%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP157^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n029.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 08:56:01 AM UTC 2026
% Result : Theorem 6.99s 2.22s
% Output : Refutation 6.99s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP157^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n029.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:28:14 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.47 thf(spec,logic,$$dhol).
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 % File : ITP157^2 : TPTP v9.2.1. Released v7.5.0.
% 0.30/0.47 % Domain : Interactive Theorem Proving
% 0.30/0.47 % Problem : Sledgehammer Preferences problem prob_86__6247862_1
% 0.30/0.47 % Version : Especial.
% 0.30/0.47 % English :
% 0.30/0.47
% 0.30/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.30/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.30/0.47 % Source : [Des21]
% 0.30/0.47 % Names : Preferences/prob_86__6247862_1 [Des21]
% 0.30/0.47
% 0.30/0.47 % Status : Theorem
% 0.30/0.47 % Rating : 0.00 v7.5.0
% 0.30/0.47 % Syntax : Number of formulae : 346 ( 152 unt; 56 typ; 0 def)
% 0.30/0.47 % Number of atoms : 645 ( 287 equ; 0 cnn)
% 0.30/0.47 % Maximal formula atoms : 7 ( 2 avg)
% 0.30/0.47 % Number of connectives : 4486 ( 48 ~; 7 |; 67 &;4129 @)
% 0.30/0.47 % ( 0 <=>; 235 =>; 0 <=; 0 <~>)
% 0.30/0.47 % Maximal formula depth : 28 ( 8 avg)
% 0.30/0.47 % Number of types : 3 ( 2 usr)
% 0.30/0.47 % Number of type conns : 301 ( 301 >; 0 *; 0 +; 0 <<)
% 0.30/0.47 % Number of symbols : 57 ( 54 usr; 6 con; 0-7 aty)
% 0.30/0.47 % Number of variables : 1409 ( 156 ^;1157 !; 16 ?;1409 :)
% 0.30/0.47 % ( 80 !>; 0 ?*; 0 @-; 0 @+)
% 0.30/0.47 % SPC : TH1_THM_EQU_NAR
% 0.30/0.47
% 0.30/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:27:40.388
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 % Could-be-implicit typings (4)
% 0.30/0.47 thf(ty_t_Product__Type_Ounit,type,
% 0.30/0.47 product_unit: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_t_Product__Type_Oprod,type,
% 0.30/0.47 product_prod: $tType > $tType > $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_t_Set_Oset,type,
% 0.30/0.47 set: $tType > $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_tf_a,type,
% 0.30/0.47 a: $tType ).
% 0.30/0.47
% 0.30/0.47 % Explicit typings (52)
% 0.30/0.47 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
% 0.30/0.47 ordere890947078_space:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_HOL_Otype,type,
% 0.30/0.47 type:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Groups_Ominus,type,
% 0.30/0.47 minus:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Obot,type,
% 0.30/0.47 bot:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Oord,type,
% 0.30/0.47 ord:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Otop,type,
% 0.30/0.47 top:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Groups_Ogroup__add,type,
% 0.30/0.47 group_add:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Groups_Oab__group__add,type,
% 0.30/0.47 ab_group_add:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Oorder__top,type,
% 0.30/0.47 order_top:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.30/0.47 ordered_ab_group_add:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.30/0.47 cancel146912293up_add:
% 0.30/0.47 !>[A: $tType] : $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_BNF__Def_OGr,type,
% 0.30/0.47 bNF_Gr:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_BNF__Def_OfstOp,type,
% 0.30/0.47 bNF_fstOp:
% 0.30/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_BNF__Def_Opick__middlep,type,
% 0.30/0.47 bNF_pick_middlep:
% 0.30/0.47 !>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_BNF__Def_OsndOp,type,
% 0.30/0.47 bNF_sndOp:
% 0.30/0.47 !>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 0.30/0.47 bNF_Greatest_image2:
% 0.30/0.47 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.30/0.47 minus_minus:
% 0.30/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_HOL_OThe,type,
% 0.30/0.47 the:
% 0.30/0.47 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_If,type,
% 0.30/0.47 if:
% 0.30/0.47 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Order__Relation_Opreorder__on,type,
% 0.30/0.47 order_preorder_on:
% 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.30/0.47 bot_bot:
% 0.30/0.47 !>[A: $tType] : A ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.30/0.47 ord_less_eq:
% 0.30/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.30/0.47 top_top:
% 0.30/0.47 !>[A: $tType] : A ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oas__good__as,type,
% 0.30/0.47 prefer951318096ood_as:
% 0.30/0.47 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oat__least__as__good,type,
% 0.30/0.47 prefer310429814s_good:
% 0.30/0.47 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Ono__better__than,type,
% 0.30/0.47 prefer1532642881r_than:
% 0.30/0.47 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Opreference,type,
% 0.30/0.47 prefer199794634erence:
% 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_OPair,type,
% 0.30/0.47 product_Pair:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_OSigma,type,
% 0.30/0.47 product_Sigma:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Ocurry,type,
% 0.30/0.47 product_curry:
% 0.30/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.30/0.47 produc2004651681e_prod:
% 0.30/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Omap__prod,type,
% 0.30/0.47 product_map_prod:
% 0.30/0.47 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
% 0.30/0.47 product_rec_bool:
% 0.30/0.47 !>[T: $tType] : ( T > T > $o > T ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Oold_Obool_Orec__set__bool,type,
% 0.30/0.47 product_rec_set_bool:
% 0.30/0.47 !>[T: $tType] : ( T > T > $o > T > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.30/0.47 product_rec_prod:
% 0.30/0.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
% 0.30/0.47 product_rec_set_prod:
% 0.30/0.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
% 0.30/0.47 product_rec_set_unit:
% 0.30/0.47 !>[T: $tType] : ( T > product_unit > T > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
% 0.30/0.47 product_rec_unit:
% 0.30/0.47 !>[T: $tType] : ( T > product_unit > T ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.30/0.47 product_fst:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 0.30/0.47 product_snd:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.30/0.47 product_swap:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Oproduct,type,
% 0.30/0.47 product_product:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Product__Type_Oscomp,type,
% 0.30/0.47 product_scomp:
% 0.30/0.47 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set_OCollect,type,
% 0.30/0.47 collect:
% 0.30/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set_Oimage,type,
% 0.30/0.47 image:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Wellfounded_Olex__prod,type,
% 0.30/0.47 lex_prod:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Wfrec_Osame__fst,type,
% 0.30/0.47 same_fst:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_member,type,
% 0.30/0.47 member:
% 0.30/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_B,type,
% 0.30/0.47 b: set @ a ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_Pr,type,
% 0.30/0.47 pr: set @ ( product_prod @ a @ a ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_y,type,
% 0.30/0.47 y: a ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_z,type,
% 0.30/0.47 z: a ).
% 0.30/0.47
% 0.30/0.47 % Relevant facts (256)
% 0.30/0.47 thf(fact_0_assms,axiom,
% 0.30/0.47 member @ a @ z @ ( prefer310429814s_good @ a @ y @ b @ pr ) ).
% 0.30/0.47
% 0.30/0.47 % assms
% 0.30/0.47 thf(fact_1_at__lst__asgd__ge,axiom,
% 0.30/0.47 ! [A: $tType,X: A,Y: A,B2: set @ A,Pr: set @ ( product_prod @ A @ A )] :
% 0.30/0.47 ( ( member @ A @ X @ ( prefer310429814s_good @ A @ Y @ B2 @ Pr ) )
% 0.30/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Pr ) ) ).
% 0.30/0.47
% 0.30/0.47 % at_lst_asgd_ge
% 0.30/0.47 thf(fact_2_prod_Oinject,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 0.30/0.47 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 0.30/0.47 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.30/0.47 = ( ( X1 = Y1 )
% 0.30/0.47 & ( X2 = Y2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod.inject
% 0.30/0.47 thf(fact_3_old_Oprod_Oinject,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A2: A,B3: B,A3: A,B4: B] :
% 0.30/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B3 )
% 0.30/0.47 = ( product_Pair @ A @ B @ A3 @ B4 ) )
% 0.30/0.47 = ( ( A2 = A3 )
% 0.30/0.47 & ( B3 = B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.prod.inject
% 0.30/0.47 thf(fact_4_surj__pair,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 0.30/0.47 ? [X3: A,Y3: B] :
% 0.30/0.47 ( P
% 0.30/0.47 = ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % surj_pair
% 0.30/0.47 thf(fact_5_prod__cases,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 0.30/0.47 ( ! [A4: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B5 ) )
% 0.30/0.47 => ( P2 @ P ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases
% 0.30/0.47 thf(fact_6_Pair__inject,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A2: A,B3: B,A3: A,B4: B] :
% 0.30/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B3 )
% 0.30/0.47 = ( product_Pair @ A @ B @ A3 @ B4 ) )
% 0.30/0.47 => ~ ( ( A2 = A3 )
% 0.30/0.47 => ( B3 != B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Pair_inject
% 0.30/0.47 thf(fact_7_prod__cases3,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.30/0.47 ~ ! [A4: A,B5: B,C2: C] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases3
% 0.30/0.47 thf(fact_8_prod__cases4,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.30/0.47 ~ ! [A4: A,B5: B,C2: C,D2: D] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases4
% 0.30/0.47 thf(fact_9_prod__cases5,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.30/0.47 ~ ! [A4: A,B5: B,C2: C,D2: D,E2: E] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases5
% 0.30/0.47 thf(fact_10_prod__cases6,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.30/0.47 ~ ! [A4: A,B5: B,C2: C,D2: D,E2: E,F2: F] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases6
% 0.30/0.47 thf(fact_11_prod__cases7,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 0.30/0.47 ~ ! [A4: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_cases7
% 0.30/0.47 thf(fact_12_prod__induct3,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.30/0.47 ( ! [A4: A,B5: B,C2: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) )
% 0.30/0.47 => ( P2 @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_induct3
% 0.30/0.47 thf(fact_13_old_Oprod_Oinducts,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.30/0.47 ( ! [A4: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B5 ) )
% 0.30/0.47 => ( P2 @ Prod ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.prod.inducts
% 0.30/0.47 thf(fact_14_old_Oprod_Oexhaust,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.30/0.47 ~ ! [A4: A,B5: B] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( product_Pair @ A @ B @ A4 @ B5 ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.prod.exhaust
% 0.30/0.47 thf(fact_15_prod__induct7,axiom,
% 0.30/0.47 ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 0.30/0.47 ( ! [A4: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
% 0.30/0.47 => ( P2 @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_induct7
% 0.30/0.47 thf(fact_16_prod__induct6,axiom,
% 0.30/0.47 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.30/0.47 ( ! [A4: A,B5: B,C2: C,D2: D,E2: E,F2: F] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
% 0.30/0.47 => ( P2 @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_induct6
% 0.30/0.47 thf(fact_17_prod__induct5,axiom,
% 0.30/0.47 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.30/0.47 ( ! [A4: A,B5: B,C2: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.30/0.47 => ( P2 @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_induct5
% 0.30/0.47 thf(fact_18_prod__induct4,axiom,
% 0.30/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.30/0.47 ( ! [A4: A,B5: B,C2: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 0.30/0.47 => ( P2 @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_induct4
% 0.30/0.47 thf(fact_19_old_Oprod_Orec,axiom,
% 0.30/0.47 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B3: B] :
% 0.30/0.47 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B3 ) )
% 0.30/0.47 = ( F1 @ A2 @ B3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.prod.rec
% 0.30/0.47 thf(fact_20_at__least__as__good__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( prefer310429814s_good @ A )
% 0.30/0.47 = ( ^ [X4: A,B6: set @ A,P3: set @ ( product_prod @ A @ A )] :
% 0.30/0.47 ( collect @ A
% 0.30/0.47 @ ^ [Y4: A] :
% 0.30/0.47 ( ( member @ A @ Y4 @ B6 )
% 0.30/0.47 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % at_least_as_good_def
% 0.30/0.47 thf(fact_21_internal__case__prod__conv,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B3: C] :
% 0.30/0.47 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B3 ) )
% 0.30/0.47 = ( C3 @ A2 @ B3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % internal_case_prod_conv
% 0.30/0.47 thf(fact_22_ssubst__Pair__rhs,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R: A,S: B,R2: set @ ( product_prod @ A @ B ),S2: B] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R2 )
% 0.30/0.47 => ( ( S2 = S )
% 0.30/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ssubst_Pair_rhs
% 0.30/0.47 thf(fact_23_curry__conv,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,B: $tType] :
% 0.30/0.47 ( ( product_curry @ B @ C @ A )
% 0.30/0.47 = ( ^ [F3: ( product_prod @ B @ C ) > A,A5: B,B7: C] : ( F3 @ ( product_Pair @ B @ C @ A5 @ B7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % curry_conv
% 0.30/0.47 thf(fact_24_curryI,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A2: A,B3: B] :
% 0.30/0.47 ( ( F4 @ ( product_Pair @ A @ B @ A2 @ B3 ) )
% 0.30/0.47 => ( product_curry @ A @ B @ $o @ F4 @ A2 @ B3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % curryI
% 0.30/0.47 thf(fact_25_preference_Oindiff__trans,axiom,
% 0.30/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 0.30/0.47 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.30/0.47 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/0.47 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 0.30/0.47 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
% 0.30/0.47 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation ) )
% 0.30/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation )
% 0.30/0.47 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % preference.indiff_trans
% 0.30/0.47 thf(fact_26_preference_Onot__outside,axiom,
% 0.30/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.30/0.47 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.30/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.30/0.47 => ( member @ A @ X @ Carrier ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % preference.not_outside
% 0.30/0.47 thf(fact_27_swap__simp,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X: B,Y: A] :
% 0.30/0.47 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 0.30/0.47 = ( product_Pair @ A @ B @ Y @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % swap_simp
% 0.30/0.47 thf(fact_28_old_Obool_Osimps_I6_J,axiom,
% 0.30/0.47 ! [T: $tType,F1: T,F22: T] :
% 0.30/0.47 ( ( product_rec_bool @ T @ F1 @ F22 @ $false )
% 0.30/0.47 = F22 ) ).
% 0.30/0.47
% 0.30/0.47 % old.bool.simps(6)
% 0.30/0.47 thf(fact_29_old_Obool_Osimps_I5_J,axiom,
% 0.30/0.47 ! [T: $tType,F1: T,F22: T] :
% 0.30/0.47 ( ( product_rec_bool @ T @ F1 @ F22 @ $true )
% 0.30/0.47 = F1 ) ).
% 0.30/0.47
% 0.30/0.47 % old.bool.simps(5)
% 0.30/0.47 thf(fact_30_swap__swap,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P: product_prod @ A @ B] :
% 0.30/0.47 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
% 0.30/0.47 = P ) ).
% 0.30/0.47
% 0.30/0.47 % swap_swap
% 0.30/0.47 thf(fact_31_curry__K,axiom,
% 0.30/0.47 ! [B: $tType,C: $tType,A: $tType,C3: C] :
% 0.30/0.47 ( ( product_curry @ A @ B @ C
% 0.30/0.47 @ ^ [X4: product_prod @ A @ B] : C3 )
% 0.30/0.47 = ( ^ [X4: A,Y4: B] : C3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % curry_K
% 0.30/0.47 thf(fact_32_curry__def,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.47 ( ( product_curry @ A @ B @ C )
% 0.30/0.47 = ( ^ [C4: ( product_prod @ A @ B ) > C,X4: A,Y4: B] : ( C4 @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % curry_def
% 0.30/0.47 thf(fact_33_curryD,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A2: A,B3: B] :
% 0.30/0.47 ( ( product_curry @ A @ B @ $o @ F4 @ A2 @ B3 )
% 0.30/0.47 => ( F4 @ ( product_Pair @ A @ B @ A2 @ B3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % curryD
% 0.30/0.47 thf(fact_34_curryE,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A2: A,B3: B] :
% 0.30/0.47 ( ( product_curry @ A @ B @ $o @ F4 @ A2 @ B3 )
% 0.30/0.47 => ( F4 @ ( product_Pair @ A @ B @ A2 @ B3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % curryE
% 0.30/0.47 thf(fact_35_no__better__than__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( prefer1532642881r_than @ A )
% 0.30/0.47 = ( ^ [X4: A,B6: set @ A,P3: set @ ( product_prod @ A @ A )] :
% 0.30/0.47 ( collect @ A
% 0.30/0.47 @ ^ [Y4: A] :
% 0.30/0.47 ( ( member @ A @ Y4 @ B6 )
% 0.30/0.47 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ P3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % no_better_than_def
% 0.30/0.47 thf(fact_36_as__good__as__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( prefer951318096ood_as @ A )
% 0.30/0.47 = ( ^ [X4: A,B6: set @ A,P3: set @ ( product_prod @ A @ A )] :
% 0.30/0.47 ( collect @ A
% 0.30/0.47 @ ^ [Y4: A] :
% 0.30/0.47 ( ( member @ A @ Y4 @ B6 )
% 0.30/0.47 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ P3 )
% 0.30/0.47 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % as_good_as_def
% 0.30/0.47 thf(fact_37_pred__equals__eq2,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 0.30/0.47 ( ( ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R2 ) )
% 0.30/0.47 = ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S3 ) ) )
% 0.30/0.47 = ( R2 = S3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % pred_equals_eq2
% 0.30/0.47 thf(fact_38_preference__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( prefer199794634erence @ A )
% 0.30/0.47 = ( ^ [Carrier2: set @ A,Relation2: set @ ( product_prod @ A @ A )] :
% 0.30/0.47 ( ! [X4: A,Y4: A] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ Relation2 )
% 0.30/0.47 => ( member @ A @ X4 @ Carrier2 ) )
% 0.30/0.47 & ! [X4: A,Y4: A] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ Relation2 )
% 0.30/0.47 => ( member @ A @ Y4 @ Carrier2 ) )
% 0.30/0.47 & ( order_preorder_on @ A @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % preference_def
% 0.30/0.47 thf(fact_39_preference_Ointro,axiom,
% 0.30/0.47 ! [A: $tType,Relation: set @ ( product_prod @ A @ A ),Carrier: set @ A] :
% 0.30/0.47 ( ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ Relation )
% 0.30/0.47 => ( member @ A @ X3 @ Carrier ) )
% 0.30/0.47 => ( ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ Relation )
% 0.30/0.47 => ( member @ A @ Y3 @ Carrier ) )
% 0.30/0.47 => ( ( order_preorder_on @ A @ Carrier @ Relation )
% 0.30/0.47 => ( prefer199794634erence @ A @ Carrier @ Relation ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % preference.intro
% 0.30/0.47 thf(fact_40_pair__in__swap__image,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,Y: A,X: B,A6: set @ ( product_prod @ B @ A )] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A6 ) )
% 0.30/0.47 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % pair_in_swap_image
% 0.30/0.47 thf(fact_41_preference_Otrans__refl,axiom,
% 0.30/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.30/0.47 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.30/0.47 => ( order_preorder_on @ A @ Carrier @ Relation ) ) ).
% 0.30/0.47
% 0.30/0.47 % preference.trans_refl
% 0.30/0.47 thf(fact_42_strict__contour__is__diff,axiom,
% 0.30/0.47 ! [A: $tType,B2: set @ A,Y: A,Pr: set @ ( product_prod @ A @ A )] :
% 0.30/0.47 ( ( collect @ A
% 0.30/0.47 @ ^ [A5: A] :
% 0.30/0.47 ( ( member @ A @ A5 @ B2 )
% 0.30/0.47 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ Y ) @ Pr )
% 0.30/0.47 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ A5 ) @ Pr ) ) )
% 0.30/0.47 = ( minus_minus @ ( set @ A ) @ ( prefer310429814s_good @ A @ Y @ B2 @ Pr ) @ ( prefer951318096ood_as @ A @ Y @ B2 @ Pr ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % strict_contour_is_diff
% 0.30/0.47 thf(fact_43_old_Orec__prod__def,axiom,
% 0.30/0.47 ! [T: $tType,B: $tType,A: $tType] :
% 0.30/0.47 ( ( product_rec_prod @ A @ B @ T )
% 0.30/0.47 = ( ^ [F12: A > B > T,X4: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.rec_prod_def
% 0.30/0.47 thf(fact_44_old_Orec__bool__def,axiom,
% 0.30/0.47 ! [T: $tType] :
% 0.30/0.47 ( ( product_rec_bool @ T )
% 0.30/0.47 = ( ^ [F12: T,F23: T,X4: $o] : ( the @ T @ ( product_rec_set_bool @ T @ F12 @ F23 @ X4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.rec_bool_def
% 0.30/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/0.47 ! [A: $tType,A2: A,P2: A > $o] :
% 0.30/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 0.30/0.47 = ( P2 @ A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % mem_Collect_eq
% 0.30/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A] :
% 0.30/0.47 ( ( collect @ A
% 0.30/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ A6 ) )
% 0.30/0.47 = A6 ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_mem_eq
% 0.30/0.47 thf(fact_47_Collect__cong,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.30/0.47 ( ! [X3: A] :
% 0.30/0.47 ( ( P2 @ X3 )
% 0.30/0.47 = ( Q @ X3 ) )
% 0.30/0.47 => ( ( collect @ A @ P2 )
% 0.30/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_cong
% 0.30/0.47 thf(fact_48_ext,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,F4: A > B,G3: A > B] :
% 0.30/0.47 ( ! [X3: A] :
% 0.30/0.47 ( ( F4 @ X3 )
% 0.30/0.47 = ( G3 @ X3 ) )
% 0.30/0.47 => ( F4 = G3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % ext
% 0.30/0.47 thf(fact_49_diff__Pair,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( minus @ B )
% 0.30/0.47 & ( minus @ A ) )
% 0.30/0.47 => ! [A2: A,B3: B,C3: A,D3: B] :
% 0.30/0.47 ( ( minus_minus @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Pair @ A @ B @ C3 @ D3 ) )
% 0.30/0.47 = ( product_Pair @ A @ B @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ B @ B3 @ D3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_Pair
% 0.30/0.47 thf(fact_50_the__sym__eq__trivial,axiom,
% 0.30/0.47 ! [A: $tType,X: A] :
% 0.30/0.47 ( ( the @ A
% 0.30/0.47 @ ( ^ [Y5: A,Z2: A] : ( Y5 = Z2 )
% 0.30/0.47 @ X ) )
% 0.30/0.47 = X ) ).
% 0.30/0.47
% 0.30/0.47 % the_sym_eq_trivial
% 0.30/0.47 thf(fact_51_the__eq__trivial,axiom,
% 0.30/0.47 ! [A: $tType,A2: A] :
% 0.30/0.47 ( ( the @ A
% 0.30/0.47 @ ^ [X4: A] : ( X4 = A2 ) )
% 0.30/0.47 = A2 ) ).
% 0.30/0.47
% 0.30/0.47 % the_eq_trivial
% 0.30/0.47 thf(fact_52_the__equality,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o,A2: A] :
% 0.30/0.47 ( ( P2 @ A2 )
% 0.30/0.47 => ( ! [X3: A] :
% 0.30/0.47 ( ( P2 @ X3 )
% 0.30/0.47 => ( X3 = A2 ) )
% 0.30/0.47 => ( ( the @ A @ P2 )
% 0.30/0.47 = A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % the_equality
% 0.30/0.47 thf(fact_53_image__ident,axiom,
% 0.30/0.47 ! [A: $tType,Y6: set @ A] :
% 0.30/0.47 ( ( image @ A @ A
% 0.30/0.47 @ ^ [X4: A] : X4
% 0.30/0.47 @ Y6 )
% 0.30/0.47 = Y6 ) ).
% 0.30/0.47
% 0.30/0.47 % image_ident
% 0.30/0.47 thf(fact_54_translation__subtract__diff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ab_group_add @ A )
% 0.30/0.47 => ! [A2: A,S: set @ A,T2: set @ A] :
% 0.30/0.47 ( ( image @ A @ A
% 0.30/0.47 @ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
% 0.30/0.47 @ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
% 0.30/0.47 = ( minus_minus @ ( set @ A )
% 0.30/0.47 @ ( image @ A @ A
% 0.30/0.47 @ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
% 0.30/0.47 @ S )
% 0.30/0.47 @ ( image @ A @ A
% 0.30/0.47 @ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
% 0.30/0.47 @ T2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % translation_subtract_diff
% 0.30/0.47 thf(fact_55_DiffI,axiom,
% 0.30/0.47 ! [A: $tType,C3: A,A6: set @ A,B2: set @ A] :
% 0.30/0.47 ( ( member @ A @ C3 @ A6 )
% 0.30/0.47 => ( ~ ( member @ A @ C3 @ B2 )
% 0.30/0.47 => ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % DiffI
% 0.30/0.47 thf(fact_56_image__eqI,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,B3: A,F4: B > A,X: B,A6: set @ B] :
% 0.30/0.47 ( ( B3
% 0.30/0.47 = ( F4 @ X ) )
% 0.30/0.47 => ( ( member @ B @ X @ A6 )
% 0.30/0.47 => ( member @ A @ B3 @ ( image @ B @ A @ F4 @ A6 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_eqI
% 0.30/0.47 thf(fact_57_Diff__idemp,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A,B2: set @ A] :
% 0.30/0.47 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) @ B2 )
% 0.30/0.47 = ( minus_minus @ ( set @ A ) @ A6 @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Diff_idemp
% 0.30/0.47 thf(fact_58_Diff__iff,axiom,
% 0.30/0.47 ! [A: $tType,C3: A,A6: set @ A,B2: set @ A] :
% 0.30/0.47 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) )
% 0.30/0.47 = ( ( member @ A @ C3 @ A6 )
% 0.30/0.47 & ~ ( member @ A @ C3 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Diff_iff
% 0.30/0.47 thf(fact_59_minus__set__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( minus_minus @ ( set @ A ) )
% 0.30/0.47 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.30/0.47 ( collect @ A
% 0.30/0.47 @ ( minus_minus @ ( A > $o )
% 0.30/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ A7 )
% 0.30/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minus_set_def
% 0.30/0.47 thf(fact_60_rev__image__eqI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X: A,A6: set @ A,B3: B,F4: A > B] :
% 0.30/0.47 ( ( member @ A @ X @ A6 )
% 0.30/0.47 => ( ( B3
% 0.30/0.47 = ( F4 @ X ) )
% 0.30/0.47 => ( member @ B @ B3 @ ( image @ A @ B @ F4 @ A6 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % rev_image_eqI
% 0.30/0.47 thf(fact_61_ball__imageD,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P2: A > $o] :
% 0.30/0.47 ( ! [X3: A] :
% 0.30/0.47 ( ( member @ A @ X3 @ ( image @ B @ A @ F4 @ A6 ) )
% 0.30/0.47 => ( P2 @ X3 ) )
% 0.30/0.47 => ! [X5: B] :
% 0.30/0.47 ( ( member @ B @ X5 @ A6 )
% 0.30/0.47 => ( P2 @ ( F4 @ X5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ball_imageD
% 0.30/0.47 thf(fact_62_image__cong,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F4: A > B,G3: A > B] :
% 0.30/0.47 ( ( M = N )
% 0.30/0.47 => ( ! [X3: A] :
% 0.30/0.47 ( ( member @ A @ X3 @ N )
% 0.30/0.47 => ( ( F4 @ X3 )
% 0.30/0.47 = ( G3 @ X3 ) ) )
% 0.30/0.47 => ( ( image @ A @ B @ F4 @ M )
% 0.30/0.47 = ( image @ A @ B @ G3 @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_cong
% 0.30/0.47 thf(fact_63_bex__imageD,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P2: A > $o] :
% 0.30/0.47 ( ? [X5: A] :
% 0.30/0.47 ( ( member @ A @ X5 @ ( image @ B @ A @ F4 @ A6 ) )
% 0.30/0.47 & ( P2 @ X5 ) )
% 0.30/0.47 => ? [X3: B] :
% 0.30/0.47 ( ( member @ B @ X3 @ A6 )
% 0.30/0.47 & ( P2 @ ( F4 @ X3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bex_imageD
% 0.30/0.47 thf(fact_64_image__iff,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,Z: A,F4: B > A,A6: set @ B] :
% 0.30/0.47 ( ( member @ A @ Z @ ( image @ B @ A @ F4 @ A6 ) )
% 0.30/0.47 = ( ? [X4: B] :
% 0.30/0.47 ( ( member @ B @ X4 @ A6 )
% 0.30/0.47 & ( Z
% 0.30/0.47 = ( F4 @ X4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_iff
% 0.30/0.47 thf(fact_65_imageI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X: A,A6: set @ A,F4: A > B] :
% 0.30/0.47 ( ( member @ A @ X @ A6 )
% 0.30/0.47 => ( member @ B @ ( F4 @ X ) @ ( image @ A @ B @ F4 @ A6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageI
% 0.30/0.47 thf(fact_66_DiffD2,axiom,
% 0.30/0.47 ! [A: $tType,C3: A,A6: set @ A,B2: set @ A] :
% 0.30/0.47 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) )
% 0.30/0.47 => ~ ( member @ A @ C3 @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % DiffD2
% 0.30/0.47 thf(fact_67_DiffD1,axiom,
% 0.30/0.47 ! [A: $tType,C3: A,A6: set @ A,B2: set @ A] :
% 0.30/0.47 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) )
% 0.30/0.47 => ( member @ A @ C3 @ A6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % DiffD1
% 0.30/0.47 thf(fact_68_DiffE,axiom,
% 0.30/0.47 ! [A: $tType,C3: A,A6: set @ A,B2: set @ A] :
% 0.30/0.47 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) )
% 0.30/0.47 => ~ ( ( member @ A @ C3 @ A6 )
% 0.30/0.47 => ( member @ A @ C3 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % DiffE
% 0.30/0.47 thf(fact_69_imageE,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,B3: A,F4: B > A,A6: set @ B] :
% 0.30/0.47 ( ( member @ A @ B3 @ ( image @ B @ A @ F4 @ A6 ) )
% 0.30/0.47 => ~ ! [X3: B] :
% 0.30/0.47 ( ( B3
% 0.30/0.47 = ( F4 @ X3 ) )
% 0.30/0.47 => ~ ( member @ B @ X3 @ A6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageE
% 0.30/0.47 thf(fact_70_image__image,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,F4: B > A,G3: C > B,A6: set @ C] :
% 0.30/0.47 ( ( image @ B @ A @ F4 @ ( image @ C @ B @ G3 @ A6 ) )
% 0.30/0.47 = ( image @ C @ A
% 0.30/0.47 @ ^ [X4: C] : ( F4 @ ( G3 @ X4 ) )
% 0.30/0.47 @ A6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_image
% 0.30/0.47 thf(fact_71_Compr__image__eq,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P2: A > $o] :
% 0.30/0.47 ( ( collect @ A
% 0.30/0.47 @ ^ [X4: A] :
% 0.30/0.47 ( ( member @ A @ X4 @ ( image @ B @ A @ F4 @ A6 ) )
% 0.30/0.47 & ( P2 @ X4 ) ) )
% 0.30/0.47 = ( image @ B @ A @ F4
% 0.30/0.47 @ ( collect @ B
% 0.30/0.47 @ ^ [X4: B] :
% 0.30/0.47 ( ( member @ B @ X4 @ A6 )
% 0.30/0.47 & ( P2 @ ( F4 @ X4 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Compr_image_eq
% 0.30/0.47 thf(fact_72_set__diff__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( minus_minus @ ( set @ A ) )
% 0.30/0.47 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.30/0.47 ( collect @ A
% 0.30/0.47 @ ^ [X4: A] :
% 0.30/0.47 ( ( member @ A @ X4 @ A7 )
% 0.30/0.47 & ~ ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % set_diff_eq
% 0.30/0.47 thf(fact_73_theI,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o,A2: A] :
% 0.30/0.47 ( ( P2 @ A2 )
% 0.30/0.47 => ( ! [X3: A] :
% 0.30/0.47 ( ( P2 @ X3 )
% 0.30/0.47 => ( X3 = A2 ) )
% 0.30/0.47 => ( P2 @ ( the @ A @ P2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % theI
% 0.30/0.47 thf(fact_74_theI_H,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o] :
% 0.30/0.47 ( ? [X5: A] :
% 0.30/0.47 ( ( P2 @ X5 )
% 0.30/0.47 & ! [Y3: A] :
% 0.30/0.47 ( ( P2 @ Y3 )
% 0.30/0.47 => ( Y3 = X5 ) ) )
% 0.30/0.47 => ( P2 @ ( the @ A @ P2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % theI'
% 0.30/0.47 thf(fact_75_theI2,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o,A2: A,Q: A > $o] :
% 0.30/0.47 ( ( P2 @ A2 )
% 0.30/0.47 => ( ! [X3: A] :
% 0.30/0.47 ( ( P2 @ X3 )
% 0.30/0.47 => ( X3 = A2 ) )
% 0.30/0.47 => ( ! [X3: A] :
% 0.30/0.47 ( ( P2 @ X3 )
% 0.30/0.47 => ( Q @ X3 ) )
% 0.30/0.47 => ( Q @ ( the @ A @ P2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % theI2
% 0.30/0.47 thf(fact_76_If__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( if @ A )
% 0.30/0.47 = ( ^ [P3: $o,X4: A,Y4: A] :
% 0.30/0.47 ( the @ A
% 0.30/0.47 @ ^ [Z3: A] :
% 0.30/0.47 ( ( P3
% 0.30/0.47 => ( Z3 = X4 ) )
% 0.30/0.47 & ( ~ P3
% 0.30/0.47 => ( Z3 = Y4 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % If_def
% 0.30/0.47 thf(fact_77_the1I2,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.30/0.47 ( ? [X5: A] :
% 0.30/0.47 ( ( P2 @ X5 )
% 0.30/0.47 & ! [Y3: A] :
% 0.30/0.47 ( ( P2 @ Y3 )
% 0.30/0.47 => ( Y3 = X5 ) ) )
% 0.30/0.47 => ( ! [X3: A] :
% 0.30/0.47 ( ( P2 @ X3 )
% 0.30/0.47 => ( Q @ X3 ) )
% 0.30/0.47 => ( Q @ ( the @ A @ P2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % the1I2
% 0.30/0.47 thf(fact_78_the1__equality,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o,A2: A] :
% 0.30/0.47 ( ? [X5: A] :
% 0.30/0.47 ( ( P2 @ X5 )
% 0.30/0.47 & ! [Y3: A] :
% 0.30/0.47 ( ( P2 @ Y3 )
% 0.30/0.47 => ( Y3 = X5 ) ) )
% 0.30/0.47 => ( ( P2 @ A2 )
% 0.30/0.47 => ( ( the @ A @ P2 )
% 0.30/0.47 = A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % the1_equality
% 0.30/0.47 thf(fact_79_minus__apply,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( minus @ B )
% 0.30/0.47 => ( ( minus_minus @ ( A > B ) )
% 0.30/0.47 = ( ^ [A7: A > B,B6: A > B,X4: A] : ( minus_minus @ B @ ( A7 @ X4 ) @ ( B6 @ X4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minus_apply
% 0.30/0.47 thf(fact_80_Inf_OINF__identity__eq,axiom,
% 0.30/0.47 ! [A: $tType,Inf: ( set @ A ) > A,A6: set @ A] :
% 0.30/0.47 ( ( Inf
% 0.30/0.47 @ ( image @ A @ A
% 0.30/0.47 @ ^ [X4: A] : X4
% 0.30/0.47 @ A6 ) )
% 0.30/0.47 = ( Inf @ A6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Inf.INF_identity_eq
% 0.30/0.47 thf(fact_81_Sup_OSUP__identity__eq,axiom,
% 0.30/0.47 ! [A: $tType,Sup: ( set @ A ) > A,A6: set @ A] :
% 0.30/0.47 ( ( Sup
% 0.30/0.47 @ ( image @ A @ A
% 0.30/0.47 @ ^ [X4: A] : X4
% 0.30/0.47 @ A6 ) )
% 0.30/0.47 = ( Sup @ A6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Sup.SUP_identity_eq
% 0.30/0.47 thf(fact_82_old_Orec__unit__def,axiom,
% 0.30/0.47 ! [T: $tType] :
% 0.30/0.47 ( ( product_rec_unit @ T )
% 0.30/0.47 = ( ^ [F12: T,X4: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.rec_unit_def
% 0.30/0.47 thf(fact_83_theI__unique,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o,X: A] :
% 0.30/0.47 ( ? [X5: A] :
% 0.30/0.47 ( ( P2 @ X5 )
% 0.30/0.47 & ! [Y3: A] :
% 0.30/0.47 ( ( P2 @ Y3 )
% 0.30/0.47 => ( Y3 = X5 ) ) )
% 0.30/0.47 => ( ( P2 @ X )
% 0.30/0.47 = ( X
% 0.30/0.47 = ( the @ A @ P2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % theI_unique
% 0.30/0.47 thf(fact_84_Nitpick_OThe__psimp,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o,X: A] :
% 0.30/0.47 ( ( P2
% 0.30/0.47 = ( ^ [Y5: A,Z2: A] : ( Y5 = Z2 )
% 0.30/0.47 @ X ) )
% 0.30/0.47 => ( ( the @ A @ P2 )
% 0.30/0.47 = X ) ) ).
% 0.30/0.47
% 0.30/0.47 % Nitpick.The_psimp
% 0.30/0.47 thf(fact_85_surj__swap,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.30/0.47 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surj_swap
% 0.30/0.47 thf(fact_86_surj__diff__right,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ab_group_add @ A )
% 0.30/0.47 => ! [A2: A] :
% 0.30/0.47 ( ( image @ A @ A
% 0.30/0.47 @ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
% 0.30/0.47 @ ( top_top @ ( set @ A ) ) )
% 0.30/0.47 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surj_diff_right
% 0.30/0.47 thf(fact_87_UNIV__I,axiom,
% 0.30/0.47 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_I
% 0.30/0.47 thf(fact_88_UNIV__witness,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_witness
% 0.30/0.47 thf(fact_89_UNIV__eq__I,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A] :
% 0.30/0.47 ( ! [X3: A] : ( member @ A @ X3 @ A6 )
% 0.30/0.47 => ( ( top_top @ ( set @ A ) )
% 0.30/0.47 = A6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_eq_I
% 0.30/0.47 thf(fact_90_UNIV__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( top_top @ ( set @ A ) )
% 0.30/0.47 = ( collect @ A
% 0.30/0.47 @ ^ [X4: A] : $true ) ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_def
% 0.30/0.47 thf(fact_91_range__eqI,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,B3: A,F4: B > A,X: B] :
% 0.30/0.47 ( ( B3
% 0.30/0.47 = ( F4 @ X ) )
% 0.30/0.47 => ( member @ A @ B3 @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % range_eqI
% 0.30/0.47 thf(fact_92_surj__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,F4: B > A] :
% 0.30/0.47 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 0.30/0.47 = ( top_top @ ( set @ A ) ) )
% 0.30/0.47 = ( ! [Y4: A] :
% 0.30/0.47 ? [X4: B] :
% 0.30/0.47 ( Y4
% 0.30/0.47 = ( F4 @ X4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surj_def
% 0.30/0.47 thf(fact_93_rangeI,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F4: B > A,X: B] : ( member @ A @ ( F4 @ X ) @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % rangeI
% 0.30/0.47 thf(fact_94_surjI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,G3: B > A,F4: A > B] :
% 0.30/0.47 ( ! [X3: A] :
% 0.30/0.47 ( ( G3 @ ( F4 @ X3 ) )
% 0.30/0.47 = X3 )
% 0.30/0.47 => ( ( image @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) )
% 0.30/0.47 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjI
% 0.30/0.47 thf(fact_95_surjE,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F4: B > A,Y: A] :
% 0.30/0.47 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 0.30/0.47 = ( top_top @ ( set @ A ) ) )
% 0.30/0.47 => ~ ! [X3: B] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( F4 @ X3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjE
% 0.30/0.47 thf(fact_96_surjD,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F4: B > A,Y: A] :
% 0.30/0.47 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 0.30/0.47 = ( top_top @ ( set @ A ) ) )
% 0.30/0.47 => ? [X3: B] :
% 0.30/0.47 ( Y
% 0.30/0.47 = ( F4 @ X3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjD
% 0.30/0.47 thf(fact_97_range__composition,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,B: $tType,F4: C > A,G3: B > C] :
% 0.30/0.47 ( ( image @ B @ A
% 0.30/0.47 @ ^ [X4: B] : ( F4 @ ( G3 @ X4 ) )
% 0.30/0.47 @ ( top_top @ ( set @ B ) ) )
% 0.30/0.47 = ( image @ C @ A @ F4 @ ( image @ B @ C @ G3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % range_composition
% 0.30/0.47 thf(fact_98_rangeE,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,B3: A,F4: B > A] :
% 0.30/0.47 ( ( member @ A @ B3 @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) )
% 0.30/0.47 => ~ ! [X3: B] :
% 0.30/0.47 ( B3
% 0.30/0.47 != ( F4 @ X3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % rangeE
% 0.30/0.47 thf(fact_99_fun__diff__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( minus @ B )
% 0.30/0.47 => ( ( minus_minus @ ( A > B ) )
% 0.30/0.47 = ( ^ [A7: A > B,B6: A > B,X4: A] : ( minus_minus @ B @ ( A7 @ X4 ) @ ( B6 @ X4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_diff_def
% 0.30/0.47 thf(fact_100_Inf_OINF__cong,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A6: set @ B,B2: set @ B,C5: B > A,D4: B > A,Inf: ( set @ A ) > A] :
% 0.30/0.47 ( ( A6 = B2 )
% 0.30/0.47 => ( ! [X3: B] :
% 0.30/0.47 ( ( member @ B @ X3 @ B2 )
% 0.30/0.47 => ( ( C5 @ X3 )
% 0.30/0.47 = ( D4 @ X3 ) ) )
% 0.30/0.47 => ( ( Inf @ ( image @ B @ A @ C5 @ A6 ) )
% 0.30/0.47 = ( Inf @ ( image @ B @ A @ D4 @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Inf.INF_cong
% 0.30/0.47 thf(fact_101_Sup_OSUP__cong,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A6: set @ B,B2: set @ B,C5: B > A,D4: B > A,Sup: ( set @ A ) > A] :
% 0.30/0.47 ( ( A6 = B2 )
% 0.30/0.47 => ( ! [X3: B] :
% 0.30/0.47 ( ( member @ B @ X3 @ B2 )
% 0.30/0.47 => ( ( C5 @ X3 )
% 0.30/0.47 = ( D4 @ X3 ) ) )
% 0.30/0.47 => ( ( Sup @ ( image @ B @ A @ C5 @ A6 ) )
% 0.30/0.47 = ( Sup @ ( image @ B @ A @ D4 @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Sup.SUP_cong
% 0.30/0.47 thf(fact_102_iso__tuple__UNIV__I,axiom,
% 0.30/0.47 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % iso_tuple_UNIV_I
% 0.30/0.47 thf(fact_103_top__apply,axiom,
% 0.30/0.47 ! [C: $tType,D: $tType] :
% 0.30/0.47 ( ( top @ C )
% 0.30/0.47 => ( ( top_top @ ( D > C ) )
% 0.30/0.47 = ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % top_apply
% 0.30/0.47 thf(fact_104_same__fstI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P2: A > $o,X: A,Y7: B,Y: B,R2: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.30/0.47 ( ( P2 @ X )
% 0.30/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y ) @ ( R2 @ X ) )
% 0.30/0.47 => ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y7 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P2 @ R2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % same_fstI
% 0.30/0.47 thf(fact_105_top__prod__def,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( top @ B )
% 0.30/0.47 & ( top @ A ) )
% 0.30/0.47 => ( ( top_top @ ( product_prod @ A @ B ) )
% 0.30/0.47 = ( product_Pair @ A @ B @ ( top_top @ A ) @ ( top_top @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % top_prod_def
% 0.30/0.47 thf(fact_106_map__prod__surj,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,D: $tType,B: $tType,F4: A > B,G3: C > D] :
% 0.30/0.47 ( ( ( image @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 0.30/0.47 = ( top_top @ ( set @ B ) ) )
% 0.30/0.47 => ( ( ( image @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
% 0.30/0.47 = ( top_top @ ( set @ D ) ) )
% 0.30/0.47 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F4 @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 0.30/0.47 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % map_prod_surj
% 0.30/0.47 thf(fact_107_range__fst,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.30/0.47 = ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % range_fst
% 0.30/0.47 thf(fact_108_map__prod__ident,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( product_map_prod @ A @ A @ B @ B
% 0.30/0.47 @ ^ [X4: A] : X4
% 0.30/0.47 @ ^ [Y4: B] : Y4 )
% 0.30/0.47 = ( ^ [Z3: product_prod @ A @ B] : Z3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % map_prod_ident
% 0.30/0.47 thf(fact_109_map__prod__simp,axiom,
% 0.30/0.47 ! [C: $tType,A: $tType,B: $tType,D: $tType,F4: C > A,G3: D > B,A2: C,B3: D] :
% 0.30/0.47 ( ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 @ ( product_Pair @ C @ D @ A2 @ B3 ) )
% 0.30/0.47 = ( product_Pair @ A @ B @ ( F4 @ A2 ) @ ( G3 @ B3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % map_prod_simp
% 0.30/0.47 thf(fact_110_fst__map__prod,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,D: $tType,C: $tType,F4: C > A,G3: D > B,X: product_prod @ C @ D] :
% 0.30/0.47 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 @ X ) )
% 0.30/0.47 = ( F4 @ ( product_fst @ C @ D @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fst_map_prod
% 0.30/0.47 thf(fact_111_map__prod__imageI,axiom,
% 0.30/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B3: B,R2: set @ ( product_prod @ A @ B ),F4: A > C,G3: B > D] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ R2 )
% 0.30/0.47 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F4 @ A2 ) @ ( G3 @ B3 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F4 @ G3 ) @ R2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % map_prod_imageI
% 0.30/0.47 thf(fact_112_fst__diff,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ( minus @ A )
% 0.30/0.47 & ( minus @ B ) )
% 0.30/0.47 => ! [X: product_prod @ A @ B,Y: product_prod @ A @ B] :
% 0.30/0.47 ( ( product_fst @ A @ B @ ( minus_minus @ ( product_prod @ A @ B ) @ X @ Y ) )
% 0.30/0.47 = ( minus_minus @ A @ ( product_fst @ A @ B @ X ) @ ( product_fst @ A @ B @ Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fst_diff
% 0.30/0.47 thf(fact_113_fst__top,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ( top @ A )
% 0.30/0.47 & ( top @ B ) )
% 0.30/0.47 => ( ( product_fst @ A @ B @ ( top_top @ ( product_prod @ A @ B ) ) )
% 0.30/0.47 = ( top_top @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fst_top
% 0.30/0.47 thf(fact_114_top__empty__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( top_top @ ( A > $o ) )
% 0.30/0.47 = ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % top_empty_eq
% 0.30/0.47 thf(fact_115_top__set__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( top_top @ ( set @ A ) )
% 0.30/0.47 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % top_set_def
% 0.30/0.47 thf(fact_116_fst__eqD,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X: A,Y: B,A2: A] :
% 0.30/0.47 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 0.30/0.47 = A2 )
% 0.30/0.47 => ( X = A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % fst_eqD
% 0.30/0.47 thf(fact_117_fst__conv,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X1: A,X2: B] :
% 0.30/0.47 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 0.30/0.47 = X1 ) ).
% 0.30/0.47
% 0.30/0.47 % fst_conv
% 0.30/0.47 thf(fact_118_prod__fun__imageE,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F4: C > A,G3: D > B,R2: set @ ( product_prod @ C @ D )] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 ) @ R2 ) )
% 0.30/0.47 => ~ ! [X3: C,Y3: D] :
% 0.30/0.47 ( ( C3
% 0.30/0.47 = ( product_Pair @ A @ B @ ( F4 @ X3 ) @ ( G3 @ Y3 ) ) )
% 0.30/0.47 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_fun_imageE
% 0.30/0.47 thf(fact_119_top__empty__eq2,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( top_top @ ( A > B > $o ) )
% 0.30/0.47 = ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % top_empty_eq2
% 0.30/0.47 thf(fact_120_eq__fst__iff,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A2: A,P: product_prod @ A @ B] :
% 0.30/0.47 ( ( A2
% 0.30/0.47 = ( product_fst @ A @ B @ P ) )
% 0.30/0.47 = ( ? [B7: B] :
% 0.30/0.47 ( P
% 0.30/0.47 = ( product_Pair @ A @ B @ A2 @ B7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eq_fst_iff
% 0.30/0.47 thf(fact_121_fstI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
% 0.30/0.47 ( ( X
% 0.30/0.47 = ( product_Pair @ A @ B @ Y @ Z ) )
% 0.30/0.47 => ( ( product_fst @ A @ B @ X )
% 0.30/0.47 = Y ) ) ).
% 0.30/0.47
% 0.30/0.47 % fstI
% 0.30/0.47 thf(fact_122_prod_Omap__ident,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 0.30/0.47 ( ( product_map_prod @ A @ A @ B @ B
% 0.30/0.47 @ ^ [X4: A] : X4
% 0.30/0.47 @ ^ [X4: B] : X4
% 0.30/0.47 @ T2 )
% 0.30/0.47 = T2 ) ).
% 0.30/0.47
% 0.30/0.47 % prod.map_ident
% 0.30/0.47 thf(fact_123_minus__prod__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ( minus @ A )
% 0.30/0.47 & ( minus @ B ) )
% 0.30/0.47 => ( ( minus_minus @ ( product_prod @ A @ B ) )
% 0.30/0.47 = ( ^ [X4: product_prod @ A @ B,Y4: product_prod @ A @ B] : ( product_Pair @ A @ B @ ( minus_minus @ A @ ( product_fst @ A @ B @ X4 ) @ ( product_fst @ A @ B @ Y4 ) ) @ ( minus_minus @ B @ ( product_snd @ A @ B @ X4 ) @ ( product_snd @ A @ B @ Y4 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minus_prod_def
% 0.30/0.47 thf(fact_124_range__snd,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.30/0.47 = ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % range_snd
% 0.30/0.47 thf(fact_125_top1I,axiom,
% 0.30/0.47 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 0.30/0.47
% 0.30/0.47 % top1I
% 0.30/0.47 thf(fact_126_top2I,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X: A,Y: B] : ( top_top @ ( A > B > $o ) @ X @ Y ) ).
% 0.30/0.47
% 0.30/0.47 % top2I
% 0.30/0.47 thf(fact_127_snd__map__prod,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,D: $tType,C: $tType,F4: C > B,G3: D > A,X: product_prod @ C @ D] :
% 0.30/0.47 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F4 @ G3 @ X ) )
% 0.30/0.47 = ( G3 @ ( product_snd @ C @ D @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % snd_map_prod
% 0.30/0.47 thf(fact_128_prod_Ocollapse,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.30/0.47 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 0.30/0.47 = Prod ) ).
% 0.30/0.47
% 0.30/0.47 % prod.collapse
% 0.30/0.47 thf(fact_129_fst__swap,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X: product_prod @ B @ A] :
% 0.30/0.47 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
% 0.30/0.47 = ( product_snd @ B @ A @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % fst_swap
% 0.30/0.47 thf(fact_130_snd__swap,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X: product_prod @ A @ B] :
% 0.30/0.47 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
% 0.30/0.47 = ( product_fst @ A @ B @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % snd_swap
% 0.30/0.47 thf(fact_131_snd__diff,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( minus @ B )
% 0.30/0.47 & ( minus @ A ) )
% 0.30/0.47 => ! [X: product_prod @ B @ A,Y: product_prod @ B @ A] :
% 0.30/0.47 ( ( product_snd @ B @ A @ ( minus_minus @ ( product_prod @ B @ A ) @ X @ Y ) )
% 0.30/0.47 = ( minus_minus @ A @ ( product_snd @ B @ A @ X ) @ ( product_snd @ B @ A @ Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % snd_diff
% 0.30/0.47 thf(fact_132_snd__top,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ( top @ A )
% 0.30/0.47 & ( top @ B ) )
% 0.30/0.47 => ( ( product_snd @ B @ A @ ( top_top @ ( product_prod @ B @ A ) ) )
% 0.30/0.47 = ( top_top @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % snd_top
% 0.30/0.47 thf(fact_133_prod__eqI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 0.30/0.47 ( ( ( product_fst @ A @ B @ P )
% 0.30/0.47 = ( product_fst @ A @ B @ Q2 ) )
% 0.30/0.47 => ( ( ( product_snd @ A @ B @ P )
% 0.30/0.47 = ( product_snd @ A @ B @ Q2 ) )
% 0.30/0.47 => ( P = Q2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_eqI
% 0.30/0.47 thf(fact_134_prod_Oexpand,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 0.30/0.47 ( ( ( ( product_fst @ A @ B @ Prod )
% 0.30/0.47 = ( product_fst @ A @ B @ Prod2 ) )
% 0.30/0.47 & ( ( product_snd @ A @ B @ Prod )
% 0.30/0.47 = ( product_snd @ A @ B @ Prod2 ) ) )
% 0.30/0.47 => ( Prod = Prod2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod.expand
% 0.30/0.47 thf(fact_135_prod__eq__iff,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ^ [Y5: product_prod @ A @ B,Z2: product_prod @ A @ B] : ( Y5 = Z2 ) )
% 0.30/0.47 = ( ^ [S4: product_prod @ A @ B,T3: product_prod @ A @ B] :
% 0.30/0.47 ( ( ( product_fst @ A @ B @ S4 )
% 0.30/0.47 = ( product_fst @ A @ B @ T3 ) )
% 0.30/0.47 & ( ( product_snd @ A @ B @ S4 )
% 0.30/0.47 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod_eq_iff
% 0.30/0.47 thf(fact_136_eq__snd__iff,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,B3: A,P: product_prod @ B @ A] :
% 0.30/0.47 ( ( B3
% 0.30/0.47 = ( product_snd @ B @ A @ P ) )
% 0.30/0.47 = ( ? [A5: B] :
% 0.30/0.47 ( P
% 0.30/0.47 = ( product_Pair @ B @ A @ A5 @ B3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eq_snd_iff
% 0.30/0.47 thf(fact_137_sndI,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
% 0.30/0.47 ( ( X
% 0.30/0.47 = ( product_Pair @ A @ B @ Y @ Z ) )
% 0.30/0.47 => ( ( product_snd @ A @ B @ X )
% 0.30/0.47 = Z ) ) ).
% 0.30/0.47
% 0.30/0.47 % sndI
% 0.30/0.47 thf(fact_138_snd__eqD,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X: B,Y: A,A2: A] :
% 0.30/0.47 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 0.30/0.47 = A2 )
% 0.30/0.47 => ( Y = A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % snd_eqD
% 0.30/0.47 thf(fact_139_snd__conv,axiom,
% 0.30/0.47 ! [Aa: $tType,A: $tType,X1: Aa,X2: A] :
% 0.30/0.47 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X2 ) )
% 0.30/0.47 = X2 ) ).
% 0.30/0.47
% 0.30/0.47 % snd_conv
% 0.30/0.47 thf(fact_140_surjective__pairing,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 0.30/0.47 ( T2
% 0.30/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjective_pairing
% 0.30/0.47 thf(fact_141_prod_Oexhaust__sel,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.30/0.47 ( Prod
% 0.30/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod.exhaust_sel
% 0.30/0.47 thf(fact_142_prod_Oswap__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( product_swap @ A @ B )
% 0.30/0.47 = ( ^ [P4: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P4 ) @ ( product_fst @ A @ B @ P4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prod.swap_def
% 0.30/0.47 thf(fact_143_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P2: A > B > $o,X: A,Y: B,A2: product_prod @ A @ B] :
% 0.30/0.47 ( ( P2 @ X @ Y )
% 0.30/0.47 => ( ( A2
% 0.30/0.47 = ( product_Pair @ A @ B @ X @ Y ) )
% 0.30/0.47 => ( P2 @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % BNF_Greatest_Fixpoint.subst_Pair
% 0.30/0.47 thf(fact_144_conjI__realizer,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,P2: A > $o,P: A,Q: B > $o,Q2: B] :
% 0.30/0.47 ( ( P2 @ P )
% 0.30/0.47 => ( ( Q @ Q2 )
% 0.30/0.47 => ( ( P2 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) )
% 0.30/0.47 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % conjI_realizer
% 0.30/0.47 thf(fact_145_exI__realizer,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,P2: A > B > $o,Y: A,X: B] :
% 0.30/0.47 ( ( P2 @ Y @ X )
% 0.30/0.47 => ( P2 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % exI_realizer
% 0.30/0.47 thf(fact_146_top__conj_I2_J,axiom,
% 0.30/0.47 ! [A: $tType,P2: $o,X: A] :
% 0.30/0.47 ( ( P2
% 0.30/0.47 & ( top_top @ ( A > $o ) @ X ) )
% 0.30/0.47 = P2 ) ).
% 0.30/0.47
% 0.30/0.47 % top_conj(2)
% 0.30/0.47 thf(fact_147_top__conj_I1_J,axiom,
% 0.30/0.47 ! [A: $tType,X: A,P2: $o] :
% 0.30/0.47 ( ( ( top_top @ ( A > $o ) @ X )
% 0.30/0.47 & P2 )
% 0.30/0.47 = P2 ) ).
% 0.30/0.47
% 0.30/0.47 % top_conj(1)
% 0.30/0.47 thf(fact_148_scomp__unfold,axiom,
% 0.30/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 0.30/0.47 ( ( product_scomp @ A @ B @ C @ D )
% 0.30/0.47 = ( ^ [F3: A > ( product_prod @ B @ C ),G4: B > C > D,X4: A] : ( G4 @ ( product_fst @ B @ C @ ( F3 @ X4 ) ) @ ( product_snd @ B @ C @ ( F3 @ X4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % scomp_unfold
% 0.30/0.47 thf(fact_149_sndOp__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,C: $tType] :
% 0.30/0.47 ( ( bNF_sndOp @ C @ A @ B )
% 0.30/0.47 = ( ^ [P3: C > A > $o,Q3: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P3 @ Q3 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sndOp_def
% 0.30/0.47 thf(fact_150_fstOp__def,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.47 ( ( bNF_fstOp @ A @ B @ C )
% 0.30/0.47 = ( ^ [P3: A > B > $o,Q3: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P3 @ Q3 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fstOp_def
% 0.30/0.47 thf(fact_151_scomp__Pair,axiom,
% 0.30/0.47 ! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
% 0.30/0.47 ( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
% 0.30/0.47 = X ) ).
% 0.30/0.47
% 0.30/0.47 % scomp_Pair
% 0.30/0.47 thf(fact_152_scomp__scomp,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: $tType,E: $tType,F4: A > ( product_prod @ E @ F ),G3: E > F > ( product_prod @ C @ D ),H: C > D > B] :
% 0.30/0.47 ( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F @ ( product_prod @ C @ D ) @ F4 @ G3 ) @ H )
% 0.30/0.47 = ( product_scomp @ A @ E @ F @ B @ F4
% 0.30/0.47 @ ^ [X4: E] : ( product_scomp @ F @ C @ D @ B @ ( G3 @ X4 ) @ H ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % scomp_scomp
% 0.30/0.47 thf(fact_153_Pair__scomp,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C: $tType,X: C,F4: C > A > B] :
% 0.30/0.47 ( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F4 )
% 0.30/0.47 = ( F4 @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % Pair_scomp
% 0.30/0.47 thf(fact_154_image2__eqI,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType,B: $tType,B3: A,F4: B > A,X: B,C3: C,G3: B > C,A6: set @ B] :
% 0.30/0.47 ( ( B3
% 0.30/0.47 = ( F4 @ X ) )
% 0.30/0.47 => ( ( C3
% 0.30/0.47 = ( G3 @ X ) )
% 0.30/0.47 => ( ( member @ B @ X @ A6 )
% 0.30/0.47 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B3 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A6 @ F4 @ G3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image2_eqI
% 0.30/0.47 thf(fact_155_map__prod__surj__on,axiom,
% 0.30/0.47 ! [B: $tType,D: $tType,C: $tType,A: $tType,F4: B > A,A6: set @ B,A8: set @ A,G3: D > C,B2: set @ D,B8: set @ C] :
% 0.30/0.47 ( ( ( image @ B @ A @ F4 @ A6 )
% 0.30/0.47 = A8 )
% 0.30/0.47 => ( ( ( image @ D @ C @ G3 @ B2 )
% 0.30/0.47 = B8 )
% 0.30/0.47 => ( ( image @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F4 @ G3 )
% 0.30/0.47 @ ( product_Sigma @ B @ D @ A6
% 0.30/0.47 @ ^ [Uu: B] : B2 ) )
% 0.30/0.47 = ( product_Sigma @ A @ C @ A8
% 0.30/0.47 @ ^ [Uu: A] : B8 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % map_prod_surj_on
% 0.30/0.47 thf(fact_156_in__lex__prod,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A2: A,B3: B,A3: A,B4: B,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
% 0.30/0.47 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Pair @ A @ B @ A3 @ B4 ) ) @ ( lex_prod @ A @ B @ R @ S ) )
% 0.30/0.47 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A3 ) @ R )
% 0.30/0.47 | ( ( A2 = A3 )
% 0.30/0.47 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B3 @ B4 ) @ S ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % in_lex_prod
% 0.30/0.47 thf(fact_157_mem__Sigma__iff,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A2: A,B3: B,A6: set @ A,B2: A > ( set @ B )] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Sigma @ A @ B @ A6 @ B2 ) )
% 0.30/0.47 = ( ( member @ A @ A2 @ A6 )
% 0.30/0.47 & ( member @ B @ B3 @ ( B2 @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mem_Sigma_iff
% 0.30/0.47 thf(fact_158_SigmaI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A2: A,A6: set @ A,B3: B,B2: A > ( set @ B )] :
% 0.30/0.47 ( ( member @ A @ A2 @ A6 )
% 0.30/0.47 => ( ( member @ B @ B3 @ ( B2 @ A2 ) )
% 0.30/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Sigma @ A @ B @ A6 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % SigmaI
% 0.30/0.47 thf(fact_159_UNIV__Times__UNIV,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 0.30/0.47 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) )
% 0.30/0.47 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_Times_UNIV
% 0.30/0.47 thf(fact_160_Times__Diff__distrib1,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A6: set @ A,B2: set @ A,C5: set @ B] :
% 0.30/0.47 ( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ A6 @ B2 )
% 0.30/0.47 @ ^ [Uu: A] : C5 )
% 0.30/0.47 = ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
% 0.30/0.47 @ ( product_Sigma @ A @ B @ A6
% 0.30/0.47 @ ^ [Uu: A] : C5 )
% 0.30/0.47 @ ( product_Sigma @ A @ B @ B2
% 0.30/0.47 @ ^ [Uu: A] : C5 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Times_Diff_distrib1
% 0.30/0.47 thf(fact_161_Sigma__Diff__distrib2,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,I: set @ A,A6: A > ( set @ B ),B2: A > ( set @ B )] :
% 0.30/0.47 ( ( product_Sigma @ A @ B @ I
% 0.30/0.47 @ ^ [I2: A] : ( minus_minus @ ( set @ B ) @ ( A6 @ I2 ) @ ( B2 @ I2 ) ) )
% 0.30/0.47 = ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I @ A6 ) @ ( product_Sigma @ A @ B @ I @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Sigma_Diff_distrib2
% 0.30/0.47 thf(fact_162_Sigma__cong,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A6: set @ A,B2: set @ A,C5: A > ( set @ B ),D4: A > ( set @ B )] :
% 0.30/0.47 ( ( A6 = B2 )
% 0.30/0.47 => ( ! [X3: A] :
% 0.30/0.47 ( ( member @ A @ X3 @ B2 )
% 0.30/0.47 => ( ( C5 @ X3 )
% 0.30/0.47 = ( D4 @ X3 ) ) )
% 0.30/0.47 => ( ( product_Sigma @ A @ B @ A6 @ C5 )
% 0.30/0.47 = ( product_Sigma @ A @ B @ B2 @ D4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Sigma_cong
% 0.30/0.47 thf(fact_163_Times__eq__cancel2,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X: A,C5: set @ A,A6: set @ B,B2: set @ B] :
% 0.30/0.47 ( ( member @ A @ X @ C5 )
% 0.30/0.47 => ( ( ( product_Sigma @ B @ A @ A6
% 0.30/0.47 @ ^ [Uu: B] : C5 )
% 0.30/0.47 = ( product_Sigma @ B @ A @ B2
% 0.30/0.47 @ ^ [Uu: B] : C5 ) )
% 0.30/0.47 = ( A6 = B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Times_eq_cancel2
% 0.30/0.47 thf(fact_164_Sigma__Diff__distrib1,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,I: set @ A,J: set @ A,C5: A > ( set @ B )] :
% 0.30/0.47 ( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ I @ J ) @ C5 )
% 0.30/0.47 = ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I @ C5 ) @ ( product_Sigma @ A @ B @ J @ C5 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Sigma_Diff_distrib1
% 0.30/0.47 thf(fact_165_SigmaE2,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A2: A,B3: B,A6: set @ A,B2: A > ( set @ B )] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Sigma @ A @ B @ A6 @ B2 ) )
% 0.30/0.47 => ~ ( ( member @ A @ A2 @ A6 )
% 0.30/0.47 => ~ ( member @ B @ B3 @ ( B2 @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % SigmaE2
% 0.30/0.47 thf(fact_166_SigmaD2,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A2: A,B3: B,A6: set @ A,B2: A > ( set @ B )] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Sigma @ A @ B @ A6 @ B2 ) )
% 0.30/0.47 => ( member @ B @ B3 @ ( B2 @ A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % SigmaD2
% 0.30/0.47 thf(fact_167_SigmaD1,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A2: A,B3: B,A6: set @ A,B2: A > ( set @ B )] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Sigma @ A @ B @ A6 @ B2 ) )
% 0.30/0.47 => ( member @ A @ A2 @ A6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % SigmaD1
% 0.30/0.47 thf(fact_168_SigmaE,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,C3: product_prod @ A @ B,A6: set @ A,B2: A > ( set @ B )] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( product_Sigma @ A @ B @ A6 @ B2 ) )
% 0.30/0.47 => ~ ! [X3: A] :
% 0.30/0.47 ( ( member @ A @ X3 @ A6 )
% 0.30/0.47 => ! [Y3: B] :
% 0.30/0.47 ( ( member @ B @ Y3 @ ( B2 @ X3 ) )
% 0.30/0.47 => ( C3
% 0.30/0.47 != ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % SigmaE
% 0.30/0.47 thf(fact_169_mem__Times__iff,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X: product_prod @ A @ B,A6: set @ A,B2: set @ B] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ X
% 0.30/0.47 @ ( product_Sigma @ A @ B @ A6
% 0.30/0.47 @ ^ [Uu: A] : B2 ) )
% 0.30/0.47 = ( ( member @ A @ ( product_fst @ A @ B @ X ) @ A6 )
% 0.30/0.47 & ( member @ B @ ( product_snd @ A @ B @ X ) @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mem_Times_iff
% 0.30/0.47 thf(fact_170_product__swap,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A6: set @ B,B2: set @ A] :
% 0.30/0.47 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
% 0.30/0.47 @ ( product_Sigma @ B @ A @ A6
% 0.30/0.47 @ ^ [Uu: B] : B2 ) )
% 0.30/0.47 = ( product_Sigma @ A @ B @ B2
% 0.30/0.47 @ ^ [Uu: A] : A6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % product_swap
% 0.30/0.47 thf(fact_171_Product__Type_Oproduct__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( product_product @ A @ B )
% 0.30/0.47 = ( ^ [A7: set @ A,B6: set @ B] :
% 0.30/0.47 ( product_Sigma @ A @ B @ A7
% 0.30/0.47 @ ^ [Uu: A] : B6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Product_Type.product_def
% 0.30/0.47 thf(fact_172_member__product,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A6: set @ A,B2: set @ B] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A6 @ B2 ) )
% 0.30/0.47 = ( member @ ( product_prod @ A @ B ) @ X
% 0.30/0.47 @ ( product_Sigma @ A @ B @ A6
% 0.30/0.47 @ ^ [Uu: A] : B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % member_product
% 0.30/0.47 thf(fact_173_subset__fst__snd,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A6: set @ ( product_prod @ A @ B )] :
% 0.30/0.47 ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6
% 0.30/0.47 @ ( product_Sigma @ A @ B @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A6 )
% 0.30/0.47 @ ^ [Uu: A] : ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_fst_snd
% 0.30/0.47 thf(fact_174_subset__antisym,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A,B2: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
% 0.30/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A6 )
% 0.30/0.47 => ( A6 = B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_antisym
% 0.30/0.47 thf(fact_175_subsetI,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A,B2: set @ A] :
% 0.30/0.47 ( ! [X3: A] :
% 0.30/0.47 ( ( member @ A @ X3 @ A6 )
% 0.30/0.47 => ( member @ A @ X3 @ B2 ) )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ A6 @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % subsetI
% 0.30/0.47 thf(fact_176_Pair__le,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( ord @ B )
% 0.30/0.47 & ( ord @ A ) )
% 0.30/0.47 => ! [A2: A,B3: B,C3: A,D3: B] :
% 0.30/0.47 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Pair @ A @ B @ C3 @ D3 ) )
% 0.30/0.47 = ( ( ord_less_eq @ A @ A2 @ C3 )
% 0.30/0.47 & ( ord_less_eq @ B @ B3 @ D3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Pair_le
% 0.30/0.47 thf(fact_177_Times__subset__cancel2,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X: A,C5: set @ A,A6: set @ B,B2: set @ B] :
% 0.30/0.47 ( ( member @ A @ X @ C5 )
% 0.30/0.47 => ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
% 0.30/0.47 @ ( product_Sigma @ B @ A @ A6
% 0.30/0.47 @ ^ [Uu: B] : C5 )
% 0.30/0.47 @ ( product_Sigma @ B @ A @ B2
% 0.30/0.47 @ ^ [Uu: B] : C5 ) )
% 0.30/0.47 = ( ord_less_eq @ ( set @ B ) @ A6 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Times_subset_cancel2
% 0.30/0.47 thf(fact_178_Sigma__mono,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A6: set @ A,C5: set @ A,B2: A > ( set @ B ),D4: A > ( set @ B )] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ C5 )
% 0.30/0.47 => ( ! [X3: A] :
% 0.30/0.47 ( ( member @ A @ X3 @ A6 )
% 0.30/0.47 => ( ord_less_eq @ ( set @ B ) @ ( B2 @ X3 ) @ ( D4 @ X3 ) ) )
% 0.30/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A6 @ B2 ) @ ( product_Sigma @ A @ B @ C5 @ D4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Sigma_mono
% 0.30/0.47 thf(fact_179_Pair__mono,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( ord @ B )
% 0.30/0.47 & ( ord @ A ) )
% 0.30/0.47 => ! [X: A,X6: A,Y: B,Y7: B] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X @ X6 )
% 0.30/0.47 => ( ( ord_less_eq @ B @ Y @ Y7 )
% 0.30/0.47 => ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( product_Pair @ A @ B @ X6 @ Y7 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Pair_mono
% 0.30/0.47 thf(fact_180_image__mono,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A6: set @ A,B2: set @ A,F4: A > B] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
% 0.30/0.47 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ ( image @ A @ B @ F4 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_mono
% 0.30/0.47 thf(fact_181_image__subsetI,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A6: set @ A,F4: A > B,B2: set @ B] :
% 0.30/0.47 ( ! [X3: A] :
% 0.30/0.47 ( ( member @ A @ X3 @ A6 )
% 0.30/0.47 => ( member @ B @ ( F4 @ X3 ) @ B2 ) )
% 0.30/0.47 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_subsetI
% 0.30/0.47 thf(fact_182_subset__imageE,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,B2: set @ A,F4: B > A,A6: set @ B] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F4 @ A6 ) )
% 0.30/0.47 => ~ ! [C6: set @ B] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ B ) @ C6 @ A6 )
% 0.30/0.47 => ( B2
% 0.30/0.47 != ( image @ B @ A @ F4 @ C6 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_imageE
% 0.30/0.47 thf(fact_183_image__subset__iff,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,B2: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ B2 )
% 0.30/0.47 = ( ! [X4: B] :
% 0.30/0.47 ( ( member @ B @ X4 @ A6 )
% 0.30/0.47 => ( member @ A @ ( F4 @ X4 ) @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_subset_iff
% 0.30/0.47 thf(fact_184_subset__image__iff,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,B2: set @ A,F4: B > A,A6: set @ B] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F4 @ A6 ) )
% 0.30/0.47 = ( ? [AA: set @ B] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ B ) @ AA @ A6 )
% 0.30/0.47 & ( B2
% 0.30/0.47 = ( image @ B @ A @ F4 @ AA ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_image_iff
% 0.30/0.47 thf(fact_185_subrelI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.30/0.47 ( ! [X3: A,Y3: B] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
% 0.30/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) )
% 0.30/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 0.30/0.47
% 0.30/0.47 % subrelI
% 0.30/0.47 thf(fact_186_Collect__mono__iff,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
% 0.30/0.47 = ( ! [X4: A] :
% 0.30/0.47 ( ( P2 @ X4 )
% 0.30/0.47 => ( Q @ X4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_mono_iff
% 0.30/0.47 thf(fact_187_set__eq__subset,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ^ [Y5: set @ A,Z2: set @ A] : ( Y5 = Z2 ) )
% 0.30/0.47 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 0.30/0.47 & ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % set_eq_subset
% 0.30/0.47 thf(fact_188_subset__trans,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A,B2: set @ A,C5: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
% 0.30/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C5 )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ A6 @ C5 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_trans
% 0.30/0.47 thf(fact_189_Collect__mono,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.30/0.47 ( ! [X3: A] :
% 0.30/0.47 ( ( P2 @ X3 )
% 0.30/0.47 => ( Q @ X3 ) )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_mono
% 0.30/0.47 thf(fact_190_subset__refl,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ A6 ) ).
% 0.30/0.47
% 0.30/0.47 % subset_refl
% 0.30/0.47 thf(fact_191_subset__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.47 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.30/0.47 ! [T3: A] :
% 0.30/0.47 ( ( member @ A @ T3 @ A7 )
% 0.30/0.47 => ( member @ A @ T3 @ B6 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_iff
% 0.30/0.47 thf(fact_192_equalityD2,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A,B2: set @ A] :
% 0.30/0.47 ( ( A6 = B2 )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ B2 @ A6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % equalityD2
% 0.30/0.47 thf(fact_193_equalityD1,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A,B2: set @ A] :
% 0.30/0.47 ( ( A6 = B2 )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ A6 @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % equalityD1
% 0.30/0.47 thf(fact_194_subset__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.47 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.30/0.47 ! [X4: A] :
% 0.30/0.47 ( ( member @ A @ X4 @ A7 )
% 0.30/0.47 => ( member @ A @ X4 @ B6 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_eq
% 0.30/0.47 thf(fact_195_equalityE,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A,B2: set @ A] :
% 0.30/0.47 ( ( A6 = B2 )
% 0.30/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
% 0.30/0.47 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % equalityE
% 0.30/0.47 thf(fact_196_subsetD,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A,B2: set @ A,C3: A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
% 0.30/0.47 => ( ( member @ A @ C3 @ A6 )
% 0.30/0.47 => ( member @ A @ C3 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subsetD
% 0.30/0.47 thf(fact_197_in__mono,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A,B2: set @ A,X: A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
% 0.30/0.47 => ( ( member @ A @ X @ A6 )
% 0.30/0.47 => ( member @ A @ X @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % in_mono
% 0.30/0.47 thf(fact_198_Diff__mono,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A,C5: set @ A,D4: set @ A,B2: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ C5 )
% 0.30/0.47 => ( ( ord_less_eq @ ( set @ A ) @ D4 @ B2 )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) @ ( minus_minus @ ( set @ A ) @ C5 @ D4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Diff_mono
% 0.30/0.47 thf(fact_199_Diff__subset,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) @ A6 ) ).
% 0.30/0.47
% 0.30/0.47 % Diff_subset
% 0.30/0.47 thf(fact_200_double__diff,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A,B2: set @ A,C5: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
% 0.30/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C5 )
% 0.30/0.47 => ( ( minus_minus @ ( set @ A ) @ B2 @ ( minus_minus @ ( set @ A ) @ C5 @ A6 ) )
% 0.30/0.47 = A6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % double_diff
% 0.30/0.47 thf(fact_201_subset__UNIV,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_UNIV
% 0.30/0.47 thf(fact_202_top_Oextremum__uniqueI,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order_top @ A )
% 0.30/0.47 => ! [A2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 0.30/0.47 => ( A2
% 0.30/0.47 = ( top_top @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % top.extremum_uniqueI
% 0.30/0.47 thf(fact_203_top_Oextremum__unique,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order_top @ A )
% 0.30/0.47 => ! [A2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 0.30/0.47 = ( A2
% 0.30/0.47 = ( top_top @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % top.extremum_unique
% 0.30/0.47 thf(fact_204_top__greatest,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order_top @ A )
% 0.30/0.47 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % top_greatest
% 0.30/0.47 thf(fact_205_prop__restrict,axiom,
% 0.30/0.47 ! [A: $tType,X: A,Z4: set @ A,X7: set @ A,P2: A > $o] :
% 0.30/0.47 ( ( member @ A @ X @ Z4 )
% 0.30/0.47 => ( ( ord_less_eq @ ( set @ A ) @ Z4
% 0.30/0.47 @ ( collect @ A
% 0.30/0.47 @ ^ [X4: A] :
% 0.30/0.47 ( ( member @ A @ X4 @ X7 )
% 0.30/0.47 & ( P2 @ X4 ) ) ) )
% 0.30/0.47 => ( P2 @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % prop_restrict
% 0.30/0.47 thf(fact_206_Collect__restrict,axiom,
% 0.30/0.47 ! [A: $tType,X7: set @ A,P2: A > $o] :
% 0.30/0.47 ( ord_less_eq @ ( set @ A )
% 0.30/0.47 @ ( collect @ A
% 0.30/0.47 @ ^ [X4: A] :
% 0.30/0.47 ( ( member @ A @ X4 @ X7 )
% 0.30/0.47 & ( P2 @ X4 ) ) )
% 0.30/0.47 @ X7 ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_restrict
% 0.30/0.47 thf(fact_207_Collect__subset,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A,P2: A > $o] :
% 0.30/0.47 ( ord_less_eq @ ( set @ A )
% 0.30/0.47 @ ( collect @ A
% 0.30/0.47 @ ^ [X4: A] :
% 0.30/0.47 ( ( member @ A @ X4 @ A6 )
% 0.30/0.47 & ( P2 @ X4 ) ) )
% 0.30/0.47 @ A6 ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_subset
% 0.30/0.47 thf(fact_208_image__Collect__subsetI,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,P2: A > $o,F4: A > B,B2: set @ B] :
% 0.30/0.47 ( ! [X3: A] :
% 0.30/0.47 ( ( P2 @ X3 )
% 0.30/0.47 => ( member @ B @ ( F4 @ X3 ) @ B2 ) )
% 0.30/0.47 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ ( collect @ A @ P2 ) ) @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_Collect_subsetI
% 0.30/0.47 thf(fact_209_subset__Collect__iff,axiom,
% 0.30/0.47 ! [A: $tType,B2: set @ A,A6: set @ A,P2: A > $o] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A6 )
% 0.30/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B2
% 0.30/0.47 @ ( collect @ A
% 0.30/0.47 @ ^ [X4: A] :
% 0.30/0.47 ( ( member @ A @ X4 @ A6 )
% 0.30/0.47 & ( P2 @ X4 ) ) ) )
% 0.30/0.47 = ( ! [X4: A] :
% 0.30/0.47 ( ( member @ A @ X4 @ B2 )
% 0.30/0.47 => ( P2 @ X4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_Collect_iff
% 0.30/0.47 thf(fact_210_subset__CollectI,axiom,
% 0.30/0.47 ! [A: $tType,B2: set @ A,A6: set @ A,Q: A > $o,P2: A > $o] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A6 )
% 0.30/0.47 => ( ! [X3: A] :
% 0.30/0.47 ( ( member @ A @ X3 @ B2 )
% 0.30/0.47 => ( ( Q @ X3 )
% 0.30/0.47 => ( P2 @ X3 ) ) )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A )
% 0.30/0.47 @ ( collect @ A
% 0.30/0.47 @ ^ [X4: A] :
% 0.30/0.47 ( ( member @ A @ X4 @ B2 )
% 0.30/0.47 & ( Q @ X4 ) ) )
% 0.30/0.47 @ ( collect @ A
% 0.30/0.47 @ ^ [X4: A] :
% 0.30/0.47 ( ( member @ A @ X4 @ A6 )
% 0.30/0.47 & ( P2 @ X4 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_CollectI
% 0.30/0.47 thf(fact_211_image__diff__subset,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,B2: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ ( image @ B @ A @ F4 @ B2 ) ) @ ( image @ B @ A @ F4 @ ( minus_minus @ ( set @ B ) @ A6 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_diff_subset
% 0.30/0.47 thf(fact_212_subset__snd__imageI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A6: set @ A,B2: set @ B,S3: set @ ( product_prod @ A @ B ),X: A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
% 0.30/0.47 @ ( product_Sigma @ A @ B @ A6
% 0.30/0.47 @ ^ [Uu: A] : B2 )
% 0.30/0.47 @ S3 )
% 0.30/0.47 => ( ( member @ A @ X @ A6 )
% 0.30/0.47 => ( ord_less_eq @ ( set @ B ) @ B2 @ ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_snd_imageI
% 0.30/0.47 thf(fact_213_subset__fst__imageI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A6: set @ A,B2: set @ B,S3: set @ ( product_prod @ A @ B ),Y: B] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
% 0.30/0.47 @ ( product_Sigma @ A @ B @ A6
% 0.30/0.47 @ ^ [Uu: A] : B2 )
% 0.30/0.47 @ S3 )
% 0.30/0.47 => ( ( member @ B @ Y @ B2 )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ A6 @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_fst_imageI
% 0.30/0.47 thf(fact_214_range__subsetD,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,F4: B > A,B2: set @ A,I3: B] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) @ B2 )
% 0.30/0.47 => ( member @ A @ ( F4 @ I3 ) @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % range_subsetD
% 0.30/0.47 thf(fact_215_pred__subset__eq,axiom,
% 0.30/0.47 ! [A: $tType,R2: set @ A,S3: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( A > $o )
% 0.30/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ R2 )
% 0.30/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ S3 ) )
% 0.30/0.47 = ( ord_less_eq @ ( set @ A ) @ R2 @ S3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % pred_subset_eq
% 0.30/0.47 thf(fact_216_less__eq__set__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.47 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.30/0.47 ( ord_less_eq @ ( A > $o )
% 0.30/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ A7 )
% 0.30/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_eq_set_def
% 0.30/0.47 thf(fact_217_pred__subset__eq2,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 0.30/0.47 ( ( ord_less_eq @ ( A > B > $o )
% 0.30/0.47 @ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R2 )
% 0.30/0.47 @ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S3 ) )
% 0.30/0.47 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % pred_subset_eq2
% 0.30/0.47 thf(fact_218_Gr__incl,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A6: set @ A,F4: A > B,B2: set @ B] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( bNF_Gr @ A @ B @ A6 @ F4 )
% 0.30/0.47 @ ( product_Sigma @ A @ B @ A6
% 0.30/0.47 @ ^ [Uu: A] : B2 ) )
% 0.30/0.47 = ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Gr_incl
% 0.30/0.47 thf(fact_219_all__subset__image,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P2: ( set @ A ) > $o] :
% 0.30/0.47 ( ( ! [B6: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) )
% 0.30/0.47 => ( P2 @ B6 ) ) )
% 0.30/0.47 = ( ! [B6: set @ B] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ B ) @ B6 @ A6 )
% 0.30/0.47 => ( P2 @ ( image @ B @ A @ F4 @ B6 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % all_subset_image
% 0.30/0.47 thf(fact_220_eq__subset,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > A > $o] :
% 0.30/0.47 ( ord_less_eq @ ( A > A > $o )
% 0.30/0.47 @ ^ [Y5: A,Z2: A] : ( Y5 = Z2 )
% 0.30/0.47 @ ^ [A5: A,B7: A] :
% 0.30/0.47 ( ( P2 @ A5 @ B7 )
% 0.30/0.47 | ( A5 = B7 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eq_subset
% 0.30/0.47 thf(fact_221_GrD1,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,X: A,Fx: B,A6: set @ A,F4: A > B] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A6 @ F4 ) )
% 0.30/0.47 => ( member @ A @ X @ A6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % GrD1
% 0.30/0.47 thf(fact_222_GrD2,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,X: A,Fx: B,A6: set @ A,F4: A > B] :
% 0.30/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A6 @ F4 ) )
% 0.30/0.47 => ( ( F4 @ X )
% 0.30/0.47 = Fx ) ) ).
% 0.30/0.47
% 0.30/0.47 % GrD2
% 0.30/0.47 thf(fact_223_diff__eq__diff__less__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [A2: A,B3: A,C3: A,D3: A] :
% 0.30/0.47 ( ( ( minus_minus @ A @ A2 @ B3 )
% 0.30/0.47 = ( minus_minus @ A @ C3 @ D3 ) )
% 0.30/0.47 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.30/0.47 = ( ord_less_eq @ A @ C3 @ D3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_eq_diff_less_eq
% 0.30/0.47 thf(fact_224_diff__right__mono,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [A2: A,B3: A,C3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B3 @ C3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_right_mono
% 0.30/0.47 thf(fact_225_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( cancel146912293up_add @ A )
% 0.30/0.47 => ! [A2: A,C3: A,B3: A] :
% 0.30/0.47 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C3 ) @ B3 )
% 0.30/0.47 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B3 ) @ C3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % cancel_ab_semigroup_add_class.diff_right_commute
% 0.30/0.47 thf(fact_226_diff__eq__diff__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [A2: A,B3: A,C3: A,D3: A] :
% 0.30/0.47 ( ( ( minus_minus @ A @ A2 @ B3 )
% 0.30/0.47 = ( minus_minus @ A @ C3 @ D3 ) )
% 0.30/0.47 => ( ( A2 = B3 )
% 0.30/0.47 = ( C3 = D3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_eq_diff_eq
% 0.30/0.47 thf(fact_227_diff__mono,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [A2: A,B3: A,D3: A,C3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ D3 @ C3 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B3 @ D3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_mono
% 0.30/0.47 thf(fact_228_diff__left__mono,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [B3: A,A2: A,C3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B3 @ A2 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ C3 @ A2 ) @ ( minus_minus @ A @ C3 @ B3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_left_mono
% 0.30/0.47 thf(fact_229_snd__image__times,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A6: set @ B,B2: set @ A] :
% 0.30/0.47 ( ( ( A6
% 0.30/0.47 = ( bot_bot @ ( set @ B ) ) )
% 0.30/0.47 => ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
% 0.30/0.47 @ ( product_Sigma @ B @ A @ A6
% 0.30/0.47 @ ^ [Uu: B] : B2 ) )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 & ( ( A6
% 0.30/0.47 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.47 => ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
% 0.30/0.47 @ ( product_Sigma @ B @ A @ A6
% 0.30/0.47 @ ^ [Uu: B] : B2 ) )
% 0.30/0.47 = B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % snd_image_times
% 0.30/0.47 thf(fact_230_fst__image__times,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,B2: set @ B,A6: set @ A] :
% 0.30/0.47 ( ( ( B2
% 0.30/0.47 = ( bot_bot @ ( set @ B ) ) )
% 0.30/0.47 => ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
% 0.30/0.47 @ ( product_Sigma @ A @ B @ A6
% 0.30/0.47 @ ^ [Uu: A] : B2 ) )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.47 & ( ( B2
% 0.30/0.47 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.47 => ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
% 0.30/0.47 @ ( product_Sigma @ A @ B @ A6
% 0.30/0.47 @ ^ [Uu: A] : B2 ) )
% 0.30/0.47 = A6 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fst_image_times
% 0.30/0.47 thf(fact_231_empty__Collect__eq,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o] :
% 0.30/0.47 ( ( ( bot_bot @ ( set @ A ) )
% 0.30/0.47 = ( collect @ A @ P2 ) )
% 0.30/0.47 = ( ! [X4: A] :
% 0.30/0.47 ~ ( P2 @ X4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % empty_Collect_eq
% 0.30/0.47 thf(fact_232_Collect__empty__eq,axiom,
% 0.30/0.47 ! [A: $tType,P2: A > $o] :
% 0.30/0.47 ( ( ( collect @ A @ P2 )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( ! [X4: A] :
% 0.30/0.47 ~ ( P2 @ X4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_empty_eq
% 0.30/0.47 thf(fact_233_all__not__in__conv,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A] :
% 0.30/0.47 ( ( ! [X4: A] :
% 0.30/0.47 ~ ( member @ A @ X4 @ A6 ) )
% 0.30/0.47 = ( A6
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % all_not_in_conv
% 0.30/0.47 thf(fact_234_empty__iff,axiom,
% 0.30/0.47 ! [A: $tType,C3: A] :
% 0.30/0.47 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % empty_iff
% 0.30/0.47 thf(fact_235_image__is__empty,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B] :
% 0.30/0.47 ( ( ( image @ B @ A @ F4 @ A6 )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( A6
% 0.30/0.47 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_is_empty
% 0.30/0.47 thf(fact_236_empty__is__image,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B] :
% 0.30/0.47 ( ( ( bot_bot @ ( set @ A ) )
% 0.30/0.47 = ( image @ B @ A @ F4 @ A6 ) )
% 0.30/0.47 = ( A6
% 0.30/0.47 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % empty_is_image
% 0.30/0.47 thf(fact_237_image__empty,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,F4: B > A] :
% 0.30/0.47 ( ( image @ B @ A @ F4 @ ( bot_bot @ ( set @ B ) ) )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_empty
% 0.30/0.47 thf(fact_238_subset__empty,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( A6
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_empty
% 0.30/0.47 thf(fact_239_empty__subsetI,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A6 ) ).
% 0.30/0.47
% 0.30/0.47 % empty_subsetI
% 0.30/0.47 thf(fact_240_Diff__empty,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A] :
% 0.30/0.47 ( ( minus_minus @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = A6 ) ).
% 0.30/0.47
% 0.30/0.47 % Diff_empty
% 0.30/0.47 thf(fact_241_empty__Diff,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A] :
% 0.30/0.47 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A6 )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % empty_Diff
% 0.30/0.47 thf(fact_242_Diff__cancel,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A] :
% 0.30/0.47 ( ( minus_minus @ ( set @ A ) @ A6 @ A6 )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Diff_cancel
% 0.30/0.47 thf(fact_243_Sigma__empty1,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,B2: A > ( set @ B )] :
% 0.30/0.47 ( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B2 )
% 0.30/0.47 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Sigma_empty1
% 0.30/0.47 thf(fact_244_Collect__const,axiom,
% 0.30/0.47 ! [A: $tType,P2: $o] :
% 0.30/0.47 ( ( P2
% 0.30/0.47 => ( ( collect @ A
% 0.30/0.47 @ ^ [S4: A] : P2 )
% 0.30/0.47 = ( top_top @ ( set @ A ) ) ) )
% 0.30/0.47 & ( ~ P2
% 0.30/0.47 => ( ( collect @ A
% 0.30/0.47 @ ^ [S4: A] : P2 )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_const
% 0.30/0.47 thf(fact_245_Times__empty,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A6: set @ A,B2: set @ B] :
% 0.30/0.47 ( ( ( product_Sigma @ A @ B @ A6
% 0.30/0.47 @ ^ [Uu: A] : B2 )
% 0.30/0.47 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.30/0.47 = ( ( A6
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 | ( B2
% 0.30/0.47 = ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Times_empty
% 0.30/0.47 thf(fact_246_Sigma__empty2,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,A6: set @ A] :
% 0.30/0.47 ( ( product_Sigma @ A @ B @ A6
% 0.30/0.47 @ ^ [Uu: A] : ( bot_bot @ ( set @ B ) ) )
% 0.30/0.47 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Sigma_empty2
% 0.30/0.47 thf(fact_247_Diff__UNIV,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A] :
% 0.30/0.47 ( ( minus_minus @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Diff_UNIV
% 0.30/0.47 thf(fact_248_Diff__eq__empty__iff,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A,B2: set @ A] :
% 0.30/0.47 ( ( ( minus_minus @ ( set @ A ) @ A6 @ B2 )
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 = ( ord_less_eq @ ( set @ A ) @ A6 @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Diff_eq_empty_iff
% 0.30/0.47 thf(fact_249_times__eq__iff,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A6: set @ A,B2: set @ B,C5: set @ A,D4: set @ B] :
% 0.30/0.47 ( ( ( product_Sigma @ A @ B @ A6
% 0.30/0.47 @ ^ [Uu: A] : B2 )
% 0.30/0.47 = ( product_Sigma @ A @ B @ C5
% 0.30/0.47 @ ^ [Uu: A] : D4 ) )
% 0.30/0.47 = ( ( ( A6 = C5 )
% 0.30/0.47 & ( B2 = D4 ) )
% 0.30/0.47 | ( ( ( A6
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 | ( B2
% 0.30/0.47 = ( bot_bot @ ( set @ B ) ) ) )
% 0.30/0.47 & ( ( C5
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 | ( D4
% 0.30/0.47 = ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % times_eq_iff
% 0.30/0.47 thf(fact_250_Sigma__empty__iff,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,I: set @ A,X7: A > ( set @ B )] :
% 0.30/0.47 ( ( ( product_Sigma @ A @ B @ I @ X7 )
% 0.30/0.47 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.30/0.47 = ( ! [X4: A] :
% 0.30/0.47 ( ( member @ A @ X4 @ I )
% 0.30/0.47 => ( ( X7 @ X4 )
% 0.30/0.47 = ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Sigma_empty_iff
% 0.30/0.47 thf(fact_251_bot__prod__def,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( bot @ B )
% 0.30/0.47 & ( bot @ A ) )
% 0.30/0.47 => ( ( bot_bot @ ( product_prod @ A @ B ) )
% 0.30/0.47 = ( product_Pair @ A @ B @ ( bot_bot @ A ) @ ( bot_bot @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bot_prod_def
% 0.30/0.47 thf(fact_252_ex__in__conv,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A] :
% 0.30/0.47 ( ( ? [X4: A] : ( member @ A @ X4 @ A6 ) )
% 0.30/0.47 = ( A6
% 0.30/0.47 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ex_in_conv
% 0.30/0.47 thf(fact_253_equals0I,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A] :
% 0.30/0.47 ( ! [Y3: A] :
% 0.30/0.47 ~ ( member @ A @ Y3 @ A6 )
% 0.30/0.47 => ( A6
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % equals0I
% 0.30/0.47 thf(fact_254_equals0D,axiom,
% 0.30/0.47 ! [A: $tType,A6: set @ A,A2: A] :
% 0.30/0.47 ( ( A6
% 0.30/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.47 => ~ ( member @ A @ A2 @ A6 ) ) ).
% 0.30/0.47
% 0.30/0.47 % equals0D
% 0.30/0.47 thf(fact_255_emptyE,axiom,
% 0.30/0.47 ! [A: $tType,A2: A] :
% 0.30/0.47 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % emptyE
% 0.30/0.47
% 0.30/0.47 % Type constructors (30)
% 0.30/0.47 thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
% 0.30/0.47 ! [A9: $tType,A10: $tType] :
% 0.30/0.47 ( ( ( ordere890947078_space @ A9 )
% 0.30/0.47 & ( ordere890947078_space @ A10 ) )
% 0.30/0.47 => ( ordere890947078_space @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.30/0.47 ! [A9: $tType,A10: $tType] :
% 0.30/0.47 ( ( order_top @ A10 )
% 0.30/0.47 => ( order_top @ ( A9 > A10 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Orderings_Otop,axiom,
% 0.30/0.47 ! [A9: $tType,A10: $tType] :
% 0.30/0.47 ( ( top @ A10 )
% 0.30/0.47 => ( top @ ( A9 > A10 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.30/0.47 ! [A9: $tType,A10: $tType] :
% 0.30/0.47 ( ( ord @ A10 )
% 0.30/0.47 => ( ord @ ( A9 > A10 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Orderings_Obot,axiom,
% 0.30/0.47 ! [A9: $tType,A10: $tType] :
% 0.30/0.47 ( ( bot @ A10 )
% 0.30/0.47 => ( bot @ ( A9 > A10 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Groups_Ominus,axiom,
% 0.30/0.47 ! [A9: $tType,A10: $tType] :
% 0.30/0.47 ( ( minus @ A10 )
% 0.30/0.47 => ( minus @ ( A9 > A10 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Orderings_Oorder__top_1,axiom,
% 0.30/0.47 ! [A9: $tType] : ( order_top @ ( set @ A9 ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Orderings_Otop_2,axiom,
% 0.30/0.47 ! [A9: $tType] : ( top @ ( set @ A9 ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
% 0.30/0.47 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Orderings_Obot_4,axiom,
% 0.30/0.47 ! [A9: $tType] : ( bot @ ( set @ A9 ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Groups_Ominus_5,axiom,
% 0.30/0.47 ! [A9: $tType] : ( minus @ ( set @ A9 ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Oorder__top_6,axiom,
% 0.30/0.47 order_top @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Otop_7,axiom,
% 0.30/0.47 top @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Oord_8,axiom,
% 0.30/0.47 ord @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Obot_9,axiom,
% 0.30/0.47 bot @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Groups_Ominus_10,axiom,
% 0.30/0.47 minus @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Oprod___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.30/0.47 ! [A9: $tType,A10: $tType] :
% 0.30/0.47 ( ( ( cancel146912293up_add @ A9 )
% 0.30/0.47 & ( cancel146912293up_add @ A10 ) )
% 0.30/0.47 => ( cancel146912293up_add @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Oprod___Groups_Oordered__ab__group__add,axiom,
% 0.30/0.47 ! [A9: $tType,A10: $tType] :
% 0.30/0.47 ( ( ( ordere890947078_space @ A9 )
% 0.30/0.47 & ( ordere890947078_space @ A10 ) )
% 0.30/0.47 => ( ordered_ab_group_add @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Oprod___Orderings_Oorder__top_11,axiom,
% 0.30/0.47 ! [A9: $tType,A10: $tType] :
% 0.30/0.47 ( ( ( order_top @ A9 )
% 0.30/0.47 & ( order_top @ A10 ) )
% 0.30/0.47 => ( order_top @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Oprod___Groups_Oab__group__add,axiom,
% 0.30/0.47 ! [A9: $tType,A10: $tType] :
% 0.30/0.47 ( ( ( ab_group_add @ A9 )
% 0.30/0.47 & ( ab_group_add @ A10 ) )
% 0.30/0.47 => ( ab_group_add @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Oprod___Groups_Ogroup__add,axiom,
% 0.30/0.47 ! [A9: $tType,A10: $tType] :
% 0.30/0.47 ( ( ( group_add @ A9 )
% 0.30/0.47 & ( group_add @ A10 ) )
% 0.30/0.47 => ( group_add @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Oprod___Orderings_Otop_12,axiom,
% 0.30/0.47 ! [A9: $tType,A10: $tType] :
% 0.30/0.47 ( ( ( top @ A9 )
% 0.30/0.47 & ( top @ A10 ) )
% 0.30/0.47 => ( top @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Oprod___Orderings_Oord_13,axiom,
% 0.30/0.47 ! [A9: $tType,A10: $tType] :
% 0.30/0.47 ( ( ( ord @ A9 )
% 0.30/0.47 & ( ord @ A10 ) )
% 0.30/0.47 => ( ord @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Oprod___Orderings_Obot_14,axiom,
% 0.30/0.47 ! [A9: $tType,A10: $tType] :
% 0.30/0.47 ( ( ( bot @ A9 )
% 0.30/0.47 & ( bot @ A10 ) )
% 0.30/0.47 => ( bot @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Oprod___Groups_Ominus_15,axiom,
% 0.30/0.47 ! [A9: $tType,A10: $tType] :
% 0.30/0.47 ( ( ( minus @ A9 )
% 0.30/0.47 & ( minus @ A10 ) )
% 0.30/0.47 => ( minus @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_16,axiom,
% 0.30/0.47 order_top @ product_unit ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Ounit___Orderings_Otop_17,axiom,
% 0.30/0.47 top @ product_unit ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Ounit___Orderings_Oord_18,axiom,
% 0.30/0.47 ord @ product_unit ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Ounit___Orderings_Obot_19,axiom,
% 0.30/0.47 bot @ product_unit ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Product__Type_Ounit___Groups_Ominus_20,axiom,
% 0.30/0.47 minus @ product_unit ).
% 0.30/0.47
% 0.30/0.47 % Helper facts (3)
% 0.30/0.47 thf(help_If_3_1_T,axiom,
% 0.30/0.47 ! [P2: $o] :
% 0.30/0.47 ( ( P2 = $true )
% 0.30/0.47 | ( P2 = $false ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(help_If_2_1_T,axiom,
% 0.30/0.47 ! [A: $tType,X: A,Y: A] :
% 0.30/0.47 ( ( if @ A @ $false @ X @ Y )
% 0.30/0.47 = Y ) ).
% 0.30/0.47
% 0.30/0.47 thf(help_If_1_1_T,axiom,
% 0.30/0.47 ! [A: $tType,X: A,Y: A] :
% 0.30/0.47 ( ( if @ A @ $true @ X @ Y )
% 0.30/0.47 = X ) ).
% 0.30/0.47
% 0.30/0.47 % Conjectures (1)
% 0.30/0.47 thf(conj_0,conjecture,
% 0.30/0.47 member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ z @ y ) @ pr ).
% 0.30/0.47
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 ------------------------
% 2.93/1.52 ---- Embedded in THF ---
% 2.93/1.53 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.93/1.53 %%% Generated on Tue Feb 24 08:28:15 EST 2026
% 2.93/1.53 %%% using '$$dhol' embedding, version 1.3.0.
% 2.93/1.53 %%% Logic specification used:
% 2.93/1.53 %%% thf(spec, logic, $$dhol).
% 2.93/1.53
% 2.93/1.53 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.9NETwVAXmU/DTF2DTF17895.p
% 2.93/1.53 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 2.93/1.53 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.93/1.53 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.93/1.53 thf(ty_tf_a, type, a: $tType).
% 2.93/1.53 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space, type, ordere890947078_space: (!> [A:$tType]: ($o))).
% 2.93/1.53 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.93/1.53 thf(sy_cl_Groups_Ominus, type, minus: (!> [A:$tType]: ($o))).
% 2.93/1.53 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 2.93/1.53 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.93/1.53 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 2.93/1.53 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.93/1.53 thf(sy_cl_Groups_Oab__group__add, type, ab_group_add: (!> [A:$tType]: ($o))).
% 2.93/1.53 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: ($o))).
% 2.93/1.53 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 2.93/1.53 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: ($o))).
% 2.93/1.53 thf(sy_c_BNF__Def_OGr, type, bNF_Gr: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (set @ ((product_prod @ A) @ B))))))).
% 2.93/1.53 thf(sy_c_BNF__Def_OfstOp, type, bNF_fstOp: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > $o)) > ((B > (C > $o)) > (((product_prod @ A) @ C) > ((product_prod @ A) @ B))))))).
% 2.93/1.53 thf(sy_c_BNF__Def_Opick__middlep, type, bNF_pick_middlep: (!> [B:$tType,A:$tType,C:$tType]: (((B > (A > $o)) > ((A > (C > $o)) > (B > (C > A))))))).
% 2.93/1.53 thf(sy_c_BNF__Def_OsndOp, type, bNF_sndOp: (!> [C:$tType,A:$tType,B:$tType]: (((C > (A > $o)) > ((A > (B > $o)) > (((product_prod @ C) @ B) > ((product_prod @ A) @ B))))))).
% 2.93/1.53 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2, type, bNF_Greatest_image2: (!> [C:$tType,A:$tType,B:$tType]: (((set @ C) > ((C > A) > ((C > B) > (set @ ((product_prod @ A) @ B)))))))).
% 2.93/1.53 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.93/1.53 thf(sy_c_HOL_OThe, type, the: (!> [A:$tType]: (((A > $o) > A)))).
% 2.93/1.53 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.93/1.53 thf(sy_c_Order__Relation_Opreorder__on, type, order_preorder_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.93/1.53 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.93/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.93/1.53 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.93/1.53 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oas__good__as, type, prefer951318096ood_as: (!> [A:$tType]: ((A > ((set @ A) > ((set @ ((product_prod @ A) @ A)) > (set @ A))))))).
% 2.93/1.53 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oat__least__as__good, type, prefer310429814s_good: (!> [A:$tType]: ((A > ((set @ A) > ((set @ ((product_prod @ A) @ A)) > (set @ A))))))).
% 2.93/1.53 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Ono__better__than, type, prefer1532642881r_than: (!> [A:$tType]: ((A > ((set @ A) > ((set @ ((product_prod @ A) @ A)) > (set @ A))))))).
% 2.93/1.53 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Opreference, type, prefer199794634erence: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.93/1.53 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.93/1.53 thf(sy_c_Product__Type_OSigma, type, product_Sigma: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (set @ B)) > (set @ ((product_prod @ A) @ B))))))).
% 2.93/1.53 thf(sy_c_Product__Type_Ocurry, type, product_curry: (!> [A:$tType,B:$tType,C:$tType]: (((((product_prod @ A) @ B) > C) > (A > (B > C)))))).
% 2.93/1.53 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.93/1.53 thf(sy_c_Product__Type_Omap__prod, type, product_map_prod: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > C) > ((B > D) > (((product_prod @ A) @ B) > ((product_prod @ C) @ D))))))).
% 2.93/1.53 thf(sy_c_Product__Type_Oold_Obool_Orec__bool, type, product_rec_bool: (!> [T:$tType]: ((T > (T > ($o > T)))))).
% 2.93/1.53 thf(sy_c_Product__Type_Oold_Obool_Orec__set__bool, type, product_rec_set_bool: (!> [T:$tType]: ((T > (T > ($o > (T > $o))))))).
% 2.93/1.53 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.93/1.53 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod, type, product_rec_set_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > (T > $o)))))).
% 2.93/1.53 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit, type, product_rec_set_unit: (!> [T:$tType]: ((T > (product_unit > (T > $o)))))).
% 2.93/1.53 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit, type, product_rec_unit: (!> [T:$tType]: ((T > (product_unit > T))))).
% 2.93/1.53 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))).
% 2.93/1.53 thf(sy_c_Product__Type_Oprod_Osnd, type, product_snd: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > B)))).
% 2.93/1.53 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 2.93/1.53 thf(sy_c_Product__Type_Oproduct, type, product_product: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ B) > (set @ ((product_prod @ A) @ B))))))).
% 2.93/1.53 thf(sy_c_Product__Type_Oscomp, type, product_scomp: (!> [A:$tType,B:$tType,C:$tType,D:$tType]: (((A > ((product_prod @ B) @ C)) > ((B > (C > D)) > (A > D)))))).
% 2.93/1.53 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.93/1.53 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 2.93/1.53 thf(sy_c_Wellfounded_Olex__prod, type, lex_prod: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ B) @ B)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 2.93/1.53 thf(sy_c_Wfrec_Osame__fst, type, same_fst: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > (set @ ((product_prod @ B) @ B))) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 2.93/1.53 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.93/1.53 thf(sy_v_B, type, b: (set @ a)).
% 2.93/1.53 thf(sy_v_Pr, type, pr: (set @ ((product_prod @ a) @ a))).
% 2.93/1.53 thf(sy_v_y, type, y: a).
% 2.93/1.53 thf(sy_v_z, type, z: a).
% 2.93/1.53 thf(fact_0_assms, axiom, (((member @ a) @ z) @ ((((prefer310429814s_good @ a) @ y) @ b) @ pr))).
% 2.93/1.53 thf(fact_1_at__lst__asgd__ge, axiom, (! [A:$tType,X:A,Y:A,B2:(set @ A),Pr:(set @ ((product_prod @ A) @ A))]: (((((member @ A) @ X) @ ((((prefer310429814s_good @ A) @ Y) @ B2) @ Pr)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Pr))))).
% 2.93/1.53 thf(fact_2_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X2:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X2) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X2 = Y2)))))).
% 2.93/1.53 thf(fact_3_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B3:B,A3:A,B4:B]: (((((((product_Pair @ A) @ B) @ A2) @ B3) = ((((product_Pair @ A) @ B) @ A3) @ B4)) = ((A2 = A3) & (B3 = B4)))))).
% 2.93/1.53 thf(fact_4_surj__pair, axiom, (! [A:$tType,B:$tType,P:((product_prod @ A) @ B)]: ((? [X3:A,Y3:B]: ((P = ((((product_Pair @ A) @ B) @ X3) @ Y3))))))).
% 2.93/1.53 thf(fact_5_prod__cases, axiom, (! [B:$tType,A:$tType,P2:(((product_prod @ A) @ B) > $o),P:((product_prod @ A) @ B)]: (((! [A4:A,B5:B]: ((P2 @ ((((product_Pair @ A) @ B) @ A4) @ B5)))) => (P2 @ P))))).
% 2.93/1.53 thf(fact_6_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B3:B,A3:A,B4:B]: (((((((product_Pair @ A) @ B) @ A2) @ B3) = ((((product_Pair @ A) @ B) @ A3) @ B4)) => (~ (((A2 = A3) => (B3 != B4)))))))).
% 2.93/1.53 thf(fact_7_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A4:A,B5:B,C2:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Pair @ B) @ C) @ B5) @ C2)))))))))).
% 2.93/1.53 thf(fact_8_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A4:A,B5:B,C2:C,D2:D]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C2) @ D2))))))))))).
% 2.93/1.53 thf(fact_9_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A4:A,B5:B,C2:C,D2:D,E2:E]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 2.93/1.53 thf(fact_10_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: ((~ ((! [A4:A,B5:B,C2:C,D2:D,E2:E,F2:F]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2))))))))))))).
% 2.93/1.53 thf(fact_11_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,G:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))))]: ((~ ((! [A4:A,B5:B,C2:C,D2:D,E2:E,F2:F,G2:G]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G)) @ E2) @ ((((product_Pair @ F) @ G) @ F2) @ G2)))))))))))))).
% 2.93/1.53 thf(fact_12_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A4:A,B5:B,C2:C]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Pair @ B) @ C) @ B5) @ C2))))) => (P2 @ X))))).
% 2.93/1.53 thf(fact_13_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P2:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A4:A,B5:B]: ((P2 @ ((((product_Pair @ A) @ B) @ A4) @ B5)))) => (P2 @ Prod))))).
% 2.93/1.53 thf(fact_14_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A4:A,B5:B]: ((Y != ((((product_Pair @ A) @ B) @ A4) @ B5))))))))).
% 2.93/1.53 thf(fact_15_prod__induct7, axiom, (! [G:$tType,F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))))]: (((! [A4:A,B5:B,C2:C,D2:D,E2:E,F2:F,G2:G]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G)) @ E2) @ ((((product_Pair @ F) @ G) @ F2) @ G2))))))))) => (P2 @ X))))).
% 2.93/1.53 thf(fact_16_prod__induct6, axiom, (! [F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: (((! [A4:A,B5:B,C2:C,D2:D,E2:E,F2:F]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2)))))))) => (P2 @ X))))).
% 2.93/1.53 thf(fact_17_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A4:A,B5:B,C2:C,D2:D,E2:E]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P2 @ X))))).
% 2.93/1.53 thf(fact_18_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A4:A,B5:B,C2:C,D2:D]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C2) @ D2)))))) => (P2 @ X))))).
% 2.93/1.53 thf(fact_19_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B3:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B3)) = ((F1 @ A2) @ B3))))).
% 2.93/1.53 thf(fact_20_at__least__as__good__def, axiom, (! [A:$tType]: (((prefer310429814s_good @ A) = (^ [X4:A,B6:(set @ A),P3:(set @ ((product_prod @ A) @ A))]: (((collect @ A) @ (^ [Y4:A]: (((((member @ A) @ Y4) @ B6) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X4)) @ P3))))))))))).
% 2.93/1.53 thf(fact_21_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C3:(B > (C > A)),A2:B,B3:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C3) @ ((((product_Pair @ B) @ C) @ A2) @ B3)) = ((C3 @ A2) @ B3))))).
% 2.93/1.53 thf(fact_22_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R:A,S:B,R2:(set @ ((product_prod @ A) @ B)),S2:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S)) @ R2) => ((S2 = S) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S2)) @ R2)))))).
% 2.93/1.53 thf(fact_23_curry__conv, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_curry @ B) @ C) @ A) = (^ [F3:(((product_prod @ B) @ C) > A),A5:B,B7:C]: ((F3 @ ((((product_Pair @ B) @ C) @ A5) @ B7)))))))).
% 2.93/1.53 thf(fact_24_curryI, axiom, (! [A:$tType,B:$tType,F4:(((product_prod @ A) @ B) > $o),A2:A,B3:B]: (((F4 @ ((((product_Pair @ A) @ B) @ A2) @ B3)) => ((((((product_curry @ A) @ B) @ $o) @ F4) @ A2) @ B3))))).
% 2.93/1.53 thf(fact_25_preference_Oindiff__trans, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A,Z:A]: (((((prefer199794634erence @ A) @ Carrier) @ Relation) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ Relation)) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ Relation) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ Y)) @ Relation)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z)) @ Relation) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ X)) @ Relation)))))))).
% 2.93/1.53 thf(fact_26_preference_Onot__outside, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer199794634erence @ A) @ Carrier) @ Relation) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) => (((member @ A) @ X) @ Carrier)))))).
% 2.93/1.53 thf(fact_27_swap__simp, axiom, (! [A:$tType,B:$tType,X:B,Y:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y)) = ((((product_Pair @ A) @ B) @ Y) @ X))))).
% 2.93/1.53 thf(fact_28_old_Obool_Osimps_I6_J, axiom, (! [T:$tType,F1:T,F22:T]: ((((((product_rec_bool @ T) @ F1) @ F22) @ $false) = F22)))).
% 2.93/1.53 thf(fact_29_old_Obool_Osimps_I5_J, axiom, (! [T:$tType,F1:T,F22:T]: ((((((product_rec_bool @ T) @ F1) @ F22) @ $true) = F1)))).
% 2.93/1.53 thf(fact_30_swap__swap, axiom, (! [B:$tType,A:$tType,P:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P)) = P)))).
% 2.93/1.53 thf(fact_31_curry__K, axiom, (! [B:$tType,C:$tType,A:$tType,C3:C]: ((((((product_curry @ A) @ B) @ C) @ (^ [X4:((product_prod @ A) @ B)]: (C3))) = (^ [X4:A,Y4:B]: (C3)))))).
% 2.93/1.53 thf(fact_32_curry__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_curry @ A) @ B) @ C) = (^ [C4:(((product_prod @ A) @ B) > C),X4:A,Y4:B]: ((C4 @ ((((product_Pair @ A) @ B) @ X4) @ Y4)))))))).
% 2.93/1.53 thf(fact_33_curryD, axiom, (! [A:$tType,B:$tType,F4:(((product_prod @ A) @ B) > $o),A2:A,B3:B]: ((((((((product_curry @ A) @ B) @ $o) @ F4) @ A2) @ B3) => (F4 @ ((((product_Pair @ A) @ B) @ A2) @ B3)))))).
% 2.93/1.53 thf(fact_34_curryE, axiom, (! [A:$tType,B:$tType,F4:(((product_prod @ A) @ B) > $o),A2:A,B3:B]: ((((((((product_curry @ A) @ B) @ $o) @ F4) @ A2) @ B3) => (F4 @ ((((product_Pair @ A) @ B) @ A2) @ B3)))))).
% 2.93/1.53 thf(fact_35_no__better__than__def, axiom, (! [A:$tType]: (((prefer1532642881r_than @ A) = (^ [X4:A,B6:(set @ A),P3:(set @ ((product_prod @ A) @ A))]: (((collect @ A) @ (^ [Y4:A]: (((((member @ A) @ Y4) @ B6) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y4)) @ P3))))))))))).
% 2.93/1.53 thf(fact_36_as__good__as__def, axiom, (! [A:$tType]: (((prefer951318096ood_as @ A) = (^ [X4:A,B6:(set @ A),P3:(set @ ((product_prod @ A) @ A))]: (((collect @ A) @ (^ [Y4:A]: (((((member @ A) @ Y4) @ B6) & ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y4)) @ P3) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X4)) @ P3)))))))))))).
% 2.93/1.53 thf(fact_37_pred__equals__eq2, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ A) @ B))]: ((((^ [X4:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y4)) @ R2))) = (^ [X4:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y4)) @ S3)))) = (R2 = S3))))).
% 2.93/1.53 thf(fact_38_preference__def, axiom, (! [A:$tType]: (((prefer199794634erence @ A) = (^ [Carrier2:(set @ A),Relation2:(set @ ((product_prod @ A) @ A))]: (((! [X4:A,Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y4)) @ Relation2) => (((member @ A) @ X4) @ Carrier2)))) & ((! [X4:A,Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y4)) @ Relation2) => (((member @ A) @ Y4) @ Carrier2)))) & (((order_preorder_on @ A) @ Carrier2) @ Relation2))))))))).
% 2.93/1.53 thf(fact_39_preference_Ointro, axiom, (! [A:$tType,Relation:(set @ ((product_prod @ A) @ A)),Carrier:(set @ A)]: (((! [X3:A,Y3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ Relation) => (((member @ A) @ X3) @ Carrier)))) => ((! [X3:A,Y3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ Relation) => (((member @ A) @ Y3) @ Carrier)))) => ((((order_preorder_on @ A) @ Carrier) @ Relation) => (((prefer199794634erence @ A) @ Carrier) @ Relation))))))).
% 2.93/1.53 thf(fact_40_pair__in__swap__image, axiom, (! [A:$tType,B:$tType,Y:A,X:B,A6:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y) @ X)) @ ((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ A6)) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X) @ Y)) @ A6))))).
% 2.93/1.53 thf(fact_41_preference_Otrans__refl, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer199794634erence @ A) @ Carrier) @ Relation) => (((order_preorder_on @ A) @ Carrier) @ Relation))))).
% 2.93/1.53 thf(fact_42_strict__contour__is__diff, axiom, (! [A:$tType,B2:(set @ A),Y:A,Pr:(set @ ((product_prod @ A) @ A))]: ((((collect @ A) @ (^ [A5:A]: (((((member @ A) @ A5) @ B2) & ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ Y)) @ Pr) & (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ A5)) @ Pr)))))))) = (((minus_minus @ (set @ A)) @ ((((prefer310429814s_good @ A) @ Y) @ B2) @ Pr)) @ ((((prefer951318096ood_as @ A) @ Y) @ B2) @ Pr)))))).
% 2.93/1.53 thf(fact_43_old_Orec__prod__def, axiom, (! [T:$tType,B:$tType,A:$tType]: (((((product_rec_prod @ A) @ B) @ T) = (^ [F12:(A > (B > T)),X4:((product_prod @ A) @ B)]: (((the @ T) @ (((((product_rec_set_prod @ A) @ B) @ T) @ F12) @ X4)))))))).
% 2.93/1.53 thf(fact_44_old_Orec__bool__def, axiom, (! [T:$tType]: (((product_rec_bool @ T) = (^ [F12:T,F23:T,X4:$o]: (((the @ T) @ ((((product_rec_set_bool @ T) @ F12) @ F23) @ X4)))))))).
% 2.93/1.53 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P2:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P2)) = (P2 @ A2))))).
% 2.93/1.53 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A6:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A6)))) = A6)))).
% 2.93/1.53 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P2 @ X3) = (Q @ X3)))) => (((collect @ A) @ P2) = ((collect @ A) @ Q)))))).
% 2.93/1.53 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F4:(A > B),G3:(A > B)]: (((! [X3:A]: (((F4 @ X3) = (G3 @ X3)))) => (F4 = G3))))).
% 2.93/1.53 thf(fact_49_diff__Pair, axiom, (! [A:$tType,B:$tType]: ((((minus @ B) & (minus @ A)) => (! [A2:A,B3:B,C3:A,D3:B]: (((((minus_minus @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B3)) @ ((((product_Pair @ A) @ B) @ C3) @ D3)) = ((((product_Pair @ A) @ B) @ (((minus_minus @ A) @ A2) @ C3)) @ (((minus_minus @ B) @ B3) @ D3))))))))).
% 2.93/1.53 thf(fact_50_the__sym__eq__trivial, axiom, (! [A:$tType,X:A]: ((((the @ A) @ ((^ [Y5:A,Z2:A]: ((Y5 = Z2))) @ X)) = X)))).
% 2.93/1.53 thf(fact_51_the__eq__trivial, axiom, (! [A:$tType,A2:A]: ((((the @ A) @ (^ [X4:A]: ((X4 = A2)))) = A2)))).
% 2.93/1.53 thf(fact_52_the__equality, axiom, (! [A:$tType,P2:(A > $o),A2:A]: (((P2 @ A2) => ((! [X3:A]: (((P2 @ X3) => (X3 = A2)))) => (((the @ A) @ P2) = A2)))))).
% 2.93/1.53 thf(fact_53_image__ident, axiom, (! [A:$tType,Y6:(set @ A)]: ((((((image @ A) @ A) @ (^ [X4:A]: (X4))) @ Y6) = Y6)))).
% 2.93/1.53 thf(fact_54_translation__subtract__diff, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,S:(set @ A),T2:(set @ A)]: ((((((image @ A) @ A) @ (^ [X4:A]: ((((minus_minus @ A) @ X4) @ A2)))) @ (((minus_minus @ (set @ A)) @ S) @ T2)) = (((minus_minus @ (set @ A)) @ ((((image @ A) @ A) @ (^ [X4:A]: ((((minus_minus @ A) @ X4) @ A2)))) @ S)) @ ((((image @ A) @ A) @ (^ [X4:A]: ((((minus_minus @ A) @ X4) @ A2)))) @ T2))))))))).
% 2.93/1.53 thf(fact_55_DiffI, axiom, (! [A:$tType,C3:A,A6:(set @ A),B2:(set @ A)]: (((((member @ A) @ C3) @ A6) => ((~ ((((member @ A) @ C3) @ B2))) => (((member @ A) @ C3) @ (((minus_minus @ (set @ A)) @ A6) @ B2))))))).
% 2.93/1.53 thf(fact_56_image__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F4:(B > A),X:B,A6:(set @ B)]: (((B3 = (F4 @ X)) => ((((member @ B) @ X) @ A6) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F4) @ A6))))))).
% 2.93/1.53 thf(fact_57_Diff__idemp, axiom, (! [A:$tType,A6:(set @ A),B2:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A6) @ B2)) @ B2) = (((minus_minus @ (set @ A)) @ A6) @ B2))))).
% 2.93/1.53 thf(fact_58_Diff__iff, axiom, (! [A:$tType,C3:A,A6:(set @ A),B2:(set @ A)]: (((((member @ A) @ C3) @ (((minus_minus @ (set @ A)) @ A6) @ B2)) = ((((member @ A) @ C3) @ A6) & (~ ((((member @ A) @ C3) @ B2)))))))).
% 2.93/1.53 thf(fact_59_minus__set__def, axiom, (! [A:$tType]: (((minus_minus @ (set @ A)) = (^ [A7:(set @ A),B6:(set @ A)]: (((collect @ A) @ (((minus_minus @ (A > $o)) @ (^ [X4:A]: ((((member @ A) @ X4) @ A7)))) @ (^ [X4:A]: ((((member @ A) @ X4) @ B6))))))))))).
% 2.93/1.53 thf(fact_60_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X:A,A6:(set @ A),B3:B,F4:(A > B)]: (((((member @ A) @ X) @ A6) => ((B3 = (F4 @ X)) => (((member @ B) @ B3) @ ((((image @ A) @ B) @ F4) @ A6))))))).
% 2.93/1.53 thf(fact_61_ball__imageD, axiom, (! [A:$tType,B:$tType,F4:(B > A),A6:(set @ B),P2:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ ((((image @ B) @ A) @ F4) @ A6)) => (P2 @ X3)))) => (! [X5:B]: (((((member @ B) @ X5) @ A6) => (P2 @ (F4 @ X5))))))))).
% 2.93/1.53 thf(fact_62_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N:(set @ A),F4:(A > B),G3:(A > B)]: (((M = N) => ((! [X3:A]: (((((member @ A) @ X3) @ N) => ((F4 @ X3) = (G3 @ X3))))) => (((((image @ A) @ B) @ F4) @ M) = ((((image @ A) @ B) @ G3) @ N))))))).
% 2.93/1.53 thf(fact_63_bex__imageD, axiom, (! [A:$tType,B:$tType,F4:(B > A),A6:(set @ B),P2:(A > $o)]: (((? [X5:A]: (((((member @ A) @ X5) @ ((((image @ B) @ A) @ F4) @ A6)) & (P2 @ X5)))) => (? [X3:B]: (((((member @ B) @ X3) @ A6) & (P2 @ (F4 @ X3))))))))).
% 2.93/1.53 thf(fact_64_image__iff, axiom, (! [A:$tType,B:$tType,Z:A,F4:(B > A),A6:(set @ B)]: (((((member @ A) @ Z) @ ((((image @ B) @ A) @ F4) @ A6)) = (? [X4:B]: (((((member @ B) @ X4) @ A6) & (Z = (F4 @ X4))))))))).
% 2.93/1.53 thf(fact_65_imageI, axiom, (! [B:$tType,A:$tType,X:A,A6:(set @ A),F4:(A > B)]: (((((member @ A) @ X) @ A6) => (((member @ B) @ (F4 @ X)) @ ((((image @ A) @ B) @ F4) @ A6)))))).
% 2.93/1.53 thf(fact_66_DiffD2, axiom, (! [A:$tType,C3:A,A6:(set @ A),B2:(set @ A)]: (((((member @ A) @ C3) @ (((minus_minus @ (set @ A)) @ A6) @ B2)) => (~ ((((member @ A) @ C3) @ B2))))))).
% 2.93/1.53 thf(fact_67_DiffD1, axiom, (! [A:$tType,C3:A,A6:(set @ A),B2:(set @ A)]: (((((member @ A) @ C3) @ (((minus_minus @ (set @ A)) @ A6) @ B2)) => (((member @ A) @ C3) @ A6))))).
% 2.93/1.53 thf(fact_68_DiffE, axiom, (! [A:$tType,C3:A,A6:(set @ A),B2:(set @ A)]: (((((member @ A) @ C3) @ (((minus_minus @ (set @ A)) @ A6) @ B2)) => (~ (((((member @ A) @ C3) @ A6) => (((member @ A) @ C3) @ B2)))))))).
% 2.93/1.53 thf(fact_69_imageE, axiom, (! [A:$tType,B:$tType,B3:A,F4:(B > A),A6:(set @ B)]: (((((member @ A) @ B3) @ ((((image @ B) @ A) @ F4) @ A6)) => (~ ((! [X3:B]: (((B3 = (F4 @ X3)) => (~ ((((member @ B) @ X3) @ A6)))))))))))).
% 2.93/1.53 thf(fact_70_image__image, axiom, (! [A:$tType,B:$tType,C:$tType,F4:(B > A),G3:(C > B),A6:(set @ C)]: ((((((image @ B) @ A) @ F4) @ ((((image @ C) @ B) @ G3) @ A6)) = ((((image @ C) @ A) @ (^ [X4:C]: ((F4 @ (G3 @ X4))))) @ A6))))).
% 2.93/1.53 thf(fact_71_Compr__image__eq, axiom, (! [A:$tType,B:$tType,F4:(B > A),A6:(set @ B),P2:(A > $o)]: ((((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ ((((image @ B) @ A) @ F4) @ A6)) & (P2 @ X4))))) = ((((image @ B) @ A) @ F4) @ ((collect @ B) @ (^ [X4:B]: (((((member @ B) @ X4) @ A6) & (P2 @ (F4 @ X4))))))))))).
% 2.93/1.53 thf(fact_72_set__diff__eq, axiom, (! [A:$tType]: (((minus_minus @ (set @ A)) = (^ [A7:(set @ A),B6:(set @ A)]: (((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ A7) & (~ ((((member @ A) @ X4) @ B6))))))))))))).
% 2.93/1.53 thf(fact_73_theI, axiom, (! [A:$tType,P2:(A > $o),A2:A]: (((P2 @ A2) => ((! [X3:A]: (((P2 @ X3) => (X3 = A2)))) => (P2 @ ((the @ A) @ P2))))))).
% 2.93/1.53 thf(fact_74_theI_H, axiom, (! [A:$tType,P2:(A > $o)]: (((? [X5:A]: (((P2 @ X5) & (! [Y3:A]: (((P2 @ Y3) => (Y3 = X5))))))) => (P2 @ ((the @ A) @ P2)))))).
% 2.93/1.53 thf(fact_75_theI2, axiom, (! [A:$tType,P2:(A > $o),A2:A,Q:(A > $o)]: (((P2 @ A2) => ((! [X3:A]: (((P2 @ X3) => (X3 = A2)))) => ((! [X3:A]: (((P2 @ X3) => (Q @ X3)))) => (Q @ ((the @ A) @ P2)))))))).
% 2.93/1.53 thf(fact_76_If__def, axiom, (! [A:$tType]: (((if @ A) = (^ [P3:$o,X4:A,Y4:A]: (((the @ A) @ (^ [Z3:A]: (((P3 => (Z3 = X4)) & ((~ (P3)) => (Z3 = Y4)))))))))))).
% 2.93/1.53 thf(fact_77_the1I2, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((? [X5:A]: (((P2 @ X5) & (! [Y3:A]: (((P2 @ Y3) => (Y3 = X5))))))) => ((! [X3:A]: (((P2 @ X3) => (Q @ X3)))) => (Q @ ((the @ A) @ P2))))))).
% 2.93/1.53 thf(fact_78_the1__equality, axiom, (! [A:$tType,P2:(A > $o),A2:A]: (((? [X5:A]: (((P2 @ X5) & (! [Y3:A]: (((P2 @ Y3) => (Y3 = X5))))))) => ((P2 @ A2) => (((the @ A) @ P2) = A2)))))).
% 2.93/1.53 thf(fact_79_minus__apply, axiom, (! [B:$tType,A:$tType]: (((minus @ B) => ((minus_minus @ (A > B)) = (^ [A7:(A > B),B6:(A > B),X4:A]: ((((minus_minus @ B) @ (A7 @ X4)) @ (B6 @ X4))))))))).
% 2.93/1.53 thf(fact_80_Inf_OINF__identity__eq, axiom, (! [A:$tType,Inf:((set @ A) > A),A6:(set @ A)]: (((Inf @ ((((image @ A) @ A) @ (^ [X4:A]: (X4))) @ A6)) = (Inf @ A6))))).
% 2.93/1.53 thf(fact_81_Sup_OSUP__identity__eq, axiom, (! [A:$tType,Sup:((set @ A) > A),A6:(set @ A)]: (((Sup @ ((((image @ A) @ A) @ (^ [X4:A]: (X4))) @ A6)) = (Sup @ A6))))).
% 2.93/1.53 thf(fact_82_old_Orec__unit__def, axiom, (! [T:$tType]: (((product_rec_unit @ T) = (^ [F12:T,X4:product_unit]: (((the @ T) @ (((product_rec_set_unit @ T) @ F12) @ X4)))))))).
% 2.93/1.53 thf(fact_83_theI__unique, axiom, (! [A:$tType,P2:(A > $o),X:A]: (((? [X5:A]: (((P2 @ X5) & (! [Y3:A]: (((P2 @ Y3) => (Y3 = X5))))))) => ((P2 @ X) = (X = ((the @ A) @ P2))))))).
% 2.93/1.53 thf(fact_84_Nitpick_OThe__psimp, axiom, (! [A:$tType,P2:(A > $o),X:A]: (((P2 = ((^ [Y5:A,Z2:A]: ((Y5 = Z2))) @ X)) => (((the @ A) @ P2) = X))))).
% 2.93/1.53 thf(fact_85_surj__swap, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ (top_top @ (set @ ((product_prod @ B) @ A)))) = (top_top @ (set @ ((product_prod @ A) @ B))))))).
% 2.93/1.53 thf(fact_86_surj__diff__right, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A]: ((((((image @ A) @ A) @ (^ [X4:A]: ((((minus_minus @ A) @ X4) @ A2)))) @ (top_top @ (set @ A))) = (top_top @ (set @ A))))))))).
% 2.93/1.53 thf(fact_87_UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 2.93/1.53 thf(fact_88_UNIV__witness, axiom, (! [A:$tType]: ((? [X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))))).
% 2.93/1.53 thf(fact_89_UNIV__eq__I, axiom, (! [A:$tType,A6:(set @ A)]: (((! [X3:A]: ((((member @ A) @ X3) @ A6))) => ((top_top @ (set @ A)) = A6))))).
% 2.93/1.53 thf(fact_90_UNIV__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (^ [X4:A]: ($true))))))).
% 2.93/1.53 thf(fact_91_range__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F4:(B > A),X:B]: (((B3 = (F4 @ X)) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F4) @ (top_top @ (set @ B)))))))).
% 2.93/1.53 thf(fact_92_surj__def, axiom, (! [B:$tType,A:$tType,F4:(B > A)]: (((((((image @ B) @ A) @ F4) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [Y4:A]: ((? [X4:B]: ((Y4 = (F4 @ X4)))))))))).
% 2.93/1.53 thf(fact_93_rangeI, axiom, (! [A:$tType,B:$tType,F4:(B > A),X:B]: ((((member @ A) @ (F4 @ X)) @ ((((image @ B) @ A) @ F4) @ (top_top @ (set @ B))))))).
% 2.93/1.53 thf(fact_94_surjI, axiom, (! [B:$tType,A:$tType,G3:(B > A),F4:(A > B)]: (((! [X3:A]: (((G3 @ (F4 @ X3)) = X3))) => (((((image @ B) @ A) @ G3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 2.93/1.53 thf(fact_95_surjE, axiom, (! [A:$tType,B:$tType,F4:(B > A),Y:A]: (((((((image @ B) @ A) @ F4) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (~ ((! [X3:B]: ((Y != (F4 @ X3)))))))))).
% 2.93/1.53 thf(fact_96_surjD, axiom, (! [A:$tType,B:$tType,F4:(B > A),Y:A]: (((((((image @ B) @ A) @ F4) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (? [X3:B]: ((Y = (F4 @ X3)))))))).
% 2.93/1.53 thf(fact_97_range__composition, axiom, (! [A:$tType,C:$tType,B:$tType,F4:(C > A),G3:(B > C)]: ((((((image @ B) @ A) @ (^ [X4:B]: ((F4 @ (G3 @ X4))))) @ (top_top @ (set @ B))) = ((((image @ C) @ A) @ F4) @ ((((image @ B) @ C) @ G3) @ (top_top @ (set @ B)))))))).
% 2.93/1.53 thf(fact_98_rangeE, axiom, (! [A:$tType,B:$tType,B3:A,F4:(B > A)]: (((((member @ A) @ B3) @ ((((image @ B) @ A) @ F4) @ (top_top @ (set @ B)))) => (~ ((! [X3:B]: ((B3 != (F4 @ X3)))))))))).
% 2.93/1.53 thf(fact_99_fun__diff__def, axiom, (! [B:$tType,A:$tType]: (((minus @ B) => ((minus_minus @ (A > B)) = (^ [A7:(A > B),B6:(A > B),X4:A]: ((((minus_minus @ B) @ (A7 @ X4)) @ (B6 @ X4))))))))).
% 2.93/1.53 thf(fact_100_Inf_OINF__cong, axiom, (! [A:$tType,B:$tType,A6:(set @ B),B2:(set @ B),C5:(B > A),D4:(B > A),Inf:((set @ A) > A)]: (((A6 = B2) => ((! [X3:B]: (((((member @ B) @ X3) @ B2) => ((C5 @ X3) = (D4 @ X3))))) => ((Inf @ ((((image @ B) @ A) @ C5) @ A6)) = (Inf @ ((((image @ B) @ A) @ D4) @ B2)))))))).
% 2.93/1.53 thf(fact_101_Sup_OSUP__cong, axiom, (! [A:$tType,B:$tType,A6:(set @ B),B2:(set @ B),C5:(B > A),D4:(B > A),Sup:((set @ A) > A)]: (((A6 = B2) => ((! [X3:B]: (((((member @ B) @ X3) @ B2) => ((C5 @ X3) = (D4 @ X3))))) => ((Sup @ ((((image @ B) @ A) @ C5) @ A6)) = (Sup @ ((((image @ B) @ A) @ D4) @ B2)))))))).
% 2.93/1.53 thf(fact_102_iso__tuple__UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 2.93/1.53 thf(fact_103_top__apply, axiom, (! [C:$tType,D:$tType]: (((top @ C) => ((top_top @ (D > C)) = (^ [X4:D]: ((top_top @ C)))))))).
% 2.93/1.53 thf(fact_104_same__fstI, axiom, (! [B:$tType,A:$tType,P2:(A > $o),X:A,Y7:B,Y:B,R2:(A > (set @ ((product_prod @ B) @ B)))]: (((P2 @ X) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y7) @ Y)) @ (R2 @ X)) => (((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y7)) @ ((((product_Pair @ A) @ B) @ X) @ Y))) @ ((((same_fst @ A) @ B) @ P2) @ R2))))))).
% 2.93/1.53 thf(fact_105_top__prod__def, axiom, (! [A:$tType,B:$tType]: ((((top @ B) & (top @ A)) => ((top_top @ ((product_prod @ A) @ B)) = ((((product_Pair @ A) @ B) @ (top_top @ A)) @ (top_top @ B))))))).
% 2.93/1.53 thf(fact_106_map__prod__surj, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F4:(A > B),G3:(C > D)]: (((((((image @ A) @ B) @ F4) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((((((image @ C) @ D) @ G3) @ (top_top @ (set @ C))) = (top_top @ (set @ D))) => (((((image @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)) @ ((((((product_map_prod @ A) @ B) @ C) @ D) @ F4) @ G3)) @ (top_top @ (set @ ((product_prod @ A) @ C)))) = (top_top @ (set @ ((product_prod @ B) @ D))))))))).
% 2.93/1.53 thf(fact_107_range__fst, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = (top_top @ (set @ A)))))).
% 2.93/1.53 thf(fact_108_map__prod__ident, axiom, (! [B:$tType,A:$tType]: ((((((((product_map_prod @ A) @ A) @ B) @ B) @ (^ [X4:A]: (X4))) @ (^ [Y4:B]: (Y4))) = (^ [Z3:((product_prod @ A) @ B)]: (Z3)))))).
% 2.93/1.53 thf(fact_109_map__prod__simp, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F4:(C > A),G3:(D > B),A2:C,B3:D]: (((((((((product_map_prod @ C) @ A) @ D) @ B) @ F4) @ G3) @ ((((product_Pair @ C) @ D) @ A2) @ B3)) = ((((product_Pair @ A) @ B) @ (F4 @ A2)) @ (G3 @ B3)))))).
% 2.93/1.53 thf(fact_110_fst__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F4:(C > A),G3:(D > B),X:((product_prod @ C) @ D)]: (((((product_fst @ A) @ B) @ (((((((product_map_prod @ C) @ A) @ D) @ B) @ F4) @ G3) @ X)) = (F4 @ (((product_fst @ C) @ D) @ X)))))).
% 2.93/1.53 thf(fact_111_map__prod__imageI, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,A2:A,B3:B,R2:(set @ ((product_prod @ A) @ B)),F4:(A > C),G3:(B > D)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B3)) @ R2) => (((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ (F4 @ A2)) @ (G3 @ B3))) @ ((((image @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ F4) @ G3)) @ R2)))))).
% 2.93/1.53 thf(fact_112_fst__diff, axiom, (! [B:$tType,A:$tType]: ((((minus @ A) & (minus @ B)) => (! [X:((product_prod @ A) @ B),Y:((product_prod @ A) @ B)]: (((((product_fst @ A) @ B) @ (((minus_minus @ ((product_prod @ A) @ B)) @ X) @ Y)) = (((minus_minus @ A) @ (((product_fst @ A) @ B) @ X)) @ (((product_fst @ A) @ B) @ Y))))))))).
% 2.93/1.53 thf(fact_113_fst__top, axiom, (! [B:$tType,A:$tType]: ((((top @ A) & (top @ B)) => ((((product_fst @ A) @ B) @ (top_top @ ((product_prod @ A) @ B))) = (top_top @ A)))))).
% 2.93/1.53 thf(fact_114_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X4:A]: ((((member @ A) @ X4) @ (top_top @ (set @ A))))))))).
% 2.93/1.53 thf(fact_115_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 2.93/1.53 thf(fact_116_fst__eqD, axiom, (! [B:$tType,A:$tType,X:A,Y:B,A2:A]: ((((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X) @ Y)) = A2) => (X = A2))))).
% 2.93/1.53 thf(fact_117_fst__conv, axiom, (! [B:$tType,A:$tType,X1:A,X2:B]: (((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X1) @ X2)) = X1)))).
% 2.93/1.53 thf(fact_118_prod__fun__imageE, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,C3:((product_prod @ A) @ B),F4:(C > A),G3:(D > B),R2:(set @ ((product_prod @ C) @ D))]: (((((member @ ((product_prod @ A) @ B)) @ C3) @ ((((image @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ C) @ A) @ D) @ B) @ F4) @ G3)) @ R2)) => (~ ((! [X3:C,Y3:D]: (((C3 = ((((product_Pair @ A) @ B) @ (F4 @ X3)) @ (G3 @ Y3))) => (~ ((((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ X3) @ Y3)) @ R2)))))))))))).
% 2.93/1.53 thf(fact_119_top__empty__eq2, axiom, (! [B:$tType,A:$tType]: (((top_top @ (A > (B > $o))) = (^ [X4:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y4)) @ (top_top @ (set @ ((product_prod @ A) @ B)))))))))).
% 2.93/1.53 thf(fact_120_eq__fst__iff, axiom, (! [A:$tType,B:$tType,A2:A,P:((product_prod @ A) @ B)]: (((A2 = (((product_fst @ A) @ B) @ P)) = (? [B7:B]: ((P = ((((product_Pair @ A) @ B) @ A2) @ B7)))))))).
% 2.93/1.53 thf(fact_121_fstI, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),Y:A,Z:B]: (((X = ((((product_Pair @ A) @ B) @ Y) @ Z)) => ((((product_fst @ A) @ B) @ X) = Y))))).
% 2.93/1.53 thf(fact_122_prod_Omap__ident, axiom, (! [B:$tType,A:$tType,T2:((product_prod @ A) @ B)]: (((((((((product_map_prod @ A) @ A) @ B) @ B) @ (^ [X4:A]: (X4))) @ (^ [X4:B]: (X4))) @ T2) = T2)))).
% 2.93/1.53 thf(fact_123_minus__prod__def, axiom, (! [B:$tType,A:$tType]: ((((minus @ A) & (minus @ B)) => ((minus_minus @ ((product_prod @ A) @ B)) = (^ [X4:((product_prod @ A) @ B),Y4:((product_prod @ A) @ B)]: (((((product_Pair @ A) @ B) @ (((minus_minus @ A) @ (((product_fst @ A) @ B) @ X4)) @ (((product_fst @ A) @ B) @ Y4))) @ (((minus_minus @ B) @ (((product_snd @ A) @ B) @ X4)) @ (((product_snd @ A) @ B) @ Y4)))))))))).
% 2.93/1.53 thf(fact_124_range__snd, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ (top_top @ (set @ ((product_prod @ B) @ A)))) = (top_top @ (set @ A)))))).
% 2.93/1.53 thf(fact_125_top1I, axiom, (! [A:$tType,X:A]: (((top_top @ (A > $o)) @ X)))).
% 2.93/1.53 thf(fact_126_top2I, axiom, (! [A:$tType,B:$tType,X:A,Y:B]: ((((top_top @ (A > (B > $o))) @ X) @ Y)))).
% 2.93/1.53 thf(fact_127_snd__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F4:(C > B),G3:(D > A),X:((product_prod @ C) @ D)]: (((((product_snd @ B) @ A) @ (((((((product_map_prod @ C) @ B) @ D) @ A) @ F4) @ G3) @ X)) = (G3 @ (((product_snd @ C) @ D) @ X)))))).
% 2.93/1.53 thf(fact_128_prod_Ocollapse, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)) = Prod)))).
% 2.93/1.53 thf(fact_129_fst__swap, axiom, (! [A:$tType,B:$tType,X:((product_prod @ B) @ A)]: (((((product_fst @ A) @ B) @ (((product_swap @ B) @ A) @ X)) = (((product_snd @ B) @ A) @ X))))).
% 2.93/1.53 thf(fact_130_snd__swap, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B)]: (((((product_snd @ B) @ A) @ (((product_swap @ A) @ B) @ X)) = (((product_fst @ A) @ B) @ X))))).
% 2.93/1.53 thf(fact_131_snd__diff, axiom, (! [A:$tType,B:$tType]: ((((minus @ B) & (minus @ A)) => (! [X:((product_prod @ B) @ A),Y:((product_prod @ B) @ A)]: (((((product_snd @ B) @ A) @ (((minus_minus @ ((product_prod @ B) @ A)) @ X) @ Y)) = (((minus_minus @ A) @ (((product_snd @ B) @ A) @ X)) @ (((product_snd @ B) @ A) @ Y))))))))).
% 2.93/1.53 thf(fact_132_snd__top, axiom, (! [B:$tType,A:$tType]: ((((top @ A) & (top @ B)) => ((((product_snd @ B) @ A) @ (top_top @ ((product_prod @ B) @ A))) = (top_top @ A)))))).
% 2.93/1.53 thf(fact_133_prod__eqI, axiom, (! [B:$tType,A:$tType,P:((product_prod @ A) @ B),Q2:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ P) = (((product_fst @ A) @ B) @ Q2)) => (((((product_snd @ A) @ B) @ P) = (((product_snd @ A) @ B) @ Q2)) => (P = Q2)))))).
% 2.93/1.53 thf(fact_134_prod_Oexpand, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B),Prod2:((product_prod @ A) @ B)]: (((((((product_fst @ A) @ B) @ Prod) = (((product_fst @ A) @ B) @ Prod2)) & ((((product_snd @ A) @ B) @ Prod) = (((product_snd @ A) @ B) @ Prod2))) => (Prod = Prod2))))).
% 2.93/1.53 thf(fact_135_prod__eq__iff, axiom, (! [B:$tType,A:$tType]: (((^ [Y5:((product_prod @ A) @ B),Z2:((product_prod @ A) @ B)]: ((Y5 = Z2))) = (^ [S4:((product_prod @ A) @ B),T3:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ S4) = (((product_fst @ A) @ B) @ T3)) & ((((product_snd @ A) @ B) @ S4) = (((product_snd @ A) @ B) @ T3))))))))).
% 2.93/1.53 thf(fact_136_eq__snd__iff, axiom, (! [A:$tType,B:$tType,B3:A,P:((product_prod @ B) @ A)]: (((B3 = (((product_snd @ B) @ A) @ P)) = (? [A5:B]: ((P = ((((product_Pair @ B) @ A) @ A5) @ B3)))))))).
% 2.93/1.53 thf(fact_137_sndI, axiom, (! [A:$tType,B:$tType,X:((product_prod @ A) @ B),Y:A,Z:B]: (((X = ((((product_Pair @ A) @ B) @ Y) @ Z)) => ((((product_snd @ A) @ B) @ X) = Z))))).
% 2.93/1.53 thf(fact_138_snd__eqD, axiom, (! [B:$tType,A:$tType,X:B,Y:A,A2:A]: ((((((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y)) = A2) => (Y = A2))))).
% 2.93/1.53 thf(fact_139_snd__conv, axiom, (! [Aa:$tType,A:$tType,X1:Aa,X2:A]: (((((product_snd @ Aa) @ A) @ ((((product_Pair @ Aa) @ A) @ X1) @ X2)) = X2)))).
% 2.93/1.53 thf(fact_140_surjective__pairing, axiom, (! [B:$tType,A:$tType,T2:((product_prod @ A) @ B)]: ((T2 = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ T2)) @ (((product_snd @ A) @ B) @ T2)))))).
% 2.93/1.53 thf(fact_141_prod_Oexhaust__sel, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))).
% 2.93/1.53 thf(fact_142_prod_Oswap__def, axiom, (! [B:$tType,A:$tType]: ((((product_swap @ A) @ B) = (^ [P4:((product_prod @ A) @ B)]: (((((product_Pair @ B) @ A) @ (((product_snd @ A) @ B) @ P4)) @ (((product_fst @ A) @ B) @ P4)))))))).
% 2.93/1.53 thf(fact_143_BNF__Greatest__Fixpoint_Osubst__Pair, axiom, (! [B:$tType,A:$tType,P2:(A > (B > $o)),X:A,Y:B,A2:((product_prod @ A) @ B)]: ((((P2 @ X) @ Y) => ((A2 = ((((product_Pair @ A) @ B) @ X) @ Y)) => ((P2 @ (((product_fst @ A) @ B) @ A2)) @ (((product_snd @ A) @ B) @ A2))))))).
% 2.93/1.53 thf(fact_144_conjI__realizer, axiom, (! [A:$tType,B:$tType,P2:(A > $o),P:A,Q:(B > $o),Q2:B]: (((P2 @ P) => ((Q @ Q2) => ((P2 @ (((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ P) @ Q2))) & (Q @ (((product_snd @ A) @ B) @ ((((product_Pair @ A) @ B) @ P) @ Q2))))))))).
% 2.93/1.53 thf(fact_145_exI__realizer, axiom, (! [B:$tType,A:$tType,P2:(A > (B > $o)),Y:A,X:B]: ((((P2 @ Y) @ X) => ((P2 @ (((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y))) @ (((product_fst @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y))))))).
% 2.93/1.53 thf(fact_146_top__conj_I2_J, axiom, (! [A:$tType,P2:$o,X:A]: (((P2 & ((top_top @ (A > $o)) @ X)) = P2)))).
% 2.93/1.53 thf(fact_147_top__conj_I1_J, axiom, (! [A:$tType,X:A,P2:$o]: (((((top_top @ (A > $o)) @ X) & P2) = P2)))).
% 2.93/1.53 thf(fact_148_scomp__unfold, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType]: ((((((product_scomp @ A) @ B) @ C) @ D) = (^ [F3:(A > ((product_prod @ B) @ C)),G4:(B > (C > D)),X4:A]: (((G4 @ (((product_fst @ B) @ C) @ (F3 @ X4))) @ (((product_snd @ B) @ C) @ (F3 @ X4))))))))).
% 2.93/1.53 thf(fact_149_sndOp__def, axiom, (! [B:$tType,A:$tType,C:$tType]: (((((bNF_sndOp @ C) @ A) @ B) = (^ [P3:(C > (A > $o)),Q3:(A > (B > $o)),Ac:((product_prod @ C) @ B)]: (((((product_Pair @ A) @ B) @ (((((((bNF_pick_middlep @ C) @ A) @ B) @ P3) @ Q3) @ (((product_fst @ C) @ B) @ Ac)) @ (((product_snd @ C) @ B) @ Ac))) @ (((product_snd @ C) @ B) @ Ac)))))))).
% 2.93/1.53 thf(fact_150_fstOp__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((bNF_fstOp @ A) @ B) @ C) = (^ [P3:(A > (B > $o)),Q3:(B > (C > $o)),Ac:((product_prod @ A) @ C)]: (((((product_Pair @ A) @ B) @ (((product_fst @ A) @ C) @ Ac)) @ (((((((bNF_pick_middlep @ A) @ B) @ C) @ P3) @ Q3) @ (((product_fst @ A) @ C) @ Ac)) @ (((product_snd @ A) @ C) @ Ac))))))))).
% 2.93/1.53 thf(fact_151_scomp__Pair, axiom, (! [C:$tType,B:$tType,A:$tType,X:(A > ((product_prod @ B) @ C))]: ((((((((product_scomp @ A) @ B) @ C) @ ((product_prod @ B) @ C)) @ X) @ ((product_Pair @ B) @ C)) = X)))).
% 2.93/1.53 thf(fact_152_scomp__scomp, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F:$tType,E:$tType,F4:(A > ((product_prod @ E) @ F)),G3:(E > (F > ((product_prod @ C) @ D))),H:(C > (D > B))]: ((((((((product_scomp @ A) @ C) @ D) @ B) @ ((((((product_scomp @ A) @ E) @ F) @ ((product_prod @ C) @ D)) @ F4) @ G3)) @ H) = ((((((product_scomp @ A) @ E) @ F) @ B) @ F4) @ (^ [X4:E]: (((((((product_scomp @ F) @ C) @ D) @ B) @ (G3 @ X4)) @ H)))))))).
% 2.93/1.53 thf(fact_153_Pair__scomp, axiom, (! [A:$tType,B:$tType,C:$tType,X:C,F4:(C > (A > B))]: ((((((((product_scomp @ A) @ C) @ A) @ B) @ (((product_Pair @ C) @ A) @ X)) @ F4) = (F4 @ X))))).
% 2.93/1.53 thf(fact_154_image2__eqI, axiom, (! [A:$tType,C:$tType,B:$tType,B3:A,F4:(B > A),X:B,C3:C,G3:(B > C),A6:(set @ B)]: (((B3 = (F4 @ X)) => ((C3 = (G3 @ X)) => ((((member @ B) @ X) @ A6) => (((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ B3) @ C3)) @ ((((((bNF_Greatest_image2 @ B) @ A) @ C) @ A6) @ F4) @ G3)))))))).
% 2.93/1.53 thf(fact_155_map__prod__surj__on, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F4:(B > A),A6:(set @ B),A8:(set @ A),G3:(D > C),B2:(set @ D),B8:(set @ C)]: (((((((image @ B) @ A) @ F4) @ A6) = A8) => ((((((image @ D) @ C) @ G3) @ B2) = B8) => (((((image @ ((product_prod @ B) @ D)) @ ((product_prod @ A) @ C)) @ ((((((product_map_prod @ B) @ A) @ D) @ C) @ F4) @ G3)) @ ((((product_Sigma @ B) @ D) @ A6) @ (^ [Uu:B]: (B2)))) = ((((product_Sigma @ A) @ C) @ A8) @ (^ [Uu:A]: (B8))))))))).
% 2.93/1.53 thf(fact_156_in__lex__prod, axiom, (! [A:$tType,B:$tType,A2:A,B3:B,A3:A,B4:B,R:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B3)) @ ((((product_Pair @ A) @ B) @ A3) @ B4))) @ ((((lex_prod @ A) @ B) @ R) @ S)) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A3)) @ R) | ((A2 = A3) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ B3) @ B4)) @ S))))))).
% 2.93/1.53 thf(fact_157_mem__Sigma__iff, axiom, (! [B:$tType,A:$tType,A2:A,B3:B,A6:(set @ A),B2:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B3)) @ ((((product_Sigma @ A) @ B) @ A6) @ B2)) = ((((member @ A) @ A2) @ A6) & (((member @ B) @ B3) @ (B2 @ A2))))))).
% 2.93/1.53 thf(fact_158_SigmaI, axiom, (! [B:$tType,A:$tType,A2:A,A6:(set @ A),B3:B,B2:(A > (set @ B))]: (((((member @ A) @ A2) @ A6) => ((((member @ B) @ B3) @ (B2 @ A2)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B3)) @ ((((product_Sigma @ A) @ B) @ A6) @ B2))))))).
% 2.93/1.53 thf(fact_159_UNIV__Times__UNIV, axiom, (! [B:$tType,A:$tType]: ((((((product_Sigma @ A) @ B) @ (top_top @ (set @ A))) @ (^ [Uu:A]: ((top_top @ (set @ B))))) = (top_top @ (set @ ((product_prod @ A) @ B))))))).
% 2.93/1.53 thf(fact_160_Times__Diff__distrib1, axiom, (! [B:$tType,A:$tType,A6:(set @ A),B2:(set @ A),C5:(set @ B)]: ((((((product_Sigma @ A) @ B) @ (((minus_minus @ (set @ A)) @ A6) @ B2)) @ (^ [Uu:A]: (C5))) = (((minus_minus @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A6) @ (^ [Uu:A]: (C5)))) @ ((((product_Sigma @ A) @ B) @ B2) @ (^ [Uu:A]: (C5)))))))).
% 2.93/1.53 thf(fact_161_Sigma__Diff__distrib2, axiom, (! [B:$tType,A:$tType,I:(set @ A),A6:(A > (set @ B)),B2:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ I) @ (^ [I2:A]: ((((minus_minus @ (set @ B)) @ (A6 @ I2)) @ (B2 @ I2))))) = (((minus_minus @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ I) @ A6)) @ ((((product_Sigma @ A) @ B) @ I) @ B2)))))).
% 2.93/1.53 thf(fact_162_Sigma__cong, axiom, (! [B:$tType,A:$tType,A6:(set @ A),B2:(set @ A),C5:(A > (set @ B)),D4:(A > (set @ B))]: (((A6 = B2) => ((! [X3:A]: (((((member @ A) @ X3) @ B2) => ((C5 @ X3) = (D4 @ X3))))) => (((((product_Sigma @ A) @ B) @ A6) @ C5) = ((((product_Sigma @ A) @ B) @ B2) @ D4))))))).
% 2.93/1.53 thf(fact_163_Times__eq__cancel2, axiom, (! [A:$tType,B:$tType,X:A,C5:(set @ A),A6:(set @ B),B2:(set @ B)]: (((((member @ A) @ X) @ C5) => ((((((product_Sigma @ B) @ A) @ A6) @ (^ [Uu:B]: (C5))) = ((((product_Sigma @ B) @ A) @ B2) @ (^ [Uu:B]: (C5)))) = (A6 = B2)))))).
% 2.93/1.53 thf(fact_164_Sigma__Diff__distrib1, axiom, (! [B:$tType,A:$tType,I:(set @ A),J:(set @ A),C5:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ (((minus_minus @ (set @ A)) @ I) @ J)) @ C5) = (((minus_minus @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ I) @ C5)) @ ((((product_Sigma @ A) @ B) @ J) @ C5)))))).
% 2.93/1.53 thf(fact_165_SigmaE2, axiom, (! [B:$tType,A:$tType,A2:A,B3:B,A6:(set @ A),B2:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B3)) @ ((((product_Sigma @ A) @ B) @ A6) @ B2)) => (~ (((((member @ A) @ A2) @ A6) => (~ ((((member @ B) @ B3) @ (B2 @ A2))))))))))).
% 2.93/1.53 thf(fact_166_SigmaD2, axiom, (! [B:$tType,A:$tType,A2:A,B3:B,A6:(set @ A),B2:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B3)) @ ((((product_Sigma @ A) @ B) @ A6) @ B2)) => (((member @ B) @ B3) @ (B2 @ A2)))))).
% 2.93/1.53 thf(fact_167_SigmaD1, axiom, (! [B:$tType,A:$tType,A2:A,B3:B,A6:(set @ A),B2:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B3)) @ ((((product_Sigma @ A) @ B) @ A6) @ B2)) => (((member @ A) @ A2) @ A6))))).
% 2.93/1.53 thf(fact_168_SigmaE, axiom, (! [A:$tType,B:$tType,C3:((product_prod @ A) @ B),A6:(set @ A),B2:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ C3) @ ((((product_Sigma @ A) @ B) @ A6) @ B2)) => (~ ((! [X3:A]: (((((member @ A) @ X3) @ A6) => (! [Y3:B]: (((((member @ B) @ Y3) @ (B2 @ X3)) => (C3 != ((((product_Pair @ A) @ B) @ X3) @ Y3)))))))))))))).
% 2.93/1.53 thf(fact_169_mem__Times__iff, axiom, (! [A:$tType,B:$tType,X:((product_prod @ A) @ B),A6:(set @ A),B2:(set @ B)]: (((((member @ ((product_prod @ A) @ B)) @ X) @ ((((product_Sigma @ A) @ B) @ A6) @ (^ [Uu:A]: (B2)))) = ((((member @ A) @ (((product_fst @ A) @ B) @ X)) @ A6) & (((member @ B) @ (((product_snd @ A) @ B) @ X)) @ B2)))))).
% 2.93/1.53 thf(fact_170_product__swap, axiom, (! [B:$tType,A:$tType,A6:(set @ B),B2:(set @ A)]: ((((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ A6) @ (^ [Uu:B]: (B2)))) = ((((product_Sigma @ A) @ B) @ B2) @ (^ [Uu:A]: (A6))))))).
% 2.93/1.53 thf(fact_171_Product__Type_Oproduct__def, axiom, (! [B:$tType,A:$tType]: ((((product_product @ A) @ B) = (^ [A7:(set @ A),B6:(set @ B)]: (((((product_Sigma @ A) @ B) @ A7) @ (^ [Uu:A]: (B6))))))))).
% 2.93/1.53 thf(fact_172_member__product, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),A6:(set @ A),B2:(set @ B)]: (((((member @ ((product_prod @ A) @ B)) @ X) @ ((((product_product @ A) @ B) @ A6) @ B2)) = (((member @ ((product_prod @ A) @ B)) @ X) @ ((((product_Sigma @ A) @ B) @ A6) @ (^ [Uu:A]: (B2)))))))).
% 2.93/1.53 thf(fact_173_subset__fst__snd, axiom, (! [B:$tType,A:$tType,A6:(set @ ((product_prod @ A) @ B))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A6) @ ((((product_Sigma @ A) @ B) @ ((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ A6)) @ (^ [Uu:A]: (((((image @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ A6)))))))).
% 2.93/1.53 thf(fact_174_subset__antisym, axiom, (! [A:$tType,A6:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A6) => (A6 = B2)))))).
% 2.93/1.53 thf(fact_175_subsetI, axiom, (! [A:$tType,A6:(set @ A),B2:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((member @ A) @ X3) @ B2)))) => (((ord_less_eq @ (set @ A)) @ A6) @ B2))))).
% 2.93/1.53 thf(fact_176_Pair__le, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B3:B,C3:A,D3:B]: (((((ord_less_eq @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B3)) @ ((((product_Pair @ A) @ B) @ C3) @ D3)) = ((((ord_less_eq @ A) @ A2) @ C3) & (((ord_less_eq @ B) @ B3) @ D3))))))))).
% 2.93/1.53 thf(fact_177_Times__subset__cancel2, axiom, (! [A:$tType,B:$tType,X:A,C5:(set @ A),A6:(set @ B),B2:(set @ B)]: (((((member @ A) @ X) @ C5) => ((((ord_less_eq @ (set @ ((product_prod @ B) @ A))) @ ((((product_Sigma @ B) @ A) @ A6) @ (^ [Uu:B]: (C5)))) @ ((((product_Sigma @ B) @ A) @ B2) @ (^ [Uu:B]: (C5)))) = (((ord_less_eq @ (set @ B)) @ A6) @ B2)))))).
% 2.93/1.53 thf(fact_178_Sigma__mono, axiom, (! [B:$tType,A:$tType,A6:(set @ A),C5:(set @ A),B2:(A > (set @ B)),D4:(A > (set @ B))]: (((((ord_less_eq @ (set @ A)) @ A6) @ C5) => ((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((ord_less_eq @ (set @ B)) @ (B2 @ X3)) @ (D4 @ X3))))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A6) @ B2)) @ ((((product_Sigma @ A) @ B) @ C5) @ D4))))))).
% 2.93/1.53 thf(fact_179_Pair__mono, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [X:A,X6:A,Y:B,Y7:B]: (((((ord_less_eq @ A) @ X) @ X6) => ((((ord_less_eq @ B) @ Y) @ Y7) => (((ord_less_eq @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ ((((product_Pair @ A) @ B) @ X6) @ Y7)))))))))).
% 2.93/1.53 thf(fact_180_image__mono, axiom, (! [B:$tType,A:$tType,A6:(set @ A),B2:(set @ A),F4:(A > B)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B2) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F4) @ A6)) @ ((((image @ A) @ B) @ F4) @ B2)))))).
% 2.93/1.53 thf(fact_181_image__subsetI, axiom, (! [A:$tType,B:$tType,A6:(set @ A),F4:(A > B),B2:(set @ B)]: (((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((member @ B) @ (F4 @ X3)) @ B2)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F4) @ A6)) @ B2))))).
% 2.93/1.53 thf(fact_182_subset__imageE, axiom, (! [A:$tType,B:$tType,B2:(set @ A),F4:(B > A),A6:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B2) @ ((((image @ B) @ A) @ F4) @ A6)) => (~ ((! [C6:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C6) @ A6) => (B2 != ((((image @ B) @ A) @ F4) @ C6))))))))))).
% 2.93/1.53 thf(fact_183_image__subset__iff, axiom, (! [A:$tType,B:$tType,F4:(B > A),A6:(set @ B),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F4) @ A6)) @ B2) = (! [X4:B]: (((((member @ B) @ X4) @ A6) => (((member @ A) @ (F4 @ X4)) @ B2)))))))).
% 2.93/1.53 thf(fact_184_subset__image__iff, axiom, (! [A:$tType,B:$tType,B2:(set @ A),F4:(B > A),A6:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B2) @ ((((image @ B) @ A) @ F4) @ A6)) = (? [AA:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ AA) @ A6) & (B2 = ((((image @ B) @ A) @ F4) @ AA))))))))).
% 2.93/1.53 thf(fact_185_subrelI, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((! [X3:A,Y3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ S)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S))))).
% 2.93/1.53 thf(fact_186_Collect__mono__iff, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P2)) @ ((collect @ A) @ Q)) = (! [X4:A]: (((P2 @ X4) => (Q @ X4)))))))).
% 2.93/1.53 thf(fact_187_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y5:(set @ A),Z2:(set @ A)]: ((Y5 = Z2))) = (^ [A7:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B6) & (((ord_less_eq @ (set @ A)) @ B6) @ A7)))))))).
% 2.93/1.53 thf(fact_188_subset__trans, axiom, (! [A:$tType,A6:(set @ A),B2:(set @ A),C5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C5) => (((ord_less_eq @ (set @ A)) @ A6) @ C5)))))).
% 2.93/1.53 thf(fact_189_Collect__mono, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P2 @ X3) => (Q @ X3)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P2)) @ ((collect @ A) @ Q)))))).
% 2.93/1.53 thf(fact_190_subset__refl, axiom, (! [A:$tType,A6:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A6) @ A6)))).
% 2.93/1.53 thf(fact_191_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B6:(set @ A)]: ((! [T3:A]: (((((member @ A) @ T3) @ A7) => (((member @ A) @ T3) @ B6)))))))))).
% 2.93/1.53 thf(fact_192_equalityD2, axiom, (! [A:$tType,A6:(set @ A),B2:(set @ A)]: (((A6 = B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A6))))).
% 2.93/1.53 thf(fact_193_equalityD1, axiom, (! [A:$tType,A6:(set @ A),B2:(set @ A)]: (((A6 = B2) => (((ord_less_eq @ (set @ A)) @ A6) @ B2))))).
% 2.93/1.53 thf(fact_194_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B6:(set @ A)]: ((! [X4:A]: (((((member @ A) @ X4) @ A7) => (((member @ A) @ X4) @ B6)))))))))).
% 2.93/1.53 thf(fact_195_equalityE, axiom, (! [A:$tType,A6:(set @ A),B2:(set @ A)]: (((A6 = B2) => (~ (((((ord_less_eq @ (set @ A)) @ A6) @ B2) => (~ ((((ord_less_eq @ (set @ A)) @ B2) @ A6)))))))))).
% 2.93/1.53 thf(fact_196_subsetD, axiom, (! [A:$tType,A6:(set @ A),B2:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A6) @ B2) => ((((member @ A) @ C3) @ A6) => (((member @ A) @ C3) @ B2)))))).
% 2.93/1.53 thf(fact_197_in__mono, axiom, (! [A:$tType,A6:(set @ A),B2:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A6) @ B2) => ((((member @ A) @ X) @ A6) => (((member @ A) @ X) @ B2)))))).
% 2.93/1.53 thf(fact_198_Diff__mono, axiom, (! [A:$tType,A6:(set @ A),C5:(set @ A),D4:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ C5) => ((((ord_less_eq @ (set @ A)) @ D4) @ B2) => (((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A6) @ B2)) @ (((minus_minus @ (set @ A)) @ C5) @ D4))))))).
% 2.93/1.53 thf(fact_199_Diff__subset, axiom, (! [A:$tType,A6:(set @ A),B2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A6) @ B2)) @ A6)))).
% 2.93/1.53 thf(fact_200_double__diff, axiom, (! [A:$tType,A6:(set @ A),B2:(set @ A),C5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C5) => ((((minus_minus @ (set @ A)) @ B2) @ (((minus_minus @ (set @ A)) @ C5) @ A6)) = A6)))))).
% 2.93/1.53 thf(fact_201_subset__UNIV, axiom, (! [A:$tType,A6:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A6) @ (top_top @ (set @ A)))))).
% 2.93/1.53 thf(fact_202_top_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A2) => (A2 = (top_top @ A))))))))).
% 2.93/1.53 thf(fact_203_top_Oextremum__unique, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A2) = (A2 = (top_top @ A))))))))).
% 2.93/1.53 thf(fact_204_top__greatest, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ (top_top @ A)))))))).
% 2.93/1.53 thf(fact_205_prop__restrict, axiom, (! [A:$tType,X:A,Z4:(set @ A),X7:(set @ A),P2:(A > $o)]: (((((member @ A) @ X) @ Z4) => ((((ord_less_eq @ (set @ A)) @ Z4) @ ((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ X7) & (P2 @ X4)))))) => (P2 @ X)))))).
% 2.93/1.53 thf(fact_206_Collect__restrict, axiom, (! [A:$tType,X7:(set @ A),P2:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ X7) & (P2 @ X4)))))) @ X7)))).
% 2.93/1.53 thf(fact_207_Collect__subset, axiom, (! [A:$tType,A6:(set @ A),P2:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ A6) & (P2 @ X4)))))) @ A6)))).
% 2.93/1.53 thf(fact_208_image__Collect__subsetI, axiom, (! [A:$tType,B:$tType,P2:(A > $o),F4:(A > B),B2:(set @ B)]: (((! [X3:A]: (((P2 @ X3) => (((member @ B) @ (F4 @ X3)) @ B2)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F4) @ ((collect @ A) @ P2))) @ B2))))).
% 2.93/1.53 thf(fact_209_subset__Collect__iff, axiom, (! [A:$tType,B2:(set @ A),A6:(set @ A),P2:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B2) @ A6) => ((((ord_less_eq @ (set @ A)) @ B2) @ ((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ A6) & (P2 @ X4)))))) = (! [X4:A]: (((((member @ A) @ X4) @ B2) => (P2 @ X4))))))))).
% 2.93/1.53 thf(fact_210_subset__CollectI, axiom, (! [A:$tType,B2:(set @ A),A6:(set @ A),Q:(A > $o),P2:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B2) @ A6) => ((! [X3:A]: (((((member @ A) @ X3) @ B2) => ((Q @ X3) => (P2 @ X3))))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ B2) & (Q @ X4)))))) @ ((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ A6) & (P2 @ X4))))))))))).
% 2.93/1.53 thf(fact_211_image__diff__subset, axiom, (! [A:$tType,B:$tType,F4:(B > A),A6:(set @ B),B2:(set @ B)]: ((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ ((((image @ B) @ A) @ F4) @ A6)) @ ((((image @ B) @ A) @ F4) @ B2))) @ ((((image @ B) @ A) @ F4) @ (((minus_minus @ (set @ B)) @ A6) @ B2)))))).
% 2.93/1.53 thf(fact_212_subset__snd__imageI, axiom, (! [B:$tType,A:$tType,A6:(set @ A),B2:(set @ B),S3:(set @ ((product_prod @ A) @ B)),X:A]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A6) @ (^ [Uu:A]: (B2)))) @ S3) => ((((member @ A) @ X) @ A6) => (((ord_less_eq @ (set @ B)) @ B2) @ ((((image @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ S3))))))).
% 2.93/1.53 thf(fact_213_subset__fst__imageI, axiom, (! [B:$tType,A:$tType,A6:(set @ A),B2:(set @ B),S3:(set @ ((product_prod @ A) @ B)),Y:B]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A6) @ (^ [Uu:A]: (B2)))) @ S3) => ((((member @ B) @ Y) @ B2) => (((ord_less_eq @ (set @ A)) @ A6) @ ((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ S3))))))).
% 2.93/1.53 thf(fact_214_range__subsetD, axiom, (! [B:$tType,A:$tType,F4:(B > A),B2:(set @ A),I3:B]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F4) @ (top_top @ (set @ B)))) @ B2) => (((member @ A) @ (F4 @ I3)) @ B2))))).
% 2.93/1.53 thf(fact_215_pred__subset__eq, axiom, (! [A:$tType,R2:(set @ A),S3:(set @ A)]: (((((ord_less_eq @ (A > $o)) @ (^ [X4:A]: ((((member @ A) @ X4) @ R2)))) @ (^ [X4:A]: ((((member @ A) @ X4) @ S3)))) = (((ord_less_eq @ (set @ A)) @ R2) @ S3))))).
% 2.93/1.53 thf(fact_216_less__eq__set__def, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B6:(set @ A)]: ((((ord_less_eq @ (A > $o)) @ (^ [X4:A]: ((((member @ A) @ X4) @ A7)))) @ (^ [X4:A]: ((((member @ A) @ X4) @ B6)))))))))).
% 2.93/1.53 thf(fact_217_pred__subset__eq2, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (A > (B > $o))) @ (^ [X4:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y4)) @ R2)))) @ (^ [X4:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y4)) @ S3)))) = (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R2) @ S3))))).
% 2.93/1.53 thf(fact_218_Gr__incl, axiom, (! [A:$tType,B:$tType,A6:(set @ A),F4:(A > B),B2:(set @ B)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((bNF_Gr @ A) @ B) @ A6) @ F4)) @ ((((product_Sigma @ A) @ B) @ A6) @ (^ [Uu:A]: (B2)))) = (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F4) @ A6)) @ B2))))).
% 2.93/1.53 thf(fact_219_all__subset__image, axiom, (! [A:$tType,B:$tType,F4:(B > A),A6:(set @ B),P2:((set @ A) > $o)]: (((! [B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B6) @ ((((image @ B) @ A) @ F4) @ A6)) => (P2 @ B6)))) = (! [B6:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ B6) @ A6) => (P2 @ ((((image @ B) @ A) @ F4) @ B6))))))))).
% 2.93/1.53 thf(fact_220_eq__subset, axiom, (! [A:$tType,P2:(A > (A > $o))]: ((((ord_less_eq @ (A > (A > $o))) @ (^ [Y5:A,Z2:A]: ((Y5 = Z2)))) @ (^ [A5:A,B7:A]: ((((P2 @ A5) @ B7) | (A5 = B7)))))))).
% 2.93/1.53 thf(fact_221_GrD1, axiom, (! [B:$tType,A:$tType,X:A,Fx:B,A6:(set @ A),F4:(A > B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Fx)) @ ((((bNF_Gr @ A) @ B) @ A6) @ F4)) => (((member @ A) @ X) @ A6))))).
% 2.93/1.53 thf(fact_222_GrD2, axiom, (! [A:$tType,B:$tType,X:A,Fx:B,A6:(set @ A),F4:(A > B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Fx)) @ ((((bNF_Gr @ A) @ B) @ A6) @ F4)) => ((F4 @ X) = Fx))))).
% 2.93/1.53 thf(fact_223_diff__eq__diff__less__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B3:A,C3:A,D3:A]: ((((((minus_minus @ A) @ A2) @ B3) = (((minus_minus @ A) @ C3) @ D3)) => ((((ord_less_eq @ A) @ A2) @ B3) = (((ord_less_eq @ A) @ C3) @ D3))))))))).
% 2.93/1.53 thf(fact_224_diff__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B3:A,C3:A]: (((((ord_less_eq @ A) @ A2) @ B3) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C3)) @ (((minus_minus @ A) @ B3) @ C3))))))))).
% 2.93/1.53 thf(fact_225_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,C3:A,B3:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C3)) @ B3) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B3)) @ C3)))))))).
% 2.93/1.53 thf(fact_226_diff__eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B3:A,C3:A,D3:A]: ((((((minus_minus @ A) @ A2) @ B3) = (((minus_minus @ A) @ C3) @ D3)) => ((A2 = B3) = (C3 = D3))))))))).
% 2.93/1.53 thf(fact_227_diff__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B3:A,D3:A,C3:A]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less_eq @ A) @ D3) @ C3) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C3)) @ (((minus_minus @ A) @ B3) @ D3)))))))))).
% 2.93/1.53 thf(fact_228_diff__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B3:A,A2:A,C3:A]: (((((ord_less_eq @ A) @ B3) @ A2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C3) @ A2)) @ (((minus_minus @ A) @ C3) @ B3))))))))).
% 2.93/1.53 thf(fact_229_snd__image__times, axiom, (! [B:$tType,A:$tType,A6:(set @ B),B2:(set @ A)]: ((((A6 = (bot_bot @ (set @ B))) => (((((image @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ A6) @ (^ [Uu:B]: (B2)))) = (bot_bot @ (set @ A)))) & ((A6 != (bot_bot @ (set @ B))) => (((((image @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ A6) @ (^ [Uu:B]: (B2)))) = B2)))))).
% 2.93/1.53 thf(fact_230_fst__image__times, axiom, (! [B:$tType,A:$tType,B2:(set @ B),A6:(set @ A)]: ((((B2 = (bot_bot @ (set @ B))) => (((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A6) @ (^ [Uu:A]: (B2)))) = (bot_bot @ (set @ A)))) & ((B2 != (bot_bot @ (set @ B))) => (((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A6) @ (^ [Uu:A]: (B2)))) = A6)))))).
% 2.93/1.53 thf(fact_231_empty__Collect__eq, axiom, (! [A:$tType,P2:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P2)) = (! [X4:A]: ((~ ((P2 @ X4))))))))).
% 2.93/1.53 thf(fact_232_Collect__empty__eq, axiom, (! [A:$tType,P2:(A > $o)]: (((((collect @ A) @ P2) = (bot_bot @ (set @ A))) = (! [X4:A]: ((~ ((P2 @ X4))))))))).
% 2.93/1.53 thf(fact_233_all__not__in__conv, axiom, (! [A:$tType,A6:(set @ A)]: (((! [X4:A]: ((~ ((((member @ A) @ X4) @ A6))))) = (A6 = (bot_bot @ (set @ A))))))).
% 2.93/1.53 thf(fact_234_empty__iff, axiom, (! [A:$tType,C3:A]: ((~ ((((member @ A) @ C3) @ (bot_bot @ (set @ A)))))))).
% 2.93/1.53 thf(fact_235_image__is__empty, axiom, (! [A:$tType,B:$tType,F4:(B > A),A6:(set @ B)]: (((((((image @ B) @ A) @ F4) @ A6) = (bot_bot @ (set @ A))) = (A6 = (bot_bot @ (set @ B))))))).
% 2.93/1.53 thf(fact_236_empty__is__image, axiom, (! [A:$tType,B:$tType,F4:(B > A),A6:(set @ B)]: ((((bot_bot @ (set @ A)) = ((((image @ B) @ A) @ F4) @ A6)) = (A6 = (bot_bot @ (set @ B))))))).
% 2.93/1.53 thf(fact_237_image__empty, axiom, (! [B:$tType,A:$tType,F4:(B > A)]: ((((((image @ B) @ A) @ F4) @ (bot_bot @ (set @ B))) = (bot_bot @ (set @ A)))))).
% 2.93/1.53 thf(fact_238_subset__empty, axiom, (! [A:$tType,A6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ (bot_bot @ (set @ A))) = (A6 = (bot_bot @ (set @ A))))))).
% 2.93/1.53 thf(fact_239_empty__subsetI, axiom, (! [A:$tType,A6:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A6)))).
% 2.93/1.53 thf(fact_240_Diff__empty, axiom, (! [A:$tType,A6:(set @ A)]: (((((minus_minus @ (set @ A)) @ A6) @ (bot_bot @ (set @ A))) = A6)))).
% 2.93/1.53 thf(fact_241_empty__Diff, axiom, (! [A:$tType,A6:(set @ A)]: (((((minus_minus @ (set @ A)) @ (bot_bot @ (set @ A))) @ A6) = (bot_bot @ (set @ A)))))).
% 2.93/1.53 thf(fact_242_Diff__cancel, axiom, (! [A:$tType,A6:(set @ A)]: (((((minus_minus @ (set @ A)) @ A6) @ A6) = (bot_bot @ (set @ A)))))).
% 2.93/1.53 thf(fact_243_Sigma__empty1, axiom, (! [B:$tType,A:$tType,B2:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ (bot_bot @ (set @ A))) @ B2) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 2.93/1.53 thf(fact_244_Collect__const, axiom, (! [A:$tType,P2:$o]: (((P2 => (((collect @ A) @ (^ [S4:A]: (P2))) = (top_top @ (set @ A)))) & ((~ (P2)) => (((collect @ A) @ (^ [S4:A]: (P2))) = (bot_bot @ (set @ A)))))))).
% 2.93/1.53 thf(fact_245_Times__empty, axiom, (! [A:$tType,B:$tType,A6:(set @ A),B2:(set @ B)]: (((((((product_Sigma @ A) @ B) @ A6) @ (^ [Uu:A]: (B2))) = (bot_bot @ (set @ ((product_prod @ A) @ B)))) = ((A6 = (bot_bot @ (set @ A))) | (B2 = (bot_bot @ (set @ B)))))))).
% 2.93/1.53 thf(fact_246_Sigma__empty2, axiom, (! [B:$tType,A:$tType,A6:(set @ A)]: ((((((product_Sigma @ A) @ B) @ A6) @ (^ [Uu:A]: ((bot_bot @ (set @ B))))) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 2.93/1.53 thf(fact_247_Diff__UNIV, axiom, (! [A:$tType,A6:(set @ A)]: (((((minus_minus @ (set @ A)) @ A6) @ (top_top @ (set @ A))) = (bot_bot @ (set @ A)))))).
% 2.93/1.53 thf(fact_248_Diff__eq__empty__iff, axiom, (! [A:$tType,A6:(set @ A),B2:(set @ A)]: ((((((minus_minus @ (set @ A)) @ A6) @ B2) = (bot_bot @ (set @ A))) = (((ord_less_eq @ (set @ A)) @ A6) @ B2))))).
% 2.93/1.53 thf(fact_249_times__eq__iff, axiom, (! [A:$tType,B:$tType,A6:(set @ A),B2:(set @ B),C5:(set @ A),D4:(set @ B)]: (((((((product_Sigma @ A) @ B) @ A6) @ (^ [Uu:A]: (B2))) = ((((product_Sigma @ A) @ B) @ C5) @ (^ [Uu:A]: (D4)))) = (((A6 = C5) & (B2 = D4)) | (((A6 = (bot_bot @ (set @ A))) | (B2 = (bot_bot @ (set @ B)))) & ((C5 = (bot_bot @ (set @ A))) | (D4 = (bot_bot @ (set @ B)))))))))).
% 2.93/1.53 thf(fact_250_Sigma__empty__iff, axiom, (! [B:$tType,A:$tType,I:(set @ A),X7:(A > (set @ B))]: (((((((product_Sigma @ A) @ B) @ I) @ X7) = (bot_bot @ (set @ ((product_prod @ A) @ B)))) = (! [X4:A]: (((((member @ A) @ X4) @ I) => ((X7 @ X4) = (bot_bot @ (set @ B)))))))))).
% 2.93/1.53 thf(fact_251_bot__prod__def, axiom, (! [A:$tType,B:$tType]: ((((bot @ B) & (bot @ A)) => ((bot_bot @ ((product_prod @ A) @ B)) = ((((product_Pair @ A) @ B) @ (bot_bot @ A)) @ (bot_bot @ B))))))).
% 2.93/1.53 thf(fact_252_ex__in__conv, axiom, (! [A:$tType,A6:(set @ A)]: (((? [X4:A]: ((((member @ A) @ X4) @ A6))) = (A6 != (bot_bot @ (set @ A))))))).
% 2.93/1.53 thf(fact_253_equals0I, axiom, (! [A:$tType,A6:(set @ A)]: (((! [Y3:A]: ((~ ((((member @ A) @ Y3) @ A6))))) => (A6 = (bot_bot @ (set @ A))))))).
% 2.93/1.53 thf(fact_254_equals0D, axiom, (! [A:$tType,A6:(set @ A),A2:A]: (((A6 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A6))))))).
% 2.93/1.53 thf(fact_255_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.93/1.53 thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space, axiom, (! [A9:$tType,A10:$tType]: ((((ordere890947078_space @ A9) & (ordere890947078_space @ A10)) => (ordere890947078_space @ ((product_prod @ A9) @ A10)))))).
% 2.93/1.53 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A9:$tType,A10:$tType]: (((order_top @ A10) => (order_top @ (A9 > A10)))))).
% 2.93/1.53 thf(tcon_fun___Orderings_Otop, axiom, (! [A9:$tType,A10:$tType]: (((top @ A10) => (top @ (A9 > A10)))))).
% 2.93/1.53 thf(tcon_fun___Orderings_Oord, axiom, (! [A9:$tType,A10:$tType]: (((ord @ A10) => (ord @ (A9 > A10)))))).
% 2.93/1.53 thf(tcon_fun___Orderings_Obot, axiom, (! [A9:$tType,A10:$tType]: (((bot @ A10) => (bot @ (A9 > A10)))))).
% 2.93/1.53 thf(tcon_fun___Groups_Ominus, axiom, (! [A9:$tType,A10:$tType]: (((minus @ A10) => (minus @ (A9 > A10)))))).
% 2.93/1.53 thf(tcon_Set_Oset___Orderings_Oorder__top_1, axiom, (! [A9:$tType]: ((order_top @ (set @ A9))))).
% 2.93/1.53 thf(tcon_Set_Oset___Orderings_Otop_2, axiom, (! [A9:$tType]: ((top @ (set @ A9))))).
% 2.93/1.53 thf(tcon_Set_Oset___Orderings_Oord_3, axiom, (! [A9:$tType]: ((ord @ (set @ A9))))).
% 2.93/1.53 thf(tcon_Set_Oset___Orderings_Obot_4, axiom, (! [A9:$tType]: ((bot @ (set @ A9))))).
% 2.93/1.53 thf(tcon_Set_Oset___Groups_Ominus_5, axiom, (! [A9:$tType]: ((minus @ (set @ A9))))).
% 2.93/1.53 thf(tcon_HOL_Obool___Orderings_Oorder__top_6, axiom, (order_top @ $o)).
% 2.93/1.53 thf(tcon_HOL_Obool___Orderings_Otop_7, axiom, (top @ $o)).
% 2.93/1.53 thf(tcon_HOL_Obool___Orderings_Oord_8, axiom, (ord @ $o)).
% 2.93/1.53 thf(tcon_HOL_Obool___Orderings_Obot_9, axiom, (bot @ $o)).
% 2.93/1.53 thf(tcon_HOL_Obool___Groups_Ominus_10, axiom, (minus @ $o)).
% 2.93/1.53 thf(tcon_Product__Type_Oprod___Groups_Ocancel__ab__semigroup__add, axiom, (! [A9:$tType,A10:$tType]: ((((cancel146912293up_add @ A9) & (cancel146912293up_add @ A10)) => (cancel146912293up_add @ ((product_prod @ A9) @ A10)))))).
% 2.93/1.53 thf(tcon_Product__Type_Oprod___Groups_Oordered__ab__group__add, axiom, (! [A9:$tType,A10:$tType]: ((((ordere890947078_space @ A9) & (ordere890947078_space @ A10)) => (ordered_ab_group_add @ ((product_prod @ A9) @ A10)))))).
% 2.93/1.53 thf(tcon_Product__Type_Oprod___Orderings_Oorder__top_11, axiom, (! [A9:$tType,A10:$tType]: ((((order_top @ A9) & (order_top @ A10)) => (order_top @ ((product_prod @ A9) @ A10)))))).
% 2.93/1.53 thf(tcon_Product__Type_Oprod___Groups_Oab__group__add, axiom, (! [A9:$tType,A10:$tType]: ((((ab_group_add @ A9) & (ab_group_add @ A10)) => (ab_group_add @ ((product_prod @ A9) @ A10)))))).
% 2.93/1.53 thf(tcon_Product__Type_Oprod___Groups_Ogroup__add, axiom, (! [A9:$tType,A10:$tType]: ((((group_add @ A9) & (group_add @ A10)) => (group_add @ ((product_prod @ A9) @ A10)))))).
% 2.93/1.53 thf(tcon_Product__Type_Oprod___Orderings_Otop_12, axiom, (! [A9:$tType,A10:$tType]: ((((top @ A9) & (top @ A10)) => (top @ ((product_prod @ A9) @ A10)))))).
% 2.93/1.53 thf(tcon_Product__Type_Oprod___Orderings_Oord_13, axiom, (! [A9:$tType,A10:$tType]: ((((ord @ A9) & (ord @ A10)) => (ord @ ((product_prod @ A9) @ A10)))))).
% 2.93/1.53 thf(tcon_Product__Type_Oprod___Orderings_Obot_14, axiom, (! [A9:$tType,A10:$tType]: ((((bot @ A9) & (bot @ A10)) => (bot @ ((product_prod @ A9) @ A10)))))).
% 2.93/1.53 thf(tcon_Product__Type_Oprod___Groups_Ominus_15, axiom, (! [A9:$tType,A10:$tType]: ((((minus @ A9) & (minus @ A10)) => (minus @ ((product_prod @ A9) @ A10)))))).
% 2.93/1.53 thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_16, axiom, (order_top @ product_unit)).
% 2.93/1.53 thf(tcon_Product__Type_Ounit___Orderings_Otop_17, axiom, (top @ product_unit)).
% 2.93/1.53 thf(tcon_Product__Type_Ounit___Orderings_Oord_18, axiom, (ord @ product_unit)).
% 2.93/1.53 thf(tcon_Product__Type_Ounit___Orderings_Obot_19, axiom, (bot @ product_unit)).
% 2.93/1.53 thf(tcon_Product__Type_Ounit___Groups_Ominus_20, axiom, (minus @ product_unit)).
% 2.93/1.53 thf(help_If_3_1_T, axiom, (! [P2:$o]: (((P2 = $true) | (P2 = $false))))).
% 2.93/1.53 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 2.93/1.53 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 2.93/1.53 thf(conj_0, conjecture, (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ z) @ y)) @ pr)).
% 2.93/1.53 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.9NETwVAXmU/DTF2DTF17895.p
% 2.93/1.53 ------------------------
% 2.93/1.56 ---- Cleaned THF ---
% 2.93/1.56 thf(ty_t_Product__Type_Ounit,type,
% 2.93/1.56 product_unit: $tType ).
% 2.93/1.56
% 2.93/1.56 thf(ty_t_Product__Type_Oprod,type,
% 2.93/1.56 product_prod: $tType > $tType > $tType ).
% 2.93/1.56
% 2.93/1.56 thf(ty_t_Set_Oset,type,
% 2.93/1.56 set: $tType > $tType ).
% 2.93/1.56
% 2.93/1.56 thf(ty_tf_a,type,
% 2.93/1.56 a: $tType ).
% 2.93/1.56
% 2.93/1.56 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
% 2.93/1.56 ordere890947078_space:
% 2.93/1.56 !>[A: $tType] : $o ).
% 2.93/1.56
% 2.93/1.56 thf(sy_cl_HOL_Otype,type,
% 2.93/1.56 type:
% 2.93/1.56 !>[A: $tType] : $o ).
% 2.93/1.56
% 2.93/1.56 thf(sy_cl_Groups_Ominus,type,
% 2.93/1.56 minus:
% 2.93/1.56 !>[A: $tType] : $o ).
% 2.93/1.56
% 2.93/1.56 thf(sy_cl_Orderings_Obot,type,
% 2.93/1.56 bot:
% 2.93/1.56 !>[A: $tType] : $o ).
% 2.93/1.56
% 2.93/1.56 thf(sy_cl_Orderings_Oord,type,
% 2.93/1.56 ord:
% 2.93/1.56 !>[A: $tType] : $o ).
% 2.93/1.56
% 2.93/1.56 thf(sy_cl_Orderings_Otop,type,
% 2.93/1.56 top:
% 2.93/1.56 !>[A: $tType] : $o ).
% 2.93/1.56
% 2.93/1.56 thf(sy_cl_Groups_Ogroup__add,type,
% 2.93/1.56 group_add:
% 2.93/1.56 !>[A: $tType] : $o ).
% 2.93/1.56
% 2.93/1.56 thf(sy_cl_Groups_Oab__group__add,type,
% 2.93/1.56 ab_group_add:
% 2.93/1.56 !>[A: $tType] : $o ).
% 2.93/1.56
% 2.93/1.56 thf(sy_cl_Orderings_Oorder__top,type,
% 2.93/1.56 order_top:
% 2.93/1.56 !>[A: $tType] : $o ).
% 2.93/1.56
% 2.93/1.56 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.93/1.56 ordered_ab_group_add:
% 2.93/1.56 !>[A: $tType] : $o ).
% 2.93/1.56
% 2.93/1.56 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.93/1.56 cancel146912293up_add:
% 2.93/1.56 !>[A: $tType] : $o ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_BNF__Def_OGr,type,
% 2.93/1.56 bNF_Gr:
% 2.93/1.56 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_BNF__Def_OfstOp,type,
% 2.93/1.56 bNF_fstOp:
% 2.93/1.56 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_BNF__Def_Opick__middlep,type,
% 2.93/1.56 bNF_pick_middlep:
% 2.93/1.56 !>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_BNF__Def_OsndOp,type,
% 2.93/1.56 bNF_sndOp:
% 2.93/1.56 !>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 2.93/1.56 bNF_Greatest_image2:
% 2.93/1.56 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.93/1.56 minus_minus:
% 2.93/1.56 !>[A: $tType] : ( A > A > A ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_HOL_OThe,type,
% 2.93/1.56 the:
% 2.93/1.56 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_If,type,
% 2.93/1.56 if:
% 2.93/1.56 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Order__Relation_Opreorder__on,type,
% 2.93/1.56 order_preorder_on:
% 2.93/1.56 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.93/1.56 bot_bot:
% 2.93/1.56 !>[A: $tType] : A ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.93/1.56 ord_less_eq:
% 2.93/1.56 !>[A: $tType] : ( A > A > $o ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.93/1.56 top_top:
% 2.93/1.56 !>[A: $tType] : A ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oas__good__as,type,
% 2.93/1.56 prefer951318096ood_as:
% 2.93/1.56 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oat__least__as__good,type,
% 2.93/1.56 prefer310429814s_good:
% 2.93/1.56 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Ono__better__than,type,
% 2.93/1.56 prefer1532642881r_than:
% 2.93/1.56 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Opreference,type,
% 2.93/1.56 prefer199794634erence:
% 2.93/1.56 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Product__Type_OPair,type,
% 2.93/1.56 product_Pair:
% 2.93/1.56 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Product__Type_OSigma,type,
% 2.93/1.56 product_Sigma:
% 2.93/1.56 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Product__Type_Ocurry,type,
% 2.93/1.56 product_curry:
% 2.93/1.56 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 2.93/1.56 produc2004651681e_prod:
% 2.93/1.56 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Product__Type_Omap__prod,type,
% 2.93/1.56 product_map_prod:
% 2.93/1.56 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
% 2.93/1.56 product_rec_bool:
% 2.93/1.56 !>[T: $tType] : ( T > T > $o > T ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Product__Type_Oold_Obool_Orec__set__bool,type,
% 2.93/1.56 product_rec_set_bool:
% 2.93/1.56 !>[T: $tType] : ( T > T > $o > T > $o ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.93/1.56 product_rec_prod:
% 2.93/1.56 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
% 2.93/1.56 product_rec_set_prod:
% 2.93/1.56 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
% 2.93/1.56 product_rec_set_unit:
% 2.93/1.56 !>[T: $tType] : ( T > product_unit > T > $o ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
% 2.93/1.56 product_rec_unit:
% 2.93/1.56 !>[T: $tType] : ( T > product_unit > T ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 2.93/1.56 product_fst:
% 2.93/1.56 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 2.93/1.56 product_snd:
% 2.93/1.56 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 2.93/1.56 product_swap:
% 2.93/1.56 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Product__Type_Oproduct,type,
% 2.93/1.56 product_product:
% 2.93/1.56 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Product__Type_Oscomp,type,
% 2.93/1.56 product_scomp:
% 2.93/1.56 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Set_OCollect,type,
% 2.93/1.56 collect:
% 2.93/1.56 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Set_Oimage,type,
% 2.93/1.56 image:
% 2.93/1.56 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Wellfounded_Olex__prod,type,
% 2.93/1.56 lex_prod:
% 2.93/1.56 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_Wfrec_Osame__fst,type,
% 2.93/1.56 same_fst:
% 2.93/1.56 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_c_member,type,
% 2.93/1.56 member:
% 2.93/1.56 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_v_B,type,
% 2.93/1.56 b: set @ a ).
% 2.93/1.56
% 2.93/1.56 thf(sy_v_Pr,type,
% 2.93/1.56 pr: set @ ( product_prod @ a @ a ) ).
% 2.93/1.56
% 2.93/1.56 thf(sy_v_y,type,
% 2.93/1.56 y: a ).
% 2.93/1.56
% 2.93/1.56 thf(sy_v_z,type,
% 2.93/1.56 z: a ).
% 2.93/1.56
% 2.93/1.56 thf(fact_0_assms,axiom,
% 2.93/1.56 member @ a @ z @ ( prefer310429814s_good @ a @ y @ b @ pr ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_1_at__lst__asgd__ge,axiom,
% 2.93/1.56 ! [A: $tType,X: A,Y: A,B2: set @ A,Pr: set @ ( product_prod @ A @ A )] :
% 2.93/1.56 ( ( member @ A @ X @ ( prefer310429814s_good @ A @ Y @ B2 @ Pr ) )
% 2.93/1.56 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Pr ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_2_prod_Oinject,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 2.93/1.56 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 2.93/1.56 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 2.93/1.56 = ( ( X1 = Y1 )
% 2.93/1.56 & ( X2 = Y2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_3_old_Oprod_Oinject,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,A2: A,B3: B,A3: A,B4: B] :
% 2.93/1.56 ( ( ( product_Pair @ A @ B @ A2 @ B3 )
% 2.93/1.56 = ( product_Pair @ A @ B @ A3 @ B4 ) )
% 2.93/1.56 = ( ( A2 = A3 )
% 2.93/1.56 & ( B3 = B4 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_4_surj__pair,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 2.93/1.56 ? [X3: A,Y3: B] :
% 2.93/1.56 ( P
% 2.93/1.56 = ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_5_prod__cases,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 2.93/1.56 ( ! [A4: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B5 ) )
% 2.93/1.56 => ( P2 @ P ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_6_Pair__inject,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,A2: A,B3: B,A3: A,B4: B] :
% 2.93/1.56 ( ( ( product_Pair @ A @ B @ A2 @ B3 )
% 2.93/1.56 = ( product_Pair @ A @ B @ A3 @ B4 ) )
% 2.93/1.56 => ~ ( ( A2 = A3 )
% 2.93/1.56 => ( B3 != B4 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_7_prod__cases3,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.93/1.56 ~ ! [A4: A,B5: B,C2: C] :
% 2.93/1.56 ( Y
% 2.93/1.56 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_8_prod__cases4,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.93/1.56 ~ ! [A4: A,B5: B,C2: C,D2: D] :
% 2.93/1.56 ( Y
% 2.93/1.56 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_9_prod__cases5,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.93/1.56 ~ ! [A4: A,B5: B,C2: C,D2: D,E2: E] :
% 2.93/1.56 ( Y
% 2.93/1.56 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_10_prod__cases6,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 2.93/1.56 ~ ! [A4: A,B5: B,C2: C,D2: D,E2: E,F2: F] :
% 2.93/1.56 ( Y
% 2.93/1.56 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_11_prod__cases7,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 2.93/1.56 ~ ! [A4: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
% 2.93/1.56 ( Y
% 2.93/1.56 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_12_prod__induct3,axiom,
% 2.93/1.56 ! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.93/1.56 ( ! [A4: A,B5: B,C2: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) )
% 2.93/1.56 => ( P2 @ X ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_13_old_Oprod_Oinducts,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.93/1.56 ( ! [A4: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B5 ) )
% 2.93/1.56 => ( P2 @ Prod ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_14_old_Oprod_Oexhaust,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 2.93/1.56 ~ ! [A4: A,B5: B] :
% 2.93/1.56 ( Y
% 2.93/1.56 != ( product_Pair @ A @ B @ A4 @ B5 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_15_prod__induct7,axiom,
% 2.93/1.56 ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 2.93/1.56 ( ! [A4: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
% 2.93/1.56 => ( P2 @ X ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_16_prod__induct6,axiom,
% 2.93/1.56 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 2.93/1.56 ( ! [A4: A,B5: B,C2: C,D2: D,E2: E,F2: F] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
% 2.93/1.56 => ( P2 @ X ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_17_prod__induct5,axiom,
% 2.93/1.56 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.93/1.56 ( ! [A4: A,B5: B,C2: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 2.93/1.56 => ( P2 @ X ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_18_prod__induct4,axiom,
% 2.93/1.56 ! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.93/1.56 ( ! [A4: A,B5: B,C2: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 2.93/1.56 => ( P2 @ X ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_19_old_Oprod_Orec,axiom,
% 2.93/1.56 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B3: B] :
% 2.93/1.56 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B3 ) )
% 2.93/1.56 = ( F1 @ A2 @ B3 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_20_at__least__as__good__def,axiom,
% 2.93/1.56 ! [A: $tType] :
% 2.93/1.56 ( ( prefer310429814s_good @ A )
% 2.93/1.56 = ( ^ [X4: A,B6: set @ A,P3: set @ ( product_prod @ A @ A )] :
% 2.93/1.56 ( collect @ A
% 2.93/1.56 @ ^ [Y4: A] :
% 2.93/1.56 ( ( member @ A @ Y4 @ B6 )
% 2.93/1.56 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_21_internal__case__prod__conv,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B3: C] :
% 2.93/1.56 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B3 ) )
% 2.93/1.56 = ( C3 @ A2 @ B3 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_22_ssubst__Pair__rhs,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,R: A,S: B,R2: set @ ( product_prod @ A @ B ),S2: B] :
% 2.93/1.56 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R2 )
% 2.93/1.56 => ( ( S2 = S )
% 2.93/1.56 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_23_curry__conv,axiom,
% 2.93/1.56 ! [A: $tType,C: $tType,B: $tType] :
% 2.93/1.56 ( ( product_curry @ B @ C @ A )
% 2.93/1.56 = ( ^ [F3: ( product_prod @ B @ C ) > A,A5: B,B7: C] : ( F3 @ ( product_Pair @ B @ C @ A5 @ B7 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_24_curryI,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A2: A,B3: B] :
% 2.93/1.56 ( ( F4 @ ( product_Pair @ A @ B @ A2 @ B3 ) )
% 2.93/1.56 => ( product_curry @ A @ B @ $o @ F4 @ A2 @ B3 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_25_preference_Oindiff__trans,axiom,
% 2.93/1.56 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 2.93/1.56 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 2.93/1.56 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 2.93/1.56 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 2.93/1.56 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
% 2.93/1.56 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation ) )
% 2.93/1.56 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation )
% 2.93/1.56 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X ) @ Relation ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_26_preference_Onot__outside,axiom,
% 2.93/1.56 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 2.93/1.56 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 2.93/1.56 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 2.93/1.56 => ( member @ A @ X @ Carrier ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_27_swap__simp,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,X: B,Y: A] :
% 2.93/1.56 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 2.93/1.56 = ( product_Pair @ A @ B @ Y @ X ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_28_old_Obool_Osimps_I6_J,axiom,
% 2.93/1.56 ! [T: $tType,F1: T,F22: T] :
% 2.93/1.56 ( ( product_rec_bool @ T @ F1 @ F22 @ $false )
% 2.93/1.56 = F22 ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_29_old_Obool_Osimps_I5_J,axiom,
% 2.93/1.56 ! [T: $tType,F1: T,F22: T] :
% 2.93/1.56 ( ( product_rec_bool @ T @ F1 @ F22 @ $true )
% 2.93/1.56 = F1 ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_30_swap__swap,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,P: product_prod @ A @ B] :
% 2.93/1.56 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
% 2.93/1.56 = P ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_31_curry__K,axiom,
% 2.93/1.56 ! [B: $tType,C: $tType,A: $tType,C3: C] :
% 2.93/1.56 ( ( product_curry @ A @ B @ C
% 2.93/1.56 @ ^ [X4: product_prod @ A @ B] : C3 )
% 2.93/1.56 = ( ^ [X4: A,Y4: B] : C3 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_32_curry__def,axiom,
% 2.93/1.56 ! [C: $tType,B: $tType,A: $tType] :
% 2.93/1.56 ( ( product_curry @ A @ B @ C )
% 2.93/1.56 = ( ^ [C4: ( product_prod @ A @ B ) > C,X4: A,Y4: B] : ( C4 @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_33_curryD,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A2: A,B3: B] :
% 2.93/1.56 ( ( product_curry @ A @ B @ $o @ F4 @ A2 @ B3 )
% 2.93/1.56 => ( F4 @ ( product_Pair @ A @ B @ A2 @ B3 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_34_curryE,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A2: A,B3: B] :
% 2.93/1.56 ( ( product_curry @ A @ B @ $o @ F4 @ A2 @ B3 )
% 2.93/1.56 => ( F4 @ ( product_Pair @ A @ B @ A2 @ B3 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_35_no__better__than__def,axiom,
% 2.93/1.56 ! [A: $tType] :
% 2.93/1.56 ( ( prefer1532642881r_than @ A )
% 2.93/1.56 = ( ^ [X4: A,B6: set @ A,P3: set @ ( product_prod @ A @ A )] :
% 2.93/1.56 ( collect @ A
% 2.93/1.56 @ ^ [Y4: A] :
% 2.93/1.56 ( ( member @ A @ Y4 @ B6 )
% 2.93/1.56 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ P3 ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_36_as__good__as__def,axiom,
% 2.93/1.56 ! [A: $tType] :
% 2.93/1.56 ( ( prefer951318096ood_as @ A )
% 2.93/1.56 = ( ^ [X4: A,B6: set @ A,P3: set @ ( product_prod @ A @ A )] :
% 2.93/1.56 ( collect @ A
% 2.93/1.56 @ ^ [Y4: A] :
% 2.93/1.56 ( ( member @ A @ Y4 @ B6 )
% 2.93/1.56 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ P3 )
% 2.93/1.56 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_37_pred__equals__eq2,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 2.93/1.56 ( ( ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R2 ) )
% 2.93/1.56 = ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S3 ) ) )
% 2.93/1.56 = ( R2 = S3 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_38_preference__def,axiom,
% 2.93/1.56 ! [A: $tType] :
% 2.93/1.56 ( ( prefer199794634erence @ A )
% 2.93/1.56 = ( ^ [Carrier2: set @ A,Relation2: set @ ( product_prod @ A @ A )] :
% 2.93/1.56 ( ! [X4: A,Y4: A] :
% 2.93/1.56 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ Relation2 )
% 2.93/1.56 => ( member @ A @ X4 @ Carrier2 ) )
% 2.93/1.56 & ! [X4: A,Y4: A] :
% 2.93/1.56 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ Relation2 )
% 2.93/1.56 => ( member @ A @ Y4 @ Carrier2 ) )
% 2.93/1.56 & ( order_preorder_on @ A @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_39_preference_Ointro,axiom,
% 2.93/1.56 ! [A: $tType,Relation: set @ ( product_prod @ A @ A ),Carrier: set @ A] :
% 2.93/1.56 ( ! [X3: A,Y3: A] :
% 2.93/1.56 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ Relation )
% 2.93/1.56 => ( member @ A @ X3 @ Carrier ) )
% 2.93/1.56 => ( ! [X3: A,Y3: A] :
% 2.93/1.56 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ Relation )
% 2.93/1.56 => ( member @ A @ Y3 @ Carrier ) )
% 2.93/1.56 => ( ( order_preorder_on @ A @ Carrier @ Relation )
% 2.93/1.56 => ( prefer199794634erence @ A @ Carrier @ Relation ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_40_pair__in__swap__image,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,Y: A,X: B,A6: set @ ( product_prod @ B @ A )] :
% 2.93/1.56 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A6 ) )
% 2.93/1.56 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A6 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_41_preference_Otrans__refl,axiom,
% 2.93/1.56 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 2.93/1.56 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 2.93/1.56 => ( order_preorder_on @ A @ Carrier @ Relation ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_42_strict__contour__is__diff,axiom,
% 2.93/1.56 ! [A: $tType,B2: set @ A,Y: A,Pr: set @ ( product_prod @ A @ A )] :
% 2.93/1.56 ( ( collect @ A
% 2.93/1.56 @ ^ [A5: A] :
% 2.93/1.56 ( ( member @ A @ A5 @ B2 )
% 2.93/1.56 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ Y ) @ Pr )
% 2.93/1.56 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ A5 ) @ Pr ) ) )
% 2.93/1.56 = ( minus_minus @ ( set @ A ) @ ( prefer310429814s_good @ A @ Y @ B2 @ Pr ) @ ( prefer951318096ood_as @ A @ Y @ B2 @ Pr ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_43_old_Orec__prod__def,axiom,
% 2.93/1.56 ! [T: $tType,B: $tType,A: $tType] :
% 2.93/1.56 ( ( product_rec_prod @ A @ B @ T )
% 2.93/1.56 = ( ^ [F12: A > B > T,X4: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X4 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_44_old_Orec__bool__def,axiom,
% 2.93/1.56 ! [T: $tType] :
% 2.93/1.56 ( ( product_rec_bool @ T )
% 2.93/1.56 = ( ^ [F12: T,F23: T,X4: $o] : ( the @ T @ ( product_rec_set_bool @ T @ F12 @ F23 @ X4 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_45_mem__Collect__eq,axiom,
% 2.93/1.56 ! [A: $tType,A2: A,P2: A > $o] :
% 2.93/1.56 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 2.93/1.56 = ( P2 @ A2 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_46_Collect__mem__eq,axiom,
% 2.93/1.56 ! [A: $tType,A6: set @ A] :
% 2.93/1.56 ( ( collect @ A
% 2.93/1.56 @ ^ [X4: A] : ( member @ A @ X4 @ A6 ) )
% 2.93/1.56 = A6 ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_47_Collect__cong,axiom,
% 2.93/1.56 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.93/1.56 ( ! [X3: A] :
% 2.93/1.56 ( ( P2 @ X3 )
% 2.93/1.56 = ( Q @ X3 ) )
% 2.93/1.56 => ( ( collect @ A @ P2 )
% 2.93/1.56 = ( collect @ A @ Q ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_48_ext,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,F4: A > B,G3: A > B] :
% 2.93/1.56 ( ! [X3: A] :
% 2.93/1.56 ( ( F4 @ X3 )
% 2.93/1.56 = ( G3 @ X3 ) )
% 2.93/1.56 => ( F4 = G3 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_49_diff__Pair,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType] :
% 2.93/1.56 ( ( ( minus @ B )
% 2.93/1.56 & ( minus @ A ) )
% 2.93/1.56 => ! [A2: A,B3: B,C3: A,D3: B] :
% 2.93/1.56 ( ( minus_minus @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Pair @ A @ B @ C3 @ D3 ) )
% 2.93/1.56 = ( product_Pair @ A @ B @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ B @ B3 @ D3 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_50_the__sym__eq__trivial,axiom,
% 2.93/1.56 ! [A: $tType,X: A] :
% 2.93/1.56 ( ( the @ A
% 2.93/1.56 @ ( ^ [Y5: A,Z2: A] : ( Y5 = Z2 )
% 2.93/1.56 @ X ) )
% 2.93/1.56 = X ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_51_the__eq__trivial,axiom,
% 2.93/1.56 ! [A: $tType,A2: A] :
% 2.93/1.56 ( ( the @ A
% 2.93/1.56 @ ^ [X4: A] : ( X4 = A2 ) )
% 2.93/1.56 = A2 ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_52_the__equality,axiom,
% 2.93/1.56 ! [A: $tType,P2: A > $o,A2: A] :
% 2.93/1.56 ( ( P2 @ A2 )
% 2.93/1.56 => ( ! [X3: A] :
% 2.93/1.56 ( ( P2 @ X3 )
% 2.93/1.56 => ( X3 = A2 ) )
% 2.93/1.56 => ( ( the @ A @ P2 )
% 2.93/1.56 = A2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_53_image__ident,axiom,
% 2.93/1.56 ! [A: $tType,Y6: set @ A] :
% 2.93/1.56 ( ( image @ A @ A
% 2.93/1.56 @ ^ [X4: A] : X4
% 2.93/1.56 @ Y6 )
% 2.93/1.56 = Y6 ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_54_translation__subtract__diff,axiom,
% 2.93/1.56 ! [A: $tType] :
% 2.93/1.56 ( ( ab_group_add @ A )
% 2.93/1.56 => ! [A2: A,S: set @ A,T2: set @ A] :
% 2.93/1.56 ( ( image @ A @ A
% 2.93/1.56 @ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
% 2.93/1.56 @ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
% 2.93/1.56 = ( minus_minus @ ( set @ A )
% 2.93/1.56 @ ( image @ A @ A
% 2.93/1.56 @ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
% 2.93/1.56 @ S )
% 2.93/1.56 @ ( image @ A @ A
% 2.93/1.56 @ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
% 2.93/1.56 @ T2 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_55_DiffI,axiom,
% 2.93/1.56 ! [A: $tType,C3: A,A6: set @ A,B2: set @ A] :
% 2.93/1.56 ( ( member @ A @ C3 @ A6 )
% 2.93/1.56 => ( ~ ( member @ A @ C3 @ B2 )
% 2.93/1.56 => ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_56_image__eqI,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,B3: A,F4: B > A,X: B,A6: set @ B] :
% 2.93/1.56 ( ( B3
% 2.93/1.56 = ( F4 @ X ) )
% 2.93/1.56 => ( ( member @ B @ X @ A6 )
% 2.93/1.56 => ( member @ A @ B3 @ ( image @ B @ A @ F4 @ A6 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_57_Diff__idemp,axiom,
% 2.93/1.56 ! [A: $tType,A6: set @ A,B2: set @ A] :
% 2.93/1.56 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) @ B2 )
% 2.93/1.56 = ( minus_minus @ ( set @ A ) @ A6 @ B2 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_58_Diff__iff,axiom,
% 2.93/1.56 ! [A: $tType,C3: A,A6: set @ A,B2: set @ A] :
% 2.93/1.56 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) )
% 2.93/1.56 = ( ( member @ A @ C3 @ A6 )
% 2.93/1.56 & ~ ( member @ A @ C3 @ B2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_59_minus__set__def,axiom,
% 2.93/1.56 ! [A: $tType] :
% 2.93/1.56 ( ( minus_minus @ ( set @ A ) )
% 2.93/1.56 = ( ^ [A7: set @ A,B6: set @ A] :
% 2.93/1.56 ( collect @ A
% 2.93/1.56 @ ( minus_minus @ ( A > $o )
% 2.93/1.56 @ ^ [X4: A] : ( member @ A @ X4 @ A7 )
% 2.93/1.56 @ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_60_rev__image__eqI,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,X: A,A6: set @ A,B3: B,F4: A > B] :
% 2.93/1.56 ( ( member @ A @ X @ A6 )
% 2.93/1.56 => ( ( B3
% 2.93/1.56 = ( F4 @ X ) )
% 2.93/1.56 => ( member @ B @ B3 @ ( image @ A @ B @ F4 @ A6 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_61_ball__imageD,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P2: A > $o] :
% 2.93/1.56 ( ! [X3: A] :
% 2.93/1.56 ( ( member @ A @ X3 @ ( image @ B @ A @ F4 @ A6 ) )
% 2.93/1.56 => ( P2 @ X3 ) )
% 2.93/1.56 => ! [X5: B] :
% 2.93/1.56 ( ( member @ B @ X5 @ A6 )
% 2.93/1.56 => ( P2 @ ( F4 @ X5 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_62_image__cong,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F4: A > B,G3: A > B] :
% 2.93/1.56 ( ( M = N )
% 2.93/1.56 => ( ! [X3: A] :
% 2.93/1.56 ( ( member @ A @ X3 @ N )
% 2.93/1.56 => ( ( F4 @ X3 )
% 2.93/1.56 = ( G3 @ X3 ) ) )
% 2.93/1.56 => ( ( image @ A @ B @ F4 @ M )
% 2.93/1.56 = ( image @ A @ B @ G3 @ N ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_63_bex__imageD,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P2: A > $o] :
% 2.93/1.56 ( ? [X5: A] :
% 2.93/1.56 ( ( member @ A @ X5 @ ( image @ B @ A @ F4 @ A6 ) )
% 2.93/1.56 & ( P2 @ X5 ) )
% 2.93/1.56 => ? [X3: B] :
% 2.93/1.56 ( ( member @ B @ X3 @ A6 )
% 2.93/1.56 & ( P2 @ ( F4 @ X3 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_64_image__iff,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,Z: A,F4: B > A,A6: set @ B] :
% 2.93/1.56 ( ( member @ A @ Z @ ( image @ B @ A @ F4 @ A6 ) )
% 2.93/1.56 = ( ? [X4: B] :
% 2.93/1.56 ( ( member @ B @ X4 @ A6 )
% 2.93/1.56 & ( Z
% 2.93/1.56 = ( F4 @ X4 ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_65_imageI,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,X: A,A6: set @ A,F4: A > B] :
% 2.93/1.56 ( ( member @ A @ X @ A6 )
% 2.93/1.56 => ( member @ B @ ( F4 @ X ) @ ( image @ A @ B @ F4 @ A6 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_66_DiffD2,axiom,
% 2.93/1.56 ! [A: $tType,C3: A,A6: set @ A,B2: set @ A] :
% 2.93/1.56 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) )
% 2.93/1.56 => ~ ( member @ A @ C3 @ B2 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_67_DiffD1,axiom,
% 2.93/1.56 ! [A: $tType,C3: A,A6: set @ A,B2: set @ A] :
% 2.93/1.56 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) )
% 2.93/1.56 => ( member @ A @ C3 @ A6 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_68_DiffE,axiom,
% 2.93/1.56 ! [A: $tType,C3: A,A6: set @ A,B2: set @ A] :
% 2.93/1.56 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) )
% 2.93/1.56 => ~ ( ( member @ A @ C3 @ A6 )
% 2.93/1.56 => ( member @ A @ C3 @ B2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_69_imageE,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,B3: A,F4: B > A,A6: set @ B] :
% 2.93/1.56 ( ( member @ A @ B3 @ ( image @ B @ A @ F4 @ A6 ) )
% 2.93/1.56 => ~ ! [X3: B] :
% 2.93/1.56 ( ( B3
% 2.93/1.56 = ( F4 @ X3 ) )
% 2.93/1.56 => ~ ( member @ B @ X3 @ A6 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_70_image__image,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,C: $tType,F4: B > A,G3: C > B,A6: set @ C] :
% 2.93/1.56 ( ( image @ B @ A @ F4 @ ( image @ C @ B @ G3 @ A6 ) )
% 2.93/1.56 = ( image @ C @ A
% 2.93/1.56 @ ^ [X4: C] : ( F4 @ ( G3 @ X4 ) )
% 2.93/1.56 @ A6 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_71_Compr__image__eq,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P2: A > $o] :
% 2.93/1.56 ( ( collect @ A
% 2.93/1.56 @ ^ [X4: A] :
% 2.93/1.56 ( ( member @ A @ X4 @ ( image @ B @ A @ F4 @ A6 ) )
% 2.93/1.56 & ( P2 @ X4 ) ) )
% 2.93/1.56 = ( image @ B @ A @ F4
% 2.93/1.56 @ ( collect @ B
% 2.93/1.56 @ ^ [X4: B] :
% 2.93/1.56 ( ( member @ B @ X4 @ A6 )
% 2.93/1.56 & ( P2 @ ( F4 @ X4 ) ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_72_set__diff__eq,axiom,
% 2.93/1.56 ! [A: $tType] :
% 2.93/1.56 ( ( minus_minus @ ( set @ A ) )
% 2.93/1.56 = ( ^ [A7: set @ A,B6: set @ A] :
% 2.93/1.56 ( collect @ A
% 2.93/1.56 @ ^ [X4: A] :
% 2.93/1.56 ( ( member @ A @ X4 @ A7 )
% 2.93/1.56 & ~ ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_73_theI,axiom,
% 2.93/1.56 ! [A: $tType,P2: A > $o,A2: A] :
% 2.93/1.56 ( ( P2 @ A2 )
% 2.93/1.56 => ( ! [X3: A] :
% 2.93/1.56 ( ( P2 @ X3 )
% 2.93/1.56 => ( X3 = A2 ) )
% 2.93/1.56 => ( P2 @ ( the @ A @ P2 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_74_theI_H,axiom,
% 2.93/1.56 ! [A: $tType,P2: A > $o] :
% 2.93/1.56 ( ? [X5: A] :
% 2.93/1.56 ( ( P2 @ X5 )
% 2.93/1.56 & ! [Y3: A] :
% 2.93/1.56 ( ( P2 @ Y3 )
% 2.93/1.56 => ( Y3 = X5 ) ) )
% 2.93/1.56 => ( P2 @ ( the @ A @ P2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_75_theI2,axiom,
% 2.93/1.56 ! [A: $tType,P2: A > $o,A2: A,Q: A > $o] :
% 2.93/1.56 ( ( P2 @ A2 )
% 2.93/1.56 => ( ! [X3: A] :
% 2.93/1.56 ( ( P2 @ X3 )
% 2.93/1.56 => ( X3 = A2 ) )
% 2.93/1.56 => ( ! [X3: A] :
% 2.93/1.56 ( ( P2 @ X3 )
% 2.93/1.56 => ( Q @ X3 ) )
% 2.93/1.56 => ( Q @ ( the @ A @ P2 ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_76_If__def,axiom,
% 2.93/1.56 ! [A: $tType] :
% 2.93/1.56 ( ( if @ A )
% 2.93/1.56 = ( ^ [P3: $o,X4: A,Y4: A] :
% 2.93/1.56 ( the @ A
% 2.93/1.56 @ ^ [Z3: A] :
% 2.93/1.56 ( ( P3
% 2.93/1.56 => ( Z3 = X4 ) )
% 2.93/1.56 & ( ~ P3
% 2.93/1.56 => ( Z3 = Y4 ) ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_77_the1I2,axiom,
% 2.93/1.56 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.93/1.56 ( ? [X5: A] :
% 2.93/1.56 ( ( P2 @ X5 )
% 2.93/1.56 & ! [Y3: A] :
% 2.93/1.56 ( ( P2 @ Y3 )
% 2.93/1.56 => ( Y3 = X5 ) ) )
% 2.93/1.56 => ( ! [X3: A] :
% 2.93/1.56 ( ( P2 @ X3 )
% 2.93/1.56 => ( Q @ X3 ) )
% 2.93/1.56 => ( Q @ ( the @ A @ P2 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_78_the1__equality,axiom,
% 2.93/1.56 ! [A: $tType,P2: A > $o,A2: A] :
% 2.93/1.56 ( ? [X5: A] :
% 2.93/1.56 ( ( P2 @ X5 )
% 2.93/1.56 & ! [Y3: A] :
% 2.93/1.56 ( ( P2 @ Y3 )
% 2.93/1.56 => ( Y3 = X5 ) ) )
% 2.93/1.56 => ( ( P2 @ A2 )
% 2.93/1.56 => ( ( the @ A @ P2 )
% 2.93/1.56 = A2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_79_minus__apply,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType] :
% 2.93/1.56 ( ( minus @ B )
% 2.93/1.56 => ( ( minus_minus @ ( A > B ) )
% 2.93/1.56 = ( ^ [A7: A > B,B6: A > B,X4: A] : ( minus_minus @ B @ ( A7 @ X4 ) @ ( B6 @ X4 ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_80_Inf_OINF__identity__eq,axiom,
% 2.93/1.56 ! [A: $tType,Inf: ( set @ A ) > A,A6: set @ A] :
% 2.93/1.56 ( ( Inf
% 2.93/1.56 @ ( image @ A @ A
% 2.93/1.56 @ ^ [X4: A] : X4
% 2.93/1.56 @ A6 ) )
% 2.93/1.56 = ( Inf @ A6 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_81_Sup_OSUP__identity__eq,axiom,
% 2.93/1.56 ! [A: $tType,Sup: ( set @ A ) > A,A6: set @ A] :
% 2.93/1.56 ( ( Sup
% 2.93/1.56 @ ( image @ A @ A
% 2.93/1.56 @ ^ [X4: A] : X4
% 2.93/1.56 @ A6 ) )
% 2.93/1.56 = ( Sup @ A6 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_82_old_Orec__unit__def,axiom,
% 2.93/1.56 ! [T: $tType] :
% 2.93/1.56 ( ( product_rec_unit @ T )
% 2.93/1.56 = ( ^ [F12: T,X4: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X4 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_83_theI__unique,axiom,
% 2.93/1.56 ! [A: $tType,P2: A > $o,X: A] :
% 2.93/1.56 ( ? [X5: A] :
% 2.93/1.56 ( ( P2 @ X5 )
% 2.93/1.56 & ! [Y3: A] :
% 2.93/1.56 ( ( P2 @ Y3 )
% 2.93/1.56 => ( Y3 = X5 ) ) )
% 2.93/1.56 => ( ( P2 @ X )
% 2.93/1.56 = ( X
% 2.93/1.56 = ( the @ A @ P2 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_84_Nitpick_OThe__psimp,axiom,
% 2.93/1.56 ! [A: $tType,P2: A > $o,X: A] :
% 2.93/1.56 ( ( P2
% 2.93/1.56 = ( ^ [Y5: A,Z2: A] : ( Y5 = Z2 )
% 2.93/1.56 @ X ) )
% 2.93/1.56 => ( ( the @ A @ P2 )
% 2.93/1.56 = X ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_85_surj__swap,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType] :
% 2.93/1.56 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.93/1.56 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_86_surj__diff__right,axiom,
% 2.93/1.56 ! [A: $tType] :
% 2.93/1.56 ( ( ab_group_add @ A )
% 2.93/1.56 => ! [A2: A] :
% 2.93/1.56 ( ( image @ A @ A
% 2.93/1.56 @ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
% 2.93/1.56 @ ( top_top @ ( set @ A ) ) )
% 2.93/1.56 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_87_UNIV__I,axiom,
% 2.93/1.56 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_88_UNIV__witness,axiom,
% 2.93/1.56 ! [A: $tType] :
% 2.93/1.56 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_89_UNIV__eq__I,axiom,
% 2.93/1.56 ! [A: $tType,A6: set @ A] :
% 2.93/1.56 ( ! [X3: A] : ( member @ A @ X3 @ A6 )
% 2.93/1.56 => ( ( top_top @ ( set @ A ) )
% 2.93/1.56 = A6 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_90_UNIV__def,axiom,
% 2.93/1.56 ! [A: $tType] :
% 2.93/1.56 ( ( top_top @ ( set @ A ) )
% 2.93/1.56 = ( collect @ A
% 2.93/1.56 @ ^ [X4: A] : $true ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_91_range__eqI,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,B3: A,F4: B > A,X: B] :
% 2.93/1.56 ( ( B3
% 2.93/1.56 = ( F4 @ X ) )
% 2.93/1.56 => ( member @ A @ B3 @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_92_surj__def,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,F4: B > A] :
% 2.93/1.56 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 2.93/1.56 = ( top_top @ ( set @ A ) ) )
% 2.93/1.56 = ( ! [Y4: A] :
% 2.93/1.56 ? [X4: B] :
% 2.93/1.56 ( Y4
% 2.93/1.56 = ( F4 @ X4 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_93_rangeI,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,F4: B > A,X: B] : ( member @ A @ ( F4 @ X ) @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_94_surjI,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,G3: B > A,F4: A > B] :
% 2.93/1.56 ( ! [X3: A] :
% 2.93/1.56 ( ( G3 @ ( F4 @ X3 ) )
% 2.93/1.56 = X3 )
% 2.93/1.56 => ( ( image @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) )
% 2.93/1.56 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_95_surjE,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,F4: B > A,Y: A] :
% 2.93/1.56 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 2.93/1.56 = ( top_top @ ( set @ A ) ) )
% 2.93/1.56 => ~ ! [X3: B] :
% 2.93/1.56 ( Y
% 2.93/1.56 != ( F4 @ X3 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_96_surjD,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,F4: B > A,Y: A] :
% 2.93/1.56 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 2.93/1.56 = ( top_top @ ( set @ A ) ) )
% 2.93/1.56 => ? [X3: B] :
% 2.93/1.56 ( Y
% 2.93/1.56 = ( F4 @ X3 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_97_range__composition,axiom,
% 2.93/1.56 ! [A: $tType,C: $tType,B: $tType,F4: C > A,G3: B > C] :
% 2.93/1.56 ( ( image @ B @ A
% 2.93/1.56 @ ^ [X4: B] : ( F4 @ ( G3 @ X4 ) )
% 2.93/1.56 @ ( top_top @ ( set @ B ) ) )
% 2.93/1.56 = ( image @ C @ A @ F4 @ ( image @ B @ C @ G3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_98_rangeE,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,B3: A,F4: B > A] :
% 2.93/1.56 ( ( member @ A @ B3 @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) )
% 2.93/1.56 => ~ ! [X3: B] :
% 2.93/1.56 ( B3
% 2.93/1.56 != ( F4 @ X3 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_99_fun__diff__def,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType] :
% 2.93/1.56 ( ( minus @ B )
% 2.93/1.56 => ( ( minus_minus @ ( A > B ) )
% 2.93/1.56 = ( ^ [A7: A > B,B6: A > B,X4: A] : ( minus_minus @ B @ ( A7 @ X4 ) @ ( B6 @ X4 ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_100_Inf_OINF__cong,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,A6: set @ B,B2: set @ B,C5: B > A,D4: B > A,Inf: ( set @ A ) > A] :
% 2.93/1.56 ( ( A6 = B2 )
% 2.93/1.56 => ( ! [X3: B] :
% 2.93/1.56 ( ( member @ B @ X3 @ B2 )
% 2.93/1.56 => ( ( C5 @ X3 )
% 2.93/1.56 = ( D4 @ X3 ) ) )
% 2.93/1.56 => ( ( Inf @ ( image @ B @ A @ C5 @ A6 ) )
% 2.93/1.56 = ( Inf @ ( image @ B @ A @ D4 @ B2 ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_101_Sup_OSUP__cong,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,A6: set @ B,B2: set @ B,C5: B > A,D4: B > A,Sup: ( set @ A ) > A] :
% 2.93/1.56 ( ( A6 = B2 )
% 2.93/1.56 => ( ! [X3: B] :
% 2.93/1.56 ( ( member @ B @ X3 @ B2 )
% 2.93/1.56 => ( ( C5 @ X3 )
% 2.93/1.56 = ( D4 @ X3 ) ) )
% 2.93/1.56 => ( ( Sup @ ( image @ B @ A @ C5 @ A6 ) )
% 2.93/1.56 = ( Sup @ ( image @ B @ A @ D4 @ B2 ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_102_iso__tuple__UNIV__I,axiom,
% 2.93/1.56 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_103_top__apply,axiom,
% 2.93/1.56 ! [C: $tType,D: $tType] :
% 2.93/1.56 ( ( top @ C )
% 2.93/1.56 => ( ( top_top @ ( D > C ) )
% 2.93/1.56 = ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_104_same__fstI,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,P2: A > $o,X: A,Y7: B,Y: B,R2: A > ( set @ ( product_prod @ B @ B ) )] :
% 2.93/1.56 ( ( P2 @ X )
% 2.93/1.56 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y ) @ ( R2 @ X ) )
% 2.93/1.56 => ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y7 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P2 @ R2 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_105_top__prod__def,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType] :
% 2.93/1.56 ( ( ( top @ B )
% 2.93/1.56 & ( top @ A ) )
% 2.93/1.56 => ( ( top_top @ ( product_prod @ A @ B ) )
% 2.93/1.56 = ( product_Pair @ A @ B @ ( top_top @ A ) @ ( top_top @ B ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_106_map__prod__surj,axiom,
% 2.93/1.56 ! [A: $tType,C: $tType,D: $tType,B: $tType,F4: A > B,G3: C > D] :
% 2.93/1.56 ( ( ( image @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 2.93/1.56 = ( top_top @ ( set @ B ) ) )
% 2.93/1.56 => ( ( ( image @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
% 2.93/1.56 = ( top_top @ ( set @ D ) ) )
% 2.93/1.56 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F4 @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 2.93/1.56 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_107_range__fst,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType] :
% 2.93/1.56 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.93/1.56 = ( top_top @ ( set @ A ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_108_map__prod__ident,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType] :
% 2.93/1.56 ( ( product_map_prod @ A @ A @ B @ B
% 2.93/1.56 @ ^ [X4: A] : X4
% 2.93/1.56 @ ^ [Y4: B] : Y4 )
% 2.93/1.56 = ( ^ [Z3: product_prod @ A @ B] : Z3 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_109_map__prod__simp,axiom,
% 2.93/1.56 ! [C: $tType,A: $tType,B: $tType,D: $tType,F4: C > A,G3: D > B,A2: C,B3: D] :
% 2.93/1.56 ( ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 @ ( product_Pair @ C @ D @ A2 @ B3 ) )
% 2.93/1.56 = ( product_Pair @ A @ B @ ( F4 @ A2 ) @ ( G3 @ B3 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_110_fst__map__prod,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,D: $tType,C: $tType,F4: C > A,G3: D > B,X: product_prod @ C @ D] :
% 2.93/1.56 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 @ X ) )
% 2.93/1.56 = ( F4 @ ( product_fst @ C @ D @ X ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_111_map__prod__imageI,axiom,
% 2.93/1.56 ! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B3: B,R2: set @ ( product_prod @ A @ B ),F4: A > C,G3: B > D] :
% 2.93/1.56 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ R2 )
% 2.93/1.56 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F4 @ A2 ) @ ( G3 @ B3 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F4 @ G3 ) @ R2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_112_fst__diff,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType] :
% 2.93/1.56 ( ( ( minus @ A )
% 2.93/1.56 & ( minus @ B ) )
% 2.93/1.56 => ! [X: product_prod @ A @ B,Y: product_prod @ A @ B] :
% 2.93/1.56 ( ( product_fst @ A @ B @ ( minus_minus @ ( product_prod @ A @ B ) @ X @ Y ) )
% 2.93/1.56 = ( minus_minus @ A @ ( product_fst @ A @ B @ X ) @ ( product_fst @ A @ B @ Y ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_113_fst__top,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType] :
% 2.93/1.56 ( ( ( top @ A )
% 2.93/1.56 & ( top @ B ) )
% 2.93/1.56 => ( ( product_fst @ A @ B @ ( top_top @ ( product_prod @ A @ B ) ) )
% 2.93/1.56 = ( top_top @ A ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_114_top__empty__eq,axiom,
% 2.93/1.56 ! [A: $tType] :
% 2.93/1.56 ( ( top_top @ ( A > $o ) )
% 2.93/1.56 = ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_115_top__set__def,axiom,
% 2.93/1.56 ! [A: $tType] :
% 2.93/1.56 ( ( top_top @ ( set @ A ) )
% 2.93/1.56 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_116_fst__eqD,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,X: A,Y: B,A2: A] :
% 2.93/1.56 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 2.93/1.56 = A2 )
% 2.93/1.56 => ( X = A2 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_117_fst__conv,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,X1: A,X2: B] :
% 2.93/1.56 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 2.93/1.56 = X1 ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_118_prod__fun__imageE,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F4: C > A,G3: D > B,R2: set @ ( product_prod @ C @ D )] :
% 2.93/1.56 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 ) @ R2 ) )
% 2.93/1.56 => ~ ! [X3: C,Y3: D] :
% 2.93/1.56 ( ( C3
% 2.93/1.56 = ( product_Pair @ A @ B @ ( F4 @ X3 ) @ ( G3 @ Y3 ) ) )
% 2.93/1.56 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_119_top__empty__eq2,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType] :
% 2.93/1.56 ( ( top_top @ ( A > B > $o ) )
% 2.93/1.56 = ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_120_eq__fst__iff,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,A2: A,P: product_prod @ A @ B] :
% 2.93/1.56 ( ( A2
% 2.93/1.56 = ( product_fst @ A @ B @ P ) )
% 2.93/1.56 = ( ? [B7: B] :
% 2.93/1.56 ( P
% 2.93/1.56 = ( product_Pair @ A @ B @ A2 @ B7 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_121_fstI,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
% 2.93/1.56 ( ( X
% 2.93/1.56 = ( product_Pair @ A @ B @ Y @ Z ) )
% 2.93/1.56 => ( ( product_fst @ A @ B @ X )
% 2.93/1.56 = Y ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_122_prod_Omap__ident,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 2.93/1.56 ( ( product_map_prod @ A @ A @ B @ B
% 2.93/1.56 @ ^ [X4: A] : X4
% 2.93/1.56 @ ^ [X4: B] : X4
% 2.93/1.56 @ T2 )
% 2.93/1.56 = T2 ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_123_minus__prod__def,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType] :
% 2.93/1.56 ( ( ( minus @ A )
% 2.93/1.56 & ( minus @ B ) )
% 2.93/1.56 => ( ( minus_minus @ ( product_prod @ A @ B ) )
% 2.93/1.56 = ( ^ [X4: product_prod @ A @ B,Y4: product_prod @ A @ B] : ( product_Pair @ A @ B @ ( minus_minus @ A @ ( product_fst @ A @ B @ X4 ) @ ( product_fst @ A @ B @ Y4 ) ) @ ( minus_minus @ B @ ( product_snd @ A @ B @ X4 ) @ ( product_snd @ A @ B @ Y4 ) ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_124_range__snd,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType] :
% 2.93/1.56 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.93/1.56 = ( top_top @ ( set @ A ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_125_top1I,axiom,
% 2.93/1.56 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_126_top2I,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,X: A,Y: B] : ( top_top @ ( A > B > $o ) @ X @ Y ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_127_snd__map__prod,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,D: $tType,C: $tType,F4: C > B,G3: D > A,X: product_prod @ C @ D] :
% 2.93/1.56 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F4 @ G3 @ X ) )
% 2.93/1.56 = ( G3 @ ( product_snd @ C @ D @ X ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_128_prod_Ocollapse,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.93/1.56 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 2.93/1.56 = Prod ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_129_fst__swap,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,X: product_prod @ B @ A] :
% 2.93/1.56 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
% 2.93/1.56 = ( product_snd @ B @ A @ X ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_130_snd__swap,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,X: product_prod @ A @ B] :
% 2.93/1.56 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
% 2.93/1.56 = ( product_fst @ A @ B @ X ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_131_snd__diff,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType] :
% 2.93/1.56 ( ( ( minus @ B )
% 2.93/1.56 & ( minus @ A ) )
% 2.93/1.56 => ! [X: product_prod @ B @ A,Y: product_prod @ B @ A] :
% 2.93/1.56 ( ( product_snd @ B @ A @ ( minus_minus @ ( product_prod @ B @ A ) @ X @ Y ) )
% 2.93/1.56 = ( minus_minus @ A @ ( product_snd @ B @ A @ X ) @ ( product_snd @ B @ A @ Y ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_132_snd__top,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType] :
% 2.93/1.56 ( ( ( top @ A )
% 2.93/1.56 & ( top @ B ) )
% 2.93/1.56 => ( ( product_snd @ B @ A @ ( top_top @ ( product_prod @ B @ A ) ) )
% 2.93/1.56 = ( top_top @ A ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_133_prod__eqI,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,P: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 2.93/1.56 ( ( ( product_fst @ A @ B @ P )
% 2.93/1.56 = ( product_fst @ A @ B @ Q2 ) )
% 2.93/1.56 => ( ( ( product_snd @ A @ B @ P )
% 2.93/1.56 = ( product_snd @ A @ B @ Q2 ) )
% 2.93/1.56 => ( P = Q2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_134_prod_Oexpand,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 2.93/1.56 ( ( ( ( product_fst @ A @ B @ Prod )
% 2.93/1.56 = ( product_fst @ A @ B @ Prod2 ) )
% 2.93/1.56 & ( ( product_snd @ A @ B @ Prod )
% 2.93/1.56 = ( product_snd @ A @ B @ Prod2 ) ) )
% 2.93/1.56 => ( Prod = Prod2 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_135_prod__eq__iff,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType] :
% 2.93/1.56 ( ( ^ [Y5: product_prod @ A @ B,Z2: product_prod @ A @ B] : ( Y5 = Z2 ) )
% 2.93/1.56 = ( ^ [S4: product_prod @ A @ B,T3: product_prod @ A @ B] :
% 2.93/1.56 ( ( ( product_fst @ A @ B @ S4 )
% 2.93/1.56 = ( product_fst @ A @ B @ T3 ) )
% 2.93/1.56 & ( ( product_snd @ A @ B @ S4 )
% 2.93/1.56 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_136_eq__snd__iff,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,B3: A,P: product_prod @ B @ A] :
% 2.93/1.56 ( ( B3
% 2.93/1.56 = ( product_snd @ B @ A @ P ) )
% 2.93/1.56 = ( ? [A5: B] :
% 2.93/1.56 ( P
% 2.93/1.56 = ( product_Pair @ B @ A @ A5 @ B3 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_137_sndI,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
% 2.93/1.56 ( ( X
% 2.93/1.56 = ( product_Pair @ A @ B @ Y @ Z ) )
% 2.93/1.56 => ( ( product_snd @ A @ B @ X )
% 2.93/1.56 = Z ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_138_snd__eqD,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,X: B,Y: A,A2: A] :
% 2.93/1.56 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 2.93/1.56 = A2 )
% 2.93/1.56 => ( Y = A2 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_139_snd__conv,axiom,
% 2.93/1.56 ! [Aa: $tType,A: $tType,X1: Aa,X2: A] :
% 2.93/1.56 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X2 ) )
% 2.93/1.56 = X2 ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_140_surjective__pairing,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 2.93/1.56 ( T2
% 2.93/1.56 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_141_prod_Oexhaust__sel,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.93/1.56 ( Prod
% 2.93/1.56 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_142_prod_Oswap__def,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType] :
% 2.93/1.56 ( ( product_swap @ A @ B )
% 2.93/1.56 = ( ^ [P4: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P4 ) @ ( product_fst @ A @ B @ P4 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_143_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,P2: A > B > $o,X: A,Y: B,A2: product_prod @ A @ B] :
% 2.93/1.56 ( ( P2 @ X @ Y )
% 2.93/1.56 => ( ( A2
% 2.93/1.56 = ( product_Pair @ A @ B @ X @ Y ) )
% 2.93/1.56 => ( P2 @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_144_conjI__realizer,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,P2: A > $o,P: A,Q: B > $o,Q2: B] :
% 2.93/1.56 ( ( P2 @ P )
% 2.93/1.56 => ( ( Q @ Q2 )
% 2.93/1.56 => ( ( P2 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) )
% 2.93/1.56 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_145_exI__realizer,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,P2: A > B > $o,Y: A,X: B] :
% 2.93/1.56 ( ( P2 @ Y @ X )
% 2.93/1.56 => ( P2 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_146_top__conj_I2_J,axiom,
% 2.93/1.56 ! [A: $tType,P2: $o,X: A] :
% 2.93/1.56 ( ( P2
% 2.93/1.56 & ( top_top @ ( A > $o ) @ X ) )
% 2.93/1.56 = P2 ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_147_top__conj_I1_J,axiom,
% 2.93/1.56 ! [A: $tType,X: A,P2: $o] :
% 2.93/1.56 ( ( ( top_top @ ( A > $o ) @ X )
% 2.93/1.56 & P2 )
% 2.93/1.56 = P2 ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_148_scomp__unfold,axiom,
% 2.93/1.56 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 2.93/1.56 ( ( product_scomp @ A @ B @ C @ D )
% 2.93/1.56 = ( ^ [F3: A > ( product_prod @ B @ C ),G4: B > C > D,X4: A] : ( G4 @ ( product_fst @ B @ C @ ( F3 @ X4 ) ) @ ( product_snd @ B @ C @ ( F3 @ X4 ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_149_sndOp__def,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,C: $tType] :
% 2.93/1.56 ( ( bNF_sndOp @ C @ A @ B )
% 2.93/1.56 = ( ^ [P3: C > A > $o,Q3: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P3 @ Q3 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_150_fstOp__def,axiom,
% 2.93/1.56 ! [C: $tType,B: $tType,A: $tType] :
% 2.93/1.56 ( ( bNF_fstOp @ A @ B @ C )
% 2.93/1.56 = ( ^ [P3: A > B > $o,Q3: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P3 @ Q3 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_151_scomp__Pair,axiom,
% 2.93/1.56 ! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
% 2.93/1.56 ( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
% 2.93/1.56 = X ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_152_scomp__scomp,axiom,
% 2.93/1.56 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: $tType,E: $tType,F4: A > ( product_prod @ E @ F ),G3: E > F > ( product_prod @ C @ D ),H: C > D > B] :
% 2.93/1.56 ( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F @ ( product_prod @ C @ D ) @ F4 @ G3 ) @ H )
% 2.93/1.56 = ( product_scomp @ A @ E @ F @ B @ F4
% 2.93/1.56 @ ^ [X4: E] : ( product_scomp @ F @ C @ D @ B @ ( G3 @ X4 ) @ H ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_153_Pair__scomp,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,C: $tType,X: C,F4: C > A > B] :
% 2.93/1.56 ( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F4 )
% 2.93/1.56 = ( F4 @ X ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_154_image2__eqI,axiom,
% 2.93/1.56 ! [A: $tType,C: $tType,B: $tType,B3: A,F4: B > A,X: B,C3: C,G3: B > C,A6: set @ B] :
% 2.93/1.56 ( ( B3
% 2.93/1.56 = ( F4 @ X ) )
% 2.93/1.56 => ( ( C3
% 2.93/1.56 = ( G3 @ X ) )
% 2.93/1.56 => ( ( member @ B @ X @ A6 )
% 2.93/1.56 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B3 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A6 @ F4 @ G3 ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_155_map__prod__surj__on,axiom,
% 2.93/1.56 ! [B: $tType,D: $tType,C: $tType,A: $tType,F4: B > A,A6: set @ B,A8: set @ A,G3: D > C,B2: set @ D,B8: set @ C] :
% 2.93/1.56 ( ( ( image @ B @ A @ F4 @ A6 )
% 2.93/1.56 = A8 )
% 2.93/1.56 => ( ( ( image @ D @ C @ G3 @ B2 )
% 2.93/1.56 = B8 )
% 2.93/1.56 => ( ( image @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F4 @ G3 )
% 2.93/1.56 @ ( product_Sigma @ B @ D @ A6
% 2.93/1.56 @ ^ [Uu: B] : B2 ) )
% 2.93/1.56 = ( product_Sigma @ A @ C @ A8
% 2.93/1.56 @ ^ [Uu: A] : B8 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_156_in__lex__prod,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,A2: A,B3: B,A3: A,B4: B,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
% 2.93/1.56 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Pair @ A @ B @ A3 @ B4 ) ) @ ( lex_prod @ A @ B @ R @ S ) )
% 2.93/1.56 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A3 ) @ R )
% 2.93/1.56 | ( ( A2 = A3 )
% 2.93/1.56 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B3 @ B4 ) @ S ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_157_mem__Sigma__iff,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,A2: A,B3: B,A6: set @ A,B2: A > ( set @ B )] :
% 2.93/1.56 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Sigma @ A @ B @ A6 @ B2 ) )
% 2.93/1.56 = ( ( member @ A @ A2 @ A6 )
% 2.93/1.56 & ( member @ B @ B3 @ ( B2 @ A2 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_158_SigmaI,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,A2: A,A6: set @ A,B3: B,B2: A > ( set @ B )] :
% 2.93/1.56 ( ( member @ A @ A2 @ A6 )
% 2.93/1.56 => ( ( member @ B @ B3 @ ( B2 @ A2 ) )
% 2.93/1.56 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Sigma @ A @ B @ A6 @ B2 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_159_UNIV__Times__UNIV,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType] :
% 2.93/1.56 ( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 2.93/1.56 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) )
% 2.93/1.56 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_160_Times__Diff__distrib1,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,A6: set @ A,B2: set @ A,C5: set @ B] :
% 2.93/1.56 ( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ A6 @ B2 )
% 2.93/1.56 @ ^ [Uu: A] : C5 )
% 2.93/1.56 = ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
% 2.93/1.56 @ ( product_Sigma @ A @ B @ A6
% 2.93/1.56 @ ^ [Uu: A] : C5 )
% 2.93/1.56 @ ( product_Sigma @ A @ B @ B2
% 2.93/1.56 @ ^ [Uu: A] : C5 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_161_Sigma__Diff__distrib2,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,I: set @ A,A6: A > ( set @ B ),B2: A > ( set @ B )] :
% 2.93/1.56 ( ( product_Sigma @ A @ B @ I
% 2.93/1.56 @ ^ [I2: A] : ( minus_minus @ ( set @ B ) @ ( A6 @ I2 ) @ ( B2 @ I2 ) ) )
% 2.93/1.56 = ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I @ A6 ) @ ( product_Sigma @ A @ B @ I @ B2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_162_Sigma__cong,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,A6: set @ A,B2: set @ A,C5: A > ( set @ B ),D4: A > ( set @ B )] :
% 2.93/1.56 ( ( A6 = B2 )
% 2.93/1.56 => ( ! [X3: A] :
% 2.93/1.56 ( ( member @ A @ X3 @ B2 )
% 2.93/1.56 => ( ( C5 @ X3 )
% 2.93/1.56 = ( D4 @ X3 ) ) )
% 2.93/1.56 => ( ( product_Sigma @ A @ B @ A6 @ C5 )
% 2.93/1.56 = ( product_Sigma @ A @ B @ B2 @ D4 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_163_Times__eq__cancel2,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,X: A,C5: set @ A,A6: set @ B,B2: set @ B] :
% 2.93/1.56 ( ( member @ A @ X @ C5 )
% 2.93/1.56 => ( ( ( product_Sigma @ B @ A @ A6
% 2.93/1.56 @ ^ [Uu: B] : C5 )
% 2.93/1.56 = ( product_Sigma @ B @ A @ B2
% 2.93/1.56 @ ^ [Uu: B] : C5 ) )
% 2.93/1.56 = ( A6 = B2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_164_Sigma__Diff__distrib1,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,I: set @ A,J: set @ A,C5: A > ( set @ B )] :
% 2.93/1.56 ( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ I @ J ) @ C5 )
% 2.93/1.56 = ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I @ C5 ) @ ( product_Sigma @ A @ B @ J @ C5 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_165_SigmaE2,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,A2: A,B3: B,A6: set @ A,B2: A > ( set @ B )] :
% 2.93/1.56 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Sigma @ A @ B @ A6 @ B2 ) )
% 2.93/1.56 => ~ ( ( member @ A @ A2 @ A6 )
% 2.93/1.56 => ~ ( member @ B @ B3 @ ( B2 @ A2 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_166_SigmaD2,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,A2: A,B3: B,A6: set @ A,B2: A > ( set @ B )] :
% 2.93/1.56 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Sigma @ A @ B @ A6 @ B2 ) )
% 2.93/1.56 => ( member @ B @ B3 @ ( B2 @ A2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_167_SigmaD1,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,A2: A,B3: B,A6: set @ A,B2: A > ( set @ B )] :
% 2.93/1.56 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Sigma @ A @ B @ A6 @ B2 ) )
% 2.93/1.56 => ( member @ A @ A2 @ A6 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_168_SigmaE,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,C3: product_prod @ A @ B,A6: set @ A,B2: A > ( set @ B )] :
% 2.93/1.56 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( product_Sigma @ A @ B @ A6 @ B2 ) )
% 2.93/1.56 => ~ ! [X3: A] :
% 2.93/1.56 ( ( member @ A @ X3 @ A6 )
% 2.93/1.56 => ! [Y3: B] :
% 2.93/1.56 ( ( member @ B @ Y3 @ ( B2 @ X3 ) )
% 2.93/1.56 => ( C3
% 2.93/1.56 != ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_169_mem__Times__iff,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,X: product_prod @ A @ B,A6: set @ A,B2: set @ B] :
% 2.93/1.56 ( ( member @ ( product_prod @ A @ B ) @ X
% 2.93/1.56 @ ( product_Sigma @ A @ B @ A6
% 2.93/1.56 @ ^ [Uu: A] : B2 ) )
% 2.93/1.56 = ( ( member @ A @ ( product_fst @ A @ B @ X ) @ A6 )
% 2.93/1.56 & ( member @ B @ ( product_snd @ A @ B @ X ) @ B2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_170_product__swap,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,A6: set @ B,B2: set @ A] :
% 2.93/1.56 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
% 2.93/1.56 @ ( product_Sigma @ B @ A @ A6
% 2.93/1.56 @ ^ [Uu: B] : B2 ) )
% 2.93/1.56 = ( product_Sigma @ A @ B @ B2
% 2.93/1.56 @ ^ [Uu: A] : A6 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_171_Product__Type_Oproduct__def,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType] :
% 2.93/1.56 ( ( product_product @ A @ B )
% 2.93/1.56 = ( ^ [A7: set @ A,B6: set @ B] :
% 2.93/1.56 ( product_Sigma @ A @ B @ A7
% 2.93/1.56 @ ^ [Uu: A] : B6 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_172_member__product,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A6: set @ A,B2: set @ B] :
% 2.93/1.56 ( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A6 @ B2 ) )
% 2.93/1.56 = ( member @ ( product_prod @ A @ B ) @ X
% 2.93/1.56 @ ( product_Sigma @ A @ B @ A6
% 2.93/1.56 @ ^ [Uu: A] : B2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_173_subset__fst__snd,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,A6: set @ ( product_prod @ A @ B )] :
% 2.93/1.56 ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6
% 2.93/1.56 @ ( product_Sigma @ A @ B @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A6 )
% 2.93/1.56 @ ^ [Uu: A] : ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A6 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_174_subset__antisym,axiom,
% 2.93/1.56 ! [A: $tType,A6: set @ A,B2: set @ A] :
% 2.93/1.56 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
% 2.93/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A6 )
% 2.93/1.56 => ( A6 = B2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_175_subsetI,axiom,
% 2.93/1.56 ! [A: $tType,A6: set @ A,B2: set @ A] :
% 2.93/1.56 ( ! [X3: A] :
% 2.93/1.56 ( ( member @ A @ X3 @ A6 )
% 2.93/1.56 => ( member @ A @ X3 @ B2 ) )
% 2.93/1.56 => ( ord_less_eq @ ( set @ A ) @ A6 @ B2 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_176_Pair__le,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType] :
% 2.93/1.56 ( ( ( ord @ B )
% 2.93/1.56 & ( ord @ A ) )
% 2.93/1.56 => ! [A2: A,B3: B,C3: A,D3: B] :
% 2.93/1.56 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Pair @ A @ B @ C3 @ D3 ) )
% 2.93/1.56 = ( ( ord_less_eq @ A @ A2 @ C3 )
% 2.93/1.56 & ( ord_less_eq @ B @ B3 @ D3 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_177_Times__subset__cancel2,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,X: A,C5: set @ A,A6: set @ B,B2: set @ B] :
% 2.93/1.56 ( ( member @ A @ X @ C5 )
% 2.93/1.56 => ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
% 2.93/1.56 @ ( product_Sigma @ B @ A @ A6
% 2.93/1.56 @ ^ [Uu: B] : C5 )
% 2.93/1.56 @ ( product_Sigma @ B @ A @ B2
% 2.93/1.56 @ ^ [Uu: B] : C5 ) )
% 2.93/1.56 = ( ord_less_eq @ ( set @ B ) @ A6 @ B2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_178_Sigma__mono,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,A6: set @ A,C5: set @ A,B2: A > ( set @ B ),D4: A > ( set @ B )] :
% 2.93/1.56 ( ( ord_less_eq @ ( set @ A ) @ A6 @ C5 )
% 2.93/1.56 => ( ! [X3: A] :
% 2.93/1.56 ( ( member @ A @ X3 @ A6 )
% 2.93/1.56 => ( ord_less_eq @ ( set @ B ) @ ( B2 @ X3 ) @ ( D4 @ X3 ) ) )
% 2.93/1.56 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A6 @ B2 ) @ ( product_Sigma @ A @ B @ C5 @ D4 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_179_Pair__mono,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType] :
% 2.93/1.56 ( ( ( ord @ B )
% 2.93/1.56 & ( ord @ A ) )
% 2.93/1.56 => ! [X: A,X6: A,Y: B,Y7: B] :
% 2.93/1.56 ( ( ord_less_eq @ A @ X @ X6 )
% 2.93/1.56 => ( ( ord_less_eq @ B @ Y @ Y7 )
% 2.93/1.56 => ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( product_Pair @ A @ B @ X6 @ Y7 ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_180_image__mono,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,A6: set @ A,B2: set @ A,F4: A > B] :
% 2.93/1.56 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
% 2.93/1.56 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ ( image @ A @ B @ F4 @ B2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_181_image__subsetI,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,A6: set @ A,F4: A > B,B2: set @ B] :
% 2.93/1.56 ( ! [X3: A] :
% 2.93/1.56 ( ( member @ A @ X3 @ A6 )
% 2.93/1.56 => ( member @ B @ ( F4 @ X3 ) @ B2 ) )
% 2.93/1.56 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ B2 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_182_subset__imageE,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,B2: set @ A,F4: B > A,A6: set @ B] :
% 2.93/1.56 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F4 @ A6 ) )
% 2.93/1.56 => ~ ! [C6: set @ B] :
% 2.93/1.56 ( ( ord_less_eq @ ( set @ B ) @ C6 @ A6 )
% 2.93/1.56 => ( B2
% 2.93/1.56 != ( image @ B @ A @ F4 @ C6 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_183_image__subset__iff,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,B2: set @ A] :
% 2.93/1.56 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ B2 )
% 2.93/1.56 = ( ! [X4: B] :
% 2.93/1.56 ( ( member @ B @ X4 @ A6 )
% 2.93/1.56 => ( member @ A @ ( F4 @ X4 ) @ B2 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_184_subset__image__iff,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,B2: set @ A,F4: B > A,A6: set @ B] :
% 2.93/1.56 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F4 @ A6 ) )
% 2.93/1.56 = ( ? [AA: set @ B] :
% 2.93/1.56 ( ( ord_less_eq @ ( set @ B ) @ AA @ A6 )
% 2.93/1.56 & ( B2
% 2.93/1.56 = ( image @ B @ A @ F4 @ AA ) ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_185_subrelI,axiom,
% 2.93/1.56 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 2.93/1.56 ( ! [X3: A,Y3: B] :
% 2.93/1.56 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
% 2.93/1.56 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) )
% 2.93/1.56 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_186_Collect__mono__iff,axiom,
% 2.93/1.56 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.93/1.56 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
% 2.93/1.56 = ( ! [X4: A] :
% 2.93/1.56 ( ( P2 @ X4 )
% 2.93/1.56 => ( Q @ X4 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_187_set__eq__subset,axiom,
% 2.93/1.56 ! [A: $tType] :
% 2.93/1.56 ( ( ^ [Y5: set @ A,Z2: set @ A] : ( Y5 = Z2 ) )
% 2.93/1.56 = ( ^ [A7: set @ A,B6: set @ A] :
% 2.93/1.56 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 2.93/1.56 & ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_188_subset__trans,axiom,
% 2.93/1.56 ! [A: $tType,A6: set @ A,B2: set @ A,C5: set @ A] :
% 2.93/1.56 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
% 2.93/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C5 )
% 2.93/1.56 => ( ord_less_eq @ ( set @ A ) @ A6 @ C5 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_189_Collect__mono,axiom,
% 2.93/1.56 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.93/1.56 ( ! [X3: A] :
% 2.93/1.56 ( ( P2 @ X3 )
% 2.93/1.56 => ( Q @ X3 ) )
% 2.93/1.56 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_190_subset__refl,axiom,
% 2.93/1.56 ! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ A6 ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_191_subset__iff,axiom,
% 2.93/1.56 ! [A: $tType] :
% 2.93/1.56 ( ( ord_less_eq @ ( set @ A ) )
% 2.93/1.56 = ( ^ [A7: set @ A,B6: set @ A] :
% 2.93/1.56 ! [T3: A] :
% 2.93/1.56 ( ( member @ A @ T3 @ A7 )
% 2.93/1.56 => ( member @ A @ T3 @ B6 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_192_equalityD2,axiom,
% 2.93/1.56 ! [A: $tType,A6: set @ A,B2: set @ A] :
% 2.93/1.56 ( ( A6 = B2 )
% 2.93/1.56 => ( ord_less_eq @ ( set @ A ) @ B2 @ A6 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_193_equalityD1,axiom,
% 2.93/1.56 ! [A: $tType,A6: set @ A,B2: set @ A] :
% 2.93/1.56 ( ( A6 = B2 )
% 2.93/1.56 => ( ord_less_eq @ ( set @ A ) @ A6 @ B2 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_194_subset__eq,axiom,
% 2.93/1.56 ! [A: $tType] :
% 2.93/1.56 ( ( ord_less_eq @ ( set @ A ) )
% 2.93/1.56 = ( ^ [A7: set @ A,B6: set @ A] :
% 2.93/1.56 ! [X4: A] :
% 2.93/1.56 ( ( member @ A @ X4 @ A7 )
% 2.93/1.56 => ( member @ A @ X4 @ B6 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_195_equalityE,axiom,
% 2.93/1.56 ! [A: $tType,A6: set @ A,B2: set @ A] :
% 2.93/1.56 ( ( A6 = B2 )
% 2.93/1.56 => ~ ( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
% 2.93/1.56 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A6 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_196_subsetD,axiom,
% 2.93/1.56 ! [A: $tType,A6: set @ A,B2: set @ A,C3: A] :
% 2.93/1.56 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
% 2.93/1.56 => ( ( member @ A @ C3 @ A6 )
% 2.93/1.56 => ( member @ A @ C3 @ B2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_197_in__mono,axiom,
% 2.93/1.56 ! [A: $tType,A6: set @ A,B2: set @ A,X: A] :
% 2.93/1.56 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
% 2.93/1.56 => ( ( member @ A @ X @ A6 )
% 2.93/1.56 => ( member @ A @ X @ B2 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_198_Diff__mono,axiom,
% 2.93/1.56 ! [A: $tType,A6: set @ A,C5: set @ A,D4: set @ A,B2: set @ A] :
% 2.93/1.56 ( ( ord_less_eq @ ( set @ A ) @ A6 @ C5 )
% 2.93/1.56 => ( ( ord_less_eq @ ( set @ A ) @ D4 @ B2 )
% 2.93/1.56 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) @ ( minus_minus @ ( set @ A ) @ C5 @ D4 ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_199_Diff__subset,axiom,
% 2.93/1.56 ! [A: $tType,A6: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) @ A6 ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_200_double__diff,axiom,
% 2.93/1.56 ! [A: $tType,A6: set @ A,B2: set @ A,C5: set @ A] :
% 2.93/1.56 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
% 2.93/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C5 )
% 2.93/1.56 => ( ( minus_minus @ ( set @ A ) @ B2 @ ( minus_minus @ ( set @ A ) @ C5 @ A6 ) )
% 2.93/1.56 = A6 ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_201_subset__UNIV,axiom,
% 2.93/1.56 ! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_202_top_Oextremum__uniqueI,axiom,
% 2.93/1.56 ! [A: $tType] :
% 2.93/1.56 ( ( order_top @ A )
% 2.93/1.56 => ! [A2: A] :
% 2.93/1.56 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 2.93/1.56 => ( A2
% 2.93/1.56 = ( top_top @ A ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_203_top_Oextremum__unique,axiom,
% 2.93/1.56 ! [A: $tType] :
% 2.93/1.56 ( ( order_top @ A )
% 2.93/1.56 => ! [A2: A] :
% 2.93/1.56 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 2.93/1.56 = ( A2
% 2.93/1.56 = ( top_top @ A ) ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_204_top__greatest,axiom,
% 2.93/1.56 ! [A: $tType] :
% 2.93/1.56 ( ( order_top @ A )
% 2.93/1.56 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_205_prop__restrict,axiom,
% 2.93/1.56 ! [A: $tType,X: A,Z4: set @ A,X7: set @ A,P2: A > $o] :
% 2.93/1.56 ( ( member @ A @ X @ Z4 )
% 2.93/1.56 => ( ( ord_less_eq @ ( set @ A ) @ Z4
% 2.93/1.56 @ ( collect @ A
% 2.93/1.56 @ ^ [X4: A] :
% 2.93/1.56 ( ( member @ A @ X4 @ X7 )
% 2.93/1.56 & ( P2 @ X4 ) ) ) )
% 2.93/1.56 => ( P2 @ X ) ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_206_Collect__restrict,axiom,
% 2.93/1.56 ! [A: $tType,X7: set @ A,P2: A > $o] :
% 2.93/1.56 ( ord_less_eq @ ( set @ A )
% 2.93/1.56 @ ( collect @ A
% 2.93/1.56 @ ^ [X4: A] :
% 2.93/1.56 ( ( member @ A @ X4 @ X7 )
% 2.93/1.56 & ( P2 @ X4 ) ) )
% 2.93/1.56 @ X7 ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_207_Collect__subset,axiom,
% 2.93/1.56 ! [A: $tType,A6: set @ A,P2: A > $o] :
% 2.93/1.56 ( ord_less_eq @ ( set @ A )
% 2.93/1.56 @ ( collect @ A
% 2.93/1.56 @ ^ [X4: A] :
% 2.93/1.56 ( ( member @ A @ X4 @ A6 )
% 2.93/1.56 & ( P2 @ X4 ) ) )
% 2.93/1.56 @ A6 ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_208_image__Collect__subsetI,axiom,
% 2.93/1.56 ! [A: $tType,B: $tType,P2: A > $o,F4: A > B,B2: set @ B] :
% 2.93/1.56 ( ! [X3: A] :
% 2.93/1.56 ( ( P2 @ X3 )
% 2.93/1.56 => ( member @ B @ ( F4 @ X3 ) @ B2 ) )
% 2.93/1.56 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ ( collect @ A @ P2 ) ) @ B2 ) ) ).
% 2.93/1.56
% 2.93/1.56 thf(fact_209_subset__Collect__iff,axiom,
% 2.93/1.56 ! [A: $tType,B2: set @ A,A6: set @ A,P2: A > $o] :
% 2.93/1.56 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A6 )
% 2.93/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B2
% 2.93/1.57 @ ( collect @ A
% 2.93/1.57 @ ^ [X4: A] :
% 2.93/1.57 ( ( member @ A @ X4 @ A6 )
% 2.93/1.57 & ( P2 @ X4 ) ) ) )
% 2.93/1.57 = ( ! [X4: A] :
% 2.93/1.57 ( ( member @ A @ X4 @ B2 )
% 2.93/1.57 => ( P2 @ X4 ) ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_210_subset__CollectI,axiom,
% 2.93/1.57 ! [A: $tType,B2: set @ A,A6: set @ A,Q: A > $o,P2: A > $o] :
% 2.93/1.57 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A6 )
% 2.93/1.57 => ( ! [X3: A] :
% 2.93/1.57 ( ( member @ A @ X3 @ B2 )
% 2.93/1.57 => ( ( Q @ X3 )
% 2.93/1.57 => ( P2 @ X3 ) ) )
% 2.93/1.57 => ( ord_less_eq @ ( set @ A )
% 2.93/1.57 @ ( collect @ A
% 2.93/1.57 @ ^ [X4: A] :
% 2.93/1.57 ( ( member @ A @ X4 @ B2 )
% 2.93/1.57 & ( Q @ X4 ) ) )
% 2.93/1.57 @ ( collect @ A
% 2.93/1.57 @ ^ [X4: A] :
% 2.93/1.57 ( ( member @ A @ X4 @ A6 )
% 2.93/1.57 & ( P2 @ X4 ) ) ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_211_image__diff__subset,axiom,
% 2.93/1.57 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,B2: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ ( image @ B @ A @ F4 @ B2 ) ) @ ( image @ B @ A @ F4 @ ( minus_minus @ ( set @ B ) @ A6 @ B2 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_212_subset__snd__imageI,axiom,
% 2.93/1.57 ! [B: $tType,A: $tType,A6: set @ A,B2: set @ B,S3: set @ ( product_prod @ A @ B ),X: A] :
% 2.93/1.57 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
% 2.93/1.57 @ ( product_Sigma @ A @ B @ A6
% 2.93/1.57 @ ^ [Uu: A] : B2 )
% 2.93/1.57 @ S3 )
% 2.93/1.57 => ( ( member @ A @ X @ A6 )
% 2.93/1.57 => ( ord_less_eq @ ( set @ B ) @ B2 @ ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S3 ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_213_subset__fst__imageI,axiom,
% 2.93/1.57 ! [B: $tType,A: $tType,A6: set @ A,B2: set @ B,S3: set @ ( product_prod @ A @ B ),Y: B] :
% 2.93/1.57 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
% 2.93/1.57 @ ( product_Sigma @ A @ B @ A6
% 2.93/1.57 @ ^ [Uu: A] : B2 )
% 2.93/1.57 @ S3 )
% 2.93/1.57 => ( ( member @ B @ Y @ B2 )
% 2.93/1.57 => ( ord_less_eq @ ( set @ A ) @ A6 @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S3 ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_214_range__subsetD,axiom,
% 2.93/1.57 ! [B: $tType,A: $tType,F4: B > A,B2: set @ A,I3: B] :
% 2.93/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) @ B2 )
% 2.93/1.57 => ( member @ A @ ( F4 @ I3 ) @ B2 ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_215_pred__subset__eq,axiom,
% 2.93/1.57 ! [A: $tType,R2: set @ A,S3: set @ A] :
% 2.93/1.57 ( ( ord_less_eq @ ( A > $o )
% 2.93/1.57 @ ^ [X4: A] : ( member @ A @ X4 @ R2 )
% 2.93/1.57 @ ^ [X4: A] : ( member @ A @ X4 @ S3 ) )
% 2.93/1.57 = ( ord_less_eq @ ( set @ A ) @ R2 @ S3 ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_216_less__eq__set__def,axiom,
% 2.93/1.57 ! [A: $tType] :
% 2.93/1.57 ( ( ord_less_eq @ ( set @ A ) )
% 2.93/1.57 = ( ^ [A7: set @ A,B6: set @ A] :
% 2.93/1.57 ( ord_less_eq @ ( A > $o )
% 2.93/1.57 @ ^ [X4: A] : ( member @ A @ X4 @ A7 )
% 2.93/1.57 @ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_217_pred__subset__eq2,axiom,
% 2.93/1.57 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 2.93/1.57 ( ( ord_less_eq @ ( A > B > $o )
% 2.93/1.57 @ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R2 )
% 2.93/1.57 @ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S3 ) )
% 2.93/1.57 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_218_Gr__incl,axiom,
% 2.93/1.57 ! [A: $tType,B: $tType,A6: set @ A,F4: A > B,B2: set @ B] :
% 2.93/1.57 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( bNF_Gr @ A @ B @ A6 @ F4 )
% 2.93/1.57 @ ( product_Sigma @ A @ B @ A6
% 2.93/1.57 @ ^ [Uu: A] : B2 ) )
% 2.93/1.57 = ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ B2 ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_219_all__subset__image,axiom,
% 2.93/1.57 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P2: ( set @ A ) > $o] :
% 2.93/1.57 ( ( ! [B6: set @ A] :
% 2.93/1.57 ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) )
% 2.93/1.57 => ( P2 @ B6 ) ) )
% 2.93/1.57 = ( ! [B6: set @ B] :
% 2.93/1.57 ( ( ord_less_eq @ ( set @ B ) @ B6 @ A6 )
% 2.93/1.57 => ( P2 @ ( image @ B @ A @ F4 @ B6 ) ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_220_eq__subset,axiom,
% 2.93/1.57 ! [A: $tType,P2: A > A > $o] :
% 2.93/1.57 ( ord_less_eq @ ( A > A > $o )
% 2.93/1.57 @ ^ [Y5: A,Z2: A] : ( Y5 = Z2 )
% 2.93/1.57 @ ^ [A5: A,B7: A] :
% 2.93/1.57 ( ( P2 @ A5 @ B7 )
% 2.93/1.57 | ( A5 = B7 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_221_GrD1,axiom,
% 2.93/1.57 ! [B: $tType,A: $tType,X: A,Fx: B,A6: set @ A,F4: A > B] :
% 2.93/1.57 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A6 @ F4 ) )
% 2.93/1.57 => ( member @ A @ X @ A6 ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_222_GrD2,axiom,
% 2.93/1.57 ! [A: $tType,B: $tType,X: A,Fx: B,A6: set @ A,F4: A > B] :
% 2.93/1.57 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A6 @ F4 ) )
% 2.93/1.57 => ( ( F4 @ X )
% 2.93/1.57 = Fx ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_223_diff__eq__diff__less__eq,axiom,
% 2.93/1.57 ! [A: $tType] :
% 2.93/1.57 ( ( ordered_ab_group_add @ A )
% 2.93/1.57 => ! [A2: A,B3: A,C3: A,D3: A] :
% 2.93/1.57 ( ( ( minus_minus @ A @ A2 @ B3 )
% 2.93/1.57 = ( minus_minus @ A @ C3 @ D3 ) )
% 2.93/1.57 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.93/1.57 = ( ord_less_eq @ A @ C3 @ D3 ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_224_diff__right__mono,axiom,
% 2.93/1.57 ! [A: $tType] :
% 2.93/1.57 ( ( ordered_ab_group_add @ A )
% 2.93/1.57 => ! [A2: A,B3: A,C3: A] :
% 2.93/1.57 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.93/1.57 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B3 @ C3 ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_225_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 2.93/1.57 ! [A: $tType] :
% 2.93/1.57 ( ( cancel146912293up_add @ A )
% 2.93/1.57 => ! [A2: A,C3: A,B3: A] :
% 2.93/1.57 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C3 ) @ B3 )
% 2.93/1.57 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B3 ) @ C3 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_226_diff__eq__diff__eq,axiom,
% 2.93/1.57 ! [A: $tType] :
% 2.93/1.57 ( ( group_add @ A )
% 2.93/1.57 => ! [A2: A,B3: A,C3: A,D3: A] :
% 2.93/1.57 ( ( ( minus_minus @ A @ A2 @ B3 )
% 2.93/1.57 = ( minus_minus @ A @ C3 @ D3 ) )
% 2.93/1.57 => ( ( A2 = B3 )
% 2.93/1.57 = ( C3 = D3 ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_227_diff__mono,axiom,
% 2.93/1.57 ! [A: $tType] :
% 2.93/1.57 ( ( ordered_ab_group_add @ A )
% 2.93/1.57 => ! [A2: A,B3: A,D3: A,C3: A] :
% 2.93/1.57 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.93/1.57 => ( ( ord_less_eq @ A @ D3 @ C3 )
% 2.93/1.57 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B3 @ D3 ) ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_228_diff__left__mono,axiom,
% 2.93/1.57 ! [A: $tType] :
% 2.93/1.57 ( ( ordered_ab_group_add @ A )
% 2.93/1.57 => ! [B3: A,A2: A,C3: A] :
% 2.93/1.57 ( ( ord_less_eq @ A @ B3 @ A2 )
% 2.93/1.57 => ( ord_less_eq @ A @ ( minus_minus @ A @ C3 @ A2 ) @ ( minus_minus @ A @ C3 @ B3 ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_229_snd__image__times,axiom,
% 2.93/1.57 ! [B: $tType,A: $tType,A6: set @ B,B2: set @ A] :
% 2.93/1.57 ( ( ( A6
% 2.93/1.57 = ( bot_bot @ ( set @ B ) ) )
% 2.93/1.57 => ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
% 2.93/1.57 @ ( product_Sigma @ B @ A @ A6
% 2.93/1.57 @ ^ [Uu: B] : B2 ) )
% 2.93/1.57 = ( bot_bot @ ( set @ A ) ) ) )
% 2.93/1.57 & ( ( A6
% 2.93/1.57 != ( bot_bot @ ( set @ B ) ) )
% 2.93/1.57 => ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
% 2.93/1.57 @ ( product_Sigma @ B @ A @ A6
% 2.93/1.57 @ ^ [Uu: B] : B2 ) )
% 2.93/1.57 = B2 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_230_fst__image__times,axiom,
% 2.93/1.57 ! [B: $tType,A: $tType,B2: set @ B,A6: set @ A] :
% 2.93/1.57 ( ( ( B2
% 2.93/1.57 = ( bot_bot @ ( set @ B ) ) )
% 2.93/1.57 => ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
% 2.93/1.57 @ ( product_Sigma @ A @ B @ A6
% 2.93/1.57 @ ^ [Uu: A] : B2 ) )
% 2.93/1.57 = ( bot_bot @ ( set @ A ) ) ) )
% 2.93/1.57 & ( ( B2
% 2.93/1.57 != ( bot_bot @ ( set @ B ) ) )
% 2.93/1.57 => ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
% 2.93/1.57 @ ( product_Sigma @ A @ B @ A6
% 2.93/1.57 @ ^ [Uu: A] : B2 ) )
% 2.93/1.57 = A6 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_231_empty__Collect__eq,axiom,
% 2.93/1.57 ! [A: $tType,P2: A > $o] :
% 2.93/1.57 ( ( ( bot_bot @ ( set @ A ) )
% 2.93/1.57 = ( collect @ A @ P2 ) )
% 2.93/1.57 = ( ! [X4: A] :
% 2.93/1.57 ~ ( P2 @ X4 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_232_Collect__empty__eq,axiom,
% 2.93/1.57 ! [A: $tType,P2: A > $o] :
% 2.93/1.57 ( ( ( collect @ A @ P2 )
% 2.93/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.93/1.57 = ( ! [X4: A] :
% 2.93/1.57 ~ ( P2 @ X4 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_233_all__not__in__conv,axiom,
% 2.93/1.57 ! [A: $tType,A6: set @ A] :
% 2.93/1.57 ( ( ! [X4: A] :
% 2.93/1.57 ~ ( member @ A @ X4 @ A6 ) )
% 2.93/1.57 = ( A6
% 2.93/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_234_empty__iff,axiom,
% 2.93/1.57 ! [A: $tType,C3: A] :
% 2.93/1.57 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_235_image__is__empty,axiom,
% 2.93/1.57 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B] :
% 2.93/1.57 ( ( ( image @ B @ A @ F4 @ A6 )
% 2.93/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.93/1.57 = ( A6
% 2.93/1.57 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_236_empty__is__image,axiom,
% 2.93/1.57 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B] :
% 2.93/1.57 ( ( ( bot_bot @ ( set @ A ) )
% 2.93/1.57 = ( image @ B @ A @ F4 @ A6 ) )
% 2.93/1.57 = ( A6
% 2.93/1.57 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_237_image__empty,axiom,
% 2.93/1.57 ! [B: $tType,A: $tType,F4: B > A] :
% 2.93/1.57 ( ( image @ B @ A @ F4 @ ( bot_bot @ ( set @ B ) ) )
% 2.93/1.57 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_238_subset__empty,axiom,
% 2.93/1.57 ! [A: $tType,A6: set @ A] :
% 2.93/1.57 ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) )
% 2.93/1.57 = ( A6
% 2.93/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_239_empty__subsetI,axiom,
% 2.93/1.57 ! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A6 ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_240_Diff__empty,axiom,
% 2.93/1.57 ! [A: $tType,A6: set @ A] :
% 2.93/1.57 ( ( minus_minus @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) )
% 2.93/1.57 = A6 ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_241_empty__Diff,axiom,
% 2.93/1.57 ! [A: $tType,A6: set @ A] :
% 2.93/1.57 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A6 )
% 2.93/1.57 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_242_Diff__cancel,axiom,
% 2.93/1.57 ! [A: $tType,A6: set @ A] :
% 2.93/1.57 ( ( minus_minus @ ( set @ A ) @ A6 @ A6 )
% 2.93/1.57 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_243_Sigma__empty1,axiom,
% 2.93/1.57 ! [B: $tType,A: $tType,B2: A > ( set @ B )] :
% 2.93/1.57 ( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B2 )
% 2.93/1.57 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_244_Collect__const,axiom,
% 2.93/1.57 ! [A: $tType,P2: $o] :
% 2.93/1.57 ( ( P2
% 2.93/1.57 => ( ( collect @ A
% 2.93/1.57 @ ^ [S4: A] : P2 )
% 2.93/1.57 = ( top_top @ ( set @ A ) ) ) )
% 2.93/1.57 & ( ~ P2
% 2.93/1.57 => ( ( collect @ A
% 2.93/1.57 @ ^ [S4: A] : P2 )
% 2.93/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_245_Times__empty,axiom,
% 2.93/1.57 ! [A: $tType,B: $tType,A6: set @ A,B2: set @ B] :
% 2.93/1.57 ( ( ( product_Sigma @ A @ B @ A6
% 2.93/1.57 @ ^ [Uu: A] : B2 )
% 2.93/1.57 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.93/1.57 = ( ( A6
% 2.93/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.93/1.57 | ( B2
% 2.93/1.57 = ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_246_Sigma__empty2,axiom,
% 2.93/1.57 ! [B: $tType,A: $tType,A6: set @ A] :
% 2.93/1.57 ( ( product_Sigma @ A @ B @ A6
% 2.93/1.57 @ ^ [Uu: A] : ( bot_bot @ ( set @ B ) ) )
% 2.93/1.57 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_247_Diff__UNIV,axiom,
% 2.93/1.57 ! [A: $tType,A6: set @ A] :
% 2.93/1.57 ( ( minus_minus @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) )
% 2.93/1.57 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_248_Diff__eq__empty__iff,axiom,
% 2.93/1.57 ! [A: $tType,A6: set @ A,B2: set @ A] :
% 2.93/1.57 ( ( ( minus_minus @ ( set @ A ) @ A6 @ B2 )
% 2.93/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.93/1.57 = ( ord_less_eq @ ( set @ A ) @ A6 @ B2 ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_249_times__eq__iff,axiom,
% 2.93/1.57 ! [A: $tType,B: $tType,A6: set @ A,B2: set @ B,C5: set @ A,D4: set @ B] :
% 2.93/1.57 ( ( ( product_Sigma @ A @ B @ A6
% 2.93/1.57 @ ^ [Uu: A] : B2 )
% 2.93/1.57 = ( product_Sigma @ A @ B @ C5
% 2.93/1.57 @ ^ [Uu: A] : D4 ) )
% 2.93/1.57 = ( ( ( A6 = C5 )
% 2.93/1.57 & ( B2 = D4 ) )
% 2.93/1.57 | ( ( ( A6
% 2.93/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.93/1.57 | ( B2
% 2.93/1.57 = ( bot_bot @ ( set @ B ) ) ) )
% 2.93/1.57 & ( ( C5
% 2.93/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.93/1.57 | ( D4
% 2.93/1.57 = ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_250_Sigma__empty__iff,axiom,
% 2.93/1.57 ! [B: $tType,A: $tType,I: set @ A,X7: A > ( set @ B )] :
% 2.93/1.57 ( ( ( product_Sigma @ A @ B @ I @ X7 )
% 2.93/1.57 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.93/1.57 = ( ! [X4: A] :
% 2.93/1.57 ( ( member @ A @ X4 @ I )
% 2.93/1.57 => ( ( X7 @ X4 )
% 2.93/1.57 = ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_251_bot__prod__def,axiom,
% 2.93/1.57 ! [A: $tType,B: $tType] :
% 2.93/1.57 ( ( ( bot @ B )
% 2.93/1.57 & ( bot @ A ) )
% 2.93/1.57 => ( ( bot_bot @ ( product_prod @ A @ B ) )
% 2.93/1.57 = ( product_Pair @ A @ B @ ( bot_bot @ A ) @ ( bot_bot @ B ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_252_ex__in__conv,axiom,
% 2.93/1.57 ! [A: $tType,A6: set @ A] :
% 2.93/1.57 ( ( ? [X4: A] : ( member @ A @ X4 @ A6 ) )
% 2.93/1.57 = ( A6
% 2.93/1.57 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_253_equals0I,axiom,
% 2.93/1.57 ! [A: $tType,A6: set @ A] :
% 2.93/1.57 ( ! [Y3: A] :
% 2.93/1.57 ~ ( member @ A @ Y3 @ A6 )
% 2.93/1.57 => ( A6
% 2.93/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_254_equals0D,axiom,
% 2.93/1.57 ! [A: $tType,A6: set @ A,A2: A] :
% 2.93/1.57 ( ( A6
% 2.93/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.93/1.57 => ~ ( member @ A @ A2 @ A6 ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(fact_255_emptyE,axiom,
% 2.93/1.57 ! [A: $tType,A2: A] :
% 2.93/1.57 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
% 2.93/1.57 ! [A9: $tType,A10: $tType] :
% 2.93/1.57 ( ( ( ordere890947078_space @ A9 )
% 2.93/1.57 & ( ordere890947078_space @ A10 ) )
% 2.93/1.57 => ( ordere890947078_space @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 2.93/1.57 ! [A9: $tType,A10: $tType] :
% 2.93/1.57 ( ( order_top @ A10 )
% 2.93/1.57 => ( order_top @ ( A9 > A10 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_fun___Orderings_Otop,axiom,
% 2.93/1.57 ! [A9: $tType,A10: $tType] :
% 2.93/1.57 ( ( top @ A10 )
% 2.93/1.57 => ( top @ ( A9 > A10 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_fun___Orderings_Oord,axiom,
% 2.93/1.57 ! [A9: $tType,A10: $tType] :
% 2.93/1.57 ( ( ord @ A10 )
% 2.93/1.57 => ( ord @ ( A9 > A10 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_fun___Orderings_Obot,axiom,
% 2.93/1.57 ! [A9: $tType,A10: $tType] :
% 2.93/1.57 ( ( bot @ A10 )
% 2.93/1.57 => ( bot @ ( A9 > A10 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_fun___Groups_Ominus,axiom,
% 2.93/1.57 ! [A9: $tType,A10: $tType] :
% 2.93/1.57 ( ( minus @ A10 )
% 2.93/1.57 => ( minus @ ( A9 > A10 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_Set_Oset___Orderings_Oorder__top_1,axiom,
% 2.93/1.57 ! [A9: $tType] : ( order_top @ ( set @ A9 ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_Set_Oset___Orderings_Otop_2,axiom,
% 2.93/1.57 ! [A9: $tType] : ( top @ ( set @ A9 ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
% 2.93/1.57 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_Set_Oset___Orderings_Obot_4,axiom,
% 2.93/1.57 ! [A9: $tType] : ( bot @ ( set @ A9 ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_Set_Oset___Groups_Ominus_5,axiom,
% 2.93/1.57 ! [A9: $tType] : ( minus @ ( set @ A9 ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_HOL_Obool___Orderings_Oorder__top_6,axiom,
% 2.93/1.57 order_top @ $o ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_HOL_Obool___Orderings_Otop_7,axiom,
% 2.93/1.57 top @ $o ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_HOL_Obool___Orderings_Oord_8,axiom,
% 2.93/1.57 ord @ $o ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_HOL_Obool___Orderings_Obot_9,axiom,
% 2.93/1.57 bot @ $o ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_HOL_Obool___Groups_Ominus_10,axiom,
% 2.93/1.57 minus @ $o ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_Product__Type_Oprod___Groups_Ocancel__ab__semigroup__add,axiom,
% 2.93/1.57 ! [A9: $tType,A10: $tType] :
% 2.93/1.57 ( ( ( cancel146912293up_add @ A9 )
% 2.93/1.57 & ( cancel146912293up_add @ A10 ) )
% 2.93/1.57 => ( cancel146912293up_add @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_Product__Type_Oprod___Groups_Oordered__ab__group__add,axiom,
% 2.93/1.57 ! [A9: $tType,A10: $tType] :
% 2.93/1.57 ( ( ( ordere890947078_space @ A9 )
% 2.93/1.57 & ( ordere890947078_space @ A10 ) )
% 2.93/1.57 => ( ordered_ab_group_add @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_Product__Type_Oprod___Orderings_Oorder__top_11,axiom,
% 2.93/1.57 ! [A9: $tType,A10: $tType] :
% 2.93/1.57 ( ( ( order_top @ A9 )
% 2.93/1.57 & ( order_top @ A10 ) )
% 2.93/1.57 => ( order_top @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_Product__Type_Oprod___Groups_Oab__group__add,axiom,
% 2.93/1.57 ! [A9: $tType,A10: $tType] :
% 2.93/1.57 ( ( ( ab_group_add @ A9 )
% 2.93/1.57 & ( ab_group_add @ A10 ) )
% 2.93/1.57 => ( ab_group_add @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_Product__Type_Oprod___Groups_Ogroup__add,axiom,
% 2.93/1.57 ! [A9: $tType,A10: $tType] :
% 2.93/1.57 ( ( ( group_add @ A9 )
% 2.93/1.57 & ( group_add @ A10 ) )
% 2.93/1.57 => ( group_add @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_Product__Type_Oprod___Orderings_Otop_12,axiom,
% 2.93/1.57 ! [A9: $tType,A10: $tType] :
% 2.93/1.57 ( ( ( top @ A9 )
% 2.93/1.57 & ( top @ A10 ) )
% 2.93/1.57 => ( top @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_Product__Type_Oprod___Orderings_Oord_13,axiom,
% 2.93/1.57 ! [A9: $tType,A10: $tType] :
% 2.93/1.57 ( ( ( ord @ A9 )
% 2.93/1.57 & ( ord @ A10 ) )
% 2.93/1.57 => ( ord @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_Product__Type_Oprod___Orderings_Obot_14,axiom,
% 2.93/1.57 ! [A9: $tType,A10: $tType] :
% 2.93/1.57 ( ( ( bot @ A9 )
% 2.93/1.57 & ( bot @ A10 ) )
% 2.93/1.57 => ( bot @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_Product__Type_Oprod___Groups_Ominus_15,axiom,
% 2.93/1.57 ! [A9: $tType,A10: $tType] :
% 2.93/1.57 ( ( ( minus @ A9 )
% 2.93/1.57 & ( minus @ A10 ) )
% 2.93/1.57 => ( minus @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_16,axiom,
% 2.93/1.57 order_top @ product_unit ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_Product__Type_Ounit___Orderings_Otop_17,axiom,
% 2.93/1.57 top @ product_unit ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_Product__Type_Ounit___Orderings_Oord_18,axiom,
% 2.93/1.57 ord @ product_unit ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_Product__Type_Ounit___Orderings_Obot_19,axiom,
% 2.93/1.57 bot @ product_unit ).
% 2.93/1.57
% 2.93/1.57 thf(tcon_Product__Type_Ounit___Groups_Ominus_20,axiom,
% 2.93/1.57 minus @ product_unit ).
% 2.93/1.57
% 2.93/1.57 thf(help_If_3_1_T,axiom,
% 2.93/1.57 ! [P2: $o] :
% 2.93/1.57 ( ( P2 = $true )
% 2.93/1.57 | ( P2 = $false ) ) ).
% 2.93/1.57
% 2.93/1.57 thf(help_If_2_1_T,axiom,
% 2.93/1.57 ! [A: $tType,X: A,Y: A] :
% 2.93/1.57 ( ( if @ A @ $false @ X @ Y )
% 2.93/1.57 = Y ) ).
% 2.93/1.57
% 2.93/1.57 thf(help_If_1_1_T,axiom,
% 2.93/1.57 ! [A: $tType,X: A,Y: A] :
% 2.93/1.57 ( ( if @ A @ $true @ X @ Y )
% 2.93/1.57 = X ) ).
% 2.93/1.57
% 2.93/1.57 thf(conj_0,conjecture,
% 2.93/1.57 member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ z @ y ) @ pr ).
% 2.93/1.57 ------------------------
% 2.93/1.58 % (18228)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_17895 for (2999ds/2Mi)
% 2.93/1.59 % (18228)Instruction limit reached!
% 2.93/1.59 % (18228)------------------------------
% 2.93/1.59 % (18228)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.59 % (18228)Termination reason: Unknown
% 2.93/1.59 % (18228)Termination phase: shuffling
% 2.93/1.59
% 2.93/1.59 % (18228)Memory used [KB]: 1407
% 2.93/1.59 % (18228)Time elapsed: 0.002 s
% 2.93/1.59 % (18228)Instructions burned: 2 (million)
% 2.93/1.59 % (18228)------------------------------
% 2.93/1.59 % (18228)------------------------------
% 2.93/1.59 % (18225)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_17895 for (2999ds/183Mi)
% 2.93/1.59 % (18231)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_17895 for (2999ds/18Mi)
% 2.93/1.59 % (18231)Instruction limit reached!
% 2.93/1.59 % (18231)------------------------------
% 2.93/1.59 % (18231)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.59 % (18231)Termination reason: Unknown
% 2.93/1.59 % (18231)Termination phase: shuffling
% 2.93/1.59
% 2.93/1.59 % (18231)Memory used [KB]: 1791
% 2.93/1.59 % (18231)Time elapsed: 0.008 s
% 2.93/1.59 % (18231)Instructions burned: 19 (million)
% 2.93/1.59 % (18231)------------------------------
% 2.93/1.59 % (18231)------------------------------
% 2.93/1.59 % (18226)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_17895 for (2999ds/4Mi)
% 2.93/1.59 % (18227)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_17895 for (2999ds/27Mi)
% 2.93/1.59 % (18229)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_17895 for (2999ds/2Mi)
% 2.93/1.59 % (18230)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_17895 for (2999ds/275Mi)
% 2.93/1.59 % (18229)Instruction limit reached!
% 2.93/1.59 % (18229)------------------------------
% 2.93/1.59 % (18229)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.59 % (18229)Termination reason: Unknown
% 2.93/1.59 % (18229)Termination phase: shuffling
% 2.93/1.59
% 2.93/1.59 % (18229)Memory used [KB]: 1407
% 2.93/1.59 % (18229)Time elapsed: 0.004 s
% 2.93/1.59 % (18229)Instructions burned: 3 (million)
% 2.93/1.59 % (18229)------------------------------
% 2.93/1.59 % (18229)------------------------------
% 2.93/1.60 % (18232)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_17895 for (2999ds/3Mi)
% 2.93/1.60 % (18226)Instruction limit reached!
% 2.93/1.60 % (18226)------------------------------
% 2.93/1.60 % (18226)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.60 % (18226)Termination reason: Unknown
% 2.93/1.60 % (18226)Termination phase: shuffling
% 2.93/1.60
% 2.93/1.60 % (18226)Memory used [KB]: 1407
% 2.93/1.60 % (18226)Time elapsed: 0.005 s
% 2.93/1.60 % (18226)Instructions burned: 5 (million)
% 2.93/1.60 % (18226)------------------------------
% 2.93/1.60 % (18226)------------------------------
% 3.02/1.60 % (18232)Instruction limit reached!
% 3.02/1.60 % (18232)------------------------------
% 3.02/1.60 % (18232)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.60 % (18232)Termination reason: Unknown
% 3.02/1.60 % (18232)Termination phase: shuffling
% 3.02/1.60
% 3.02/1.60 % (18232)Memory used [KB]: 1407
% 3.02/1.60 % (18232)Time elapsed: 0.003 s
% 3.02/1.60 % (18232)Instructions burned: 5 (million)
% 3.02/1.60 % (18232)------------------------------
% 3.02/1.60 % (18232)------------------------------
% 3.02/1.60 % (18233)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_17895 for (2999ds/37Mi)
% 3.02/1.61 % (18236)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_17895 for (2999ds/1041Mi)
% 3.02/1.61 % (18227)Instruction limit reached!
% 3.02/1.61 % (18227)------------------------------
% 3.02/1.61 % (18227)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.61 % (18227)Termination reason: Unknown
% 3.02/1.61 % (18227)Termination phase: Property scanning
% 3.02/1.61
% 3.02/1.61 % (18227)Memory used [KB]: 1918
% 3.02/1.61 % (18227)Time elapsed: 0.019 s
% 3.02/1.61 % (18227)Instructions burned: 27 (million)
% 3.02/1.61 % (18227)------------------------------
% 3.02/1.61 % (18227)------------------------------
% 3.02/1.61 % (18234)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_17895 for (2999ds/15Mi)
% 3.02/1.61 % (18235)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_17895 for (2999ds/3Mi)
% 3.02/1.61 % (18235)Instruction limit reached!
% 3.02/1.61 % (18235)------------------------------
% 3.02/1.61 % (18235)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.61 % (18235)Termination reason: Unknown
% 3.02/1.61 % (18235)Termination phase: shuffling
% 3.02/1.61
% 3.02/1.61 % (18235)Memory used [KB]: 1407
% 3.02/1.61 % (18235)Time elapsed: 0.004 s
% 3.02/1.61 % (18235)Instructions burned: 3 (million)
% 3.02/1.61 % (18235)------------------------------
% 3.02/1.61 % (18235)------------------------------
% 3.02/1.62 % (18233)Instruction limit reached!
% 3.02/1.62 % (18233)------------------------------
% 3.02/1.62 % (18233)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.62 % (18233)Termination reason: Unknown
% 3.02/1.62 % (18233)Termination phase: shuffling
% 3.02/1.62
% 3.02/1.62 % (18233)Memory used [KB]: 2174
% 3.02/1.62 % (18233)Time elapsed: 0.013 s
% 3.02/1.62 % (18233)Instructions burned: 40 (million)
% 3.02/1.62 % (18233)------------------------------
% 3.02/1.62 % (18233)------------------------------
% 3.02/1.62 % (18234)Instruction limit reached!
% 3.02/1.62 % (18234)------------------------------
% 3.02/1.62 % (18234)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.62 % (18234)Termination reason: Unknown
% 3.02/1.62 % (18234)Termination phase: shuffling
% 3.02/1.62
% 3.02/1.62 % (18234)Memory used [KB]: 1791
% 3.02/1.62 % (18234)Time elapsed: 0.012 s
% 3.02/1.62 % (18234)Instructions burned: 16 (million)
% 3.02/1.62 % (18234)------------------------------
% 3.02/1.62 % (18234)------------------------------
% 3.02/1.62 % (18239)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_17895 for (2999ds/3Mi)
% 3.02/1.63 % (18230)Refutation not found, incomplete strategy
% 3.02/1.63 % (18230)------------------------------
% 3.02/1.63 % (18230)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.63 % (18230)Termination reason: Refutation not found, incomplete strategy
% 3.02/1.63
% 3.02/1.63
% 3.02/1.63 % (18230)Memory used [KB]: 6652
% 3.02/1.63 % (18230)Time elapsed: 0.034 s
% 3.02/1.63 % (18230)Instructions burned: 54 (million)
% 3.02/1.63 % (18230)------------------------------
% 3.02/1.63 % (18230)------------------------------
% 3.02/1.63 % (18239)Instruction limit reached!
% 3.02/1.63 % (18239)------------------------------
% 3.02/1.63 % (18239)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.63 % (18239)Termination reason: Unknown
% 3.02/1.63 % (18239)Termination phase: shuffling
% 3.02/1.63
% 3.02/1.63 % (18239)Memory used [KB]: 1407
% 3.02/1.63 % (18239)Time elapsed: 0.002 s
% 3.02/1.63 % (18239)Instructions burned: 3 (million)
% 3.02/1.63 % (18239)------------------------------
% 3.02/1.63 % (18239)------------------------------
% 3.02/1.63 % (18237)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_17895 for (2999ds/7Mi)
% 3.02/1.63 % (18237)Instruction limit reached!
% 3.02/1.63 % (18237)------------------------------
% 3.02/1.63 % (18237)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.63 % (18237)Termination reason: Unknown
% 3.02/1.63 % (18237)Termination phase: shuffling
% 3.02/1.63
% 3.02/1.63 % (18237)Memory used [KB]: 1535
% 3.02/1.63 % (18237)Time elapsed: 0.007 s
% 3.02/1.63 % (18237)Instructions burned: 8 (million)
% 3.02/1.63 % (18237)------------------------------
% 3.02/1.63 % (18237)------------------------------
% 3.02/1.63 % (18238)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_17895 for (2999ds/16Mi)
% 3.02/1.63 % (18241)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_17895 for (2999ds/7Mi)
% 3.02/1.64 % (18240)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_17895 for (2999ds/3Mi)
% 3.02/1.64 % (18240)Instruction limit reached!
% 3.02/1.64 % (18240)------------------------------
% 3.02/1.64 % (18240)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.64 % (18240)Termination reason: Unknown
% 3.02/1.64 % (18240)Termination phase: shuffling
% 3.02/1.64
% 3.02/1.64 % (18240)Memory used [KB]: 1407
% 3.02/1.64 % (18240)Time elapsed: 0.002 s
% 3.02/1.64 % (18240)Instructions burned: 3 (million)
% 3.02/1.64 % (18240)------------------------------
% 3.02/1.64 % (18240)------------------------------
% 3.02/1.64 % (18241)Instruction limit reached!
% 3.02/1.64 % (18241)------------------------------
% 3.02/1.64 % (18241)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.64 % (18241)Termination reason: Unknown
% 3.02/1.64 % (18241)Termination phase: shuffling
% 3.02/1.64
% 3.02/1.64 % (18241)Memory used [KB]: 1535
% 3.02/1.64 % (18241)Time elapsed: 0.004 s
% 3.02/1.64 % (18241)Instructions burned: 9 (million)
% 3.02/1.64 % (18241)------------------------------
% 3.02/1.64 % (18241)------------------------------
% 3.02/1.64 % (18242)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_17895 for (2999ds/3Mi)
% 3.02/1.64 % (18242)Instruction limit reached!
% 3.02/1.64 % (18242)------------------------------
% 3.02/1.64 % (18242)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.64 % (18242)Termination reason: Unknown
% 3.02/1.64 % (18242)Termination phase: shuffling
% 3.02/1.64
% 3.02/1.64 % (18242)Memory used [KB]: 1407
% 3.02/1.64 % (18242)Time elapsed: 0.002 s
% 3.02/1.64 % (18242)Instructions burned: 3 (million)
% 3.02/1.64 % (18242)------------------------------
% 3.02/1.64 % (18242)------------------------------
% 3.02/1.64 % (18238)Instruction limit reached!
% 3.02/1.64 % (18238)------------------------------
% 3.02/1.64 % (18238)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.64 % (18238)Termination reason: Unknown
% 3.02/1.64 % (18238)Termination phase: shuffling
% 3.02/1.64
% 3.02/1.64 % (18238)Memory used [KB]: 1791
% 3.02/1.64 % (18238)Time elapsed: 0.012 s
% 3.02/1.64 % (18238)Instructions burned: 16 (million)
% 3.02/1.64 % (18238)------------------------------
% 3.02/1.64 % (18238)------------------------------
% 3.02/1.65 % (18244)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_17895 for (2999ds/18Mi)
% 3.02/1.65 % (18225)Instruction limit reached!
% 3.02/1.65 % (18225)------------------------------
% 3.02/1.65 % (18225)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.65 % (18225)Termination reason: Unknown
% 3.02/1.65 % (18225)Termination phase: Saturation
% 3.02/1.65
% 3.02/1.65 % (18225)Memory used [KB]: 7419
% 3.02/1.65 % (18225)Time elapsed: 0.062 s
% 3.02/1.65 % (18225)Instructions burned: 185 (million)
% 3.02/1.65 % (18225)------------------------------
% 3.02/1.65 % (18225)------------------------------
% 3.02/1.65 % (18245)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_17895 for (2999ds/710Mi)
% 3.02/1.65 % (18246)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_17895 for (2999ds/6Mi)
% 3.02/1.65 % (18243)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_17895 for (2999ds/4Mi)
% 3.02/1.65 % (18246)Instruction limit reached!
% 3.02/1.65 % (18246)------------------------------
% 3.02/1.65 % (18246)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.65 % (18246)Termination reason: Unknown
% 3.02/1.65 % (18246)Termination phase: shuffling
% 3.02/1.65
% 3.02/1.65 % (18246)Memory used [KB]: 1535
% 3.02/1.65 % (18246)Time elapsed: 0.004 s
% 3.02/1.65 % (18246)Instructions burned: 6 (million)
% 3.02/1.65 % (18246)------------------------------
% 3.02/1.65 % (18246)------------------------------
% 3.02/1.65 % (18244)Instruction limit reached!
% 3.02/1.65 % (18244)------------------------------
% 3.02/1.65 % (18244)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.65 % (18244)Termination reason: Unknown
% 3.02/1.65 % (18244)Termination phase: shuffling
% 3.02/1.65
% 3.02/1.65 % (18244)Memory used [KB]: 1791
% 3.02/1.65 % (18244)Time elapsed: 0.007 s
% 3.02/1.65 % (18244)Instructions burned: 18 (million)
% 3.02/1.65 % (18244)------------------------------
% 3.02/1.65 % (18244)------------------------------
% 3.02/1.65 % (18243)Instruction limit reached!
% 3.02/1.65 % (18243)------------------------------
% 3.02/1.65 % (18243)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.65 % (18243)Termination reason: Unknown
% 3.02/1.65 % (18243)Termination phase: shuffling
% 3.02/1.65
% 3.02/1.65 % (18243)Memory used [KB]: 1407
% 3.02/1.65 % (18243)Time elapsed: 0.005 s
% 3.02/1.65 % (18243)Instructions burned: 5 (million)
% 3.02/1.65 % (18243)------------------------------
% 3.02/1.65 % (18243)------------------------------
% 3.02/1.66 % (18248)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_17895 for (2999ds/21Mi)
% 3.02/1.66 % (18247)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_17895 for (2999ds/902Mi)
% 3.02/1.66 % (18250)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_17895 for (2999ds/6Mi)
% 3.02/1.66 % (18249)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_17895 for (2999ds/5Mi)
% 3.02/1.66 % (18250)Instruction limit reached!
% 3.02/1.66 % (18250)------------------------------
% 3.02/1.66 % (18250)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.66 % (18250)Termination reason: Unknown
% 3.02/1.66 % (18250)Termination phase: shuffling
% 3.02/1.66
% 3.02/1.66 % (18250)Memory used [KB]: 1407
% 3.02/1.66 % (18250)Time elapsed: 0.003 s
% 3.02/1.66 % (18250)Instructions burned: 6 (million)
% 3.02/1.66 % (18250)------------------------------
% 3.02/1.66 % (18250)------------------------------
% 3.02/1.66 % (18249)Instruction limit reached!
% 3.02/1.66 % (18249)------------------------------
% 3.02/1.66 % (18249)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.66 % (18249)Termination reason: Unknown
% 3.02/1.66 % (18249)Termination phase: shuffling
% 3.02/1.66
% 3.02/1.66 % (18249)Memory used [KB]: 1535
% 3.02/1.66 % (18249)Time elapsed: 0.004 s
% 3.02/1.66 % (18249)Instructions burned: 7 (million)
% 3.02/1.66 % (18249)------------------------------
% 3.02/1.66 % (18249)------------------------------
% 3.02/1.66 % (18248)Instruction limit reached!
% 3.02/1.66 % (18248)------------------------------
% 3.02/1.66 % (18248)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.66 % (18248)Termination reason: Unknown
% 3.02/1.66 % (18248)Termination phase: shuffling
% 3.02/1.66
% 3.02/1.66 % (18248)Memory used [KB]: 1918
% 3.02/1.66 % (18248)Time elapsed: 0.010 s
% 3.02/1.66 % (18248)Instructions burned: 23 (million)
% 3.02/1.66 % (18248)------------------------------
% 3.02/1.66 % (18248)------------------------------
% 3.02/1.67 % (18251)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_17895 for (2999ds/377Mi)
% 3.02/1.67 % (18252)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_17895 for (2999ds/779Mi)
% 3.02/1.67 % (18253)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_17895 for (2999ds/19Mi)
% 3.02/1.68 % (18254)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_17895 for (2999ds/879Mi)
% 3.02/1.68 % (18253)Instruction limit reached!
% 3.02/1.68 % (18253)------------------------------
% 3.02/1.68 % (18253)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.68 % (18253)Termination reason: Unknown
% 3.02/1.68 % (18253)Termination phase: shuffling
% 3.02/1.68
% 3.02/1.68 % (18253)Memory used [KB]: 1791
% 3.02/1.68 % (18253)Time elapsed: 0.008 s
% 3.02/1.68 % (18253)Instructions burned: 20 (million)
% 3.02/1.68 % (18253)------------------------------
% 3.02/1.68 % (18253)------------------------------
% 3.02/1.69 % (18255)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_17895 for (2998ds/17Mi)
% 3.02/1.70 % (18255)Instruction limit reached!
% 3.02/1.70 % (18255)------------------------------
% 3.02/1.70 % (18255)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.70 % (18255)Termination reason: Unknown
% 3.02/1.70 % (18255)Termination phase: shuffling
% 3.02/1.70
% 3.02/1.70 % (18255)Memory used [KB]: 1791
% 3.02/1.70 % (18255)Time elapsed: 0.008 s
% 3.02/1.70 % (18255)Instructions burned: 17 (million)
% 3.02/1.70 % (18255)------------------------------
% 3.02/1.70 % (18255)------------------------------
% 3.02/1.71 % (18247)Refutation not found, incomplete strategy
% 3.02/1.71 % (18247)------------------------------
% 3.02/1.71 % (18247)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.71 % (18247)Termination reason: Refutation not found, incomplete strategy
% 3.02/1.71
% 3.02/1.71
% 3.02/1.71 % (18247)Memory used [KB]: 7547
% 3.02/1.71 % (18247)Time elapsed: 0.048 s
% 3.02/1.71 % (18247)Instructions burned: 135 (million)
% 3.02/1.71 % (18247)------------------------------
% 3.02/1.71 % (18247)------------------------------
% 3.02/1.71 % (18256)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_17895 for (2998ds/3Mi)
% 3.02/1.71 % (18256)Instruction limit reached!
% 3.02/1.71 % (18256)------------------------------
% 3.02/1.71 % (18256)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.71 % (18256)Termination reason: Unknown
% 3.02/1.71 % (18256)Termination phase: shuffling
% 3.02/1.71
% 3.02/1.71 % (18256)Memory used [KB]: 1407
% 3.02/1.71 % (18256)Time elapsed: 0.003 s
% 3.02/1.71 % (18256)Instructions burned: 3 (million)
% 3.02/1.71 % (18256)------------------------------
% 3.02/1.71 % (18256)------------------------------
% 3.02/1.72 % (18257)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_17895 for (2998ds/30Mi)
% 3.02/1.72 % (18258)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_17895 for (2998ds/127Mi)
% 3.02/1.73 % (18257)Instruction limit reached!
% 3.02/1.73 % (18257)------------------------------
% 3.02/1.73 % (18257)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.73 % (18257)Termination reason: Unknown
% 3.02/1.73 % (18257)Termination phase: Property scanning
% 3.02/1.73
% 3.02/1.73 % (18257)Memory used [KB]: 2046
% 3.02/1.73 % (18257)Time elapsed: 0.013 s
% 3.02/1.73 % (18257)Instructions burned: 31 (million)
% 3.02/1.73 % (18257)------------------------------
% 3.02/1.73 % (18257)------------------------------
% 3.02/1.74 % (18259)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_17895 for (2998ds/100Mi)
% 3.02/1.76 % (18258)Instruction limit reached!
% 3.02/1.76 % (18258)------------------------------
% 3.02/1.76 % (18258)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.76 % (18258)Termination reason: Unknown
% 3.02/1.76 % (18258)Termination phase: Saturation
% 3.02/1.76
% 3.02/1.76 % (18258)Memory used [KB]: 7164
% 3.02/1.76 % (18258)Time elapsed: 0.043 s
% 3.02/1.76 % (18258)Instructions burned: 127 (million)
% 3.02/1.76 % (18258)------------------------------
% 3.02/1.76 % (18258)------------------------------
% 3.02/1.77 % (18259)Instruction limit reached!
% 3.02/1.77 % (18259)------------------------------
% 3.02/1.77 % (18259)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.77 % (18259)Termination reason: Unknown
% 3.02/1.77 % (18259)Termination phase: Property scanning
% 3.02/1.77
% 3.02/1.77 % (18259)Memory used [KB]: 2686
% 3.02/1.77 % (18259)Time elapsed: 0.033 s
% 3.02/1.77 % (18259)Instructions burned: 103 (million)
% 3.02/1.77 % (18259)------------------------------
% 3.02/1.77 % (18259)------------------------------
% 3.02/1.77 % (18260)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_17895 for (2998ds/3Mi)
% 3.02/1.77 % (18260)Instruction limit reached!
% 3.02/1.77 % (18260)------------------------------
% 3.02/1.77 % (18260)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.77 % (18260)Termination reason: Unknown
% 3.02/1.77 % (18260)Termination phase: shuffling
% 3.02/1.77
% 3.02/1.77 % (18260)Memory used [KB]: 1407
% 3.02/1.77 % (18260)Time elapsed: 0.003 s
% 3.02/1.77 % (18260)Instructions burned: 4 (million)
% 3.02/1.77 % (18260)------------------------------
% 3.02/1.77 % (18260)------------------------------
% 3.02/1.78 % (18261)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_17895 for (2998ds/20Mi)
% 3.02/1.78 % (18262)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_17895 for (2998ds/86Mi)
% 3.02/1.79 % (18261)Instruction limit reached!
% 3.02/1.79 % (18261)------------------------------
% 3.02/1.79 % (18261)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.79 % (18261)Termination reason: Unknown
% 3.02/1.79 % (18261)Termination phase: shuffling
% 3.02/1.79
% 3.02/1.79 % (18261)Memory used [KB]: 1791
% 3.02/1.79 % (18261)Time elapsed: 0.009 s
% 3.02/1.79 % (18261)Instructions burned: 20 (million)
% 3.02/1.79 % (18261)------------------------------
% 3.02/1.79 % (18261)------------------------------
% 3.02/1.80 % (18263)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_17895 for (2997ds/107Mi)
% 3.02/1.80 % (18251)Instruction limit reached!
% 3.02/1.80 % (18251)------------------------------
% 3.02/1.80 % (18251)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.80 % (18251)Termination reason: Unknown
% 3.02/1.80 % (18251)Termination phase: Saturation
% 3.02/1.80
% 3.02/1.80 % (18251)Memory used [KB]: 8571
% 3.02/1.80 % (18251)Time elapsed: 0.133 s
% 3.02/1.80 % (18251)Instructions burned: 379 (million)
% 3.02/1.80 % (18251)------------------------------
% 3.02/1.80 % (18251)------------------------------
% 3.02/1.81 % (18262)Instruction limit reached!
% 3.02/1.81 % (18262)------------------------------
% 3.02/1.81 % (18262)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.81 % (18262)Termination reason: Unknown
% 3.02/1.81 % (18262)Termination phase: Property scanning
% 3.02/1.81
% 3.02/1.81 % (18262)Memory used [KB]: 2302
% 3.02/1.81 % (18262)Time elapsed: 0.030 s
% 3.02/1.81 % (18262)Instructions burned: 88 (million)
% 3.02/1.81 % (18262)------------------------------
% 3.02/1.81 % (18262)------------------------------
% 3.02/1.81 % (18264)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_17895 for (2997ds/39Mi)
% 3.02/1.82 % (18265)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_17895 for (2997ds/448Mi)
% 3.02/1.82 % (18264)Instruction limit reached!
% 3.02/1.82 % (18264)------------------------------
% 3.02/1.82 % (18264)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.82 % (18264)Termination reason: Unknown
% 3.02/1.82 % (18264)Termination phase: shuffling
% 3.02/1.82
% 3.02/1.82 % (18264)Memory used [KB]: 2174
% 3.02/1.82 % (18264)Time elapsed: 0.014 s
% 3.02/1.82 % (18264)Instructions burned: 40 (million)
% 3.02/1.82 % (18264)------------------------------
% 3.02/1.82 % (18264)------------------------------
% 3.02/1.83 % (18263)Instruction limit reached!
% 3.02/1.83 % (18263)------------------------------
% 3.02/1.83 % (18263)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.83 % (18263)Termination reason: Unknown
% 3.02/1.83 % (18263)Termination phase: Saturation
% 3.02/1.83
% 3.02/1.83 % (18263)Memory used [KB]: 7036
% 3.02/1.83 % (18263)Time elapsed: 0.038 s
% 3.02/1.83 % (18263)Instructions burned: 109 (million)
% 3.02/1.83 % (18263)------------------------------
% 3.02/1.83 % (18263)------------------------------
% 3.02/1.83 % (18266)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_17895 for (2997ds/46Mi)
% 3.02/1.84 % (18267)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_17895 for (2997ds/98Mi)
% 3.02/1.85 % (18266)Instruction limit reached!
% 3.02/1.85 % (18266)------------------------------
% 3.02/1.85 % (18266)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.85 % (18266)Termination reason: Unknown
% 3.02/1.85 % (18266)Termination phase: Preprocessing 2
% 3.02/1.85
% 3.02/1.85 % (18266)Memory used [KB]: 2046
% 3.02/1.85 % (18266)Time elapsed: 0.018 s
% 3.02/1.85 % (18266)Instructions burned: 48 (million)
% 3.02/1.85 % (18266)------------------------------
% 3.02/1.85 % (18266)------------------------------
% 3.02/1.86 % (18268)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_17895 for (2997ds/507Mi)
% 3.02/1.87 % (18245)Instruction limit reached!
% 3.02/1.87 % (18245)------------------------------
% 3.02/1.87 % (18245)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.87 % (18245)Termination reason: Unknown
% 3.02/1.87 % (18245)Termination phase: Saturation
% 3.02/1.87
% 3.02/1.87 % (18245)Memory used [KB]: 8571
% 3.02/1.87 % (18245)Time elapsed: 0.220 s
% 3.02/1.87 % (18245)Instructions burned: 713 (million)
% 3.02/1.87 % (18245)------------------------------
% 3.02/1.87 % (18245)------------------------------
% 3.02/1.87 % (18267)Instruction limit reached!
% 3.02/1.87 % (18267)------------------------------
% 3.02/1.87 % (18267)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.87 % (18267)Termination reason: Unknown
% 3.02/1.87 % (18267)Termination phase: Property scanning
% 3.02/1.87
% 3.02/1.87 % (18267)Memory used [KB]: 2686
% 3.02/1.87 % (18267)Time elapsed: 0.032 s
% 3.02/1.87 % (18267)Instructions burned: 99 (million)
% 3.02/1.87 % (18267)------------------------------
% 3.02/1.87 % (18267)------------------------------
% 3.02/1.88 % (18270)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_17895 for (2996ds/39Mi)
% 3.02/1.88 % (18269)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_17895 for (2996ds/149Mi)
% 3.02/1.90 % (18270)Instruction limit reached!
% 3.02/1.90 % (18270)------------------------------
% 3.02/1.90 % (18270)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.90 % (18270)Termination reason: Unknown
% 3.02/1.90 % (18270)Termination phase: Preprocessing 1
% 3.02/1.90
% 3.02/1.90 % (18270)Memory used [KB]: 2046
% 3.02/1.90 % (18270)Time elapsed: 0.016 s
% 3.02/1.90 % (18270)Instructions burned: 41 (million)
% 3.02/1.90 % (18270)------------------------------
% 3.02/1.90 % (18270)------------------------------
% 3.02/1.91 % (18271)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_17895 for (2996ds/6Mi)
% 3.02/1.91 % (18271)Instruction limit reached!
% 3.02/1.91 % (18271)------------------------------
% 3.02/1.91 % (18271)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.91 % (18271)Termination reason: Unknown
% 3.02/1.91 % (18271)Termination phase: shuffling
% 3.02/1.91
% 3.02/1.91 % (18271)Memory used [KB]: 1535
% 3.02/1.91 % (18271)Time elapsed: 0.004 s
% 3.02/1.91 % (18271)Instructions burned: 7 (million)
% 3.02/1.91 % (18271)------------------------------
% 3.02/1.91 % (18271)------------------------------
% 3.02/1.91 % (18252)Instruction limit reached!
% 3.02/1.91 % (18252)------------------------------
% 3.02/1.91 % (18252)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.91 % (18252)Termination reason: Unknown
% 3.02/1.91 % (18252)Termination phase: Saturation
% 3.02/1.91
% 3.02/1.91 % (18252)Memory used [KB]: 9083
% 3.02/1.91 % (18252)Time elapsed: 0.243 s
% 3.02/1.91 % (18252)Instructions burned: 779 (million)
% 3.02/1.91 % (18252)------------------------------
% 3.02/1.91 % (18252)------------------------------
% 3.02/1.92 % (18272)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_17895 for (2996ds/1087Mi)
% 3.02/1.92 % (18273)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_17895 for (2996ds/47Mi)
% 3.02/1.93 % (18236)Instruction limit reached!
% 3.02/1.93 % (18236)------------------------------
% 3.02/1.93 % (18236)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.93 % (18236)Termination reason: Unknown
% 3.02/1.93 % (18236)Termination phase: Saturation
% 3.02/1.93
% 3.02/1.93 % (18236)Memory used [KB]: 10234
% 3.02/1.93 % (18236)Time elapsed: 0.322 s
% 3.02/1.93 % (18236)Instructions burned: 1041 (million)
% 3.02/1.93 % (18236)------------------------------
% 3.02/1.93 % (18236)------------------------------
% 3.02/1.94 % (18274)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_17895 for (2996ds/44Mi)
% 3.02/1.94 % (18273)Instruction limit reached!
% 3.02/1.94 % (18273)------------------------------
% 3.02/1.94 % (18273)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.94 % (18273)Termination reason: Unknown
% 3.02/1.94 % (18273)Termination phase: Preprocessing 3
% 3.02/1.94
% 3.02/1.94 % (18273)Memory used [KB]: 2174
% 3.02/1.94 % (18273)Time elapsed: 0.017 s
% 3.02/1.94 % (18273)Instructions burned: 48 (million)
% 3.02/1.94 % (18273)------------------------------
% 3.02/1.94 % (18273)------------------------------
% 3.02/1.95 % (18275)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_17895 for (2996ds/432Mi)
% 3.02/1.95 % (18274)Instruction limit reached!
% 3.02/1.95 % (18274)------------------------------
% 3.02/1.95 % (18274)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.95 % (18274)Termination reason: Unknown
% 3.02/1.95 % (18274)Termination phase: Property scanning
% 3.02/1.95
% 3.02/1.95 % (18274)Memory used [KB]: 2302
% 3.02/1.95 % (18274)Time elapsed: 0.014 s
% 3.02/1.95 % (18274)Instructions burned: 45 (million)
% 3.02/1.95 % (18274)------------------------------
% 3.02/1.95 % (18274)------------------------------
% 3.02/1.95 % (18254)Instruction limit reached!
% 3.02/1.95 % (18254)------------------------------
% 3.02/1.95 % (18254)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.95 % (18254)Termination reason: Unknown
% 3.02/1.95 % (18254)Termination phase: Saturation
% 3.02/1.95
% 3.02/1.95 % (18254)Memory used [KB]: 8315
% 3.02/1.95 % (18254)Time elapsed: 0.278 s
% 3.02/1.95 % (18254)Instructions burned: 880 (million)
% 3.02/1.95 % (18254)------------------------------
% 3.02/1.95 % (18254)------------------------------
% 3.02/1.96 % (18276)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_17895 for (2996ds/62Mi)
% 3.02/1.96 % (18277)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_17895 for (2996ds/31Mi)
% 3.02/1.96 % (18265)Instruction limit reached!
% 3.02/1.96 % (18265)------------------------------
% 3.02/1.96 % (18265)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.96 % (18265)Termination reason: Unknown
% 3.02/1.96 % (18265)Termination phase: Saturation
% 3.02/1.96
% 3.02/1.96 % (18265)Memory used [KB]: 8955
% 3.02/1.96 % (18265)Time elapsed: 0.147 s
% 3.02/1.96 % (18265)Instructions burned: 448 (million)
% 3.02/1.96 % (18265)------------------------------
% 3.02/1.96 % (18265)------------------------------
% 3.02/1.97 % (18269)Instruction limit reached!
% 3.02/1.97 % (18269)------------------------------
% 3.02/1.97 % (18269)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.97 % (18269)Termination reason: Unknown
% 3.02/1.97 % (18269)Termination phase: Saturation
% 3.02/1.97
% 3.02/1.97 % (18269)Memory used [KB]: 7291
% 3.02/1.97 % (18269)Time elapsed: 0.084 s
% 3.02/1.97 % (18269)Instructions burned: 149 (million)
% 3.02/1.97 % (18269)------------------------------
% 3.02/1.97 % (18269)------------------------------
% 3.02/1.97 % (18277)Instruction limit reached!
% 3.02/1.97 % (18277)------------------------------
% 3.02/1.97 % (18277)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.97 % (18277)Termination reason: Unknown
% 3.02/1.97 % (18277)Termination phase: Property scanning
% 3.02/1.97
% 3.02/1.97 % (18277)Memory used [KB]: 1918
% 3.02/1.97 % (18277)Time elapsed: 0.012 s
% 3.02/1.97 % (18277)Instructions burned: 34 (million)
% 3.02/1.97 % (18277)------------------------------
% 3.02/1.97 % (18277)------------------------------
% 3.02/1.97 % (18278)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_17895 for (2996ds/96Mi)
% 3.02/1.98 % (18279)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_17895 for (2996ds/453Mi)
% 3.02/1.98 % (18276)Instruction limit reached!
% 3.02/1.98 % (18276)------------------------------
% 3.02/1.98 % (18276)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.98 % (18276)Termination reason: Unknown
% 3.02/1.98 % (18276)Termination phase: Property scanning
% 3.02/1.98
% 3.02/1.98 % (18276)Memory used [KB]: 2302
% 3.02/1.98 % (18276)Time elapsed: 0.019 s
% 3.02/1.98 % (18276)Instructions burned: 64 (million)
% 3.02/1.98 % (18276)------------------------------
% 3.02/1.98 % (18276)------------------------------
% 3.02/1.98 % (18280)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_17895 for (2995ds/71Mi)
% 3.02/1.99 % (18281)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_17895 for (2995ds/83Mi)
% 3.02/2.00 % (18280)Instruction limit reached!
% 3.02/2.00 % (18280)------------------------------
% 3.02/2.00 % (18280)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/2.00 % (18280)Termination reason: Unknown
% 3.02/2.00 % (18280)Termination phase: Function definition elimination
% 3.02/2.00
% 3.02/2.00 % (18280)Memory used [KB]: 2302
% 3.02/2.00 % (18280)Time elapsed: 0.023 s
% 3.02/2.00 % (18280)Instructions burned: 73 (million)
% 3.02/2.00 % (18280)------------------------------
% 3.02/2.00 % (18280)------------------------------
% 3.02/2.00 % (18278)Instruction limit reached!
% 3.02/2.00 % (18278)------------------------------
% 3.02/2.00 % (18278)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/2.00 % (18278)Termination reason: Unknown
% 3.02/2.00 % (18278)Termination phase: Saturation
% 3.02/2.00
% 3.02/2.00 % (18278)Memory used [KB]: 6908
% 3.02/2.00 % (18278)Time elapsed: 0.030 s
% 3.02/2.00 % (18278)Instructions burned: 97 (million)
% 3.02/2.00 % (18278)------------------------------
% 3.02/2.00 % (18278)------------------------------
% 3.02/2.01 % (18281)Instruction limit reached!
% 3.02/2.01 % (18281)------------------------------
% 3.02/2.01 % (18281)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/2.01 % (18281)Termination reason: Unknown
% 3.02/2.01 % (18281)Termination phase: Saturation
% 3.02/2.01
% 3.02/2.01 % (18281)Memory used [KB]: 6780
% 3.02/2.01 % (18281)Time elapsed: 0.027 s
% 3.02/2.01 % (18281)Instructions burned: 83 (million)
% 3.02/2.01 % (18281)------------------------------
% 3.02/2.01 % (18281)------------------------------
% 6.02/2.01 % (18282)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_17895 for (2995ds/11Mi)
% 6.02/2.01 % (18283)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_17895 for (2995ds/41Mi)
% 6.02/2.02 % (18282)Instruction limit reached!
% 6.02/2.02 % (18282)------------------------------
% 6.02/2.02 % (18282)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.02/2.02 % (18282)Termination reason: Unknown
% 6.02/2.02 % (18282)Termination phase: shuffling
% 6.02/2.02
% 6.02/2.02 % (18282)Memory used [KB]: 1663
% 6.02/2.02 % (18282)Time elapsed: 0.005 s
% 6.02/2.02 % (18282)Instructions burned: 13 (million)
% 6.02/2.02 % (18282)------------------------------
% 6.02/2.02 % (18282)------------------------------
% 6.02/2.02 % (18284)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_17895 for (2995ds/6Mi)
% 6.02/2.02 % (18284)Instruction limit reached!
% 6.02/2.02 % (18284)------------------------------
% 6.02/2.02 % (18284)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.02/2.02 % (18284)Termination reason: Unknown
% 6.02/2.02 % (18284)Termination phase: shuffling
% 6.02/2.02
% 6.02/2.02 % (18284)Memory used [KB]: 1535
% 6.02/2.02 % (18284)Time elapsed: 0.003 s
% 6.02/2.02 % (18284)Instructions burned: 6 (million)
% 6.02/2.02 % (18284)------------------------------
% 6.02/2.02 % (18284)------------------------------
% 6.02/2.03 % (18283)Instruction limit reached!
% 6.02/2.03 % (18283)------------------------------
% 6.02/2.03 % (18283)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.02/2.03 % (18283)Termination reason: Unknown
% 6.02/2.03 % (18283)Termination phase: shuffling
% 6.02/2.03
% 6.02/2.03 % (18283)Memory used [KB]: 2302
% 6.02/2.03 % (18283)Time elapsed: 0.013 s
% 6.02/2.03 % (18283)Instructions burned: 42 (million)
% 6.02/2.03 % (18283)------------------------------
% 6.02/2.03 % (18283)------------------------------
% 6.02/2.03 % (18285)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_17895 for (2995ds/150Mi)
% 6.02/2.03 % (18268)Instruction limit reached!
% 6.02/2.03 % (18268)------------------------------
% 6.02/2.03 % (18268)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.02/2.03 % (18268)Termination reason: Unknown
% 6.02/2.03 % (18268)Termination phase: Saturation
% 6.02/2.03
% 6.02/2.03 % (18268)Memory used [KB]: 8827
% 6.02/2.03 % (18268)Time elapsed: 0.171 s
% 6.02/2.03 % (18268)Instructions burned: 509 (million)
% 6.02/2.03 % (18268)------------------------------
% 6.02/2.03 % (18268)------------------------------
% 6.02/2.03 % (18286)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_17895 for (2995ds/343Mi)
% 6.02/2.03 % (18287)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_17895 for (2995ds/109Mi)
% 6.02/2.04 % (18288)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_17895 for (2995ds/253Mi)
% 6.02/2.06 % (18288)Refutation not found, incomplete strategy
% 6.02/2.06 % (18288)------------------------------
% 6.02/2.06 % (18288)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.02/2.06 % (18288)Termination reason: Refutation not found, incomplete strategy
% 6.02/2.06
% 6.02/2.06
% 6.02/2.06 % (18288)Memory used [KB]: 7036
% 6.02/2.06 % (18288)Time elapsed: 0.023 s
% 6.02/2.06 % (18288)Instructions burned: 65 (million)
% 6.02/2.06 % (18288)------------------------------
% 6.02/2.06 % (18288)------------------------------
% 6.02/2.07 % (18287)Instruction limit reached!
% 6.02/2.07 % (18287)------------------------------
% 6.02/2.07 % (18287)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.02/2.07 % (18287)Termination reason: Unknown
% 6.02/2.07 % (18287)Termination phase: Saturation
% 6.02/2.07
% 6.02/2.07 % (18287)Memory used [KB]: 7036
% 6.02/2.07 % (18287)Time elapsed: 0.034 s
% 6.02/2.07 % (18287)Instructions burned: 110 (million)
% 6.02/2.07 % (18287)------------------------------
% 6.02/2.07 % (18287)------------------------------
% 6.02/2.07 % (18289)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_17895 for (2995ds/140Mi)
% 6.02/2.07 % (18285)Instruction limit reached!
% 6.02/2.07 % (18285)------------------------------
% 6.02/2.07 % (18285)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.02/2.07 % (18285)Termination reason: Unknown
% 6.02/2.07 % (18285)Termination phase: Saturation
% 6.02/2.07
% 6.02/2.07 % (18285)Memory used [KB]: 7291
% 6.02/2.07 % (18285)Time elapsed: 0.048 s
% 6.02/2.07 % (18285)Instructions burned: 151 (million)
% 6.02/2.07 % (18285)------------------------------
% 6.02/2.07 % (18285)------------------------------
% 6.02/2.08 % (18290)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_17895 for (2995ds/614Mi)
% 6.02/2.08 % (18291)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_17895 for (2995ds/202Mi)
% 6.02/2.08 % (18275)Instruction limit reached!
% 6.02/2.08 % (18275)------------------------------
% 6.02/2.08 % (18275)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.02/2.08 % (18275)Termination reason: Unknown
% 6.02/2.08 % (18275)Termination phase: Saturation
% 6.02/2.08
% 6.02/2.08 % (18275)Memory used [KB]: 4733
% 6.02/2.08 % (18275)Time elapsed: 0.136 s
% 6.02/2.08 % (18275)Instructions burned: 435 (million)
% 6.02/2.08 % (18275)------------------------------
% 6.02/2.08 % (18275)------------------------------
% 6.02/2.09 % (18292)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_17895 for (2994ds/339Mi)
% 6.02/2.11 % (18291)Refutation not found, incomplete strategy
% 6.02/2.11 % (18291)------------------------------
% 6.02/2.11 % (18291)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.02/2.11 % (18291)Termination reason: Refutation not found, incomplete strategy
% 6.02/2.11
% 6.02/2.11
% 6.02/2.11 % (18291)Memory used [KB]: 7164
% 6.02/2.11 % (18291)Time elapsed: 0.026 s
% 6.02/2.11 % (18291)Instructions burned: 77 (million)
% 6.02/2.11 % (18291)------------------------------
% 6.02/2.11 % (18291)------------------------------
% 6.54/2.12 % (18289)Instruction limit reached!
% 6.54/2.12 % (18289)------------------------------
% 6.54/2.12 % (18289)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.54/2.12 % (18289)Termination reason: Unknown
% 6.54/2.12 % (18289)Termination phase: Saturation
% 6.54/2.12
% 6.54/2.12 % (18289)Memory used [KB]: 7291
% 6.54/2.12 % (18289)Time elapsed: 0.046 s
% 6.54/2.12 % (18289)Instructions burned: 140 (million)
% 6.54/2.12 % (18289)------------------------------
% 6.54/2.12 % (18289)------------------------------
% 6.54/2.12 % (18293)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_17895 for (2994ds/283Mi)
% 6.54/2.12 % (18279)Instruction limit reached!
% 6.54/2.12 % (18279)------------------------------
% 6.54/2.12 % (18279)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.54/2.12 % (18279)Termination reason: Unknown
% 6.54/2.12 % (18279)Termination phase: Saturation
% 6.54/2.12
% 6.54/2.12 % (18279)Memory used [KB]: 8955
% 6.54/2.12 % (18279)Time elapsed: 0.145 s
% 6.54/2.12 % (18279)Instructions burned: 454 (million)
% 6.54/2.12 % (18279)------------------------------
% 6.54/2.12 % (18279)------------------------------
% 6.54/2.13 % (18294)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_17895 for (2994ds/134Mi)
% 6.54/2.13 % (18295)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_17895 for (2994ds/298Mi)
% 6.54/2.14 % (18286)Instruction limit reached!
% 6.54/2.14 % (18286)------------------------------
% 6.54/2.14 % (18286)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.54/2.14 % (18286)Termination reason: Unknown
% 6.54/2.14 % (18286)Termination phase: Saturation
% 6.54/2.14
% 6.54/2.14 % (18286)Memory used [KB]: 8699
% 6.54/2.14 % (18286)Time elapsed: 0.109 s
% 6.54/2.14 % (18286)Instructions burned: 344 (million)
% 6.54/2.14 % (18286)------------------------------
% 6.54/2.14 % (18286)------------------------------
% 6.54/2.15 % (18296)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_17895 for (2994ds/192Mi)
% 6.54/2.17 % (18294)Instruction limit reached!
% 6.54/2.17 % (18294)------------------------------
% 6.54/2.17 % (18294)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.54/2.17 % (18294)Termination reason: Unknown
% 6.54/2.17 % (18294)Termination phase: Saturation
% 6.54/2.17
% 6.54/2.17 % (18294)Memory used [KB]: 7164
% 6.54/2.17 % (18294)Time elapsed: 0.043 s
% 6.54/2.17 % (18294)Instructions burned: 135 (million)
% 6.54/2.17 % (18294)------------------------------
% 6.54/2.17 % (18294)------------------------------
% 6.54/2.18 % (18297)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_17895 for (2993ds/258Mi)
% 6.54/2.20 % (18292)Instruction limit reached!
% 6.54/2.20 % (18292)------------------------------
% 6.54/2.20 % (18292)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.54/2.20 % (18292)Termination reason: Unknown
% 6.54/2.20 % (18292)Termination phase: Saturation
% 6.54/2.20
% 6.54/2.20 % (18292)Memory used [KB]: 8059
% 6.54/2.20 % (18292)Time elapsed: 0.102 s
% 6.54/2.20 % (18292)Instructions burned: 341 (million)
% 6.54/2.20 % (18292)------------------------------
% 6.54/2.20 % (18292)------------------------------
% 6.99/2.20 % (18298)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_17895 for (2993ds/90Mi)
% 6.99/2.21 % (18293)Instruction limit reached!
% 6.99/2.21 % (18293)------------------------------
% 6.99/2.21 % (18293)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.99/2.21 % (18293)Termination reason: Unknown
% 6.99/2.21 % (18293)Termination phase: Saturation
% 6.99/2.21
% 6.99/2.21 % (18293)Memory used [KB]: 7931
% 6.99/2.21 % (18293)Time elapsed: 0.112 s
% 6.99/2.21 % (18293)Instructions burned: 283 (million)
% 6.99/2.21 % (18293)------------------------------
% 6.99/2.21 % (18293)------------------------------
% 6.99/2.21 % (18296)Instruction limit reached!
% 6.99/2.21 % (18296)------------------------------
% 6.99/2.21 % (18296)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.99/2.21 % (18296)Termination reason: Unknown
% 6.99/2.21 % (18296)Termination phase: Saturation
% 6.99/2.21
% 6.99/2.21 % (18296)Memory used [KB]: 8443
% 6.99/2.21 % (18296)Time elapsed: 0.061 s
% 6.99/2.21 % (18296)Instructions burned: 193 (million)
% 6.99/2.21 % (18296)------------------------------
% 6.99/2.21 % (18296)------------------------------
% 6.99/2.21 % (18297)First to succeed.
% 6.99/2.22 % (18299)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_17895 for (2993ds/231Mi)
% 6.99/2.22 % (18297)Refutation found. Thanks to Tanya!
% 6.99/2.22 % SZS status Theorem for DTF2THF_17895
% 6.99/2.22 % SZS output start Proof for DTF2THF_17895
% 6.99/2.22 thf(type_def_5, type, set: ($tType) > $tType).
% 6.99/2.22 thf(type_def_7, type, product_prod: ($tType * $tType) > $tType).
% 6.99/2.22 thf(type_def_8, type, product_unit: $tType).
% 6.99/2.22 thf(type_def_9, type, a: $tType).
% 6.99/2.22 thf(func_def_0, type, product_unit: $tType).
% 6.99/2.22 thf(func_def_1, type, product_prod: ($tType * $tType) > $tType).
% 6.99/2.22 thf(func_def_2, type, set: ($tType) > $tType).
% 6.99/2.22 thf(func_def_3, type, a: $tType).
% 6.99/2.22 thf(func_def_4, type, ordere890947078_space: !>[X0: $tType]:($o)).
% 6.99/2.22 thf(func_def_5, type, type: !>[X0: $tType]:($o)).
% 6.99/2.22 thf(func_def_6, type, minus: !>[X0: $tType]:($o)).
% 6.99/2.22 thf(func_def_7, type, bot: !>[X0: $tType]:($o)).
% 6.99/2.22 thf(func_def_8, type, ord: !>[X0: $tType]:($o)).
% 6.99/2.22 thf(func_def_9, type, top: !>[X0: $tType]:($o)).
% 6.99/2.22 thf(func_def_10, type, group_add: !>[X0: $tType]:($o)).
% 6.99/2.22 thf(func_def_11, type, ab_group_add: !>[X0: $tType]:($o)).
% 6.99/2.22 thf(func_def_12, type, order_top: !>[X0: $tType]:($o)).
% 6.99/2.22 thf(func_def_13, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 6.99/2.22 thf(func_def_14, type, cancel146912293up_add: !>[X0: $tType]:($o)).
% 6.99/2.22 thf(func_def_15, type, bNF_Gr: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > set @ product_prod @ X0 @ X1)).
% 6.99/2.22 thf(func_def_16, type, bNF_fstOp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > product_prod @ X0 @ X2 > product_prod @ X0 @ X1)).
% 6.99/2.22 thf(func_def_17, type, bNF_pick_middlep: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > X0 > X2 > X1)).
% 6.99/2.22 thf(func_def_18, type, bNF_sndOp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > product_prod @ X0 @ X2 > product_prod @ X1 @ X2)).
% 6.99/2.22 thf(func_def_19, type, bNF_Greatest_image2: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ X0 > (X0 > X1) > (X0 > X2) > set @ product_prod @ X1 @ X2)).
% 6.99/2.22 thf(func_def_20, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 6.99/2.22 thf(func_def_21, type, the: !>[X0: $tType]:((X0 > $o) > X0)).
% 6.99/2.22 thf(func_def_22, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 6.99/2.22 thf(func_def_23, type, order_preorder_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 6.99/2.22 thf(func_def_24, type, bot_bot: !>[X0: $tType]:(X0)).
% 6.99/2.22 thf(func_def_25, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 6.99/2.22 thf(func_def_26, type, top_top: !>[X0: $tType]:(X0)).
% 6.99/2.22 thf(func_def_27, type, prefer951318096ood_as: !>[X0: $tType]:(X0 > set @ X0 > set @ product_prod @ X0 @ X0 > set @ X0)).
% 6.99/2.22 thf(func_def_28, type, prefer310429814s_good: !>[X0: $tType]:(X0 > set @ X0 > set @ product_prod @ X0 @ X0 > set @ X0)).
% 6.99/2.22 thf(func_def_29, type, prefer1532642881r_than: !>[X0: $tType]:(X0 > set @ X0 > set @ product_prod @ X0 @ X0 > set @ X0)).
% 6.99/2.22 thf(func_def_30, type, prefer199794634erence: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 6.99/2.22 thf(func_def_31, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 6.99/2.22 thf(func_def_32, type, product_Sigma: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > set @ X1) > set @ product_prod @ X0 @ X1)).
% 6.99/2.22 thf(func_def_33, type, product_curry: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X0 @ X1 > X2) > X0 > X1 > X2)).
% 6.99/2.22 thf(func_def_34, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 6.99/2.22 thf(func_def_35, type, product_map_prod: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > product_prod @ X0 @ X2 > product_prod @ X1 @ X3)).
% 6.99/2.22 thf(func_def_36, type, product_rec_bool: !>[X0: $tType]:(X0 > X0 > $o > X0)).
% 6.99/2.22 thf(func_def_37, type, product_rec_set_bool: !>[X0: $tType]:(X0 > X0 > $o > X0 > $o)).
% 6.99/2.22 thf(func_def_38, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 6.99/2.22 thf(func_def_39, type, product_rec_set_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2 > $o)).
% 6.99/2.22 thf(func_def_40, type, product_rec_set_unit: !>[X0: $tType]:(X0 > product_unit > X0 > $o)).
% 6.99/2.22 thf(func_def_41, type, product_rec_unit: !>[X0: $tType]:(X0 > product_unit > X0)).
% 6.99/2.22 thf(func_def_42, type, product_fst: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 6.99/2.22 thf(func_def_43, type, product_snd: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 6.99/2.22 thf(func_def_44, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 6.99/2.22 thf(func_def_45, type, product_product: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > set @ product_prod @ X0 @ X1)).
% 6.99/2.22 thf(func_def_46, type, product_scomp: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > product_prod @ X1 @ X2) > (X1 > X2 > X3) > X0 > X3)).
% 6.99/2.22 thf(func_def_47, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 6.99/2.22 thf(func_def_48, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 6.99/2.22 thf(func_def_49, type, lex_prod: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X1 @ X1 > set @ product_prod @ product_prod @ X0 @ X1 @ product_prod @ X0 @ X1)).
% 6.99/2.22 thf(func_def_50, type, same_fst: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > set @ product_prod @ X1 @ X1) > set @ product_prod @ product_prod @ X0 @ X1 @ product_prod @ X0 @ X1)).
% 6.99/2.22 thf(func_def_51, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 6.99/2.22 thf(func_def_52, type, b: set @ a).
% 6.99/2.22 thf(func_def_53, type, pr: set @ product_prod @ a @ a).
% 6.99/2.22 thf(func_def_54, type, y: a).
% 6.99/2.22 thf(func_def_55, type, z: a).
% 6.99/2.22 thf(func_def_71, type, sK0: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 6.99/2.22 thf(func_def_72, type, sK1: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 6.99/2.22 thf(func_def_73, type, sK2: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 6.99/2.22 thf(func_def_74, type, sK3: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 6.99/2.22 thf(func_def_75, type, sK4: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 6.99/2.22 thf(func_def_76, type, sK5: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 6.99/2.22 thf(func_def_77, type, sK6: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 6.99/2.22 thf(func_def_78, type, sK7: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 6.99/2.22 thf(func_def_79, type, sK8: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 6.99/2.22 thf(func_def_80, type, sK9: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 6.99/2.22 thf(func_def_81, type, sK10: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 6.99/2.22 thf(func_def_82, type, sK11: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 6.99/2.22 thf(func_def_83, type, sK12: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X0)).
% 6.99/2.22 thf(func_def_84, type, sK13: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X1)).
% 6.99/2.22 thf(func_def_85, type, sK14: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X2)).
% 6.99/2.22 thf(func_def_86, type, sK15: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X3)).
% 6.99/2.22 thf(func_def_87, type, sK16: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X4)).
% 6.99/2.22 thf(func_def_88, type, sK17: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X5)).
% 6.99/2.22 thf(func_def_89, type, sK18: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X6)).
% 6.99/2.22 thf(func_def_90, type, sK19: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 6.99/2.22 thf(func_def_91, type, sK20: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 6.99/2.22 thf(func_def_92, type, sK21: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 6.99/2.22 thf(func_def_93, type, sK22: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 6.99/2.22 thf(func_def_94, type, sK23: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 6.99/2.22 thf(func_def_95, type, sK24: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 6.99/2.22 thf(func_def_96, type, sK25: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 6.99/2.22 thf(func_def_97, type, sK26: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 6.99/2.22 thf(func_def_98, type, sK27: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X6)).
% 6.99/2.22 thf(func_def_99, type, sK28: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 6.99/2.22 thf(func_def_100, type, sK29: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 6.99/2.22 thf(func_def_101, type, sK30: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 6.99/2.22 thf(func_def_102, type, sK31: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 6.99/2.22 thf(func_def_103, type, sK32: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 6.99/2.22 thf(func_def_104, type, sK33: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 6.99/2.22 thf(func_def_105, type, sK34: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 6.99/2.22 thf(func_def_106, type, sK35: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 6.99/2.22 thf(func_def_107, type, sK36: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 6.99/2.22 thf(func_def_108, type, sK37: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 6.99/2.22 thf(func_def_109, type, sK38: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 6.99/2.22 thf(func_def_110, type, sK39: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 6.99/2.22 thf(func_def_111, type, sK40: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 6.99/2.22 thf(func_def_112, type, sK41: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 6.99/2.22 thf(func_def_113, type, sK42: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 6.99/2.22 thf(func_def_114, type, sK43: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 6.99/2.22 thf(func_def_115, type, sK44: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 6.99/2.22 thf(func_def_116, type, sK45: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 6.99/2.22 thf(func_def_117, type, sK46: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 6.99/2.22 thf(func_def_118, type, sK47: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 6.99/2.22 thf(func_def_119, type, sK48: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 6.99/2.22 thf(func_def_120, type, sK49: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 6.99/2.22 thf(func_def_121, type, sK50: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 6.99/2.22 thf(func_def_122, type, sK51: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 6.99/2.22 thf(func_def_123, type, sK52: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 6.99/2.22 thf(func_def_124, type, sK53: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X0)).
% 6.99/2.22 thf(func_def_125, type, sK54: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X1)).
% 6.99/2.22 thf(func_def_126, type, sK55: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X2)).
% 6.99/2.22 thf(func_def_127, type, sK56: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X3)).
% 6.99/2.22 thf(func_def_128, type, sK57: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X4)).
% 6.99/2.22 thf(func_def_129, type, sK58: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X5)).
% 6.99/2.22 thf(func_def_131, type, ph60: !>[X0: $tType]:(X0)).
% 6.99/2.22 thf(func_def_132, type, sK61: set @ a > a).
% 6.99/2.22 thf(func_def_133, type, sK62: a > set @ product_prod @ a @ a > set @ a > set @ a > a).
% 6.99/2.22 thf(f948,plain,(
% 6.99/2.22 $false),
% 6.99/2.22 inference(trivial_inequality_removal,[status(thm)],[f945])).
% 6.99/2.22 thf(f945,plain,(
% 6.99/2.22 ($true = $false)),
% 6.99/2.22 inference(superposition,[status(thm)],[f879,f932])).
% 6.99/2.22 thf(f932,plain,(
% 6.99/2.22 ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ b) @ y) @ pr) = $false)),
% 6.99/2.22 inference(trivial_inequality_removal,[status(thm)],[f931])).
% 6.99/2.22 thf(f931,plain,(
% 6.99/2.22 ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ b) @ y) @ pr) = $false) | ($true = $false)),
% 6.99/2.22 inference(forward_demodulation,[status(thm)],[f922,f905])).
% 6.99/2.22 thf(f905,plain,(
% 6.99/2.22 ($true = (member @ a @ (sK61 @ b) @ b))),
% 6.99/2.22 inference(trivial_inequality_removal,[status(thm)],[f902])).
% 6.99/2.22 thf(f902,plain,(
% 6.99/2.22 ($true != $true) | ($true = (member @ a @ (sK61 @ b) @ b))),
% 6.99/2.22 inference(equality_factoring,[status(thm)],[f876])).
% 6.99/2.22 thf(f876,plain,(
% 6.99/2.22 ( ! [X0 : set @ a] : (($true = (member @ a @ (sK61 @ X0) @ b)) | ($true = (member @ a @ (sK61 @ X0) @ X0))) )),
% 6.99/2.22 inference(binary_proxy_clausification,[status(thm)],[f874])).
% 6.99/2.22 thf(f874,plain,(
% 6.99/2.22 ( ! [X0 : set @ a] : (($true = (member @ a @ (sK61 @ X0) @ b)) | ($true = ((member @ a @ (sK61 @ X0) @ X0) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ X0) @ y) @ pr)))) )),
% 6.99/2.22 inference(binary_proxy_clausification,[status(thm)],[f868])).
% 6.99/2.22 thf(f868,plain,(
% 6.99/2.22 ( ! [X0 : set @ a] : (($true = ((member @ a @ (sK61 @ X0) @ b) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ X0) @ y) @ pr))) | ($true = ((member @ a @ (sK61 @ X0) @ X0) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ X0) @ y) @ pr)))) )),
% 6.99/2.22 inference(binary_proxy_clausification,[status(thm)],[f867])).
% 6.99/2.22 thf(f867,plain,(
% 6.99/2.22 ( ! [X0 : set @ a] : ((((member @ a @ (sK61 @ X0) @ X0) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ X0) @ y) @ pr)) != ((member @ a @ (sK61 @ X0) @ b) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ X0) @ y) @ pr)))) )),
% 6.99/2.22 inference(beta_eta_normalization,[status(thm)],[f866])).
% 6.99/2.22 thf(f866,plain,(
% 6.99/2.22 ( ! [X0 : set @ a] : ((((^[Y0 : a]: ((member @ a @ Y0 @ X0) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ Y0 @ y) @ pr))) @ (sK61 @ X0)) != ((^[Y0 : a]: ((member @ a @ Y0 @ b) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ Y0 @ y) @ pr))) @ (sK61 @ X0)))) )),
% 6.99/2.22 inference(negative_extensionality,[status(thm)],[f865])).
% 6.99/2.22 thf(f865,plain,(
% 6.99/2.22 ( ! [X0 : set @ a] : (((^[Y0 : a]: ((member @ a @ Y0 @ b) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ Y0 @ y) @ pr))) != (^[Y0 : a]: ((member @ a @ Y0 @ X0) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ Y0 @ y) @ pr))))) )),
% 6.99/2.22 inference(trivial_inequality_removal,[status(thm)],[f864])).
% 6.99/2.22 thf(f864,plain,(
% 6.99/2.22 ( ! [X0 : set @ a] : (($true != $true) | ((^[Y0 : a]: ((member @ a @ Y0 @ b) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ Y0 @ y) @ pr))) != (^[Y0 : a]: ((member @ a @ Y0 @ X0) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ Y0 @ y) @ pr))))) )),
% 6.99/2.22 inference(constrained_superposition,[status(thm)],[f862,f860])).
% 6.99/2.22 thf(f860,plain,(
% 6.99/2.22 ($true = (member @ a @ z @ (collect @ a @ (^[Y0 : a]: ((member @ a @ Y0 @ b) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ Y0 @ y) @ pr))))))),
% 6.99/2.22 inference(beta_eta_normalization,[status(thm)],[f855])).
% 6.99/2.22 thf(f855,plain,(
% 6.99/2.22 ($true = (member @ a @ z @ ((^[Y0 : a]: ((^[Y1 : set @ a]: ((^[Y2 : set @ product_prod @ a @ a]: (collect @ a @ (^[Y3 : a]: ((member @ a @ Y3 @ Y1) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ Y3 @ Y0) @ Y2))))))))) @ y @ b @ pr)))),
% 6.99/2.22 inference(definition_unfolding,[status(thm)],[f850,f839])).
% 6.99/2.22 thf(f839,plain,(
% 6.99/2.22 ( ! [X0 : $tType] : ((prefer310429814s_good @ X0 = (^[Y0 : X0]: ((^[Y1 : set @ X0]: ((^[Y2 : set @ product_prod @ X0 @ X0]: (collect @ X0 @ (^[Y3 : X0]: ((member @ X0 @ Y3 @ Y1) & (member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ Y3 @ Y0) @ Y2))))))))))) )),
% 6.99/2.22 inference(cnf_transformation,[status(thm)],[f416])).
% 6.99/2.22 thf(f416,plain,(
% 6.99/2.22 ! [X0 : $tType] : (prefer310429814s_good @ X0 = (^[Y0 : X0]: ((^[Y1 : set @ X0]: ((^[Y2 : set @ product_prod @ X0 @ X0]: (collect @ X0 @ (^[Y3 : X0]: ((member @ X0 @ Y3 @ Y1) & (member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ Y3 @ Y0) @ Y2))))))))))),
% 6.99/2.22 inference(fool_elimination,[status(thm)],[f415])).
% 6.99/2.22 thf(f415,plain,(
% 6.99/2.22 ! [X0 : $tType] : (prefer310429814s_good @ X0 = (^[X1 : X0, X2 : set @ X0, X3 : set @ product_prod @ X0 @ X0] : (collect @ X0 @ (^[X4 : X0] : ((member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X4 @ X1) @ X3) & (member @ X0 @ X4 @ X2))))))),
% 6.99/2.22 inference(rectify,[status(thm)],[f21])).
% 6.99/2.22 thf(f21,axiom,(
% 6.99/2.22 ! [X0 : $tType] : (prefer310429814s_good @ X0 = (^[X33 : X0, X34 : set @ X0, X35 : set @ product_prod @ X0 @ X0] : (collect @ X0 @ (^[X36 : X0] : ((member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X36 @ X33) @ X35) & (member @ X0 @ X36 @ X34))))))),
% 6.99/2.22 file('/export/starexec/sandbox/tmp/tmp.9NETwVAXmU/DTF2THF_17895.p',fact_20_at__least__as__good__def)).
% 6.99/2.22 thf(f850,plain,(
% 6.99/2.22 ((member @ a @ z @ (prefer310429814s_good @ a @ y @ b @ pr)) = $true)),
% 6.99/2.22 inference(cnf_transformation,[status(thm)],[f344])).
% 6.99/2.22 thf(f344,plain,(
% 6.99/2.22 ((member @ a @ z @ (prefer310429814s_good @ a @ y @ b @ pr)) = $true)),
% 6.99/2.22 inference(fool_elimination,[status(thm)],[f343])).
% 6.99/2.22 thf(f343,plain,(
% 6.99/2.22 (member @ a @ z @ (prefer310429814s_good @ a @ y @ b @ pr))),
% 6.99/2.22 inference(rectify,[status(thm)],[f1])).
% 6.99/2.22 thf(f1,axiom,(
% 6.99/2.22 (member @ a @ z @ (prefer310429814s_good @ a @ y @ b @ pr))),
% 6.99/2.22 file('/export/starexec/sandbox/tmp/tmp.9NETwVAXmU/DTF2THF_17895.p',fact_0_assms)).
% 6.99/2.22 thf(f862,plain,(
% 6.99/2.22 ( ! [X0 : set @ a] : (($true != (member @ a @ z @ (collect @ a @ (^[Y0 : a]: ((member @ a @ Y0 @ X0) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ Y0 @ y) @ pr))))))) )),
% 6.99/2.22 inference(trivial_inequality_removal,[status(thm)],[f861])).
% 6.99/2.22 thf(f861,plain,(
% 6.99/2.22 ( ! [X0 : set @ a] : (($true != (member @ a @ z @ (collect @ a @ (^[Y0 : a]: ((member @ a @ Y0 @ X0) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ Y0 @ y) @ pr)))))) | ($true != $true)) )),
% 6.99/2.22 inference(superposition,[status(thm)],[f843,f858])).
% 6.99/2.22 thf(f858,plain,(
% 6.99/2.22 ( ! [X0 : $tType,X2 : X0,X3 : set @ X0,X1 : X0,X4 : set @ product_prod @ X0 @ X0] : (($true != (member @ X0 @ X1 @ (collect @ X0 @ (^[Y0 : X0]: ((member @ X0 @ Y0 @ X3) & (member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ Y0 @ X2) @ X4)))))) | ((member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X1 @ X2) @ X4) = $true)) )),
% 6.99/2.22 inference(beta_eta_normalization,[status(thm)],[f854])).
% 6.99/2.22 thf(f854,plain,(
% 6.99/2.22 ( ! [X0 : $tType,X2 : X0,X3 : set @ X0,X1 : X0,X4 : set @ product_prod @ X0 @ X0] : (($true != (member @ X0 @ X1 @ ((^[Y0 : X0]: ((^[Y1 : set @ X0]: ((^[Y2 : set @ product_prod @ X0 @ X0]: (collect @ X0 @ (^[Y3 : X0]: ((member @ X0 @ Y3 @ Y1) & (member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ Y3 @ Y0) @ Y2))))))))) @ X2 @ X3 @ X4))) | ((member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X1 @ X2) @ X4) = $true)) )),
% 6.99/2.22 inference(definition_unfolding,[status(thm)],[f845,f839])).
% 6.99/2.22 thf(f845,plain,(
% 6.99/2.22 ( ! [X0 : $tType,X2 : X0,X3 : set @ X0,X1 : X0,X4 : set @ product_prod @ X0 @ X0] : (((member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X1 @ X2) @ X4) = $true) | ((member @ X0 @ X1 @ (prefer310429814s_good @ X0 @ X2 @ X3 @ X4)) != $true)) )),
% 6.99/2.22 inference(cnf_transformation,[status(thm)],[f795])).
% 6.99/2.22 thf(f795,plain,(
% 6.99/2.22 ! [X0 : $tType,X1 : X0,X2 : X0,X3 : set @ X0,X4 : set @ product_prod @ X0 @ X0] : (((member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X1 @ X2) @ X4) = $true) | ((member @ X0 @ X1 @ (prefer310429814s_good @ X0 @ X2 @ X3 @ X4)) != $true))),
% 6.99/2.22 inference(ennf_transformation,[status(thm)],[f692])).
% 6.99/2.22 thf(f692,plain,(
% 6.99/2.22 ! [X0 : $tType,X1 : X0,X2 : X0,X3 : set @ X0,X4 : set @ product_prod @ X0 @ X0] : (((member @ X0 @ X1 @ (prefer310429814s_good @ X0 @ X2 @ X3 @ X4)) = $true) => ((member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X1 @ X2) @ X4) = $true))),
% 6.99/2.22 inference(fool_elimination,[status(thm)],[f691])).
% 6.99/2.22 thf(f691,plain,(
% 6.99/2.22 ! [X0 : $tType,X1 : X0,X2 : X0,X3 : set @ X0,X4 : set @ product_prod @ X0 @ X0] : ((member @ X0 @ X1 @ (prefer310429814s_good @ X0 @ X2 @ X3 @ X4)) => (member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X1 @ X2) @ X4))),
% 6.99/2.22 inference(rectify,[status(thm)],[f2])).
% 6.99/2.22 thf(f2,axiom,(
% 6.99/2.22 ! [X0 : $tType,X5 : X0,X6 : X0,X7 : set @ X0,X8 : set @ product_prod @ X0 @ X0] : ((member @ X0 @ X5 @ (prefer310429814s_good @ X0 @ X6 @ X7 @ X8)) => (member @ product_prod @ X0 @ X0 @ (product_Pair @ X0 @ X0 @ X5 @ X6) @ X8))),
% 6.99/2.22 file('/export/starexec/sandbox/tmp/tmp.9NETwVAXmU/DTF2THF_17895.p',fact_1_at__lst__asgd__ge)).
% 6.99/2.22 thf(f843,plain,(
% 6.99/2.22 ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ z @ y) @ pr) != $true)),
% 6.99/2.22 inference(cnf_transformation,[status(thm)],[f773])).
% 6.99/2.22 thf(f773,plain,(
% 6.99/2.22 ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ z @ y) @ pr) != $true)),
% 6.99/2.22 inference(flattening,[status(thm)],[f590])).
% 6.99/2.22 thf(f590,plain,(
% 6.99/2.22 ~((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ z @ y) @ pr) = $true)),
% 6.99/2.22 inference(fool_elimination,[status(thm)],[f589])).
% 6.99/2.22 thf(f589,plain,(
% 6.99/2.22 ~(member @ product_prod @ a @ a @ (product_Pair @ a @ a @ z @ y) @ pr)),
% 6.99/2.22 inference(rectify,[status(thm)],[f291])).
% 6.99/2.22 thf(f291,negated_conjecture,(
% 6.99/2.22 ~(member @ product_prod @ a @ a @ (product_Pair @ a @ a @ z @ y) @ pr)),
% 6.99/2.22 inference(negated_conjecture,[status(cth)],[f290])).
% 6.99/2.22 thf(f290,conjecture,(
% 6.99/2.22 (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ z @ y) @ pr)),
% 6.99/2.22 file('/export/starexec/sandbox/tmp/tmp.9NETwVAXmU/DTF2THF_17895.p',conj_0)).
% 6.99/2.22 thf(f922,plain,(
% 6.99/2.22 ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ b) @ y) @ pr) = $false) | ($false = (member @ a @ (sK61 @ b) @ b))),
% 6.99/2.22 inference(trivial_inequality_removal,[status(thm)],[f913])).
% 6.99/2.22 thf(f913,plain,(
% 6.99/2.22 ($false = (member @ a @ (sK61 @ b) @ b)) | ($false != $false) | ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ b) @ y) @ pr) = $false)),
% 6.99/2.22 inference(equality_factoring,[status(thm)],[f872])).
% 6.99/2.22 thf(f872,plain,(
% 6.99/2.22 ( ! [X0 : set @ a] : (($false = (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ X0) @ y) @ pr)) | ($false = (member @ a @ (sK61 @ X0) @ X0)) | ($false = (member @ a @ (sK61 @ X0) @ b))) )),
% 6.99/2.22 inference(duplicate_literal_removal,[status(thm)],[f871])).
% 6.99/2.22 thf(f871,plain,(
% 6.99/2.22 ( ! [X0 : set @ a] : (($false = (member @ a @ (sK61 @ X0) @ X0)) | ($false = (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ X0) @ y) @ pr)) | ($false = (member @ a @ (sK61 @ X0) @ b)) | ($false = (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ X0) @ y) @ pr))) )),
% 6.99/2.22 inference(binary_proxy_clausification,[status(thm)],[f870])).
% 6.99/2.22 thf(f870,plain,(
% 6.99/2.22 ( ! [X0 : set @ a] : (($false = (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ X0) @ y) @ pr)) | ($false = ((member @ a @ (sK61 @ X0) @ b) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ X0) @ y) @ pr))) | ($false = (member @ a @ (sK61 @ X0) @ X0))) )),
% 6.99/2.22 inference(binary_proxy_clausification,[status(thm)],[f869])).
% 6.99/2.22 thf(f869,plain,(
% 6.99/2.22 ( ! [X0 : set @ a] : (($false = ((member @ a @ (sK61 @ X0) @ X0) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ X0) @ y) @ pr))) | ($false = ((member @ a @ (sK61 @ X0) @ b) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ X0) @ y) @ pr)))) )),
% 6.99/2.22 inference(binary_proxy_clausification,[status(thm)],[f867])).
% 6.99/2.22 thf(f879,plain,(
% 6.99/2.22 ( ! [X0 : set @ a] : (($true = (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ X0) @ y) @ pr))) )),
% 6.99/2.22 inference(duplicate_literal_removal,[status(thm)],[f877])).
% 6.99/2.22 thf(f877,plain,(
% 6.99/2.22 ( ! [X0 : set @ a] : (($true = (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ X0) @ y) @ pr)) | ($true = (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ X0) @ y) @ pr))) )),
% 6.99/2.22 inference(binary_proxy_clausification,[status(thm)],[f873])).
% 6.99/2.22 thf(f873,plain,(
% 6.99/2.22 ( ! [X0 : set @ a] : (($true = ((member @ a @ (sK61 @ X0) @ X0) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ X0) @ y) @ pr))) | ($true = (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ (sK61 @ X0) @ y) @ pr))) )),
% 6.99/2.22 inference(binary_proxy_clausification,[status(thm)],[f868])).
% 6.99/2.22 % SZS output end Proof for DTF2THF_17895
% 6.99/2.22 % (18297)------------------------------
% 6.99/2.22 % (18297)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.99/2.22 % (18297)Termination reason: Refutation
% 6.99/2.22
% 6.99/2.22 % (18297)Memory used [KB]: 7036
% 6.99/2.22 % (18297)Time elapsed: 0.038 s
% 6.99/2.22 % (18297)Instructions burned: 117 (million)
% 6.99/2.22 % (18297)------------------------------
% 6.99/2.22 % (18297)------------------------------
% 6.99/2.22 % (18224)Success in time 0.653 s
%------------------------------------------------------------------------------