%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP109^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n006.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:37 AM UTC 2026
% Result : Theorem 9.88s 2.70s
% Output : Refutation 9.88s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP109^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.33 % Computer : n006.cluster.edu
% 0.17/0.33 % Model : x86_64 x86_64
% 0.17/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.33 % Memory : 8042.1875MB
% 0.17/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.33 % CPULimit : 300
% 0.17/0.33 % WCLimit : 300
% 0.17/0.33 % DateTime : Tue Feb 24 08:26:00 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.20/0.46 ---- Original DTF file ---
% 0.20/0.46 thf(spec,logic,$$dhol).
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 % File : ITP109^2 : TPTP v9.2.1. Released v7.5.0.
% 0.20/0.46 % Domain : Interactive Theorem Proving
% 0.20/0.46 % Problem : Sledgehammer Lower_Semicontinuous problem prob_1098__6259092_1
% 0.20/0.46 % Version : Especial.
% 0.20/0.46 % English :
% 0.20/0.46
% 0.20/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.20/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.20/0.46 % Source : [Des21]
% 0.20/0.46 % Names : Lower_Semicontinuous/prob_1098__6259092_1 [Des21]
% 0.20/0.46
% 0.20/0.46 % Status : Theorem
% 0.20/0.46 % Rating : 0.33 v8.1.0, 0.25 v7.5.0
% 0.20/0.46 % Syntax : Number of formulae : 417 ( 131 unt; 65 typ; 0 def)
% 0.20/0.46 % Number of atoms : 812 ( 325 equ; 0 cnn)
% 0.20/0.46 % Maximal formula atoms : 12 ( 2 avg)
% 0.20/0.46 % Number of connectives : 4226 ( 74 ~; 10 |; 45 &;3738 @)
% 0.20/0.46 % ( 0 <=>; 359 =>; 0 <=; 0 <~>)
% 0.20/0.46 % Maximal formula depth : 22 ( 6 avg)
% 0.20/0.46 % Number of types : 6 ( 5 usr)
% 0.20/0.46 % Number of type conns : 129 ( 129 >; 0 *; 0 +; 0 <<)
% 0.20/0.46 % Number of symbols : 63 ( 60 usr; 6 con; 0-4 aty)
% 0.20/0.46 % Number of variables : 1058 ( 87 ^; 903 !; 16 ?;1058 :)
% 0.20/0.46 % ( 52 !>; 0 ?*; 0 @-; 0 @+)
% 0.20/0.46 % SPC : TH1_THM_EQU_NAR
% 0.20/0.46
% 0.20/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:28:51.464
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 % Could-be-implicit typings (6)
% 0.20/0.46 thf(ty_t_Extended__Real_Oereal,type,
% 0.20/0.46 extended_ereal: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Real_Oreal,type,
% 0.20/0.46 real: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Set_Oset,type,
% 0.20/0.46 set: $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Nat_Onat,type,
% 0.20/0.46 nat: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_tf_b,type,
% 0.20/0.46 b: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_tf_a,type,
% 0.20/0.46 a: $tType ).
% 0.20/0.46
% 0.20/0.46 % Explicit typings (59)
% 0.20/0.46 thf(sy_cl_HOL_Otype,type,
% 0.20/0.46 type:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Oone,type,
% 0.20/0.46 one:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Oring,type,
% 0.20/0.46 ring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Oplus,type,
% 0.20/0.46 plus:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Fields_Ofield,type,
% 0.20/0.46 field:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ominus,type,
% 0.20/0.46 minus:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Otimes,type,
% 0.20/0.46 times:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Oring__1,type,
% 0.20/0.46 ring_1:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Osemiring,type,
% 0.20/0.46 semiring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Ocomm__ring,type,
% 0.20/0.46 comm_ring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ogroup__add,type,
% 0.20/0.46 group_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Osemiring__0,type,
% 0.20/0.46 semiring_0:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.20/0.46 finite_finite:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Omonoid__mult,type,
% 0.20/0.46 monoid_mult:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Oab__group__add,type,
% 0.20/0.46 ab_group_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Ocomm__semiring,type,
% 0.20/0.46 comm_semiring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Fields_Odivision__ring,type,
% 0.20/0.46 division_ring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.20/0.46 semigroup_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Parity_Osemiring__bits,type,
% 0.20/0.46 semiring_bits:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Osemidom__divide,type,
% 0.20/0.46 semidom_divide:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Osemigroup__mult,type,
% 0.20/0.46 semigroup_mult:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.20/0.46 comm_monoid_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.20/0.46 ab_semigroup_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ocomm__monoid__mult,type,
% 0.20/0.46 comm_monoid_mult:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Oab__semigroup__mult,type,
% 0.20/0.46 ab_semigroup_mult:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Real__Vector__Spaces_OscaleR,type,
% 0.20/0.46 real_Vector_scaleR:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.20/0.46 cancel_semigroup_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
% 0.20/0.46 comm_s1003936772cancel:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.20/0.46 cancel1352612707id_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.20/0.46 cancel146912293up_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type,
% 0.20/0.46 real_V1076094709vector:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.20/0.46 ordere779506340up_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Real__Vector__Spaces_Oreal__algebra,type,
% 0.20/0.46 real_V148923926lgebra:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Real__Vector__Spaces_Oreal__div__algebra,type,
% 0.20/0.46 real_V1015715145lgebra:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Convex_Oconvex,type,
% 0.20/0.46 convex:
% 0.20/0.46 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fields_Oinverse__class_Oinverse,type,
% 0.20/0.46 inverse_inverse:
% 0.20/0.46 !>[A: $tType] : ( A > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Finite__Set_Ofinite,type,
% 0.20/0.46 finite_finite2:
% 0.20/0.46 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.20/0.46 minus_minus:
% 0.20/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.20/0.46 one_one:
% 0.20/0.46 !>[A: $tType] : A ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.20/0.46 plus_plus:
% 0.20/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Groups_Otimes__class_Otimes,type,
% 0.20/0.46 times_times:
% 0.20/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
% 0.20/0.46 groups1340683514dd_sum:
% 0.20/0.46 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_If,type,
% 0.20/0.46 if:
% 0.20/0.46 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Oconvex__on,type,
% 0.20/0.46 lower_673667120vex_on:
% 0.20/0.46 !>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.20/0.46 bot_bot:
% 0.20/0.46 !>[A: $tType] : A ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.20/0.46 ord_less_eq:
% 0.20/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
% 0.20/0.46 real_V1908273582scaleR:
% 0.20/0.46 !>[A: $tType] : ( real > A > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 0.20/0.46 divide_divide:
% 0.20/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_OCollect,type,
% 0.20/0.46 collect:
% 0.20/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_Oinsert,type,
% 0.20/0.46 insert:
% 0.20/0.46 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_member,type,
% 0.20/0.46 member:
% 0.20/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_C,type,
% 0.20/0.46 c: set @ b ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_a,type,
% 0.20/0.46 a2: a > real ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_aa____,type,
% 0.20/0.46 aa: a > real ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_f,type,
% 0.20/0.46 f: b > extended_ereal ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_i____,type,
% 0.20/0.46 i: a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_s,type,
% 0.20/0.46 s: set @ a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_sa____,type,
% 0.20/0.46 sa: set @ a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_y,type,
% 0.20/0.46 y: a > b ).
% 0.20/0.46
% 0.20/0.46 % Relevant facts (255)
% 0.20/0.46 thf(fact_0_insert_Ohyps_I3_J,axiom,
% 0.20/0.46 ~ ( member @ a @ i @ sa ) ).
% 0.20/0.46
% 0.20/0.46 % insert.hyps(3)
% 0.20/0.46 thf(fact_1_asm,axiom,
% 0.20/0.46 ( ( aa @ i )
% 0.20/0.46 != ( one_one @ real ) ) ).
% 0.20/0.46
% 0.20/0.46 % asm
% 0.20/0.46 thf(fact_2__092_060open_062_I_092_060Sum_062x_092_060in_062s_O_Aa_Ax_A_K_092_060_094sub_062R_Ay_Ax_A_P_092_060_094sub_062R_A_I1_A_N_Aa_Ai_J_J_A_061_A_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_P_092_060_094sub_062R_A_I1_A_N_Aa_Ai_J_092_060close_062,axiom,
% 0.20/0.46 ( ( groups1340683514dd_sum @ a @ b
% 0.20/0.46 @ ^ [X: a] : ( real_V1908273582scaleR @ b @ ( inverse_inverse @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) @ ( real_V1908273582scaleR @ b @ ( aa @ X ) @ ( y @ X ) ) )
% 0.20/0.46 @ sa )
% 0.20/0.46 = ( real_V1908273582scaleR @ b @ ( inverse_inverse @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) )
% 0.20/0.46 @ ( groups1340683514dd_sum @ a @ b
% 0.20/0.46 @ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
% 0.20/0.46 @ sa ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % \<open>(\<Sum>x\<in>s. a x *\<^sub>R y x /\<^sub>R (1 - a i)) = (\<Sum>j\<in>s. a j *\<^sub>R y j) /\<^sub>R (1 - a i)\<close>
% 0.20/0.46 thf(fact_3_insert_Ohyps_I2_J,axiom,
% 0.20/0.46 ( sa
% 0.20/0.46 != ( bot_bot @ ( set @ a ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % insert.hyps(2)
% 0.20/0.46 thf(fact_4_insert_Ohyps_I1_J,axiom,
% 0.20/0.46 finite_finite2 @ a @ sa ).
% 0.20/0.46
% 0.20/0.46 % insert.hyps(1)
% 0.20/0.46 thf(fact_5__092_060open_062f_A_I_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_L_Aa_Ai_A_K_092_060_094sub_062R_Ay_Ai_J_A_061_Af_A_I_I_I1_A_N_Aa_Ai_J_A_K_Ainverse_A_I1_A_N_Aa_Ai_J_J_A_K_092_060_094sub_062R_A_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_L_Aa_Ai_A_K_092_060_094sub_062R_Ay_Ai_J_092_060close_062,axiom,
% 0.20/0.46 ( ( f
% 0.20/0.46 @ ( plus_plus @ b
% 0.20/0.46 @ ( groups1340683514dd_sum @ a @ b
% 0.20/0.46 @ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
% 0.20/0.46 @ sa )
% 0.20/0.46 @ ( real_V1908273582scaleR @ b @ ( aa @ i ) @ ( y @ i ) ) ) )
% 0.20/0.46 = ( f
% 0.20/0.46 @ ( plus_plus @ b
% 0.20/0.46 @ ( real_V1908273582scaleR @ b @ ( times_times @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) @ ( inverse_inverse @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) )
% 0.20/0.46 @ ( groups1340683514dd_sum @ a @ b
% 0.20/0.46 @ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
% 0.20/0.46 @ sa ) )
% 0.20/0.46 @ ( real_V1908273582scaleR @ b @ ( aa @ i ) @ ( y @ i ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % \<open>f ((\<Sum>j\<in>s. a j *\<^sub>R y j) + a i *\<^sub>R y i) = f (((1 - a i) * inverse (1 - a i)) *\<^sub>R (\<Sum>j\<in>s. a j *\<^sub>R y j) + a i *\<^sub>R y i)\<close>
% 0.20/0.46 thf(fact_6_calculation,axiom,
% 0.20/0.46 ( ( f
% 0.20/0.46 @ ( groups1340683514dd_sum @ a @ b
% 0.20/0.46 @ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
% 0.20/0.46 @ ( insert @ a @ i @ sa ) ) )
% 0.20/0.46 = ( f
% 0.20/0.46 @ ( plus_plus @ b
% 0.20/0.46 @ ( real_V1908273582scaleR @ b @ ( times_times @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) @ ( inverse_inverse @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) )
% 0.20/0.46 @ ( groups1340683514dd_sum @ a @ b
% 0.20/0.46 @ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
% 0.20/0.46 @ sa ) )
% 0.20/0.46 @ ( real_V1908273582scaleR @ b @ ( aa @ i ) @ ( y @ i ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % calculation
% 0.20/0.46 thf(fact_7__092_060open_062f_A_I_092_060Sum_062j_092_060in_062insert_Ai_As_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_061_Af_A_I_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_L_Aa_Ai_A_K_092_060_094sub_062R_Ay_Ai_J_092_060close_062,axiom,
% 0.20/0.46 ( ( f
% 0.20/0.46 @ ( groups1340683514dd_sum @ a @ b
% 0.20/0.46 @ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
% 0.20/0.46 @ ( insert @ a @ i @ sa ) ) )
% 0.20/0.46 = ( f
% 0.20/0.46 @ ( plus_plus @ b
% 0.20/0.46 @ ( groups1340683514dd_sum @ a @ b
% 0.20/0.46 @ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
% 0.20/0.46 @ sa )
% 0.20/0.46 @ ( real_V1908273582scaleR @ b @ ( aa @ i ) @ ( y @ i ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % \<open>f (\<Sum>j\<in>insert i s. a j *\<^sub>R y j) = f ((\<Sum>j\<in>s. a j *\<^sub>R y j) + a i *\<^sub>R y i)\<close>
% 0.20/0.46 thf(fact_8_scaleR__collapse,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( real_V1076094709vector @ A )
% 0.20/0.46 => ! [U: real,A2: A] :
% 0.20/0.46 ( ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ ( minus_minus @ real @ ( one_one @ real ) @ U ) @ A2 ) @ ( real_V1908273582scaleR @ A @ U @ A2 ) )
% 0.20/0.46 = A2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % scaleR_collapse
% 0.20/0.46 thf(fact_9_scaleR__eq__iff,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( real_V1076094709vector @ A )
% 0.20/0.46 => ! [B2: A,U: real,A2: A] :
% 0.20/0.46 ( ( ( plus_plus @ A @ B2 @ ( real_V1908273582scaleR @ A @ U @ A2 ) )
% 0.20/0.46 = ( plus_plus @ A @ A2 @ ( real_V1908273582scaleR @ A @ U @ B2 ) ) )
% 0.20/0.46 = ( ( A2 = B2 )
% 0.20/0.46 | ( U
% 0.20/0.46 = ( one_one @ real ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % scaleR_eq_iff
% 0.20/0.46 thf(fact_10_yai_I1_J,axiom,
% 0.20/0.46 member @ b @ ( y @ i ) @ c ).
% 0.20/0.46
% 0.20/0.46 % yai(1)
% 0.20/0.46 thf(fact_11_scaleR__scaleR,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( real_V1076094709vector @ A )
% 0.20/0.46 => ! [A2: real,B2: real,X2: A] :
% 0.20/0.46 ( ( real_V1908273582scaleR @ A @ A2 @ ( real_V1908273582scaleR @ A @ B2 @ X2 ) )
% 0.20/0.46 = ( real_V1908273582scaleR @ A @ ( times_times @ real @ A2 @ B2 ) @ X2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % scaleR_scaleR
% 0.20/0.46 thf(fact_12_scaleR__one,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( real_V1076094709vector @ A )
% 0.20/0.46 => ! [X2: A] :
% 0.20/0.46 ( ( real_V1908273582scaleR @ A @ ( one_one @ real ) @ X2 )
% 0.20/0.46 = X2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % scaleR_one
% 0.20/0.46 thf(fact_13__092_060open_062sum_Aa_As_A_061_A1_A_N_Aa_Ai_092_060close_062,axiom,
% 0.20/0.46 ( ( groups1340683514dd_sum @ a @ real @ aa @ sa )
% 0.20/0.46 = ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % \<open>sum a s = 1 - a i\<close>
% 0.20/0.46 thf(fact_14_inverse__eq__1__iff,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( field @ A )
% 0.20/0.46 => ! [X2: A] :
% 0.20/0.46 ( ( ( inverse_inverse @ A @ X2 )
% 0.20/0.46 = ( one_one @ A ) )
% 0.20/0.46 = ( X2
% 0.20/0.46 = ( one_one @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inverse_eq_1_iff
% 0.20/0.46 thf(fact_15_inverse__1,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( division_ring @ A )
% 0.20/0.46 => ( ( inverse_inverse @ A @ ( one_one @ A ) )
% 0.20/0.46 = ( one_one @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inverse_1
% 0.20/0.46 thf(fact_16_inverse__mult__distrib,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( field @ A )
% 0.20/0.46 => ! [A2: A,B2: A] :
% 0.20/0.46 ( ( inverse_inverse @ A @ ( times_times @ A @ A2 @ B2 ) )
% 0.20/0.46 = ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inverse_mult_distrib
% 0.20/0.46 thf(fact_17_mult__scaleR__left,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( real_V148923926lgebra @ A )
% 0.20/0.46 => ! [A2: real,X2: A,Y: A] :
% 0.20/0.46 ( ( times_times @ A @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) @ Y )
% 0.20/0.46 = ( real_V1908273582scaleR @ A @ A2 @ ( times_times @ A @ X2 @ Y ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % mult_scaleR_left
% 0.20/0.46 thf(fact_18_assms_I1_J,axiom,
% 0.20/0.46 finite_finite2 @ a @ s ).
% 0.20/0.46
% 0.20/0.46 % assms(1)
% 0.20/0.46 thf(fact_19_assms_I2_J,axiom,
% 0.20/0.46 ( s
% 0.20/0.46 != ( bot_bot @ ( set @ a ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % assms(2)
% 0.20/0.46 thf(fact_20_assms_I7_J,axiom,
% 0.20/0.46 ! [I: a] :
% 0.20/0.46 ( ( member @ a @ I @ s )
% 0.20/0.46 => ( member @ b @ ( y @ I ) @ c ) ) ).
% 0.20/0.46
% 0.20/0.46 % assms(7)
% 0.20/0.46 thf(fact_21_insert_Oprems_I3_J,axiom,
% 0.20/0.46 ! [I: a] :
% 0.20/0.46 ( ( member @ a @ I @ ( insert @ a @ i @ sa ) )
% 0.20/0.46 => ( member @ b @ ( y @ I ) @ c ) ) ).
% 0.20/0.46
% 0.20/0.46 % insert.prems(3)
% 0.20/0.46 thf(fact_22_inverse__inverse__eq,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( division_ring @ A )
% 0.20/0.46 => ! [A2: A] :
% 0.20/0.46 ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
% 0.20/0.46 = A2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % inverse_inverse_eq
% 0.20/0.46 thf(fact_23_inverse__eq__iff__eq,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( division_ring @ A )
% 0.20/0.46 => ! [A2: A,B2: A] :
% 0.20/0.46 ( ( ( inverse_inverse @ A @ A2 )
% 0.20/0.46 = ( inverse_inverse @ A @ B2 ) )
% 0.20/0.46 = ( A2 = B2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inverse_eq_iff_eq
% 0.20/0.46 thf(fact_24_insert_Oprems_I1_J,axiom,
% 0.20/0.46 ( ( groups1340683514dd_sum @ a @ real @ aa @ ( insert @ a @ i @ sa ) )
% 0.20/0.46 = ( one_one @ real ) ) ).
% 0.20/0.46
% 0.20/0.46 % insert.prems(1)
% 0.20/0.46 thf(fact_25_assms_I3_J,axiom,
% 0.20/0.46 lower_673667120vex_on @ b @ c @ f ).
% 0.20/0.46
% 0.20/0.46 % assms(3)
% 0.20/0.46 thf(fact_26_fis,axiom,
% 0.20/0.46 finite_finite2 @ a @ ( insert @ a @ i @ sa ) ).
% 0.20/0.46
% 0.20/0.46 % fis
% 0.20/0.46 thf(fact_27_mult__scaleR__right,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( real_V148923926lgebra @ A )
% 0.20/0.46 => ! [X2: A,A2: real,Y: A] :
% 0.20/0.46 ( ( times_times @ A @ X2 @ ( real_V1908273582scaleR @ A @ A2 @ Y ) )
% 0.20/0.46 = ( real_V1908273582scaleR @ A @ A2 @ ( times_times @ A @ X2 @ Y ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % mult_scaleR_right
% 0.20/0.46 thf(fact_28_assms_I4_J,axiom,
% 0.20/0.46 convex @ b @ c ).
% 0.20/0.46
% 0.20/0.46 % assms(4)
% 0.20/0.46 thf(fact_29_assms_I5_J,axiom,
% 0.20/0.46 ( ( groups1340683514dd_sum @ a @ real @ a2 @ s )
% 0.20/0.46 = ( one_one @ real ) ) ).
% 0.20/0.46
% 0.20/0.46 % assms(5)
% 0.20/0.46 thf(fact_30_real__scaleR__def,axiom,
% 0.20/0.46 ( ( real_V1908273582scaleR @ real )
% 0.20/0.46 = ( times_times @ real ) ) ).
% 0.20/0.46
% 0.20/0.46 % real_scaleR_def
% 0.20/0.46 thf(fact_31_scale__left__commute,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( real_V1076094709vector @ A )
% 0.20/0.46 => ! [A2: real,B2: real,X2: A] :
% 0.20/0.46 ( ( real_V1908273582scaleR @ A @ A2 @ ( real_V1908273582scaleR @ A @ B2 @ X2 ) )
% 0.20/0.46 = ( real_V1908273582scaleR @ A @ B2 @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % scale_left_commute
% 0.20/0.46 thf(fact_32_inverse__eq__imp__eq,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( division_ring @ A )
% 0.20/0.46 => ! [A2: A,B2: A] :
% 0.20/0.46 ( ( ( inverse_inverse @ A @ A2 )
% 0.20/0.46 = ( inverse_inverse @ A @ B2 ) )
% 0.20/0.46 => ( A2 = B2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inverse_eq_imp_eq
% 0.20/0.46 thf(fact_33_scaleR__add__right,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( real_V1076094709vector @ A )
% 0.20/0.46 => ! [A2: real,X2: A,Y: A] :
% 0.20/0.46 ( ( real_V1908273582scaleR @ A @ A2 @ ( plus_plus @ A @ X2 @ Y ) )
% 0.20/0.46 = ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) @ ( real_V1908273582scaleR @ A @ A2 @ Y ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % scaleR_add_right
% 0.20/0.46 thf(fact_34_scaleR__add__left,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( real_V1076094709vector @ A )
% 0.20/0.46 => ! [A2: real,B2: real,X2: A] :
% 0.20/0.46 ( ( real_V1908273582scaleR @ A @ ( plus_plus @ real @ A2 @ B2 ) @ X2 )
% 0.20/0.46 = ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) @ ( real_V1908273582scaleR @ A @ B2 @ X2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % scaleR_add_left
% 0.20/0.46 thf(fact_35_scaleR__left_Oadd,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( real_V1076094709vector @ A )
% 0.20/0.46 => ! [X2: real,Y: real,Xa: A] :
% 0.20/0.46 ( ( real_V1908273582scaleR @ A @ ( plus_plus @ real @ X2 @ Y ) @ Xa )
% 0.20/0.46 = ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ X2 @ Xa ) @ ( real_V1908273582scaleR @ A @ Y @ Xa ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % scaleR_left.add
% 0.20/0.46 thf(fact_36_scale__right__diff__distrib,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( real_V1076094709vector @ A )
% 0.20/0.46 => ! [A2: real,X2: A,Y: A] :
% 0.20/0.46 ( ( real_V1908273582scaleR @ A @ A2 @ ( minus_minus @ A @ X2 @ Y ) )
% 0.20/0.46 = ( minus_minus @ A @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) @ ( real_V1908273582scaleR @ A @ A2 @ Y ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % scale_right_diff_distrib
% 0.20/0.46 thf(fact_37_mult__commute__imp__mult__inverse__commute,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( division_ring @ A )
% 0.20/0.46 => ! [Y: A,X2: A] :
% 0.20/0.46 ( ( ( times_times @ A @ Y @ X2 )
% 0.20/0.46 = ( times_times @ A @ X2 @ Y ) )
% 0.20/0.46 => ( ( times_times @ A @ ( inverse_inverse @ A @ Y ) @ X2 )
% 0.20/0.46 = ( times_times @ A @ X2 @ ( inverse_inverse @ A @ Y ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % mult_commute_imp_mult_inverse_commute
% 0.20/0.46 thf(fact_38_scale__sum__right,axiom,
% 0.20/0.46 ! [A: $tType,C: $tType] :
% 0.20/0.46 ( ( real_V1076094709vector @ A )
% 0.20/0.46 => ! [A2: real,F: C > A,A3: set @ C] :
% 0.20/0.46 ( ( real_V1908273582scaleR @ A @ A2 @ ( groups1340683514dd_sum @ C @ A @ F @ A3 ) )
% 0.20/0.46 = ( groups1340683514dd_sum @ C @ A
% 0.20/0.46 @ ^ [X: C] : ( real_V1908273582scaleR @ A @ A2 @ ( F @ X ) )
% 0.20/0.46 @ A3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % scale_sum_right
% 0.20/0.46 thf(fact_39_scale__sum__left,axiom,
% 0.20/0.46 ! [A: $tType,C: $tType] :
% 0.20/0.46 ( ( real_V1076094709vector @ A )
% 0.20/0.46 => ! [F: C > real,A3: set @ C,X2: A] :
% 0.20/0.46 ( ( real_V1908273582scaleR @ A @ ( groups1340683514dd_sum @ C @ real @ F @ A3 ) @ X2 )
% 0.20/0.46 = ( groups1340683514dd_sum @ C @ A
% 0.20/0.46 @ ^ [A4: C] : ( real_V1908273582scaleR @ A @ ( F @ A4 ) @ X2 )
% 0.20/0.46 @ A3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % scale_sum_left
% 0.20/0.46 thf(fact_40_scaleR__right_Osum,axiom,
% 0.20/0.46 ! [A: $tType,C: $tType] :
% 0.20/0.46 ( ( real_V1076094709vector @ A )
% 0.20/0.46 => ! [A2: real,G: C > A,A3: set @ C] :
% 0.20/0.46 ( ( real_V1908273582scaleR @ A @ A2 @ ( groups1340683514dd_sum @ C @ A @ G @ A3 ) )
% 0.20/0.46 = ( groups1340683514dd_sum @ C @ A
% 0.20/0.46 @ ^ [X: C] : ( real_V1908273582scaleR @ A @ A2 @ ( G @ X ) )
% 0.20/0.46 @ A3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % scaleR_right.sum
% 0.20/0.46 thf(fact_41_scaleR__left_Osum,axiom,
% 0.20/0.46 ! [A: $tType,C: $tType] :
% 0.20/0.46 ( ( real_V1076094709vector @ A )
% 0.20/0.46 => ! [G: C > real,A3: set @ C,X2: A] :
% 0.20/0.46 ( ( real_V1908273582scaleR @ A @ ( groups1340683514dd_sum @ C @ real @ G @ A3 ) @ X2 )
% 0.20/0.46 = ( groups1340683514dd_sum @ C @ A
% 0.20/0.46 @ ^ [X: C] : ( real_V1908273582scaleR @ A @ ( G @ X ) @ X2 )
% 0.20/0.46 @ A3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % scaleR_left.sum
% 0.20/0.46 thf(fact_42_inverse__unique,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( division_ring @ A )
% 0.20/0.46 => ! [A2: A,B2: A] :
% 0.20/0.46 ( ( ( times_times @ A @ A2 @ B2 )
% 0.20/0.46 = ( one_one @ A ) )
% 0.20/0.46 => ( ( inverse_inverse @ A @ A2 )
% 0.20/0.46 = B2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inverse_unique
% 0.20/0.46 thf(fact_43_scale__left__diff__distrib,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( real_V1076094709vector @ A )
% 0.20/0.46 => ! [A2: real,B2: real,X2: A] :
% 0.20/0.46 ( ( real_V1908273582scaleR @ A @ ( minus_minus @ real @ A2 @ B2 ) @ X2 )
% 0.20/0.46 = ( minus_minus @ A @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) @ ( real_V1908273582scaleR @ A @ B2 @ X2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % scale_left_diff_distrib
% 0.20/0.46 thf(fact_44_scaleR__left_Odiff,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( real_V1076094709vector @ A )
% 0.20/0.46 => ! [X2: real,Y: real,Xa: A] :
% 0.20/0.46 ( ( real_V1908273582scaleR @ A @ ( minus_minus @ real @ X2 @ Y ) @ Xa )
% 0.20/0.46 = ( minus_minus @ A @ ( real_V1908273582scaleR @ A @ X2 @ Xa ) @ ( real_V1908273582scaleR @ A @ Y @ Xa ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % scaleR_left.diff
% 0.20/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,P: A > $o] :
% 0.20/0.46 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.20/0.46 = ( P @ A2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % mem_Collect_eq
% 0.20/0.46 thf(fact_46_Collect__mem__eq,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A] :
% 0.20/0.46 ( ( collect @ A
% 0.20/0.46 @ ^ [X: A] : ( member @ A @ X @ A3 ) )
% 0.20/0.46 = A3 ) ).
% 0.20/0.46
% 0.20/0.46 % Collect_mem_eq
% 0.20/0.46 thf(fact_47_Collect__cong,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.20/0.46 ( ! [X3: A] :
% 0.20/0.46 ( ( P @ X3 )
% 0.20/0.46 = ( Q @ X3 ) )
% 0.20/0.46 => ( ( collect @ A @ P )
% 0.20/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Collect_cong
% 0.20/0.46 thf(fact_48_ext,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.20/0.46 ( ! [X3: A] :
% 0.20/0.46 ( ( F @ X3 )
% 0.20/0.46 = ( G @ X3 ) )
% 0.20/0.46 => ( F = G ) ) ).
% 0.20/0.46
% 0.20/0.46 % ext
% 0.20/0.46 thf(fact_49_inverse__scaleR__distrib,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( real_V1015715145lgebra @ A )
% 0.20/0.46 => ! [A2: real,X2: A] :
% 0.20/0.46 ( ( inverse_inverse @ A @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) )
% 0.20/0.46 = ( real_V1908273582scaleR @ A @ ( inverse_inverse @ real @ A2 ) @ ( inverse_inverse @ A @ X2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inverse_scaleR_distrib
% 0.20/0.46 thf(fact_50_asum,axiom,
% 0.20/0.46 ( member @ b
% 0.20/0.46 @ ( groups1340683514dd_sum @ a @ b
% 0.20/0.46 @ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( divide_divide @ real @ ( aa @ J ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) @ ( y @ J ) )
% 0.20/0.46 @ sa )
% 0.20/0.46 @ c ) ).
% 0.20/0.46
% 0.20/0.46 % asum
% 0.20/0.46 thf(fact_51_sum_Oinsert,axiom,
% 0.20/0.46 ! [A: $tType,B: $tType] :
% 0.20/0.46 ( ( comm_monoid_add @ A )
% 0.20/0.46 => ! [A3: set @ B,X2: B,G: B > A] :
% 0.20/0.46 ( ( finite_finite2 @ B @ A3 )
% 0.20/0.46 => ( ~ ( member @ B @ X2 @ A3 )
% 0.20/0.46 => ( ( groups1340683514dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
% 0.20/0.46 = ( plus_plus @ A @ ( G @ X2 ) @ ( groups1340683514dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum.insert
% 0.20/0.46 thf(fact_52_affine__hull__finite__step,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( real_V1076094709vector @ A )
% 0.20/0.46 => ! [S: set @ A,A2: A,W: real,Y: A] :
% 0.20/0.46 ( ( finite_finite2 @ A @ S )
% 0.20/0.46 => ( ( ? [U2: A > real] :
% 0.20/0.46 ( ( ( groups1340683514dd_sum @ A @ real @ U2 @ ( insert @ A @ A2 @ S ) )
% 0.20/0.46 = W )
% 0.20/0.46 & ( ( groups1340683514dd_sum @ A @ A
% 0.20/0.46 @ ^ [X: A] : ( real_V1908273582scaleR @ A @ ( U2 @ X ) @ X )
% 0.20/0.46 @ ( insert @ A @ A2 @ S ) )
% 0.20/0.46 = Y ) ) )
% 0.20/0.46 = ( ? [V: real,U2: A > real] :
% 0.20/0.46 ( ( ( groups1340683514dd_sum @ A @ real @ U2 @ S )
% 0.20/0.46 = ( minus_minus @ real @ W @ V ) )
% 0.20/0.46 & ( ( groups1340683514dd_sum @ A @ A
% 0.20/0.46 @ ^ [X: A] : ( real_V1908273582scaleR @ A @ ( U2 @ X ) @ X )
% 0.20/0.46 @ S )
% 0.20/0.46 = ( minus_minus @ A @ Y @ ( real_V1908273582scaleR @ A @ V @ A2 ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % affine_hull_finite_step
% 0.20/0.46 thf(fact_53_a1,axiom,
% 0.20/0.46 ( ( groups1340683514dd_sum @ a @ real
% 0.20/0.46 @ ^ [J: a] : ( divide_divide @ real @ ( aa @ J ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) )
% 0.20/0.46 @ sa )
% 0.20/0.46 = ( one_one @ real ) ) ).
% 0.20/0.46
% 0.20/0.46 % a1
% 0.20/0.46 thf(fact_54__092_060open_062sum_Aa_As_A_P_A_I1_A_N_Aa_Ai_J_A_061_A1_092_060close_062,axiom,
% 0.20/0.46 ( ( divide_divide @ real @ ( groups1340683514dd_sum @ a @ real @ aa @ sa ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) )
% 0.20/0.46 = ( one_one @ real ) ) ).
% 0.20/0.46
% 0.20/0.46 % \<open>sum a s / (1 - a i) = 1\<close>
% 0.20/0.46 thf(fact_55_singleton__conv,axiom,
% 0.20/0.46 ! [A: $tType,A2: A] :
% 0.20/0.46 ( ( collect @ A
% 0.20/0.46 @ ^ [X: A] : ( X = A2 ) )
% 0.20/0.46 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % singleton_conv
% 0.20/0.46 thf(fact_56_singleton__conv2,axiom,
% 0.20/0.46 ! [A: $tType,A2: A] :
% 0.20/0.46 ( ( collect @ A
% 0.20/0.46 @ ( ^ [Y2: A,Z: A] : ( Y2 = Z )
% 0.20/0.46 @ A2 ) )
% 0.20/0.46 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % singleton_conv2
% 0.20/0.46 thf(fact_57_finite__insert,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,A3: set @ A] :
% 0.20/0.46 ( ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) )
% 0.20/0.46 = ( finite_finite2 @ A @ A3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % finite_insert
% 0.20/0.46 thf(fact_58_sum__diff1,axiom,
% 0.20/0.46 ! [A: $tType,B: $tType] :
% 0.20/0.46 ( ( ab_group_add @ A )
% 0.20/0.46 => ! [A3: set @ B,A2: B,F: B > A] :
% 0.20/0.46 ( ( finite_finite2 @ B @ A3 )
% 0.20/0.46 => ( ( ( member @ B @ A2 @ A3 )
% 0.20/0.46 => ( ( groups1340683514dd_sum @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 0.20/0.46 = ( minus_minus @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 0.20/0.46 & ( ~ ( member @ B @ A2 @ A3 )
% 0.20/0.46 => ( ( groups1340683514dd_sum @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 0.20/0.46 = ( groups1340683514dd_sum @ B @ A @ F @ A3 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum_diff1
% 0.20/0.46 thf(fact_59_singletonI,axiom,
% 0.20/0.46 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % singletonI
% 0.20/0.46 thf(fact_60_empty__Collect__eq,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o] :
% 0.20/0.46 ( ( ( bot_bot @ ( set @ A ) )
% 0.20/0.46 = ( collect @ A @ P ) )
% 0.20/0.46 = ( ! [X: A] :
% 0.20/0.46 ~ ( P @ X ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % empty_Collect_eq
% 0.20/0.46 thf(fact_61_Collect__empty__eq,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o] :
% 0.20/0.46 ( ( ( collect @ A @ P )
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.20/0.46 = ( ! [X: A] :
% 0.20/0.46 ~ ( P @ X ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Collect_empty_eq
% 0.20/0.46 thf(fact_62_all__not__in__conv,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A] :
% 0.20/0.46 ( ( ! [X: A] :
% 0.20/0.46 ~ ( member @ A @ X @ A3 ) )
% 0.20/0.46 = ( A3
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % all_not_in_conv
% 0.20/0.46 thf(fact_63_empty__iff,axiom,
% 0.20/0.46 ! [A: $tType,C2: A] :
% 0.20/0.46 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % empty_iff
% 0.20/0.46 thf(fact_64_finite__code,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( finite_finite @ A )
% 0.20/0.46 => ( ( finite_finite2 @ A )
% 0.20/0.46 = ( ^ [A5: set @ A] : $true ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % finite_code
% 0.20/0.46 thf(fact_65_insert__absorb2,axiom,
% 0.20/0.46 ! [A: $tType,X2: A,A3: set @ A] :
% 0.20/0.46 ( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A3 ) )
% 0.20/0.46 = ( insert @ A @ X2 @ A3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % insert_absorb2
% 0.20/0.46 thf(fact_66_insert__iff,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,B2: A,A3: set @ A] :
% 0.20/0.46 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
% 0.20/0.46 = ( ( A2 = B2 )
% 0.20/0.46 | ( member @ A @ A2 @ A3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % insert_iff
% 0.20/0.46 thf(fact_67_insertCI,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,B3: set @ A,B2: A] :
% 0.20/0.46 ( ( ~ ( member @ A @ A2 @ B3 )
% 0.20/0.46 => ( A2 = B2 ) )
% 0.20/0.46 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % insertCI
% 0.20/0.46 thf(fact_68_Diff__idemp,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ B3 )
% 0.20/0.46 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % Diff_idemp
% 0.20/0.46 thf(fact_69_Diff__iff,axiom,
% 0.20/0.46 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 0.20/0.46 = ( ( member @ A @ C2 @ A3 )
% 0.20/0.46 & ~ ( member @ A @ C2 @ B3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Diff_iff
% 0.20/0.46 thf(fact_70_DiffI,axiom,
% 0.20/0.46 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( member @ A @ C2 @ A3 )
% 0.20/0.46 => ( ~ ( member @ A @ C2 @ B3 )
% 0.20/0.46 => ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % DiffI
% 0.20/0.46 thf(fact_71_finite__Collect__disjI,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.20/0.46 ( ( finite_finite2 @ A
% 0.20/0.46 @ ( collect @ A
% 0.20/0.46 @ ^ [X: A] :
% 0.20/0.46 ( ( P @ X )
% 0.20/0.46 | ( Q @ X ) ) ) )
% 0.20/0.46 = ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.20/0.46 & ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % finite_Collect_disjI
% 0.20/0.46 thf(fact_72_finite__Collect__conjI,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.20/0.46 ( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.20/0.46 | ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
% 0.20/0.46 => ( finite_finite2 @ A
% 0.20/0.46 @ ( collect @ A
% 0.20/0.46 @ ^ [X: A] :
% 0.20/0.46 ( ( P @ X )
% 0.20/0.46 & ( Q @ X ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % finite_Collect_conjI
% 0.20/0.46 thf(fact_73_times__divide__eq__left,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( field @ A )
% 0.20/0.46 => ! [B2: A,C2: A,A2: A] :
% 0.20/0.46 ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
% 0.20/0.46 = ( divide_divide @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % times_divide_eq_left
% 0.20/0.46 thf(fact_74_divide__divide__eq__left,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( field @ A )
% 0.20/0.46 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.46 ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
% 0.20/0.46 = ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % divide_divide_eq_left
% 0.20/0.46 thf(fact_75_divide__divide__eq__right,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( field @ A )
% 0.20/0.46 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.46 ( ( divide_divide @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.20/0.46 = ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % divide_divide_eq_right
% 0.20/0.46 thf(fact_76_times__divide__eq__right,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( division_ring @ A )
% 0.20/0.46 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.46 ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.20/0.46 = ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % times_divide_eq_right
% 0.20/0.46 thf(fact_77_Diff__cancel,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A] :
% 0.20/0.46 ( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Diff_cancel
% 0.20/0.46 thf(fact_78_empty__Diff,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A] :
% 0.20/0.46 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % empty_Diff
% 0.20/0.46 thf(fact_79_Diff__empty,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A] :
% 0.20/0.46 ( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.20/0.46 = A3 ) ).
% 0.20/0.46
% 0.20/0.46 % Diff_empty
% 0.20/0.46 thf(fact_80_finite__Diff2,axiom,
% 0.20/0.46 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.20/0.46 ( ( finite_finite2 @ A @ B3 )
% 0.20/0.46 => ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 0.20/0.46 = ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % finite_Diff2
% 0.20/0.46 thf(fact_81_finite__Diff,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( finite_finite2 @ A @ A3 )
% 0.20/0.46 => ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % finite_Diff
% 0.20/0.46 thf(fact_82_insert__Diff1,axiom,
% 0.20/0.46 ! [A: $tType,X2: A,B3: set @ A,A3: set @ A] :
% 0.20/0.46 ( ( member @ A @ X2 @ B3 )
% 0.20/0.46 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B3 )
% 0.20/0.46 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % insert_Diff1
% 0.20/0.46 thf(fact_83_Diff__insert0,axiom,
% 0.20/0.46 ! [A: $tType,X2: A,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ~ ( member @ A @ X2 @ A3 )
% 0.20/0.46 => ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ B3 ) )
% 0.20/0.46 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Diff_insert0
% 0.20/0.46 thf(fact_84_inverse__divide,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( field @ A )
% 0.20/0.46 => ! [A2: A,B2: A] :
% 0.20/0.46 ( ( inverse_inverse @ A @ ( divide_divide @ A @ A2 @ B2 ) )
% 0.20/0.46 = ( divide_divide @ A @ B2 @ A2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inverse_divide
% 0.20/0.46 thf(fact_85_insert__Diff__single,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,A3: set @ A] :
% 0.20/0.46 ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.20/0.46 = ( insert @ A @ A2 @ A3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % insert_Diff_single
% 0.20/0.46 thf(fact_86_finite__Diff__insert,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
% 0.20/0.46 ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) )
% 0.20/0.46 = ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % finite_Diff_insert
% 0.20/0.46 thf(fact_87_sum__diff1__nat,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,A3: set @ A,F: A > nat] :
% 0.20/0.46 ( ( ( member @ A @ A2 @ A3 )
% 0.20/0.46 => ( ( groups1340683514dd_sum @ A @ nat @ F @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.20/0.46 = ( minus_minus @ nat @ ( groups1340683514dd_sum @ A @ nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 0.20/0.46 & ( ~ ( member @ A @ A2 @ A3 )
% 0.20/0.46 => ( ( groups1340683514dd_sum @ A @ nat @ F @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.20/0.46 = ( groups1340683514dd_sum @ A @ nat @ F @ A3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum_diff1_nat
% 0.20/0.46 thf(fact_88_minus__set__def,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( minus_minus @ ( set @ A ) )
% 0.20/0.46 = ( ^ [A5: set @ A,B4: set @ A] :
% 0.20/0.46 ( collect @ A
% 0.20/0.46 @ ( minus_minus @ ( A > $o )
% 0.20/0.46 @ ^ [X: A] : ( member @ A @ X @ A5 )
% 0.20/0.46 @ ^ [X: A] : ( member @ A @ X @ B4 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % minus_set_def
% 0.20/0.46 thf(fact_89_set__diff__eq,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( minus_minus @ ( set @ A ) )
% 0.20/0.46 = ( ^ [A5: set @ A,B4: set @ A] :
% 0.20/0.46 ( collect @ A
% 0.20/0.46 @ ^ [X: A] :
% 0.20/0.46 ( ( member @ A @ X @ A5 )
% 0.20/0.46 & ~ ( member @ A @ X @ B4 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % set_diff_eq
% 0.20/0.46 thf(fact_90_DiffD2,axiom,
% 0.20/0.46 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 0.20/0.46 => ~ ( member @ A @ C2 @ B3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % DiffD2
% 0.20/0.46 thf(fact_91_DiffD1,axiom,
% 0.20/0.46 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 0.20/0.46 => ( member @ A @ C2 @ A3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % DiffD1
% 0.20/0.46 thf(fact_92_DiffE,axiom,
% 0.20/0.46 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 0.20/0.46 => ~ ( ( member @ A @ C2 @ A3 )
% 0.20/0.46 => ( member @ A @ C2 @ B3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % DiffE
% 0.20/0.46 thf(fact_93_Diff__infinite__finite,axiom,
% 0.20/0.46 ! [A: $tType,T: set @ A,S: set @ A] :
% 0.20/0.46 ( ( finite_finite2 @ A @ T )
% 0.20/0.46 => ( ~ ( finite_finite2 @ A @ S )
% 0.20/0.46 => ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S @ T ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Diff_infinite_finite
% 0.20/0.46 thf(fact_94_insert__Diff__if,axiom,
% 0.20/0.46 ! [A: $tType,X2: A,B3: set @ A,A3: set @ A] :
% 0.20/0.46 ( ( ( member @ A @ X2 @ B3 )
% 0.20/0.46 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B3 )
% 0.20/0.46 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) )
% 0.20/0.46 & ( ~ ( member @ A @ X2 @ B3 )
% 0.20/0.46 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B3 )
% 0.20/0.46 = ( insert @ A @ X2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % insert_Diff_if
% 0.20/0.46 thf(fact_95_sum__divide__distrib,axiom,
% 0.20/0.46 ! [A: $tType,B: $tType] :
% 0.20/0.46 ( ( field @ A )
% 0.20/0.46 => ! [F: B > A,A3: set @ B,R: A] :
% 0.20/0.46 ( ( divide_divide @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ R )
% 0.20/0.46 = ( groups1340683514dd_sum @ B @ A
% 0.20/0.46 @ ^ [N: B] : ( divide_divide @ A @ ( F @ N ) @ R )
% 0.20/0.46 @ A3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum_divide_distrib
% 0.20/0.46 thf(fact_96_convex__ereal__add,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( real_V1076094709vector @ A )
% 0.20/0.46 => ! [S2: set @ A,F: A > extended_ereal,G: A > extended_ereal] :
% 0.20/0.46 ( ( lower_673667120vex_on @ A @ S2 @ F )
% 0.20/0.46 => ( ( lower_673667120vex_on @ A @ S2 @ G )
% 0.20/0.46 => ( lower_673667120vex_on @ A @ S2
% 0.20/0.46 @ ^ [X: A] : ( plus_plus @ extended_ereal @ ( F @ X ) @ ( G @ X ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % convex_ereal_add
% 0.20/0.46 thf(fact_97_times__divide__times__eq,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( field @ A )
% 0.20/0.46 => ! [X2: A,Y: A,Z2: A,W: A] :
% 0.20/0.46 ( ( times_times @ A @ ( divide_divide @ A @ X2 @ Y ) @ ( divide_divide @ A @ Z2 @ W ) )
% 0.20/0.46 = ( divide_divide @ A @ ( times_times @ A @ X2 @ Z2 ) @ ( times_times @ A @ Y @ W ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % times_divide_times_eq
% 0.20/0.46 thf(fact_98_divide__divide__times__eq,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( field @ A )
% 0.20/0.46 => ! [X2: A,Y: A,Z2: A,W: A] :
% 0.20/0.46 ( ( divide_divide @ A @ ( divide_divide @ A @ X2 @ Y ) @ ( divide_divide @ A @ Z2 @ W ) )
% 0.20/0.46 = ( divide_divide @ A @ ( times_times @ A @ X2 @ W ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % divide_divide_times_eq
% 0.20/0.46 thf(fact_99_divide__divide__eq__left_H,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( division_ring @ A )
% 0.20/0.46 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.46 ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
% 0.20/0.46 = ( divide_divide @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % divide_divide_eq_left'
% 0.20/0.46 thf(fact_100_add__divide__distrib,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( division_ring @ A )
% 0.20/0.46 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.46 ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 0.20/0.46 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % add_divide_distrib
% 0.20/0.46 thf(fact_101_diff__divide__distrib,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( division_ring @ A )
% 0.20/0.46 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.46 ( ( divide_divide @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
% 0.20/0.46 = ( minus_minus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_divide_distrib
% 0.20/0.46 thf(fact_102_Diff__insert__absorb,axiom,
% 0.20/0.46 ! [A: $tType,X2: A,A3: set @ A] :
% 0.20/0.46 ( ~ ( member @ A @ X2 @ A3 )
% 0.20/0.46 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.46 = A3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % Diff_insert_absorb
% 0.20/0.46 thf(fact_103_Diff__insert2,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
% 0.20/0.46 ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
% 0.20/0.46 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % Diff_insert2
% 0.20/0.46 thf(fact_104_insert__Diff,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,A3: set @ A] :
% 0.20/0.46 ( ( member @ A @ A2 @ A3 )
% 0.20/0.46 => ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.20/0.46 = A3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % insert_Diff
% 0.20/0.46 thf(fact_105_Diff__insert,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
% 0.20/0.46 ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
% 0.20/0.46 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Diff_insert
% 0.20/0.46 thf(fact_106_finite__empty__induct,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
% 0.20/0.46 ( ( finite_finite2 @ A @ A3 )
% 0.20/0.46 => ( ( P @ A3 )
% 0.20/0.46 => ( ! [A6: A,A7: set @ A] :
% 0.20/0.46 ( ( finite_finite2 @ A @ A7 )
% 0.20/0.46 => ( ( member @ A @ A6 @ A7 )
% 0.20/0.46 => ( ( P @ A7 )
% 0.20/0.46 => ( P @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
% 0.20/0.46 => ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % finite_empty_induct
% 0.20/0.46 thf(fact_107_infinite__coinduct,axiom,
% 0.20/0.46 ! [A: $tType,X4: ( set @ A ) > $o,A3: set @ A] :
% 0.20/0.46 ( ( X4 @ A3 )
% 0.20/0.46 => ( ! [A7: set @ A] :
% 0.20/0.46 ( ( X4 @ A7 )
% 0.20/0.46 => ? [X5: A] :
% 0.20/0.46 ( ( member @ A @ X5 @ A7 )
% 0.20/0.46 & ( ( X4 @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.20/0.46 | ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
% 0.20/0.46 => ~ ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % infinite_coinduct
% 0.20/0.46 thf(fact_108_infinite__remove,axiom,
% 0.20/0.46 ! [A: $tType,S: set @ A,A2: A] :
% 0.20/0.46 ( ~ ( finite_finite2 @ A @ S )
% 0.20/0.46 => ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % infinite_remove
% 0.20/0.46 thf(fact_109_field__class_Ofield__divide__inverse,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( field @ A )
% 0.20/0.46 => ( ( divide_divide @ A )
% 0.20/0.46 = ( ^ [A4: A,B5: A] : ( times_times @ A @ A4 @ ( inverse_inverse @ A @ B5 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % field_class.field_divide_inverse
% 0.20/0.46 thf(fact_110_divide__inverse,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( division_ring @ A )
% 0.20/0.46 => ( ( divide_divide @ A )
% 0.20/0.46 = ( ^ [A4: A,B5: A] : ( times_times @ A @ A4 @ ( inverse_inverse @ A @ B5 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % divide_inverse
% 0.20/0.46 thf(fact_111_divide__inverse__commute,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( field @ A )
% 0.20/0.46 => ( ( divide_divide @ A )
% 0.20/0.46 = ( ^ [A4: A,B5: A] : ( times_times @ A @ ( inverse_inverse @ A @ B5 ) @ A4 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % divide_inverse_commute
% 0.20/0.46 thf(fact_112_inverse__eq__divide,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( division_ring @ A )
% 0.20/0.46 => ( ( inverse_inverse @ A )
% 0.20/0.46 = ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inverse_eq_divide
% 0.20/0.46 thf(fact_113_ex__in__conv,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A] :
% 0.20/0.46 ( ( ? [X: A] : ( member @ A @ X @ A3 ) )
% 0.20/0.46 = ( A3
% 0.20/0.46 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % ex_in_conv
% 0.20/0.46 thf(fact_114_equals0I,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A] :
% 0.20/0.46 ( ! [Y3: A] :
% 0.20/0.46 ~ ( member @ A @ Y3 @ A3 )
% 0.20/0.46 => ( A3
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % equals0I
% 0.20/0.46 thf(fact_115_equals0D,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,A2: A] :
% 0.20/0.46 ( ( A3
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.20/0.46 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % equals0D
% 0.20/0.46 thf(fact_116_emptyE,axiom,
% 0.20/0.46 ! [A: $tType,A2: A] :
% 0.20/0.46 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % emptyE
% 0.20/0.46 thf(fact_117_finite,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( finite_finite @ A )
% 0.20/0.46 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % finite
% 0.20/0.46 thf(fact_118_finite__set__choice,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
% 0.20/0.46 ( ( finite_finite2 @ A @ A3 )
% 0.20/0.46 => ( ! [X3: A] :
% 0.20/0.46 ( ( member @ A @ X3 @ A3 )
% 0.20/0.46 => ? [X_1: B] : ( P @ X3 @ X_1 ) )
% 0.20/0.46 => ? [F2: A > B] :
% 0.20/0.46 ! [X5: A] :
% 0.20/0.46 ( ( member @ A @ X5 @ A3 )
% 0.20/0.46 => ( P @ X5 @ ( F2 @ X5 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % finite_set_choice
% 0.20/0.46 thf(fact_119_mk__disjoint__insert,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,A3: set @ A] :
% 0.20/0.46 ( ( member @ A @ A2 @ A3 )
% 0.20/0.46 => ? [B6: set @ A] :
% 0.20/0.46 ( ( A3
% 0.20/0.46 = ( insert @ A @ A2 @ B6 ) )
% 0.20/0.46 & ~ ( member @ A @ A2 @ B6 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % mk_disjoint_insert
% 0.20/0.46 thf(fact_120_insert__commute,axiom,
% 0.20/0.46 ! [A: $tType,X2: A,Y: A,A3: set @ A] :
% 0.20/0.46 ( ( insert @ A @ X2 @ ( insert @ A @ Y @ A3 ) )
% 0.20/0.46 = ( insert @ A @ Y @ ( insert @ A @ X2 @ A3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % insert_commute
% 0.20/0.46 thf(fact_121_insert__eq__iff,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,A3: set @ A,B2: A,B3: set @ A] :
% 0.20/0.46 ( ~ ( member @ A @ A2 @ A3 )
% 0.20/0.46 => ( ~ ( member @ A @ B2 @ B3 )
% 0.20/0.46 => ( ( ( insert @ A @ A2 @ A3 )
% 0.20/0.46 = ( insert @ A @ B2 @ B3 ) )
% 0.20/0.46 = ( ( ( A2 = B2 )
% 0.20/0.46 => ( A3 = B3 ) )
% 0.20/0.46 & ( ( A2 != B2 )
% 0.20/0.46 => ? [C3: set @ A] :
% 0.20/0.46 ( ( A3
% 0.20/0.46 = ( insert @ A @ B2 @ C3 ) )
% 0.20/0.46 & ~ ( member @ A @ B2 @ C3 )
% 0.20/0.46 & ( B3
% 0.20/0.46 = ( insert @ A @ A2 @ C3 ) )
% 0.20/0.46 & ~ ( member @ A @ A2 @ C3 ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % insert_eq_iff
% 0.20/0.46 thf(fact_122_insert__absorb,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,A3: set @ A] :
% 0.20/0.46 ( ( member @ A @ A2 @ A3 )
% 0.20/0.46 => ( ( insert @ A @ A2 @ A3 )
% 0.20/0.46 = A3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % insert_absorb
% 0.20/0.46 thf(fact_123_insert__ident,axiom,
% 0.20/0.46 ! [A: $tType,X2: A,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ~ ( member @ A @ X2 @ A3 )
% 0.20/0.46 => ( ~ ( member @ A @ X2 @ B3 )
% 0.20/0.46 => ( ( ( insert @ A @ X2 @ A3 )
% 0.20/0.46 = ( insert @ A @ X2 @ B3 ) )
% 0.20/0.46 = ( A3 = B3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % insert_ident
% 0.20/0.46 thf(fact_124_Set_Oset__insert,axiom,
% 0.20/0.46 ! [A: $tType,X2: A,A3: set @ A] :
% 0.20/0.46 ( ( member @ A @ X2 @ A3 )
% 0.20/0.46 => ~ ! [B6: set @ A] :
% 0.20/0.46 ( ( A3
% 0.20/0.46 = ( insert @ A @ X2 @ B6 ) )
% 0.20/0.46 => ( member @ A @ X2 @ B6 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Set.set_insert
% 0.20/0.46 thf(fact_125_insertI2,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,B3: set @ A,B2: A] :
% 0.20/0.46 ( ( member @ A @ A2 @ B3 )
% 0.20/0.46 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % insertI2
% 0.20/0.46 thf(fact_126_insertI1,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,B3: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % insertI1
% 0.20/0.46 thf(fact_127_insertE,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,B2: A,A3: set @ A] :
% 0.20/0.46 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
% 0.20/0.46 => ( ( A2 != B2 )
% 0.20/0.46 => ( member @ A @ A2 @ A3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % insertE
% 0.20/0.46 thf(fact_128_sum_Ocong,axiom,
% 0.20/0.46 ! [A: $tType,B: $tType] :
% 0.20/0.46 ( ( comm_monoid_add @ A )
% 0.20/0.46 => ! [A3: set @ B,B3: set @ B,G: B > A,H: B > A] :
% 0.20/0.46 ( ( A3 = B3 )
% 0.20/0.46 => ( ! [X3: B] :
% 0.20/0.46 ( ( member @ B @ X3 @ B3 )
% 0.20/0.46 => ( ( G @ X3 )
% 0.20/0.46 = ( H @ X3 ) ) )
% 0.20/0.46 => ( ( groups1340683514dd_sum @ B @ A @ G @ A3 )
% 0.20/0.46 = ( groups1340683514dd_sum @ B @ A @ H @ B3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum.cong
% 0.20/0.46 thf(fact_129_sum_Oeq__general,axiom,
% 0.20/0.46 ! [C: $tType,A: $tType,B: $tType] :
% 0.20/0.46 ( ( comm_monoid_add @ A )
% 0.20/0.46 => ! [B3: set @ B,A3: set @ C,H: C > B,Gamma: B > A,Phi: C > A] :
% 0.20/0.46 ( ! [Y3: B] :
% 0.20/0.46 ( ( member @ B @ Y3 @ B3 )
% 0.20/0.46 => ? [X5: C] :
% 0.20/0.46 ( ( member @ C @ X5 @ A3 )
% 0.20/0.46 & ( ( H @ X5 )
% 0.20/0.46 = Y3 )
% 0.20/0.46 & ! [Ya: C] :
% 0.20/0.46 ( ( ( member @ C @ Ya @ A3 )
% 0.20/0.46 & ( ( H @ Ya )
% 0.20/0.46 = Y3 ) )
% 0.20/0.46 => ( Ya = X5 ) ) ) )
% 0.20/0.46 => ( ! [X3: C] :
% 0.20/0.46 ( ( member @ C @ X3 @ A3 )
% 0.20/0.46 => ( ( member @ B @ ( H @ X3 ) @ B3 )
% 0.20/0.46 & ( ( Gamma @ ( H @ X3 ) )
% 0.20/0.46 = ( Phi @ X3 ) ) ) )
% 0.20/0.46 => ( ( groups1340683514dd_sum @ C @ A @ Phi @ A3 )
% 0.20/0.46 = ( groups1340683514dd_sum @ B @ A @ Gamma @ B3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum.eq_general
% 0.20/0.46 thf(fact_130_sum_Oeq__general__inverses,axiom,
% 0.20/0.46 ! [C: $tType,A: $tType,B: $tType] :
% 0.20/0.46 ( ( comm_monoid_add @ A )
% 0.20/0.46 => ! [B3: set @ B,K: B > C,A3: set @ C,H: C > B,Gamma: B > A,Phi: C > A] :
% 0.20/0.46 ( ! [Y3: B] :
% 0.20/0.46 ( ( member @ B @ Y3 @ B3 )
% 0.20/0.46 => ( ( member @ C @ ( K @ Y3 ) @ A3 )
% 0.20/0.46 & ( ( H @ ( K @ Y3 ) )
% 0.20/0.46 = Y3 ) ) )
% 0.20/0.46 => ( ! [X3: C] :
% 0.20/0.46 ( ( member @ C @ X3 @ A3 )
% 0.20/0.46 => ( ( member @ B @ ( H @ X3 ) @ B3 )
% 0.20/0.46 & ( ( K @ ( H @ X3 ) )
% 0.20/0.46 = X3 )
% 0.20/0.46 & ( ( Gamma @ ( H @ X3 ) )
% 0.20/0.46 = ( Phi @ X3 ) ) ) )
% 0.20/0.46 => ( ( groups1340683514dd_sum @ C @ A @ Phi @ A3 )
% 0.20/0.46 = ( groups1340683514dd_sum @ B @ A @ Gamma @ B3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum.eq_general_inverses
% 0.20/0.46 thf(fact_131_sum_Oreindex__bij__witness,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType,C: $tType] :
% 0.20/0.46 ( ( comm_monoid_add @ A )
% 0.20/0.46 => ! [S: set @ B,I2: C > B,J2: B > C,T: set @ C,H: C > A,G: B > A] :
% 0.20/0.46 ( ! [A6: B] :
% 0.20/0.46 ( ( member @ B @ A6 @ S )
% 0.20/0.46 => ( ( I2 @ ( J2 @ A6 ) )
% 0.20/0.46 = A6 ) )
% 0.20/0.46 => ( ! [A6: B] :
% 0.20/0.46 ( ( member @ B @ A6 @ S )
% 0.20/0.46 => ( member @ C @ ( J2 @ A6 ) @ T ) )
% 0.20/0.46 => ( ! [B7: C] :
% 0.20/0.46 ( ( member @ C @ B7 @ T )
% 0.20/0.46 => ( ( J2 @ ( I2 @ B7 ) )
% 0.20/0.46 = B7 ) )
% 0.20/0.46 => ( ! [B7: C] :
% 0.20/0.46 ( ( member @ C @ B7 @ T )
% 0.20/0.46 => ( member @ B @ ( I2 @ B7 ) @ S ) )
% 0.20/0.46 => ( ! [A6: B] :
% 0.20/0.46 ( ( member @ B @ A6 @ S )
% 0.20/0.46 => ( ( H @ ( J2 @ A6 ) )
% 0.20/0.46 = ( G @ A6 ) ) )
% 0.20/0.46 => ( ( groups1340683514dd_sum @ B @ A @ G @ S )
% 0.20/0.46 = ( groups1340683514dd_sum @ C @ A @ H @ T ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum.reindex_bij_witness
% 0.20/0.46 thf(fact_132_if__smult,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( real_Vector_scaleR @ A )
% 0.20/0.46 => ! [P: $o,X2: real,Y: real,V2: A] :
% 0.20/0.46 ( ( P
% 0.20/0.46 => ( ( real_V1908273582scaleR @ A @ ( if @ real @ P @ X2 @ Y ) @ V2 )
% 0.20/0.46 = ( real_V1908273582scaleR @ A @ X2 @ V2 ) ) )
% 0.20/0.46 & ( ~ P
% 0.20/0.46 => ( ( real_V1908273582scaleR @ A @ ( if @ real @ P @ X2 @ Y ) @ V2 )
% 0.20/0.46 = ( real_V1908273582scaleR @ A @ Y @ V2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % if_smult
% 0.20/0.46 thf(fact_133_sum_Oinsert__remove,axiom,
% 0.20/0.46 ! [A: $tType,B: $tType] :
% 0.20/0.46 ( ( comm_monoid_add @ A )
% 0.20/0.46 => ! [A3: set @ B,G: B > A,X2: B] :
% 0.20/0.46 ( ( finite_finite2 @ B @ A3 )
% 0.20/0.46 => ( ( groups1340683514dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
% 0.20/0.46 = ( plus_plus @ A @ ( G @ X2 ) @ ( groups1340683514dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum.insert_remove
% 0.20/0.46 thf(fact_134_sum_Oremove,axiom,
% 0.20/0.46 ! [A: $tType,B: $tType] :
% 0.20/0.46 ( ( comm_monoid_add @ A )
% 0.20/0.46 => ! [A3: set @ B,X2: B,G: B > A] :
% 0.20/0.46 ( ( finite_finite2 @ B @ A3 )
% 0.20/0.46 => ( ( member @ B @ X2 @ A3 )
% 0.20/0.46 => ( ( groups1340683514dd_sum @ B @ A @ G @ A3 )
% 0.20/0.46 = ( plus_plus @ A @ ( G @ X2 ) @ ( groups1340683514dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum.remove
% 0.20/0.46 thf(fact_135_sum_Odelta__remove,axiom,
% 0.20/0.46 ! [A: $tType,B: $tType] :
% 0.20/0.46 ( ( comm_monoid_add @ A )
% 0.20/0.46 => ! [S: set @ B,A2: B,B2: B > A,C2: B > A] :
% 0.20/0.46 ( ( finite_finite2 @ B @ S )
% 0.20/0.46 => ( ( ( member @ B @ A2 @ S )
% 0.20/0.46 => ( ( groups1340683514dd_sum @ B @ A
% 0.20/0.46 @ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( C2 @ K2 ) )
% 0.20/0.46 @ S )
% 0.20/0.46 = ( plus_plus @ A @ ( B2 @ A2 ) @ ( groups1340683514dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
% 0.20/0.46 & ( ~ ( member @ B @ A2 @ S )
% 0.20/0.46 => ( ( groups1340683514dd_sum @ B @ A
% 0.20/0.46 @ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( C2 @ K2 ) )
% 0.20/0.46 @ S )
% 0.20/0.46 = ( groups1340683514dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum.delta_remove
% 0.20/0.46 thf(fact_136_empty__def,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( bot_bot @ ( set @ A ) )
% 0.20/0.46 = ( collect @ A
% 0.20/0.46 @ ^ [X: A] : $false ) ) ).
% 0.20/0.46
% 0.20/0.46 % empty_def
% 0.20/0.46 thf(fact_137_pigeonhole__infinite__rel,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,R2: A > B > $o] :
% 0.20/0.46 ( ~ ( finite_finite2 @ A @ A3 )
% 0.20/0.46 => ( ( finite_finite2 @ B @ B3 )
% 0.20/0.46 => ( ! [X3: A] :
% 0.20/0.46 ( ( member @ A @ X3 @ A3 )
% 0.20/0.46 => ? [Xa2: B] :
% 0.20/0.46 ( ( member @ B @ Xa2 @ B3 )
% 0.20/0.46 & ( R2 @ X3 @ Xa2 ) ) )
% 0.20/0.46 => ? [X3: B] :
% 0.20/0.46 ( ( member @ B @ X3 @ B3 )
% 0.20/0.46 & ~ ( finite_finite2 @ A
% 0.20/0.46 @ ( collect @ A
% 0.20/0.46 @ ^ [A4: A] :
% 0.20/0.46 ( ( member @ A @ A4 @ A3 )
% 0.20/0.46 & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % pigeonhole_infinite_rel
% 0.20/0.46 thf(fact_138_not__finite__existsD,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o] :
% 0.20/0.46 ( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.20/0.46 => ? [X_12: A] : ( P @ X_12 ) ) ).
% 0.20/0.46
% 0.20/0.46 % not_finite_existsD
% 0.20/0.46 thf(fact_139_insert__Collect,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,P: A > $o] :
% 0.20/0.46 ( ( insert @ A @ A2 @ ( collect @ A @ P ) )
% 0.20/0.46 = ( collect @ A
% 0.20/0.46 @ ^ [U2: A] :
% 0.20/0.46 ( ( U2 != A2 )
% 0.20/0.46 => ( P @ U2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % insert_Collect
% 0.20/0.46 thf(fact_140_insert__compr,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( insert @ A )
% 0.20/0.46 = ( ^ [A4: A,B4: set @ A] :
% 0.20/0.46 ( collect @ A
% 0.20/0.46 @ ^ [X: A] :
% 0.20/0.46 ( ( X = A4 )
% 0.20/0.46 | ( member @ A @ X @ B4 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % insert_compr
% 0.20/0.46 thf(fact_141_sum__delta__notmem_I4_J,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType] :
% 0.20/0.46 ( ( comm_monoid_add @ B )
% 0.20/0.46 => ! [X2: A,S2: set @ A,P: A > B,Q: A > B] :
% 0.20/0.46 ( ~ ( member @ A @ X2 @ S2 )
% 0.20/0.46 => ( ( groups1340683514dd_sum @ A @ B
% 0.20/0.46 @ ^ [Y4: A] : ( if @ B @ ( X2 = Y4 ) @ ( P @ Y4 ) @ ( Q @ Y4 ) )
% 0.20/0.46 @ S2 )
% 0.20/0.46 = ( groups1340683514dd_sum @ A @ B @ Q @ S2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum_delta_notmem(4)
% 0.20/0.46 thf(fact_142_sum__delta__notmem_I3_J,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType] :
% 0.20/0.46 ( ( comm_monoid_add @ B )
% 0.20/0.46 => ! [X2: A,S2: set @ A,P: A > B,Q: A > B] :
% 0.20/0.46 ( ~ ( member @ A @ X2 @ S2 )
% 0.20/0.46 => ( ( groups1340683514dd_sum @ A @ B
% 0.20/0.46 @ ^ [Y4: A] : ( if @ B @ ( Y4 = X2 ) @ ( P @ Y4 ) @ ( Q @ Y4 ) )
% 0.20/0.46 @ S2 )
% 0.20/0.46 = ( groups1340683514dd_sum @ A @ B @ Q @ S2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum_delta_notmem(3)
% 0.20/0.46 thf(fact_143_sum__delta__notmem_I2_J,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType] :
% 0.20/0.46 ( ( comm_monoid_add @ B )
% 0.20/0.46 => ! [X2: A,S2: set @ A,P: A > B,Q: A > B] :
% 0.20/0.46 ( ~ ( member @ A @ X2 @ S2 )
% 0.20/0.46 => ( ( groups1340683514dd_sum @ A @ B
% 0.20/0.46 @ ^ [Y4: A] : ( if @ B @ ( X2 = Y4 ) @ ( P @ X2 ) @ ( Q @ Y4 ) )
% 0.20/0.46 @ S2 )
% 0.20/0.46 = ( groups1340683514dd_sum @ A @ B @ Q @ S2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum_delta_notmem(2)
% 0.20/0.46 thf(fact_144_sum__delta__notmem_I1_J,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType] :
% 0.20/0.46 ( ( comm_monoid_add @ B )
% 0.20/0.46 => ! [X2: A,S2: set @ A,P: A > B,Q: A > B] :
% 0.20/0.46 ( ~ ( member @ A @ X2 @ S2 )
% 0.20/0.46 => ( ( groups1340683514dd_sum @ A @ B
% 0.20/0.46 @ ^ [Y4: A] : ( if @ B @ ( Y4 = X2 ) @ ( P @ X2 ) @ ( Q @ Y4 ) )
% 0.20/0.46 @ S2 )
% 0.20/0.46 = ( groups1340683514dd_sum @ A @ B @ Q @ S2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum_delta_notmem(1)
% 0.20/0.46 thf(fact_145_sum_Oswap,axiom,
% 0.20/0.46 ! [A: $tType,B: $tType,C: $tType] :
% 0.20/0.46 ( ( comm_monoid_add @ A )
% 0.20/0.46 => ! [G: B > C > A,B3: set @ C,A3: set @ B] :
% 0.20/0.46 ( ( groups1340683514dd_sum @ B @ A
% 0.20/0.46 @ ^ [I3: B] : ( groups1340683514dd_sum @ C @ A @ ( G @ I3 ) @ B3 )
% 0.20/0.46 @ A3 )
% 0.20/0.46 = ( groups1340683514dd_sum @ C @ A
% 0.20/0.46 @ ^ [J: C] :
% 0.20/0.46 ( groups1340683514dd_sum @ B @ A
% 0.20/0.46 @ ^ [I3: B] : ( G @ I3 @ J )
% 0.20/0.46 @ A3 )
% 0.20/0.46 @ B3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum.swap
% 0.20/0.46 thf(fact_146_infinite__imp__nonempty,axiom,
% 0.20/0.46 ! [A: $tType,S: set @ A] :
% 0.20/0.46 ( ~ ( finite_finite2 @ A @ S )
% 0.20/0.46 => ( S
% 0.20/0.46 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % infinite_imp_nonempty
% 0.20/0.46 thf(fact_147_finite_OemptyI,axiom,
% 0.20/0.46 ! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % finite.emptyI
% 0.20/0.46 thf(fact_148_singleton__inject,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,B2: A] :
% 0.20/0.46 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.20/0.46 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.46 => ( A2 = B2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % singleton_inject
% 0.20/0.46 thf(fact_149_insert__not__empty,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,A3: set @ A] :
% 0.20/0.46 ( ( insert @ A @ A2 @ A3 )
% 0.20/0.46 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % insert_not_empty
% 0.20/0.46 thf(fact_150_doubleton__eq__iff,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,B2: A,C2: A,D: A] :
% 0.20/0.46 ( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.46 = ( insert @ A @ C2 @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.20/0.46 = ( ( ( A2 = C2 )
% 0.20/0.46 & ( B2 = D ) )
% 0.20/0.46 | ( ( A2 = D )
% 0.20/0.46 & ( B2 = C2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % doubleton_eq_iff
% 0.20/0.46 thf(fact_151_singleton__iff,axiom,
% 0.20/0.46 ! [A: $tType,B2: A,A2: A] :
% 0.20/0.46 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.46 = ( B2 = A2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % singleton_iff
% 0.20/0.46 thf(fact_152_singletonD,axiom,
% 0.20/0.46 ! [A: $tType,B2: A,A2: A] :
% 0.20/0.46 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.46 => ( B2 = A2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % singletonD
% 0.20/0.46 thf(fact_153_finite_OinsertI,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,A2: A] :
% 0.20/0.46 ( ( finite_finite2 @ A @ A3 )
% 0.20/0.46 => ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % finite.insertI
% 0.20/0.46 thf(fact_154_sum__product,axiom,
% 0.20/0.46 ! [B: $tType,C: $tType,A: $tType] :
% 0.20/0.46 ( ( semiring_0 @ B )
% 0.20/0.46 => ! [F: A > B,A3: set @ A,G: C > B,B3: set @ C] :
% 0.20/0.46 ( ( times_times @ B @ ( groups1340683514dd_sum @ A @ B @ F @ A3 ) @ ( groups1340683514dd_sum @ C @ B @ G @ B3 ) )
% 0.20/0.46 = ( groups1340683514dd_sum @ A @ B
% 0.20/0.46 @ ^ [I3: A] :
% 0.20/0.46 ( groups1340683514dd_sum @ C @ B
% 0.20/0.46 @ ^ [J: C] : ( times_times @ B @ ( F @ I3 ) @ ( G @ J ) )
% 0.20/0.46 @ B3 )
% 0.20/0.46 @ A3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum_product
% 0.20/0.46 thf(fact_155_sum__distrib__left,axiom,
% 0.20/0.46 ! [A: $tType,B: $tType] :
% 0.20/0.46 ( ( semiring_0 @ A )
% 0.20/0.46 => ! [R: A,F: B > A,A3: set @ B] :
% 0.20/0.46 ( ( times_times @ A @ R @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) )
% 0.20/0.46 = ( groups1340683514dd_sum @ B @ A
% 0.20/0.46 @ ^ [N: B] : ( times_times @ A @ R @ ( F @ N ) )
% 0.20/0.46 @ A3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum_distrib_left
% 0.20/0.46 thf(fact_156_sum__distrib__right,axiom,
% 0.20/0.46 ! [A: $tType,B: $tType] :
% 0.20/0.46 ( ( semiring_0 @ A )
% 0.20/0.46 => ! [F: B > A,A3: set @ B,R: A] :
% 0.20/0.46 ( ( times_times @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ R )
% 0.20/0.46 = ( groups1340683514dd_sum @ B @ A
% 0.20/0.46 @ ^ [N: B] : ( times_times @ A @ ( F @ N ) @ R )
% 0.20/0.46 @ A3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum_distrib_right
% 0.20/0.46 thf(fact_157_sum_Odistrib,axiom,
% 0.20/0.46 ! [A: $tType,B: $tType] :
% 0.20/0.46 ( ( comm_monoid_add @ A )
% 0.20/0.46 => ! [G: B > A,H: B > A,A3: set @ B] :
% 0.20/0.46 ( ( groups1340683514dd_sum @ B @ A
% 0.20/0.46 @ ^ [X: B] : ( plus_plus @ A @ ( G @ X ) @ ( H @ X ) )
% 0.20/0.46 @ A3 )
% 0.20/0.46 = ( plus_plus @ A @ ( groups1340683514dd_sum @ B @ A @ G @ A3 ) @ ( groups1340683514dd_sum @ B @ A @ H @ A3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum.distrib
% 0.20/0.46 thf(fact_158_Collect__conv__if2,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o,A2: A] :
% 0.20/0.46 ( ( ( P @ A2 )
% 0.20/0.46 => ( ( collect @ A
% 0.20/0.46 @ ^ [X: A] :
% 0.20/0.46 ( ( A2 = X )
% 0.20/0.46 & ( P @ X ) ) )
% 0.20/0.46 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.20/0.46 & ( ~ ( P @ A2 )
% 0.20/0.46 => ( ( collect @ A
% 0.20/0.46 @ ^ [X: A] :
% 0.20/0.46 ( ( A2 = X )
% 0.20/0.46 & ( P @ X ) ) )
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Collect_conv_if2
% 0.20/0.46 thf(fact_159_Collect__conv__if,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o,A2: A] :
% 0.20/0.46 ( ( ( P @ A2 )
% 0.20/0.46 => ( ( collect @ A
% 0.20/0.46 @ ^ [X: A] :
% 0.20/0.46 ( ( X = A2 )
% 0.20/0.46 & ( P @ X ) ) )
% 0.20/0.46 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.20/0.46 & ( ~ ( P @ A2 )
% 0.20/0.46 => ( ( collect @ A
% 0.20/0.46 @ ^ [X: A] :
% 0.20/0.46 ( ( X = A2 )
% 0.20/0.46 & ( P @ X ) ) )
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Collect_conv_if
% 0.20/0.46 thf(fact_160_sum__subtractf,axiom,
% 0.20/0.46 ! [A: $tType,B: $tType] :
% 0.20/0.46 ( ( ab_group_add @ A )
% 0.20/0.46 => ! [F: B > A,G: B > A,A3: set @ B] :
% 0.20/0.46 ( ( groups1340683514dd_sum @ B @ A
% 0.20/0.46 @ ^ [X: B] : ( minus_minus @ A @ ( F @ X ) @ ( G @ X ) )
% 0.20/0.46 @ A3 )
% 0.20/0.46 = ( minus_minus @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ ( groups1340683514dd_sum @ B @ A @ G @ A3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sum_subtractf
% 0.20/0.46 thf(fact_161_sum_Oswap__restrict,axiom,
% 0.20/0.46 ! [A: $tType,B: $tType,C: $tType] :
% 0.20/0.46 ( ( comm_monoid_add @ A )
% 0.20/0.46 => ! [A3: set @ B,B3: set @ C,G: B > C > A,R2: B > C > $o] :
% 0.20/0.46 ( ( finite_finite2 @ B @ A3 )
% 0.20/0.46 => ( ( finite_finite2 @ C @ B3 )
% 0.20/0.46 => ( ( groups1340683514dd_sum @ B @ A
% 0.20/0.46 @ ^ [X: B] :
% 0.20/0.46 ( groups1340683514dd_sum @ C @ A @ ( G @ X )
% 0.20/0.46 @ ( collect @ C
% 0.20/0.47 @ ^ [Y4: C] :
% 0.20/0.47 ( ( member @ C @ Y4 @ B3 )
% 0.20/0.47 & ( R2 @ X @ Y4 ) ) ) )
% 0.20/0.47 @ A3 )
% 0.20/0.47 = ( groups1340683514dd_sum @ C @ A
% 0.20/0.47 @ ^ [Y4: C] :
% 0.20/0.47 ( groups1340683514dd_sum @ B @ A
% 0.20/0.47 @ ^ [X: B] : ( G @ X @ Y4 )
% 0.20/0.47 @ ( collect @ B
% 0.20/0.47 @ ^ [X: B] :
% 0.20/0.47 ( ( member @ B @ X @ A3 )
% 0.20/0.47 & ( R2 @ X @ Y4 ) ) ) )
% 0.20/0.47 @ B3 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % sum.swap_restrict
% 0.20/0.47 thf(fact_162_infinite__finite__induct,axiom,
% 0.20/0.47 ! [A: $tType,P: ( set @ A ) > $o,A3: set @ A] :
% 0.20/0.47 ( ! [A7: set @ A] :
% 0.20/0.47 ( ~ ( finite_finite2 @ A @ A7 )
% 0.20/0.47 => ( P @ A7 ) )
% 0.20/0.47 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.20/0.47 => ( ! [X3: A,F3: set @ A] :
% 0.20/0.47 ( ( finite_finite2 @ A @ F3 )
% 0.20/0.47 => ( ~ ( member @ A @ X3 @ F3 )
% 0.20/0.47 => ( ( P @ F3 )
% 0.20/0.47 => ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) )
% 0.20/0.47 => ( P @ A3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % infinite_finite_induct
% 0.20/0.47 thf(fact_163_finite__ne__induct,axiom,
% 0.20/0.47 ! [A: $tType,F4: set @ A,P: ( set @ A ) > $o] :
% 0.20/0.47 ( ( finite_finite2 @ A @ F4 )
% 0.20/0.47 => ( ( F4
% 0.20/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.20/0.47 => ( ! [X3: A] : ( P @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.47 => ( ! [X3: A,F3: set @ A] :
% 0.20/0.47 ( ( finite_finite2 @ A @ F3 )
% 0.20/0.47 => ( ( F3
% 0.20/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.20/0.47 => ( ~ ( member @ A @ X3 @ F3 )
% 0.20/0.47 => ( ( P @ F3 )
% 0.20/0.47 => ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) ) )
% 0.20/0.47 => ( P @ F4 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % finite_ne_induct
% 0.20/0.47 thf(fact_164_finite_Oinducts,axiom,
% 0.20/0.47 ! [A: $tType,X2: set @ A,P: ( set @ A ) > $o] :
% 0.20/0.47 ( ( finite_finite2 @ A @ X2 )
% 0.20/0.47 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.20/0.47 => ( ! [A7: set @ A,A6: A] :
% 0.20/0.47 ( ( finite_finite2 @ A @ A7 )
% 0.20/0.47 => ( ( P @ A7 )
% 0.20/0.47 => ( P @ ( insert @ A @ A6 @ A7 ) ) ) )
% 0.20/0.47 => ( P @ X2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % finite.inducts
% 0.20/0.47 thf(fact_165_finite__induct,axiom,
% 0.20/0.47 ! [A: $tType,F4: set @ A,P: ( set @ A ) > $o] :
% 0.20/0.47 ( ( finite_finite2 @ A @ F4 )
% 0.20/0.47 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.20/0.47 => ( ! [X3: A,F3: set @ A] :
% 0.20/0.47 ( ( finite_finite2 @ A @ F3 )
% 0.20/0.47 => ( ~ ( member @ A @ X3 @ F3 )
% 0.20/0.47 => ( ( P @ F3 )
% 0.20/0.47 => ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) )
% 0.20/0.47 => ( P @ F4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % finite_induct
% 0.20/0.47 thf(fact_166_finite_Osimps,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( finite_finite2 @ A )
% 0.20/0.47 = ( ^ [A4: set @ A] :
% 0.20/0.47 ( ( A4
% 0.20/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.20/0.47 | ? [A5: set @ A,B5: A] :
% 0.20/0.47 ( ( A4
% 0.20/0.47 = ( insert @ A @ B5 @ A5 ) )
% 0.20/0.47 & ( finite_finite2 @ A @ A5 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % finite.simps
% 0.20/0.47 thf(fact_167_finite_Ocases,axiom,
% 0.20/0.47 ! [A: $tType,A2: set @ A] :
% 0.20/0.47 ( ( finite_finite2 @ A @ A2 )
% 0.20/0.47 => ( ( A2
% 0.20/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.20/0.47 => ~ ! [A7: set @ A] :
% 0.20/0.47 ( ? [A6: A] :
% 0.20/0.47 ( A2
% 0.20/0.47 = ( insert @ A @ A6 @ A7 ) )
% 0.20/0.47 => ~ ( finite_finite2 @ A @ A7 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % finite.cases
% 0.20/0.47 thf(fact_168_prod_Ofinite__Collect__op,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType] :
% 0.20/0.47 ( ( comm_monoid_mult @ A )
% 0.20/0.47 => ! [I4: set @ B,X2: B > A,Y: B > A] :
% 0.20/0.47 ( ( finite_finite2 @ B
% 0.20/0.47 @ ( collect @ B
% 0.20/0.47 @ ^ [I3: B] :
% 0.20/0.47 ( ( member @ B @ I3 @ I4 )
% 0.20/0.47 & ( ( X2 @ I3 )
% 0.20/0.47 != ( one_one @ A ) ) ) ) )
% 0.20/0.47 => ( ( finite_finite2 @ B
% 0.20/0.47 @ ( collect @ B
% 0.20/0.47 @ ^ [I3: B] :
% 0.20/0.47 ( ( member @ B @ I3 @ I4 )
% 0.20/0.47 & ( ( Y @ I3 )
% 0.20/0.47 != ( one_one @ A ) ) ) ) )
% 0.20/0.47 => ( finite_finite2 @ B
% 0.20/0.47 @ ( collect @ B
% 0.20/0.47 @ ^ [I3: B] :
% 0.20/0.47 ( ( member @ B @ I3 @ I4 )
% 0.20/0.47 & ( ( times_times @ A @ ( X2 @ I3 ) @ ( Y @ I3 ) )
% 0.20/0.47 != ( one_one @ A ) ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod.finite_Collect_op
% 0.20/0.47 thf(fact_169_sum_Oinsert__if,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType] :
% 0.20/0.47 ( ( comm_monoid_add @ A )
% 0.20/0.47 => ! [A3: set @ B,X2: B,G: B > A] :
% 0.20/0.47 ( ( finite_finite2 @ B @ A3 )
% 0.20/0.47 => ( ( ( member @ B @ X2 @ A3 )
% 0.20/0.47 => ( ( groups1340683514dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
% 0.20/0.47 = ( groups1340683514dd_sum @ B @ A @ G @ A3 ) ) )
% 0.20/0.47 & ( ~ ( member @ B @ X2 @ A3 )
% 0.20/0.47 => ( ( groups1340683514dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
% 0.20/0.47 = ( plus_plus @ A @ ( G @ X2 ) @ ( groups1340683514dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % sum.insert_if
% 0.20/0.47 thf(fact_170_convex__singleton,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( real_V1076094709vector @ A )
% 0.20/0.47 => ! [A2: A] : ( convex @ A @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % convex_singleton
% 0.20/0.47 thf(fact_171_real__divide__square__eq,axiom,
% 0.20/0.47 ! [R: real,A2: real] :
% 0.20/0.47 ( ( divide_divide @ real @ ( times_times @ real @ R @ A2 ) @ ( times_times @ real @ R @ R ) )
% 0.20/0.47 = ( divide_divide @ real @ A2 @ R ) ) ).
% 0.20/0.47
% 0.20/0.47 % real_divide_square_eq
% 0.20/0.47 thf(fact_172_convex__empty,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( real_V1076094709vector @ A )
% 0.20/0.47 => ( convex @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % convex_empty
% 0.20/0.47 thf(fact_173_div__by__1,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semidom_divide @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
% 0.20/0.47 = A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % div_by_1
% 0.20/0.47 thf(fact_174_bits__div__by__1,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring_bits @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
% 0.20/0.47 = A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % bits_div_by_1
% 0.20/0.47 thf(fact_175_add__diff__cancel,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ! [A2: A,B2: A] :
% 0.20/0.47 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 0.20/0.47 = A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_diff_cancel
% 0.20/0.47 thf(fact_176_diff__add__cancel,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ! [A2: A,B2: A] :
% 0.20/0.47 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
% 0.20/0.47 = A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % diff_add_cancel
% 0.20/0.47 thf(fact_177_add__right__cancel,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( cancel_semigroup_add @ A )
% 0.20/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.20/0.47 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.20/0.47 = ( plus_plus @ A @ C2 @ A2 ) )
% 0.20/0.47 = ( B2 = C2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_right_cancel
% 0.20/0.47 thf(fact_178_add__left__cancel,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( cancel_semigroup_add @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.20/0.47 = ( plus_plus @ A @ A2 @ C2 ) )
% 0.20/0.47 = ( B2 = C2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_left_cancel
% 0.20/0.47 thf(fact_179_mult_Oright__neutral,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( monoid_mult @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( times_times @ A @ A2 @ ( one_one @ A ) )
% 0.20/0.47 = A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % mult.right_neutral
% 0.20/0.47 thf(fact_180_mult_Oleft__neutral,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( monoid_mult @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( times_times @ A @ ( one_one @ A ) @ A2 )
% 0.20/0.47 = A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % mult.left_neutral
% 0.20/0.47 thf(fact_181_add__diff__cancel__right_H,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( cancel146912293up_add @ A )
% 0.20/0.47 => ! [A2: A,B2: A] :
% 0.20/0.47 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 0.20/0.47 = A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_diff_cancel_right'
% 0.20/0.47 thf(fact_182_add__diff__cancel__right,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( cancel146912293up_add @ A )
% 0.20/0.47 => ! [A2: A,C2: A,B2: A] :
% 0.20/0.47 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.20/0.47 = ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_diff_cancel_right
% 0.20/0.47 thf(fact_183_add__diff__cancel__left_H,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( cancel146912293up_add @ A )
% 0.20/0.47 => ! [A2: A,B2: A] :
% 0.20/0.47 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 )
% 0.20/0.47 = B2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_diff_cancel_left'
% 0.20/0.47 thf(fact_184_add__diff__cancel__left,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( cancel146912293up_add @ A )
% 0.20/0.47 => ! [C2: A,A2: A,B2: A] :
% 0.20/0.47 ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 0.20/0.47 = ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_diff_cancel_left
% 0.20/0.47 thf(fact_185_mult_Oleft__commute,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ab_semigroup_mult @ A )
% 0.20/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.20/0.47 ( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) )
% 0.20/0.47 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % mult.left_commute
% 0.20/0.47 thf(fact_186_mult_Ocommute,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ab_semigroup_mult @ A )
% 0.20/0.47 => ( ( times_times @ A )
% 0.20/0.47 = ( ^ [A4: A,B5: A] : ( times_times @ A @ B5 @ A4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % mult.commute
% 0.20/0.47 thf(fact_187_mult_Oassoc,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semigroup_mult @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
% 0.20/0.47 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % mult.assoc
% 0.20/0.47 thf(fact_188_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ab_semigroup_mult @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
% 0.20/0.47 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ab_semigroup_mult_class.mult_ac(1)
% 0.20/0.47 thf(fact_189_add__right__imp__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( cancel_semigroup_add @ A )
% 0.20/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.20/0.47 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.20/0.47 = ( plus_plus @ A @ C2 @ A2 ) )
% 0.20/0.47 => ( B2 = C2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_right_imp_eq
% 0.20/0.47 thf(fact_190_add__left__imp__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( cancel_semigroup_add @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.20/0.47 = ( plus_plus @ A @ A2 @ C2 ) )
% 0.20/0.47 => ( B2 = C2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_left_imp_eq
% 0.20/0.47 thf(fact_191_add_Oleft__commute,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ab_semigroup_add @ A )
% 0.20/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.20/0.47 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
% 0.20/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add.left_commute
% 0.20/0.47 thf(fact_192_add_Ocommute,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ab_semigroup_add @ A )
% 0.20/0.47 => ( ( plus_plus @ A )
% 0.20/0.47 = ( ^ [A4: A,B5: A] : ( plus_plus @ A @ B5 @ A4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add.commute
% 0.20/0.47 thf(fact_193_add_Oright__cancel,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.20/0.47 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.20/0.47 = ( plus_plus @ A @ C2 @ A2 ) )
% 0.20/0.47 = ( B2 = C2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add.right_cancel
% 0.20/0.47 thf(fact_194_add_Oleft__cancel,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.20/0.47 = ( plus_plus @ A @ A2 @ C2 ) )
% 0.20/0.47 = ( B2 = C2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add.left_cancel
% 0.20/0.47 thf(fact_195_add_Oassoc,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semigroup_add @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 0.20/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add.assoc
% 0.20/0.47 thf(fact_196_group__cancel_Oadd2,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( comm_monoid_add @ A )
% 0.20/0.47 => ! [B3: A,K: A,B2: A,A2: A] :
% 0.20/0.47 ( ( B3
% 0.20/0.47 = ( plus_plus @ A @ K @ B2 ) )
% 0.20/0.47 => ( ( plus_plus @ A @ A2 @ B3 )
% 0.20/0.47 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % group_cancel.add2
% 0.20/0.47 thf(fact_197_group__cancel_Oadd1,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( comm_monoid_add @ A )
% 0.20/0.47 => ! [A3: A,K: A,A2: A,B2: A] :
% 0.20/0.47 ( ( A3
% 0.20/0.47 = ( plus_plus @ A @ K @ A2 ) )
% 0.20/0.47 => ( ( plus_plus @ A @ A3 @ B2 )
% 0.20/0.47 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % group_cancel.add1
% 0.20/0.47 thf(fact_198_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ordere779506340up_add @ A )
% 0.20/0.47 => ! [I2: A,J2: A,K: A,L: A] :
% 0.20/0.47 ( ( ( I2 = J2 )
% 0.20/0.47 & ( K = L ) )
% 0.20/0.47 => ( ( plus_plus @ A @ I2 @ K )
% 0.20/0.47 = ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_mono_thms_linordered_semiring(4)
% 0.20/0.47 thf(fact_199_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ab_semigroup_add @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 0.20/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ab_semigroup_add_class.add_ac(1)
% 0.20/0.47 thf(fact_200_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( cancel146912293up_add @ A )
% 0.20/0.47 => ! [A2: A,C2: A,B2: A] :
% 0.20/0.47 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 )
% 0.20/0.47 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % cancel_ab_semigroup_add_class.diff_right_commute
% 0.20/0.47 thf(fact_201_diff__eq__diff__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A,D: A] :
% 0.20/0.47 ( ( ( minus_minus @ A @ A2 @ B2 )
% 0.20/0.47 = ( minus_minus @ A @ C2 @ D ) )
% 0.20/0.47 => ( ( A2 = B2 )
% 0.20/0.47 = ( C2 = D ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % diff_eq_diff_eq
% 0.20/0.47 thf(fact_202_one__reorient,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( one @ A )
% 0.20/0.47 => ! [X2: A] :
% 0.20/0.47 ( ( ( one_one @ A )
% 0.20/0.47 = X2 )
% 0.20/0.47 = ( X2
% 0.20/0.47 = ( one_one @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % one_reorient
% 0.20/0.47 thf(fact_203_convex__set__plus,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( real_V1076094709vector @ A )
% 0.20/0.47 => ! [S: set @ A,T: set @ A] :
% 0.20/0.47 ( ( convex @ A @ S )
% 0.20/0.47 => ( ( convex @ A @ T )
% 0.20/0.47 => ( convex @ A @ ( plus_plus @ ( set @ A ) @ S @ T ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % convex_set_plus
% 0.20/0.47 thf(fact_204_finite__set__sum,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( comm_monoid_add @ B )
% 0.20/0.47 => ! [A3: set @ A,B3: A > ( set @ B )] :
% 0.20/0.47 ( ( finite_finite2 @ A @ A3 )
% 0.20/0.47 => ( ! [X3: A] :
% 0.20/0.47 ( ( member @ A @ X3 @ A3 )
% 0.20/0.47 => ( finite_finite2 @ B @ ( B3 @ X3 ) ) )
% 0.20/0.47 => ( finite_finite2 @ B @ ( groups1340683514dd_sum @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % finite_set_sum
% 0.20/0.47 thf(fact_205_convex__set__sum,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( real_V1076094709vector @ B )
% 0.20/0.47 => ! [A3: set @ A,B3: A > ( set @ B )] :
% 0.20/0.47 ( ! [I5: A] :
% 0.20/0.47 ( ( member @ A @ I5 @ A3 )
% 0.20/0.47 => ( convex @ B @ ( B3 @ I5 ) ) )
% 0.20/0.47 => ( convex @ B @ ( groups1340683514dd_sum @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % convex_set_sum
% 0.20/0.47 thf(fact_206_combine__common__factor,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring @ A )
% 0.20/0.47 => ! [A2: A,E: A,B2: A,C2: A] :
% 0.20/0.47 ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ C2 ) )
% 0.20/0.47 = ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ E ) @ C2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % combine_common_factor
% 0.20/0.47 thf(fact_207_distrib__right,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 0.20/0.47 = ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % distrib_right
% 0.20/0.47 thf(fact_208_distrib__left,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.20/0.47 = ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % distrib_left
% 0.20/0.47 thf(fact_209_comm__semiring__class_Odistrib,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( comm_semiring @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 0.20/0.47 = ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comm_semiring_class.distrib
% 0.20/0.47 thf(fact_210_ring__class_Oring__distribs_I1_J,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ring @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.20/0.47 = ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ring_class.ring_distribs(1)
% 0.20/0.47 thf(fact_211_ring__class_Oring__distribs_I2_J,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ring @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 0.20/0.47 = ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ring_class.ring_distribs(2)
% 0.20/0.47 thf(fact_212_right__diff__distrib_H,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( comm_s1003936772cancel @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
% 0.20/0.47 = ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % right_diff_distrib'
% 0.20/0.47 thf(fact_213_left__diff__distrib_H,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( comm_s1003936772cancel @ A )
% 0.20/0.47 => ! [B2: A,C2: A,A2: A] :
% 0.20/0.47 ( ( times_times @ A @ ( minus_minus @ A @ B2 @ C2 ) @ A2 )
% 0.20/0.47 = ( minus_minus @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C2 @ A2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % left_diff_distrib'
% 0.20/0.47 thf(fact_214_right__diff__distrib,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ring @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
% 0.20/0.47 = ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % right_diff_distrib
% 0.20/0.47 thf(fact_215_left__diff__distrib,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ring @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
% 0.20/0.47 = ( minus_minus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % left_diff_distrib
% 0.20/0.47 thf(fact_216_mult_Ocomm__neutral,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( comm_monoid_mult @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( times_times @ A @ A2 @ ( one_one @ A ) )
% 0.20/0.47 = A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % mult.comm_neutral
% 0.20/0.47 thf(fact_217_comm__monoid__mult__class_Omult__1,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( comm_monoid_mult @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( times_times @ A @ ( one_one @ A ) @ A2 )
% 0.20/0.47 = A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % comm_monoid_mult_class.mult_1
% 0.20/0.47 thf(fact_218_add__implies__diff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( cancel1352612707id_add @ A )
% 0.20/0.47 => ! [C2: A,B2: A,A2: A] :
% 0.20/0.47 ( ( ( plus_plus @ A @ C2 @ B2 )
% 0.20/0.47 = A2 )
% 0.20/0.47 => ( C2
% 0.20/0.47 = ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_implies_diff
% 0.20/0.47 thf(fact_219_diff__diff__add,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( cancel146912293up_add @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
% 0.20/0.47 = ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % diff_diff_add
% 0.20/0.47 thf(fact_220_diff__add__eq__diff__diff__swap,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.20/0.47 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % diff_add_eq_diff_diff_swap
% 0.20/0.47 thf(fact_221_diff__add__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ab_group_add @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
% 0.20/0.47 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % diff_add_eq
% 0.20/0.47 thf(fact_222_diff__diff__eq2,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
% 0.20/0.47 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % diff_diff_eq2
% 0.20/0.47 thf(fact_223_add__diff__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
% 0.20/0.47 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_diff_eq
% 0.20/0.47 thf(fact_224_eq__diff__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ! [A2: A,C2: A,B2: A] :
% 0.20/0.47 ( ( A2
% 0.20/0.47 = ( minus_minus @ A @ C2 @ B2 ) )
% 0.20/0.47 = ( ( plus_plus @ A @ A2 @ B2 )
% 0.20/0.47 = C2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % eq_diff_eq
% 0.20/0.47 thf(fact_225_diff__eq__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.20/0.47 ( ( ( minus_minus @ A @ A2 @ B2 )
% 0.20/0.47 = C2 )
% 0.20/0.47 = ( A2
% 0.20/0.47 = ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % diff_eq_eq
% 0.20/0.47 thf(fact_226_group__cancel_Osub1,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ab_group_add @ A )
% 0.20/0.47 => ! [A3: A,K: A,A2: A,B2: A] :
% 0.20/0.47 ( ( A3
% 0.20/0.47 = ( plus_plus @ A @ K @ A2 ) )
% 0.20/0.47 => ( ( minus_minus @ A @ A3 @ B2 )
% 0.20/0.47 = ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % group_cancel.sub1
% 0.20/0.47 thf(fact_227_lambda__one,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( monoid_mult @ A )
% 0.20/0.47 => ( ( ^ [X: A] : X )
% 0.20/0.47 = ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % lambda_one
% 0.20/0.47 thf(fact_228_square__diff__square__factored,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( comm_ring @ A )
% 0.20/0.47 => ! [X2: A,Y: A] :
% 0.20/0.47 ( ( minus_minus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y @ Y ) )
% 0.20/0.47 = ( times_times @ A @ ( plus_plus @ A @ X2 @ Y ) @ ( minus_minus @ A @ X2 @ Y ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % square_diff_square_factored
% 0.20/0.47 thf(fact_229_eq__add__iff2,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ring @ A )
% 0.20/0.47 => ! [A2: A,E: A,C2: A,B2: A,D: A] :
% 0.20/0.47 ( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 )
% 0.20/0.47 = ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D ) )
% 0.20/0.47 = ( C2
% 0.20/0.47 = ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E ) @ D ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % eq_add_iff2
% 0.20/0.47 thf(fact_230_eq__add__iff1,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ring @ A )
% 0.20/0.47 => ! [A2: A,E: A,C2: A,B2: A,D: A] :
% 0.20/0.47 ( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 )
% 0.20/0.47 = ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D ) )
% 0.20/0.47 = ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E ) @ C2 )
% 0.20/0.47 = D ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % eq_add_iff1
% 0.20/0.47 thf(fact_231_square__diff__one__factored,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ring_1 @ A )
% 0.20/0.47 => ! [X2: A] :
% 0.20/0.47 ( ( minus_minus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( one_one @ A ) )
% 0.20/0.47 = ( times_times @ A @ ( plus_plus @ A @ X2 @ ( one_one @ A ) ) @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % square_diff_one_factored
% 0.20/0.47 thf(fact_232_minus__apply,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( minus @ B )
% 0.20/0.47 => ( ( minus_minus @ ( A > B ) )
% 0.20/0.47 = ( ^ [A5: A > B,B4: A > B,X: A] : ( minus_minus @ B @ ( A5 @ X ) @ ( B4 @ X ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % minus_apply
% 0.20/0.47 thf(fact_233_vector__space__over__itself_Oscale__one,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( field @ A )
% 0.20/0.47 => ! [X2: A] :
% 0.20/0.47 ( ( times_times @ A @ ( one_one @ A ) @ X2 )
% 0.20/0.47 = X2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % vector_space_over_itself.scale_one
% 0.20/0.47 thf(fact_234_segment__degen__1,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( real_V1076094709vector @ A )
% 0.20/0.47 => ! [U: real,A2: A,B2: A] :
% 0.20/0.47 ( ( ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ ( minus_minus @ real @ ( one_one @ real ) @ U ) @ A2 ) @ ( real_V1908273582scaleR @ A @ U @ B2 ) )
% 0.20/0.47 = B2 )
% 0.20/0.47 = ( ( A2 = B2 )
% 0.20/0.47 | ( U
% 0.20/0.47 = ( one_one @ real ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % segment_degen_1
% 0.20/0.47 thf(fact_235_vector__space__over__itself_Oscale__scale,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( field @ A )
% 0.20/0.47 => ! [A2: A,B2: A,X2: A] :
% 0.20/0.47 ( ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ X2 ) )
% 0.20/0.47 = ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ X2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % vector_space_over_itself.scale_scale
% 0.20/0.47 thf(fact_236_vector__space__over__itself_Oscale__left__commute,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( field @ A )
% 0.20/0.47 => ! [A2: A,B2: A,X2: A] :
% 0.20/0.47 ( ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ X2 ) )
% 0.20/0.47 = ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ X2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % vector_space_over_itself.scale_left_commute
% 0.20/0.47 thf(fact_237_vector__space__over__itself_Oscale__left__distrib,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( field @ A )
% 0.20/0.47 => ! [A2: A,B2: A,X2: A] :
% 0.20/0.47 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ X2 )
% 0.20/0.47 = ( plus_plus @ A @ ( times_times @ A @ A2 @ X2 ) @ ( times_times @ A @ B2 @ X2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % vector_space_over_itself.scale_left_distrib
% 0.20/0.47 thf(fact_238_vector__space__over__itself_Oscale__right__distrib,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( field @ A )
% 0.20/0.47 => ! [A2: A,X2: A,Y: A] :
% 0.20/0.47 ( ( times_times @ A @ A2 @ ( plus_plus @ A @ X2 @ Y ) )
% 0.20/0.47 = ( plus_plus @ A @ ( times_times @ A @ A2 @ X2 ) @ ( times_times @ A @ A2 @ Y ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % vector_space_over_itself.scale_right_distrib
% 0.20/0.47 thf(fact_239_vector__space__over__itself_Oscale__left__diff__distrib,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( field @ A )
% 0.20/0.47 => ! [A2: A,B2: A,X2: A] :
% 0.20/0.47 ( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ X2 )
% 0.20/0.47 = ( minus_minus @ A @ ( times_times @ A @ A2 @ X2 ) @ ( times_times @ A @ B2 @ X2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % vector_space_over_itself.scale_left_diff_distrib
% 0.20/0.47 thf(fact_240_vector__space__over__itself_Oscale__right__diff__distrib,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( field @ A )
% 0.20/0.47 => ! [A2: A,X2: A,Y: A] :
% 0.20/0.47 ( ( times_times @ A @ A2 @ ( minus_minus @ A @ X2 @ Y ) )
% 0.20/0.47 = ( minus_minus @ A @ ( times_times @ A @ A2 @ X2 ) @ ( times_times @ A @ A2 @ Y ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % vector_space_over_itself.scale_right_diff_distrib
% 0.20/0.47 thf(fact_241_add__scaleR__degen,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( real_V1076094709vector @ A )
% 0.20/0.47 => ! [U: real,B2: A,V2: real,A2: A] :
% 0.20/0.47 ( ( ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ U @ B2 ) @ ( real_V1908273582scaleR @ A @ V2 @ A2 ) )
% 0.20/0.47 = ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ U @ A2 ) @ ( real_V1908273582scaleR @ A @ V2 @ B2 ) ) )
% 0.20/0.47 => ( ( U != V2 )
% 0.20/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_scaleR_degen
% 0.20/0.47 thf(fact_242_vector__space__over__itself_Oscale__sum__left,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType] :
% 0.20/0.47 ( ( field @ A )
% 0.20/0.47 => ! [F: C > A,A3: set @ C,X2: A] :
% 0.20/0.47 ( ( times_times @ A @ ( groups1340683514dd_sum @ C @ A @ F @ A3 ) @ X2 )
% 0.20/0.47 = ( groups1340683514dd_sum @ C @ A
% 0.20/0.47 @ ^ [A4: C] : ( times_times @ A @ ( F @ A4 ) @ X2 )
% 0.20/0.47 @ A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % vector_space_over_itself.scale_sum_left
% 0.20/0.47 thf(fact_243_vector__space__over__itself_Oscale__sum__right,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType] :
% 0.20/0.47 ( ( field @ A )
% 0.20/0.47 => ! [A2: A,F: C > A,A3: set @ C] :
% 0.20/0.47 ( ( times_times @ A @ A2 @ ( groups1340683514dd_sum @ C @ A @ F @ A3 ) )
% 0.20/0.47 = ( groups1340683514dd_sum @ C @ A
% 0.20/0.47 @ ^ [X: C] : ( times_times @ A @ A2 @ ( F @ X ) )
% 0.20/0.47 @ A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % vector_space_over_itself.scale_sum_right
% 0.20/0.47 thf(fact_244_fun__diff__def,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( minus @ B )
% 0.20/0.47 => ( ( minus_minus @ ( A > B ) )
% 0.20/0.47 = ( ^ [A5: A > B,B4: A > B,X: A] : ( minus_minus @ B @ ( A5 @ X ) @ ( B4 @ X ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % fun_diff_def
% 0.20/0.47 thf(fact_245_vector__space__over__itself_Ofinite__Basis,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( field @ A )
% 0.20/0.47 => ( finite_finite2 @ A @ ( insert @ A @ ( one_one @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % vector_space_over_itself.finite_Basis
% 0.20/0.47 thf(fact_246_set__plus__intro,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( plus @ A )
% 0.20/0.47 => ! [A2: A,C4: set @ A,B2: A,D2: set @ A] :
% 0.20/0.47 ( ( member @ A @ A2 @ C4 )
% 0.20/0.47 => ( ( member @ A @ B2 @ D2 )
% 0.20/0.47 => ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ ( set @ A ) @ C4 @ D2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % set_plus_intro
% 0.20/0.47 thf(fact_247_sum__clauses_I2_J,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType] :
% 0.20/0.47 ( ( comm_monoid_add @ A )
% 0.20/0.47 => ! [S: set @ B,X2: B,F: B > A] :
% 0.20/0.47 ( ( finite_finite2 @ B @ S )
% 0.20/0.47 => ( ( ( member @ B @ X2 @ S )
% 0.20/0.47 => ( ( groups1340683514dd_sum @ B @ A @ F @ ( insert @ B @ X2 @ S ) )
% 0.20/0.47 = ( groups1340683514dd_sum @ B @ A @ F @ S ) ) )
% 0.20/0.47 & ( ~ ( member @ B @ X2 @ S )
% 0.20/0.47 => ( ( groups1340683514dd_sum @ B @ A @ F @ ( insert @ B @ X2 @ S ) )
% 0.20/0.47 = ( plus_plus @ A @ ( F @ X2 ) @ ( groups1340683514dd_sum @ B @ A @ F @ S ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % sum_clauses(2)
% 0.20/0.47 thf(fact_248_set__times__intro,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( times @ A )
% 0.20/0.47 => ! [A2: A,C4: set @ A,B2: A,D2: set @ A] :
% 0.20/0.47 ( ( member @ A @ A2 @ C4 )
% 0.20/0.47 => ( ( member @ A @ B2 @ D2 )
% 0.20/0.47 => ( member @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ ( set @ A ) @ C4 @ D2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % set_times_intro
% 0.20/0.47 thf(fact_249_set__times__elim,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( times @ A )
% 0.20/0.47 => ! [X2: A,A3: set @ A,B3: set @ A] :
% 0.20/0.47 ( ( member @ A @ X2 @ ( times_times @ ( set @ A ) @ A3 @ B3 ) )
% 0.20/0.47 => ~ ! [A6: A,B7: A] :
% 0.20/0.47 ( ( X2
% 0.20/0.47 = ( times_times @ A @ A6 @ B7 ) )
% 0.20/0.47 => ( ( member @ A @ A6 @ A3 )
% 0.20/0.47 => ~ ( member @ A @ B7 @ B3 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % set_times_elim
% 0.20/0.47 thf(fact_250_finite__set__times,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( times @ A )
% 0.20/0.47 => ! [S2: set @ A,T2: set @ A] :
% 0.20/0.47 ( ( finite_finite2 @ A @ S2 )
% 0.20/0.47 => ( ( finite_finite2 @ A @ T2 )
% 0.20/0.47 => ( finite_finite2 @ A @ ( times_times @ ( set @ A ) @ S2 @ T2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % finite_set_times
% 0.20/0.47 thf(fact_251_set__plus__elim,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( plus @ A )
% 0.20/0.47 => ! [X2: A,A3: set @ A,B3: set @ A] :
% 0.20/0.47 ( ( member @ A @ X2 @ ( plus_plus @ ( set @ A ) @ A3 @ B3 ) )
% 0.20/0.47 => ~ ! [A6: A,B7: A] :
% 0.20/0.47 ( ( X2
% 0.20/0.47 = ( plus_plus @ A @ A6 @ B7 ) )
% 0.20/0.47 => ( ( member @ A @ A6 @ A3 )
% 0.20/0.47 => ~ ( member @ A @ B7 @ B3 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % set_plus_elim
% 0.20/0.47 thf(fact_252_finite__set__plus,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( plus @ A )
% 0.20/0.47 => ! [S2: set @ A,T2: set @ A] :
% 0.20/0.47 ( ( finite_finite2 @ A @ S2 )
% 0.20/0.47 => ( ( finite_finite2 @ A @ T2 )
% 0.20/0.47 => ( finite_finite2 @ A @ ( plus_plus @ ( set @ A ) @ S2 @ T2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % finite_set_plus
% 0.20/0.47 thf(fact_253_set__one,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( one @ A )
% 0.20/0.47 => ( ( one_one @ ( set @ A ) )
% 0.20/0.47 = ( insert @ A @ ( one_one @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % set_one
% 0.20/0.47 thf(fact_254_ai1,axiom,
% 0.20/0.47 ord_less_eq @ real @ ( aa @ i ) @ ( one_one @ real ) ).
% 0.20/0.47
% 0.20/0.47 % ai1
% 0.20/0.47
% 0.20/0.47 % Subclasses (12)
% 0.20/0.47 thf(subcl_Real__Vector__Spaces_Oreal__vector___HOL_Otype,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( real_V1076094709vector @ A )
% 0.20/0.47 => ( type @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Oplus,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( real_V1076094709vector @ A )
% 0.20/0.47 => ( plus @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ominus,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( real_V1076094709vector @ A )
% 0.20/0.47 => ( minus @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ogroup__add,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( real_V1076094709vector @ A )
% 0.20/0.47 => ( group_add @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Oab__group__add,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( real_V1076094709vector @ A )
% 0.20/0.47 => ( ab_group_add @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Osemigroup__add,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( real_V1076094709vector @ A )
% 0.20/0.47 => ( semigroup_add @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ocomm__monoid__add,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( real_V1076094709vector @ A )
% 0.20/0.47 => ( comm_monoid_add @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Oab__semigroup__add,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( real_V1076094709vector @ A )
% 0.20/0.47 => ( ab_semigroup_add @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subcl_Real__Vector__Spaces_Oreal__vector___Real__Vector__Spaces_OscaleR,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( real_V1076094709vector @ A )
% 0.20/0.47 => ( real_Vector_scaleR @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ocancel__semigroup__add,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( real_V1076094709vector @ A )
% 0.20/0.47 => ( cancel_semigroup_add @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ocancel__comm__monoid__add,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( real_V1076094709vector @ A )
% 0.20/0.47 => ( cancel1352612707id_add @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( real_V1076094709vector @ A )
% 0.20/0.47 => ( cancel146912293up_add @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 % Type constructors (80)
% 0.20/0.47 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.20/0.47 ! [A8: $tType,A9: $tType] :
% 0.20/0.47 ( ( ( finite_finite @ A8 )
% 0.20/0.47 & ( finite_finite @ A9 ) )
% 0.20/0.47 => ( finite_finite @ ( A8 > A9 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Groups_Ominus,axiom,
% 0.20/0.47 ! [A8: $tType,A9: $tType] :
% 0.20/0.47 ( ( minus @ A9 )
% 0.20/0.47 => ( minus @ ( A8 > A9 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 0.20/0.47 ordere779506340up_add @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.20/0.47 cancel146912293up_add @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 0.20/0.47 cancel1352612707id_add @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel,axiom,
% 0.20/0.47 comm_s1003936772cancel @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 0.20/0.47 cancel_semigroup_add @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
% 0.20/0.47 ab_semigroup_mult @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
% 0.20/0.47 comm_monoid_mult @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 0.20/0.47 ab_semigroup_add @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 0.20/0.47 comm_monoid_add @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Groups_Osemigroup__mult,axiom,
% 0.20/0.47 semigroup_mult @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Rings_Osemidom__divide,axiom,
% 0.20/0.47 semidom_divide @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Parity_Osemiring__bits,axiom,
% 0.20/0.47 semiring_bits @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 0.20/0.47 semigroup_add @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Rings_Ocomm__semiring,axiom,
% 0.20/0.47 comm_semiring @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Groups_Omonoid__mult,axiom,
% 0.20/0.47 monoid_mult @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Rings_Osemiring__0,axiom,
% 0.20/0.47 semiring_0 @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Rings_Osemiring,axiom,
% 0.20/0.47 semiring @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Groups_Otimes,axiom,
% 0.20/0.47 times @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Groups_Ominus_1,axiom,
% 0.20/0.47 minus @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Groups_Oplus,axiom,
% 0.20/0.47 plus @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Groups_Oone,axiom,
% 0.20/0.47 one @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Groups_Oab__semigroup__mult_2,axiom,
% 0.20/0.47 ! [A8: $tType] :
% 0.20/0.47 ( ( ab_semigroup_mult @ A8 )
% 0.20/0.47 => ( ab_semigroup_mult @ ( set @ A8 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Groups_Ocomm__monoid__mult_3,axiom,
% 0.20/0.47 ! [A8: $tType] :
% 0.20/0.47 ( ( comm_monoid_mult @ A8 )
% 0.20/0.47 => ( comm_monoid_mult @ ( set @ A8 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Groups_Oab__semigroup__add_4,axiom,
% 0.20/0.47 ! [A8: $tType] :
% 0.20/0.47 ( ( ab_semigroup_add @ A8 )
% 0.20/0.47 => ( ab_semigroup_add @ ( set @ A8 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Groups_Ocomm__monoid__add_5,axiom,
% 0.20/0.47 ! [A8: $tType] :
% 0.20/0.47 ( ( comm_monoid_add @ A8 )
% 0.20/0.47 => ( comm_monoid_add @ ( set @ A8 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Groups_Osemigroup__mult_6,axiom,
% 0.20/0.47 ! [A8: $tType] :
% 0.20/0.47 ( ( semigroup_mult @ A8 )
% 0.20/0.47 => ( semigroup_mult @ ( set @ A8 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Groups_Osemigroup__add_7,axiom,
% 0.20/0.47 ! [A8: $tType] :
% 0.20/0.47 ( ( semigroup_add @ A8 )
% 0.20/0.47 => ( semigroup_add @ ( set @ A8 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Groups_Omonoid__mult_8,axiom,
% 0.20/0.47 ! [A8: $tType] :
% 0.20/0.47 ( ( monoid_mult @ A8 )
% 0.20/0.47 => ( monoid_mult @ ( set @ A8 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Finite__Set_Ofinite_9,axiom,
% 0.20/0.47 ! [A8: $tType] :
% 0.20/0.47 ( ( finite_finite @ A8 )
% 0.20/0.47 => ( finite_finite @ ( set @ A8 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Groups_Otimes_10,axiom,
% 0.20/0.47 ! [A8: $tType] :
% 0.20/0.47 ( ( times @ A8 )
% 0.20/0.47 => ( times @ ( set @ A8 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Groups_Ominus_11,axiom,
% 0.20/0.47 ! [A8: $tType] : ( minus @ ( set @ A8 ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Groups_Oplus_12,axiom,
% 0.20/0.47 ! [A8: $tType] :
% 0.20/0.47 ( ( plus @ A8 )
% 0.20/0.47 => ( plus @ ( set @ A8 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Groups_Oone_13,axiom,
% 0.20/0.47 ! [A8: $tType] :
% 0.20/0.47 ( ( one @ A8 )
% 0.20/0.47 => ( one @ ( set @ A8 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_HOL_Obool___Finite__Set_Ofinite_14,axiom,
% 0.20/0.47 finite_finite @ $o ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_HOL_Obool___Groups_Ominus_15,axiom,
% 0.20/0.47 minus @ $o ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra,axiom,
% 0.20/0.47 real_V1015715145lgebra @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra,axiom,
% 0.20/0.47 real_V148923926lgebra @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_16,axiom,
% 0.20/0.47 ordere779506340up_add @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector,axiom,
% 0.20/0.47 real_V1076094709vector @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_17,axiom,
% 0.20/0.47 cancel146912293up_add @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_18,axiom,
% 0.20/0.47 cancel1352612707id_add @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_19,axiom,
% 0.20/0.47 comm_s1003936772cancel @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_20,axiom,
% 0.20/0.47 cancel_semigroup_add @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Real__Vector__Spaces_OscaleR,axiom,
% 0.20/0.47 real_Vector_scaleR @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_21,axiom,
% 0.20/0.47 ab_semigroup_mult @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_22,axiom,
% 0.20/0.47 comm_monoid_mult @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_23,axiom,
% 0.20/0.47 ab_semigroup_add @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_24,axiom,
% 0.20/0.47 comm_monoid_add @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Osemigroup__mult_25,axiom,
% 0.20/0.47 semigroup_mult @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Rings_Osemidom__divide_26,axiom,
% 0.20/0.47 semidom_divide @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Osemigroup__add_27,axiom,
% 0.20/0.47 semigroup_add @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
% 0.20/0.47 division_ring @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Rings_Ocomm__semiring_28,axiom,
% 0.20/0.47 comm_semiring @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Oab__group__add,axiom,
% 0.20/0.47 ab_group_add @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Omonoid__mult_29,axiom,
% 0.20/0.47 monoid_mult @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Rings_Osemiring__0_30,axiom,
% 0.20/0.47 semiring_0 @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
% 0.20/0.47 group_add @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Rings_Ocomm__ring,axiom,
% 0.20/0.47 comm_ring @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Rings_Osemiring_31,axiom,
% 0.20/0.47 semiring @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Rings_Oring__1,axiom,
% 0.20/0.47 ring_1 @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Otimes_32,axiom,
% 0.20/0.47 times @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Ominus_33,axiom,
% 0.20/0.47 minus @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Fields_Ofield,axiom,
% 0.20/0.47 field @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Oplus_34,axiom,
% 0.20/0.47 plus @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Rings_Oring,axiom,
% 0.20/0.47 ring @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Oone_35,axiom,
% 0.20/0.47 one @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Extended__Real_Oereal___Groups_Oordered__ab__semigroup__add_36,axiom,
% 0.20/0.47 ordere779506340up_add @ extended_ereal ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Extended__Real_Oereal___Groups_Oab__semigroup__mult_37,axiom,
% 0.20/0.47 ab_semigroup_mult @ extended_ereal ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Extended__Real_Oereal___Groups_Ocomm__monoid__mult_38,axiom,
% 0.20/0.47 comm_monoid_mult @ extended_ereal ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Extended__Real_Oereal___Groups_Oab__semigroup__add_39,axiom,
% 0.20/0.47 ab_semigroup_add @ extended_ereal ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Extended__Real_Oereal___Groups_Ocomm__monoid__add_40,axiom,
% 0.20/0.47 comm_monoid_add @ extended_ereal ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Extended__Real_Oereal___Groups_Osemigroup__mult_41,axiom,
% 0.20/0.47 semigroup_mult @ extended_ereal ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Extended__Real_Oereal___Groups_Osemigroup__add_42,axiom,
% 0.20/0.47 semigroup_add @ extended_ereal ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Extended__Real_Oereal___Groups_Omonoid__mult_43,axiom,
% 0.20/0.47 monoid_mult @ extended_ereal ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Extended__Real_Oereal___Groups_Otimes_44,axiom,
% 0.20/0.47 times @ extended_ereal ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Extended__Real_Oereal___Groups_Ominus_45,axiom,
% 0.20/0.47 minus @ extended_ereal ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Extended__Real_Oereal___Groups_Oplus_46,axiom,
% 0.20/0.47 plus @ extended_ereal ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Extended__Real_Oereal___Groups_Oone_47,axiom,
% 0.20/0.47 one @ extended_ereal ).
% 0.20/0.47
% 0.20/0.47 % Helper facts (3)
% 0.20/0.47 thf(help_If_3_1_T,axiom,
% 0.20/0.47 ! [P: $o] :
% 0.20/0.47 ( ( P = $true )
% 0.20/0.47 | ( P = $false ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(help_If_2_1_T,axiom,
% 0.20/0.47 ! [A: $tType,X2: A,Y: A] :
% 0.20/0.47 ( ( if @ A @ $false @ X2 @ Y )
% 0.20/0.47 = Y ) ).
% 0.20/0.47
% 0.20/0.47 thf(help_If_1_1_T,axiom,
% 0.20/0.47 ! [A: $tType,X2: A,Y: A] :
% 0.20/0.47 ( ( if @ A @ $true @ X2 @ Y )
% 0.20/0.47 = X2 ) ).
% 0.20/0.47
% 0.20/0.47 % Free types (1)
% 0.20/0.47 thf(tfree_0,hypothesis,
% 0.20/0.47 real_V1076094709vector @ b ).
% 0.20/0.47
% 0.20/0.47 % Conjectures (1)
% 0.20/0.47 thf(conj_0,conjecture,
% 0.20/0.47 ( ( f
% 0.20/0.47 @ ( plus_plus @ b
% 0.20/0.47 @ ( real_V1908273582scaleR @ b @ ( times_times @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) @ ( inverse_inverse @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) )
% 0.20/0.47 @ ( groups1340683514dd_sum @ a @ b
% 0.20/0.47 @ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
% 0.20/0.47 @ sa ) )
% 0.20/0.47 @ ( real_V1908273582scaleR @ b @ ( aa @ i ) @ ( y @ i ) ) ) )
% 0.20/0.47 = ( f
% 0.20/0.47 @ ( plus_plus @ b
% 0.20/0.47 @ ( real_V1908273582scaleR @ b @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) )
% 0.20/0.47 @ ( groups1340683514dd_sum @ a @ b
% 0.20/0.47 @ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( times_times @ real @ ( aa @ J ) @ ( inverse_inverse @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) ) @ ( y @ J ) )
% 0.20/0.47 @ sa ) )
% 0.20/0.47 @ ( real_V1908273582scaleR @ b @ ( aa @ i ) @ ( y @ i ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 %------------------------------------------------------------------------------
% 0.20/0.47 ------------------------
% 2.58/1.49 ---- Embedded in THF ---
% 2.58/1.49 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.58/1.49 %%% Generated on Tue Feb 24 08:26:01 EST 2026
% 2.58/1.49 %%% using '$$dhol' embedding, version 1.3.0.
% 2.58/1.49 %%% Logic specification used:
% 2.58/1.49 %%% thf(spec, logic, $$dhol).
% 2.58/1.49
% 2.58/1.49 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.a0uzsVYvBA/DTF2DTF4788.p
% 2.58/1.49 thf(ty_t_Extended__Real_Oereal, type, extended_ereal: $tType).
% 2.58/1.49 thf(ty_t_Real_Oreal, type, real: $tType).
% 2.58/1.49 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.58/1.49 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.58/1.49 thf(ty_tf_b, type, b: $tType).
% 2.58/1.49 thf(ty_tf_a, type, a: $tType).
% 2.58/1.49 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Groups_Oone, type, one: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Rings_Oring, type, ring: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Groups_Oplus, type, plus: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Fields_Ofield, type, field: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Groups_Ominus, type, minus: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Groups_Otimes, type, times: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Rings_Oring__1, type, ring_1: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Rings_Osemiring, type, semiring: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Rings_Ocomm__ring, type, comm_ring: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Rings_Osemiring__0, type, semiring_0: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Groups_Omonoid__mult, type, monoid_mult: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Groups_Oab__group__add, type, ab_group_add: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Rings_Ocomm__semiring, type, comm_semiring: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Fields_Odivision__ring, type, division_ring: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Parity_Osemiring__bits, type, semiring_bits: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Rings_Osemidom__divide, type, semidom_divide: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Groups_Osemigroup__mult, type, semigroup_mult: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Groups_Ocomm__monoid__mult, type, comm_monoid_mult: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Groups_Oab__semigroup__mult, type, ab_semigroup_mult: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Real__Vector__Spaces_OscaleR, type, real_Vector_scaleR: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Rings_Ocomm__semiring__1__cancel, type, comm_s1003936772cancel: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Real__Vector__Spaces_Oreal__vector, type, real_V1076094709vector: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Real__Vector__Spaces_Oreal__algebra, type, real_V148923926lgebra: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_cl_Real__Vector__Spaces_Oreal__div__algebra, type, real_V1015715145lgebra: (!> [A:$tType]: ($o))).
% 2.58/1.49 thf(sy_c_Convex_Oconvex, type, convex: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.58/1.49 thf(sy_c_Fields_Oinverse__class_Oinverse, type, inverse_inverse: (!> [A:$tType]: ((A > A)))).
% 2.58/1.49 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.58/1.49 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.58/1.49 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))).
% 2.58/1.49 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 2.58/1.49 thf(sy_c_Groups_Otimes__class_Otimes, type, times_times: (!> [A:$tType]: ((A > (A > A))))).
% 2.58/1.49 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum, type, groups1340683514dd_sum: (!> [B:$tType,A:$tType]: (((B > A) > ((set @ B) > A))))).
% 2.58/1.49 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.58/1.49 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Oconvex__on, type, lower_673667120vex_on: (!> [A:$tType]: (((set @ A) > ((A > extended_ereal) > $o))))).
% 2.58/1.49 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.58/1.49 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.58/1.49 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR, type, real_V1908273582scaleR: (!> [A:$tType]: ((real > (A > A))))).
% 2.58/1.49 thf(sy_c_Rings_Odivide__class_Odivide, type, divide_divide: (!> [A:$tType]: ((A > (A > A))))).
% 2.58/1.49 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.58/1.49 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.58/1.49 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.58/1.49 thf(sy_v_C, type, c: (set @ b)).
% 2.58/1.49 thf(sy_v_a, type, a2: (a > real)).
% 2.58/1.49 thf(sy_v_aa____, type, aa: (a > real)).
% 2.58/1.49 thf(sy_v_f, type, f: (b > extended_ereal)).
% 2.58/1.49 thf(sy_v_i____, type, i: a).
% 2.58/1.49 thf(sy_v_s, type, s: (set @ a)).
% 2.58/1.49 thf(sy_v_sa____, type, sa: (set @ a)).
% 2.58/1.49 thf(sy_v_y, type, y: (a > b)).
% 2.58/1.49 thf(fact_0_insert_Ohyps_I3_J, axiom, (~ ((((member @ a) @ i) @ sa)))).
% 2.58/1.49 thf(fact_1_asm, axiom, ((aa @ i) != (one_one @ real))).
% 2.58/1.49 thf(fact_2__092_060open_062_I_092_060Sum_062x_092_060in_062s_O_Aa_Ax_A_K_092_060_094sub_062R_Ay_Ax_A_P_092_060_094sub_062R_A_I1_A_N_Aa_Ai_J_J_A_061_A_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_P_092_060_094sub_062R_A_I1_A_N_Aa_Ai_J_092_060close_062, axiom, (((((groups1340683514dd_sum @ a) @ b) @ (^ [X:a]: ((((real_V1908273582scaleR @ b) @ ((inverse_inverse @ real) @ (((minus_minus @ real) @ (one_one @ real)) @ (aa @ i)))) @ (((real_V1908273582scaleR @ b) @ (aa @ X)) @ (y @ X)))))) @ sa) = (((real_V1908273582scaleR @ b) @ ((inverse_inverse @ real) @ (((minus_minus @ real) @ (one_one @ real)) @ (aa @ i)))) @ ((((groups1340683514dd_sum @ a) @ b) @ (^ [J:a]: ((((real_V1908273582scaleR @ b) @ (aa @ J)) @ (y @ J))))) @ sa)))).
% 2.58/1.49 thf(fact_3_insert_Ohyps_I2_J, axiom, (sa != (bot_bot @ (set @ a)))).
% 2.58/1.49 thf(fact_4_insert_Ohyps_I1_J, axiom, ((finite_finite2 @ a) @ sa)).
% 2.58/1.49 thf(fact_5__092_060open_062f_A_I_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_L_Aa_Ai_A_K_092_060_094sub_062R_Ay_Ai_J_A_061_Af_A_I_I_I1_A_N_Aa_Ai_J_A_K_Ainverse_A_I1_A_N_Aa_Ai_J_J_A_K_092_060_094sub_062R_A_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_L_Aa_Ai_A_K_092_060_094sub_062R_Ay_Ai_J_092_060close_062, axiom, ((f @ (((plus_plus @ b) @ ((((groups1340683514dd_sum @ a) @ b) @ (^ [J:a]: ((((real_V1908273582scaleR @ b) @ (aa @ J)) @ (y @ J))))) @ sa)) @ (((real_V1908273582scaleR @ b) @ (aa @ i)) @ (y @ i)))) = (f @ (((plus_plus @ b) @ (((real_V1908273582scaleR @ b) @ (((times_times @ real) @ (((minus_minus @ real) @ (one_one @ real)) @ (aa @ i))) @ ((inverse_inverse @ real) @ (((minus_minus @ real) @ (one_one @ real)) @ (aa @ i))))) @ ((((groups1340683514dd_sum @ a) @ b) @ (^ [J:a]: ((((real_V1908273582scaleR @ b) @ (aa @ J)) @ (y @ J))))) @ sa))) @ (((real_V1908273582scaleR @ b) @ (aa @ i)) @ (y @ i)))))).
% 2.58/1.49 thf(fact_6_calculation, axiom, ((f @ ((((groups1340683514dd_sum @ a) @ b) @ (^ [J:a]: ((((real_V1908273582scaleR @ b) @ (aa @ J)) @ (y @ J))))) @ (((insert @ a) @ i) @ sa))) = (f @ (((plus_plus @ b) @ (((real_V1908273582scaleR @ b) @ (((times_times @ real) @ (((minus_minus @ real) @ (one_one @ real)) @ (aa @ i))) @ ((inverse_inverse @ real) @ (((minus_minus @ real) @ (one_one @ real)) @ (aa @ i))))) @ ((((groups1340683514dd_sum @ a) @ b) @ (^ [J:a]: ((((real_V1908273582scaleR @ b) @ (aa @ J)) @ (y @ J))))) @ sa))) @ (((real_V1908273582scaleR @ b) @ (aa @ i)) @ (y @ i)))))).
% 2.58/1.49 thf(fact_7__092_060open_062f_A_I_092_060Sum_062j_092_060in_062insert_Ai_As_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_061_Af_A_I_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_L_Aa_Ai_A_K_092_060_094sub_062R_Ay_Ai_J_092_060close_062, axiom, ((f @ ((((groups1340683514dd_sum @ a) @ b) @ (^ [J:a]: ((((real_V1908273582scaleR @ b) @ (aa @ J)) @ (y @ J))))) @ (((insert @ a) @ i) @ sa))) = (f @ (((plus_plus @ b) @ ((((groups1340683514dd_sum @ a) @ b) @ (^ [J:a]: ((((real_V1908273582scaleR @ b) @ (aa @ J)) @ (y @ J))))) @ sa)) @ (((real_V1908273582scaleR @ b) @ (aa @ i)) @ (y @ i)))))).
% 2.58/1.49 thf(fact_8_scaleR__collapse, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [U:real,A2:A]: (((((plus_plus @ A) @ (((real_V1908273582scaleR @ A) @ (((minus_minus @ real) @ (one_one @ real)) @ U)) @ A2)) @ (((real_V1908273582scaleR @ A) @ U) @ A2)) = A2))))))).
% 2.58/1.49 thf(fact_9_scaleR__eq__iff, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [B2:A,U:real,A2:A]: ((((((plus_plus @ A) @ B2) @ (((real_V1908273582scaleR @ A) @ U) @ A2)) = (((plus_plus @ A) @ A2) @ (((real_V1908273582scaleR @ A) @ U) @ B2))) = ((A2 = B2) | (U = (one_one @ real)))))))))).
% 2.58/1.49 thf(fact_10_yai_I1_J, axiom, (((member @ b) @ (y @ i)) @ c)).
% 2.58/1.49 thf(fact_11_scaleR__scaleR, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [A2:real,B2:real,X2:A]: (((((real_V1908273582scaleR @ A) @ A2) @ (((real_V1908273582scaleR @ A) @ B2) @ X2)) = (((real_V1908273582scaleR @ A) @ (((times_times @ real) @ A2) @ B2)) @ X2)))))))).
% 2.58/1.49 thf(fact_12_scaleR__one, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [X2:A]: (((((real_V1908273582scaleR @ A) @ (one_one @ real)) @ X2) = X2))))))).
% 2.58/1.49 thf(fact_13__092_060open_062sum_Aa_As_A_061_A1_A_N_Aa_Ai_092_060close_062, axiom, (((((groups1340683514dd_sum @ a) @ real) @ aa) @ sa) = (((minus_minus @ real) @ (one_one @ real)) @ (aa @ i)))).
% 2.58/1.49 thf(fact_14_inverse__eq__1__iff, axiom, (! [A:$tType]: (((field @ A) => (! [X2:A]: (((((inverse_inverse @ A) @ X2) = (one_one @ A)) = (X2 = (one_one @ A))))))))).
% 2.58/1.49 thf(fact_15_inverse__1, axiom, (! [A:$tType]: (((division_ring @ A) => (((inverse_inverse @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 2.58/1.49 thf(fact_16_inverse__mult__distrib, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A]: ((((inverse_inverse @ A) @ (((times_times @ A) @ A2) @ B2)) = (((times_times @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2))))))))).
% 2.58/1.49 thf(fact_17_mult__scaleR__left, axiom, (! [A:$tType]: (((real_V148923926lgebra @ A) => (! [A2:real,X2:A,Y:A]: (((((times_times @ A) @ (((real_V1908273582scaleR @ A) @ A2) @ X2)) @ Y) = (((real_V1908273582scaleR @ A) @ A2) @ (((times_times @ A) @ X2) @ Y))))))))).
% 2.58/1.49 thf(fact_18_assms_I1_J, axiom, ((finite_finite2 @ a) @ s)).
% 2.58/1.49 thf(fact_19_assms_I2_J, axiom, (s != (bot_bot @ (set @ a)))).
% 2.58/1.49 thf(fact_20_assms_I7_J, axiom, (! [I:a]: (((((member @ a) @ I) @ s) => (((member @ b) @ (y @ I)) @ c))))).
% 2.58/1.49 thf(fact_21_insert_Oprems_I3_J, axiom, (! [I:a]: (((((member @ a) @ I) @ (((insert @ a) @ i) @ sa)) => (((member @ b) @ (y @ I)) @ c))))).
% 2.58/1.49 thf(fact_22_inverse__inverse__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: ((((inverse_inverse @ A) @ ((inverse_inverse @ A) @ A2)) = A2))))))).
% 2.58/1.49 thf(fact_23_inverse__eq__iff__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A]: (((((inverse_inverse @ A) @ A2) = ((inverse_inverse @ A) @ B2)) = (A2 = B2)))))))).
% 2.58/1.49 thf(fact_24_insert_Oprems_I1_J, axiom, (((((groups1340683514dd_sum @ a) @ real) @ aa) @ (((insert @ a) @ i) @ sa)) = (one_one @ real))).
% 2.58/1.49 thf(fact_25_assms_I3_J, axiom, (((lower_673667120vex_on @ b) @ c) @ f)).
% 2.58/1.49 thf(fact_26_fis, axiom, ((finite_finite2 @ a) @ (((insert @ a) @ i) @ sa))).
% 2.58/1.49 thf(fact_27_mult__scaleR__right, axiom, (! [A:$tType]: (((real_V148923926lgebra @ A) => (! [X2:A,A2:real,Y:A]: (((((times_times @ A) @ X2) @ (((real_V1908273582scaleR @ A) @ A2) @ Y)) = (((real_V1908273582scaleR @ A) @ A2) @ (((times_times @ A) @ X2) @ Y))))))))).
% 2.58/1.49 thf(fact_28_assms_I4_J, axiom, ((convex @ b) @ c)).
% 2.58/1.49 thf(fact_29_assms_I5_J, axiom, (((((groups1340683514dd_sum @ a) @ real) @ a2) @ s) = (one_one @ real))).
% 2.58/1.49 thf(fact_30_real__scaleR__def, axiom, ((real_V1908273582scaleR @ real) = (times_times @ real))).
% 2.58/1.49 thf(fact_31_scale__left__commute, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [A2:real,B2:real,X2:A]: (((((real_V1908273582scaleR @ A) @ A2) @ (((real_V1908273582scaleR @ A) @ B2) @ X2)) = (((real_V1908273582scaleR @ A) @ B2) @ (((real_V1908273582scaleR @ A) @ A2) @ X2))))))))).
% 2.58/1.49 thf(fact_32_inverse__eq__imp__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A]: (((((inverse_inverse @ A) @ A2) = ((inverse_inverse @ A) @ B2)) => (A2 = B2)))))))).
% 2.58/1.49 thf(fact_33_scaleR__add__right, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [A2:real,X2:A,Y:A]: (((((real_V1908273582scaleR @ A) @ A2) @ (((plus_plus @ A) @ X2) @ Y)) = (((plus_plus @ A) @ (((real_V1908273582scaleR @ A) @ A2) @ X2)) @ (((real_V1908273582scaleR @ A) @ A2) @ Y))))))))).
% 2.58/1.49 thf(fact_34_scaleR__add__left, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [A2:real,B2:real,X2:A]: (((((real_V1908273582scaleR @ A) @ (((plus_plus @ real) @ A2) @ B2)) @ X2) = (((plus_plus @ A) @ (((real_V1908273582scaleR @ A) @ A2) @ X2)) @ (((real_V1908273582scaleR @ A) @ B2) @ X2))))))))).
% 2.58/1.49 thf(fact_35_scaleR__left_Oadd, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [X2:real,Y:real,Xa:A]: (((((real_V1908273582scaleR @ A) @ (((plus_plus @ real) @ X2) @ Y)) @ Xa) = (((plus_plus @ A) @ (((real_V1908273582scaleR @ A) @ X2) @ Xa)) @ (((real_V1908273582scaleR @ A) @ Y) @ Xa))))))))).
% 2.58/1.49 thf(fact_36_scale__right__diff__distrib, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [A2:real,X2:A,Y:A]: (((((real_V1908273582scaleR @ A) @ A2) @ (((minus_minus @ A) @ X2) @ Y)) = (((minus_minus @ A) @ (((real_V1908273582scaleR @ A) @ A2) @ X2)) @ (((real_V1908273582scaleR @ A) @ A2) @ Y))))))))).
% 2.58/1.49 thf(fact_37_mult__commute__imp__mult__inverse__commute, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Y:A,X2:A]: ((((((times_times @ A) @ Y) @ X2) = (((times_times @ A) @ X2) @ Y)) => ((((times_times @ A) @ ((inverse_inverse @ A) @ Y)) @ X2) = (((times_times @ A) @ X2) @ ((inverse_inverse @ A) @ Y)))))))))).
% 2.58/1.49 thf(fact_38_scale__sum__right, axiom, (! [A:$tType,C:$tType]: (((real_V1076094709vector @ A) => (! [A2:real,F:(C > A),A3:(set @ C)]: (((((real_V1908273582scaleR @ A) @ A2) @ ((((groups1340683514dd_sum @ C) @ A) @ F) @ A3)) = ((((groups1340683514dd_sum @ C) @ A) @ (^ [X:C]: ((((real_V1908273582scaleR @ A) @ A2) @ (F @ X))))) @ A3)))))))).
% 2.58/1.49 thf(fact_39_scale__sum__left, axiom, (! [A:$tType,C:$tType]: (((real_V1076094709vector @ A) => (! [F:(C > real),A3:(set @ C),X2:A]: (((((real_V1908273582scaleR @ A) @ ((((groups1340683514dd_sum @ C) @ real) @ F) @ A3)) @ X2) = ((((groups1340683514dd_sum @ C) @ A) @ (^ [A4:C]: ((((real_V1908273582scaleR @ A) @ (F @ A4)) @ X2)))) @ A3)))))))).
% 2.58/1.49 thf(fact_40_scaleR__right_Osum, axiom, (! [A:$tType,C:$tType]: (((real_V1076094709vector @ A) => (! [A2:real,G:(C > A),A3:(set @ C)]: (((((real_V1908273582scaleR @ A) @ A2) @ ((((groups1340683514dd_sum @ C) @ A) @ G) @ A3)) = ((((groups1340683514dd_sum @ C) @ A) @ (^ [X:C]: ((((real_V1908273582scaleR @ A) @ A2) @ (G @ X))))) @ A3)))))))).
% 2.58/1.49 thf(fact_41_scaleR__left_Osum, axiom, (! [A:$tType,C:$tType]: (((real_V1076094709vector @ A) => (! [G:(C > real),A3:(set @ C),X2:A]: (((((real_V1908273582scaleR @ A) @ ((((groups1340683514dd_sum @ C) @ real) @ G) @ A3)) @ X2) = ((((groups1340683514dd_sum @ C) @ A) @ (^ [X:C]: ((((real_V1908273582scaleR @ A) @ (G @ X)) @ X2)))) @ A3)))))))).
% 2.58/1.49 thf(fact_42_inverse__unique, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A]: ((((((times_times @ A) @ A2) @ B2) = (one_one @ A)) => (((inverse_inverse @ A) @ A2) = B2)))))))).
% 2.58/1.49 thf(fact_43_scale__left__diff__distrib, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [A2:real,B2:real,X2:A]: (((((real_V1908273582scaleR @ A) @ (((minus_minus @ real) @ A2) @ B2)) @ X2) = (((minus_minus @ A) @ (((real_V1908273582scaleR @ A) @ A2) @ X2)) @ (((real_V1908273582scaleR @ A) @ B2) @ X2))))))))).
% 2.58/1.49 thf(fact_44_scaleR__left_Odiff, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [X2:real,Y:real,Xa:A]: (((((real_V1908273582scaleR @ A) @ (((minus_minus @ real) @ X2) @ Y)) @ Xa) = (((minus_minus @ A) @ (((real_V1908273582scaleR @ A) @ X2) @ Xa)) @ (((real_V1908273582scaleR @ A) @ Y) @ Xa))))))))).
% 2.58/1.49 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.58/1.49 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A3)))) = A3)))).
% 2.58/1.49 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.58/1.49 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.58/1.49 thf(fact_49_inverse__scaleR__distrib, axiom, (! [A:$tType]: (((real_V1015715145lgebra @ A) => (! [A2:real,X2:A]: ((((inverse_inverse @ A) @ (((real_V1908273582scaleR @ A) @ A2) @ X2)) = (((real_V1908273582scaleR @ A) @ ((inverse_inverse @ real) @ A2)) @ ((inverse_inverse @ A) @ X2))))))))).
% 2.58/1.49 thf(fact_50_asum, axiom, (((member @ b) @ ((((groups1340683514dd_sum @ a) @ b) @ (^ [J:a]: ((((real_V1908273582scaleR @ b) @ (((divide_divide @ real) @ (aa @ J)) @ (((minus_minus @ real) @ (one_one @ real)) @ (aa @ i)))) @ (y @ J))))) @ sa)) @ c)).
% 2.58/1.49 thf(fact_51_sum_Oinsert, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),X2:B,G:(B > A)]: ((((finite_finite2 @ B) @ A3) => ((~ ((((member @ B) @ X2) @ A3))) => (((((groups1340683514dd_sum @ B) @ A) @ G) @ (((insert @ B) @ X2) @ A3)) = (((plus_plus @ A) @ (G @ X2)) @ ((((groups1340683514dd_sum @ B) @ A) @ G) @ A3))))))))))).
% 2.58/1.49 thf(fact_52_affine__hull__finite__step, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [S:(set @ A),A2:A,W:real,Y:A]: ((((finite_finite2 @ A) @ S) => ((? [U2:(A > real)]: (((((((groups1340683514dd_sum @ A) @ real) @ U2) @ (((insert @ A) @ A2) @ S)) = W) & (((((groups1340683514dd_sum @ A) @ A) @ (^ [X:A]: ((((real_V1908273582scaleR @ A) @ (U2 @ X)) @ X)))) @ (((insert @ A) @ A2) @ S)) = Y)))) = (? [V:real,U2:(A > real)]: (((((((groups1340683514dd_sum @ A) @ real) @ U2) @ S) = (((minus_minus @ real) @ W) @ V)) & (((((groups1340683514dd_sum @ A) @ A) @ (^ [X:A]: ((((real_V1908273582scaleR @ A) @ (U2 @ X)) @ X)))) @ S) = (((minus_minus @ A) @ Y) @ (((real_V1908273582scaleR @ A) @ V) @ A2)))))))))))))).
% 2.58/1.49 thf(fact_53_a1, axiom, (((((groups1340683514dd_sum @ a) @ real) @ (^ [J:a]: ((((divide_divide @ real) @ (aa @ J)) @ (((minus_minus @ real) @ (one_one @ real)) @ (aa @ i)))))) @ sa) = (one_one @ real))).
% 2.58/1.49 thf(fact_54__092_060open_062sum_Aa_As_A_P_A_I1_A_N_Aa_Ai_J_A_061_A1_092_060close_062, axiom, ((((divide_divide @ real) @ ((((groups1340683514dd_sum @ a) @ real) @ aa) @ sa)) @ (((minus_minus @ real) @ (one_one @ real)) @ (aa @ i))) = (one_one @ real))).
% 2.58/1.49 thf(fact_55_singleton__conv, axiom, (! [A:$tType,A2:A]: ((((collect @ A) @ (^ [X:A]: ((X = A2)))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.58/1.49 thf(fact_56_singleton__conv2, axiom, (! [A:$tType,A2:A]: ((((collect @ A) @ ((^ [Y2:A,Z:A]: ((Y2 = Z))) @ A2)) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.58/1.49 thf(fact_57_finite__insert, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: ((((finite_finite2 @ A) @ (((insert @ A) @ A2) @ A3)) = ((finite_finite2 @ A) @ A3))))).
% 2.58/1.49 thf(fact_58_sum__diff1, axiom, (! [A:$tType,B:$tType]: (((ab_group_add @ A) => (! [A3:(set @ B),A2:B,F:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((((member @ B) @ A2) @ A3) => (((((groups1340683514dd_sum @ B) @ A) @ F) @ (((minus_minus @ (set @ B)) @ A3) @ (((insert @ B) @ A2) @ (bot_bot @ (set @ B))))) = (((minus_minus @ A) @ ((((groups1340683514dd_sum @ B) @ A) @ F) @ A3)) @ (F @ A2)))) & ((~ ((((member @ B) @ A2) @ A3))) => (((((groups1340683514dd_sum @ B) @ A) @ F) @ (((minus_minus @ (set @ B)) @ A3) @ (((insert @ B) @ A2) @ (bot_bot @ (set @ B))))) = ((((groups1340683514dd_sum @ B) @ A) @ F) @ A3))))))))))).
% 2.58/1.49 thf(fact_59_singletonI, axiom, (! [A:$tType,A2:A]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.58/1.49 thf(fact_60_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X:A]: ((~ ((P @ X))))))))).
% 2.58/1.49 thf(fact_61_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X:A]: ((~ ((P @ X))))))))).
% 2.58/1.49 thf(fact_62_all__not__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X:A]: ((~ ((((member @ A) @ X) @ A3))))) = (A3 = (bot_bot @ (set @ A))))))).
% 2.58/1.49 thf(fact_63_empty__iff, axiom, (! [A:$tType,C2:A]: ((~ ((((member @ A) @ C2) @ (bot_bot @ (set @ A)))))))).
% 2.58/1.49 thf(fact_64_finite__code, axiom, (! [A:$tType]: (((finite_finite @ A) => ((finite_finite2 @ A) = (^ [A5:(set @ A)]: ($true))))))).
% 2.58/1.49 thf(fact_65_insert__absorb2, axiom, (! [A:$tType,X2:A,A3:(set @ A)]: (((((insert @ A) @ X2) @ (((insert @ A) @ X2) @ A3)) = (((insert @ A) @ X2) @ A3))))).
% 2.58/1.49 thf(fact_66_insert__iff, axiom, (! [A:$tType,A2:A,B2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B2) @ A3)) = ((A2 = B2) | (((member @ A) @ A2) @ A3)))))).
% 2.58/1.49 thf(fact_67_insertCI, axiom, (! [A:$tType,A2:A,B3:(set @ A),B2:A]: ((((~ ((((member @ A) @ A2) @ B3))) => (A2 = B2)) => (((member @ A) @ A2) @ (((insert @ A) @ B2) @ B3)))))).
% 2.58/1.49 thf(fact_68_Diff__idemp, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) @ B3) = (((minus_minus @ (set @ A)) @ A3) @ B3))))).
% 2.58/1.49 thf(fact_69_Diff__iff, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) = ((((member @ A) @ C2) @ A3) & (~ ((((member @ A) @ C2) @ B3)))))))).
% 2.58/1.49 thf(fact_70_DiffI, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ A3) => ((~ ((((member @ A) @ C2) @ B3))) => (((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A3) @ B3))))))).
% 2.58/1.49 thf(fact_71_finite__Collect__disjI, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: ((((finite_finite2 @ A) @ ((collect @ A) @ (^ [X:A]: (((P @ X) | (Q @ X)))))) = (((finite_finite2 @ A) @ ((collect @ A) @ P)) & ((finite_finite2 @ A) @ ((collect @ A) @ Q))))))).
% 2.58/1.49 thf(fact_72_finite__Collect__conjI, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((finite_finite2 @ A) @ ((collect @ A) @ P)) | ((finite_finite2 @ A) @ ((collect @ A) @ Q))) => ((finite_finite2 @ A) @ ((collect @ A) @ (^ [X:A]: (((P @ X) & (Q @ X)))))))))).
% 2.58/1.49 thf(fact_73_times__divide__eq__left, axiom, (! [A:$tType]: (((field @ A) => (! [B2:A,C2:A,A2:A]: (((((times_times @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ A2) = (((divide_divide @ A) @ (((times_times @ A) @ B2) @ A2)) @ C2)))))))).
% 2.58/1.49 thf(fact_74_divide__divide__eq__left, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A,C2:A]: (((((divide_divide @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ C2) = (((divide_divide @ A) @ A2) @ (((times_times @ A) @ B2) @ C2))))))))).
% 2.58/1.49 thf(fact_75_divide__divide__eq__right, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A,C2:A]: (((((divide_divide @ A) @ A2) @ (((divide_divide @ A) @ B2) @ C2)) = (((divide_divide @ A) @ (((times_times @ A) @ A2) @ C2)) @ B2)))))))).
% 2.58/1.49 thf(fact_76_times__divide__eq__right, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ A2) @ (((divide_divide @ A) @ B2) @ C2)) = (((divide_divide @ A) @ (((times_times @ A) @ A2) @ B2)) @ C2)))))))).
% 2.58/1.49 thf(fact_77_Diff__cancel, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ A3) = (bot_bot @ (set @ A)))))).
% 2.58/1.49 thf(fact_78_empty__Diff, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ (bot_bot @ (set @ A))) @ A3) = (bot_bot @ (set @ A)))))).
% 2.58/1.49 thf(fact_79_Diff__empty, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = A3)))).
% 2.58/1.49 thf(fact_80_finite__Diff2, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => (((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) = ((finite_finite2 @ A) @ A3)))))).
% 2.58/1.49 thf(fact_81_finite__Diff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A3) @ B3)))))).
% 2.58/1.49 thf(fact_82_insert__Diff1, axiom, (! [A:$tType,X2:A,B3:(set @ A),A3:(set @ A)]: (((((member @ A) @ X2) @ B3) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X2) @ A3)) @ B3) = (((minus_minus @ (set @ A)) @ A3) @ B3)))))).
% 2.58/1.49 thf(fact_83_Diff__insert0, axiom, (! [A:$tType,X2:A,A3:(set @ A),B3:(set @ A)]: (((~ ((((member @ A) @ X2) @ A3))) => ((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X2) @ B3)) = (((minus_minus @ (set @ A)) @ A3) @ B3)))))).
% 2.58/1.49 thf(fact_84_inverse__divide, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A]: ((((inverse_inverse @ A) @ (((divide_divide @ A) @ A2) @ B2)) = (((divide_divide @ A) @ B2) @ A2)))))))).
% 2.58/1.49 thf(fact_85_insert__Diff__single, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((insert @ A) @ A2) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = (((insert @ A) @ A2) @ A3))))).
% 2.58/1.49 thf(fact_86_finite__Diff__insert, axiom, (! [A:$tType,A3:(set @ A),A2:A,B3:(set @ A)]: ((((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ B3))) = ((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A3) @ B3)))))).
% 2.58/1.49 thf(fact_87_sum__diff1__nat, axiom, (! [A:$tType,A2:A,A3:(set @ A),F:(A > nat)]: ((((((member @ A) @ A2) @ A3) => (((((groups1340683514dd_sum @ A) @ nat) @ F) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = (((minus_minus @ nat) @ ((((groups1340683514dd_sum @ A) @ nat) @ F) @ A3)) @ (F @ A2)))) & ((~ ((((member @ A) @ A2) @ A3))) => (((((groups1340683514dd_sum @ A) @ nat) @ F) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = ((((groups1340683514dd_sum @ A) @ nat) @ F) @ A3))))))).
% 2.58/1.49 thf(fact_88_minus__set__def, axiom, (! [A:$tType]: (((minus_minus @ (set @ A)) = (^ [A5:(set @ A),B4:(set @ A)]: (((collect @ A) @ (((minus_minus @ (A > $o)) @ (^ [X:A]: ((((member @ A) @ X) @ A5)))) @ (^ [X:A]: ((((member @ A) @ X) @ B4))))))))))).
% 2.58/1.49 thf(fact_89_set__diff__eq, axiom, (! [A:$tType]: (((minus_minus @ (set @ A)) = (^ [A5:(set @ A),B4:(set @ A)]: (((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ A5) & (~ ((((member @ A) @ X) @ B4))))))))))))).
% 2.58/1.49 thf(fact_90_DiffD2, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) => (~ ((((member @ A) @ C2) @ B3))))))).
% 2.58/1.49 thf(fact_91_DiffD1, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) => (((member @ A) @ C2) @ A3))))).
% 2.58/1.49 thf(fact_92_DiffE, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) => (~ (((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ B3)))))))).
% 2.58/1.49 thf(fact_93_Diff__infinite__finite, axiom, (! [A:$tType,T:(set @ A),S:(set @ A)]: ((((finite_finite2 @ A) @ T) => ((~ (((finite_finite2 @ A) @ S))) => (~ (((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ S) @ T))))))))).
% 2.58/1.49 thf(fact_94_insert__Diff__if, axiom, (! [A:$tType,X2:A,B3:(set @ A),A3:(set @ A)]: ((((((member @ A) @ X2) @ B3) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X2) @ A3)) @ B3) = (((minus_minus @ (set @ A)) @ A3) @ B3))) & ((~ ((((member @ A) @ X2) @ B3))) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X2) @ A3)) @ B3) = (((insert @ A) @ X2) @ (((minus_minus @ (set @ A)) @ A3) @ B3)))))))).
% 2.58/1.49 thf(fact_95_sum__divide__distrib, axiom, (! [A:$tType,B:$tType]: (((field @ A) => (! [F:(B > A),A3:(set @ B),R:A]: (((((divide_divide @ A) @ ((((groups1340683514dd_sum @ B) @ A) @ F) @ A3)) @ R) = ((((groups1340683514dd_sum @ B) @ A) @ (^ [N:B]: ((((divide_divide @ A) @ (F @ N)) @ R)))) @ A3)))))))).
% 2.58/1.49 thf(fact_96_convex__ereal__add, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [S2:(set @ A),F:(A > extended_ereal),G:(A > extended_ereal)]: (((((lower_673667120vex_on @ A) @ S2) @ F) => ((((lower_673667120vex_on @ A) @ S2) @ G) => (((lower_673667120vex_on @ A) @ S2) @ (^ [X:A]: ((((plus_plus @ extended_ereal) @ (F @ X)) @ (G @ X))))))))))))).
% 2.58/1.49 thf(fact_97_times__divide__times__eq, axiom, (! [A:$tType]: (((field @ A) => (! [X2:A,Y:A,Z2:A,W:A]: (((((times_times @ A) @ (((divide_divide @ A) @ X2) @ Y)) @ (((divide_divide @ A) @ Z2) @ W)) = (((divide_divide @ A) @ (((times_times @ A) @ X2) @ Z2)) @ (((times_times @ A) @ Y) @ W))))))))).
% 2.58/1.49 thf(fact_98_divide__divide__times__eq, axiom, (! [A:$tType]: (((field @ A) => (! [X2:A,Y:A,Z2:A,W:A]: (((((divide_divide @ A) @ (((divide_divide @ A) @ X2) @ Y)) @ (((divide_divide @ A) @ Z2) @ W)) = (((divide_divide @ A) @ (((times_times @ A) @ X2) @ W)) @ (((times_times @ A) @ Y) @ Z2))))))))).
% 2.58/1.49 thf(fact_99_divide__divide__eq__left_H, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A,C2:A]: (((((divide_divide @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ C2) = (((divide_divide @ A) @ A2) @ (((times_times @ A) @ C2) @ B2))))))))).
% 2.58/1.49 thf(fact_100_add__divide__distrib, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A,C2:A]: (((((divide_divide @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A2) @ C2)) @ (((divide_divide @ A) @ B2) @ C2))))))))).
% 2.58/1.49 thf(fact_101_diff__divide__distrib, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A,C2:A]: (((((divide_divide @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2) = (((minus_minus @ A) @ (((divide_divide @ A) @ A2) @ C2)) @ (((divide_divide @ A) @ B2) @ C2))))))))).
% 2.58/1.49 thf(fact_102_Diff__insert__absorb, axiom, (! [A:$tType,X2:A,A3:(set @ A)]: (((~ ((((member @ A) @ X2) @ A3))) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X2) @ A3)) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) = A3))))).
% 2.58/1.49 thf(fact_103_Diff__insert2, axiom, (! [A:$tType,A3:(set @ A),A2:A,B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ B3)) = (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) @ B3))))).
% 2.58/1.49 thf(fact_104_insert__Diff, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ A3) => ((((insert @ A) @ A2) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = A3))))).
% 2.58/1.49 thf(fact_105_Diff__insert, axiom, (! [A:$tType,A3:(set @ A),A2:A,B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ B3)) = (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.58/1.49 thf(fact_106_finite__empty__induct, axiom, (! [A:$tType,A3:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ A3) => ((P @ A3) => ((! [A6:A,A7:(set @ A)]: ((((finite_finite2 @ A) @ A7) => ((((member @ A) @ A6) @ A7) => ((P @ A7) => (P @ (((minus_minus @ (set @ A)) @ A7) @ (((insert @ A) @ A6) @ (bot_bot @ (set @ A)))))))))) => (P @ (bot_bot @ (set @ A))))))))).
% 2.58/1.49 thf(fact_107_infinite__coinduct, axiom, (! [A:$tType,X4:((set @ A) > $o),A3:(set @ A)]: (((X4 @ A3) => ((! [A7:(set @ A)]: (((X4 @ A7) => (? [X5:A]: (((((member @ A) @ X5) @ A7) & ((X4 @ (((minus_minus @ (set @ A)) @ A7) @ (((insert @ A) @ X5) @ (bot_bot @ (set @ A))))) | (~ (((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A7) @ (((insert @ A) @ X5) @ (bot_bot @ (set @ A)))))))))))))) => (~ (((finite_finite2 @ A) @ A3)))))))).
% 2.58/1.49 thf(fact_108_infinite__remove, axiom, (! [A:$tType,S:(set @ A),A2:A]: (((~ (((finite_finite2 @ A) @ S))) => (~ (((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ S) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))))))).
% 2.58/1.49 thf(fact_109_field__class_Ofield__divide__inverse, axiom, (! [A:$tType]: (((field @ A) => ((divide_divide @ A) = (^ [A4:A,B5:A]: ((((times_times @ A) @ A4) @ ((inverse_inverse @ A) @ B5))))))))).
% 2.58/1.49 thf(fact_110_divide__inverse, axiom, (! [A:$tType]: (((division_ring @ A) => ((divide_divide @ A) = (^ [A4:A,B5:A]: ((((times_times @ A) @ A4) @ ((inverse_inverse @ A) @ B5))))))))).
% 2.58/1.49 thf(fact_111_divide__inverse__commute, axiom, (! [A:$tType]: (((field @ A) => ((divide_divide @ A) = (^ [A4:A,B5:A]: ((((times_times @ A) @ ((inverse_inverse @ A) @ B5)) @ A4)))))))).
% 2.58/1.49 thf(fact_112_inverse__eq__divide, axiom, (! [A:$tType]: (((division_ring @ A) => ((inverse_inverse @ A) = ((divide_divide @ A) @ (one_one @ A))))))).
% 2.58/1.49 thf(fact_113_ex__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((? [X:A]: ((((member @ A) @ X) @ A3))) = (A3 != (bot_bot @ (set @ A))))))).
% 2.58/1.49 thf(fact_114_equals0I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [Y3:A]: ((~ ((((member @ A) @ Y3) @ A3))))) => (A3 = (bot_bot @ (set @ A))))))).
% 2.58/1.49 thf(fact_115_equals0D, axiom, (! [A:$tType,A3:(set @ A),A2:A]: (((A3 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A3))))))).
% 2.58/1.49 thf(fact_116_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.58/1.49 thf(fact_117_finite, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A3:(set @ A)]: (((finite_finite2 @ A) @ A3))))))).
% 2.58/1.49 thf(fact_118_finite__set__choice, axiom, (! [B:$tType,A:$tType,A3:(set @ A),P:(A > (B > $o))]: ((((finite_finite2 @ A) @ A3) => ((! [X3:A]: (((((member @ A) @ X3) @ A3) => (? [X_1:B]: (((P @ X3) @ X_1)))))) => (? [F2:(A > B)]: ((! [X5:A]: (((((member @ A) @ X5) @ A3) => ((P @ X5) @ (F2 @ X5)))))))))))).
% 2.58/1.49 thf(fact_119_mk__disjoint__insert, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ A3) => (? [B6:(set @ A)]: (((A3 = (((insert @ A) @ A2) @ B6)) & (~ ((((member @ A) @ A2) @ B6)))))))))).
% 2.58/1.49 thf(fact_120_insert__commute, axiom, (! [A:$tType,X2:A,Y:A,A3:(set @ A)]: (((((insert @ A) @ X2) @ (((insert @ A) @ Y) @ A3)) = (((insert @ A) @ Y) @ (((insert @ A) @ X2) @ A3)))))).
% 2.58/1.49 thf(fact_121_insert__eq__iff, axiom, (! [A:$tType,A2:A,A3:(set @ A),B2:A,B3:(set @ A)]: (((~ ((((member @ A) @ A2) @ A3))) => ((~ ((((member @ A) @ B2) @ B3))) => (((((insert @ A) @ A2) @ A3) = (((insert @ A) @ B2) @ B3)) = (((A2 = B2) => (A3 = B3)) & ((A2 != B2) => (? [C3:(set @ A)]: (((A3 = (((insert @ A) @ B2) @ C3)) & ((~ ((((member @ A) @ B2) @ C3))) & ((B3 = (((insert @ A) @ A2) @ C3)) & (~ ((((member @ A) @ A2) @ C3)))))))))))))))).
% 2.58/1.49 thf(fact_122_insert__absorb, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ A3) => ((((insert @ A) @ A2) @ A3) = A3))))).
% 2.58/1.49 thf(fact_123_insert__ident, axiom, (! [A:$tType,X2:A,A3:(set @ A),B3:(set @ A)]: (((~ ((((member @ A) @ X2) @ A3))) => ((~ ((((member @ A) @ X2) @ B3))) => (((((insert @ A) @ X2) @ A3) = (((insert @ A) @ X2) @ B3)) = (A3 = B3))))))).
% 2.58/1.49 thf(fact_124_Set_Oset__insert, axiom, (! [A:$tType,X2:A,A3:(set @ A)]: (((((member @ A) @ X2) @ A3) => (~ ((! [B6:(set @ A)]: (((A3 = (((insert @ A) @ X2) @ B6)) => (((member @ A) @ X2) @ B6)))))))))).
% 2.58/1.49 thf(fact_125_insertI2, axiom, (! [A:$tType,A2:A,B3:(set @ A),B2:A]: (((((member @ A) @ A2) @ B3) => (((member @ A) @ A2) @ (((insert @ A) @ B2) @ B3)))))).
% 2.58/1.49 thf(fact_126_insertI1, axiom, (! [A:$tType,A2:A,B3:(set @ A)]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ B3))))).
% 2.58/1.49 thf(fact_127_insertE, axiom, (! [A:$tType,A2:A,B2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B2) @ A3)) => ((A2 != B2) => (((member @ A) @ A2) @ A3)))))).
% 2.58/1.49 thf(fact_128_sum_Ocong, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),B3:(set @ B),G:(B > A),H:(B > A)]: (((A3 = B3) => ((! [X3:B]: (((((member @ B) @ X3) @ B3) => ((G @ X3) = (H @ X3))))) => (((((groups1340683514dd_sum @ B) @ A) @ G) @ A3) = ((((groups1340683514dd_sum @ B) @ A) @ H) @ B3)))))))))).
% 2.58/1.49 thf(fact_129_sum_Oeq__general, axiom, (! [C:$tType,A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [B3:(set @ B),A3:(set @ C),H:(C > B),Gamma:(B > A),Phi:(C > A)]: (((! [Y3:B]: (((((member @ B) @ Y3) @ B3) => (? [X5:C]: (((((member @ C) @ X5) @ A3) & (((H @ X5) = Y3) & (! [Ya:C]: ((((((member @ C) @ Ya) @ A3) & ((H @ Ya) = Y3)) => (Ya = X5))))))))))) => ((! [X3:C]: (((((member @ C) @ X3) @ A3) => ((((member @ B) @ (H @ X3)) @ B3) & ((Gamma @ (H @ X3)) = (Phi @ X3)))))) => (((((groups1340683514dd_sum @ C) @ A) @ Phi) @ A3) = ((((groups1340683514dd_sum @ B) @ A) @ Gamma) @ B3)))))))))).
% 2.58/1.49 thf(fact_130_sum_Oeq__general__inverses, axiom, (! [C:$tType,A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [B3:(set @ B),K:(B > C),A3:(set @ C),H:(C > B),Gamma:(B > A),Phi:(C > A)]: (((! [Y3:B]: (((((member @ B) @ Y3) @ B3) => ((((member @ C) @ (K @ Y3)) @ A3) & ((H @ (K @ Y3)) = Y3))))) => ((! [X3:C]: (((((member @ C) @ X3) @ A3) => ((((member @ B) @ (H @ X3)) @ B3) & (((K @ (H @ X3)) = X3) & ((Gamma @ (H @ X3)) = (Phi @ X3))))))) => (((((groups1340683514dd_sum @ C) @ A) @ Phi) @ A3) = ((((groups1340683514dd_sum @ B) @ A) @ Gamma) @ B3)))))))))).
% 2.58/1.49 thf(fact_131_sum_Oreindex__bij__witness, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comm_monoid_add @ A) => (! [S:(set @ B),I2:(C > B),J2:(B > C),T:(set @ C),H:(C > A),G:(B > A)]: (((! [A6:B]: (((((member @ B) @ A6) @ S) => ((I2 @ (J2 @ A6)) = A6)))) => ((! [A6:B]: (((((member @ B) @ A6) @ S) => (((member @ C) @ (J2 @ A6)) @ T)))) => ((! [B7:C]: (((((member @ C) @ B7) @ T) => ((J2 @ (I2 @ B7)) = B7)))) => ((! [B7:C]: (((((member @ C) @ B7) @ T) => (((member @ B) @ (I2 @ B7)) @ S)))) => ((! [A6:B]: (((((member @ B) @ A6) @ S) => ((H @ (J2 @ A6)) = (G @ A6))))) => (((((groups1340683514dd_sum @ B) @ A) @ G) @ S) = ((((groups1340683514dd_sum @ C) @ A) @ H) @ T))))))))))))).
% 2.58/1.49 thf(fact_132_if__smult, axiom, (! [A:$tType]: (((real_Vector_scaleR @ A) => (! [P:$o,X2:real,Y:real,V2:A]: (((P => ((((real_V1908273582scaleR @ A) @ ((((if @ real) @ P) @ X2) @ Y)) @ V2) = (((real_V1908273582scaleR @ A) @ X2) @ V2))) & ((~ (P)) => ((((real_V1908273582scaleR @ A) @ ((((if @ real) @ P) @ X2) @ Y)) @ V2) = (((real_V1908273582scaleR @ A) @ Y) @ V2)))))))))).
% 2.58/1.49 thf(fact_133_sum_Oinsert__remove, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),G:(B > A),X2:B]: ((((finite_finite2 @ B) @ A3) => (((((groups1340683514dd_sum @ B) @ A) @ G) @ (((insert @ B) @ X2) @ A3)) = (((plus_plus @ A) @ (G @ X2)) @ ((((groups1340683514dd_sum @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A3) @ (((insert @ B) @ X2) @ (bot_bot @ (set @ B)))))))))))))).
% 2.58/1.49 thf(fact_134_sum_Oremove, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),X2:B,G:(B > A)]: ((((finite_finite2 @ B) @ A3) => ((((member @ B) @ X2) @ A3) => (((((groups1340683514dd_sum @ B) @ A) @ G) @ A3) = (((plus_plus @ A) @ (G @ X2)) @ ((((groups1340683514dd_sum @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A3) @ (((insert @ B) @ X2) @ (bot_bot @ (set @ B))))))))))))))).
% 2.58/1.49 thf(fact_135_sum_Odelta__remove, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [S:(set @ B),A2:B,B2:(B > A),C2:(B > A)]: ((((finite_finite2 @ B) @ S) => (((((member @ B) @ A2) @ S) => (((((groups1340683514dd_sum @ B) @ A) @ (^ [K2:B]: (((((if @ A) @ (K2 = A2)) @ (B2 @ K2)) @ (C2 @ K2))))) @ S) = (((plus_plus @ A) @ (B2 @ A2)) @ ((((groups1340683514dd_sum @ B) @ A) @ C2) @ (((minus_minus @ (set @ B)) @ S) @ (((insert @ B) @ A2) @ (bot_bot @ (set @ B)))))))) & ((~ ((((member @ B) @ A2) @ S))) => (((((groups1340683514dd_sum @ B) @ A) @ (^ [K2:B]: (((((if @ A) @ (K2 = A2)) @ (B2 @ K2)) @ (C2 @ K2))))) @ S) = ((((groups1340683514dd_sum @ B) @ A) @ C2) @ (((minus_minus @ (set @ B)) @ S) @ (((insert @ B) @ A2) @ (bot_bot @ (set @ B))))))))))))))).
% 2.58/1.49 thf(fact_136_empty__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (^ [X:A]: ($false))))))).
% 2.58/1.49 thf(fact_137_pigeonhole__infinite__rel, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ B),R2:(A > (B > $o))]: (((~ (((finite_finite2 @ A) @ A3))) => (((finite_finite2 @ B) @ B3) => ((! [X3:A]: (((((member @ A) @ X3) @ A3) => (? [Xa2:B]: (((((member @ B) @ Xa2) @ B3) & ((R2 @ X3) @ Xa2))))))) => (? [X3:B]: (((((member @ B) @ X3) @ B3) & (~ (((finite_finite2 @ A) @ ((collect @ A) @ (^ [A4:A]: (((((member @ A) @ A4) @ A3) & ((R2 @ A4) @ X3))))))))))))))))).
% 2.58/1.49 thf(fact_138_not__finite__existsD, axiom, (! [A:$tType,P:(A > $o)]: (((~ (((finite_finite2 @ A) @ ((collect @ A) @ P)))) => (? [X_12:A]: ((P @ X_12))))))).
% 2.58/1.49 thf(fact_139_insert__Collect, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((insert @ A) @ A2) @ ((collect @ A) @ P)) = ((collect @ A) @ (^ [U2:A]: (((U2 != A2) => (P @ U2))))))))).
% 2.58/1.49 thf(fact_140_insert__compr, axiom, (! [A:$tType]: (((insert @ A) = (^ [A4:A,B4:(set @ A)]: (((collect @ A) @ (^ [X:A]: (((X = A4) | (((member @ A) @ X) @ B4))))))))))).
% 2.58/1.49 thf(fact_141_sum__delta__notmem_I4_J, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ B) => (! [X2:A,S2:(set @ A),P:(A > B),Q:(A > B)]: (((~ ((((member @ A) @ X2) @ S2))) => (((((groups1340683514dd_sum @ A) @ B) @ (^ [Y4:A]: (((((if @ B) @ (X2 = Y4)) @ (P @ Y4)) @ (Q @ Y4))))) @ S2) = ((((groups1340683514dd_sum @ A) @ B) @ Q) @ S2))))))))).
% 2.58/1.49 thf(fact_142_sum__delta__notmem_I3_J, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ B) => (! [X2:A,S2:(set @ A),P:(A > B),Q:(A > B)]: (((~ ((((member @ A) @ X2) @ S2))) => (((((groups1340683514dd_sum @ A) @ B) @ (^ [Y4:A]: (((((if @ B) @ (Y4 = X2)) @ (P @ Y4)) @ (Q @ Y4))))) @ S2) = ((((groups1340683514dd_sum @ A) @ B) @ Q) @ S2))))))))).
% 2.58/1.49 thf(fact_143_sum__delta__notmem_I2_J, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ B) => (! [X2:A,S2:(set @ A),P:(A > B),Q:(A > B)]: (((~ ((((member @ A) @ X2) @ S2))) => (((((groups1340683514dd_sum @ A) @ B) @ (^ [Y4:A]: (((((if @ B) @ (X2 = Y4)) @ (P @ X2)) @ (Q @ Y4))))) @ S2) = ((((groups1340683514dd_sum @ A) @ B) @ Q) @ S2))))))))).
% 2.58/1.49 thf(fact_144_sum__delta__notmem_I1_J, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ B) => (! [X2:A,S2:(set @ A),P:(A > B),Q:(A > B)]: (((~ ((((member @ A) @ X2) @ S2))) => (((((groups1340683514dd_sum @ A) @ B) @ (^ [Y4:A]: (((((if @ B) @ (Y4 = X2)) @ (P @ X2)) @ (Q @ Y4))))) @ S2) = ((((groups1340683514dd_sum @ A) @ B) @ Q) @ S2))))))))).
% 2.58/1.49 thf(fact_145_sum_Oswap, axiom, (! [A:$tType,B:$tType,C:$tType]: (((comm_monoid_add @ A) => (! [G:(B > (C > A)),B3:(set @ C),A3:(set @ B)]: ((((((groups1340683514dd_sum @ B) @ A) @ (^ [I3:B]: (((((groups1340683514dd_sum @ C) @ A) @ (G @ I3)) @ B3)))) @ A3) = ((((groups1340683514dd_sum @ C) @ A) @ (^ [J:C]: (((((groups1340683514dd_sum @ B) @ A) @ (^ [I3:B]: (((G @ I3) @ J)))) @ A3)))) @ B3)))))))).
% 2.58/1.49 thf(fact_146_infinite__imp__nonempty, axiom, (! [A:$tType,S:(set @ A)]: (((~ (((finite_finite2 @ A) @ S))) => (S != (bot_bot @ (set @ A))))))).
% 2.58/1.49 thf(fact_147_finite_OemptyI, axiom, (! [A:$tType]: (((finite_finite2 @ A) @ (bot_bot @ (set @ A)))))).
% 2.58/1.49 thf(fact_148_singleton__inject, axiom, (! [A:$tType,A2:A,B2:A]: ((((((insert @ A) @ A2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) => (A2 = B2))))).
% 2.58/1.49 thf(fact_149_insert__not__empty, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((insert @ A) @ A2) @ A3) != (bot_bot @ (set @ A)))))).
% 2.58/1.49 thf(fact_150_doubleton__eq__iff, axiom, (! [A:$tType,A2:A,B2:A,C2:A,D:A]: ((((((insert @ A) @ A2) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C2) @ (((insert @ A) @ D) @ (bot_bot @ (set @ A))))) = (((A2 = C2) & (B2 = D)) | ((A2 = D) & (B2 = C2))))))).
% 2.58/1.49 thf(fact_151_singleton__iff, axiom, (! [A:$tType,B2:A,A2:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = (B2 = A2))))).
% 2.58/1.49 thf(fact_152_singletonD, axiom, (! [A:$tType,B2:A,A2:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) => (B2 = A2))))).
% 2.58/1.49 thf(fact_153_finite_OinsertI, axiom, (! [A:$tType,A3:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A3) => ((finite_finite2 @ A) @ (((insert @ A) @ A2) @ A3)))))).
% 2.58/1.49 thf(fact_154_sum__product, axiom, (! [B:$tType,C:$tType,A:$tType]: (((semiring_0 @ B) => (! [F:(A > B),A3:(set @ A),G:(C > B),B3:(set @ C)]: (((((times_times @ B) @ ((((groups1340683514dd_sum @ A) @ B) @ F) @ A3)) @ ((((groups1340683514dd_sum @ C) @ B) @ G) @ B3)) = ((((groups1340683514dd_sum @ A) @ B) @ (^ [I3:A]: (((((groups1340683514dd_sum @ C) @ B) @ (^ [J:C]: ((((times_times @ B) @ (F @ I3)) @ (G @ J))))) @ B3)))) @ A3)))))))).
% 2.58/1.49 thf(fact_155_sum__distrib__left, axiom, (! [A:$tType,B:$tType]: (((semiring_0 @ A) => (! [R:A,F:(B > A),A3:(set @ B)]: (((((times_times @ A) @ R) @ ((((groups1340683514dd_sum @ B) @ A) @ F) @ A3)) = ((((groups1340683514dd_sum @ B) @ A) @ (^ [N:B]: ((((times_times @ A) @ R) @ (F @ N))))) @ A3)))))))).
% 2.58/1.49 thf(fact_156_sum__distrib__right, axiom, (! [A:$tType,B:$tType]: (((semiring_0 @ A) => (! [F:(B > A),A3:(set @ B),R:A]: (((((times_times @ A) @ ((((groups1340683514dd_sum @ B) @ A) @ F) @ A3)) @ R) = ((((groups1340683514dd_sum @ B) @ A) @ (^ [N:B]: ((((times_times @ A) @ (F @ N)) @ R)))) @ A3)))))))).
% 2.58/1.49 thf(fact_157_sum_Odistrib, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [G:(B > A),H:(B > A),A3:(set @ B)]: ((((((groups1340683514dd_sum @ B) @ A) @ (^ [X:B]: ((((plus_plus @ A) @ (G @ X)) @ (H @ X))))) @ A3) = (((plus_plus @ A) @ ((((groups1340683514dd_sum @ B) @ A) @ G) @ A3)) @ ((((groups1340683514dd_sum @ B) @ A) @ H) @ A3))))))))).
% 2.58/1.49 thf(fact_158_Collect__conv__if2, axiom, (! [A:$tType,P:(A > $o),A2:A]: ((((P @ A2) => (((collect @ A) @ (^ [X:A]: (((A2 = X) & (P @ X))))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) & ((~ ((P @ A2))) => (((collect @ A) @ (^ [X:A]: (((A2 = X) & (P @ X))))) = (bot_bot @ (set @ A)))))))).
% 2.58/1.49 thf(fact_159_Collect__conv__if, axiom, (! [A:$tType,P:(A > $o),A2:A]: ((((P @ A2) => (((collect @ A) @ (^ [X:A]: (((X = A2) & (P @ X))))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) & ((~ ((P @ A2))) => (((collect @ A) @ (^ [X:A]: (((X = A2) & (P @ X))))) = (bot_bot @ (set @ A)))))))).
% 2.58/1.49 thf(fact_160_sum__subtractf, axiom, (! [A:$tType,B:$tType]: (((ab_group_add @ A) => (! [F:(B > A),G:(B > A),A3:(set @ B)]: ((((((groups1340683514dd_sum @ B) @ A) @ (^ [X:B]: ((((minus_minus @ A) @ (F @ X)) @ (G @ X))))) @ A3) = (((minus_minus @ A) @ ((((groups1340683514dd_sum @ B) @ A) @ F) @ A3)) @ ((((groups1340683514dd_sum @ B) @ A) @ G) @ A3))))))))).
% 2.58/1.49 thf(fact_161_sum_Oswap__restrict, axiom, (! [A:$tType,B:$tType,C:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),B3:(set @ C),G:(B > (C > A)),R2:(B > (C > $o))]: ((((finite_finite2 @ B) @ A3) => (((finite_finite2 @ C) @ B3) => (((((groups1340683514dd_sum @ B) @ A) @ (^ [X:B]: (((((groups1340683514dd_sum @ C) @ A) @ (G @ X)) @ ((collect @ C) @ (^ [Y4:C]: (((((member @ C) @ Y4) @ B3) & ((R2 @ X) @ Y4))))))))) @ A3) = ((((groups1340683514dd_sum @ C) @ A) @ (^ [Y4:C]: (((((groups1340683514dd_sum @ B) @ A) @ (^ [X:B]: (((G @ X) @ Y4)))) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ A3) & ((R2 @ X) @ Y4))))))))) @ B3)))))))))).
% 2.58/1.49 thf(fact_162_infinite__finite__induct, axiom, (! [A:$tType,P:((set @ A) > $o),A3:(set @ A)]: (((! [A7:(set @ A)]: (((~ (((finite_finite2 @ A) @ A7))) => (P @ A7)))) => ((P @ (bot_bot @ (set @ A))) => ((! [X3:A,F3:(set @ A)]: ((((finite_finite2 @ A) @ F3) => ((~ ((((member @ A) @ X3) @ F3))) => ((P @ F3) => (P @ (((insert @ A) @ X3) @ F3))))))) => (P @ A3))))))).
% 2.58/1.49 thf(fact_163_finite__ne__induct, axiom, (! [A:$tType,F4:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ F4) => ((F4 != (bot_bot @ (set @ A))) => ((! [X3:A]: ((P @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))) => ((! [X3:A,F3:(set @ A)]: ((((finite_finite2 @ A) @ F3) => ((F3 != (bot_bot @ (set @ A))) => ((~ ((((member @ A) @ X3) @ F3))) => ((P @ F3) => (P @ (((insert @ A) @ X3) @ F3)))))))) => (P @ F4)))))))).
% 2.58/1.49 thf(fact_164_finite_Oinducts, axiom, (! [A:$tType,X2:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ X2) => ((P @ (bot_bot @ (set @ A))) => ((! [A7:(set @ A),A6:A]: ((((finite_finite2 @ A) @ A7) => ((P @ A7) => (P @ (((insert @ A) @ A6) @ A7)))))) => (P @ X2))))))).
% 2.58/1.49 thf(fact_165_finite__induct, axiom, (! [A:$tType,F4:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ F4) => ((P @ (bot_bot @ (set @ A))) => ((! [X3:A,F3:(set @ A)]: ((((finite_finite2 @ A) @ F3) => ((~ ((((member @ A) @ X3) @ F3))) => ((P @ F3) => (P @ (((insert @ A) @ X3) @ F3))))))) => (P @ F4))))))).
% 2.58/1.49 thf(fact_166_finite_Osimps, axiom, (! [A:$tType]: (((finite_finite2 @ A) = (^ [A4:(set @ A)]: (((A4 = (bot_bot @ (set @ A))) | (? [A5:(set @ A),B5:A]: (((A4 = (((insert @ A) @ B5) @ A5)) & ((finite_finite2 @ A) @ A5))))))))))).
% 2.58/1.49 thf(fact_167_finite_Ocases, axiom, (! [A:$tType,A2:(set @ A)]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => (~ ((! [A7:(set @ A)]: (((? [A6:A]: ((A2 = (((insert @ A) @ A6) @ A7)))) => (~ (((finite_finite2 @ A) @ A7))))))))))))).
% 2.58/1.49 thf(fact_168_prod_Ofinite__Collect__op, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [I4:(set @ B),X2:(B > A),Y:(B > A)]: ((((finite_finite2 @ B) @ ((collect @ B) @ (^ [I3:B]: (((((member @ B) @ I3) @ I4) & ((X2 @ I3) != (one_one @ A))))))) => (((finite_finite2 @ B) @ ((collect @ B) @ (^ [I3:B]: (((((member @ B) @ I3) @ I4) & ((Y @ I3) != (one_one @ A))))))) => ((finite_finite2 @ B) @ ((collect @ B) @ (^ [I3:B]: (((((member @ B) @ I3) @ I4) & ((((times_times @ A) @ (X2 @ I3)) @ (Y @ I3)) != (one_one @ A))))))))))))))).
% 2.58/1.49 thf(fact_169_sum_Oinsert__if, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),X2:B,G:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((((member @ B) @ X2) @ A3) => (((((groups1340683514dd_sum @ B) @ A) @ G) @ (((insert @ B) @ X2) @ A3)) = ((((groups1340683514dd_sum @ B) @ A) @ G) @ A3))) & ((~ ((((member @ B) @ X2) @ A3))) => (((((groups1340683514dd_sum @ B) @ A) @ G) @ (((insert @ B) @ X2) @ A3)) = (((plus_plus @ A) @ (G @ X2)) @ ((((groups1340683514dd_sum @ B) @ A) @ G) @ A3)))))))))))).
% 2.58/1.49 thf(fact_170_convex__singleton, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [A2:A]: (((convex @ A) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))))))))).
% 2.58/1.49 thf(fact_171_real__divide__square__eq, axiom, (! [R:real,A2:real]: (((((divide_divide @ real) @ (((times_times @ real) @ R) @ A2)) @ (((times_times @ real) @ R) @ R)) = (((divide_divide @ real) @ A2) @ R))))).
% 2.58/1.49 thf(fact_172_convex__empty, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => ((convex @ A) @ (bot_bot @ (set @ A))))))).
% 2.58/1.49 thf(fact_173_div__by__1, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (one_one @ A)) = A2))))))).
% 2.58/1.49 thf(fact_174_bits__div__by__1, axiom, (! [A:$tType]: (((semiring_bits @ A) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (one_one @ A)) = A2))))))).
% 2.58/1.49 thf(fact_175_add__diff__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = A2))))))).
% 2.58/1.49 thf(fact_176_diff__add__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ B2) = A2))))))).
% 2.58/1.49 thf(fact_177_add__right__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A2:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C2) @ A2)) = (B2 = C2)))))))).
% 2.58/1.49 thf(fact_178_add__left__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C2)) = (B2 = C2)))))))).
% 2.58/1.49 thf(fact_179_mult_Oright__neutral, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A2:A]: (((((times_times @ A) @ A2) @ (one_one @ A)) = A2))))))).
% 2.58/1.49 thf(fact_180_mult_Oleft__neutral, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A2:A]: (((((times_times @ A) @ (one_one @ A)) @ A2) = A2))))))).
% 2.58/1.49 thf(fact_181_add__diff__cancel__right_H, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = A2))))))).
% 2.58/1.49 thf(fact_182_add__diff__cancel__right, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,C2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ A2) @ B2)))))))).
% 2.58/1.49 thf(fact_183_add__diff__cancel__left_H, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ A2) = B2))))))).
% 2.58/1.49 thf(fact_184_add__diff__cancel__left, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [C2:A,A2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2)) = (((minus_minus @ A) @ A2) @ B2)))))))).
% 2.58/1.49 thf(fact_185_mult_Oleft__commute, axiom, (! [A:$tType]: (((ab_semigroup_mult @ A) => (! [B2:A,A2:A,C2:A]: (((((times_times @ A) @ B2) @ (((times_times @ A) @ A2) @ C2)) = (((times_times @ A) @ A2) @ (((times_times @ A) @ B2) @ C2))))))))).
% 2.58/1.49 thf(fact_186_mult_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_mult @ A) => ((times_times @ A) = (^ [A4:A,B5:A]: ((((times_times @ A) @ B5) @ A4)))))))).
% 2.58/1.49 thf(fact_187_mult_Oassoc, axiom, (! [A:$tType]: (((semigroup_mult @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ (((times_times @ A) @ A2) @ B2)) @ C2) = (((times_times @ A) @ A2) @ (((times_times @ A) @ B2) @ C2))))))))).
% 2.58/1.49 thf(fact_188_ab__semigroup__mult__class_Omult__ac_I1_J, axiom, (! [A:$tType]: (((ab_semigroup_mult @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ (((times_times @ A) @ A2) @ B2)) @ C2) = (((times_times @ A) @ A2) @ (((times_times @ A) @ B2) @ C2))))))))).
% 2.58/1.49 thf(fact_189_add__right__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A2:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C2) @ A2)) => (B2 = C2)))))))).
% 2.58/1.49 thf(fact_190_add__left__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C2)) => (B2 = C2)))))))).
% 2.58/1.49 thf(fact_191_add_Oleft__commute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [B2:A,A2:A,C2:A]: (((((plus_plus @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C2)) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 2.58/1.49 thf(fact_192_add_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => ((plus_plus @ A) = (^ [A4:A,B5:A]: ((((plus_plus @ A) @ B5) @ A4)))))))).
% 2.58/1.49 thf(fact_193_add_Oright__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [B2:A,A2:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C2) @ A2)) = (B2 = C2)))))))).
% 2.58/1.49 thf(fact_194_add_Oleft__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C2)) = (B2 = C2)))))))).
% 2.58/1.49 thf(fact_195_add_Oassoc, axiom, (! [A:$tType]: (((semigroup_add @ A) => (! [A2:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 2.58/1.49 thf(fact_196_group__cancel_Oadd2, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [B3:A,K:A,B2:A,A2:A]: (((B3 = (((plus_plus @ A) @ K) @ B2)) => ((((plus_plus @ A) @ A2) @ B3) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A2) @ B2)))))))))).
% 2.58/1.49 thf(fact_197_group__cancel_Oadd1, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A3:A,K:A,A2:A,B2:A]: (((A3 = (((plus_plus @ A) @ K) @ A2)) => ((((plus_plus @ A) @ A3) @ B2) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A2) @ B2)))))))))).
% 2.58/1.49 thf(fact_198_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I2:A,J2:A,K:A,L:A]: ((((I2 = J2) & (K = L)) => ((((plus_plus @ A) @ I2) @ K) = (((plus_plus @ A) @ J2) @ L))))))))).
% 2.58/1.49 thf(fact_199_ab__semigroup__add__class_Oadd__ac_I1_J, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [A2:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 2.58/1.49 thf(fact_200_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,C2:A,B2:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ B2) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2)))))))).
% 2.58/1.49 thf(fact_201_diff__eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A,D:A]: ((((((minus_minus @ A) @ A2) @ B2) = (((minus_minus @ A) @ C2) @ D)) => ((A2 = B2) = (C2 = D))))))))).
% 2.58/1.49 thf(fact_202_one__reorient, axiom, (! [A:$tType]: (((one @ A) => (! [X2:A]: ((((one_one @ A) = X2) = (X2 = (one_one @ A))))))))).
% 2.58/1.49 thf(fact_203_convex__set__plus, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [S:(set @ A),T:(set @ A)]: ((((convex @ A) @ S) => (((convex @ A) @ T) => ((convex @ A) @ (((plus_plus @ (set @ A)) @ S) @ T)))))))))).
% 2.58/1.49 thf(fact_204_finite__set__sum, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ B) => (! [A3:(set @ A),B3:(A > (set @ B))]: ((((finite_finite2 @ A) @ A3) => ((! [X3:A]: (((((member @ A) @ X3) @ A3) => ((finite_finite2 @ B) @ (B3 @ X3))))) => ((finite_finite2 @ B) @ ((((groups1340683514dd_sum @ A) @ (set @ B)) @ B3) @ A3)))))))))).
% 2.58/1.49 thf(fact_205_convex__set__sum, axiom, (! [B:$tType,A:$tType]: (((real_V1076094709vector @ B) => (! [A3:(set @ A),B3:(A > (set @ B))]: (((! [I5:A]: (((((member @ A) @ I5) @ A3) => ((convex @ B) @ (B3 @ I5))))) => ((convex @ B) @ ((((groups1340683514dd_sum @ A) @ (set @ B)) @ B3) @ A3))))))))).
% 2.58/1.49 thf(fact_206_combine__common__factor, axiom, (! [A:$tType]: (((semiring @ A) => (! [A2:A,E:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((times_times @ A) @ A2) @ E)) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E)) @ C2)) = (((plus_plus @ A) @ (((times_times @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ E)) @ C2)))))))).
% 2.58/1.49 thf(fact_207_distrib__right, axiom, (! [A:$tType]: (((semiring @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2))))))))).
% 2.58/1.49 thf(fact_208_distrib__left, axiom, (! [A:$tType]: (((semiring @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2)) = (((plus_plus @ A) @ (((times_times @ A) @ A2) @ B2)) @ (((times_times @ A) @ A2) @ C2))))))))).
% 2.58/1.49 thf(fact_209_comm__semiring__class_Odistrib, axiom, (! [A:$tType]: (((comm_semiring @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2))))))))).
% 2.58/1.49 thf(fact_210_ring__class_Oring__distribs_I1_J, axiom, (! [A:$tType]: (((ring @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2)) = (((plus_plus @ A) @ (((times_times @ A) @ A2) @ B2)) @ (((times_times @ A) @ A2) @ C2))))))))).
% 2.58/1.49 thf(fact_211_ring__class_Oring__distribs_I2_J, axiom, (! [A:$tType]: (((ring @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2))))))))).
% 2.58/1.49 thf(fact_212_right__diff__distrib_H, axiom, (! [A:$tType]: (((comm_s1003936772cancel @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ A2) @ (((minus_minus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((times_times @ A) @ A2) @ B2)) @ (((times_times @ A) @ A2) @ C2))))))))).
% 2.58/1.49 thf(fact_213_left__diff__distrib_H, axiom, (! [A:$tType]: (((comm_s1003936772cancel @ A) => (! [B2:A,C2:A,A2:A]: (((((times_times @ A) @ (((minus_minus @ A) @ B2) @ C2)) @ A2) = (((minus_minus @ A) @ (((times_times @ A) @ B2) @ A2)) @ (((times_times @ A) @ C2) @ A2))))))))).
% 2.58/1.49 thf(fact_214_right__diff__distrib, axiom, (! [A:$tType]: (((ring @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ A2) @ (((minus_minus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((times_times @ A) @ A2) @ B2)) @ (((times_times @ A) @ A2) @ C2))))))))).
% 2.58/1.49 thf(fact_215_left__diff__distrib, axiom, (! [A:$tType]: (((ring @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2) = (((minus_minus @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2))))))))).
% 2.58/1.49 thf(fact_216_mult_Ocomm__neutral, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A2:A]: (((((times_times @ A) @ A2) @ (one_one @ A)) = A2))))))).
% 2.58/1.49 thf(fact_217_comm__monoid__mult__class_Omult__1, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A2:A]: (((((times_times @ A) @ (one_one @ A)) @ A2) = A2))))))).
% 2.58/1.49 thf(fact_218_add__implies__diff, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [C2:A,B2:A,A2:A]: ((((((plus_plus @ A) @ C2) @ B2) = A2) => (C2 = (((minus_minus @ A) @ A2) @ B2))))))))).
% 2.58/1.49 thf(fact_219_diff__diff__add, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,B2:A,C2:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2) = (((minus_minus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 2.58/1.49 thf(fact_220_diff__add__eq__diff__diff__swap, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((minus_minus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ B2)))))))).
% 2.58/1.49 thf(fact_221_diff__add__eq, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2) = (((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ B2)))))))).
% 2.58/1.49 thf(fact_222_diff__diff__eq2, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((minus_minus @ A) @ A2) @ (((minus_minus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ B2)))))))).
% 2.58/1.49 thf(fact_223_add__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((plus_plus @ A) @ A2) @ (((minus_minus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2)))))))).
% 2.58/1.49 thf(fact_224_eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,C2:A,B2:A]: (((A2 = (((minus_minus @ A) @ C2) @ B2)) = ((((plus_plus @ A) @ A2) @ B2) = C2)))))))).
% 2.58/1.49 thf(fact_225_diff__eq__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: ((((((minus_minus @ A) @ A2) @ B2) = C2) = (A2 = (((plus_plus @ A) @ C2) @ B2))))))))).
% 2.58/1.49 thf(fact_226_group__cancel_Osub1, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A,K:A,A2:A,B2:A]: (((A3 = (((plus_plus @ A) @ K) @ A2)) => ((((minus_minus @ A) @ A3) @ B2) = (((plus_plus @ A) @ K) @ (((minus_minus @ A) @ A2) @ B2)))))))))).
% 2.58/1.49 thf(fact_227_lambda__one, axiom, (! [A:$tType]: (((monoid_mult @ A) => ((^ [X:A]: (X)) = ((times_times @ A) @ (one_one @ A))))))).
% 2.58/1.49 thf(fact_228_square__diff__square__factored, axiom, (! [A:$tType]: (((comm_ring @ A) => (! [X2:A,Y:A]: (((((minus_minus @ A) @ (((times_times @ A) @ X2) @ X2)) @ (((times_times @ A) @ Y) @ Y)) = (((times_times @ A) @ (((plus_plus @ A) @ X2) @ Y)) @ (((minus_minus @ A) @ X2) @ Y))))))))).
% 2.58/1.49 thf(fact_229_eq__add__iff2, axiom, (! [A:$tType]: (((ring @ A) => (! [A2:A,E:A,C2:A,B2:A,D:A]: ((((((plus_plus @ A) @ (((times_times @ A) @ A2) @ E)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E)) @ D)) = (C2 = (((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ B2) @ A2)) @ E)) @ D))))))))).
% 2.58/1.49 thf(fact_230_eq__add__iff1, axiom, (! [A:$tType]: (((ring @ A) => (! [A2:A,E:A,C2:A,B2:A,D:A]: ((((((plus_plus @ A) @ (((times_times @ A) @ A2) @ E)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E)) @ D)) = ((((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ E)) @ C2) = D)))))))).
% 2.58/1.49 thf(fact_231_square__diff__one__factored, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X2:A]: (((((minus_minus @ A) @ (((times_times @ A) @ X2) @ X2)) @ (one_one @ A)) = (((times_times @ A) @ (((plus_plus @ A) @ X2) @ (one_one @ A))) @ (((minus_minus @ A) @ X2) @ (one_one @ A)))))))))).
% 2.58/1.49 thf(fact_232_minus__apply, axiom, (! [B:$tType,A:$tType]: (((minus @ B) => ((minus_minus @ (A > B)) = (^ [A5:(A > B),B4:(A > B),X:A]: ((((minus_minus @ B) @ (A5 @ X)) @ (B4 @ X))))))))).
% 2.58/1.49 thf(fact_233_vector__space__over__itself_Oscale__one, axiom, (! [A:$tType]: (((field @ A) => (! [X2:A]: (((((times_times @ A) @ (one_one @ A)) @ X2) = X2))))))).
% 2.58/1.49 thf(fact_234_segment__degen__1, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [U:real,A2:A,B2:A]: ((((((plus_plus @ A) @ (((real_V1908273582scaleR @ A) @ (((minus_minus @ real) @ (one_one @ real)) @ U)) @ A2)) @ (((real_V1908273582scaleR @ A) @ U) @ B2)) = B2) = ((A2 = B2) | (U = (one_one @ real)))))))))).
% 2.58/1.49 thf(fact_235_vector__space__over__itself_Oscale__scale, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A,X2:A]: (((((times_times @ A) @ A2) @ (((times_times @ A) @ B2) @ X2)) = (((times_times @ A) @ (((times_times @ A) @ A2) @ B2)) @ X2)))))))).
% 2.58/1.49 thf(fact_236_vector__space__over__itself_Oscale__left__commute, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A,X2:A]: (((((times_times @ A) @ A2) @ (((times_times @ A) @ B2) @ X2)) = (((times_times @ A) @ B2) @ (((times_times @ A) @ A2) @ X2))))))))).
% 2.58/1.49 thf(fact_237_vector__space__over__itself_Oscale__left__distrib, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A,X2:A]: (((((times_times @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ X2) = (((plus_plus @ A) @ (((times_times @ A) @ A2) @ X2)) @ (((times_times @ A) @ B2) @ X2))))))))).
% 2.58/1.49 thf(fact_238_vector__space__over__itself_Oscale__right__distrib, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,X2:A,Y:A]: (((((times_times @ A) @ A2) @ (((plus_plus @ A) @ X2) @ Y)) = (((plus_plus @ A) @ (((times_times @ A) @ A2) @ X2)) @ (((times_times @ A) @ A2) @ Y))))))))).
% 2.58/1.49 thf(fact_239_vector__space__over__itself_Oscale__left__diff__distrib, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A,X2:A]: (((((times_times @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ X2) = (((minus_minus @ A) @ (((times_times @ A) @ A2) @ X2)) @ (((times_times @ A) @ B2) @ X2))))))))).
% 2.58/1.49 thf(fact_240_vector__space__over__itself_Oscale__right__diff__distrib, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,X2:A,Y:A]: (((((times_times @ A) @ A2) @ (((minus_minus @ A) @ X2) @ Y)) = (((minus_minus @ A) @ (((times_times @ A) @ A2) @ X2)) @ (((times_times @ A) @ A2) @ Y))))))))).
% 2.58/1.49 thf(fact_241_add__scaleR__degen, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (! [U:real,B2:A,V2:real,A2:A]: ((((((plus_plus @ A) @ (((real_V1908273582scaleR @ A) @ U) @ B2)) @ (((real_V1908273582scaleR @ A) @ V2) @ A2)) = (((plus_plus @ A) @ (((real_V1908273582scaleR @ A) @ U) @ A2)) @ (((real_V1908273582scaleR @ A) @ V2) @ B2))) => ((U != V2) => (A2 = B2))))))))).
% 2.58/1.49 thf(fact_242_vector__space__over__itself_Oscale__sum__left, axiom, (! [A:$tType,C:$tType]: (((field @ A) => (! [F:(C > A),A3:(set @ C),X2:A]: (((((times_times @ A) @ ((((groups1340683514dd_sum @ C) @ A) @ F) @ A3)) @ X2) = ((((groups1340683514dd_sum @ C) @ A) @ (^ [A4:C]: ((((times_times @ A) @ (F @ A4)) @ X2)))) @ A3)))))))).
% 2.58/1.49 thf(fact_243_vector__space__over__itself_Oscale__sum__right, axiom, (! [A:$tType,C:$tType]: (((field @ A) => (! [A2:A,F:(C > A),A3:(set @ C)]: (((((times_times @ A) @ A2) @ ((((groups1340683514dd_sum @ C) @ A) @ F) @ A3)) = ((((groups1340683514dd_sum @ C) @ A) @ (^ [X:C]: ((((times_times @ A) @ A2) @ (F @ X))))) @ A3)))))))).
% 2.58/1.49 thf(fact_244_fun__diff__def, axiom, (! [B:$tType,A:$tType]: (((minus @ B) => ((minus_minus @ (A > B)) = (^ [A5:(A > B),B4:(A > B),X:A]: ((((minus_minus @ B) @ (A5 @ X)) @ (B4 @ X))))))))).
% 2.58/1.49 thf(fact_245_vector__space__over__itself_Ofinite__Basis, axiom, (! [A:$tType]: (((field @ A) => ((finite_finite2 @ A) @ (((insert @ A) @ (one_one @ A)) @ (bot_bot @ (set @ A)))))))).
% 2.58/1.49 thf(fact_246_set__plus__intro, axiom, (! [A:$tType]: (((plus @ A) => (! [A2:A,C4:(set @ A),B2:A,D2:(set @ A)]: (((((member @ A) @ A2) @ C4) => ((((member @ A) @ B2) @ D2) => (((member @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ (((plus_plus @ (set @ A)) @ C4) @ D2)))))))))).
% 2.58/1.49 thf(fact_247_sum__clauses_I2_J, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [S:(set @ B),X2:B,F:(B > A)]: ((((finite_finite2 @ B) @ S) => (((((member @ B) @ X2) @ S) => (((((groups1340683514dd_sum @ B) @ A) @ F) @ (((insert @ B) @ X2) @ S)) = ((((groups1340683514dd_sum @ B) @ A) @ F) @ S))) & ((~ ((((member @ B) @ X2) @ S))) => (((((groups1340683514dd_sum @ B) @ A) @ F) @ (((insert @ B) @ X2) @ S)) = (((plus_plus @ A) @ (F @ X2)) @ ((((groups1340683514dd_sum @ B) @ A) @ F) @ S)))))))))))).
% 2.58/1.49 thf(fact_248_set__times__intro, axiom, (! [A:$tType]: (((times @ A) => (! [A2:A,C4:(set @ A),B2:A,D2:(set @ A)]: (((((member @ A) @ A2) @ C4) => ((((member @ A) @ B2) @ D2) => (((member @ A) @ (((times_times @ A) @ A2) @ B2)) @ (((times_times @ (set @ A)) @ C4) @ D2)))))))))).
% 2.58/1.49 thf(fact_249_set__times__elim, axiom, (! [A:$tType]: (((times @ A) => (! [X2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ X2) @ (((times_times @ (set @ A)) @ A3) @ B3)) => (~ ((! [A6:A,B7:A]: (((X2 = (((times_times @ A) @ A6) @ B7)) => ((((member @ A) @ A6) @ A3) => (~ ((((member @ A) @ B7) @ B3)))))))))))))))).
% 2.58/1.49 thf(fact_250_finite__set__times, axiom, (! [A:$tType]: (((times @ A) => (! [S2:(set @ A),T2:(set @ A)]: ((((finite_finite2 @ A) @ S2) => (((finite_finite2 @ A) @ T2) => ((finite_finite2 @ A) @ (((times_times @ (set @ A)) @ S2) @ T2)))))))))).
% 2.58/1.49 thf(fact_251_set__plus__elim, axiom, (! [A:$tType]: (((plus @ A) => (! [X2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ X2) @ (((plus_plus @ (set @ A)) @ A3) @ B3)) => (~ ((! [A6:A,B7:A]: (((X2 = (((plus_plus @ A) @ A6) @ B7)) => ((((member @ A) @ A6) @ A3) => (~ ((((member @ A) @ B7) @ B3)))))))))))))))).
% 2.58/1.49 thf(fact_252_finite__set__plus, axiom, (! [A:$tType]: (((plus @ A) => (! [S2:(set @ A),T2:(set @ A)]: ((((finite_finite2 @ A) @ S2) => (((finite_finite2 @ A) @ T2) => ((finite_finite2 @ A) @ (((plus_plus @ (set @ A)) @ S2) @ T2)))))))))).
% 2.58/1.49 thf(fact_253_set__one, axiom, (! [A:$tType]: (((one @ A) => ((one_one @ (set @ A)) = (((insert @ A) @ (one_one @ A)) @ (bot_bot @ (set @ A)))))))).
% 2.58/1.49 thf(fact_254_ai1, axiom, (((ord_less_eq @ real) @ (aa @ i)) @ (one_one @ real))).
% 2.58/1.49 thf(subcl_Real__Vector__Spaces_Oreal__vector___HOL_Otype, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (type @ A))))).
% 2.58/1.49 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Oplus, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (plus @ A))))).
% 2.58/1.49 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ominus, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (minus @ A))))).
% 2.58/1.49 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ogroup__add, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (group_add @ A))))).
% 2.58/1.49 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Oab__group__add, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (ab_group_add @ A))))).
% 2.58/1.49 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Osemigroup__add, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (semigroup_add @ A))))).
% 2.58/1.49 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ocomm__monoid__add, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (comm_monoid_add @ A))))).
% 2.58/1.49 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Oab__semigroup__add, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (ab_semigroup_add @ A))))).
% 2.58/1.49 thf(subcl_Real__Vector__Spaces_Oreal__vector___Real__Vector__Spaces_OscaleR, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (real_Vector_scaleR @ A))))).
% 2.58/1.49 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ocancel__semigroup__add, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (cancel_semigroup_add @ A))))).
% 2.58/1.49 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ocancel__comm__monoid__add, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (cancel1352612707id_add @ A))))).
% 2.58/1.49 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ocancel__ab__semigroup__add, axiom, (! [A:$tType]: (((real_V1076094709vector @ A) => (cancel146912293up_add @ A))))).
% 2.58/1.49 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A8:$tType,A9:$tType]: ((((finite_finite @ A8) & (finite_finite @ A9)) => (finite_finite @ (A8 > A9)))))).
% 2.58/1.49 thf(tcon_fun___Groups_Ominus, axiom, (! [A8:$tType,A9:$tType]: (((minus @ A9) => (minus @ (A8 > A9)))))).
% 2.58/1.49 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add, axiom, (ordere779506340up_add @ nat)).
% 2.58/1.49 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add, axiom, (cancel146912293up_add @ nat)).
% 2.58/1.49 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add, axiom, (cancel1352612707id_add @ nat)).
% 2.58/1.49 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel, axiom, (comm_s1003936772cancel @ nat)).
% 2.58/1.49 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add, axiom, (cancel_semigroup_add @ nat)).
% 2.58/1.49 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult, axiom, (ab_semigroup_mult @ nat)).
% 2.58/1.49 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult, axiom, (comm_monoid_mult @ nat)).
% 2.58/1.49 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add, axiom, (ab_semigroup_add @ nat)).
% 2.58/1.49 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add, axiom, (comm_monoid_add @ nat)).
% 2.58/1.49 thf(tcon_Nat_Onat___Groups_Osemigroup__mult, axiom, (semigroup_mult @ nat)).
% 2.58/1.49 thf(tcon_Nat_Onat___Rings_Osemidom__divide, axiom, (semidom_divide @ nat)).
% 2.58/1.49 thf(tcon_Nat_Onat___Parity_Osemiring__bits, axiom, (semiring_bits @ nat)).
% 2.58/1.49 thf(tcon_Nat_Onat___Groups_Osemigroup__add, axiom, (semigroup_add @ nat)).
% 2.58/1.49 thf(tcon_Nat_Onat___Rings_Ocomm__semiring, axiom, (comm_semiring @ nat)).
% 2.58/1.49 thf(tcon_Nat_Onat___Groups_Omonoid__mult, axiom, (monoid_mult @ nat)).
% 2.58/1.49 thf(tcon_Nat_Onat___Rings_Osemiring__0, axiom, (semiring_0 @ nat)).
% 2.58/1.49 thf(tcon_Nat_Onat___Rings_Osemiring, axiom, (semiring @ nat)).
% 2.58/1.49 thf(tcon_Nat_Onat___Groups_Otimes, axiom, (times @ nat)).
% 2.58/1.49 thf(tcon_Nat_Onat___Groups_Ominus_1, axiom, (minus @ nat)).
% 2.58/1.49 thf(tcon_Nat_Onat___Groups_Oplus, axiom, (plus @ nat)).
% 2.58/1.49 thf(tcon_Nat_Onat___Groups_Oone, axiom, (one @ nat)).
% 2.58/1.49 thf(tcon_Set_Oset___Groups_Oab__semigroup__mult_2, axiom, (! [A8:$tType]: (((ab_semigroup_mult @ A8) => (ab_semigroup_mult @ (set @ A8)))))).
% 2.58/1.49 thf(tcon_Set_Oset___Groups_Ocomm__monoid__mult_3, axiom, (! [A8:$tType]: (((comm_monoid_mult @ A8) => (comm_monoid_mult @ (set @ A8)))))).
% 2.58/1.49 thf(tcon_Set_Oset___Groups_Oab__semigroup__add_4, axiom, (! [A8:$tType]: (((ab_semigroup_add @ A8) => (ab_semigroup_add @ (set @ A8)))))).
% 2.58/1.49 thf(tcon_Set_Oset___Groups_Ocomm__monoid__add_5, axiom, (! [A8:$tType]: (((comm_monoid_add @ A8) => (comm_monoid_add @ (set @ A8)))))).
% 2.58/1.49 thf(tcon_Set_Oset___Groups_Osemigroup__mult_6, axiom, (! [A8:$tType]: (((semigroup_mult @ A8) => (semigroup_mult @ (set @ A8)))))).
% 2.58/1.49 thf(tcon_Set_Oset___Groups_Osemigroup__add_7, axiom, (! [A8:$tType]: (((semigroup_add @ A8) => (semigroup_add @ (set @ A8)))))).
% 2.58/1.49 thf(tcon_Set_Oset___Groups_Omonoid__mult_8, axiom, (! [A8:$tType]: (((monoid_mult @ A8) => (monoid_mult @ (set @ A8)))))).
% 2.58/1.49 thf(tcon_Set_Oset___Finite__Set_Ofinite_9, axiom, (! [A8:$tType]: (((finite_finite @ A8) => (finite_finite @ (set @ A8)))))).
% 2.58/1.49 thf(tcon_Set_Oset___Groups_Otimes_10, axiom, (! [A8:$tType]: (((times @ A8) => (times @ (set @ A8)))))).
% 2.58/1.49 thf(tcon_Set_Oset___Groups_Ominus_11, axiom, (! [A8:$tType]: ((minus @ (set @ A8))))).
% 2.58/1.49 thf(tcon_Set_Oset___Groups_Oplus_12, axiom, (! [A8:$tType]: (((plus @ A8) => (plus @ (set @ A8)))))).
% 2.58/1.49 thf(tcon_Set_Oset___Groups_Oone_13, axiom, (! [A8:$tType]: (((one @ A8) => (one @ (set @ A8)))))).
% 2.58/1.49 thf(tcon_HOL_Obool___Finite__Set_Ofinite_14, axiom, (finite_finite @ $o)).
% 2.58/1.49 thf(tcon_HOL_Obool___Groups_Ominus_15, axiom, (minus @ $o)).
% 2.58/1.49 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra, axiom, (real_V1015715145lgebra @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra, axiom, (real_V148923926lgebra @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_16, axiom, (ordere779506340up_add @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector, axiom, (real_V1076094709vector @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_17, axiom, (cancel146912293up_add @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_18, axiom, (cancel1352612707id_add @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_19, axiom, (comm_s1003936772cancel @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_20, axiom, (cancel_semigroup_add @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Real__Vector__Spaces_OscaleR, axiom, (real_Vector_scaleR @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_21, axiom, (ab_semigroup_mult @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_22, axiom, (comm_monoid_mult @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_23, axiom, (ab_semigroup_add @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_24, axiom, (comm_monoid_add @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Groups_Osemigroup__mult_25, axiom, (semigroup_mult @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Rings_Osemidom__divide_26, axiom, (semidom_divide @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Groups_Osemigroup__add_27, axiom, (semigroup_add @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Fields_Odivision__ring, axiom, (division_ring @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Rings_Ocomm__semiring_28, axiom, (comm_semiring @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Groups_Oab__group__add, axiom, (ab_group_add @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Groups_Omonoid__mult_29, axiom, (monoid_mult @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Rings_Osemiring__0_30, axiom, (semiring_0 @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Groups_Ogroup__add, axiom, (group_add @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Rings_Ocomm__ring, axiom, (comm_ring @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Rings_Osemiring_31, axiom, (semiring @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Rings_Oring__1, axiom, (ring_1 @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Groups_Otimes_32, axiom, (times @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Groups_Ominus_33, axiom, (minus @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Fields_Ofield, axiom, (field @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Groups_Oplus_34, axiom, (plus @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Rings_Oring, axiom, (ring @ real)).
% 2.58/1.49 thf(tcon_Real_Oreal___Groups_Oone_35, axiom, (one @ real)).
% 2.58/1.49 thf(tcon_Extended__Real_Oereal___Groups_Oordered__ab__semigroup__add_36, axiom, (ordere779506340up_add @ extended_ereal)).
% 2.58/1.49 thf(tcon_Extended__Real_Oereal___Groups_Oab__semigroup__mult_37, axiom, (ab_semigroup_mult @ extended_ereal)).
% 2.58/1.49 thf(tcon_Extended__Real_Oereal___Groups_Ocomm__monoid__mult_38, axiom, (comm_monoid_mult @ extended_ereal)).
% 2.58/1.49 thf(tcon_Extended__Real_Oereal___Groups_Oab__semigroup__add_39, axiom, (ab_semigroup_add @ extended_ereal)).
% 2.58/1.49 thf(tcon_Extended__Real_Oereal___Groups_Ocomm__monoid__add_40, axiom, (comm_monoid_add @ extended_ereal)).
% 2.58/1.49 thf(tcon_Extended__Real_Oereal___Groups_Osemigroup__mult_41, axiom, (semigroup_mult @ extended_ereal)).
% 2.58/1.49 thf(tcon_Extended__Real_Oereal___Groups_Osemigroup__add_42, axiom, (semigroup_add @ extended_ereal)).
% 2.58/1.49 thf(tcon_Extended__Real_Oereal___Groups_Omonoid__mult_43, axiom, (monoid_mult @ extended_ereal)).
% 2.58/1.49 thf(tcon_Extended__Real_Oereal___Groups_Otimes_44, axiom, (times @ extended_ereal)).
% 2.58/1.49 thf(tcon_Extended__Real_Oereal___Groups_Ominus_45, axiom, (minus @ extended_ereal)).
% 2.58/1.49 thf(tcon_Extended__Real_Oereal___Groups_Oplus_46, axiom, (plus @ extended_ereal)).
% 2.58/1.49 thf(tcon_Extended__Real_Oereal___Groups_Oone_47, axiom, (one @ extended_ereal)).
% 2.58/1.49 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.58/1.49 thf(help_If_2_1_T, axiom, (! [A:$tType,X2:A,Y:A]: ((((((if @ A) @ $false) @ X2) @ Y) = Y)))).
% 2.58/1.49 thf(help_If_1_1_T, axiom, (! [A:$tType,X2:A,Y:A]: ((((((if @ A) @ $true) @ X2) @ Y) = X2)))).
% 2.58/1.49 thf(tfree_0, hypothesis, (real_V1076094709vector @ b)).
% 2.58/1.49 thf(conj_0, conjecture, ((f @ (((plus_plus @ b) @ (((real_V1908273582scaleR @ b) @ (((times_times @ real) @ (((minus_minus @ real) @ (one_one @ real)) @ (aa @ i))) @ ((inverse_inverse @ real) @ (((minus_minus @ real) @ (one_one @ real)) @ (aa @ i))))) @ ((((groups1340683514dd_sum @ a) @ b) @ (^ [J:a]: ((((real_V1908273582scaleR @ b) @ (aa @ J)) @ (y @ J))))) @ sa))) @ (((real_V1908273582scaleR @ b) @ (aa @ i)) @ (y @ i)))) = (f @ (((plus_plus @ b) @ (((real_V1908273582scaleR @ b) @ (((minus_minus @ real) @ (one_one @ real)) @ (aa @ i))) @ ((((groups1340683514dd_sum @ a) @ b) @ (^ [J:a]: ((((real_V1908273582scaleR @ b) @ (((times_times @ real) @ (aa @ J)) @ ((inverse_inverse @ real) @ (((minus_minus @ real) @ (one_one @ real)) @ (aa @ i))))) @ (y @ J))))) @ sa))) @ (((real_V1908273582scaleR @ b) @ (aa @ i)) @ (y @ i)))))).
% 2.58/1.49 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.a0uzsVYvBA/DTF2DTF4788.p
% 2.58/1.49 ------------------------
% 2.94/1.53 ---- Cleaned THF ---
% 2.94/1.53 thf(ty_t_Extended__Real_Oereal,type,
% 2.94/1.53 extended_ereal: $tType ).
% 2.94/1.53
% 2.94/1.53 thf(ty_t_Real_Oreal,type,
% 2.94/1.53 real: $tType ).
% 2.94/1.53
% 2.94/1.53 thf(ty_t_Set_Oset,type,
% 2.94/1.53 set: $tType > $tType ).
% 2.94/1.53
% 2.94/1.53 thf(ty_t_Nat_Onat,type,
% 2.94/1.53 nat: $tType ).
% 2.94/1.53
% 2.94/1.53 thf(ty_tf_b,type,
% 2.94/1.53 b: $tType ).
% 2.94/1.53
% 2.94/1.53 thf(ty_tf_a,type,
% 2.94/1.53 a: $tType ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_HOL_Otype,type,
% 2.94/1.53 type:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Groups_Oone,type,
% 2.94/1.53 one:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Rings_Oring,type,
% 2.94/1.53 ring:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Groups_Oplus,type,
% 2.94/1.53 plus:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Fields_Ofield,type,
% 2.94/1.53 field:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Groups_Ominus,type,
% 2.94/1.53 minus:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Groups_Otimes,type,
% 2.94/1.53 times:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Rings_Oring__1,type,
% 2.94/1.53 ring_1:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Rings_Osemiring,type,
% 2.94/1.53 semiring:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Rings_Ocomm__ring,type,
% 2.94/1.53 comm_ring:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Groups_Ogroup__add,type,
% 2.94/1.53 group_add:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Rings_Osemiring__0,type,
% 2.94/1.53 semiring_0:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Finite__Set_Ofinite,type,
% 2.94/1.53 finite_finite:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Groups_Omonoid__mult,type,
% 2.94/1.53 monoid_mult:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Groups_Oab__group__add,type,
% 2.94/1.53 ab_group_add:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Rings_Ocomm__semiring,type,
% 2.94/1.53 comm_semiring:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Fields_Odivision__ring,type,
% 2.94/1.53 division_ring:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Groups_Osemigroup__add,type,
% 2.94/1.53 semigroup_add:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Parity_Osemiring__bits,type,
% 2.94/1.53 semiring_bits:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Rings_Osemidom__divide,type,
% 2.94/1.53 semidom_divide:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Groups_Osemigroup__mult,type,
% 2.94/1.53 semigroup_mult:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 2.94/1.53 comm_monoid_add:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 2.94/1.53 ab_semigroup_add:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Groups_Ocomm__monoid__mult,type,
% 2.94/1.53 comm_monoid_mult:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Groups_Oab__semigroup__mult,type,
% 2.94/1.53 ab_semigroup_mult:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Real__Vector__Spaces_OscaleR,type,
% 2.94/1.53 real_Vector_scaleR:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 2.94/1.53 cancel_semigroup_add:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
% 2.94/1.53 comm_s1003936772cancel:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 2.94/1.53 cancel1352612707id_add:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.94/1.53 cancel146912293up_add:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type,
% 2.94/1.53 real_V1076094709vector:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 2.94/1.53 ordere779506340up_add:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Real__Vector__Spaces_Oreal__algebra,type,
% 2.94/1.53 real_V148923926lgebra:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_cl_Real__Vector__Spaces_Oreal__div__algebra,type,
% 2.94/1.53 real_V1015715145lgebra:
% 2.94/1.53 !>[A: $tType] : $o ).
% 2.94/1.53
% 2.94/1.53 thf(sy_c_Convex_Oconvex,type,
% 2.94/1.53 convex:
% 2.94/1.53 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.94/1.53
% 2.94/1.53 thf(sy_c_Fields_Oinverse__class_Oinverse,type,
% 2.94/1.53 inverse_inverse:
% 2.94/1.53 !>[A: $tType] : ( A > A ) ).
% 2.94/1.53
% 2.94/1.53 thf(sy_c_Finite__Set_Ofinite,type,
% 2.94/1.53 finite_finite2:
% 2.94/1.53 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.94/1.53
% 2.94/1.53 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.94/1.53 minus_minus:
% 2.94/1.53 !>[A: $tType] : ( A > A > A ) ).
% 2.94/1.53
% 2.94/1.53 thf(sy_c_Groups_Oone__class_Oone,type,
% 2.94/1.53 one_one:
% 2.94/1.53 !>[A: $tType] : A ).
% 2.94/1.53
% 2.94/1.53 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 2.94/1.53 plus_plus:
% 2.94/1.53 !>[A: $tType] : ( A > A > A ) ).
% 2.94/1.53
% 2.94/1.53 thf(sy_c_Groups_Otimes__class_Otimes,type,
% 2.94/1.53 times_times:
% 2.94/1.53 !>[A: $tType] : ( A > A > A ) ).
% 2.94/1.53
% 2.94/1.53 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
% 2.94/1.53 groups1340683514dd_sum:
% 2.94/1.53 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
% 2.94/1.53
% 2.94/1.53 thf(sy_c_If,type,
% 2.94/1.53 if:
% 2.94/1.53 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.94/1.53
% 2.94/1.53 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Oconvex__on,type,
% 2.94/1.53 lower_673667120vex_on:
% 2.94/1.53 !>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > $o ) ).
% 2.94/1.53
% 2.94/1.53 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.94/1.53 bot_bot:
% 2.94/1.53 !>[A: $tType] : A ).
% 2.94/1.53
% 2.94/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.94/1.53 ord_less_eq:
% 2.94/1.53 !>[A: $tType] : ( A > A > $o ) ).
% 2.94/1.53
% 2.94/1.53 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
% 2.94/1.53 real_V1908273582scaleR:
% 2.94/1.53 !>[A: $tType] : ( real > A > A ) ).
% 2.94/1.53
% 2.94/1.53 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 2.94/1.53 divide_divide:
% 2.94/1.53 !>[A: $tType] : ( A > A > A ) ).
% 2.94/1.53
% 2.94/1.53 thf(sy_c_Set_OCollect,type,
% 2.94/1.53 collect:
% 2.94/1.53 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(sy_c_Set_Oinsert,type,
% 2.94/1.53 insert:
% 2.94/1.53 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(sy_c_member,type,
% 2.94/1.53 member:
% 2.94/1.53 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.94/1.53
% 2.94/1.53 thf(sy_v_C,type,
% 2.94/1.53 c: set @ b ).
% 2.94/1.53
% 2.94/1.53 thf(sy_v_a,type,
% 2.94/1.53 a2: a > real ).
% 2.94/1.53
% 2.94/1.53 thf(sy_v_aa____,type,
% 2.94/1.53 aa: a > real ).
% 2.94/1.53
% 2.94/1.53 thf(sy_v_f,type,
% 2.94/1.53 f: b > extended_ereal ).
% 2.94/1.53
% 2.94/1.53 thf(sy_v_i____,type,
% 2.94/1.53 i: a ).
% 2.94/1.53
% 2.94/1.53 thf(sy_v_s,type,
% 2.94/1.53 s: set @ a ).
% 2.94/1.53
% 2.94/1.53 thf(sy_v_sa____,type,
% 2.94/1.53 sa: set @ a ).
% 2.94/1.53
% 2.94/1.53 thf(sy_v_y,type,
% 2.94/1.53 y: a > b ).
% 2.94/1.53
% 2.94/1.53 thf(fact_0_insert_Ohyps_I3_J,axiom,
% 2.94/1.53 ~ ( member @ a @ i @ sa ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_1_asm,axiom,
% 2.94/1.53 ( ( aa @ i )
% 2.94/1.53 != ( one_one @ real ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_2__092_060open_062_I_092_060Sum_062x_092_060in_062s_O_Aa_Ax_A_K_092_060_094sub_062R_Ay_Ax_A_P_092_060_094sub_062R_A_I1_A_N_Aa_Ai_J_J_A_061_A_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_P_092_060_094sub_062R_A_I1_A_N_Aa_Ai_J_092_060close_062,axiom,
% 2.94/1.53 ( ( groups1340683514dd_sum @ a @ b
% 2.94/1.53 @ ^ [X: a] : ( real_V1908273582scaleR @ b @ ( inverse_inverse @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) @ ( real_V1908273582scaleR @ b @ ( aa @ X ) @ ( y @ X ) ) )
% 2.94/1.53 @ sa )
% 2.94/1.53 = ( real_V1908273582scaleR @ b @ ( inverse_inverse @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) )
% 2.94/1.53 @ ( groups1340683514dd_sum @ a @ b
% 2.94/1.53 @ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
% 2.94/1.53 @ sa ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_3_insert_Ohyps_I2_J,axiom,
% 2.94/1.53 ( sa
% 2.94/1.53 != ( bot_bot @ ( set @ a ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_4_insert_Ohyps_I1_J,axiom,
% 2.94/1.53 finite_finite2 @ a @ sa ).
% 2.94/1.53
% 2.94/1.53 thf(fact_5__092_060open_062f_A_I_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_L_Aa_Ai_A_K_092_060_094sub_062R_Ay_Ai_J_A_061_Af_A_I_I_I1_A_N_Aa_Ai_J_A_K_Ainverse_A_I1_A_N_Aa_Ai_J_J_A_K_092_060_094sub_062R_A_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_L_Aa_Ai_A_K_092_060_094sub_062R_Ay_Ai_J_092_060close_062,axiom,
% 2.94/1.53 ( ( f
% 2.94/1.53 @ ( plus_plus @ b
% 2.94/1.53 @ ( groups1340683514dd_sum @ a @ b
% 2.94/1.53 @ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
% 2.94/1.53 @ sa )
% 2.94/1.53 @ ( real_V1908273582scaleR @ b @ ( aa @ i ) @ ( y @ i ) ) ) )
% 2.94/1.53 = ( f
% 2.94/1.53 @ ( plus_plus @ b
% 2.94/1.53 @ ( real_V1908273582scaleR @ b @ ( times_times @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) @ ( inverse_inverse @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) )
% 2.94/1.53 @ ( groups1340683514dd_sum @ a @ b
% 2.94/1.53 @ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
% 2.94/1.53 @ sa ) )
% 2.94/1.53 @ ( real_V1908273582scaleR @ b @ ( aa @ i ) @ ( y @ i ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_6_calculation,axiom,
% 2.94/1.53 ( ( f
% 2.94/1.53 @ ( groups1340683514dd_sum @ a @ b
% 2.94/1.53 @ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
% 2.94/1.53 @ ( insert @ a @ i @ sa ) ) )
% 2.94/1.53 = ( f
% 2.94/1.53 @ ( plus_plus @ b
% 2.94/1.53 @ ( real_V1908273582scaleR @ b @ ( times_times @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) @ ( inverse_inverse @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) )
% 2.94/1.53 @ ( groups1340683514dd_sum @ a @ b
% 2.94/1.53 @ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
% 2.94/1.53 @ sa ) )
% 2.94/1.53 @ ( real_V1908273582scaleR @ b @ ( aa @ i ) @ ( y @ i ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_7__092_060open_062f_A_I_092_060Sum_062j_092_060in_062insert_Ai_As_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_061_Af_A_I_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_L_Aa_Ai_A_K_092_060_094sub_062R_Ay_Ai_J_092_060close_062,axiom,
% 2.94/1.53 ( ( f
% 2.94/1.53 @ ( groups1340683514dd_sum @ a @ b
% 2.94/1.53 @ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
% 2.94/1.53 @ ( insert @ a @ i @ sa ) ) )
% 2.94/1.53 = ( f
% 2.94/1.53 @ ( plus_plus @ b
% 2.94/1.53 @ ( groups1340683514dd_sum @ a @ b
% 2.94/1.53 @ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
% 2.94/1.53 @ sa )
% 2.94/1.53 @ ( real_V1908273582scaleR @ b @ ( aa @ i ) @ ( y @ i ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_8_scaleR__collapse,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( real_V1076094709vector @ A )
% 2.94/1.53 => ! [U: real,A2: A] :
% 2.94/1.53 ( ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ ( minus_minus @ real @ ( one_one @ real ) @ U ) @ A2 ) @ ( real_V1908273582scaleR @ A @ U @ A2 ) )
% 2.94/1.53 = A2 ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_9_scaleR__eq__iff,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( real_V1076094709vector @ A )
% 2.94/1.53 => ! [B2: A,U: real,A2: A] :
% 2.94/1.53 ( ( ( plus_plus @ A @ B2 @ ( real_V1908273582scaleR @ A @ U @ A2 ) )
% 2.94/1.53 = ( plus_plus @ A @ A2 @ ( real_V1908273582scaleR @ A @ U @ B2 ) ) )
% 2.94/1.53 = ( ( A2 = B2 )
% 2.94/1.53 | ( U
% 2.94/1.53 = ( one_one @ real ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_10_yai_I1_J,axiom,
% 2.94/1.53 member @ b @ ( y @ i ) @ c ).
% 2.94/1.53
% 2.94/1.53 thf(fact_11_scaleR__scaleR,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( real_V1076094709vector @ A )
% 2.94/1.53 => ! [A2: real,B2: real,X2: A] :
% 2.94/1.53 ( ( real_V1908273582scaleR @ A @ A2 @ ( real_V1908273582scaleR @ A @ B2 @ X2 ) )
% 2.94/1.53 = ( real_V1908273582scaleR @ A @ ( times_times @ real @ A2 @ B2 ) @ X2 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_12_scaleR__one,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( real_V1076094709vector @ A )
% 2.94/1.53 => ! [X2: A] :
% 2.94/1.53 ( ( real_V1908273582scaleR @ A @ ( one_one @ real ) @ X2 )
% 2.94/1.53 = X2 ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_13__092_060open_062sum_Aa_As_A_061_A1_A_N_Aa_Ai_092_060close_062,axiom,
% 2.94/1.53 ( ( groups1340683514dd_sum @ a @ real @ aa @ sa )
% 2.94/1.53 = ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_14_inverse__eq__1__iff,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( field @ A )
% 2.94/1.53 => ! [X2: A] :
% 2.94/1.53 ( ( ( inverse_inverse @ A @ X2 )
% 2.94/1.53 = ( one_one @ A ) )
% 2.94/1.53 = ( X2
% 2.94/1.53 = ( one_one @ A ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_15_inverse__1,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( division_ring @ A )
% 2.94/1.53 => ( ( inverse_inverse @ A @ ( one_one @ A ) )
% 2.94/1.53 = ( one_one @ A ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_16_inverse__mult__distrib,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( field @ A )
% 2.94/1.53 => ! [A2: A,B2: A] :
% 2.94/1.53 ( ( inverse_inverse @ A @ ( times_times @ A @ A2 @ B2 ) )
% 2.94/1.53 = ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_17_mult__scaleR__left,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( real_V148923926lgebra @ A )
% 2.94/1.53 => ! [A2: real,X2: A,Y: A] :
% 2.94/1.53 ( ( times_times @ A @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) @ Y )
% 2.94/1.53 = ( real_V1908273582scaleR @ A @ A2 @ ( times_times @ A @ X2 @ Y ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_18_assms_I1_J,axiom,
% 2.94/1.53 finite_finite2 @ a @ s ).
% 2.94/1.53
% 2.94/1.53 thf(fact_19_assms_I2_J,axiom,
% 2.94/1.53 ( s
% 2.94/1.53 != ( bot_bot @ ( set @ a ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_20_assms_I7_J,axiom,
% 2.94/1.53 ! [I: a] :
% 2.94/1.53 ( ( member @ a @ I @ s )
% 2.94/1.53 => ( member @ b @ ( y @ I ) @ c ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_21_insert_Oprems_I3_J,axiom,
% 2.94/1.53 ! [I: a] :
% 2.94/1.53 ( ( member @ a @ I @ ( insert @ a @ i @ sa ) )
% 2.94/1.53 => ( member @ b @ ( y @ I ) @ c ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_22_inverse__inverse__eq,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( division_ring @ A )
% 2.94/1.53 => ! [A2: A] :
% 2.94/1.53 ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
% 2.94/1.53 = A2 ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_23_inverse__eq__iff__eq,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( division_ring @ A )
% 2.94/1.53 => ! [A2: A,B2: A] :
% 2.94/1.53 ( ( ( inverse_inverse @ A @ A2 )
% 2.94/1.53 = ( inverse_inverse @ A @ B2 ) )
% 2.94/1.53 = ( A2 = B2 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_24_insert_Oprems_I1_J,axiom,
% 2.94/1.53 ( ( groups1340683514dd_sum @ a @ real @ aa @ ( insert @ a @ i @ sa ) )
% 2.94/1.53 = ( one_one @ real ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_25_assms_I3_J,axiom,
% 2.94/1.53 lower_673667120vex_on @ b @ c @ f ).
% 2.94/1.53
% 2.94/1.53 thf(fact_26_fis,axiom,
% 2.94/1.53 finite_finite2 @ a @ ( insert @ a @ i @ sa ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_27_mult__scaleR__right,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( real_V148923926lgebra @ A )
% 2.94/1.53 => ! [X2: A,A2: real,Y: A] :
% 2.94/1.53 ( ( times_times @ A @ X2 @ ( real_V1908273582scaleR @ A @ A2 @ Y ) )
% 2.94/1.53 = ( real_V1908273582scaleR @ A @ A2 @ ( times_times @ A @ X2 @ Y ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_28_assms_I4_J,axiom,
% 2.94/1.53 convex @ b @ c ).
% 2.94/1.53
% 2.94/1.53 thf(fact_29_assms_I5_J,axiom,
% 2.94/1.53 ( ( groups1340683514dd_sum @ a @ real @ a2 @ s )
% 2.94/1.53 = ( one_one @ real ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_30_real__scaleR__def,axiom,
% 2.94/1.53 ( ( real_V1908273582scaleR @ real )
% 2.94/1.53 = ( times_times @ real ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_31_scale__left__commute,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( real_V1076094709vector @ A )
% 2.94/1.53 => ! [A2: real,B2: real,X2: A] :
% 2.94/1.53 ( ( real_V1908273582scaleR @ A @ A2 @ ( real_V1908273582scaleR @ A @ B2 @ X2 ) )
% 2.94/1.53 = ( real_V1908273582scaleR @ A @ B2 @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_32_inverse__eq__imp__eq,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( division_ring @ A )
% 2.94/1.53 => ! [A2: A,B2: A] :
% 2.94/1.53 ( ( ( inverse_inverse @ A @ A2 )
% 2.94/1.53 = ( inverse_inverse @ A @ B2 ) )
% 2.94/1.53 => ( A2 = B2 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_33_scaleR__add__right,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( real_V1076094709vector @ A )
% 2.94/1.53 => ! [A2: real,X2: A,Y: A] :
% 2.94/1.53 ( ( real_V1908273582scaleR @ A @ A2 @ ( plus_plus @ A @ X2 @ Y ) )
% 2.94/1.53 = ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) @ ( real_V1908273582scaleR @ A @ A2 @ Y ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_34_scaleR__add__left,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( real_V1076094709vector @ A )
% 2.94/1.53 => ! [A2: real,B2: real,X2: A] :
% 2.94/1.53 ( ( real_V1908273582scaleR @ A @ ( plus_plus @ real @ A2 @ B2 ) @ X2 )
% 2.94/1.53 = ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) @ ( real_V1908273582scaleR @ A @ B2 @ X2 ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_35_scaleR__left_Oadd,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( real_V1076094709vector @ A )
% 2.94/1.53 => ! [X2: real,Y: real,Xa: A] :
% 2.94/1.53 ( ( real_V1908273582scaleR @ A @ ( plus_plus @ real @ X2 @ Y ) @ Xa )
% 2.94/1.53 = ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ X2 @ Xa ) @ ( real_V1908273582scaleR @ A @ Y @ Xa ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_36_scale__right__diff__distrib,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( real_V1076094709vector @ A )
% 2.94/1.53 => ! [A2: real,X2: A,Y: A] :
% 2.94/1.53 ( ( real_V1908273582scaleR @ A @ A2 @ ( minus_minus @ A @ X2 @ Y ) )
% 2.94/1.53 = ( minus_minus @ A @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) @ ( real_V1908273582scaleR @ A @ A2 @ Y ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_37_mult__commute__imp__mult__inverse__commute,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( division_ring @ A )
% 2.94/1.53 => ! [Y: A,X2: A] :
% 2.94/1.53 ( ( ( times_times @ A @ Y @ X2 )
% 2.94/1.53 = ( times_times @ A @ X2 @ Y ) )
% 2.94/1.53 => ( ( times_times @ A @ ( inverse_inverse @ A @ Y ) @ X2 )
% 2.94/1.53 = ( times_times @ A @ X2 @ ( inverse_inverse @ A @ Y ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_38_scale__sum__right,axiom,
% 2.94/1.53 ! [A: $tType,C: $tType] :
% 2.94/1.53 ( ( real_V1076094709vector @ A )
% 2.94/1.53 => ! [A2: real,F: C > A,A3: set @ C] :
% 2.94/1.53 ( ( real_V1908273582scaleR @ A @ A2 @ ( groups1340683514dd_sum @ C @ A @ F @ A3 ) )
% 2.94/1.53 = ( groups1340683514dd_sum @ C @ A
% 2.94/1.53 @ ^ [X: C] : ( real_V1908273582scaleR @ A @ A2 @ ( F @ X ) )
% 2.94/1.53 @ A3 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_39_scale__sum__left,axiom,
% 2.94/1.53 ! [A: $tType,C: $tType] :
% 2.94/1.53 ( ( real_V1076094709vector @ A )
% 2.94/1.53 => ! [F: C > real,A3: set @ C,X2: A] :
% 2.94/1.53 ( ( real_V1908273582scaleR @ A @ ( groups1340683514dd_sum @ C @ real @ F @ A3 ) @ X2 )
% 2.94/1.53 = ( groups1340683514dd_sum @ C @ A
% 2.94/1.53 @ ^ [A4: C] : ( real_V1908273582scaleR @ A @ ( F @ A4 ) @ X2 )
% 2.94/1.53 @ A3 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_40_scaleR__right_Osum,axiom,
% 2.94/1.53 ! [A: $tType,C: $tType] :
% 2.94/1.53 ( ( real_V1076094709vector @ A )
% 2.94/1.53 => ! [A2: real,G: C > A,A3: set @ C] :
% 2.94/1.53 ( ( real_V1908273582scaleR @ A @ A2 @ ( groups1340683514dd_sum @ C @ A @ G @ A3 ) )
% 2.94/1.53 = ( groups1340683514dd_sum @ C @ A
% 2.94/1.53 @ ^ [X: C] : ( real_V1908273582scaleR @ A @ A2 @ ( G @ X ) )
% 2.94/1.53 @ A3 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_41_scaleR__left_Osum,axiom,
% 2.94/1.53 ! [A: $tType,C: $tType] :
% 2.94/1.53 ( ( real_V1076094709vector @ A )
% 2.94/1.53 => ! [G: C > real,A3: set @ C,X2: A] :
% 2.94/1.53 ( ( real_V1908273582scaleR @ A @ ( groups1340683514dd_sum @ C @ real @ G @ A3 ) @ X2 )
% 2.94/1.53 = ( groups1340683514dd_sum @ C @ A
% 2.94/1.53 @ ^ [X: C] : ( real_V1908273582scaleR @ A @ ( G @ X ) @ X2 )
% 2.94/1.53 @ A3 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_42_inverse__unique,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( division_ring @ A )
% 2.94/1.53 => ! [A2: A,B2: A] :
% 2.94/1.53 ( ( ( times_times @ A @ A2 @ B2 )
% 2.94/1.53 = ( one_one @ A ) )
% 2.94/1.53 => ( ( inverse_inverse @ A @ A2 )
% 2.94/1.53 = B2 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_43_scale__left__diff__distrib,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( real_V1076094709vector @ A )
% 2.94/1.53 => ! [A2: real,B2: real,X2: A] :
% 2.94/1.53 ( ( real_V1908273582scaleR @ A @ ( minus_minus @ real @ A2 @ B2 ) @ X2 )
% 2.94/1.53 = ( minus_minus @ A @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) @ ( real_V1908273582scaleR @ A @ B2 @ X2 ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_44_scaleR__left_Odiff,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( real_V1076094709vector @ A )
% 2.94/1.53 => ! [X2: real,Y: real,Xa: A] :
% 2.94/1.53 ( ( real_V1908273582scaleR @ A @ ( minus_minus @ real @ X2 @ Y ) @ Xa )
% 2.94/1.53 = ( minus_minus @ A @ ( real_V1908273582scaleR @ A @ X2 @ Xa ) @ ( real_V1908273582scaleR @ A @ Y @ Xa ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_45_mem__Collect__eq,axiom,
% 2.94/1.53 ! [A: $tType,A2: A,P: A > $o] :
% 2.94/1.53 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.94/1.53 = ( P @ A2 ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_46_Collect__mem__eq,axiom,
% 2.94/1.53 ! [A: $tType,A3: set @ A] :
% 2.94/1.53 ( ( collect @ A
% 2.94/1.53 @ ^ [X: A] : ( member @ A @ X @ A3 ) )
% 2.94/1.53 = A3 ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_47_Collect__cong,axiom,
% 2.94/1.53 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.94/1.53 ( ! [X3: A] :
% 2.94/1.53 ( ( P @ X3 )
% 2.94/1.53 = ( Q @ X3 ) )
% 2.94/1.53 => ( ( collect @ A @ P )
% 2.94/1.53 = ( collect @ A @ Q ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_48_ext,axiom,
% 2.94/1.53 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.94/1.53 ( ! [X3: A] :
% 2.94/1.53 ( ( F @ X3 )
% 2.94/1.53 = ( G @ X3 ) )
% 2.94/1.53 => ( F = G ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_49_inverse__scaleR__distrib,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( real_V1015715145lgebra @ A )
% 2.94/1.53 => ! [A2: real,X2: A] :
% 2.94/1.53 ( ( inverse_inverse @ A @ ( real_V1908273582scaleR @ A @ A2 @ X2 ) )
% 2.94/1.53 = ( real_V1908273582scaleR @ A @ ( inverse_inverse @ real @ A2 ) @ ( inverse_inverse @ A @ X2 ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_50_asum,axiom,
% 2.94/1.53 ( member @ b
% 2.94/1.53 @ ( groups1340683514dd_sum @ a @ b
% 2.94/1.53 @ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( divide_divide @ real @ ( aa @ J ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) @ ( y @ J ) )
% 2.94/1.53 @ sa )
% 2.94/1.53 @ c ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_51_sum_Oinsert,axiom,
% 2.94/1.53 ! [A: $tType,B: $tType] :
% 2.94/1.53 ( ( comm_monoid_add @ A )
% 2.94/1.53 => ! [A3: set @ B,X2: B,G: B > A] :
% 2.94/1.53 ( ( finite_finite2 @ B @ A3 )
% 2.94/1.53 => ( ~ ( member @ B @ X2 @ A3 )
% 2.94/1.53 => ( ( groups1340683514dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
% 2.94/1.53 = ( plus_plus @ A @ ( G @ X2 ) @ ( groups1340683514dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_52_affine__hull__finite__step,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( real_V1076094709vector @ A )
% 2.94/1.53 => ! [S: set @ A,A2: A,W: real,Y: A] :
% 2.94/1.53 ( ( finite_finite2 @ A @ S )
% 2.94/1.53 => ( ( ? [U2: A > real] :
% 2.94/1.53 ( ( ( groups1340683514dd_sum @ A @ real @ U2 @ ( insert @ A @ A2 @ S ) )
% 2.94/1.53 = W )
% 2.94/1.53 & ( ( groups1340683514dd_sum @ A @ A
% 2.94/1.53 @ ^ [X: A] : ( real_V1908273582scaleR @ A @ ( U2 @ X ) @ X )
% 2.94/1.53 @ ( insert @ A @ A2 @ S ) )
% 2.94/1.53 = Y ) ) )
% 2.94/1.53 = ( ? [V: real,U2: A > real] :
% 2.94/1.53 ( ( ( groups1340683514dd_sum @ A @ real @ U2 @ S )
% 2.94/1.53 = ( minus_minus @ real @ W @ V ) )
% 2.94/1.53 & ( ( groups1340683514dd_sum @ A @ A
% 2.94/1.53 @ ^ [X: A] : ( real_V1908273582scaleR @ A @ ( U2 @ X ) @ X )
% 2.94/1.53 @ S )
% 2.94/1.53 = ( minus_minus @ A @ Y @ ( real_V1908273582scaleR @ A @ V @ A2 ) ) ) ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_53_a1,axiom,
% 2.94/1.53 ( ( groups1340683514dd_sum @ a @ real
% 2.94/1.53 @ ^ [J: a] : ( divide_divide @ real @ ( aa @ J ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) )
% 2.94/1.53 @ sa )
% 2.94/1.53 = ( one_one @ real ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_54__092_060open_062sum_Aa_As_A_P_A_I1_A_N_Aa_Ai_J_A_061_A1_092_060close_062,axiom,
% 2.94/1.53 ( ( divide_divide @ real @ ( groups1340683514dd_sum @ a @ real @ aa @ sa ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) )
% 2.94/1.53 = ( one_one @ real ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_55_singleton__conv,axiom,
% 2.94/1.53 ! [A: $tType,A2: A] :
% 2.94/1.53 ( ( collect @ A
% 2.94/1.53 @ ^ [X: A] : ( X = A2 ) )
% 2.94/1.53 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_56_singleton__conv2,axiom,
% 2.94/1.53 ! [A: $tType,A2: A] :
% 2.94/1.53 ( ( collect @ A
% 2.94/1.53 @ ( ^ [Y2: A,Z: A] : ( Y2 = Z )
% 2.94/1.53 @ A2 ) )
% 2.94/1.53 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_57_finite__insert,axiom,
% 2.94/1.53 ! [A: $tType,A2: A,A3: set @ A] :
% 2.94/1.53 ( ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) )
% 2.94/1.53 = ( finite_finite2 @ A @ A3 ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_58_sum__diff1,axiom,
% 2.94/1.53 ! [A: $tType,B: $tType] :
% 2.94/1.53 ( ( ab_group_add @ A )
% 2.94/1.53 => ! [A3: set @ B,A2: B,F: B > A] :
% 2.94/1.53 ( ( finite_finite2 @ B @ A3 )
% 2.94/1.53 => ( ( ( member @ B @ A2 @ A3 )
% 2.94/1.53 => ( ( groups1340683514dd_sum @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 2.94/1.53 = ( minus_minus @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 2.94/1.53 & ( ~ ( member @ B @ A2 @ A3 )
% 2.94/1.53 => ( ( groups1340683514dd_sum @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 2.94/1.53 = ( groups1340683514dd_sum @ B @ A @ F @ A3 ) ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_59_singletonI,axiom,
% 2.94/1.53 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_60_empty__Collect__eq,axiom,
% 2.94/1.53 ! [A: $tType,P: A > $o] :
% 2.94/1.53 ( ( ( bot_bot @ ( set @ A ) )
% 2.94/1.53 = ( collect @ A @ P ) )
% 2.94/1.53 = ( ! [X: A] :
% 2.94/1.53 ~ ( P @ X ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_61_Collect__empty__eq,axiom,
% 2.94/1.53 ! [A: $tType,P: A > $o] :
% 2.94/1.53 ( ( ( collect @ A @ P )
% 2.94/1.53 = ( bot_bot @ ( set @ A ) ) )
% 2.94/1.53 = ( ! [X: A] :
% 2.94/1.53 ~ ( P @ X ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_62_all__not__in__conv,axiom,
% 2.94/1.53 ! [A: $tType,A3: set @ A] :
% 2.94/1.53 ( ( ! [X: A] :
% 2.94/1.53 ~ ( member @ A @ X @ A3 ) )
% 2.94/1.53 = ( A3
% 2.94/1.53 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_63_empty__iff,axiom,
% 2.94/1.53 ! [A: $tType,C2: A] :
% 2.94/1.53 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_64_finite__code,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( finite_finite @ A )
% 2.94/1.53 => ( ( finite_finite2 @ A )
% 2.94/1.53 = ( ^ [A5: set @ A] : $true ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_65_insert__absorb2,axiom,
% 2.94/1.53 ! [A: $tType,X2: A,A3: set @ A] :
% 2.94/1.53 ( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A3 ) )
% 2.94/1.53 = ( insert @ A @ X2 @ A3 ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_66_insert__iff,axiom,
% 2.94/1.53 ! [A: $tType,A2: A,B2: A,A3: set @ A] :
% 2.94/1.53 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
% 2.94/1.53 = ( ( A2 = B2 )
% 2.94/1.53 | ( member @ A @ A2 @ A3 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_67_insertCI,axiom,
% 2.94/1.53 ! [A: $tType,A2: A,B3: set @ A,B2: A] :
% 2.94/1.53 ( ( ~ ( member @ A @ A2 @ B3 )
% 2.94/1.53 => ( A2 = B2 ) )
% 2.94/1.53 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_68_Diff__idemp,axiom,
% 2.94/1.53 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.94/1.53 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ B3 )
% 2.94/1.53 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_69_Diff__iff,axiom,
% 2.94/1.53 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 2.94/1.53 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 2.94/1.53 = ( ( member @ A @ C2 @ A3 )
% 2.94/1.53 & ~ ( member @ A @ C2 @ B3 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_70_DiffI,axiom,
% 2.94/1.53 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 2.94/1.53 ( ( member @ A @ C2 @ A3 )
% 2.94/1.53 => ( ~ ( member @ A @ C2 @ B3 )
% 2.94/1.53 => ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_71_finite__Collect__disjI,axiom,
% 2.94/1.53 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.94/1.53 ( ( finite_finite2 @ A
% 2.94/1.53 @ ( collect @ A
% 2.94/1.53 @ ^ [X: A] :
% 2.94/1.53 ( ( P @ X )
% 2.94/1.53 | ( Q @ X ) ) ) )
% 2.94/1.53 = ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 2.94/1.53 & ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_72_finite__Collect__conjI,axiom,
% 2.94/1.53 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.94/1.53 ( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 2.94/1.53 | ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
% 2.94/1.53 => ( finite_finite2 @ A
% 2.94/1.53 @ ( collect @ A
% 2.94/1.53 @ ^ [X: A] :
% 2.94/1.53 ( ( P @ X )
% 2.94/1.53 & ( Q @ X ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_73_times__divide__eq__left,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( field @ A )
% 2.94/1.53 => ! [B2: A,C2: A,A2: A] :
% 2.94/1.53 ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
% 2.94/1.53 = ( divide_divide @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_74_divide__divide__eq__left,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( field @ A )
% 2.94/1.53 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.53 ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
% 2.94/1.53 = ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_75_divide__divide__eq__right,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( field @ A )
% 2.94/1.53 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.53 ( ( divide_divide @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
% 2.94/1.53 = ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_76_times__divide__eq__right,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( division_ring @ A )
% 2.94/1.53 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.53 ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
% 2.94/1.53 = ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_77_Diff__cancel,axiom,
% 2.94/1.53 ! [A: $tType,A3: set @ A] :
% 2.94/1.53 ( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
% 2.94/1.53 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_78_empty__Diff,axiom,
% 2.94/1.53 ! [A: $tType,A3: set @ A] :
% 2.94/1.53 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
% 2.94/1.53 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_79_Diff__empty,axiom,
% 2.94/1.53 ! [A: $tType,A3: set @ A] :
% 2.94/1.53 ( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 2.94/1.53 = A3 ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_80_finite__Diff2,axiom,
% 2.94/1.53 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 2.94/1.53 ( ( finite_finite2 @ A @ B3 )
% 2.94/1.53 => ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 2.94/1.53 = ( finite_finite2 @ A @ A3 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_81_finite__Diff,axiom,
% 2.94/1.53 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.94/1.53 ( ( finite_finite2 @ A @ A3 )
% 2.94/1.53 => ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_82_insert__Diff1,axiom,
% 2.94/1.53 ! [A: $tType,X2: A,B3: set @ A,A3: set @ A] :
% 2.94/1.53 ( ( member @ A @ X2 @ B3 )
% 2.94/1.53 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B3 )
% 2.94/1.53 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_83_Diff__insert0,axiom,
% 2.94/1.53 ! [A: $tType,X2: A,A3: set @ A,B3: set @ A] :
% 2.94/1.53 ( ~ ( member @ A @ X2 @ A3 )
% 2.94/1.53 => ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ B3 ) )
% 2.94/1.53 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_84_inverse__divide,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( field @ A )
% 2.94/1.53 => ! [A2: A,B2: A] :
% 2.94/1.53 ( ( inverse_inverse @ A @ ( divide_divide @ A @ A2 @ B2 ) )
% 2.94/1.53 = ( divide_divide @ A @ B2 @ A2 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_85_insert__Diff__single,axiom,
% 2.94/1.53 ! [A: $tType,A2: A,A3: set @ A] :
% 2.94/1.53 ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.94/1.53 = ( insert @ A @ A2 @ A3 ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_86_finite__Diff__insert,axiom,
% 2.94/1.53 ! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
% 2.94/1.53 ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) )
% 2.94/1.53 = ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_87_sum__diff1__nat,axiom,
% 2.94/1.53 ! [A: $tType,A2: A,A3: set @ A,F: A > nat] :
% 2.94/1.53 ( ( ( member @ A @ A2 @ A3 )
% 2.94/1.53 => ( ( groups1340683514dd_sum @ A @ nat @ F @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.94/1.53 = ( minus_minus @ nat @ ( groups1340683514dd_sum @ A @ nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 2.94/1.53 & ( ~ ( member @ A @ A2 @ A3 )
% 2.94/1.53 => ( ( groups1340683514dd_sum @ A @ nat @ F @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.94/1.53 = ( groups1340683514dd_sum @ A @ nat @ F @ A3 ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_88_minus__set__def,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( minus_minus @ ( set @ A ) )
% 2.94/1.53 = ( ^ [A5: set @ A,B4: set @ A] :
% 2.94/1.53 ( collect @ A
% 2.94/1.53 @ ( minus_minus @ ( A > $o )
% 2.94/1.53 @ ^ [X: A] : ( member @ A @ X @ A5 )
% 2.94/1.53 @ ^ [X: A] : ( member @ A @ X @ B4 ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_89_set__diff__eq,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( minus_minus @ ( set @ A ) )
% 2.94/1.53 = ( ^ [A5: set @ A,B4: set @ A] :
% 2.94/1.53 ( collect @ A
% 2.94/1.53 @ ^ [X: A] :
% 2.94/1.53 ( ( member @ A @ X @ A5 )
% 2.94/1.53 & ~ ( member @ A @ X @ B4 ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_90_DiffD2,axiom,
% 2.94/1.53 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 2.94/1.53 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 2.94/1.53 => ~ ( member @ A @ C2 @ B3 ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_91_DiffD1,axiom,
% 2.94/1.53 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 2.94/1.53 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 2.94/1.53 => ( member @ A @ C2 @ A3 ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_92_DiffE,axiom,
% 2.94/1.53 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 2.94/1.53 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 2.94/1.53 => ~ ( ( member @ A @ C2 @ A3 )
% 2.94/1.53 => ( member @ A @ C2 @ B3 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_93_Diff__infinite__finite,axiom,
% 2.94/1.53 ! [A: $tType,T: set @ A,S: set @ A] :
% 2.94/1.53 ( ( finite_finite2 @ A @ T )
% 2.94/1.53 => ( ~ ( finite_finite2 @ A @ S )
% 2.94/1.53 => ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S @ T ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_94_insert__Diff__if,axiom,
% 2.94/1.53 ! [A: $tType,X2: A,B3: set @ A,A3: set @ A] :
% 2.94/1.53 ( ( ( member @ A @ X2 @ B3 )
% 2.94/1.53 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B3 )
% 2.94/1.53 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) )
% 2.94/1.53 & ( ~ ( member @ A @ X2 @ B3 )
% 2.94/1.53 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B3 )
% 2.94/1.53 = ( insert @ A @ X2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_95_sum__divide__distrib,axiom,
% 2.94/1.53 ! [A: $tType,B: $tType] :
% 2.94/1.53 ( ( field @ A )
% 2.94/1.53 => ! [F: B > A,A3: set @ B,R: A] :
% 2.94/1.53 ( ( divide_divide @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ R )
% 2.94/1.53 = ( groups1340683514dd_sum @ B @ A
% 2.94/1.53 @ ^ [N: B] : ( divide_divide @ A @ ( F @ N ) @ R )
% 2.94/1.53 @ A3 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_96_convex__ereal__add,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( real_V1076094709vector @ A )
% 2.94/1.53 => ! [S2: set @ A,F: A > extended_ereal,G: A > extended_ereal] :
% 2.94/1.53 ( ( lower_673667120vex_on @ A @ S2 @ F )
% 2.94/1.53 => ( ( lower_673667120vex_on @ A @ S2 @ G )
% 2.94/1.53 => ( lower_673667120vex_on @ A @ S2
% 2.94/1.53 @ ^ [X: A] : ( plus_plus @ extended_ereal @ ( F @ X ) @ ( G @ X ) ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_97_times__divide__times__eq,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( field @ A )
% 2.94/1.53 => ! [X2: A,Y: A,Z2: A,W: A] :
% 2.94/1.53 ( ( times_times @ A @ ( divide_divide @ A @ X2 @ Y ) @ ( divide_divide @ A @ Z2 @ W ) )
% 2.94/1.53 = ( divide_divide @ A @ ( times_times @ A @ X2 @ Z2 ) @ ( times_times @ A @ Y @ W ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_98_divide__divide__times__eq,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( field @ A )
% 2.94/1.53 => ! [X2: A,Y: A,Z2: A,W: A] :
% 2.94/1.53 ( ( divide_divide @ A @ ( divide_divide @ A @ X2 @ Y ) @ ( divide_divide @ A @ Z2 @ W ) )
% 2.94/1.53 = ( divide_divide @ A @ ( times_times @ A @ X2 @ W ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_99_divide__divide__eq__left_H,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( division_ring @ A )
% 2.94/1.53 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.53 ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
% 2.94/1.53 = ( divide_divide @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_100_add__divide__distrib,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( division_ring @ A )
% 2.94/1.53 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.53 ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 2.94/1.53 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_101_diff__divide__distrib,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( division_ring @ A )
% 2.94/1.53 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.53 ( ( divide_divide @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
% 2.94/1.53 = ( minus_minus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_102_Diff__insert__absorb,axiom,
% 2.94/1.53 ! [A: $tType,X2: A,A3: set @ A] :
% 2.94/1.53 ( ~ ( member @ A @ X2 @ A3 )
% 2.94/1.53 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.94/1.53 = A3 ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_103_Diff__insert2,axiom,
% 2.94/1.53 ! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
% 2.94/1.53 ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
% 2.94/1.53 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_104_insert__Diff,axiom,
% 2.94/1.53 ! [A: $tType,A2: A,A3: set @ A] :
% 2.94/1.53 ( ( member @ A @ A2 @ A3 )
% 2.94/1.53 => ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.94/1.53 = A3 ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_105_Diff__insert,axiom,
% 2.94/1.53 ! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
% 2.94/1.53 ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
% 2.94/1.53 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_106_finite__empty__induct,axiom,
% 2.94/1.53 ! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
% 2.94/1.53 ( ( finite_finite2 @ A @ A3 )
% 2.94/1.53 => ( ( P @ A3 )
% 2.94/1.53 => ( ! [A6: A,A7: set @ A] :
% 2.94/1.53 ( ( finite_finite2 @ A @ A7 )
% 2.94/1.53 => ( ( member @ A @ A6 @ A7 )
% 2.94/1.53 => ( ( P @ A7 )
% 2.94/1.53 => ( P @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
% 2.94/1.53 => ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_107_infinite__coinduct,axiom,
% 2.94/1.53 ! [A: $tType,X4: ( set @ A ) > $o,A3: set @ A] :
% 2.94/1.53 ( ( X4 @ A3 )
% 2.94/1.53 => ( ! [A7: set @ A] :
% 2.94/1.53 ( ( X4 @ A7 )
% 2.94/1.53 => ? [X5: A] :
% 2.94/1.53 ( ( member @ A @ X5 @ A7 )
% 2.94/1.53 & ( ( X4 @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.94/1.53 | ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
% 2.94/1.53 => ~ ( finite_finite2 @ A @ A3 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_108_infinite__remove,axiom,
% 2.94/1.53 ! [A: $tType,S: set @ A,A2: A] :
% 2.94/1.53 ( ~ ( finite_finite2 @ A @ S )
% 2.94/1.53 => ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_109_field__class_Ofield__divide__inverse,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( field @ A )
% 2.94/1.53 => ( ( divide_divide @ A )
% 2.94/1.53 = ( ^ [A4: A,B5: A] : ( times_times @ A @ A4 @ ( inverse_inverse @ A @ B5 ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_110_divide__inverse,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( division_ring @ A )
% 2.94/1.53 => ( ( divide_divide @ A )
% 2.94/1.53 = ( ^ [A4: A,B5: A] : ( times_times @ A @ A4 @ ( inverse_inverse @ A @ B5 ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_111_divide__inverse__commute,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( field @ A )
% 2.94/1.53 => ( ( divide_divide @ A )
% 2.94/1.53 = ( ^ [A4: A,B5: A] : ( times_times @ A @ ( inverse_inverse @ A @ B5 ) @ A4 ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_112_inverse__eq__divide,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( division_ring @ A )
% 2.94/1.53 => ( ( inverse_inverse @ A )
% 2.94/1.53 = ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_113_ex__in__conv,axiom,
% 2.94/1.53 ! [A: $tType,A3: set @ A] :
% 2.94/1.53 ( ( ? [X: A] : ( member @ A @ X @ A3 ) )
% 2.94/1.53 = ( A3
% 2.94/1.53 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_114_equals0I,axiom,
% 2.94/1.53 ! [A: $tType,A3: set @ A] :
% 2.94/1.53 ( ! [Y3: A] :
% 2.94/1.53 ~ ( member @ A @ Y3 @ A3 )
% 2.94/1.53 => ( A3
% 2.94/1.53 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_115_equals0D,axiom,
% 2.94/1.53 ! [A: $tType,A3: set @ A,A2: A] :
% 2.94/1.53 ( ( A3
% 2.94/1.53 = ( bot_bot @ ( set @ A ) ) )
% 2.94/1.53 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_116_emptyE,axiom,
% 2.94/1.53 ! [A: $tType,A2: A] :
% 2.94/1.53 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_117_finite,axiom,
% 2.94/1.53 ! [A: $tType] :
% 2.94/1.53 ( ( finite_finite @ A )
% 2.94/1.53 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_118_finite__set__choice,axiom,
% 2.94/1.53 ! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
% 2.94/1.53 ( ( finite_finite2 @ A @ A3 )
% 2.94/1.53 => ( ! [X3: A] :
% 2.94/1.53 ( ( member @ A @ X3 @ A3 )
% 2.94/1.53 => ? [X_1: B] : ( P @ X3 @ X_1 ) )
% 2.94/1.53 => ? [F2: A > B] :
% 2.94/1.53 ! [X5: A] :
% 2.94/1.53 ( ( member @ A @ X5 @ A3 )
% 2.94/1.53 => ( P @ X5 @ ( F2 @ X5 ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_119_mk__disjoint__insert,axiom,
% 2.94/1.53 ! [A: $tType,A2: A,A3: set @ A] :
% 2.94/1.53 ( ( member @ A @ A2 @ A3 )
% 2.94/1.53 => ? [B6: set @ A] :
% 2.94/1.53 ( ( A3
% 2.94/1.53 = ( insert @ A @ A2 @ B6 ) )
% 2.94/1.53 & ~ ( member @ A @ A2 @ B6 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_120_insert__commute,axiom,
% 2.94/1.53 ! [A: $tType,X2: A,Y: A,A3: set @ A] :
% 2.94/1.53 ( ( insert @ A @ X2 @ ( insert @ A @ Y @ A3 ) )
% 2.94/1.53 = ( insert @ A @ Y @ ( insert @ A @ X2 @ A3 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_121_insert__eq__iff,axiom,
% 2.94/1.53 ! [A: $tType,A2: A,A3: set @ A,B2: A,B3: set @ A] :
% 2.94/1.53 ( ~ ( member @ A @ A2 @ A3 )
% 2.94/1.53 => ( ~ ( member @ A @ B2 @ B3 )
% 2.94/1.53 => ( ( ( insert @ A @ A2 @ A3 )
% 2.94/1.53 = ( insert @ A @ B2 @ B3 ) )
% 2.94/1.53 = ( ( ( A2 = B2 )
% 2.94/1.53 => ( A3 = B3 ) )
% 2.94/1.53 & ( ( A2 != B2 )
% 2.94/1.53 => ? [C3: set @ A] :
% 2.94/1.53 ( ( A3
% 2.94/1.53 = ( insert @ A @ B2 @ C3 ) )
% 2.94/1.53 & ~ ( member @ A @ B2 @ C3 )
% 2.94/1.53 & ( B3
% 2.94/1.53 = ( insert @ A @ A2 @ C3 ) )
% 2.94/1.53 & ~ ( member @ A @ A2 @ C3 ) ) ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_122_insert__absorb,axiom,
% 2.94/1.53 ! [A: $tType,A2: A,A3: set @ A] :
% 2.94/1.53 ( ( member @ A @ A2 @ A3 )
% 2.94/1.53 => ( ( insert @ A @ A2 @ A3 )
% 2.94/1.53 = A3 ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_123_insert__ident,axiom,
% 2.94/1.53 ! [A: $tType,X2: A,A3: set @ A,B3: set @ A] :
% 2.94/1.53 ( ~ ( member @ A @ X2 @ A3 )
% 2.94/1.53 => ( ~ ( member @ A @ X2 @ B3 )
% 2.94/1.53 => ( ( ( insert @ A @ X2 @ A3 )
% 2.94/1.53 = ( insert @ A @ X2 @ B3 ) )
% 2.94/1.53 = ( A3 = B3 ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_124_Set_Oset__insert,axiom,
% 2.94/1.53 ! [A: $tType,X2: A,A3: set @ A] :
% 2.94/1.53 ( ( member @ A @ X2 @ A3 )
% 2.94/1.53 => ~ ! [B6: set @ A] :
% 2.94/1.53 ( ( A3
% 2.94/1.53 = ( insert @ A @ X2 @ B6 ) )
% 2.94/1.53 => ( member @ A @ X2 @ B6 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_125_insertI2,axiom,
% 2.94/1.53 ! [A: $tType,A2: A,B3: set @ A,B2: A] :
% 2.94/1.53 ( ( member @ A @ A2 @ B3 )
% 2.94/1.53 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_126_insertI1,axiom,
% 2.94/1.53 ! [A: $tType,A2: A,B3: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B3 ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_127_insertE,axiom,
% 2.94/1.53 ! [A: $tType,A2: A,B2: A,A3: set @ A] :
% 2.94/1.53 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
% 2.94/1.53 => ( ( A2 != B2 )
% 2.94/1.53 => ( member @ A @ A2 @ A3 ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_128_sum_Ocong,axiom,
% 2.94/1.53 ! [A: $tType,B: $tType] :
% 2.94/1.53 ( ( comm_monoid_add @ A )
% 2.94/1.53 => ! [A3: set @ B,B3: set @ B,G: B > A,H: B > A] :
% 2.94/1.53 ( ( A3 = B3 )
% 2.94/1.53 => ( ! [X3: B] :
% 2.94/1.53 ( ( member @ B @ X3 @ B3 )
% 2.94/1.53 => ( ( G @ X3 )
% 2.94/1.53 = ( H @ X3 ) ) )
% 2.94/1.53 => ( ( groups1340683514dd_sum @ B @ A @ G @ A3 )
% 2.94/1.53 = ( groups1340683514dd_sum @ B @ A @ H @ B3 ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_129_sum_Oeq__general,axiom,
% 2.94/1.53 ! [C: $tType,A: $tType,B: $tType] :
% 2.94/1.53 ( ( comm_monoid_add @ A )
% 2.94/1.53 => ! [B3: set @ B,A3: set @ C,H: C > B,Gamma: B > A,Phi: C > A] :
% 2.94/1.53 ( ! [Y3: B] :
% 2.94/1.53 ( ( member @ B @ Y3 @ B3 )
% 2.94/1.53 => ? [X5: C] :
% 2.94/1.53 ( ( member @ C @ X5 @ A3 )
% 2.94/1.53 & ( ( H @ X5 )
% 2.94/1.53 = Y3 )
% 2.94/1.53 & ! [Ya: C] :
% 2.94/1.53 ( ( ( member @ C @ Ya @ A3 )
% 2.94/1.53 & ( ( H @ Ya )
% 2.94/1.53 = Y3 ) )
% 2.94/1.53 => ( Ya = X5 ) ) ) )
% 2.94/1.53 => ( ! [X3: C] :
% 2.94/1.53 ( ( member @ C @ X3 @ A3 )
% 2.94/1.53 => ( ( member @ B @ ( H @ X3 ) @ B3 )
% 2.94/1.53 & ( ( Gamma @ ( H @ X3 ) )
% 2.94/1.53 = ( Phi @ X3 ) ) ) )
% 2.94/1.53 => ( ( groups1340683514dd_sum @ C @ A @ Phi @ A3 )
% 2.94/1.53 = ( groups1340683514dd_sum @ B @ A @ Gamma @ B3 ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_130_sum_Oeq__general__inverses,axiom,
% 2.94/1.53 ! [C: $tType,A: $tType,B: $tType] :
% 2.94/1.53 ( ( comm_monoid_add @ A )
% 2.94/1.53 => ! [B3: set @ B,K: B > C,A3: set @ C,H: C > B,Gamma: B > A,Phi: C > A] :
% 2.94/1.53 ( ! [Y3: B] :
% 2.94/1.53 ( ( member @ B @ Y3 @ B3 )
% 2.94/1.53 => ( ( member @ C @ ( K @ Y3 ) @ A3 )
% 2.94/1.53 & ( ( H @ ( K @ Y3 ) )
% 2.94/1.53 = Y3 ) ) )
% 2.94/1.53 => ( ! [X3: C] :
% 2.94/1.53 ( ( member @ C @ X3 @ A3 )
% 2.94/1.53 => ( ( member @ B @ ( H @ X3 ) @ B3 )
% 2.94/1.53 & ( ( K @ ( H @ X3 ) )
% 2.94/1.53 = X3 )
% 2.94/1.53 & ( ( Gamma @ ( H @ X3 ) )
% 2.94/1.53 = ( Phi @ X3 ) ) ) )
% 2.94/1.53 => ( ( groups1340683514dd_sum @ C @ A @ Phi @ A3 )
% 2.94/1.53 = ( groups1340683514dd_sum @ B @ A @ Gamma @ B3 ) ) ) ) ) ).
% 2.94/1.53
% 2.94/1.53 thf(fact_131_sum_Oreindex__bij__witness,axiom,
% 2.94/1.53 ! [B: $tType,A: $tType,C: $tType] :
% 2.94/1.53 ( ( comm_monoid_add @ A )
% 2.94/1.53 => ! [S: set @ B,I2: C > B,J2: B > C,T: set @ C,H: C > A,G: B > A] :
% 2.94/1.54 ( ! [A6: B] :
% 2.94/1.54 ( ( member @ B @ A6 @ S )
% 2.94/1.54 => ( ( I2 @ ( J2 @ A6 ) )
% 2.94/1.54 = A6 ) )
% 2.94/1.54 => ( ! [A6: B] :
% 2.94/1.54 ( ( member @ B @ A6 @ S )
% 2.94/1.54 => ( member @ C @ ( J2 @ A6 ) @ T ) )
% 2.94/1.54 => ( ! [B7: C] :
% 2.94/1.54 ( ( member @ C @ B7 @ T )
% 2.94/1.54 => ( ( J2 @ ( I2 @ B7 ) )
% 2.94/1.54 = B7 ) )
% 2.94/1.54 => ( ! [B7: C] :
% 2.94/1.54 ( ( member @ C @ B7 @ T )
% 2.94/1.54 => ( member @ B @ ( I2 @ B7 ) @ S ) )
% 2.94/1.54 => ( ! [A6: B] :
% 2.94/1.54 ( ( member @ B @ A6 @ S )
% 2.94/1.54 => ( ( H @ ( J2 @ A6 ) )
% 2.94/1.54 = ( G @ A6 ) ) )
% 2.94/1.54 => ( ( groups1340683514dd_sum @ B @ A @ G @ S )
% 2.94/1.54 = ( groups1340683514dd_sum @ C @ A @ H @ T ) ) ) ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_132_if__smult,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( real_Vector_scaleR @ A )
% 2.94/1.54 => ! [P: $o,X2: real,Y: real,V2: A] :
% 2.94/1.54 ( ( P
% 2.94/1.54 => ( ( real_V1908273582scaleR @ A @ ( if @ real @ P @ X2 @ Y ) @ V2 )
% 2.94/1.54 = ( real_V1908273582scaleR @ A @ X2 @ V2 ) ) )
% 2.94/1.54 & ( ~ P
% 2.94/1.54 => ( ( real_V1908273582scaleR @ A @ ( if @ real @ P @ X2 @ Y ) @ V2 )
% 2.94/1.54 = ( real_V1908273582scaleR @ A @ Y @ V2 ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_133_sum_Oinsert__remove,axiom,
% 2.94/1.54 ! [A: $tType,B: $tType] :
% 2.94/1.54 ( ( comm_monoid_add @ A )
% 2.94/1.54 => ! [A3: set @ B,G: B > A,X2: B] :
% 2.94/1.54 ( ( finite_finite2 @ B @ A3 )
% 2.94/1.54 => ( ( groups1340683514dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
% 2.94/1.54 = ( plus_plus @ A @ ( G @ X2 ) @ ( groups1340683514dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_134_sum_Oremove,axiom,
% 2.94/1.54 ! [A: $tType,B: $tType] :
% 2.94/1.54 ( ( comm_monoid_add @ A )
% 2.94/1.54 => ! [A3: set @ B,X2: B,G: B > A] :
% 2.94/1.54 ( ( finite_finite2 @ B @ A3 )
% 2.94/1.54 => ( ( member @ B @ X2 @ A3 )
% 2.94/1.54 => ( ( groups1340683514dd_sum @ B @ A @ G @ A3 )
% 2.94/1.54 = ( plus_plus @ A @ ( G @ X2 ) @ ( groups1340683514dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_135_sum_Odelta__remove,axiom,
% 2.94/1.54 ! [A: $tType,B: $tType] :
% 2.94/1.54 ( ( comm_monoid_add @ A )
% 2.94/1.54 => ! [S: set @ B,A2: B,B2: B > A,C2: B > A] :
% 2.94/1.54 ( ( finite_finite2 @ B @ S )
% 2.94/1.54 => ( ( ( member @ B @ A2 @ S )
% 2.94/1.54 => ( ( groups1340683514dd_sum @ B @ A
% 2.94/1.54 @ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( C2 @ K2 ) )
% 2.94/1.54 @ S )
% 2.94/1.54 = ( plus_plus @ A @ ( B2 @ A2 ) @ ( groups1340683514dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
% 2.94/1.54 & ( ~ ( member @ B @ A2 @ S )
% 2.94/1.54 => ( ( groups1340683514dd_sum @ B @ A
% 2.94/1.54 @ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( C2 @ K2 ) )
% 2.94/1.54 @ S )
% 2.94/1.54 = ( groups1340683514dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_136_empty__def,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( bot_bot @ ( set @ A ) )
% 2.94/1.54 = ( collect @ A
% 2.94/1.54 @ ^ [X: A] : $false ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_137_pigeonhole__infinite__rel,axiom,
% 2.94/1.54 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,R2: A > B > $o] :
% 2.94/1.54 ( ~ ( finite_finite2 @ A @ A3 )
% 2.94/1.54 => ( ( finite_finite2 @ B @ B3 )
% 2.94/1.54 => ( ! [X3: A] :
% 2.94/1.54 ( ( member @ A @ X3 @ A3 )
% 2.94/1.54 => ? [Xa2: B] :
% 2.94/1.54 ( ( member @ B @ Xa2 @ B3 )
% 2.94/1.54 & ( R2 @ X3 @ Xa2 ) ) )
% 2.94/1.54 => ? [X3: B] :
% 2.94/1.54 ( ( member @ B @ X3 @ B3 )
% 2.94/1.54 & ~ ( finite_finite2 @ A
% 2.94/1.54 @ ( collect @ A
% 2.94/1.54 @ ^ [A4: A] :
% 2.94/1.54 ( ( member @ A @ A4 @ A3 )
% 2.94/1.54 & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_138_not__finite__existsD,axiom,
% 2.94/1.54 ! [A: $tType,P: A > $o] :
% 2.94/1.54 ( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 2.94/1.54 => ? [X_12: A] : ( P @ X_12 ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_139_insert__Collect,axiom,
% 2.94/1.54 ! [A: $tType,A2: A,P: A > $o] :
% 2.94/1.54 ( ( insert @ A @ A2 @ ( collect @ A @ P ) )
% 2.94/1.54 = ( collect @ A
% 2.94/1.54 @ ^ [U2: A] :
% 2.94/1.54 ( ( U2 != A2 )
% 2.94/1.54 => ( P @ U2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_140_insert__compr,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( insert @ A )
% 2.94/1.54 = ( ^ [A4: A,B4: set @ A] :
% 2.94/1.54 ( collect @ A
% 2.94/1.54 @ ^ [X: A] :
% 2.94/1.54 ( ( X = A4 )
% 2.94/1.54 | ( member @ A @ X @ B4 ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_141_sum__delta__notmem_I4_J,axiom,
% 2.94/1.54 ! [B: $tType,A: $tType] :
% 2.94/1.54 ( ( comm_monoid_add @ B )
% 2.94/1.54 => ! [X2: A,S2: set @ A,P: A > B,Q: A > B] :
% 2.94/1.54 ( ~ ( member @ A @ X2 @ S2 )
% 2.94/1.54 => ( ( groups1340683514dd_sum @ A @ B
% 2.94/1.54 @ ^ [Y4: A] : ( if @ B @ ( X2 = Y4 ) @ ( P @ Y4 ) @ ( Q @ Y4 ) )
% 2.94/1.54 @ S2 )
% 2.94/1.54 = ( groups1340683514dd_sum @ A @ B @ Q @ S2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_142_sum__delta__notmem_I3_J,axiom,
% 2.94/1.54 ! [B: $tType,A: $tType] :
% 2.94/1.54 ( ( comm_monoid_add @ B )
% 2.94/1.54 => ! [X2: A,S2: set @ A,P: A > B,Q: A > B] :
% 2.94/1.54 ( ~ ( member @ A @ X2 @ S2 )
% 2.94/1.54 => ( ( groups1340683514dd_sum @ A @ B
% 2.94/1.54 @ ^ [Y4: A] : ( if @ B @ ( Y4 = X2 ) @ ( P @ Y4 ) @ ( Q @ Y4 ) )
% 2.94/1.54 @ S2 )
% 2.94/1.54 = ( groups1340683514dd_sum @ A @ B @ Q @ S2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_143_sum__delta__notmem_I2_J,axiom,
% 2.94/1.54 ! [B: $tType,A: $tType] :
% 2.94/1.54 ( ( comm_monoid_add @ B )
% 2.94/1.54 => ! [X2: A,S2: set @ A,P: A > B,Q: A > B] :
% 2.94/1.54 ( ~ ( member @ A @ X2 @ S2 )
% 2.94/1.54 => ( ( groups1340683514dd_sum @ A @ B
% 2.94/1.54 @ ^ [Y4: A] : ( if @ B @ ( X2 = Y4 ) @ ( P @ X2 ) @ ( Q @ Y4 ) )
% 2.94/1.54 @ S2 )
% 2.94/1.54 = ( groups1340683514dd_sum @ A @ B @ Q @ S2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_144_sum__delta__notmem_I1_J,axiom,
% 2.94/1.54 ! [B: $tType,A: $tType] :
% 2.94/1.54 ( ( comm_monoid_add @ B )
% 2.94/1.54 => ! [X2: A,S2: set @ A,P: A > B,Q: A > B] :
% 2.94/1.54 ( ~ ( member @ A @ X2 @ S2 )
% 2.94/1.54 => ( ( groups1340683514dd_sum @ A @ B
% 2.94/1.54 @ ^ [Y4: A] : ( if @ B @ ( Y4 = X2 ) @ ( P @ X2 ) @ ( Q @ Y4 ) )
% 2.94/1.54 @ S2 )
% 2.94/1.54 = ( groups1340683514dd_sum @ A @ B @ Q @ S2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_145_sum_Oswap,axiom,
% 2.94/1.54 ! [A: $tType,B: $tType,C: $tType] :
% 2.94/1.54 ( ( comm_monoid_add @ A )
% 2.94/1.54 => ! [G: B > C > A,B3: set @ C,A3: set @ B] :
% 2.94/1.54 ( ( groups1340683514dd_sum @ B @ A
% 2.94/1.54 @ ^ [I3: B] : ( groups1340683514dd_sum @ C @ A @ ( G @ I3 ) @ B3 )
% 2.94/1.54 @ A3 )
% 2.94/1.54 = ( groups1340683514dd_sum @ C @ A
% 2.94/1.54 @ ^ [J: C] :
% 2.94/1.54 ( groups1340683514dd_sum @ B @ A
% 2.94/1.54 @ ^ [I3: B] : ( G @ I3 @ J )
% 2.94/1.54 @ A3 )
% 2.94/1.54 @ B3 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_146_infinite__imp__nonempty,axiom,
% 2.94/1.54 ! [A: $tType,S: set @ A] :
% 2.94/1.54 ( ~ ( finite_finite2 @ A @ S )
% 2.94/1.54 => ( S
% 2.94/1.54 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_147_finite_OemptyI,axiom,
% 2.94/1.54 ! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_148_singleton__inject,axiom,
% 2.94/1.54 ! [A: $tType,A2: A,B2: A] :
% 2.94/1.54 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 2.94/1.54 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.94/1.54 => ( A2 = B2 ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_149_insert__not__empty,axiom,
% 2.94/1.54 ! [A: $tType,A2: A,A3: set @ A] :
% 2.94/1.54 ( ( insert @ A @ A2 @ A3 )
% 2.94/1.54 != ( bot_bot @ ( set @ A ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_150_doubleton__eq__iff,axiom,
% 2.94/1.54 ! [A: $tType,A2: A,B2: A,C2: A,D: A] :
% 2.94/1.54 ( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.94/1.54 = ( insert @ A @ C2 @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.94/1.54 = ( ( ( A2 = C2 )
% 2.94/1.54 & ( B2 = D ) )
% 2.94/1.54 | ( ( A2 = D )
% 2.94/1.54 & ( B2 = C2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_151_singleton__iff,axiom,
% 2.94/1.54 ! [A: $tType,B2: A,A2: A] :
% 2.94/1.54 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.94/1.54 = ( B2 = A2 ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_152_singletonD,axiom,
% 2.94/1.54 ! [A: $tType,B2: A,A2: A] :
% 2.94/1.54 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.94/1.54 => ( B2 = A2 ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_153_finite_OinsertI,axiom,
% 2.94/1.54 ! [A: $tType,A3: set @ A,A2: A] :
% 2.94/1.54 ( ( finite_finite2 @ A @ A3 )
% 2.94/1.54 => ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_154_sum__product,axiom,
% 2.94/1.54 ! [B: $tType,C: $tType,A: $tType] :
% 2.94/1.54 ( ( semiring_0 @ B )
% 2.94/1.54 => ! [F: A > B,A3: set @ A,G: C > B,B3: set @ C] :
% 2.94/1.54 ( ( times_times @ B @ ( groups1340683514dd_sum @ A @ B @ F @ A3 ) @ ( groups1340683514dd_sum @ C @ B @ G @ B3 ) )
% 2.94/1.54 = ( groups1340683514dd_sum @ A @ B
% 2.94/1.54 @ ^ [I3: A] :
% 2.94/1.54 ( groups1340683514dd_sum @ C @ B
% 2.94/1.54 @ ^ [J: C] : ( times_times @ B @ ( F @ I3 ) @ ( G @ J ) )
% 2.94/1.54 @ B3 )
% 2.94/1.54 @ A3 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_155_sum__distrib__left,axiom,
% 2.94/1.54 ! [A: $tType,B: $tType] :
% 2.94/1.54 ( ( semiring_0 @ A )
% 2.94/1.54 => ! [R: A,F: B > A,A3: set @ B] :
% 2.94/1.54 ( ( times_times @ A @ R @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) )
% 2.94/1.54 = ( groups1340683514dd_sum @ B @ A
% 2.94/1.54 @ ^ [N: B] : ( times_times @ A @ R @ ( F @ N ) )
% 2.94/1.54 @ A3 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_156_sum__distrib__right,axiom,
% 2.94/1.54 ! [A: $tType,B: $tType] :
% 2.94/1.54 ( ( semiring_0 @ A )
% 2.94/1.54 => ! [F: B > A,A3: set @ B,R: A] :
% 2.94/1.54 ( ( times_times @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ R )
% 2.94/1.54 = ( groups1340683514dd_sum @ B @ A
% 2.94/1.54 @ ^ [N: B] : ( times_times @ A @ ( F @ N ) @ R )
% 2.94/1.54 @ A3 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_157_sum_Odistrib,axiom,
% 2.94/1.54 ! [A: $tType,B: $tType] :
% 2.94/1.54 ( ( comm_monoid_add @ A )
% 2.94/1.54 => ! [G: B > A,H: B > A,A3: set @ B] :
% 2.94/1.54 ( ( groups1340683514dd_sum @ B @ A
% 2.94/1.54 @ ^ [X: B] : ( plus_plus @ A @ ( G @ X ) @ ( H @ X ) )
% 2.94/1.54 @ A3 )
% 2.94/1.54 = ( plus_plus @ A @ ( groups1340683514dd_sum @ B @ A @ G @ A3 ) @ ( groups1340683514dd_sum @ B @ A @ H @ A3 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_158_Collect__conv__if2,axiom,
% 2.94/1.54 ! [A: $tType,P: A > $o,A2: A] :
% 2.94/1.54 ( ( ( P @ A2 )
% 2.94/1.54 => ( ( collect @ A
% 2.94/1.54 @ ^ [X: A] :
% 2.94/1.54 ( ( A2 = X )
% 2.94/1.54 & ( P @ X ) ) )
% 2.94/1.54 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.94/1.54 & ( ~ ( P @ A2 )
% 2.94/1.54 => ( ( collect @ A
% 2.94/1.54 @ ^ [X: A] :
% 2.94/1.54 ( ( A2 = X )
% 2.94/1.54 & ( P @ X ) ) )
% 2.94/1.54 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_159_Collect__conv__if,axiom,
% 2.94/1.54 ! [A: $tType,P: A > $o,A2: A] :
% 2.94/1.54 ( ( ( P @ A2 )
% 2.94/1.54 => ( ( collect @ A
% 2.94/1.54 @ ^ [X: A] :
% 2.94/1.54 ( ( X = A2 )
% 2.94/1.54 & ( P @ X ) ) )
% 2.94/1.54 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.94/1.54 & ( ~ ( P @ A2 )
% 2.94/1.54 => ( ( collect @ A
% 2.94/1.54 @ ^ [X: A] :
% 2.94/1.54 ( ( X = A2 )
% 2.94/1.54 & ( P @ X ) ) )
% 2.94/1.54 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_160_sum__subtractf,axiom,
% 2.94/1.54 ! [A: $tType,B: $tType] :
% 2.94/1.54 ( ( ab_group_add @ A )
% 2.94/1.54 => ! [F: B > A,G: B > A,A3: set @ B] :
% 2.94/1.54 ( ( groups1340683514dd_sum @ B @ A
% 2.94/1.54 @ ^ [X: B] : ( minus_minus @ A @ ( F @ X ) @ ( G @ X ) )
% 2.94/1.54 @ A3 )
% 2.94/1.54 = ( minus_minus @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ ( groups1340683514dd_sum @ B @ A @ G @ A3 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_161_sum_Oswap__restrict,axiom,
% 2.94/1.54 ! [A: $tType,B: $tType,C: $tType] :
% 2.94/1.54 ( ( comm_monoid_add @ A )
% 2.94/1.54 => ! [A3: set @ B,B3: set @ C,G: B > C > A,R2: B > C > $o] :
% 2.94/1.54 ( ( finite_finite2 @ B @ A3 )
% 2.94/1.54 => ( ( finite_finite2 @ C @ B3 )
% 2.94/1.54 => ( ( groups1340683514dd_sum @ B @ A
% 2.94/1.54 @ ^ [X: B] :
% 2.94/1.54 ( groups1340683514dd_sum @ C @ A @ ( G @ X )
% 2.94/1.54 @ ( collect @ C
% 2.94/1.54 @ ^ [Y4: C] :
% 2.94/1.54 ( ( member @ C @ Y4 @ B3 )
% 2.94/1.54 & ( R2 @ X @ Y4 ) ) ) )
% 2.94/1.54 @ A3 )
% 2.94/1.54 = ( groups1340683514dd_sum @ C @ A
% 2.94/1.54 @ ^ [Y4: C] :
% 2.94/1.54 ( groups1340683514dd_sum @ B @ A
% 2.94/1.54 @ ^ [X: B] : ( G @ X @ Y4 )
% 2.94/1.54 @ ( collect @ B
% 2.94/1.54 @ ^ [X: B] :
% 2.94/1.54 ( ( member @ B @ X @ A3 )
% 2.94/1.54 & ( R2 @ X @ Y4 ) ) ) )
% 2.94/1.54 @ B3 ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_162_infinite__finite__induct,axiom,
% 2.94/1.54 ! [A: $tType,P: ( set @ A ) > $o,A3: set @ A] :
% 2.94/1.54 ( ! [A7: set @ A] :
% 2.94/1.54 ( ~ ( finite_finite2 @ A @ A7 )
% 2.94/1.54 => ( P @ A7 ) )
% 2.94/1.54 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.94/1.54 => ( ! [X3: A,F3: set @ A] :
% 2.94/1.54 ( ( finite_finite2 @ A @ F3 )
% 2.94/1.54 => ( ~ ( member @ A @ X3 @ F3 )
% 2.94/1.54 => ( ( P @ F3 )
% 2.94/1.54 => ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) )
% 2.94/1.54 => ( P @ A3 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_163_finite__ne__induct,axiom,
% 2.94/1.54 ! [A: $tType,F4: set @ A,P: ( set @ A ) > $o] :
% 2.94/1.54 ( ( finite_finite2 @ A @ F4 )
% 2.94/1.54 => ( ( F4
% 2.94/1.54 != ( bot_bot @ ( set @ A ) ) )
% 2.94/1.54 => ( ! [X3: A] : ( P @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.94/1.54 => ( ! [X3: A,F3: set @ A] :
% 2.94/1.54 ( ( finite_finite2 @ A @ F3 )
% 2.94/1.54 => ( ( F3
% 2.94/1.54 != ( bot_bot @ ( set @ A ) ) )
% 2.94/1.54 => ( ~ ( member @ A @ X3 @ F3 )
% 2.94/1.54 => ( ( P @ F3 )
% 2.94/1.54 => ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) ) )
% 2.94/1.54 => ( P @ F4 ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_164_finite_Oinducts,axiom,
% 2.94/1.54 ! [A: $tType,X2: set @ A,P: ( set @ A ) > $o] :
% 2.94/1.54 ( ( finite_finite2 @ A @ X2 )
% 2.94/1.54 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.94/1.54 => ( ! [A7: set @ A,A6: A] :
% 2.94/1.54 ( ( finite_finite2 @ A @ A7 )
% 2.94/1.54 => ( ( P @ A7 )
% 2.94/1.54 => ( P @ ( insert @ A @ A6 @ A7 ) ) ) )
% 2.94/1.54 => ( P @ X2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_165_finite__induct,axiom,
% 2.94/1.54 ! [A: $tType,F4: set @ A,P: ( set @ A ) > $o] :
% 2.94/1.54 ( ( finite_finite2 @ A @ F4 )
% 2.94/1.54 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.94/1.54 => ( ! [X3: A,F3: set @ A] :
% 2.94/1.54 ( ( finite_finite2 @ A @ F3 )
% 2.94/1.54 => ( ~ ( member @ A @ X3 @ F3 )
% 2.94/1.54 => ( ( P @ F3 )
% 2.94/1.54 => ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) )
% 2.94/1.54 => ( P @ F4 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_166_finite_Osimps,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( finite_finite2 @ A )
% 2.94/1.54 = ( ^ [A4: set @ A] :
% 2.94/1.54 ( ( A4
% 2.94/1.54 = ( bot_bot @ ( set @ A ) ) )
% 2.94/1.54 | ? [A5: set @ A,B5: A] :
% 2.94/1.54 ( ( A4
% 2.94/1.54 = ( insert @ A @ B5 @ A5 ) )
% 2.94/1.54 & ( finite_finite2 @ A @ A5 ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_167_finite_Ocases,axiom,
% 2.94/1.54 ! [A: $tType,A2: set @ A] :
% 2.94/1.54 ( ( finite_finite2 @ A @ A2 )
% 2.94/1.54 => ( ( A2
% 2.94/1.54 != ( bot_bot @ ( set @ A ) ) )
% 2.94/1.54 => ~ ! [A7: set @ A] :
% 2.94/1.54 ( ? [A6: A] :
% 2.94/1.54 ( A2
% 2.94/1.54 = ( insert @ A @ A6 @ A7 ) )
% 2.94/1.54 => ~ ( finite_finite2 @ A @ A7 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_168_prod_Ofinite__Collect__op,axiom,
% 2.94/1.54 ! [A: $tType,B: $tType] :
% 2.94/1.54 ( ( comm_monoid_mult @ A )
% 2.94/1.54 => ! [I4: set @ B,X2: B > A,Y: B > A] :
% 2.94/1.54 ( ( finite_finite2 @ B
% 2.94/1.54 @ ( collect @ B
% 2.94/1.54 @ ^ [I3: B] :
% 2.94/1.54 ( ( member @ B @ I3 @ I4 )
% 2.94/1.54 & ( ( X2 @ I3 )
% 2.94/1.54 != ( one_one @ A ) ) ) ) )
% 2.94/1.54 => ( ( finite_finite2 @ B
% 2.94/1.54 @ ( collect @ B
% 2.94/1.54 @ ^ [I3: B] :
% 2.94/1.54 ( ( member @ B @ I3 @ I4 )
% 2.94/1.54 & ( ( Y @ I3 )
% 2.94/1.54 != ( one_one @ A ) ) ) ) )
% 2.94/1.54 => ( finite_finite2 @ B
% 2.94/1.54 @ ( collect @ B
% 2.94/1.54 @ ^ [I3: B] :
% 2.94/1.54 ( ( member @ B @ I3 @ I4 )
% 2.94/1.54 & ( ( times_times @ A @ ( X2 @ I3 ) @ ( Y @ I3 ) )
% 2.94/1.54 != ( one_one @ A ) ) ) ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_169_sum_Oinsert__if,axiom,
% 2.94/1.54 ! [A: $tType,B: $tType] :
% 2.94/1.54 ( ( comm_monoid_add @ A )
% 2.94/1.54 => ! [A3: set @ B,X2: B,G: B > A] :
% 2.94/1.54 ( ( finite_finite2 @ B @ A3 )
% 2.94/1.54 => ( ( ( member @ B @ X2 @ A3 )
% 2.94/1.54 => ( ( groups1340683514dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
% 2.94/1.54 = ( groups1340683514dd_sum @ B @ A @ G @ A3 ) ) )
% 2.94/1.54 & ( ~ ( member @ B @ X2 @ A3 )
% 2.94/1.54 => ( ( groups1340683514dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
% 2.94/1.54 = ( plus_plus @ A @ ( G @ X2 ) @ ( groups1340683514dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_170_convex__singleton,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( real_V1076094709vector @ A )
% 2.94/1.54 => ! [A2: A] : ( convex @ A @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_171_real__divide__square__eq,axiom,
% 2.94/1.54 ! [R: real,A2: real] :
% 2.94/1.54 ( ( divide_divide @ real @ ( times_times @ real @ R @ A2 ) @ ( times_times @ real @ R @ R ) )
% 2.94/1.54 = ( divide_divide @ real @ A2 @ R ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_172_convex__empty,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( real_V1076094709vector @ A )
% 2.94/1.54 => ( convex @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_173_div__by__1,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( semidom_divide @ A )
% 2.94/1.54 => ! [A2: A] :
% 2.94/1.54 ( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
% 2.94/1.54 = A2 ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_174_bits__div__by__1,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( semiring_bits @ A )
% 2.94/1.54 => ! [A2: A] :
% 2.94/1.54 ( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
% 2.94/1.54 = A2 ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_175_add__diff__cancel,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( group_add @ A )
% 2.94/1.54 => ! [A2: A,B2: A] :
% 2.94/1.54 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 2.94/1.54 = A2 ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_176_diff__add__cancel,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( group_add @ A )
% 2.94/1.54 => ! [A2: A,B2: A] :
% 2.94/1.54 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
% 2.94/1.54 = A2 ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_177_add__right__cancel,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( cancel_semigroup_add @ A )
% 2.94/1.54 => ! [B2: A,A2: A,C2: A] :
% 2.94/1.54 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.94/1.54 = ( plus_plus @ A @ C2 @ A2 ) )
% 2.94/1.54 = ( B2 = C2 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_178_add__left__cancel,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( cancel_semigroup_add @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.94/1.54 = ( plus_plus @ A @ A2 @ C2 ) )
% 2.94/1.54 = ( B2 = C2 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_179_mult_Oright__neutral,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( monoid_mult @ A )
% 2.94/1.54 => ! [A2: A] :
% 2.94/1.54 ( ( times_times @ A @ A2 @ ( one_one @ A ) )
% 2.94/1.54 = A2 ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_180_mult_Oleft__neutral,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( monoid_mult @ A )
% 2.94/1.54 => ! [A2: A] :
% 2.94/1.54 ( ( times_times @ A @ ( one_one @ A ) @ A2 )
% 2.94/1.54 = A2 ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_181_add__diff__cancel__right_H,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( cancel146912293up_add @ A )
% 2.94/1.54 => ! [A2: A,B2: A] :
% 2.94/1.54 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 2.94/1.54 = A2 ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_182_add__diff__cancel__right,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( cancel146912293up_add @ A )
% 2.94/1.54 => ! [A2: A,C2: A,B2: A] :
% 2.94/1.54 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 2.94/1.54 = ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_183_add__diff__cancel__left_H,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( cancel146912293up_add @ A )
% 2.94/1.54 => ! [A2: A,B2: A] :
% 2.94/1.54 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 )
% 2.94/1.54 = B2 ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_184_add__diff__cancel__left,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( cancel146912293up_add @ A )
% 2.94/1.54 => ! [C2: A,A2: A,B2: A] :
% 2.94/1.54 ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 2.94/1.54 = ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_185_mult_Oleft__commute,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( ab_semigroup_mult @ A )
% 2.94/1.54 => ! [B2: A,A2: A,C2: A] :
% 2.94/1.54 ( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) )
% 2.94/1.54 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_186_mult_Ocommute,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( ab_semigroup_mult @ A )
% 2.94/1.54 => ( ( times_times @ A )
% 2.94/1.54 = ( ^ [A4: A,B5: A] : ( times_times @ A @ B5 @ A4 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_187_mult_Oassoc,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( semigroup_mult @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
% 2.94/1.54 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_188_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( ab_semigroup_mult @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
% 2.94/1.54 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_189_add__right__imp__eq,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( cancel_semigroup_add @ A )
% 2.94/1.54 => ! [B2: A,A2: A,C2: A] :
% 2.94/1.54 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.94/1.54 = ( plus_plus @ A @ C2 @ A2 ) )
% 2.94/1.54 => ( B2 = C2 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_190_add__left__imp__eq,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( cancel_semigroup_add @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.94/1.54 = ( plus_plus @ A @ A2 @ C2 ) )
% 2.94/1.54 => ( B2 = C2 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_191_add_Oleft__commute,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( ab_semigroup_add @ A )
% 2.94/1.54 => ! [B2: A,A2: A,C2: A] :
% 2.94/1.54 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
% 2.94/1.54 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_192_add_Ocommute,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( ab_semigroup_add @ A )
% 2.94/1.54 => ( ( plus_plus @ A )
% 2.94/1.54 = ( ^ [A4: A,B5: A] : ( plus_plus @ A @ B5 @ A4 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_193_add_Oright__cancel,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( group_add @ A )
% 2.94/1.54 => ! [B2: A,A2: A,C2: A] :
% 2.94/1.54 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.94/1.54 = ( plus_plus @ A @ C2 @ A2 ) )
% 2.94/1.54 = ( B2 = C2 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_194_add_Oleft__cancel,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( group_add @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.94/1.54 = ( plus_plus @ A @ A2 @ C2 ) )
% 2.94/1.54 = ( B2 = C2 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_195_add_Oassoc,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( semigroup_add @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 2.94/1.54 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_196_group__cancel_Oadd2,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( comm_monoid_add @ A )
% 2.94/1.54 => ! [B3: A,K: A,B2: A,A2: A] :
% 2.94/1.54 ( ( B3
% 2.94/1.54 = ( plus_plus @ A @ K @ B2 ) )
% 2.94/1.54 => ( ( plus_plus @ A @ A2 @ B3 )
% 2.94/1.54 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_197_group__cancel_Oadd1,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( comm_monoid_add @ A )
% 2.94/1.54 => ! [A3: A,K: A,A2: A,B2: A] :
% 2.94/1.54 ( ( A3
% 2.94/1.54 = ( plus_plus @ A @ K @ A2 ) )
% 2.94/1.54 => ( ( plus_plus @ A @ A3 @ B2 )
% 2.94/1.54 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_198_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( ordere779506340up_add @ A )
% 2.94/1.54 => ! [I2: A,J2: A,K: A,L: A] :
% 2.94/1.54 ( ( ( I2 = J2 )
% 2.94/1.54 & ( K = L ) )
% 2.94/1.54 => ( ( plus_plus @ A @ I2 @ K )
% 2.94/1.54 = ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_199_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( ab_semigroup_add @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 2.94/1.54 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_200_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( cancel146912293up_add @ A )
% 2.94/1.54 => ! [A2: A,C2: A,B2: A] :
% 2.94/1.54 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 )
% 2.94/1.54 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_201_diff__eq__diff__eq,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( group_add @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A,D: A] :
% 2.94/1.54 ( ( ( minus_minus @ A @ A2 @ B2 )
% 2.94/1.54 = ( minus_minus @ A @ C2 @ D ) )
% 2.94/1.54 => ( ( A2 = B2 )
% 2.94/1.54 = ( C2 = D ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_202_one__reorient,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( one @ A )
% 2.94/1.54 => ! [X2: A] :
% 2.94/1.54 ( ( ( one_one @ A )
% 2.94/1.54 = X2 )
% 2.94/1.54 = ( X2
% 2.94/1.54 = ( one_one @ A ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_203_convex__set__plus,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( real_V1076094709vector @ A )
% 2.94/1.54 => ! [S: set @ A,T: set @ A] :
% 2.94/1.54 ( ( convex @ A @ S )
% 2.94/1.54 => ( ( convex @ A @ T )
% 2.94/1.54 => ( convex @ A @ ( plus_plus @ ( set @ A ) @ S @ T ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_204_finite__set__sum,axiom,
% 2.94/1.54 ! [B: $tType,A: $tType] :
% 2.94/1.54 ( ( comm_monoid_add @ B )
% 2.94/1.54 => ! [A3: set @ A,B3: A > ( set @ B )] :
% 2.94/1.54 ( ( finite_finite2 @ A @ A3 )
% 2.94/1.54 => ( ! [X3: A] :
% 2.94/1.54 ( ( member @ A @ X3 @ A3 )
% 2.94/1.54 => ( finite_finite2 @ B @ ( B3 @ X3 ) ) )
% 2.94/1.54 => ( finite_finite2 @ B @ ( groups1340683514dd_sum @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_205_convex__set__sum,axiom,
% 2.94/1.54 ! [B: $tType,A: $tType] :
% 2.94/1.54 ( ( real_V1076094709vector @ B )
% 2.94/1.54 => ! [A3: set @ A,B3: A > ( set @ B )] :
% 2.94/1.54 ( ! [I5: A] :
% 2.94/1.54 ( ( member @ A @ I5 @ A3 )
% 2.94/1.54 => ( convex @ B @ ( B3 @ I5 ) ) )
% 2.94/1.54 => ( convex @ B @ ( groups1340683514dd_sum @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_206_combine__common__factor,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( semiring @ A )
% 2.94/1.54 => ! [A2: A,E: A,B2: A,C2: A] :
% 2.94/1.54 ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ C2 ) )
% 2.94/1.54 = ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ E ) @ C2 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_207_distrib__right,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( semiring @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 2.94/1.54 = ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_208_distrib__left,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( semiring @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
% 2.94/1.54 = ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_209_comm__semiring__class_Odistrib,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( comm_semiring @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 2.94/1.54 = ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_210_ring__class_Oring__distribs_I1_J,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( ring @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
% 2.94/1.54 = ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_211_ring__class_Oring__distribs_I2_J,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( ring @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
% 2.94/1.54 = ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_212_right__diff__distrib_H,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( comm_s1003936772cancel @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
% 2.94/1.54 = ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_213_left__diff__distrib_H,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( comm_s1003936772cancel @ A )
% 2.94/1.54 => ! [B2: A,C2: A,A2: A] :
% 2.94/1.54 ( ( times_times @ A @ ( minus_minus @ A @ B2 @ C2 ) @ A2 )
% 2.94/1.54 = ( minus_minus @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C2 @ A2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_214_right__diff__distrib,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( ring @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
% 2.94/1.54 = ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_215_left__diff__distrib,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( ring @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
% 2.94/1.54 = ( minus_minus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_216_mult_Ocomm__neutral,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( comm_monoid_mult @ A )
% 2.94/1.54 => ! [A2: A] :
% 2.94/1.54 ( ( times_times @ A @ A2 @ ( one_one @ A ) )
% 2.94/1.54 = A2 ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_217_comm__monoid__mult__class_Omult__1,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( comm_monoid_mult @ A )
% 2.94/1.54 => ! [A2: A] :
% 2.94/1.54 ( ( times_times @ A @ ( one_one @ A ) @ A2 )
% 2.94/1.54 = A2 ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_218_add__implies__diff,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( cancel1352612707id_add @ A )
% 2.94/1.54 => ! [C2: A,B2: A,A2: A] :
% 2.94/1.54 ( ( ( plus_plus @ A @ C2 @ B2 )
% 2.94/1.54 = A2 )
% 2.94/1.54 => ( C2
% 2.94/1.54 = ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_219_diff__diff__add,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( cancel146912293up_add @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
% 2.94/1.54 = ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_220_diff__add__eq__diff__diff__swap,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( group_add @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
% 2.94/1.54 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_221_diff__add__eq,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( ab_group_add @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
% 2.94/1.54 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_222_diff__diff__eq2,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( group_add @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
% 2.94/1.54 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_223_add__diff__eq,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( group_add @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
% 2.94/1.54 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_224_eq__diff__eq,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( group_add @ A )
% 2.94/1.54 => ! [A2: A,C2: A,B2: A] :
% 2.94/1.54 ( ( A2
% 2.94/1.54 = ( minus_minus @ A @ C2 @ B2 ) )
% 2.94/1.54 = ( ( plus_plus @ A @ A2 @ B2 )
% 2.94/1.54 = C2 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_225_diff__eq__eq,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( group_add @ A )
% 2.94/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.94/1.54 ( ( ( minus_minus @ A @ A2 @ B2 )
% 2.94/1.54 = C2 )
% 2.94/1.54 = ( A2
% 2.94/1.54 = ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_226_group__cancel_Osub1,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( ab_group_add @ A )
% 2.94/1.54 => ! [A3: A,K: A,A2: A,B2: A] :
% 2.94/1.54 ( ( A3
% 2.94/1.54 = ( plus_plus @ A @ K @ A2 ) )
% 2.94/1.54 => ( ( minus_minus @ A @ A3 @ B2 )
% 2.94/1.54 = ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_227_lambda__one,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( monoid_mult @ A )
% 2.94/1.54 => ( ( ^ [X: A] : X )
% 2.94/1.54 = ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_228_square__diff__square__factored,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( comm_ring @ A )
% 2.94/1.54 => ! [X2: A,Y: A] :
% 2.94/1.54 ( ( minus_minus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y @ Y ) )
% 2.94/1.54 = ( times_times @ A @ ( plus_plus @ A @ X2 @ Y ) @ ( minus_minus @ A @ X2 @ Y ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_229_eq__add__iff2,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( ring @ A )
% 2.94/1.54 => ! [A2: A,E: A,C2: A,B2: A,D: A] :
% 2.94/1.54 ( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 )
% 2.94/1.54 = ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D ) )
% 2.94/1.54 = ( C2
% 2.94/1.54 = ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E ) @ D ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_230_eq__add__iff1,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( ring @ A )
% 2.94/1.54 => ! [A2: A,E: A,C2: A,B2: A,D: A] :
% 2.94/1.54 ( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 )
% 2.94/1.54 = ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D ) )
% 2.94/1.54 = ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E ) @ C2 )
% 2.94/1.54 = D ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_231_square__diff__one__factored,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( ring_1 @ A )
% 2.94/1.54 => ! [X2: A] :
% 2.94/1.54 ( ( minus_minus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( one_one @ A ) )
% 2.94/1.54 = ( times_times @ A @ ( plus_plus @ A @ X2 @ ( one_one @ A ) ) @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_232_minus__apply,axiom,
% 2.94/1.54 ! [B: $tType,A: $tType] :
% 2.94/1.54 ( ( minus @ B )
% 2.94/1.54 => ( ( minus_minus @ ( A > B ) )
% 2.94/1.54 = ( ^ [A5: A > B,B4: A > B,X: A] : ( minus_minus @ B @ ( A5 @ X ) @ ( B4 @ X ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_233_vector__space__over__itself_Oscale__one,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( field @ A )
% 2.94/1.54 => ! [X2: A] :
% 2.94/1.54 ( ( times_times @ A @ ( one_one @ A ) @ X2 )
% 2.94/1.54 = X2 ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_234_segment__degen__1,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( real_V1076094709vector @ A )
% 2.94/1.54 => ! [U: real,A2: A,B2: A] :
% 2.94/1.54 ( ( ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ ( minus_minus @ real @ ( one_one @ real ) @ U ) @ A2 ) @ ( real_V1908273582scaleR @ A @ U @ B2 ) )
% 2.94/1.54 = B2 )
% 2.94/1.54 = ( ( A2 = B2 )
% 2.94/1.54 | ( U
% 2.94/1.54 = ( one_one @ real ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_235_vector__space__over__itself_Oscale__scale,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( field @ A )
% 2.94/1.54 => ! [A2: A,B2: A,X2: A] :
% 2.94/1.54 ( ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ X2 ) )
% 2.94/1.54 = ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ X2 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_236_vector__space__over__itself_Oscale__left__commute,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( field @ A )
% 2.94/1.54 => ! [A2: A,B2: A,X2: A] :
% 2.94/1.54 ( ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ X2 ) )
% 2.94/1.54 = ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ X2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_237_vector__space__over__itself_Oscale__left__distrib,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( field @ A )
% 2.94/1.54 => ! [A2: A,B2: A,X2: A] :
% 2.94/1.54 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ X2 )
% 2.94/1.54 = ( plus_plus @ A @ ( times_times @ A @ A2 @ X2 ) @ ( times_times @ A @ B2 @ X2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_238_vector__space__over__itself_Oscale__right__distrib,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( field @ A )
% 2.94/1.54 => ! [A2: A,X2: A,Y: A] :
% 2.94/1.54 ( ( times_times @ A @ A2 @ ( plus_plus @ A @ X2 @ Y ) )
% 2.94/1.54 = ( plus_plus @ A @ ( times_times @ A @ A2 @ X2 ) @ ( times_times @ A @ A2 @ Y ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_239_vector__space__over__itself_Oscale__left__diff__distrib,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( field @ A )
% 2.94/1.54 => ! [A2: A,B2: A,X2: A] :
% 2.94/1.54 ( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ X2 )
% 2.94/1.54 = ( minus_minus @ A @ ( times_times @ A @ A2 @ X2 ) @ ( times_times @ A @ B2 @ X2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_240_vector__space__over__itself_Oscale__right__diff__distrib,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( field @ A )
% 2.94/1.54 => ! [A2: A,X2: A,Y: A] :
% 2.94/1.54 ( ( times_times @ A @ A2 @ ( minus_minus @ A @ X2 @ Y ) )
% 2.94/1.54 = ( minus_minus @ A @ ( times_times @ A @ A2 @ X2 ) @ ( times_times @ A @ A2 @ Y ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_241_add__scaleR__degen,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( real_V1076094709vector @ A )
% 2.94/1.54 => ! [U: real,B2: A,V2: real,A2: A] :
% 2.94/1.54 ( ( ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ U @ B2 ) @ ( real_V1908273582scaleR @ A @ V2 @ A2 ) )
% 2.94/1.54 = ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ U @ A2 ) @ ( real_V1908273582scaleR @ A @ V2 @ B2 ) ) )
% 2.94/1.54 => ( ( U != V2 )
% 2.94/1.54 => ( A2 = B2 ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_242_vector__space__over__itself_Oscale__sum__left,axiom,
% 2.94/1.54 ! [A: $tType,C: $tType] :
% 2.94/1.54 ( ( field @ A )
% 2.94/1.54 => ! [F: C > A,A3: set @ C,X2: A] :
% 2.94/1.54 ( ( times_times @ A @ ( groups1340683514dd_sum @ C @ A @ F @ A3 ) @ X2 )
% 2.94/1.54 = ( groups1340683514dd_sum @ C @ A
% 2.94/1.54 @ ^ [A4: C] : ( times_times @ A @ ( F @ A4 ) @ X2 )
% 2.94/1.54 @ A3 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_243_vector__space__over__itself_Oscale__sum__right,axiom,
% 2.94/1.54 ! [A: $tType,C: $tType] :
% 2.94/1.54 ( ( field @ A )
% 2.94/1.54 => ! [A2: A,F: C > A,A3: set @ C] :
% 2.94/1.54 ( ( times_times @ A @ A2 @ ( groups1340683514dd_sum @ C @ A @ F @ A3 ) )
% 2.94/1.54 = ( groups1340683514dd_sum @ C @ A
% 2.94/1.54 @ ^ [X: C] : ( times_times @ A @ A2 @ ( F @ X ) )
% 2.94/1.54 @ A3 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_244_fun__diff__def,axiom,
% 2.94/1.54 ! [B: $tType,A: $tType] :
% 2.94/1.54 ( ( minus @ B )
% 2.94/1.54 => ( ( minus_minus @ ( A > B ) )
% 2.94/1.54 = ( ^ [A5: A > B,B4: A > B,X: A] : ( minus_minus @ B @ ( A5 @ X ) @ ( B4 @ X ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_245_vector__space__over__itself_Ofinite__Basis,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( field @ A )
% 2.94/1.54 => ( finite_finite2 @ A @ ( insert @ A @ ( one_one @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_246_set__plus__intro,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( plus @ A )
% 2.94/1.54 => ! [A2: A,C4: set @ A,B2: A,D2: set @ A] :
% 2.94/1.54 ( ( member @ A @ A2 @ C4 )
% 2.94/1.54 => ( ( member @ A @ B2 @ D2 )
% 2.94/1.54 => ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ ( set @ A ) @ C4 @ D2 ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_247_sum__clauses_I2_J,axiom,
% 2.94/1.54 ! [A: $tType,B: $tType] :
% 2.94/1.54 ( ( comm_monoid_add @ A )
% 2.94/1.54 => ! [S: set @ B,X2: B,F: B > A] :
% 2.94/1.54 ( ( finite_finite2 @ B @ S )
% 2.94/1.54 => ( ( ( member @ B @ X2 @ S )
% 2.94/1.54 => ( ( groups1340683514dd_sum @ B @ A @ F @ ( insert @ B @ X2 @ S ) )
% 2.94/1.54 = ( groups1340683514dd_sum @ B @ A @ F @ S ) ) )
% 2.94/1.54 & ( ~ ( member @ B @ X2 @ S )
% 2.94/1.54 => ( ( groups1340683514dd_sum @ B @ A @ F @ ( insert @ B @ X2 @ S ) )
% 2.94/1.54 = ( plus_plus @ A @ ( F @ X2 ) @ ( groups1340683514dd_sum @ B @ A @ F @ S ) ) ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_248_set__times__intro,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( times @ A )
% 2.94/1.54 => ! [A2: A,C4: set @ A,B2: A,D2: set @ A] :
% 2.94/1.54 ( ( member @ A @ A2 @ C4 )
% 2.94/1.54 => ( ( member @ A @ B2 @ D2 )
% 2.94/1.54 => ( member @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ ( set @ A ) @ C4 @ D2 ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_249_set__times__elim,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( times @ A )
% 2.94/1.54 => ! [X2: A,A3: set @ A,B3: set @ A] :
% 2.94/1.54 ( ( member @ A @ X2 @ ( times_times @ ( set @ A ) @ A3 @ B3 ) )
% 2.94/1.54 => ~ ! [A6: A,B7: A] :
% 2.94/1.54 ( ( X2
% 2.94/1.54 = ( times_times @ A @ A6 @ B7 ) )
% 2.94/1.54 => ( ( member @ A @ A6 @ A3 )
% 2.94/1.54 => ~ ( member @ A @ B7 @ B3 ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_250_finite__set__times,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( times @ A )
% 2.94/1.54 => ! [S2: set @ A,T2: set @ A] :
% 2.94/1.54 ( ( finite_finite2 @ A @ S2 )
% 2.94/1.54 => ( ( finite_finite2 @ A @ T2 )
% 2.94/1.54 => ( finite_finite2 @ A @ ( times_times @ ( set @ A ) @ S2 @ T2 ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_251_set__plus__elim,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( plus @ A )
% 2.94/1.54 => ! [X2: A,A3: set @ A,B3: set @ A] :
% 2.94/1.54 ( ( member @ A @ X2 @ ( plus_plus @ ( set @ A ) @ A3 @ B3 ) )
% 2.94/1.54 => ~ ! [A6: A,B7: A] :
% 2.94/1.54 ( ( X2
% 2.94/1.54 = ( plus_plus @ A @ A6 @ B7 ) )
% 2.94/1.54 => ( ( member @ A @ A6 @ A3 )
% 2.94/1.54 => ~ ( member @ A @ B7 @ B3 ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_252_finite__set__plus,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( plus @ A )
% 2.94/1.54 => ! [S2: set @ A,T2: set @ A] :
% 2.94/1.54 ( ( finite_finite2 @ A @ S2 )
% 2.94/1.54 => ( ( finite_finite2 @ A @ T2 )
% 2.94/1.54 => ( finite_finite2 @ A @ ( plus_plus @ ( set @ A ) @ S2 @ T2 ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_253_set__one,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( one @ A )
% 2.94/1.54 => ( ( one_one @ ( set @ A ) )
% 2.94/1.54 = ( insert @ A @ ( one_one @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(fact_254_ai1,axiom,
% 2.94/1.54 ord_less_eq @ real @ ( aa @ i ) @ ( one_one @ real ) ).
% 2.94/1.54
% 2.94/1.54 thf(subcl_Real__Vector__Spaces_Oreal__vector___HOL_Otype,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( real_V1076094709vector @ A )
% 2.94/1.54 => ( type @ A ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Oplus,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( real_V1076094709vector @ A )
% 2.94/1.54 => ( plus @ A ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ominus,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( real_V1076094709vector @ A )
% 2.94/1.54 => ( minus @ A ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ogroup__add,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( real_V1076094709vector @ A )
% 2.94/1.54 => ( group_add @ A ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Oab__group__add,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( real_V1076094709vector @ A )
% 2.94/1.54 => ( ab_group_add @ A ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Osemigroup__add,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( real_V1076094709vector @ A )
% 2.94/1.54 => ( semigroup_add @ A ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ocomm__monoid__add,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( real_V1076094709vector @ A )
% 2.94/1.54 => ( comm_monoid_add @ A ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Oab__semigroup__add,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( real_V1076094709vector @ A )
% 2.94/1.54 => ( ab_semigroup_add @ A ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(subcl_Real__Vector__Spaces_Oreal__vector___Real__Vector__Spaces_OscaleR,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( real_V1076094709vector @ A )
% 2.94/1.54 => ( real_Vector_scaleR @ A ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ocancel__semigroup__add,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( real_V1076094709vector @ A )
% 2.94/1.54 => ( cancel_semigroup_add @ A ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ocancel__comm__monoid__add,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( real_V1076094709vector @ A )
% 2.94/1.54 => ( cancel1352612707id_add @ A ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(subcl_Real__Vector__Spaces_Oreal__vector___Groups_Ocancel__ab__semigroup__add,axiom,
% 2.94/1.54 ! [A: $tType] :
% 2.94/1.54 ( ( real_V1076094709vector @ A )
% 2.94/1.54 => ( cancel146912293up_add @ A ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 2.94/1.54 ! [A8: $tType,A9: $tType] :
% 2.94/1.54 ( ( ( finite_finite @ A8 )
% 2.94/1.54 & ( finite_finite @ A9 ) )
% 2.94/1.54 => ( finite_finite @ ( A8 > A9 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_fun___Groups_Ominus,axiom,
% 2.94/1.54 ! [A8: $tType,A9: $tType] :
% 2.94/1.54 ( ( minus @ A9 )
% 2.94/1.54 => ( minus @ ( A8 > A9 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 2.94/1.54 ordere779506340up_add @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 2.94/1.54 cancel146912293up_add @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 2.94/1.54 cancel1352612707id_add @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel,axiom,
% 2.94/1.54 comm_s1003936772cancel @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 2.94/1.54 cancel_semigroup_add @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
% 2.94/1.54 ab_semigroup_mult @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
% 2.94/1.54 comm_monoid_mult @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 2.94/1.54 ab_semigroup_add @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 2.94/1.54 comm_monoid_add @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Groups_Osemigroup__mult,axiom,
% 2.94/1.54 semigroup_mult @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Rings_Osemidom__divide,axiom,
% 2.94/1.54 semidom_divide @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Parity_Osemiring__bits,axiom,
% 2.94/1.54 semiring_bits @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 2.94/1.54 semigroup_add @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Rings_Ocomm__semiring,axiom,
% 2.94/1.54 comm_semiring @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Groups_Omonoid__mult,axiom,
% 2.94/1.54 monoid_mult @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Rings_Osemiring__0,axiom,
% 2.94/1.54 semiring_0 @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Rings_Osemiring,axiom,
% 2.94/1.54 semiring @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Groups_Otimes,axiom,
% 2.94/1.54 times @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Groups_Ominus_1,axiom,
% 2.94/1.54 minus @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Groups_Oplus,axiom,
% 2.94/1.54 plus @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Nat_Onat___Groups_Oone,axiom,
% 2.94/1.54 one @ nat ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Set_Oset___Groups_Oab__semigroup__mult_2,axiom,
% 2.94/1.54 ! [A8: $tType] :
% 2.94/1.54 ( ( ab_semigroup_mult @ A8 )
% 2.94/1.54 => ( ab_semigroup_mult @ ( set @ A8 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Set_Oset___Groups_Ocomm__monoid__mult_3,axiom,
% 2.94/1.54 ! [A8: $tType] :
% 2.94/1.54 ( ( comm_monoid_mult @ A8 )
% 2.94/1.54 => ( comm_monoid_mult @ ( set @ A8 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Set_Oset___Groups_Oab__semigroup__add_4,axiom,
% 2.94/1.54 ! [A8: $tType] :
% 2.94/1.54 ( ( ab_semigroup_add @ A8 )
% 2.94/1.54 => ( ab_semigroup_add @ ( set @ A8 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Set_Oset___Groups_Ocomm__monoid__add_5,axiom,
% 2.94/1.54 ! [A8: $tType] :
% 2.94/1.54 ( ( comm_monoid_add @ A8 )
% 2.94/1.54 => ( comm_monoid_add @ ( set @ A8 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Set_Oset___Groups_Osemigroup__mult_6,axiom,
% 2.94/1.54 ! [A8: $tType] :
% 2.94/1.54 ( ( semigroup_mult @ A8 )
% 2.94/1.54 => ( semigroup_mult @ ( set @ A8 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Set_Oset___Groups_Osemigroup__add_7,axiom,
% 2.94/1.54 ! [A8: $tType] :
% 2.94/1.54 ( ( semigroup_add @ A8 )
% 2.94/1.54 => ( semigroup_add @ ( set @ A8 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Set_Oset___Groups_Omonoid__mult_8,axiom,
% 2.94/1.54 ! [A8: $tType] :
% 2.94/1.54 ( ( monoid_mult @ A8 )
% 2.94/1.54 => ( monoid_mult @ ( set @ A8 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Set_Oset___Finite__Set_Ofinite_9,axiom,
% 2.94/1.54 ! [A8: $tType] :
% 2.94/1.54 ( ( finite_finite @ A8 )
% 2.94/1.54 => ( finite_finite @ ( set @ A8 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Set_Oset___Groups_Otimes_10,axiom,
% 2.94/1.54 ! [A8: $tType] :
% 2.94/1.54 ( ( times @ A8 )
% 2.94/1.54 => ( times @ ( set @ A8 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Set_Oset___Groups_Ominus_11,axiom,
% 2.94/1.54 ! [A8: $tType] : ( minus @ ( set @ A8 ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Set_Oset___Groups_Oplus_12,axiom,
% 2.94/1.54 ! [A8: $tType] :
% 2.94/1.54 ( ( plus @ A8 )
% 2.94/1.54 => ( plus @ ( set @ A8 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Set_Oset___Groups_Oone_13,axiom,
% 2.94/1.54 ! [A8: $tType] :
% 2.94/1.54 ( ( one @ A8 )
% 2.94/1.54 => ( one @ ( set @ A8 ) ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_HOL_Obool___Finite__Set_Ofinite_14,axiom,
% 2.94/1.54 finite_finite @ $o ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_HOL_Obool___Groups_Ominus_15,axiom,
% 2.94/1.54 minus @ $o ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra,axiom,
% 2.94/1.54 real_V1015715145lgebra @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra,axiom,
% 2.94/1.54 real_V148923926lgebra @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_16,axiom,
% 2.94/1.54 ordere779506340up_add @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector,axiom,
% 2.94/1.54 real_V1076094709vector @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_17,axiom,
% 2.94/1.54 cancel146912293up_add @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_18,axiom,
% 2.94/1.54 cancel1352612707id_add @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_19,axiom,
% 2.94/1.54 comm_s1003936772cancel @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_20,axiom,
% 2.94/1.54 cancel_semigroup_add @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Real__Vector__Spaces_OscaleR,axiom,
% 2.94/1.54 real_Vector_scaleR @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_21,axiom,
% 2.94/1.54 ab_semigroup_mult @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_22,axiom,
% 2.94/1.54 comm_monoid_mult @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_23,axiom,
% 2.94/1.54 ab_semigroup_add @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_24,axiom,
% 2.94/1.54 comm_monoid_add @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Groups_Osemigroup__mult_25,axiom,
% 2.94/1.54 semigroup_mult @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Rings_Osemidom__divide_26,axiom,
% 2.94/1.54 semidom_divide @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Groups_Osemigroup__add_27,axiom,
% 2.94/1.54 semigroup_add @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
% 2.94/1.54 division_ring @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Rings_Ocomm__semiring_28,axiom,
% 2.94/1.54 comm_semiring @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Groups_Oab__group__add,axiom,
% 2.94/1.54 ab_group_add @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Groups_Omonoid__mult_29,axiom,
% 2.94/1.54 monoid_mult @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Rings_Osemiring__0_30,axiom,
% 2.94/1.54 semiring_0 @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
% 2.94/1.54 group_add @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Rings_Ocomm__ring,axiom,
% 2.94/1.54 comm_ring @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Rings_Osemiring_31,axiom,
% 2.94/1.54 semiring @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Rings_Oring__1,axiom,
% 2.94/1.54 ring_1 @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Groups_Otimes_32,axiom,
% 2.94/1.54 times @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Groups_Ominus_33,axiom,
% 2.94/1.54 minus @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Fields_Ofield,axiom,
% 2.94/1.54 field @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Groups_Oplus_34,axiom,
% 2.94/1.54 plus @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Rings_Oring,axiom,
% 2.94/1.54 ring @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Real_Oreal___Groups_Oone_35,axiom,
% 2.94/1.54 one @ real ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Extended__Real_Oereal___Groups_Oordered__ab__semigroup__add_36,axiom,
% 2.94/1.54 ordere779506340up_add @ extended_ereal ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Extended__Real_Oereal___Groups_Oab__semigroup__mult_37,axiom,
% 2.94/1.54 ab_semigroup_mult @ extended_ereal ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Extended__Real_Oereal___Groups_Ocomm__monoid__mult_38,axiom,
% 2.94/1.54 comm_monoid_mult @ extended_ereal ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Extended__Real_Oereal___Groups_Oab__semigroup__add_39,axiom,
% 2.94/1.54 ab_semigroup_add @ extended_ereal ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Extended__Real_Oereal___Groups_Ocomm__monoid__add_40,axiom,
% 2.94/1.54 comm_monoid_add @ extended_ereal ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Extended__Real_Oereal___Groups_Osemigroup__mult_41,axiom,
% 2.94/1.54 semigroup_mult @ extended_ereal ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Extended__Real_Oereal___Groups_Osemigroup__add_42,axiom,
% 2.94/1.54 semigroup_add @ extended_ereal ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Extended__Real_Oereal___Groups_Omonoid__mult_43,axiom,
% 2.94/1.54 monoid_mult @ extended_ereal ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Extended__Real_Oereal___Groups_Otimes_44,axiom,
% 2.94/1.54 times @ extended_ereal ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Extended__Real_Oereal___Groups_Ominus_45,axiom,
% 2.94/1.54 minus @ extended_ereal ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Extended__Real_Oereal___Groups_Oplus_46,axiom,
% 2.94/1.54 plus @ extended_ereal ).
% 2.94/1.54
% 2.94/1.54 thf(tcon_Extended__Real_Oereal___Groups_Oone_47,axiom,
% 2.94/1.54 one @ extended_ereal ).
% 2.94/1.54
% 2.94/1.54 thf(help_If_3_1_T,axiom,
% 2.94/1.54 ! [P: $o] :
% 2.94/1.54 ( ( P = $true )
% 2.94/1.54 | ( P = $false ) ) ).
% 2.94/1.54
% 2.94/1.54 thf(help_If_2_1_T,axiom,
% 2.94/1.54 ! [A: $tType,X2: A,Y: A] :
% 2.94/1.54 ( ( if @ A @ $false @ X2 @ Y )
% 2.94/1.54 = Y ) ).
% 2.94/1.54
% 2.94/1.54 thf(help_If_1_1_T,axiom,
% 2.94/1.54 ! [A: $tType,X2: A,Y: A] :
% 2.94/1.54 ( ( if @ A @ $true @ X2 @ Y )
% 2.94/1.54 = X2 ) ).
% 2.94/1.54
% 2.94/1.54 thf(tfree_0,hypothesis,
% 2.94/1.54 real_V1076094709vector @ b ).
% 2.94/1.54
% 2.94/1.54 thf(conj_0,conjecture,
% 2.94/1.54 ( ( f
% 2.94/1.54 @ ( plus_plus @ b
% 2.94/1.54 @ ( real_V1908273582scaleR @ b @ ( times_times @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) @ ( inverse_inverse @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) )
% 2.94/1.54 @ ( groups1340683514dd_sum @ a @ b
% 2.94/1.54 @ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( aa @ J ) @ ( y @ J ) )
% 2.94/1.54 @ sa ) )
% 2.94/1.54 @ ( real_V1908273582scaleR @ b @ ( aa @ i ) @ ( y @ i ) ) ) )
% 2.94/1.54 = ( f
% 2.94/1.54 @ ( plus_plus @ b
% 2.94/1.54 @ ( real_V1908273582scaleR @ b @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) )
% 2.94/1.54 @ ( groups1340683514dd_sum @ a @ b
% 2.94/1.54 @ ^ [J: a] : ( real_V1908273582scaleR @ b @ ( times_times @ real @ ( aa @ J ) @ ( inverse_inverse @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( aa @ i ) ) ) ) @ ( y @ J ) )
% 2.94/1.54 @ sa ) )
% 2.94/1.54 @ ( real_V1908273582scaleR @ b @ ( aa @ i ) @ ( y @ i ) ) ) ) ) ).
% 2.94/1.54 ------------------------
% 2.94/1.56 % (4946)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/27Mi)
% 2.94/1.56 % (4948)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/2Mi)
% 2.94/1.56 % (4950)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/18Mi)
% 2.94/1.56 % (4947)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/2Mi)
% 2.94/1.56 % (4947)Instruction limit reached!
% 2.94/1.56 % (4947)------------------------------
% 2.94/1.56 % (4947)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.56 % (4947)Termination reason: Unknown
% 2.94/1.56 % (4947)Termination phase: shuffling
% 2.94/1.56
% 2.94/1.56 % (4948)Instruction limit reached!
% 2.94/1.56 % (4948)------------------------------
% 2.94/1.56 % (4948)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.56 % (4948)Termination reason: Unknown
% 2.94/1.56 % (4948)Termination phase: shuffling
% 2.94/1.56
% 2.94/1.56 % (4948)Memory used [KB]: 1407
% 2.94/1.56 % (4948)Time elapsed: 0.002 s
% 2.94/1.56 % (4948)Instructions burned: 3 (million)
% 2.94/1.56 % (4948)------------------------------
% 2.94/1.56 % (4948)------------------------------
% 2.94/1.56 % (4947)Memory used [KB]: 1407
% 2.94/1.56 % (4947)Time elapsed: 0.002 s
% 2.94/1.56 % (4947)Instructions burned: 2 (million)
% 2.94/1.56 % (4947)------------------------------
% 2.94/1.56 % (4947)------------------------------
% 2.94/1.56 % (4944)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/183Mi)
% 2.94/1.56 % (4949)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/275Mi)
% 2.94/1.56 % (4945)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/4Mi)
% 2.94/1.56 % (4945)Instruction limit reached!
% 2.94/1.56 % (4945)------------------------------
% 2.94/1.56 % (4945)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.56 % (4945)Termination reason: Unknown
% 2.94/1.56 % (4945)Termination phase: shuffling
% 2.94/1.56
% 2.94/1.56 % (4945)Memory used [KB]: 1407
% 2.94/1.56 % (4945)Time elapsed: 0.003 s
% 2.94/1.56 % (4945)Instructions burned: 6 (million)
% 2.94/1.56 % (4945)------------------------------
% 2.94/1.56 % (4945)------------------------------
% 2.94/1.56 % (4950)Instruction limit reached!
% 2.94/1.56 % (4950)------------------------------
% 2.94/1.56 % (4950)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.56 % (4950)Termination reason: Unknown
% 2.94/1.56 % (4950)Termination phase: shuffling
% 2.94/1.56
% 2.94/1.56 % (4950)Memory used [KB]: 1791
% 2.94/1.56 % (4950)Time elapsed: 0.007 s
% 2.94/1.56 % (4950)Instructions burned: 18 (million)
% 2.94/1.56 % (4950)------------------------------
% 2.94/1.56 % (4950)------------------------------
% 2.94/1.56 % (4946)Instruction limit reached!
% 2.94/1.56 % (4946)------------------------------
% 2.94/1.56 % (4946)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.56 % (4946)Termination reason: Unknown
% 2.94/1.56 % (4946)Termination phase: Property scanning
% 2.94/1.56
% 2.94/1.56 % (4946)Memory used [KB]: 1918
% 2.94/1.56 % (4946)Time elapsed: 0.010 s
% 2.94/1.56 % (4946)Instructions burned: 27 (million)
% 2.94/1.56 % (4946)------------------------------
% 2.94/1.56 % (4946)------------------------------
% 2.94/1.57 % (4952)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/3Mi)
% 2.94/1.57 % (4953)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/37Mi)
% 2.94/1.57 % (4952)Instruction limit reached!
% 2.94/1.57 % (4952)------------------------------
% 2.94/1.57 % (4952)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.57 % (4952)Termination reason: Unknown
% 2.94/1.57 % (4952)Termination phase: shuffling
% 2.94/1.57
% 2.94/1.57 % (4952)Memory used [KB]: 1407
% 2.94/1.57 % (4952)Time elapsed: 0.003 s
% 2.94/1.57 % (4952)Instructions burned: 3 (million)
% 2.94/1.57 % (4952)------------------------------
% 2.94/1.57 % (4952)------------------------------
% 2.94/1.57 % (4955)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/3Mi)
% 2.94/1.57 % (4954)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/15Mi)
% 2.94/1.57 % (4955)Instruction limit reached!
% 2.94/1.57 % (4955)------------------------------
% 2.94/1.57 % (4955)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.57 % (4955)Termination reason: Unknown
% 2.94/1.57 % (4955)Termination phase: shuffling
% 2.94/1.57
% 2.94/1.57 % (4955)Memory used [KB]: 1407
% 2.94/1.57 % (4955)Time elapsed: 0.002 s
% 2.94/1.57 % (4955)Instructions burned: 3 (million)
% 2.94/1.57 % (4955)------------------------------
% 2.94/1.57 % (4955)------------------------------
% 2.94/1.58 % (4954)Instruction limit reached!
% 2.94/1.58 % (4954)------------------------------
% 2.94/1.58 % (4954)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.58 % (4954)Termination reason: Unknown
% 2.94/1.58 % (4954)Termination phase: shuffling
% 2.94/1.58
% 2.94/1.58 % (4954)Memory used [KB]: 1791
% 2.94/1.58 % (4954)Time elapsed: 0.010 s
% 2.94/1.58 % (4954)Instructions burned: 16 (million)
% 2.94/1.58 % (4954)------------------------------
% 2.94/1.58 % (4954)------------------------------
% 2.94/1.58 % (4963)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/1041Mi)
% 2.94/1.58 % (4964)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/7Mi)
% 2.94/1.58 % (4965)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/16Mi)
% 2.94/1.58 % (4953)Instruction limit reached!
% 2.94/1.58 % (4953)------------------------------
% 2.94/1.58 % (4953)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.58 % (4953)Termination reason: Unknown
% 2.94/1.58 % (4953)Termination phase: shuffling
% 2.94/1.58
% 2.94/1.58 % (4953)Memory used [KB]: 2174
% 2.94/1.58 % (4953)Time elapsed: 0.019 s
% 2.94/1.58 % (4953)Instructions burned: 38 (million)
% 2.94/1.58 % (4953)------------------------------
% 2.94/1.58 % (4953)------------------------------
% 2.94/1.59 % (4964)Instruction limit reached!
% 2.94/1.59 % (4964)------------------------------
% 2.94/1.59 % (4964)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.59 % (4964)Termination reason: Unknown
% 2.94/1.59 % (4964)Termination phase: shuffling
% 2.94/1.59
% 2.94/1.59 % (4964)Memory used [KB]: 1535
% 2.94/1.59 % (4964)Time elapsed: 0.005 s
% 2.94/1.59 % (4964)Instructions burned: 7 (million)
% 2.94/1.59 % (4964)------------------------------
% 2.94/1.59 % (4964)------------------------------
% 2.94/1.59 % (4965)Instruction limit reached!
% 2.94/1.59 % (4965)------------------------------
% 2.94/1.59 % (4965)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.59 % (4965)Termination reason: Unknown
% 2.94/1.59 % (4965)Termination phase: shuffling
% 2.94/1.59
% 2.94/1.59 % (4965)Memory used [KB]: 1791
% 2.94/1.59 % (4965)Time elapsed: 0.009 s
% 2.94/1.59 % (4965)Instructions burned: 17 (million)
% 2.94/1.59 % (4965)------------------------------
% 2.94/1.59 % (4965)------------------------------
% 2.94/1.60 % (4968)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/3Mi)
% 2.98/1.60 % (4968)Instruction limit reached!
% 2.98/1.60 % (4968)------------------------------
% 2.98/1.60 % (4968)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.60 % (4968)Termination reason: Unknown
% 2.98/1.60 % (4968)Termination phase: shuffling
% 2.98/1.60
% 2.98/1.60 % (4968)Memory used [KB]: 1407
% 2.98/1.60 % (4968)Time elapsed: 0.003 s
% 2.98/1.60 % (4968)Instructions burned: 4 (million)
% 2.98/1.60 % (4968)------------------------------
% 2.98/1.60 % (4968)------------------------------
% 2.98/1.60 % (4971)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/7Mi)
% 2.98/1.61 % (4970)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/3Mi)
% 2.98/1.61 % (4971)Instruction limit reached!
% 2.98/1.61 % (4971)------------------------------
% 2.98/1.61 % (4971)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.61 % (4970)Instruction limit reached!
% 2.98/1.61 % (4970)------------------------------
% 2.98/1.61 % (4970)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.61 % (4970)Termination reason: Unknown
% 2.98/1.61 % (4970)Termination phase: shuffling
% 2.98/1.61
% 2.98/1.61 % (4970)Memory used [KB]: 1407
% 2.98/1.61 % (4970)Time elapsed: 0.002 s
% 2.98/1.61 % (4970)Instructions burned: 3 (million)
% 2.98/1.61 % (4970)------------------------------
% 2.98/1.61 % (4970)------------------------------
% 2.98/1.61 % (4971)Termination reason: Unknown
% 2.98/1.61 % (4971)Termination phase: shuffling
% 2.98/1.61
% 2.98/1.61 % (4971)Memory used [KB]: 1535
% 2.98/1.61 % (4971)Time elapsed: 0.005 s
% 2.98/1.61 % (4971)Instructions burned: 8 (million)
% 2.98/1.61 % (4971)------------------------------
% 2.98/1.61 % (4971)------------------------------
% 2.98/1.61 % (4974)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/3Mi)
% 2.98/1.61 % (4976)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/4Mi)
% 2.98/1.61 % (4974)Instruction limit reached!
% 2.98/1.61 % (4974)------------------------------
% 2.98/1.61 % (4974)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.61 % (4974)Termination reason: Unknown
% 2.98/1.61 % (4974)Termination phase: shuffling
% 2.98/1.61
% 2.98/1.61 % (4974)Memory used [KB]: 1407
% 2.98/1.61 % (4974)Time elapsed: 0.003 s
% 2.98/1.61 % (4974)Instructions burned: 4 (million)
% 2.98/1.61 % (4974)------------------------------
% 2.98/1.61 % (4974)------------------------------
% 2.98/1.61 % (4976)Instruction limit reached!
% 2.98/1.61 % (4976)------------------------------
% 2.98/1.61 % (4976)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.61 % (4976)Termination reason: Unknown
% 2.98/1.61 % (4976)Termination phase: shuffling
% 2.98/1.61
% 2.98/1.61 % (4976)Memory used [KB]: 1407
% 2.98/1.61 % (4976)Time elapsed: 0.004 s
% 2.98/1.61 % (4976)Instructions burned: 4 (million)
% 2.98/1.61 % (4976)------------------------------
% 2.98/1.61 % (4976)------------------------------
% 2.98/1.62 % (4979)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/18Mi)
% 2.98/1.62 % (4981)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/6Mi)
% 2.98/1.62 % (4980)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/710Mi)
% 2.98/1.63 % (4983)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/902Mi)
% 2.98/1.63 % (4979)Instruction limit reached!
% 2.98/1.63 % (4979)------------------------------
% 2.98/1.63 % (4979)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.63 % (4979)Termination reason: Unknown
% 2.98/1.63 % (4979)Termination phase: shuffling
% 2.98/1.63
% 2.98/1.63 % (4979)Memory used [KB]: 1791
% 2.98/1.63 % (4979)Time elapsed: 0.011 s
% 2.98/1.63 % (4979)Instructions burned: 19 (million)
% 2.98/1.63 % (4979)------------------------------
% 2.98/1.63 % (4979)------------------------------
% 2.98/1.63 % (4981)Instruction limit reached!
% 2.98/1.63 % (4981)------------------------------
% 2.98/1.63 % (4981)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.63 % (4981)Termination reason: Unknown
% 2.98/1.63 % (4981)Termination phase: shuffling
% 2.98/1.63
% 2.98/1.63 % (4981)Memory used [KB]: 1407
% 2.98/1.63 % (4981)Time elapsed: 0.005 s
% 2.98/1.63 % (4981)Instructions burned: 6 (million)
% 2.98/1.63 % (4981)------------------------------
% 2.98/1.63 % (4981)------------------------------
% 2.98/1.63 % (4944)Instruction limit reached!
% 2.98/1.63 % (4944)------------------------------
% 2.98/1.63 % (4944)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.63 % (4944)Termination reason: Unknown
% 2.98/1.63 % (4944)Termination phase: Saturation
% 2.98/1.63
% 2.98/1.63 % (4944)Memory used [KB]: 7164
% 2.98/1.63 % (4944)Time elapsed: 0.072 s
% 2.98/1.63 % (4944)Instructions burned: 183 (million)
% 2.98/1.63 % (4944)------------------------------
% 2.98/1.63 % (4944)------------------------------
% 2.98/1.64 % (4988)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/21Mi)
% 2.98/1.64 % (4991)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/6Mi)
% 2.98/1.64 % (4990)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/5Mi)
% 2.98/1.64 % (4991)Instruction limit reached!
% 2.98/1.64 % (4991)------------------------------
% 2.98/1.64 % (4991)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.64 % (4991)Termination reason: Unknown
% 2.98/1.64 % (4991)Termination phase: shuffling
% 2.98/1.64
% 2.98/1.64 % (4991)Memory used [KB]: 1535
% 2.98/1.64 % (4991)Time elapsed: 0.005 s
% 2.98/1.64 % (4991)Instructions burned: 7 (million)
% 2.98/1.64 % (4991)------------------------------
% 2.98/1.64 % (4991)------------------------------
% 2.98/1.64 % (4990)Instruction limit reached!
% 2.98/1.64 % (4990)------------------------------
% 2.98/1.64 % (4990)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.64 % (4990)Termination reason: Unknown
% 2.98/1.64 % (4990)Termination phase: shuffling
% 2.98/1.64
% 2.98/1.64 % (4990)Memory used [KB]: 1407
% 2.98/1.64 % (4990)Time elapsed: 0.004 s
% 2.98/1.64 % (4990)Instructions burned: 5 (million)
% 2.98/1.64 % (4990)------------------------------
% 2.98/1.64 % (4990)------------------------------
% 2.98/1.65 % (4988)Instruction limit reached!
% 2.98/1.65 % (4988)------------------------------
% 2.98/1.65 % (4988)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.65 % (4988)Termination reason: Unknown
% 2.98/1.65 % (4988)Termination phase: shuffling
% 2.98/1.65
% 2.98/1.65 % (4988)Memory used [KB]: 1918
% 2.98/1.65 % (4988)Time elapsed: 0.009 s
% 2.98/1.65 % (4988)Instructions burned: 23 (million)
% 2.98/1.65 % (4988)------------------------------
% 2.98/1.65 % (4988)------------------------------
% 2.98/1.66 % (4998)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/19Mi)
% 2.98/1.66 % (4995)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/377Mi)
% 2.98/1.66 % (4996)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_4788 for (2999ds/779Mi)
% 2.98/1.66 % (4949)Refutation not found, incomplete strategy
% 2.98/1.66 % (4949)------------------------------
% 2.98/1.66 % (4949)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.66 % (4949)Termination reason: Refutation not found, incomplete strategy
% 2.98/1.66
% 2.98/1.66
% 2.98/1.66 % (4949)Memory used [KB]: 7291
% 2.98/1.66 % (4949)Time elapsed: 0.105 s
% 2.98/1.66 % (4949)Instructions burned: 235 (million)
% 2.98/1.66 % (4949)------------------------------
% 2.98/1.66 % (4949)------------------------------
% 2.98/1.66 % (4998)Instruction limit reached!
% 2.98/1.66 % (4998)------------------------------
% 2.98/1.66 % (4998)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.66 % (4998)Termination reason: Unknown
% 2.98/1.66 % (4998)Termination phase: shuffling
% 2.98/1.66
% 2.98/1.66 % (4998)Memory used [KB]: 1791
% 2.98/1.66 % (4998)Time elapsed: 0.007 s
% 2.98/1.66 % (4998)Instructions burned: 21 (million)
% 2.98/1.66 % (4998)------------------------------
% 2.98/1.66 % (4998)------------------------------
% 2.98/1.67 % (5002)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_4788 for (2998ds/879Mi)
% 2.98/1.68 % (5003)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_4788 for (2998ds/17Mi)
% 2.98/1.69 % (5003)Instruction limit reached!
% 2.98/1.69 % (5003)------------------------------
% 2.98/1.69 % (5003)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.69 % (5003)Termination reason: Unknown
% 2.98/1.69 % (5003)Termination phase: shuffling
% 2.98/1.69
% 2.98/1.69 % (5003)Memory used [KB]: 1791
% 2.98/1.69 % (5003)Time elapsed: 0.009 s
% 2.98/1.69 % (5003)Instructions burned: 19 (million)
% 2.98/1.69 % (5003)------------------------------
% 2.98/1.69 % (5003)------------------------------
% 2.98/1.70 % (5006)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_4788 for (2998ds/3Mi)
% 2.98/1.70 % (5006)Instruction limit reached!
% 2.98/1.70 % (5006)------------------------------
% 2.98/1.70 % (5006)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.70 % (5006)Termination reason: Unknown
% 2.98/1.70 % (5006)Termination phase: shuffling
% 2.98/1.70
% 2.98/1.70 % (5006)Memory used [KB]: 1407
% 2.98/1.70 % (5006)Time elapsed: 0.003 s
% 2.98/1.70 % (5006)Instructions burned: 5 (million)
% 2.98/1.70 % (5006)------------------------------
% 2.98/1.70 % (5006)------------------------------
% 2.98/1.71 % (5008)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_4788 for (2998ds/30Mi)
% 2.98/1.72 % (5008)Instruction limit reached!
% 2.98/1.72 % (5008)------------------------------
% 2.98/1.72 % (5008)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.72 % (5008)Termination reason: Unknown
% 2.98/1.72 % (5008)Termination phase: Property scanning
% 2.98/1.72
% 2.98/1.72 % (5008)Memory used [KB]: 1918
% 2.98/1.72 % (5008)Time elapsed: 0.011 s
% 2.98/1.72 % (5008)Instructions burned: 32 (million)
% 2.98/1.72 % (5008)------------------------------
% 2.98/1.72 % (5008)------------------------------
% 2.98/1.73 % (5010)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_4788 for (2998ds/127Mi)
% 2.98/1.77 % (5010)Instruction limit reached!
% 2.98/1.77 % (5010)------------------------------
% 2.98/1.77 % (5010)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.77 % (5010)Termination reason: Unknown
% 2.98/1.77 % (5010)Termination phase: Saturation
% 2.98/1.77
% 2.98/1.77 % (5010)Memory used [KB]: 6780
% 2.98/1.77 % (5010)Time elapsed: 0.045 s
% 2.98/1.77 % (5010)Instructions burned: 128 (million)
% 2.98/1.77 % (5010)------------------------------
% 2.98/1.77 % (5010)------------------------------
% 2.98/1.80 % (5014)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_4788 for (2997ds/100Mi)
% 2.98/1.84 % (4995)Instruction limit reached!
% 2.98/1.84 % (4995)------------------------------
% 2.98/1.84 % (4995)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.84 % (4995)Termination reason: Unknown
% 2.98/1.84 % (4995)Termination phase: Saturation
% 2.98/1.84
% 2.98/1.84 % (4995)Memory used [KB]: 7931
% 2.98/1.84 % (4995)Time elapsed: 0.181 s
% 2.98/1.84 % (4995)Instructions burned: 378 (million)
% 2.98/1.84 % (4995)------------------------------
% 2.98/1.84 % (4995)------------------------------
% 2.98/1.84 % (5014)Instruction limit reached!
% 2.98/1.84 % (5014)------------------------------
% 2.98/1.84 % (5014)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.84 % (5014)Termination reason: Unknown
% 2.98/1.84 % (5014)Termination phase: Property scanning
% 2.98/1.84
% 2.98/1.84 % (5014)Memory used [KB]: 2430
% 2.98/1.84 % (5014)Time elapsed: 0.047 s
% 2.98/1.84 % (5014)Instructions burned: 101 (million)
% 2.98/1.84 % (5014)------------------------------
% 2.98/1.84 % (5014)------------------------------
% 2.98/1.86 % (5019)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_4788 for (2997ds/3Mi)
% 2.98/1.86 % (5019)Instruction limit reached!
% 2.98/1.86 % (5019)------------------------------
% 2.98/1.86 % (5019)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.86 % (5019)Termination reason: Unknown
% 2.98/1.86 % (5019)Termination phase: shuffling
% 2.98/1.86
% 2.98/1.86 % (5019)Memory used [KB]: 1407
% 2.98/1.86 % (5019)Time elapsed: 0.003 s
% 2.98/1.86 % (5019)Instructions burned: 3 (million)
% 2.98/1.86 % (5019)------------------------------
% 2.98/1.86 % (5019)------------------------------
% 2.98/1.86 % (5020)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_4788 for (2997ds/20Mi)
% 2.98/1.87 % (5022)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_4788 for (2996ds/86Mi)
% 2.98/1.88 % (5020)Instruction limit reached!
% 2.98/1.88 % (5020)------------------------------
% 2.98/1.88 % (5020)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.88 % (5020)Termination reason: Unknown
% 2.98/1.88 % (5020)Termination phase: shuffling
% 2.98/1.88
% 2.98/1.88 % (5020)Memory used [KB]: 1791
% 2.98/1.88 % (5020)Time elapsed: 0.010 s
% 2.98/1.88 % (5020)Instructions burned: 20 (million)
% 2.98/1.88 % (5020)------------------------------
% 2.98/1.88 % (5020)------------------------------
% 2.98/1.89 % (5024)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_4788 for (2996ds/107Mi)
% 2.98/1.91 % (4983)Refutation not found, incomplete strategy
% 2.98/1.91 % (4983)------------------------------
% 2.98/1.91 % (4983)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.91 % (4983)Termination reason: Refutation not found, incomplete strategy
% 2.98/1.91
% 2.98/1.91
% 2.98/1.91 % (4983)Memory used [KB]: 7291
% 2.98/1.91 % (4983)Time elapsed: 0.285 s
% 2.98/1.91 % (4983)Instructions burned: 636 (million)
% 2.98/1.91 % (4983)------------------------------
% 2.98/1.91 % (4983)------------------------------
% 2.98/1.91 % (5022)Instruction limit reached!
% 2.98/1.91 % (5022)------------------------------
% 2.98/1.91 % (5022)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.91 % (5022)Termination reason: Unknown
% 2.98/1.91 % (5022)Termination phase: Property scanning
% 2.98/1.91
% 2.98/1.91 % (5022)Memory used [KB]: 2302
% 2.98/1.91 % (5022)Time elapsed: 0.042 s
% 2.98/1.91 % (5022)Instructions burned: 86 (million)
% 2.98/1.91 % (5022)------------------------------
% 2.98/1.91 % (5022)------------------------------
% 2.98/1.93 % (5029)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_4788 for (2996ds/448Mi)
% 2.98/1.93 % (5028)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_4788 for (2996ds/39Mi)
% 2.98/1.94 % (5028)Instruction limit reached!
% 2.98/1.94 % (5028)------------------------------
% 2.98/1.94 % (5028)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.94 % (5028)Termination reason: Unknown
% 2.98/1.94 % (5028)Termination phase: shuffling
% 2.98/1.94
% 2.98/1.94 % (5028)Memory used [KB]: 2174
% 2.98/1.94 % (5028)Time elapsed: 0.013 s
% 2.98/1.94 % (5028)Instructions burned: 39 (million)
% 2.98/1.94 % (5028)------------------------------
% 2.98/1.94 % (5028)------------------------------
% 2.98/1.95 % (5024)Instruction limit reached!
% 2.98/1.95 % (5024)------------------------------
% 2.98/1.95 % (5024)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.95 % (5024)Termination reason: Unknown
% 2.98/1.95 % (5024)Termination phase: Saturation
% 2.98/1.95
% 2.98/1.95 % (5024)Memory used [KB]: 6780
% 2.98/1.95 % (5024)Time elapsed: 0.053 s
% 2.98/1.95 % (5024)Instructions burned: 108 (million)
% 2.98/1.95 % (5024)------------------------------
% 2.98/1.95 % (5024)------------------------------
% 2.98/1.95 % (5032)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_4788 for (2995ds/46Mi)
% 2.98/1.96 % (5033)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_4788 for (2995ds/98Mi)
% 2.98/1.96 % (4980)Instruction limit reached!
% 2.98/1.96 % (4980)------------------------------
% 2.98/1.96 % (4980)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.96 % (4980)Termination reason: Unknown
% 2.98/1.96 % (4980)Termination phase: Saturation
% 2.98/1.96
% 2.98/1.96 % (4980)Memory used [KB]: 7803
% 2.98/1.96 % (4980)Time elapsed: 0.340 s
% 2.98/1.96 % (4980)Instructions burned: 712 (million)
% 2.98/1.96 % (4980)------------------------------
% 2.98/1.96 % (4980)------------------------------
% 2.98/1.97 % (5032)Instruction limit reached!
% 2.98/1.97 % (5032)------------------------------
% 2.98/1.97 % (5032)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.97 % (5032)Termination reason: Unknown
% 2.98/1.97 % (5032)Termination phase: SInE selection
% 2.98/1.97
% 2.98/1.97 % (5032)Memory used [KB]: 1918
% 2.98/1.97 % (5032)Time elapsed: 0.015 s
% 2.98/1.97 % (5032)Instructions burned: 46 (million)
% 2.98/1.97 % (5032)------------------------------
% 2.98/1.97 % (5032)------------------------------
% 2.98/1.97 % (5034)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_4788 for (2995ds/507Mi)
% 2.98/1.98 % (5035)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_4788 for (2995ds/149Mi)
% 2.98/1.98 % (5033)Instruction limit reached!
% 2.98/1.98 % (5033)------------------------------
% 2.98/1.98 % (5033)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.98 % (5033)Termination reason: Unknown
% 2.98/1.98 % (5033)Termination phase: Property scanning
% 2.98/1.98
% 2.98/1.98 % (5033)Memory used [KB]: 2430
% 2.98/1.98 % (5033)Time elapsed: 0.028 s
% 2.98/1.98 % (5033)Instructions burned: 100 (million)
% 2.98/1.98 % (5033)------------------------------
% 2.98/1.98 % (5033)------------------------------
% 2.98/1.99 % (5038)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_4788 for (2995ds/39Mi)
% 2.98/2.00 % (4996)Instruction limit reached!
% 2.98/2.00 % (4996)------------------------------
% 2.98/2.00 % (4996)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/2.00 % (4996)Termination reason: Unknown
% 2.98/2.00 % (4996)Termination phase: Saturation
% 2.98/2.00
% 2.98/2.00 % (4996)Memory used [KB]: 7931
% 2.98/2.00 % (4996)Time elapsed: 0.341 s
% 2.98/2.00 % (4996)Instructions burned: 781 (million)
% 2.98/2.00 % (4996)------------------------------
% 2.98/2.00 % (4996)------------------------------
% 2.98/2.00 % (4963)Instruction limit reached!
% 2.98/2.00 % (4963)------------------------------
% 2.98/2.00 % (4963)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/2.00 % (4963)Termination reason: Unknown
% 2.98/2.00 % (4963)Termination phase: Saturation
% 2.98/2.00
% 2.98/2.00 % (4963)Memory used [KB]: 9978
% 2.98/2.00 % (4963)Time elapsed: 0.410 s
% 2.98/2.00 % (4963)Instructions burned: 1041 (million)
% 2.98/2.00 % (4963)------------------------------
% 2.98/2.00 % (4963)------------------------------
% 2.98/2.00 % (5038)Instruction limit reached!
% 2.98/2.00 % (5038)------------------------------
% 2.98/2.00 % (5038)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/2.00 % (5038)Termination reason: Unknown
% 2.98/2.00 % (5038)Termination phase: Preprocessing 1
% 2.98/2.00
% 2.98/2.00 % (5038)Memory used [KB]: 1918
% 2.98/2.00 % (5038)Time elapsed: 0.014 s
% 2.98/2.00 % (5038)Instructions burned: 41 (million)
% 2.98/2.00 % (5038)------------------------------
% 2.98/2.00 % (5038)------------------------------
% 2.98/2.01 % (5046)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_4788 for (2995ds/6Mi)
% 2.98/2.01 % (5046)Instruction limit reached!
% 2.98/2.01 % (5046)------------------------------
% 2.98/2.01 % (5046)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/2.01 % (5046)Termination reason: Unknown
% 2.98/2.01 % (5046)Termination phase: shuffling
% 2.98/2.01
% 2.98/2.01 % (5046)Memory used [KB]: 1535
% 2.98/2.01 % (5046)Time elapsed: 0.003 s
% 2.98/2.01 % (5046)Instructions burned: 7 (million)
% 2.98/2.01 % (5046)------------------------------
% 2.98/2.01 % (5046)------------------------------
% 6.11/2.01 % (5048)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_4788 for (2995ds/1087Mi)
% 6.11/2.01 % (5049)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_4788 for (2995ds/47Mi)
% 6.11/2.02 % (5035)Instruction limit reached!
% 6.11/2.02 % (5035)------------------------------
% 6.11/2.02 % (5035)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.11/2.02 % (5035)Termination reason: Unknown
% 6.11/2.02 % (5035)Termination phase: Saturation
% 6.11/2.02
% 6.11/2.02 % (5035)Memory used [KB]: 6908
% 6.11/2.02 % (5035)Time elapsed: 0.045 s
% 6.11/2.02 % (5035)Instructions burned: 151 (million)
% 6.11/2.02 % (5035)------------------------------
% 6.11/2.02 % (5035)------------------------------
% 6.11/2.02 % (5053)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_4788 for (2995ds/44Mi)
% 6.11/2.03 % (5049)Instruction limit reached!
% 6.11/2.03 % (5049)------------------------------
% 6.11/2.03 % (5049)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.11/2.03 % (5049)Termination reason: Unknown
% 6.11/2.03 % (5049)Termination phase: Preprocessing 3
% 6.11/2.03
% 6.11/2.03 % (5049)Memory used [KB]: 2046
% 6.11/2.03 % (5049)Time elapsed: 0.016 s
% 6.11/2.03 % (5049)Instructions burned: 48 (million)
% 6.11/2.03 % (5049)------------------------------
% 6.11/2.03 % (5049)------------------------------
% 6.11/2.03 % (5058)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_4788 for (2995ds/432Mi)
% 6.11/2.03 % (5053)Instruction limit reached!
% 6.11/2.03 % (5053)------------------------------
% 6.11/2.03 % (5053)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.11/2.03 % (5053)Termination reason: Unknown
% 6.11/2.03 % (5053)Termination phase: Property scanning
% 6.11/2.03
% 6.11/2.03 % (5053)Memory used [KB]: 2174
% 6.11/2.03 % (5053)Time elapsed: 0.014 s
% 6.11/2.03 % (5053)Instructions burned: 45 (million)
% 6.11/2.03 % (5053)------------------------------
% 6.11/2.03 % (5053)------------------------------
% 6.11/2.04 % (5002)Instruction limit reached!
% 6.11/2.04 % (5002)------------------------------
% 6.11/2.04 % (5002)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.11/2.04 % (5002)Termination reason: Unknown
% 6.11/2.04 % (5002)Termination phase: Saturation
% 6.11/2.04
% 6.11/2.04 % (5002)Memory used [KB]: 8443
% 6.11/2.04 % (5002)Time elapsed: 0.367 s
% 6.11/2.04 % (5002)Instructions burned: 880 (million)
% 6.11/2.04 % (5002)------------------------------
% 6.11/2.04 % (5002)------------------------------
% 6.11/2.04 % (5062)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_4788 for (2995ds/62Mi)
% 6.11/2.04 % (5065)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_4788 for (2995ds/31Mi)
% 6.11/2.05 % (5067)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_4788 for (2995ds/96Mi)
% 6.11/2.05 % (5065)Instruction limit reached!
% 6.11/2.05 % (5065)------------------------------
% 6.11/2.05 % (5065)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.11/2.05 % (5065)Termination reason: Unknown
% 6.11/2.05 % (5065)Termination phase: Property scanning
% 6.11/2.05
% 6.11/2.05 % (5065)Memory used [KB]: 1918
% 6.11/2.05 % (5065)Time elapsed: 0.011 s
% 6.11/2.05 % (5065)Instructions burned: 32 (million)
% 6.11/2.05 % (5065)------------------------------
% 6.11/2.05 % (5065)------------------------------
% 6.11/2.06 % (5062)Instruction limit reached!
% 6.11/2.06 % (5062)------------------------------
% 6.11/2.06 % (5062)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.11/2.06 % (5062)Termination reason: Unknown
% 6.11/2.06 % (5062)Termination phase: Property scanning
% 6.11/2.06
% 6.11/2.06 % (5062)Memory used [KB]: 2174
% 6.11/2.06 % (5062)Time elapsed: 0.018 s
% 6.11/2.06 % (5062)Instructions burned: 63 (million)
% 6.11/2.06 % (5062)------------------------------
% 6.11/2.06 % (5062)------------------------------
% 6.11/2.06 % (5074)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_4788 for (2994ds/453Mi)
% 6.11/2.06 % (5077)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_4788 for (2994ds/71Mi)
% 6.11/2.07 % (5029)Instruction limit reached!
% 6.11/2.07 % (5029)------------------------------
% 6.11/2.07 % (5029)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.11/2.07 % (5029)Termination reason: Unknown
% 6.11/2.07 % (5029)Termination phase: Saturation
% 6.11/2.07
% 6.11/2.07 % (5029)Memory used [KB]: 7931
% 6.11/2.07 % (5029)Time elapsed: 0.138 s
% 6.11/2.07 % (5029)Instructions burned: 449 (million)
% 6.11/2.07 % (5029)------------------------------
% 6.11/2.07 % (5029)------------------------------
% 6.11/2.08 % (5067)Instruction limit reached!
% 6.11/2.08 % (5067)------------------------------
% 6.11/2.08 % (5067)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.11/2.08 % (5067)Termination reason: Unknown
% 6.11/2.08 % (5067)Termination phase: Property scanning
% 6.11/2.08
% 6.11/2.08 % (5067)Memory used [KB]: 2302
% 6.11/2.08 % (5067)Time elapsed: 0.029 s
% 6.11/2.08 % (5067)Instructions burned: 98 (million)
% 6.11/2.08 % (5067)------------------------------
% 6.11/2.08 % (5067)------------------------------
% 6.11/2.08 % (5084)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_4788 for (2994ds/83Mi)
% 6.11/2.08 % (5088)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_4788 for (2994ds/11Mi)
% 6.11/2.09 % (5077)Instruction limit reached!
% 6.11/2.09 % (5077)------------------------------
% 6.11/2.09 % (5077)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.11/2.09 % (5077)Termination reason: Unknown
% 6.11/2.09 % (5077)Termination phase: Function definition elimination
% 6.11/2.09
% 6.11/2.09 % (5077)Memory used [KB]: 2302
% 6.11/2.09 % (5077)Time elapsed: 0.023 s
% 6.11/2.09 % (5077)Instructions burned: 74 (million)
% 6.11/2.09 % (5077)------------------------------
% 6.11/2.09 % (5077)------------------------------
% 6.11/2.09 % (5088)Instruction limit reached!
% 6.11/2.09 % (5088)------------------------------
% 6.11/2.09 % (5088)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.11/2.09 % (5088)Termination reason: Unknown
% 6.11/2.09 % (5088)Termination phase: shuffling
% 6.11/2.09
% 6.11/2.09 % (5088)Memory used [KB]: 1663
% 6.11/2.09 % (5088)Time elapsed: 0.005 s
% 6.11/2.09 % (5088)Instructions burned: 13 (million)
% 6.11/2.09 % (5088)------------------------------
% 6.11/2.09 % (5088)------------------------------
% 6.11/2.10 % (5092)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_4788 for (2994ds/41Mi)
% 6.11/2.10 % (5094)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_4788 for (2994ds/6Mi)
% 6.11/2.10 % (5094)Instruction limit reached!
% 6.11/2.10 % (5094)------------------------------
% 6.11/2.10 % (5094)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.11/2.10 % (5094)Termination reason: Unknown
% 6.11/2.10 % (5094)Termination phase: shuffling
% 6.11/2.10
% 6.11/2.10 % (5094)Memory used [KB]: 1535
% 6.11/2.10 % (5094)Time elapsed: 0.003 s
% 6.11/2.10 % (5094)Instructions burned: 6 (million)
% 6.11/2.10 % (5094)------------------------------
% 6.11/2.10 % (5094)------------------------------
% 6.11/2.10 % (5084)Instruction limit reached!
% 6.11/2.10 % (5084)------------------------------
% 6.11/2.10 % (5084)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.11/2.10 % (5084)Termination reason: Unknown
% 6.11/2.10 % (5084)Termination phase: Saturation
% 6.11/2.10
% 6.11/2.10 % (5084)Memory used [KB]: 6652
% 6.11/2.10 % (5084)Time elapsed: 0.027 s
% 6.11/2.10 % (5084)Instructions burned: 86 (million)
% 6.11/2.10 % (5084)------------------------------
% 6.11/2.10 % (5084)------------------------------
% 6.11/2.11 % (5092)Instruction limit reached!
% 6.11/2.11 % (5092)------------------------------
% 6.11/2.11 % (5092)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.11/2.11 % (5092)Termination reason: Unknown
% 6.11/2.11 % (5092)Termination phase: Property scanning
% 6.11/2.11
% 6.11/2.11 % (5092)Memory used [KB]: 2174
% 6.11/2.11 % (5092)Time elapsed: 0.013 s
% 6.11/2.11 % (5092)Instructions burned: 43 (million)
% 6.11/2.11 % (5092)------------------------------
% 6.11/2.11 % (5092)------------------------------
% 6.11/2.11 % (5099)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_4788 for (2994ds/150Mi)
% 6.58/2.11 % (5102)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_4788 for (2994ds/343Mi)
% 6.58/2.12 % (5105)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_4788 for (2994ds/109Mi)
% 6.58/2.12 % (5034)Instruction limit reached!
% 6.58/2.12 % (5034)------------------------------
% 6.58/2.12 % (5034)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.58/2.12 % (5034)Termination reason: Unknown
% 6.58/2.12 % (5034)Termination phase: Saturation
% 6.58/2.12
% 6.58/2.12 % (5034)Memory used [KB]: 7803
% 6.58/2.12 % (5034)Time elapsed: 0.148 s
% 6.58/2.12 % (5034)Instructions burned: 508 (million)
% 6.58/2.12 % (5034)------------------------------
% 6.58/2.12 % (5034)------------------------------
% 6.58/2.13 % (5110)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_4788 for (2994ds/253Mi)
% 6.58/2.15 % (5105)Instruction limit reached!
% 6.58/2.15 % (5105)------------------------------
% 6.58/2.15 % (5105)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.58/2.15 % (5105)Termination reason: Unknown
% 6.58/2.15 % (5105)Termination phase: Saturation
% 6.58/2.15
% 6.58/2.15 % (5105)Memory used [KB]: 6780
% 6.58/2.15 % (5105)Time elapsed: 0.035 s
% 6.58/2.15 % (5105)Instructions burned: 112 (million)
% 6.58/2.15 % (5105)------------------------------
% 6.58/2.15 % (5105)------------------------------
% 6.58/2.15 % (5099)Instruction limit reached!
% 6.58/2.15 % (5099)------------------------------
% 6.58/2.15 % (5099)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.58/2.15 % (5099)Termination reason: Unknown
% 6.58/2.15 % (5099)Termination phase: Saturation
% 6.58/2.15
% 6.58/2.15 % (5099)Memory used [KB]: 6908
% 6.58/2.15 % (5099)Time elapsed: 0.046 s
% 6.58/2.15 % (5099)Instructions burned: 151 (million)
% 6.58/2.15 % (5099)------------------------------
% 6.58/2.15 % (5099)------------------------------
% 6.58/2.16 % (5121)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_4788 for (2993ds/140Mi)
% 6.58/2.16 % (5122)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_4788 for (2993ds/614Mi)
% 6.58/2.17 % (5058)Instruction limit reached!
% 6.58/2.17 % (5058)------------------------------
% 6.58/2.17 % (5058)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.58/2.17 % (5058)Termination reason: Unknown
% 6.58/2.17 % (5058)Termination phase: Saturation
% 6.58/2.17
% 6.58/2.17 % (5058)Memory used [KB]: 3709
% 6.58/2.17 % (5058)Time elapsed: 0.138 s
% 6.58/2.17 % (5058)Instructions burned: 433 (million)
% 6.58/2.17 % (5058)------------------------------
% 6.58/2.17 % (5058)------------------------------
% 6.58/2.18 % (5127)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_4788 for (2993ds/202Mi)
% 6.85/2.20 % (5074)Instruction limit reached!
% 6.85/2.20 % (5074)------------------------------
% 6.85/2.20 % (5074)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.85/2.20 % (5074)Termination reason: Unknown
% 6.85/2.20 % (5074)Termination phase: Saturation
% 6.85/2.20
% 6.85/2.20 % (5074)Memory used [KB]: 8059
% 6.85/2.20 % (5074)Time elapsed: 0.138 s
% 6.85/2.20 % (5074)Instructions burned: 454 (million)
% 6.85/2.20 % (5074)------------------------------
% 6.85/2.20 % (5074)------------------------------
% 6.85/2.21 % (5121)Instruction limit reached!
% 6.85/2.21 % (5121)------------------------------
% 6.85/2.21 % (5121)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.85/2.21 % (5121)Termination reason: Unknown
% 6.85/2.21 % (5121)Termination phase: Saturation
% 6.85/2.21
% 6.85/2.21 % (5121)Memory used [KB]: 6780
% 6.85/2.21 % (5121)Time elapsed: 0.046 s
% 6.85/2.21 % (5121)Instructions burned: 142 (million)
% 6.85/2.21 % (5121)------------------------------
% 6.85/2.21 % (5121)------------------------------
% 6.85/2.21 % (5110)Instruction limit reached!
% 6.85/2.21 % (5110)------------------------------
% 6.85/2.21 % (5110)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.85/2.21 % (5110)Termination reason: Unknown
% 6.85/2.21 % (5110)Termination phase: Saturation
% 6.85/2.21
% 6.85/2.21 % (5110)Memory used [KB]: 7675
% 6.85/2.21 % (5110)Time elapsed: 0.079 s
% 6.85/2.21 % (5110)Instructions burned: 256 (million)
% 6.85/2.21 % (5110)------------------------------
% 6.85/2.21 % (5110)------------------------------
% 6.85/2.21 % (5144)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_4788 for (2993ds/339Mi)
% 6.85/2.22 % (5145)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=283:si=on:rtra=on_0 on DTF2THF_4788 for (2993ds/283Mi)
% 6.85/2.22 % (5146)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_4788 for (2993ds/134Mi)
% 6.85/2.22 % (5102)Instruction limit reached!
% 6.85/2.22 % (5102)------------------------------
% 6.85/2.22 % (5102)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.85/2.22 % (5102)Termination reason: Unknown
% 6.85/2.22 % (5102)Termination phase: Saturation
% 6.85/2.22
% 6.85/2.22 % (5102)Memory used [KB]: 7675
% 6.85/2.22 % (5102)Time elapsed: 0.110 s
% 6.85/2.22 % (5102)Instructions burned: 343 (million)
% 6.85/2.22 % (5102)------------------------------
% 6.85/2.22 % (5102)------------------------------
% 6.85/2.23 % (5158)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_4788 for (2993ds/298Mi)
% 6.85/2.24 % (5127)Instruction limit reached!
% 6.85/2.24 % (5127)------------------------------
% 6.85/2.24 % (5127)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.85/2.24 % (5127)Termination reason: Unknown
% 6.85/2.24 % (5127)Termination phase: Saturation
% 6.85/2.24
% 6.85/2.24 % (5127)Memory used [KB]: 7419
% 6.85/2.24 % (5127)Time elapsed: 0.063 s
% 6.85/2.24 % (5127)Instructions burned: 203 (million)
% 6.85/2.24 % (5127)------------------------------
% 6.85/2.24 % (5127)------------------------------
% 6.85/2.25 % (5170)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_4788 for (2993ds/192Mi)
% 6.85/2.26 % (5146)Instruction limit reached!
% 6.85/2.26 % (5146)------------------------------
% 6.85/2.26 % (5146)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.85/2.26 % (5146)Termination reason: Unknown
% 6.85/2.26 % (5146)Termination phase: Saturation
% 6.85/2.26
% 6.85/2.26 % (5146)Memory used [KB]: 6908
% 6.85/2.26 % (5146)Time elapsed: 0.040 s
% 6.85/2.26 % (5146)Instructions burned: 137 (million)
% 6.85/2.26 % (5146)------------------------------
% 6.85/2.26 % (5146)------------------------------
% 6.85/2.27 % (5173)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_4788 for (2992ds/258Mi)
% 7.38/2.30 % (5145)Instruction limit reached!
% 7.38/2.30 % (5145)------------------------------
% 7.38/2.30 % (5145)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.30 % (5145)Termination reason: Unknown
% 7.38/2.30 % (5145)Termination phase: Saturation
% 7.38/2.30
% 7.38/2.30 % (5145)Memory used [KB]: 7675
% 7.38/2.30 % (5145)Time elapsed: 0.088 s
% 7.38/2.30 % (5145)Instructions burned: 283 (million)
% 7.38/2.30 % (5145)------------------------------
% 7.38/2.30 % (5145)------------------------------
% 7.38/2.30 % (5144)Instruction limit reached!
% 7.38/2.30 % (5144)------------------------------
% 7.38/2.30 % (5144)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.30 % (5144)Termination reason: Unknown
% 7.38/2.30 % (5144)Termination phase: Saturation
% 7.38/2.30
% 7.38/2.30 % (5144)Memory used [KB]: 7164
% 7.38/2.30 % (5144)Time elapsed: 0.096 s
% 7.38/2.30 % (5144)Instructions burned: 340 (million)
% 7.38/2.30 % (5144)------------------------------
% 7.38/2.30 % (5144)------------------------------
% 7.38/2.31 % (5170)Instruction limit reached!
% 7.38/2.31 % (5170)------------------------------
% 7.38/2.31 % (5170)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.31 % (5170)Termination reason: Unknown
% 7.38/2.31 % (5170)Termination phase: Saturation
% 7.38/2.31
% 7.38/2.31 % (5170)Memory used [KB]: 7931
% 7.38/2.31 % (5170)Time elapsed: 0.061 s
% 7.38/2.31 % (5170)Instructions burned: 193 (million)
% 7.38/2.31 % (5170)------------------------------
% 7.38/2.31 % (5170)------------------------------
% 7.38/2.31 % (5197)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_4788 for (2992ds/90Mi)
% 7.38/2.31 % (5199)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_4788 for (2992ds/231Mi)
% 7.38/2.32 % (5158)Instruction limit reached!
% 7.38/2.32 % (5158)------------------------------
% 7.38/2.32 % (5158)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.32 % (5158)Termination reason: Unknown
% 7.38/2.32 % (5158)Termination phase: Saturation
% 7.38/2.32
% 7.38/2.32 % (5158)Memory used [KB]: 7931
% 7.38/2.32 % (5158)Time elapsed: 0.087 s
% 7.38/2.32 % (5158)Instructions burned: 299 (million)
% 7.38/2.32 % (5158)------------------------------
% 7.38/2.32 % (5158)------------------------------
% 7.38/2.32 % (5203)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_4788 for (2992ds/458Mi)
% 7.38/2.33 % (5207)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_4788 for (2992ds/31Mi)
% 7.38/2.34 % (5207)Instruction limit reached!
% 7.38/2.34 % (5207)------------------------------
% 7.38/2.34 % (5207)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.34 % (5207)Termination reason: Unknown
% 7.38/2.34 % (5207)Termination phase: Property scanning
% 7.38/2.34
% 7.38/2.34 % (5207)Memory used [KB]: 1918
% 7.38/2.34 % (5207)Time elapsed: 0.012 s
% 7.38/2.34 % (5207)Instructions burned: 32 (million)
% 7.38/2.34 % (5207)------------------------------
% 7.38/2.34 % (5207)------------------------------
% 7.38/2.34 % (5197)Instruction limit reached!
% 7.38/2.34 % (5197)------------------------------
% 7.38/2.34 % (5197)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.34 % (5197)Termination reason: Unknown
% 7.38/2.34 % (5197)Termination phase: Property scanning
% 7.38/2.34
% 7.38/2.34 % (5197)Memory used [KB]: 2302
% 7.38/2.34 % (5197)Time elapsed: 0.028 s
% 7.38/2.34 % (5197)Instructions burned: 90 (million)
% 7.38/2.34 % (5197)------------------------------
% 7.38/2.34 % (5197)------------------------------
% 7.38/2.34 % (5173)Instruction limit reached!
% 7.38/2.34 % (5173)------------------------------
% 7.38/2.34 % (5173)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.34 % (5173)Termination reason: Unknown
% 7.38/2.34 % (5173)Termination phase: Saturation
% 7.38/2.34
% 7.38/2.34 % (5173)Memory used [KB]: 7675
% 7.38/2.34 % (5173)Time elapsed: 0.080 s
% 7.38/2.34 % (5173)Instructions burned: 259 (million)
% 7.38/2.34 % (5173)------------------------------
% 7.38/2.34 % (5173)------------------------------
% 7.38/2.35 % (5215)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_4788 for (2992ds/297Mi)
% 7.38/2.35 % (5216)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_4788 for (2992ds/987Mi)
% 7.38/2.35 % (5122)Instruction limit reached!
% 7.38/2.35 % (5122)------------------------------
% 7.38/2.35 % (5122)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.35 % (5122)Termination reason: Unknown
% 7.38/2.35 % (5122)Termination phase: Saturation
% 7.38/2.35
% 7.38/2.35 % (5122)Memory used [KB]: 8571
% 7.38/2.35 % (5122)Time elapsed: 0.188 s
% 7.38/2.35 % (5122)Instructions burned: 615 (million)
% 7.38/2.35 % (5122)------------------------------
% 7.38/2.35 % (5122)------------------------------
% 7.38/2.35 % (5218)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_4788 for (2991ds/182Mi)
% 7.38/2.36 % (5048)Instruction limit reached!
% 7.38/2.36 % (5048)------------------------------
% 7.38/2.36 % (5048)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.36 % (5048)Termination reason: Unknown
% 7.38/2.36 % (5048)Termination phase: Saturation
% 7.38/2.36
% 7.38/2.36 % (5048)Memory used [KB]: 8315
% 7.38/2.36 % (5048)Time elapsed: 0.344 s
% 7.38/2.36 % (5048)Instructions burned: 1089 (million)
% 7.38/2.36 % (5048)------------------------------
% 7.38/2.36 % (5048)------------------------------
% 7.38/2.36 % (5221)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_4788 for (2991ds/182Mi)
% 7.38/2.37 % (5224)dis+10_1:8_cha=on:cnfonf=lazy_pi_sigma_gen:fe=axiom:hud=5:plsq=on:plsqr=32,1:prag=on:sos=all:i=154:si=on:rtra=on_0 on DTF2THF_4788 for (2991ds/154Mi)
% 7.38/2.38 % (5199)Instruction limit reached!
% 7.38/2.38 % (5199)------------------------------
% 7.38/2.38 % (5199)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.38 % (5199)Termination reason: Unknown
% 7.38/2.38 % (5199)Termination phase: Saturation
% 7.38/2.38
% 7.38/2.38 % (5199)Memory used [KB]: 7036
% 7.38/2.38 % (5199)Time elapsed: 0.063 s
% 7.38/2.38 % (5199)Instructions burned: 232 (million)
% 7.38/2.38 % (5199)------------------------------
% 7.38/2.38 % (5199)------------------------------
% 7.38/2.39 % (5231)dis+1003_1:64_atotf=0.4:au=on:fsr=off:hflw=5:hfsq=on:hfsqc=20:hfsql=off:plsq=on:plsqc=1:plsqr=32,1:i=79:si=on:rtra=on_0 on DTF2THF_4788 for (2991ds/79Mi)
% 8.50/2.41 % (5218)Instruction limit reached!
% 8.50/2.41 % (5218)------------------------------
% 8.50/2.41 % (5218)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.50/2.41 % (5218)Termination reason: Unknown
% 8.50/2.41 % (5218)Termination phase: Saturation
% 8.50/2.41
% 8.50/2.41 % (5218)Memory used [KB]: 7164
% 8.50/2.41 % (5218)Time elapsed: 0.054 s
% 8.50/2.41 % (5218)Instructions burned: 182 (million)
% 8.50/2.41 % (5218)------------------------------
% 8.50/2.41 % (5218)------------------------------
% 8.50/2.41 % (5224)Instruction limit reached!
% 8.50/2.41 % (5224)------------------------------
% 8.50/2.41 % (5224)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.50/2.41 % (5224)Termination reason: Unknown
% 8.50/2.41 % (5224)Termination phase: Saturation
% 8.50/2.41
% 8.50/2.41 % (5224)Memory used [KB]: 2942
% 8.50/2.41 % (5224)Time elapsed: 0.044 s
% 8.50/2.41 % (5224)Instructions burned: 155 (million)
% 8.50/2.41 % (5224)------------------------------
% 8.50/2.41 % (5224)------------------------------
% 8.50/2.41 % (5231)Instruction limit reached!
% 8.50/2.41 % (5231)------------------------------
% 8.50/2.41 % (5231)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.50/2.41 % (5231)Termination reason: Unknown
% 8.50/2.41 % (5231)Termination phase: Property scanning
% 8.50/2.41
% 8.50/2.41 % (5231)Memory used [KB]: 2302
% 8.50/2.41 % (5231)Time elapsed: 0.025 s
% 8.50/2.41 % (5231)Instructions burned: 81 (million)
% 8.50/2.41 % (5231)------------------------------
% 8.50/2.41 % (5231)------------------------------
% 8.50/2.42 % (5244)dis+1010_5:2_anc=all_dependent:awrs=converge:bd=off:bet=on:c=on:cha=on:cnfonf=lazy_not_gen_be_off:fe=axiom:hud=5:kws=precedence:piset=and:prag=on:s2a=on:sac=on:slsq=on:slsqc=1:sp=reverse_arity:ss=axioms:st=1.5:i=216:si=on:rtra=on_0 on DTF2THF_4788 for (2991ds/216Mi)
% 8.50/2.42 % (5221)Instruction limit reached!
% 8.50/2.42 % (5221)------------------------------
% 8.50/2.42 % (5221)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.50/2.42 % (5221)Termination reason: Unknown
% 8.50/2.42 % (5221)Termination phase: Saturation
% 8.50/2.42
% 8.50/2.42 % (5221)Memory used [KB]: 7291
% 8.50/2.42 % (5221)Time elapsed: 0.058 s
% 8.50/2.42 % (5221)Instructions burned: 183 (million)
% 8.50/2.42 % (5221)------------------------------
% 8.50/2.42 % (5221)------------------------------
% 8.50/2.42 % (5245)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_4788 for (2991ds/114Mi)
% 8.50/2.42 % (5247)dis+1002_1:1_acc=on:au=on:bd=off:e2e=on:sgt=8:ss=axioms:i=922:si=on:rtra=on_0 on DTF2THF_4788 for (2991ds/922Mi)
% 8.50/2.43 % (5249)lrs+10_1:128_au=on:bd=off:cnfonf=lazy_not_gen_be_off:nwc=10.0:piset=and:i=182:si=on:rtra=on_0 on DTF2THF_4788 for (2991ds/182Mi)
% 8.50/2.44 % (5215)Instruction limit reached!
% 8.50/2.44 % (5215)------------------------------
% 8.50/2.44 % (5215)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.50/2.44 % (5215)Termination reason: Unknown
% 8.50/2.44 % (5215)Termination phase: Saturation
% 8.50/2.44
% 8.50/2.44 % (5215)Memory used [KB]: 7675
% 8.50/2.44 % (5215)Time elapsed: 0.092 s
% 8.50/2.44 % (5215)Instructions burned: 298 (million)
% 8.50/2.44 % (5215)------------------------------
% 8.50/2.44 % (5215)------------------------------
% 8.50/2.45 % (5254)ott+1010_1:1024_au=on:av=off:c=on:cnfonf=conj_eager:piset=equals:s2a=on:sp=occurrence:spb=units:ss=axioms:i=164:si=on:rtra=on_0 on DTF2THF_4788 for (2991ds/164Mi)
% 8.50/2.45 % (5245)Instruction limit reached!
% 8.50/2.45 % (5245)------------------------------
% 8.50/2.45 % (5245)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.50/2.45 % (5245)Termination reason: Unknown
% 8.50/2.45 % (5245)Termination phase: Saturation
% 8.50/2.45
% 8.50/2.45 % (5245)Memory used [KB]: 6780
% 8.50/2.45 % (5245)Time elapsed: 0.035 s
% 8.50/2.45 % (5245)Instructions burned: 116 (million)
% 8.50/2.45 % (5245)------------------------------
% 8.50/2.45 % (5245)------------------------------
% 8.50/2.46 % (5261)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=50:si=on:rtra=on_0 on DTF2THF_4788 for (2990ds/50Mi)
% 8.50/2.46 % (5203)Instruction limit reached!
% 8.50/2.46 % (5203)------------------------------
% 8.50/2.46 % (5203)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.50/2.46 % (5203)Termination reason: Unknown
% 8.50/2.46 % (5203)Termination phase: Saturation
% 8.50/2.46
% 8.50/2.46 % (5203)Memory used [KB]: 7931
% 8.50/2.46 % (5203)Time elapsed: 0.144 s
% 8.50/2.46 % (5203)Instructions burned: 458 (million)
% 8.50/2.46 % (5203)------------------------------
% 8.50/2.46 % (5203)------------------------------
% 8.50/2.47 % (5264)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_4788 for (2990ds/368Mi)
% 8.50/2.48 % (5261)Instruction limit reached!
% 8.50/2.48 % (5261)------------------------------
% 8.50/2.48 % (5261)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.50/2.48 % (5261)Termination reason: Unknown
% 8.50/2.48 % (5261)Termination phase: Property scanning
% 8.50/2.48
% 8.50/2.48 % (5261)Memory used [KB]: 2174
% 8.50/2.48 % (5261)Time elapsed: 0.015 s
% 8.50/2.48 % (5261)Instructions burned: 51 (million)
% 8.50/2.48 % (5261)------------------------------
% 8.50/2.48 % (5261)------------------------------
% 8.50/2.48 % (5249)Instruction limit reached!
% 8.50/2.48 % (5249)------------------------------
% 8.50/2.48 % (5249)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.50/2.48 % (5249)Termination reason: Unknown
% 8.50/2.48 % (5249)Termination phase: Saturation
% 8.50/2.48
% 8.50/2.48 % (5249)Memory used [KB]: 7291
% 8.50/2.48 % (5249)Time elapsed: 0.052 s
% 8.50/2.48 % (5249)Instructions burned: 183 (million)
% 8.50/2.48 % (5249)------------------------------
% 8.50/2.48 % (5249)------------------------------
% 8.50/2.48 % (5244)Instruction limit reached!
% 8.50/2.48 % (5244)------------------------------
% 8.50/2.48 % (5244)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.50/2.48 % (5244)Termination reason: Unknown
% 8.50/2.48 % (5244)Termination phase: Saturation
% 8.50/2.48
% 8.50/2.48 % (5244)Memory used [KB]: 7419
% 8.50/2.48 % (5244)Time elapsed: 0.064 s
% 8.50/2.48 % (5244)Instructions burned: 216 (million)
% 8.50/2.48 % (5244)------------------------------
% 8.50/2.48 % (5244)------------------------------
% 8.50/2.48 % (5270)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1021:si=on:rtra=on_0 on DTF2THF_4788 for (2990ds/1021Mi)
% 8.50/2.49 % (5271)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=267:si=on:rtra=on_0 on DTF2THF_4788 for (2990ds/267Mi)
% 8.50/2.49 % (5272)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=204:si=on:rtra=on_0 on DTF2THF_4788 for (2990ds/204Mi)
% 8.50/2.50 % (5254)Instruction limit reached!
% 8.50/2.50 % (5254)------------------------------
% 8.50/2.50 % (5254)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.50/2.50 % (5254)Termination reason: Unknown
% 8.50/2.50 % (5254)Termination phase: Saturation
% 8.50/2.50
% 8.50/2.50 % (5254)Memory used [KB]: 2814
% 8.50/2.50 % (5254)Time elapsed: 0.049 s
% 8.50/2.50 % (5254)Instructions burned: 167 (million)
% 8.50/2.50 % (5254)------------------------------
% 8.50/2.50 % (5254)------------------------------
% 9.23/2.51 % (5277)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_4788 for (2990ds/26Mi)
% 9.23/2.51 % (5277)Instruction limit reached!
% 9.23/2.51 % (5277)------------------------------
% 9.23/2.51 % (5277)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.23/2.51 % (5277)Termination reason: Unknown
% 9.23/2.51 % (5277)Termination phase: shuffling
% 9.23/2.51
% 9.23/2.51 % (5277)Memory used [KB]: 2046
% 9.23/2.51 % (5277)Time elapsed: 0.009 s
% 9.23/2.51 % (5277)Instructions burned: 29 (million)
% 9.23/2.51 % (5277)------------------------------
% 9.23/2.51 % (5277)------------------------------
% 9.23/2.52 % (5286)dis+1002_1:1_bd=preordered:c=on:cbe=off:cnfonf=lazy_not_gen_be_off:hud=10:ixr=off:prag=on:i=53:si=on:rtra=on_0 on DTF2THF_4788 for (2990ds/53Mi)
% 9.23/2.54 % (5286)Instruction limit reached!
% 9.23/2.54 % (5286)------------------------------
% 9.23/2.54 % (5286)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.23/2.54 % (5286)Termination reason: Unknown
% 9.23/2.54 % (5286)Termination phase: Property scanning
% 9.23/2.54
% 9.23/2.54 % (5286)Memory used [KB]: 2174
% 9.23/2.54 % (5286)Time elapsed: 0.017 s
% 9.23/2.54 % (5286)Instructions burned: 54 (million)
% 9.23/2.54 % (5286)------------------------------
% 9.23/2.54 % (5286)------------------------------
% 9.23/2.55 % (5301)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=4436:si=on:rtra=on_0 on DTF2THF_4788 for (2990ds/4436Mi)
% 9.23/2.55 % (5272)Instruction limit reached!
% 9.23/2.55 % (5272)------------------------------
% 9.23/2.55 % (5272)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.23/2.55 % (5272)Termination reason: Unknown
% 9.23/2.55 % (5272)Termination phase: Saturation
% 9.23/2.55
% 9.23/2.55 % (5272)Memory used [KB]: 7164
% 9.23/2.55 % (5272)Time elapsed: 0.064 s
% 9.23/2.55 % (5272)Instructions burned: 206 (million)
% 9.23/2.55 % (5272)------------------------------
% 9.23/2.55 % (5272)------------------------------
% 9.23/2.56 % (5310)dis+1010_1:1_acc=model:avsq=on:ixr=off:prag=on:s2a=on:sac=on:i=1539:si=on:rtra=on_0 on DTF2THF_4788 for (2989ds/1539Mi)
% 9.23/2.57 % (5271)Instruction limit reached!
% 9.23/2.57 % (5271)------------------------------
% 9.23/2.57 % (5271)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.23/2.57 % (5271)Termination reason: Unknown
% 9.23/2.57 % (5271)Termination phase: Saturation
% 9.23/2.57
% 9.23/2.57 % (5271)Memory used [KB]: 7675
% 9.23/2.57 % (5271)Time elapsed: 0.084 s
% 9.23/2.57 % (5271)Instructions burned: 267 (million)
% 9.23/2.57 % (5271)------------------------------
% 9.23/2.57 % (5271)------------------------------
% 9.23/2.58 % (5264)Instruction limit reached!
% 9.23/2.58 % (5264)------------------------------
% 9.23/2.58 % (5264)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.23/2.58 % (5264)Termination reason: Unknown
% 9.23/2.58 % (5264)Termination phase: Saturation
% 9.23/2.58
% 9.23/2.58 % (5264)Memory used [KB]: 8187
% 9.23/2.58 % (5264)Time elapsed: 0.107 s
% 9.23/2.58 % (5264)Instructions burned: 368 (million)
% 9.23/2.58 % (5264)------------------------------
% 9.23/2.58 % (5264)------------------------------
% 9.23/2.58 % (5316)lrs+2_1:1_bd=off:e2e=on:fde=none:kws=inv_precedence:plsq=on:plsqr=32,1:sos=on:i=408:si=on:rtra=on_0 on DTF2THF_4788 for (2989ds/408Mi)
% 9.23/2.59 % (5319)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=5205:si=on:rtra=on_0 on DTF2THF_4788 for (2989ds/5205Mi)
% 9.56/2.66 % (5216)Instruction limit reached!
% 9.56/2.66 % (5216)------------------------------
% 9.56/2.66 % (5216)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.56/2.66 % (5216)Termination reason: Unknown
% 9.56/2.66 % (5216)Termination phase: Saturation
% 9.56/2.66
% 9.56/2.66 % (5216)Memory used [KB]: 9083
% 9.56/2.66 % (5216)Time elapsed: 0.330 s
% 9.56/2.66 % (5216)Instructions burned: 989 (million)
% 9.56/2.66 % (5216)------------------------------
% 9.56/2.66 % (5216)------------------------------
% 9.56/2.67 % (5323)dis+1010_1:4_au=on:av=off:cbe=off:nwc=10.0:s2a=on:s2at=3.0:sd=2:ss=axioms:st=5.0:i=1913:si=on:rtra=on_0 on DTF2THF_4788 for (2988ds/1913Mi)
% 9.88/2.70 % (5310)First to succeed.
% 9.88/2.70 % (5310)Refutation found. Thanks to Tanya!
% 9.88/2.70 % SZS status Theorem for DTF2THF_4788
% 9.88/2.70 % SZS output start Proof for DTF2THF_4788
% 9.88/2.70 thf(type_def_5, type, set: ($tType) > $tType).
% 9.88/2.70 thf(type_def_7, type, extended_ereal: $tType).
% 9.88/2.70 thf(type_def_8, type, real: $tType).
% 9.88/2.70 thf(type_def_9, type, b: $tType).
% 9.88/2.70 thf(type_def_10, type, a: $tType).
% 9.88/2.70 thf(type_def_11, type, nat: $tType).
% 9.88/2.70 thf(func_def_0, type, extended_ereal: $tType).
% 9.88/2.70 thf(func_def_1, type, real: $tType).
% 9.88/2.70 thf(func_def_2, type, set: ($tType) > $tType).
% 9.88/2.70 thf(func_def_3, type, nat: $tType).
% 9.88/2.70 thf(func_def_4, type, b: $tType).
% 9.88/2.70 thf(func_def_5, type, a: $tType).
% 9.88/2.70 thf(func_def_6, type, type: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_7, type, one: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_8, type, ring: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_9, type, plus: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_10, type, field: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_11, type, minus: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_12, type, times: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_13, type, ring_1: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_14, type, semiring: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_15, type, comm_ring: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_16, type, group_add: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_17, type, semiring_0: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_18, type, finite_finite: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_19, type, monoid_mult: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_20, type, ab_group_add: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_21, type, comm_semiring: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_22, type, division_ring: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_23, type, semigroup_add: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_24, type, semiring_bits: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_25, type, semidom_divide: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_26, type, semigroup_mult: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_27, type, comm_monoid_add: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_28, type, ab_semigroup_add: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_29, type, comm_monoid_mult: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_30, type, ab_semigroup_mult: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_31, type, real_Vector_scaleR: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_32, type, cancel_semigroup_add: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_33, type, comm_s1003936772cancel: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_34, type, cancel1352612707id_add: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_35, type, cancel146912293up_add: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_36, type, real_V1076094709vector: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_37, type, ordere779506340up_add: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_38, type, real_V148923926lgebra: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_39, type, real_V1015715145lgebra: !>[X0: $tType]:($o)).
% 9.88/2.70 thf(func_def_40, type, convex: !>[X0: $tType]:(set @ X0 > $o)).
% 9.88/2.70 thf(func_def_41, type, inverse_inverse: !>[X0: $tType]:(X0 > X0)).
% 9.88/2.70 thf(func_def_42, type, finite_finite2: !>[X0: $tType]:(set @ X0 > $o)).
% 9.88/2.70 thf(func_def_43, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 9.88/2.70 thf(func_def_44, type, one_one: !>[X0: $tType]:(X0)).
% 9.88/2.70 thf(func_def_45, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 9.88/2.70 thf(func_def_46, type, times_times: !>[X0: $tType]:(X0 > X0 > X0)).
% 9.88/2.70 thf(func_def_47, type, groups1340683514dd_sum: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X1)).
% 9.88/2.70 thf(func_def_48, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 9.88/2.70 thf(func_def_49, type, lower_673667120vex_on: !>[X0: $tType]:(set @ X0 > (X0 > extended_ereal) > $o)).
% 9.88/2.70 thf(func_def_50, type, bot_bot: !>[X0: $tType]:(X0)).
% 9.88/2.70 thf(func_def_51, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 9.88/2.70 thf(func_def_52, type, real_V1908273582scaleR: !>[X0: $tType]:(real > X0 > X0)).
% 9.88/2.70 thf(func_def_53, type, divide_divide: !>[X0: $tType]:(X0 > X0 > X0)).
% 9.88/2.70 thf(func_def_54, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 9.88/2.70 thf(func_def_55, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 9.88/2.70 thf(func_def_56, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 9.88/2.70 thf(func_def_57, type, c: set @ b).
% 9.88/2.70 thf(func_def_58, type, a2: a > real).
% 9.88/2.70 thf(func_def_59, type, aa: a > real).
% 9.88/2.70 thf(func_def_60, type, f: b > extended_ereal).
% 9.88/2.70 thf(func_def_61, type, i: a).
% 9.88/2.70 thf(func_def_62, type, s: set @ a).
% 9.88/2.70 thf(func_def_63, type, sa: set @ a).
% 9.88/2.70 thf(func_def_64, type, y: a > b).
% 9.88/2.70 thf(func_def_79, type, sP0: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > set @ X0 > $o)).
% 9.88/2.70 thf(func_def_80, type, sP1: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X0 > X1) > set @ X0 > $o)).
% 9.88/2.70 thf(func_def_81, type, sK2: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 9.88/2.70 thf(func_def_82, type, sK3: !>[X0: $tType]:((set @ X0 > $o) > X0)).
% 9.88/2.70 thf(func_def_83, type, sK4: !>[X0: $tType]:((set @ X0 > $o) > X0)).
% 9.88/2.70 thf(func_def_84, type, sK5: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 9.88/2.70 thf(func_def_85, type, sK6: !>[X0: $tType, X1: $tType]:((X1 > set @ X0) > set @ X1 > X1)).
% 9.88/2.70 thf(func_def_86, type, sK7: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1)).
% 9.88/2.70 thf(func_def_87, type, sK8: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1 > X0)).
% 9.88/2.70 thf(func_def_88, type, sK9: !>[X0: $tType]:(set @ X0 > X0 > set @ X0)).
% 9.88/2.70 thf(func_def_89, type, sK10: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X0 > X1) > set @ X0 > X0)).
% 9.88/2.70 thf(func_def_90, type, sK11: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > set @ X0 > X1)).
% 9.88/2.70 thf(func_def_91, type, sK12: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X1 > X0) > set @ X0 > X0)).
% 9.88/2.70 thf(func_def_92, type, sK13: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > set @ X0 > X1)).
% 9.88/2.70 thf(func_def_93, type, sK14: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X1) > (X0 > X2) > set @ X0 > X0)).
% 9.88/2.70 thf(func_def_94, type, sK15: !>[X0: $tType, X1: $tType]:((X1 > set @ X0) > set @ X1 > X1)).
% 9.88/2.70 thf(func_def_95, type, sK16: !>[X0: $tType]:((set @ X0 > $o) > X0)).
% 9.88/2.70 thf(func_def_96, type, sK17: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 9.88/2.70 thf(func_def_97, type, sK18: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 9.88/2.70 thf(func_def_98, type, sK19: !>[X0: $tType]:((set @ X0 > $o) > X0)).
% 9.88/2.70 thf(func_def_99, type, sK20: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 9.88/2.70 thf(func_def_100, type, sK21: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 9.88/2.70 thf(func_def_101, type, sK22: !>[X0: $tType]:(set @ X0 > set @ X0 > X0 > X0)).
% 9.88/2.70 thf(func_def_102, type, sK23: !>[X0: $tType]:(set @ X0 > set @ X0 > X0 > X0)).
% 9.88/2.70 thf(func_def_103, type, sK24: !>[X0: $tType]:(set @ X0 > X0 > set @ X0)).
% 9.88/2.70 thf(func_def_104, type, sK25: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 9.88/2.70 thf(func_def_105, type, sK26: !>[X0: $tType]:(set @ X0 > set @ X0)).
% 9.88/2.70 thf(func_def_106, type, sK27: !>[X0: $tType]:(set @ X0 > X0)).
% 9.88/2.70 thf(func_def_107, type, sK28: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X0 > set @ X1 > X1)).
% 9.88/2.70 thf(func_def_108, type, sK29: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X0 > set @ X1 > X0)).
% 9.88/2.70 thf(func_def_109, type, sK30: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 9.88/2.70 thf(func_def_110, type, sK31: !>[X0: $tType]:((set @ X0 > $o) > X0)).
% 9.88/2.70 thf(func_def_111, type, sK32: !>[X0: $tType]:((set @ X0 > $o) > X0)).
% 9.88/2.70 thf(func_def_112, type, sK33: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 9.88/2.70 thf(func_def_113, type, sK34: !>[X0: $tType]:(set @ X0 > set @ X0 > X0 > X0)).
% 9.88/2.70 thf(func_def_114, type, sK35: !>[X0: $tType]:(set @ X0 > set @ X0 > X0 > X0)).
% 9.88/2.70 thf(func_def_115, type, sK36: !>[X0: $tType]:((X0 > $o) > X0)).
% 9.88/2.70 thf(func_def_116, type, sK37: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X1) > (X0 > X2) > set @ X0 > set @ X2 > X0)).
% 9.88/2.70 thf(func_def_117, type, sK38: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1 > X1)).
% 9.88/2.70 thf(func_def_118, type, sK39: !>[X0: $tType, X1: $tType]:(X0 > (X0 > X1) > set @ X0 > set @ X1 > X0)).
% 9.88/2.70 thf(func_def_119, type, sK40: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 9.88/2.70 thf(func_def_120, type, sK41: !>[X0: $tType]:(set @ X0 > X0)).
% 9.88/2.70 thf(func_def_121, type, sK42: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X1) > (X0 > X2) > set @ X0 > (X2 > X0) > set @ X2 > X0)).
% 9.88/2.70 thf(func_def_122, type, sK43: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > (X1 > X0) > set @ X1 > X1)).
% 9.88/2.70 thf(func_def_124, type, sK45: !>[X0: $tType]:(set @ X0 > X0)).
% 9.88/2.70 thf(func_def_125, type, sK46: !>[X0: $tType]:(X0 > X0 > set @ X0 > set @ X0 > set @ X0)).
% 9.88/2.70 thf(func_def_126, type, sK47: !>[X0: $tType]:(set @ X0 > X0)).
% 9.88/2.70 thf(func_def_127, type, sK48: !>[X0: $tType]:((X0 > $o) > X0)).
% 9.88/2.70 thf(func_def_128, type, sK49: !>[X0: $tType]:(set @ X0 > real > X0 > X0 > X0 > real)).
% 9.88/2.70 thf(func_def_129, type, sK50: !>[X0: $tType]:(set @ X0 > real > X0 > X0 > X0 > real)).
% 9.88/2.70 thf(func_def_130, type, sK51: !>[X0: $tType]:(set @ X0 > real > X0 > X0 > real)).
% 9.88/2.70 thf(func_def_131, type, sK52: !>[X0: $tType]:((X0 > $o) > X0)).
% 9.88/2.70 thf(func_def_133, type, ph53: !>[X0: $tType]:(X0)).
% 9.88/2.70 thf(func_def_134, type, sK54: a).
% 9.88/2.70 thf(func_def_135, type, sK55: a).
% 9.88/2.70 thf(f2986,plain,(
% 9.88/2.70 $false),
% 9.88/2.70 inference(avatar_sat_refutation,[status(thm)],[f2263,f2273,f2403,f2784,f2974,f2985])).
% 9.88/2.70 thf(f2985,plain,(
% 9.88/2.70 ~spl44_2 | spl44_68),
% 9.88/2.70 inference(avatar_contradiction_clause,[status(thm)],[f2984])).
% 9.88/2.70 thf(f2984,plain,(
% 9.88/2.70 $false | (~spl44_2 | spl44_68)),
% 9.88/2.70 inference(subsumption_resolution,[status(thm)],[f2983,f2378])).
% 9.88/2.70 thf(f2378,plain,(
% 9.88/2.70 ( ! [X2 : b,X0 : real,X1 : real] : (((real_V1908273582scaleR @ b @ (times_times @ real @ X0 @ X1) @ X2) = (real_V1908273582scaleR @ b @ X0 @ (real_V1908273582scaleR @ b @ X1 @ X2)))) ) | ~spl44_2),
% 9.88/2.70 inference(trivial_inequality_removal,[status(thm)],[f2376])).
% 9.88/2.70 thf(f2376,plain,(
% 9.88/2.70 ( ! [X2 : b,X0 : real,X1 : real] : (((real_V1908273582scaleR @ b @ (times_times @ real @ X0 @ X1) @ X2) = (real_V1908273582scaleR @ b @ X0 @ (real_V1908273582scaleR @ b @ X1 @ X2))) | ($true != $true)) ) | ~spl44_2),
% 9.88/2.70 inference(superposition,[status(thm)],[f1507,f2262])).
% 9.88/2.70 thf(f2262,plain,(
% 9.88/2.70 (real_V1076094709vector @ b = $true) | ~spl44_2),
% 9.88/2.70 inference(avatar_component_clause,[status(thm)],[f2260])).
% 9.88/2.70 thf(f2260,definition,(
% 9.88/2.70 spl44_2 <=> (real_V1076094709vector @ b = $true)),
% 9.88/2.70 introduced(definition,[new_symbols(naming,[spl44_2])],[avatar_definition])).
% 9.88/2.70 thf(f1507,plain,(
% 9.88/2.70 ( ! [X0 : $tType,X2 : real,X3 : X0,X1 : real] : ((real_V1076094709vector @ X0 != $true) | ((real_V1908273582scaleR @ X0 @ (times_times @ real @ X1 @ X2) @ X3) = (real_V1908273582scaleR @ X0 @ X1 @ (real_V1908273582scaleR @ X0 @ X2 @ X3)))) )),
% 9.88/2.70 inference(cnf_transformation,[status(thm)],[f1155])).
% 9.88/2.70 thf(f1155,plain,(
% 9.88/2.70 ! [X0 : $tType] : (! [X1 : real,X2 : real,X3 : X0] : ((real_V1908273582scaleR @ X0 @ (times_times @ real @ X1 @ X2) @ X3) = (real_V1908273582scaleR @ X0 @ X1 @ (real_V1908273582scaleR @ X0 @ X2 @ X3))) | (real_V1076094709vector @ X0 != $true))),
% 9.88/2.70 inference(ennf_transformation,[status(thm)],[f888])).
% 9.88/2.70 thf(f888,plain,(
% 9.88/2.70 ! [X0 : $tType] : ((real_V1076094709vector @ X0 = $true) => ! [X1 : real,X2 : real,X3 : X0] : ((real_V1908273582scaleR @ X0 @ (times_times @ real @ X1 @ X2) @ X3) = (real_V1908273582scaleR @ X0 @ X1 @ (real_V1908273582scaleR @ X0 @ X2 @ X3))))),
% 9.88/2.70 inference(fool_elimination,[status(thm)],[f887])).
% 9.88/2.70 thf(f887,plain,(
% 9.88/2.70 ! [X0 : $tType] : (real_V1076094709vector @ X0 => ! [X1 : real,X2 : real,X3 : X0] : ((real_V1908273582scaleR @ X0 @ (times_times @ real @ X1 @ X2) @ X3) = (real_V1908273582scaleR @ X0 @ X1 @ (real_V1908273582scaleR @ X0 @ X2 @ X3))))),
% 9.88/2.70 inference(rectify,[status(thm)],[f12])).
% 9.88/2.70 thf(f12,axiom,(
% 9.88/2.70 ! [X0 : $tType] : (real_V1076094709vector @ X0 => ! [X5 : real,X6 : real,X7 : X0] : ((real_V1908273582scaleR @ X0 @ X5 @ (real_V1908273582scaleR @ X0 @ X6 @ X7)) = (real_V1908273582scaleR @ X0 @ (times_times @ real @ X5 @ X6) @ X7)))),
% 9.88/2.70 file('/export/starexec/sandbox2/tmp/tmp.a0uzsVYvBA/DTF2THF_4788.p',fact_11_scaleR__scaleR)).
% 9.88/2.70 thf(f2983,plain,(
% 9.88/2.70 ((real_V1908273582scaleR @ b @ (times_times @ real @ (aa @ sK55) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (y @ sK55)) != (real_V1908273582scaleR @ b @ (aa @ sK55) @ (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (y @ sK55)))) | (~spl44_2 | spl44_68)),
% 9.88/2.70 inference(forward_demodulation,[status(thm)],[f2982,f2388])).
% 9.88/2.70 thf(f2388,plain,(
% 9.88/2.70 ( ! [X2 : b,X0 : real,X1 : real] : (((real_V1908273582scaleR @ b @ X0 @ (real_V1908273582scaleR @ b @ X1 @ X2)) = (real_V1908273582scaleR @ b @ X1 @ (real_V1908273582scaleR @ b @ X0 @ X2)))) ) | ~spl44_2),
% 9.88/2.70 inference(trivial_inequality_removal,[status(thm)],[f2385])).
% 9.88/2.70 thf(f2385,plain,(
% 9.88/2.70 ( ! [X2 : b,X0 : real,X1 : real] : (((real_V1908273582scaleR @ b @ X0 @ (real_V1908273582scaleR @ b @ X1 @ X2)) = (real_V1908273582scaleR @ b @ X1 @ (real_V1908273582scaleR @ b @ X0 @ X2))) | ($true != $true)) ) | ~spl44_2),
% 9.88/2.70 inference(superposition,[status(thm)],[f1800,f2262])).
% 9.88/2.70 thf(f1800,plain,(
% 9.88/2.70 ( ! [X0 : $tType,X2 : real,X3 : X0,X1 : real] : ((real_V1076094709vector @ X0 != $true) | ((real_V1908273582scaleR @ X0 @ X2 @ (real_V1908273582scaleR @ X0 @ X1 @ X3)) = (real_V1908273582scaleR @ X0 @ X1 @ (real_V1908273582scaleR @ X0 @ X2 @ X3)))) )),
% 9.88/2.70 inference(cnf_transformation,[status(thm)],[f1213])).
% 9.88/2.70 thf(f1213,plain,(
% 9.88/2.70 ! [X0 : $tType] : ((real_V1076094709vector @ X0 != $true) | ! [X1 : real,X2 : real,X3 : X0] : ((real_V1908273582scaleR @ X0 @ X2 @ (real_V1908273582scaleR @ X0 @ X1 @ X3)) = (real_V1908273582scaleR @ X0 @ X1 @ (real_V1908273582scaleR @ X0 @ X2 @ X3))))),
% 9.88/2.70 inference(ennf_transformation,[status(thm)],[f434])).
% 9.88/2.70 thf(f434,plain,(
% 9.88/2.70 ! [X0 : $tType] : ((real_V1076094709vector @ X0 = $true) => ! [X1 : real,X2 : real,X3 : X0] : ((real_V1908273582scaleR @ X0 @ X2 @ (real_V1908273582scaleR @ X0 @ X1 @ X3)) = (real_V1908273582scaleR @ X0 @ X1 @ (real_V1908273582scaleR @ X0 @ X2 @ X3))))),
% 9.88/2.70 inference(fool_elimination,[status(thm)],[f433])).
% 9.88/2.70 thf(f433,plain,(
% 9.88/2.70 ! [X0 : $tType] : (real_V1076094709vector @ X0 => ! [X1 : real,X2 : real,X3 : X0] : ((real_V1908273582scaleR @ X0 @ X2 @ (real_V1908273582scaleR @ X0 @ X1 @ X3)) = (real_V1908273582scaleR @ X0 @ X1 @ (real_V1908273582scaleR @ X0 @ X2 @ X3))))),
% 9.88/2.70 inference(rectify,[status(thm)],[f32])).
% 9.88/2.70 thf(f32,axiom,(
% 9.88/2.70 ! [X0 : $tType] : (real_V1076094709vector @ X0 => ! [X5 : real,X6 : real,X7 : X0] : ((real_V1908273582scaleR @ X0 @ X5 @ (real_V1908273582scaleR @ X0 @ X6 @ X7)) = (real_V1908273582scaleR @ X0 @ X6 @ (real_V1908273582scaleR @ X0 @ X5 @ X7))))),
% 9.88/2.70 file('/export/starexec/sandbox2/tmp/tmp.a0uzsVYvBA/DTF2THF_4788.p',fact_31_scale__left__commute)).
% 9.88/2.70 thf(f2982,plain,(
% 9.88/2.70 ((real_V1908273582scaleR @ b @ (times_times @ real @ (aa @ sK55) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (y @ sK55)) != (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (real_V1908273582scaleR @ b @ (aa @ sK55) @ (y @ sK55)))) | spl44_68),
% 9.88/2.70 inference(beta_eta_normalization,[status(thm)],[f2981])).
% 9.88/2.70 thf(f2981,plain,(
% 9.88/2.70 (((^[Y0 : a]: (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0)))) @ sK55) != ((^[Y0 : a]: (real_V1908273582scaleR @ b @ (times_times @ real @ (aa @ Y0) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (y @ Y0))) @ sK55)) | spl44_68),
% 9.88/2.70 inference(negative_extensionality,[status(thm)],[f2973])).
% 9.88/2.70 thf(f2973,plain,(
% 9.88/2.70 ((^[Y0 : a]: (real_V1908273582scaleR @ b @ (times_times @ real @ (aa @ Y0) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (y @ Y0))) != (^[Y0 : a]: (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0))))) | spl44_68),
% 9.88/2.70 inference(avatar_component_clause,[status(thm)],[f2971])).
% 9.88/2.70 thf(f2971,definition,(
% 9.88/2.70 spl44_68 <=> ((^[Y0 : a]: (real_V1908273582scaleR @ b @ (times_times @ real @ (aa @ Y0) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (y @ Y0))) = (^[Y0 : a]: (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0)))))),
% 9.88/2.70 introduced(definition,[new_symbols(naming,[spl44_68])],[avatar_definition])).
% 9.88/2.70 thf(f2974,plain,(
% 9.88/2.70 ~spl44_68 | spl44_25 | ~spl44_46),
% 9.88/2.70 inference(avatar_split_clause,[status(thm)],[f2786,f2781,f2400,f2971])).
% 9.88/2.70 thf(f2400,definition,(
% 9.88/2.70 spl44_25 <=> ((f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (times_times @ real @ (aa @ Y0) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (y @ Y0))) @ sa)) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i)))) = (f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0))) @ sa))) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i)))))),
% 9.88/2.70 introduced(definition,[new_symbols(naming,[spl44_25])],[avatar_definition])).
% 9.88/2.70 thf(f2781,definition,(
% 9.88/2.70 spl44_46 <=> ((groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0)))) @ sa) = (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0))) @ sa)))),
% 9.88/2.70 introduced(definition,[new_symbols(naming,[spl44_46])],[avatar_definition])).
% 9.88/2.70 thf(f2786,plain,(
% 9.88/2.70 ((^[Y0 : a]: (real_V1908273582scaleR @ b @ (times_times @ real @ (aa @ Y0) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (y @ Y0))) != (^[Y0 : a]: (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0))))) | (spl44_25 | ~spl44_46)),
% 9.88/2.70 inference(trivial_inequality_removal,[status(thm)],[f2785])).
% 9.88/2.70 thf(f2785,plain,(
% 9.88/2.70 ((f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0))) @ sa))) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i)))) != (f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0))) @ sa))) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i))))) | ((^[Y0 : a]: (real_V1908273582scaleR @ b @ (times_times @ real @ (aa @ Y0) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (y @ Y0))) != (^[Y0 : a]: (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0))))) | (spl44_25 | ~spl44_46)),
% 9.88/2.70 inference(constrained_superposition,[status(thm)],[f2402,f2783])).
% 9.88/2.70 thf(f2783,plain,(
% 9.88/2.70 ((groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0)))) @ sa) = (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0))) @ sa))) | ~spl44_46),
% 9.88/2.70 inference(avatar_component_clause,[status(thm)],[f2781])).
% 9.88/2.70 thf(f2402,plain,(
% 9.88/2.70 ((f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (times_times @ real @ (aa @ Y0) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (y @ Y0))) @ sa)) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i)))) != (f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0))) @ sa))) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i))))) | spl44_25),
% 9.88/2.70 inference(avatar_component_clause,[status(thm)],[f2400])).
% 9.88/2.70 thf(f2784,plain,(
% 9.88/2.70 spl44_46),
% 9.88/2.70 inference(avatar_split_clause,[status(thm)],[f1708,f2781])).
% 9.88/2.70 thf(f1708,plain,(
% 9.88/2.70 ((groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0)))) @ sa) = (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0))) @ sa)))),
% 9.88/2.70 inference(cnf_transformation,[status(thm)],[f624])).
% 9.88/2.70 thf(f624,plain,(
% 9.88/2.70 ((groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0)))) @ sa) = (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0))) @ sa)))),
% 9.88/2.70 inference(fool_elimination,[status(thm)],[f623])).
% 9.88/2.70 thf(f623,plain,(
% 9.88/2.70 ((groups1340683514dd_sum @ a @ b @ (^[X0 : a] : (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (real_V1908273582scaleR @ b @ (aa @ X0) @ (y @ X0)))) @ sa) = (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (groups1340683514dd_sum @ a @ b @ (^[X1 : a] : (real_V1908273582scaleR @ b @ (aa @ X1) @ (y @ X1))) @ sa)))),
% 9.88/2.70 inference(rectify,[status(thm)],[f3])).
% 9.88/2.70 thf(f3,axiom,(
% 9.88/2.70 ((groups1340683514dd_sum @ a @ b @ (^[X2 : a] : (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (real_V1908273582scaleR @ b @ (aa @ X2) @ (y @ X2)))) @ sa) = (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (groups1340683514dd_sum @ a @ b @ (^[X3 : a] : (real_V1908273582scaleR @ b @ (aa @ X3) @ (y @ X3))) @ sa)))),
% 9.88/2.70 file('/export/starexec/sandbox2/tmp/tmp.a0uzsVYvBA/DTF2THF_4788.p',fact_2__092_060open_062_I_092_060Sum_062x_092_060in_062s_O_Aa_Ax_A_K_092_060_094sub_062R_Ay_Ax_A_P_092_060_094sub_062R_A_I1_A_N_Aa_Ai_J_J_A_061_A_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_P_092_060_094sub_062R_A_I1_A_N_Aa_Ai_J_092_060close_062)).
% 9.88/2.70 thf(f2403,plain,(
% 9.88/2.70 ~spl44_25 | ~spl44_2 | spl44_4),
% 9.88/2.70 inference(avatar_split_clause,[status(thm)],[f2389,f2270,f2260,f2400])).
% 9.88/2.70 thf(f2270,definition,(
% 9.88/2.70 spl44_4 <=> ((f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (times_times @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0))) @ sa)) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i)))) = (f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (times_times @ real @ (aa @ Y0) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (y @ Y0))) @ sa)) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i)))))),
% 9.88/2.70 introduced(definition,[new_symbols(naming,[spl44_4])],[avatar_definition])).
% 9.88/2.70 thf(f2389,plain,(
% 9.88/2.70 ((f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (times_times @ real @ (aa @ Y0) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (y @ Y0))) @ sa)) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i)))) != (f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (real_V1908273582scaleR @ b @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i))) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0))) @ sa))) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i))))) | (~spl44_2 | spl44_4)),
% 9.88/2.70 inference(backward_demodulation,[status(thm)],[f2272,f2378])).
% 9.88/2.70 thf(f2272,plain,(
% 9.88/2.70 ((f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (times_times @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0))) @ sa)) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i)))) != (f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (times_times @ real @ (aa @ Y0) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (y @ Y0))) @ sa)) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i))))) | spl44_4),
% 9.88/2.70 inference(avatar_component_clause,[status(thm)],[f2270])).
% 9.88/2.70 thf(f2273,plain,(
% 9.88/2.70 ~spl44_4),
% 9.88/2.70 inference(avatar_split_clause,[status(thm)],[f1415,f2270])).
% 9.88/2.70 thf(f1415,plain,(
% 9.88/2.70 ((f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (times_times @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0))) @ sa)) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i)))) != (f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (times_times @ real @ (aa @ Y0) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (y @ Y0))) @ sa)) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i)))))),
% 9.88/2.70 inference(cnf_transformation,[status(thm)],[f1018])).
% 9.88/2.70 thf(f1018,plain,(
% 9.88/2.70 ((f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (times_times @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0))) @ sa)) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i)))) != (f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (times_times @ real @ (aa @ Y0) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (y @ Y0))) @ sa)) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i)))))),
% 9.88/2.70 inference(flattening,[status(thm)],[f494])).
% 9.88/2.70 thf(f494,plain,(
% 9.88/2.70 ~((f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (times_times @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (aa @ Y0) @ (y @ Y0))) @ sa)) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i)))) = (f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (groups1340683514dd_sum @ a @ b @ (^[Y0 : a]: (real_V1908273582scaleR @ b @ (times_times @ real @ (aa @ Y0) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (y @ Y0))) @ sa)) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i)))))),
% 9.88/2.70 inference(fool_elimination,[status(thm)],[f493])).
% 9.88/2.70 thf(f493,plain,(
% 9.88/2.70 ~((f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (groups1340683514dd_sum @ a @ b @ (^[X0 : a] : (real_V1908273582scaleR @ b @ (times_times @ real @ (aa @ X0) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (y @ X0))) @ sa)) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i)))) = (f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (times_times @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (groups1340683514dd_sum @ a @ b @ (^[X1 : a] : (real_V1908273582scaleR @ b @ (aa @ X1) @ (y @ X1))) @ sa)) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i)))))),
% 9.88/2.70 inference(rectify,[status(thm)],[f353])).
% 9.88/2.70 thf(f353,negated_conjecture,(
% 9.88/2.70 ~((f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (groups1340683514dd_sum @ a @ b @ (^[X3 : a] : (real_V1908273582scaleR @ b @ (times_times @ real @ (aa @ X3) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (y @ X3))) @ sa)) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i)))) = (f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (times_times @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (groups1340683514dd_sum @ a @ b @ (^[X3 : a] : (real_V1908273582scaleR @ b @ (aa @ X3) @ (y @ X3))) @ sa)) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i)))))),
% 9.88/2.70 inference(negated_conjecture,[status(cth)],[f352])).
% 9.88/2.70 thf(f352,conjecture,(
% 9.88/2.70 ((f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (groups1340683514dd_sum @ a @ b @ (^[X3 : a] : (real_V1908273582scaleR @ b @ (times_times @ real @ (aa @ X3) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (y @ X3))) @ sa)) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i)))) = (f @ (plus_plus @ b @ (real_V1908273582scaleR @ b @ (times_times @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)) @ (inverse_inverse @ real @ (minus_minus @ real @ one_one @ real @ (aa @ i)))) @ (groups1340683514dd_sum @ a @ b @ (^[X3 : a] : (real_V1908273582scaleR @ b @ (aa @ X3) @ (y @ X3))) @ sa)) @ (real_V1908273582scaleR @ b @ (aa @ i) @ (y @ i)))))),
% 9.88/2.70 file('/export/starexec/sandbox2/tmp/tmp.a0uzsVYvBA/DTF2THF_4788.p',conj_0)).
% 9.88/2.70 thf(f2263,plain,(
% 9.88/2.70 spl44_2),
% 9.88/2.70 inference(avatar_split_clause,[status(thm)],[f1401,f2260])).
% 9.88/2.70 thf(f1401,plain,(
% 9.88/2.70 (real_V1076094709vector @ b = $true)),
% 9.88/2.70 inference(cnf_transformation,[status(thm)],[f980])).
% 9.88/2.70 thf(f980,plain,(
% 9.88/2.70 (real_V1076094709vector @ b = $true)),
% 9.88/2.70 inference(fool_elimination,[status(thm)],[f979])).
% 9.88/2.70 thf(f979,plain,(
% 9.88/2.70 real_V1076094709vector @ b),
% 9.88/2.70 inference(rectify,[status(thm)],[f351])).
% 9.88/2.70 thf(f351,axiom,(
% 9.88/2.70 real_V1076094709vector @ b),
% 9.88/2.70 file('/export/starexec/sandbox2/tmp/tmp.a0uzsVYvBA/DTF2THF_4788.p',tfree_0)).
% 9.88/2.70 % SZS output end Proof for DTF2THF_4788
% 9.88/2.70 % (5310)------------------------------
% 9.88/2.70 % (5310)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.88/2.70 % (5310)Termination reason: Refutation
% 9.88/2.70
% 9.88/2.70 % (5310)Memory used [KB]: 8059
% 9.88/2.70 % (5310)Time elapsed: 0.140 s
% 9.88/2.70 % (5310)Instructions burned: 440 (million)
% 9.88/2.70 % (5310)------------------------------
% 9.88/2.70 % (5310)------------------------------
% 9.88/2.70 % (4942)Success in time 1.153 s
%------------------------------------------------------------------------------