat (@ (@ tptp.minus_minus_nat (@ tptp.suc M)) N)) (@ tptp.suc K)) (@ (@ tptp.minus_minus_nat (@ (@ tptp.minus_minus_nat M) N)) K))) (forall ((M tptp.nat) (N tptp.nat)) (= (@ (@ tptp.minus_minus_nat (@ tptp.suc M)) (@ tptp.suc N)) (@ (@ tptp.minus_minus_nat M) N))) (forall ((M tptp.nat) (N tptp.nat)) (let ((_let_1 (@ tptp.plus_plus_nat M))) (= (@ _let_1 (@ tptp.suc N)) (@ tptp.suc (@ _let_1 N))))) (forall ((X2 tptp.nat) (Y2 tptp.nat)) (= (= (@ tptp.suc X2) (@ tptp.suc Y2)) (= X2 Y2))) (forall ((Nat tptp.nat) (Nat2 tptp.nat)) (= (= (@ tptp.suc Nat) (@ tptp.suc Nat2)) (= Nat Nat2))) (forall ((X tptp.nat) (Y tptp.nat)) (=> (= (@ tptp.suc X) (@ tptp.suc Y)) (= X Y))) (forall ((N tptp.nat)) (not (= N (@ tptp.suc N)))) (forall ((I tptp.nat) (J tptp.nat) (K tptp.nat)) (let ((_let_1 (@ tptp.minus_minus_nat I))) (= (@ (@ tptp.minus_minus_nat (@ _let_1 J)) K) (@ (@ tptp.minus_minus_nat (@ _let_1 K)) J)))) (forall ((M tptp.nat) (N tptp.nat)) (= (@ (@ tptp.plus_plus_nat (@ tptp.suc M)) N) (@ tptp.suc (@ (@ tptp.plus_plus_nat M) N)))) (forall ((A3 tptp.nat) (K tptp.nat) (A tptp.nat)) (let ((_let_1 (@ tptp.plus_plus_nat K))) (=> (= A3 (@ _let_1 A)) (= (@ tptp.suc A3) (@ _let_1 (@ tptp.suc A)))))) (forall ((M tptp.nat) (N tptp.nat)) (= (@ (@ tptp.plus_plus_nat (@ tptp.suc M)) N) (@ (@ tptp.plus_plus_nat M) (@ tptp.suc N)))) (forall ((P (-> tptp.nat Bool)) (K tptp.nat) (I tptp.nat)) (=> (@ P K) (=> (forall ((N2 tptp.nat)) (=> (@ P (@ tptp.suc N2)) (@ P N2))) (@ P (@ (@ tptp.minus_minus_nat K) I))))) (forall ((M tptp.nat) (N tptp.nat)) (= (@ (@ tptp.minus_minus_nat (@ (@ tptp.plus_plus_nat M) N)) N) M)) (forall ((N tptp.nat) (M tptp.nat)) (= (@ (@ tptp.minus_minus_nat (@ (@ tptp.plus_plus_nat N) M)) N) M)) (forall ((M tptp.nat) (K tptp.nat) (N tptp.nat)) (= (@ (@ tptp.minus_minus_nat (@ (@ tptp.plus_plus_nat M) K)) (@ (@ tptp.plus_plus_nat N) K)) (@ (@ tptp.minus_minus_nat M) N))) (forall ((K tptp.nat) (M tptp.nat) (N tptp.nat)) (let ((_let_1 (@ tptp.plus_plus_nat K))) (= (@ (@ tptp.minus_minus_nat (@ _let_1 M)) (@ _let_1 N)) (@ (@ tptp.minus_minus_nat M) N)))) (forall ((A tptp.nat) (B tptp.nat)) (= (@ (@ tptp.minus_minus_nat (@ (@ tptp.plus_plus_nat A) B)) B) A)) (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) B)) B) A)) (forall ((A tptp.nat) (C tptp.nat) (B tptp.nat)) (= (@ (@ tptp.minus_minus_nat (@ (@ tptp.plus_plus_nat A) C)) (@ (@ tptp.plus_plus_nat B) C)) (@ (@ tptp.minus_minus_nat A) B))) (forall ((A tptp.int) (C tptp.int) (B tptp.int)) (= (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) C)) (@ (@ tptp.plus_plus_int B) C)) (@ (@ tptp.minus_minus_int A) B))) (forall ((A tptp.nat) (B tptp.nat)) (= (@ (@ tptp.minus_minus_nat (@ (@ tptp.plus_plus_nat A) B)) A) B)) (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) B)) A) B)) (forall ((C tptp.nat) (A tptp.nat) (B tptp.nat)) (let ((_let_1 (@ tptp.plus_plus_nat C))) (= (@ (@ tptp.minus_minus_nat (@ _let_1 A)) (@ _let_1 B)) (@ (@ tptp.minus_minus_nat A) B)))) (forall ((C tptp.int) (A tptp.int) (B tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int C))) (= (@ (@ tptp.minus_minus_int (@ _let_1 A)) (@ _let_1 B)) (@ (@ tptp.minus_minus_int A) B)))) (forall ((B tptp.int) (A tptp.int) (C tptp.int)) (= (= (@ (@ tptp.plus_plus_int B) A) (@ (@ tptp.plus_plus_int C) A)) (= B C))) (forall ((B tptp.nat) (A tptp.nat) (C tptp.nat)) (= (= (@ (@ tptp.plus_plus_nat B) A) (@ (@ tptp.plus_plus_nat C) A)) (= B C))) (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (= (= (@ _let_1 B) (@ _let_1 C)) (= B C)))) (forall ((A tptp.nat) (B tptp.nat) (C tptp.nat)) (let ((_let_1 (@ tptp.plus_plus_nat A))) (= (= (@ _let_1 B) (@ _let_1 C)) (= B C)))) (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) B)) B) A)) (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.plus_plus_int (@ (@ tptp.minus_minus_int A) B)) B) A)) (forall ((B tptp.int) (A tptp.int) (C tptp.int)) (=> (= (@ (@ tptp.plus_plus_int B) A) (@ (@ tptp.plus_plus_int C) A)) (= B C))) (forall ((B tptp.nat) (A tptp.nat) (C tptp.nat)) (=> (= (@ (@ tptp.plus_plus_nat B) A) (@ (@ tptp.plus_plus_nat C) A)) (= B C))) (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (=> (= (@ _let_1 B) (@ _let_1 C)) (= B C)))) (forall ((A tptp.nat) (B tptp.nat) (C tptp.nat)) (let ((_let_1 (@ tptp.plus_plus_nat A))) (=> (= (@ _let_1 B) (@ _let_1 C)) (= B C)))) (forall ((B tptp.int) (A tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int B))) (let ((_let_2 (@ tptp.plus_plus_int A))) (= (@ _let_1 (@ _let_2 C)) (@ _let_2 (@ _let_1 C)))))) (forall ((B tptp.nat) (A tptp.nat) (C tptp.nat)) (let ((_let_1 (@ tptp.plus_plus_nat B))) (let ((_let_2 (@ tptp.plus_plus_nat A))) (= (@ _let_1 (@ _let_2 C)) (@ _let_2 (@ _let_1 C)))))) (= tptp.plus_plus_int (lambda ((A5 tptp.int) (B4 tptp.int)) (@ (@ tptp.plus_plus_int B4) A5))) (= tptp.plus_plus_nat (lambda ((A5 tptp.nat) (B4 tptp.nat)) (@ (@ tptp.plus_plus_nat B4) A5))) (forall ((B tptp.int) (A tptp.int) (C tptp.int)) (= (= (@ (@ tptp.plus_plus_int B) A) (@ (@ tptp.plus_plus_int C) A)) (= B C))) (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (= (= (@ _let_1 B) (@ _let_1 C)) (= B C)))) (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (= (@ (@ tptp.plus_plus_int (@ _let_1 B)) C) (@ _let_1 (@ (@ tptp.plus_plus_int B) C))))) (forall ((A tptp.nat) (B tptp.nat) (C tptp.nat)) (let ((_let_1 (@ tptp.plus_plus_nat A))) (= (@ (@ tptp.plus_plus_nat (@ 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(@ (@ _let_1 A) C))))))))) (forall ((A tptp.a) (B tptp.a) (C tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_2) (=> (@ (@ tptp.member_a B) _let_2) (=> (@ (@ tptp.member_a C) _let_2) (= (= (@ (@ _let_1 (@ (@ tptp.a_inv_a_b tptp.r) B)) C) A) (= C (@ (@ _let_1 B) A))))))))) (forall ((A tptp.a) (B tptp.a) (C tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_2) (=> (@ (@ tptp.member_a B) _let_2) (=> (@ (@ tptp.member_a C) _let_2) (= (= A (@ (@ _let_1 (@ (@ tptp.a_inv_a_b tptp.r) B)) C)) (= C (@ (@ _let_1 B) A))))))))) (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.a_inv_a_b tptp.r))) (let ((_let_2 (@ tptp.add_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ _let_1 (@ (@ _let_2 X) Y)) (@ (@ _let_2 (@ _let_1 Y)) (@ _let_1 X))))))))) (forall ((X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (= (@ (@ _let_1 X) Y) Z) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (=> (@ (@ tptp.member_a Z) _let_2) (= (@ (@ _let_1 (@ (@ tptp.a_inv_a_b tptp.r) X)) Z) Y)))))))) (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) X))) (let ((_let_2 (@ tptp.a_inv_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ _let_1 (@ _let_2 Y)) (@ _let_2 (@ _let_1 Y))))))))) (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ tptp.a_inv_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ (@ _let_1 (@ _let_2 X)) Y) (@ _let_2 (@ (@ _let_1 X) Y))))))))) (forall ((X tptp.a) (I tptp.nat)) (let ((_let_1 (@ (@ tptp.add_pow_a_b_nat tptp.r) I))) (let ((_let_2 (@ tptp.a_inv_a_b tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ _let_1 (@ _let_2 X)) (@ _let_2 (@ _let_1 X))))))) (forall ((X tptp.a) (I tptp.int)) (let ((_let_1 (@ (@ tptp.add_pow_a_b_int tptp.r) I))) (let ((_let_2 (@ tptp.a_inv_a_b tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ _let_1 (@ _let_2 X)) (@ _let_2 (@ _let_1 X))))))) (forall ((X tptp.a) (Y tptp.a)) (= (@ (@ (@ tptp.a_minus_a_b tptp.r) X) Y) (@ (@ (@ tptp.add_a_b tptp.r) X) (@ (@ tptp.a_inv_a_b tptp.r) Y)))) (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (=> (= (@ (@ (@ tptp.add_a_b tptp.r) X) Y) (@ tptp.zero_a_b tptp.r)) (= X (@ (@ tptp.a_inv_a_b tptp.r) Y))))))) (forall ((X tptp.a)) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.add_a_b tptp.r) X) (@ (@ tptp.a_inv_a_b tptp.r) X)) (@ tptp.zero_a_b tptp.r)))) (forall ((Y tptp.a) (X tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (= (@ (@ (@ tptp.add_a_b tptp.r) Y) X) (@ tptp.zero_a_b tptp.r)) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (= (@ (@ tptp.a_inv_a_b tptp.r) X) Y)))))) (forall ((X tptp.a)) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.add_a_b tptp.r) (@ (@ tptp.a_inv_a_b tptp.r) X)) X) (@ tptp.zero_a_b tptp.r)))) (forall ((X tptp.a) (N tptp.int) (M tptp.int)) (let ((_let_1 (@ tptp.add_pow_a_b_int tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ _let_1 (@ (@ tptp.minus_minus_int N) M)) X) (@ (@ (@ tptp.add_a_b tptp.r) (@ (@ _let_1 N) X)) (@ (@ tptp.a_inv_a_b tptp.r) (@ (@ _let_1 M) X))))))) (forall ((X tptp.a)) (let ((_let_1 (@ tptp.a_inv_a_b tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ _let_1 (@ _let_1 X)) X)))) (forall ((X tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (@ (@ tptp.member_a (@ (@ tptp.a_inv_a_b tptp.r) X)) _let_1)))) (= (@ _let_19 _let_20) _let_20) (forall ((X tptp.a)) (let ((_let_1 (@ tptp.zero_a_b tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (= (@ (@ tptp.a_inv_a_b tptp.r) X) _let_1) (= X _let_1))))) (@ (@ tptp.member_a (@ _let_19 _let_18)) _let_17) (forall ((G tptp.partia96731725t_unit)) (= (@ tptp.mult_a_Product_unit (@ tptp.units_873712258t_unit G)) (@ tptp.mult_a_Product_unit G))) (forall ((G tptp.partia1833973666xt_a_b)) (= (@ tptp.mult_a_Product_unit (@ tptp.units_1411277569xt_a_b G)) (@ tptp.mult_a_ring_ext_a_b G))) (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a X) _let_1) (@ (@ tptp.member_a (@ (@ tptp.a_inv_a_b G) X)) _let_1))))) (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a)) (let ((_let_1 (@ tptp.a_inv_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b G)) (= (@ _let_1 (@ _let_1 X)) X))))) (= tptp.a_minus_a_b (lambda ((R4 tptp.partia1833973666xt_a_b) (X4 tptp.a) (Y5 tptp.a)) (@ (@ (@ tptp.add_a_b R4) X4) (@ (@ tptp.a_inv_a_b R4) Y5)))) (forall ((G tptp.partia96731725t_unit)) (= (@ tptp.one_a_Product_unit (@ tptp.units_873712258t_unit G)) (@ tptp.one_a_Product_unit G))) (forall ((G tptp.partia1833973666xt_a_b)) (= (@ tptp.one_a_Product_unit (@ tptp.units_1411277569xt_a_b G)) (@ tptp.one_a_ring_ext_a_b G))) (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.a_inv_a_b G))) (let ((_let_2 (@ tptp.add_a_b G))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ _let_1 (@ (@ _let_2 X) Y)) (@ (@ _let_2 (@ _let_1 X)) (@ _let_1 Y)))))))))) (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.add_a_b G))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (= (@ (@ _let_1 X) (@ (@ _let_1 (@ (@ tptp.a_inv_a_b G) X)) Y)) Y))))))) (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.add_a_b G))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (= (@ (@ _let_1 (@ (@ tptp.a_inv_a_b G) X)) (@ (@ _let_1 X) Y)) Y))))))) (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (=> (@ tptp.abelian_group_a_b G) (= (@ (@ (@ tptp.a_minus_a_b G) X) Y) (@ (@ (@ tptp.add_a_b G) X) (@ (@ tptp.a_inv_a_b G) Y))))) (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a)) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b G)) (= (@ (@ (@ tptp.add_a_b G) (@ (@ tptp.a_inv_a_b G) X)) X) (@ tptp.zero_a_b G))))) (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a)) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b G)) (= (@ (@ (@ tptp.add_a_b G) X) (@ (@ tptp.a_inv_a_b G) X)) (@ tptp.zero_a_b G))))) (forall ((G tptp.partia1833973666xt_a_b) (Y tptp.a) (X tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (= (@ (@ (@ tptp.add_a_b G) Y) X) (@ tptp.zero_a_b G)) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (= (@ (@ tptp.a_inv_a_b G) X) Y))))))) (forall ((G tptp.partia1833973666xt_a_b) (A tptp.a) (B tptp.a)) (let ((_let_1 (@ tptp.a_inv_a_b G))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (@ (@ tptp.member_a A) _let_2) (=> (@ (@ tptp.member_a B) _let_2) (= (@ _let_1 (@ (@ (@ tptp.add_a_b G) A) B)) (@ (@ (@ tptp.a_minus_a_b G) (@ _let_1 A)) B)))))))) (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.add_a_b G))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.abelian_group_a_b G) (=> (= (@ (@ _let_1 X) Y) Z) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (=> (@ (@ tptp.member_a Z) _let_2) (= (@ (@ _let_1 (@ (@ tptp.a_inv_a_b G) X)) Z) Y))))))))) (forall ((X tptp.a) (I tptp.int)) (let ((_let_1 (@ tptp.add_pow_a_b_int tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ _let_1 (@ tptp.uminus_uminus_int I)) X) (@ (@ tptp.a_inv_a_b tptp.r) (@ (@ _let_1 I) X)))))) (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.plus_plus_int A) (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int A)) B)) B)) (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int A)) (@ (@ tptp.plus_plus_int A) B)) B)) (forall ((A tptp.int) (B tptp.int)) (= (@ tptp.uminus_uminus_int (@ (@ tptp.plus_plus_int A) B)) (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int A)) (@ tptp.uminus_uminus_int B)))) (forall ((A tptp.int) (B tptp.int)) (= (@ tptp.uminus_uminus_int (@ (@ tptp.minus_minus_int A) B)) (@ (@ tptp.minus_minus_int B) A))) (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.minus_minus_int A) (@ tptp.uminus_uminus_int B)) (@ (@ tptp.plus_plus_int A) B))) (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int A)) B) (@ (@ tptp.minus_minus_int B) A))) (forall ((B5 tptp.int) (K tptp.int) (B tptp.int) (A tptp.int)) (let ((_let_1 (@ tptp.minus_minus_int A))) (=> (= B5 (@ (@ tptp.plus_plus_int K) B)) (= (@ _let_1 B5) (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int K)) (@ _let_1 B)))))) (= tptp.minus_minus_int (lambda ((A5 tptp.int) (B4 tptp.int)) (@ (@ tptp.plus_plus_int A5) (@ tptp.uminus_uminus_int B4)))) (= tptp.minus_minus_int (lambda ((A5 tptp.int) (B4 tptp.int)) (@ (@ tptp.plus_plus_int A5) (@ tptp.uminus_uminus_int B4)))) (forall ((A3 tptp.int) (K tptp.int) (A tptp.int)) (=> (= A3 (@ (@ tptp.plus_plus_int K) A)) (= (@ tptp.uminus_uminus_int A3) (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int K)) (@ tptp.uminus_uminus_int A))))) (forall ((A tptp.int) (B tptp.int)) (= (@ tptp.uminus_uminus_int (@ (@ tptp.plus_plus_int A) B)) (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int B)) (@ tptp.uminus_uminus_int A)))) (forall ((B tptp.int) (A tptp.int)) (= (@ (@ tptp.minus_minus_int (@ tptp.uminus_uminus_int B)) A) (@ (@ tptp.minus_minus_int (@ tptp.uminus_uminus_int A)) B))) (forall ((X tptp.a) (N tptp.nat)) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.add_pow_a_b_int tptp.r) (@ tptp.uminus_uminus_int (@ tptp.semiri2019852685at_int N))) X) (@ (@ tptp.a_inv_a_b tptp.r) (@ (@ (@ tptp.add_pow_a_b_nat tptp.r) N) X))))) (@ tptp.comm_g1684316527t_unit _let_16) (forall ((M tptp.nat) (N tptp.nat)) (= (= (@ tptp.semiri1382578993at_nat M) (@ tptp.semiri1382578993at_nat N)) (= M N))) (forall ((M tptp.nat) (N tptp.nat)) (= (= (@ tptp.semiri2019852685at_int M) (@ tptp.semiri2019852685at_int N)) (= M N))) (forall ((M tptp.nat) (N tptp.nat)) (= (@ tptp.semiri1382578993at_nat (@ (@ tptp.plus_plus_nat M) N)) (@ (@ tptp.plus_plus_nat (@ tptp.semiri1382578993at_nat M)) (@ tptp.semiri1382578993at_nat N)))) (forall ((M tptp.nat) (N tptp.nat)) (= (@ tptp.semiri2019852685at_int (@ (@ tptp.plus_plus_nat M) N)) (@ (@ tptp.plus_plus_int (@ tptp.semiri2019852685at_int M)) (@ tptp.semiri2019852685at_int N)))) (forall ((G tptp.partia96731725t_unit) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.mult_a_Product_unit G))) (let ((_let_2 (@ tptp.partia1955795460t_unit G))) (=> (@ tptp.comm_g1684316527t_unit G) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (= (@ (@ _let_1 X) Y) (@ (@ _let_1 Y) X)))))))) (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b G))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.comm_g791708116xt_a_b G) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (= (@ (@ _let_1 X) Y) (@ (@ _let_1 Y) X)))))))) (forall ((G tptp.partia96731725t_unit) (X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.mult_a_Product_unit G))) (let ((_let_2 (@ _let_1 X))) (let ((_let_3 (@ tptp.partia1955795460t_unit G))) (=> (@ tptp.comm_g1684316527t_unit G) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ (@ _let_1 (@ _let_2 Y)) Z) (@ _let_2 (@ (@ _let_1 Y) Z))))))))))) (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b G))) (let ((_let_2 (@ _let_1 X))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.comm_g791708116xt_a_b G) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ (@ _let_1 (@ _let_2 Y)) Z) (@ _let_2 (@ (@ _let_1 Y) Z))))))))))) (forall ((G tptp.partia96731725t_unit) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1955795460t_unit G))) (=> (@ tptp.comm_g1684316527t_unit G) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_Product_unit G) X) Y)) _let_1)))))) (forall ((G tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b G))) (=> (@ tptp.comm_g791708116xt_a_b G) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_ring_ext_a_b G) X) Y)) _let_1)))))) (forall ((G tptp.partia96731725t_unit)) (=> (@ tptp.comm_g1684316527t_unit G) (@ (@ tptp.member_a (@ tptp.one_a_Product_unit G)) (@ tptp.partia1955795460t_unit G)))) (forall ((G tptp.partia1833973666xt_a_b)) (=> (@ tptp.comm_g791708116xt_a_b G) (@ (@ tptp.member_a (@ tptp.one_a_ring_ext_a_b G)) (@ tptp.partia1066395285xt_a_b G)))) _let_15 true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
% 0.48/0.86 )
% 0.48/0.86 % SZS output end Proof for ITP106^1
% 0.48/0.86 % cvc5---1.0.5 exiting
% 0.48/0.86 % cvc5---1.0.5 exiting
%------------------------------------------------------------------------------