%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP154^2 : 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 : n023.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:59 AM UTC 2026
% Result : Theorem 2.96s 1.70s
% Output : Refutation 2.96s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP154^2 : 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 : n023.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 08:28:49 EST 2026
% 0.17/0.35 % CPUTime :
% 0.17/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.27/0.47 ---- Original DTF file ---
% 0.27/0.47 thf(spec,logic,$$dhol).
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 % File : ITP154^2 : TPTP v9.2.1. Released v7.5.0.
% 0.27/0.47 % Domain : Interactive Theorem Proving
% 0.27/0.47 % Problem : Sledgehammer Preferences problem prob_205__6250810_1
% 0.27/0.47 % Version : Especial.
% 0.27/0.47 % English :
% 0.27/0.47
% 0.27/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.27/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.27/0.47 % Source : [Des21]
% 0.27/0.47 % Names : Preferences/prob_205__6250810_1 [Des21]
% 0.27/0.47
% 0.27/0.47 % Status : Theorem
% 0.27/0.47 % Rating : 0.00 v7.5.0
% 0.27/0.47 % Syntax : Number of formulae : 352 ( 87 unt; 50 typ; 0 def)
% 0.27/0.47 % Number of atoms : 1011 ( 194 equ; 0 cnn)
% 0.27/0.47 % Maximal formula atoms : 13 ( 3 avg)
% 0.27/0.47 % Number of connectives : 5267 ( 112 ~; 14 |; 90 &;4508 @)
% 0.27/0.47 % ( 0 <=>; 543 =>; 0 <=; 0 <~>)
% 0.27/0.47 % Maximal formula depth : 28 ( 9 avg)
% 0.27/0.47 % Number of types : 2 ( 1 usr)
% 0.27/0.47 % Number of type conns : 134 ( 134 >; 0 *; 0 +; 0 <<)
% 0.27/0.47 % Number of symbols : 51 ( 49 usr; 5 con; 0-5 aty)
% 0.27/0.47 % Number of variables : 1125 ( 39 ^;1019 !; 17 ?;1125 :)
% 0.27/0.47 % ( 50 !>; 0 ?*; 0 @-; 0 @+)
% 0.27/0.47 % SPC : TH1_THM_EQU_NAR
% 0.27/0.47
% 0.27/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:29:01.254
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 % Could-be-implicit typings (3)
% 0.27/0.47 thf(ty_t_Product__Type_Oprod,type,
% 0.27/0.47 product_prod: $tType > $tType > $tType ).
% 0.27/0.47
% 0.27/0.47 thf(ty_t_Set_Oset,type,
% 0.27/0.47 set: $tType > $tType ).
% 0.27/0.47
% 0.27/0.47 thf(ty_tf_a,type,
% 0.27/0.47 a: $tType ).
% 0.27/0.47
% 0.27/0.47 % Explicit typings (47)
% 0.27/0.47 thf(sy_cl_HOL_Otype,type,
% 0.27/0.47 type:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Lattices_Oinf,type,
% 0.27/0.47 inf:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Lattices_Osup,type,
% 0.27/0.47 sup:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Orderings_Obot,type,
% 0.27/0.47 bot:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Orderings_Oord,type,
% 0.27/0.47 ord:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.27/0.47 order:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Lattices_Olattice,type,
% 0.27/0.47 lattice:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.27/0.47 finite_finite:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Orderings_Olinorder,type,
% 0.27/0.47 linorder:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Orderings_Opreorder,type,
% 0.27/0.47 preorder:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.27/0.47 order_bot:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 0.27/0.47 semilattice_inf:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.27/0.47 semilattice_sup:
% 0.27/0.47 !>[A: $tType] : $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.27/0.47 bNF_Ca1785829860lChain:
% 0.27/0.47 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Finite__Set_Ofinite,type,
% 0.27/0.47 finite_finite2:
% 0.27/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 0.27/0.47 inf_inf:
% 0.27/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.27/0.47 sup_sup:
% 0.27/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
% 0.27/0.47 lattic929149872er_Max:
% 0.27/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Lattices__Big_Olinorder__class_OMin,type,
% 0.27/0.47 lattic929674654er_Min:
% 0.27/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
% 0.27/0.47 lattic1704895705min_on:
% 0.27/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
% 0.27/0.47 lattic2065218050nf_fin:
% 0.27/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
% 0.27/0.47 lattic477160up_fin:
% 0.27/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Order__Relation_Opreorder__on,type,
% 0.27/0.47 order_preorder_on:
% 0.27/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Order__Relation_Ounder,type,
% 0.27/0.47 order_under:
% 0.27/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.27/0.47 bot_bot:
% 0.27/0.47 !>[A: $tType] : A ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.27/0.47 ord_less_eq:
% 0.27/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 0.27/0.47 order_Greatest:
% 0.27/0.47 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oas__good__as,type,
% 0.27/0.47 prefer951318096ood_as:
% 0.27/0.47 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oat__least__as__good,type,
% 0.27/0.47 prefer310429814s_good:
% 0.27/0.47 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Ono__better__than,type,
% 0.27/0.47 prefer1532642881r_than:
% 0.27/0.47 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Opreference,type,
% 0.27/0.47 prefer199794634erence:
% 0.27/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference,type,
% 0.27/0.47 prefer1997167224erence:
% 0.27/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference__axioms,type,
% 0.27/0.47 prefer1801827867axioms:
% 0.27/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Product__Type_OPair,type,
% 0.27/0.47 product_Pair:
% 0.27/0.47 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.27/0.47 produc2004651681e_prod:
% 0.27/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.27/0.47 product_rec_prod:
% 0.27/0.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Relation_OId__on,type,
% 0.27/0.47 id_on:
% 0.27/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Relation_Orefl__on,type,
% 0.27/0.47 refl_on:
% 0.27/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Relation_Ototal__on,type,
% 0.27/0.47 total_on:
% 0.27/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Relation_Otrans,type,
% 0.27/0.47 trans:
% 0.27/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Set_OCollect,type,
% 0.27/0.47 collect:
% 0.27/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Set_Ois__empty,type,
% 0.27/0.47 is_empty:
% 0.27/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_member,type,
% 0.27/0.47 member:
% 0.27/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_v_carrier,type,
% 0.27/0.47 carrier: set @ a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_v_relation,type,
% 0.27/0.47 relation: set @ ( product_prod @ a @ a ) ).
% 0.27/0.47
% 0.27/0.47 thf(sy_v_x,type,
% 0.27/0.47 x: a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_v_y,type,
% 0.27/0.47 y: a ).
% 0.27/0.47
% 0.27/0.47 % Relevant facts (256)
% 0.27/0.47 thf(fact_0_assms_I3_J,axiom,
% 0.27/0.47 member @ a @ y @ carrier ).
% 0.27/0.47
% 0.27/0.47 % assms(3)
% 0.27/0.47 thf(fact_1_assms_I2_J,axiom,
% 0.27/0.47 member @ a @ x @ carrier ).
% 0.27/0.47
% 0.27/0.47 % assms(2)
% 0.27/0.47 thf(fact_2_rational__preference__axioms,axiom,
% 0.27/0.47 prefer1997167224erence @ a @ carrier @ relation ).
% 0.27/0.47
% 0.27/0.47 % rational_preference_axioms
% 0.27/0.47 thf(fact_3_nbt__nest,axiom,
% 0.27/0.47 ! [Y: a,X: a] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) )
% 0.27/0.47 | ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % nbt_nest
% 0.27/0.47 thf(fact_4_assms_I1_J,axiom,
% 0.27/0.47 finite_finite2 @ a @ carrier ).
% 0.27/0.47
% 0.27/0.47 % assms(1)
% 0.27/0.47 thf(fact_5_preference__axioms,axiom,
% 0.27/0.47 prefer199794634erence @ a @ carrier @ relation ).
% 0.27/0.47
% 0.27/0.47 % preference_axioms
% 0.27/0.47 thf(fact_6_trans__refl,axiom,
% 0.27/0.47 order_preorder_on @ a @ carrier @ relation ).
% 0.27/0.47
% 0.27/0.47 % trans_refl
% 0.27/0.47 thf(fact_7_no__better__subset__pref,axiom,
% 0.27/0.47 ! [X: a,Y: a] :
% 0.27/0.47 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.27/0.47 => ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % no_better_subset_pref
% 0.27/0.47 thf(fact_8_no__better__thansubset__rel,axiom,
% 0.27/0.47 ! [X: a,Y: a] :
% 0.27/0.47 ( ( member @ a @ X @ carrier )
% 0.27/0.47 => ( ( member @ a @ Y @ carrier )
% 0.27/0.47 => ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) )
% 0.27/0.47 => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % no_better_thansubset_rel
% 0.27/0.47 thf(fact_9_no__better__than__nonepty,axiom,
% 0.27/0.47 ! [X: a] :
% 0.27/0.47 ( ( carrier
% 0.27/0.47 != ( bot_bot @ ( set @ a ) ) )
% 0.27/0.47 => ( ( member @ a @ X @ carrier )
% 0.27/0.47 => ( ( prefer1532642881r_than @ a @ X @ carrier @ relation )
% 0.27/0.47 != ( bot_bot @ ( set @ a ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % no_better_than_nonepty
% 0.27/0.47 thf(fact_10_total,axiom,
% 0.27/0.47 total_on @ a @ carrier @ relation ).
% 0.27/0.47
% 0.27/0.47 % total
% 0.27/0.47 thf(fact_11_worse__in__no__better,axiom,
% 0.27/0.47 ! [X: a,Y: a] :
% 0.27/0.47 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.27/0.47 => ( member @ a @ Y @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % worse_in_no_better
% 0.27/0.47 thf(fact_12_reflexivity,axiom,
% 0.27/0.47 refl_on @ a @ carrier @ relation ).
% 0.27/0.47
% 0.27/0.47 % reflexivity
% 0.27/0.47 thf(fact_13_compl,axiom,
% 0.27/0.47 ! [X2: a] :
% 0.27/0.47 ( ( member @ a @ X2 @ carrier )
% 0.27/0.47 => ! [Xa: a] :
% 0.27/0.47 ( ( member @ a @ Xa @ carrier )
% 0.27/0.47 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Xa ) @ relation )
% 0.27/0.47 | ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Xa @ X2 ) @ relation ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % compl
% 0.27/0.47 thf(fact_14_completeD,axiom,
% 0.27/0.47 ! [X: a,Y: a] :
% 0.27/0.47 ( ( member @ a @ X @ carrier )
% 0.27/0.47 => ( ( member @ a @ Y @ carrier )
% 0.27/0.47 => ( ( X != Y )
% 0.27/0.47 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.27/0.47 | ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % completeD
% 0.27/0.47 thf(fact_15_not__outside,axiom,
% 0.27/0.47 ! [X: a,Y: a] :
% 0.27/0.47 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.27/0.47 => ( member @ a @ X @ carrier ) ) ).
% 0.27/0.47
% 0.27/0.47 % not_outside
% 0.27/0.47 thf(fact_16_strict__is__neg__transitive,axiom,
% 0.27/0.47 ! [X: a,Y: a,Z: a] :
% 0.27/0.47 ( ( ( member @ a @ X @ carrier )
% 0.27/0.47 & ( member @ a @ Y @ carrier )
% 0.27/0.47 & ( member @ a @ Z @ carrier ) )
% 0.27/0.47 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.27/0.47 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
% 0.27/0.47 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation )
% 0.27/0.47 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X ) @ relation ) )
% 0.27/0.47 | ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation )
% 0.27/0.47 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % strict_is_neg_transitive
% 0.27/0.47 thf(fact_17_weak__is__transitive,axiom,
% 0.27/0.47 ! [X: a,Y: a,Z: a] :
% 0.27/0.47 ( ( ( member @ a @ X @ carrier )
% 0.27/0.47 & ( member @ a @ Y @ carrier )
% 0.27/0.47 & ( member @ a @ Z @ carrier ) )
% 0.27/0.47 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.27/0.47 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
% 0.27/0.47 => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % weak_is_transitive
% 0.27/0.47 thf(fact_18_strict__trans,axiom,
% 0.27/0.47 ! [X: a,Y: a,Z: a] :
% 0.27/0.47 ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.27/0.47 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
% 0.27/0.47 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
% 0.27/0.47 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation ) )
% 0.27/0.47 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation )
% 0.27/0.47 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X ) @ relation ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % strict_trans
% 0.27/0.47 thf(fact_19_indiff__trans,axiom,
% 0.27/0.47 ! [X: a,Y: a,Z: a] :
% 0.27/0.47 ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.27/0.47 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
% 0.27/0.47 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
% 0.27/0.47 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation ) )
% 0.27/0.47 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation )
% 0.27/0.47 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X ) @ relation ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % indiff_trans
% 0.27/0.47 thf(fact_20_strict__not__refl__weak,axiom,
% 0.27/0.47 ! [X: a,Y: a] :
% 0.27/0.47 ( ( ( member @ a @ X @ carrier )
% 0.27/0.47 & ( member @ a @ Y @ carrier ) )
% 0.27/0.47 => ( ( ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
% 0.27/0.47 = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.27/0.47 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % strict_not_refl_weak
% 0.27/0.47 thf(fact_21_rational__preference_Oaxioms_I1_J,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.27/0.47 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( prefer199794634erence @ A @ Carrier @ Relation ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference.axioms(1)
% 0.27/0.47 thf(fact_22_preference__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( prefer199794634erence @ A )
% 0.27/0.47 = ( ^ [Carrier2: set @ A,Relation2: set @ ( product_prod @ A @ A )] :
% 0.27/0.47 ( ! [X3: A,Y2: A] :
% 0.27/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ Relation2 )
% 0.27/0.47 => ( member @ A @ X3 @ Carrier2 ) )
% 0.27/0.47 & ! [X3: A,Y2: A] :
% 0.27/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ Relation2 )
% 0.27/0.47 => ( member @ A @ Y2 @ Carrier2 ) )
% 0.27/0.47 & ( order_preorder_on @ A @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % preference_def
% 0.27/0.47 thf(fact_23_preference_Ointro,axiom,
% 0.27/0.47 ! [A: $tType,Relation: set @ ( product_prod @ A @ A ),Carrier: set @ A] :
% 0.27/0.47 ( ! [X4: A,Y3: A] :
% 0.27/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ Relation )
% 0.27/0.47 => ( member @ A @ X4 @ Carrier ) )
% 0.27/0.47 => ( ! [X4: A,Y3: A] :
% 0.27/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ Relation )
% 0.27/0.47 => ( member @ A @ Y3 @ Carrier ) )
% 0.27/0.47 => ( ( order_preorder_on @ A @ Carrier @ Relation )
% 0.27/0.47 => ( prefer199794634erence @ A @ Carrier @ Relation ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % preference.intro
% 0.27/0.47 thf(fact_24_preference_Otrans__refl,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.27/0.47 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( order_preorder_on @ A @ Carrier @ Relation ) ) ).
% 0.27/0.47
% 0.27/0.47 % preference.trans_refl
% 0.27/0.47 thf(fact_25_preference_Onot__outside,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.27/0.47 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.27/0.47 => ( member @ A @ X @ Carrier ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % preference.not_outside
% 0.27/0.47 thf(fact_26_preference_Oreflexivity,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.27/0.47 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( refl_on @ A @ Carrier @ Relation ) ) ).
% 0.27/0.47
% 0.27/0.47 % preference.reflexivity
% 0.27/0.47 thf(fact_27_preference_Oindiff__trans,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 0.27/0.47 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.27/0.47 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 0.27/0.47 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
% 0.27/0.47 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation ) )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation )
% 0.27/0.47 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % preference.indiff_trans
% 0.27/0.47 thf(fact_28_rational__preference_Ocompl,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.27/0.47 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ! [X2: A] :
% 0.27/0.47 ( ( member @ A @ X2 @ Carrier )
% 0.27/0.47 => ! [Xa: A] :
% 0.27/0.47 ( ( member @ A @ Xa @ Carrier )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa ) @ Relation )
% 0.27/0.47 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference.compl
% 0.27/0.47 thf(fact_29_rational__preference_Ototal,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.27/0.47 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( total_on @ A @ Carrier @ Relation ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference.total
% 0.27/0.47 thf(fact_30_rational__preference_OcompleteD,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.27/0.47 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( ( member @ A @ X @ Carrier )
% 0.27/0.47 => ( ( member @ A @ Y @ Carrier )
% 0.27/0.47 => ( ( X != Y )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.27/0.47 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference.completeD
% 0.27/0.47 thf(fact_31_rational__preference_Ostrict__trans,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 0.27/0.47 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.27/0.47 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 0.27/0.47 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
% 0.27/0.47 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation ) )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation )
% 0.27/0.47 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference.strict_trans
% 0.27/0.47 thf(fact_32_rational__preference_Oweak__is__transitive,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 0.27/0.47 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( ( ( member @ A @ X @ Carrier )
% 0.27/0.47 & ( member @ A @ Y @ Carrier )
% 0.27/0.47 & ( member @ A @ Z @ Carrier ) )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
% 0.27/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference.weak_is_transitive
% 0.27/0.47 thf(fact_33_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.27/0.47 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( ( ( member @ A @ X @ Carrier )
% 0.27/0.47 & ( member @ A @ Y @ Carrier ) )
% 0.27/0.47 => ( ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 0.27/0.47 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.27/0.47 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference.strict_not_refl_weak
% 0.27/0.47 thf(fact_34_rational__preference_Ostrict__is__neg__transitive,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 0.27/0.47 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( ( ( member @ A @ X @ Carrier )
% 0.27/0.47 & ( member @ A @ Y @ Carrier )
% 0.27/0.47 & ( member @ A @ Z @ Carrier ) )
% 0.27/0.47 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.27/0.47 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 0.27/0.47 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation )
% 0.27/0.47 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X ) @ Relation ) )
% 0.27/0.47 | ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation )
% 0.27/0.47 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference.strict_is_neg_transitive
% 0.27/0.47 thf(fact_35_rational__preference_Oworse__in__no__better,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.27/0.47 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.27/0.47 => ( member @ A @ Y @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference.worse_in_no_better
% 0.27/0.47 thf(fact_36_rational__preference_Ono__better__than__nonepty,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A] :
% 0.27/0.47 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( ( Carrier
% 0.27/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 => ( ( member @ A @ X @ Carrier )
% 0.27/0.47 => ( ( prefer1532642881r_than @ A @ X @ Carrier @ Relation )
% 0.27/0.47 != ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference.no_better_than_nonepty
% 0.27/0.47 thf(fact_37_rational__preference_Ono__better__thansubset__rel,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.27/0.47 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( ( member @ A @ X @ Carrier )
% 0.27/0.47 => ( ( member @ A @ Y @ Carrier )
% 0.27/0.47 => ( ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) )
% 0.27/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference.no_better_thansubset_rel
% 0.27/0.47 thf(fact_38_rational__preference_Ono__better__subset__pref,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.27/0.47 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.27/0.47 => ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference.no_better_subset_pref
% 0.27/0.47 thf(fact_39_rational__preference_Onbt__nest,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),Y: A,X: A] :
% 0.27/0.47 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) )
% 0.27/0.47 | ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference.nbt_nest
% 0.27/0.47 thf(fact_40_at__lst__asgd__not__ge,axiom,
% 0.27/0.47 ! [X: a,Y: a] :
% 0.27/0.47 ( ( carrier
% 0.27/0.47 != ( bot_bot @ ( set @ a ) ) )
% 0.27/0.47 => ( ( member @ a @ X @ carrier )
% 0.27/0.47 => ( ( member @ a @ Y @ carrier )
% 0.27/0.47 => ( ~ ( member @ a @ X @ ( prefer310429814s_good @ a @ Y @ carrier @ relation ) )
% 0.27/0.47 => ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % at_lst_asgd_not_ge
% 0.27/0.47 thf(fact_41_same__at__least__as__equal,axiom,
% 0.27/0.47 ! [Z: a,Y: a] :
% 0.27/0.47 ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation )
% 0.27/0.47 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation ) )
% 0.27/0.47 => ( ( prefer310429814s_good @ a @ Z @ carrier @ relation )
% 0.27/0.47 = ( prefer310429814s_good @ a @ Y @ carrier @ relation ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % same_at_least_as_equal
% 0.27/0.47 thf(fact_42_pref__in__at__least__as,axiom,
% 0.27/0.47 ! [X: a,Y: a] :
% 0.27/0.47 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.27/0.47 => ( member @ a @ X @ ( prefer310429814s_good @ a @ Y @ carrier @ relation ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % pref_in_at_least_as
% 0.27/0.47 thf(fact_43_empty__subsetI,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% 0.27/0.47
% 0.27/0.47 % empty_subsetI
% 0.27/0.47 thf(fact_44_subset__empty,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 = ( A2
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_empty
% 0.27/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.27/0.47 ! [A: $tType,A3: A,P: A > $o] :
% 0.27/0.47 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.27/0.47 = ( P @ A3 ) ) ).
% 0.27/0.47
% 0.27/0.47 % mem_Collect_eq
% 0.27/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A] :
% 0.27/0.47 ( ( collect @ A
% 0.27/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
% 0.27/0.47 = A2 ) ).
% 0.27/0.47
% 0.27/0.47 % Collect_mem_eq
% 0.27/0.47 thf(fact_47_Collect__cong,axiom,
% 0.27/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.27/0.47 ( ! [X4: A] :
% 0.27/0.47 ( ( P @ X4 )
% 0.27/0.47 = ( Q @ X4 ) )
% 0.27/0.47 => ( ( collect @ A @ P )
% 0.27/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Collect_cong
% 0.27/0.47 thf(fact_48_ext,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.27/0.47 ( ! [X4: A] :
% 0.27/0.47 ( ( F @ X4 )
% 0.27/0.47 = ( G @ X4 ) )
% 0.27/0.47 => ( F = G ) ) ).
% 0.27/0.47
% 0.27/0.47 % ext
% 0.27/0.47 thf(fact_49_Pair__le,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType] :
% 0.27/0.47 ( ( ( ord @ B )
% 0.27/0.47 & ( ord @ A ) )
% 0.27/0.47 => ! [A3: A,B2: B,C2: A,D: B] :
% 0.27/0.47 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D ) )
% 0.27/0.47 = ( ( ord_less_eq @ A @ A3 @ C2 )
% 0.27/0.47 & ( ord_less_eq @ B @ B2 @ D ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Pair_le
% 0.27/0.47 thf(fact_50_as__good__as__sameIff,axiom,
% 0.27/0.47 ! [Z: a,Y: a] :
% 0.27/0.47 ( ( member @ a @ Z @ ( prefer951318096ood_as @ a @ Y @ carrier @ relation ) )
% 0.27/0.47 = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation )
% 0.27/0.47 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % as_good_as_sameIff
% 0.27/0.47 thf(fact_51_as__good__asIff,axiom,
% 0.27/0.47 ! [X: a,Y: a] :
% 0.27/0.47 ( ( member @ a @ X @ ( prefer951318096ood_as @ a @ Y @ carrier @ relation ) )
% 0.27/0.47 = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 0.27/0.47 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % as_good_asIff
% 0.27/0.47 thf(fact_52_finite__has__maximal,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [A2: set @ A] :
% 0.27/0.47 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.47 => ( ( A2
% 0.27/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 => ? [X4: A] :
% 0.27/0.47 ( ( member @ A @ X4 @ A2 )
% 0.27/0.47 & ! [Xa: A] :
% 0.27/0.47 ( ( member @ A @ Xa @ A2 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ X4 @ Xa )
% 0.27/0.47 => ( X4 = Xa ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % finite_has_maximal
% 0.27/0.47 thf(fact_53_finite__has__minimal,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [A2: set @ A] :
% 0.27/0.47 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.47 => ( ( A2
% 0.27/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 => ? [X4: A] :
% 0.27/0.47 ( ( member @ A @ X4 @ A2 )
% 0.27/0.47 & ! [Xa: A] :
% 0.27/0.47 ( ( member @ A @ Xa @ A2 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ Xa @ X4 )
% 0.27/0.47 => ( X4 = Xa ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % finite_has_minimal
% 0.27/0.47 thf(fact_54_empty__Collect__eq,axiom,
% 0.27/0.47 ! [A: $tType,P: A > $o] :
% 0.27/0.47 ( ( ( bot_bot @ ( set @ A ) )
% 0.27/0.47 = ( collect @ A @ P ) )
% 0.27/0.47 = ( ! [X3: A] :
% 0.27/0.47 ~ ( P @ X3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % empty_Collect_eq
% 0.27/0.47 thf(fact_55_Collect__empty__eq,axiom,
% 0.27/0.47 ! [A: $tType,P: A > $o] :
% 0.27/0.47 ( ( ( collect @ A @ P )
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 = ( ! [X3: A] :
% 0.27/0.47 ~ ( P @ X3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Collect_empty_eq
% 0.27/0.47 thf(fact_56_all__not__in__conv,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A] :
% 0.27/0.47 ( ( ! [X3: A] :
% 0.27/0.47 ~ ( member @ A @ X3 @ A2 ) )
% 0.27/0.47 = ( A2
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % all_not_in_conv
% 0.27/0.47 thf(fact_57_empty__iff,axiom,
% 0.27/0.47 ! [A: $tType,C2: A] :
% 0.27/0.47 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % empty_iff
% 0.27/0.47 thf(fact_58_subsetI,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.27/0.47 ( ! [X4: A] :
% 0.27/0.47 ( ( member @ A @ X4 @ A2 )
% 0.27/0.47 => ( member @ A @ X4 @ B3 ) )
% 0.27/0.47 => ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% 0.27/0.47
% 0.27/0.47 % subsetI
% 0.27/0.47 thf(fact_59_subset__antisym,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.27/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
% 0.27/0.47 => ( A2 = B3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_antisym
% 0.27/0.47 thf(fact_60_finite__code,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( finite_finite @ A )
% 0.27/0.47 => ( ( finite_finite2 @ A )
% 0.27/0.47 = ( ^ [A4: set @ A] : $true ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % finite_code
% 0.27/0.47 thf(fact_61_at__least__as__goodD,axiom,
% 0.27/0.47 ! [A: $tType,Z: A,Y: A,B3: set @ A,Pr: set @ ( product_prod @ A @ A )] :
% 0.27/0.47 ( ( member @ A @ Z @ ( prefer310429814s_good @ A @ Y @ B3 @ Pr ) )
% 0.27/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Pr ) ) ).
% 0.27/0.47
% 0.27/0.47 % at_least_as_goodD
% 0.27/0.47 thf(fact_62_at__lst__asgd__ge,axiom,
% 0.27/0.47 ! [A: $tType,X: A,Y: A,B3: set @ A,Pr: set @ ( product_prod @ A @ A )] :
% 0.27/0.47 ( ( member @ A @ X @ ( prefer310429814s_good @ A @ Y @ B3 @ Pr ) )
% 0.27/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Pr ) ) ).
% 0.27/0.47
% 0.27/0.47 % at_lst_asgd_ge
% 0.27/0.47 thf(fact_63_ex__in__conv,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A] :
% 0.27/0.47 ( ( ? [X3: A] : ( member @ A @ X3 @ A2 ) )
% 0.27/0.47 = ( A2
% 0.27/0.47 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ex_in_conv
% 0.27/0.47 thf(fact_64_equals0I,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A] :
% 0.27/0.47 ( ! [Y3: A] :
% 0.27/0.47 ~ ( member @ A @ Y3 @ A2 )
% 0.27/0.47 => ( A2
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % equals0I
% 0.27/0.47 thf(fact_65_equals0D,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,A3: A] :
% 0.27/0.47 ( ( A2
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 => ~ ( member @ A @ A3 @ A2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % equals0D
% 0.27/0.47 thf(fact_66_emptyE,axiom,
% 0.27/0.47 ! [A: $tType,A3: A] :
% 0.27/0.47 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % emptyE
% 0.27/0.47 thf(fact_67_in__mono,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B3: set @ A,X: A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.27/0.47 => ( ( member @ A @ X @ A2 )
% 0.27/0.47 => ( member @ A @ X @ B3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % in_mono
% 0.27/0.47 thf(fact_68_subsetD,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B3: set @ A,C2: A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.27/0.47 => ( ( member @ A @ C2 @ A2 )
% 0.27/0.47 => ( member @ A @ C2 @ B3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subsetD
% 0.27/0.47 thf(fact_69_equalityE,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.27/0.47 ( ( A2 = B3 )
% 0.27/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.27/0.47 => ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % equalityE
% 0.27/0.47 thf(fact_70_subset__eq,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.27/0.47 = ( ^ [A4: set @ A,B4: set @ A] :
% 0.27/0.47 ! [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ A4 )
% 0.27/0.47 => ( member @ A @ X3 @ B4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_eq
% 0.27/0.47 thf(fact_71_equalityD1,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.27/0.47 ( ( A2 = B3 )
% 0.27/0.47 => ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% 0.27/0.47
% 0.27/0.47 % equalityD1
% 0.27/0.47 thf(fact_72_equalityD2,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.27/0.47 ( ( A2 = B3 )
% 0.27/0.47 => ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % equalityD2
% 0.27/0.47 thf(fact_73_subset__iff,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.27/0.47 = ( ^ [A4: set @ A,B4: set @ A] :
% 0.27/0.47 ! [T2: A] :
% 0.27/0.47 ( ( member @ A @ T2 @ A4 )
% 0.27/0.47 => ( member @ A @ T2 @ B4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_iff
% 0.27/0.47 thf(fact_74_subset__refl,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 0.27/0.47
% 0.27/0.47 % subset_refl
% 0.27/0.47 thf(fact_75_Collect__mono,axiom,
% 0.27/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.27/0.47 ( ! [X4: A] :
% 0.27/0.47 ( ( P @ X4 )
% 0.27/0.47 => ( Q @ X4 ) )
% 0.27/0.47 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Collect_mono
% 0.27/0.47 thf(fact_76_subset__trans,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B3: set @ A,C3: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.27/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
% 0.27/0.47 => ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_trans
% 0.27/0.47 thf(fact_77_set__eq__subset,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ^ [Y4: set @ A,Z2: set @ A] : ( Y4 = Z2 ) )
% 0.27/0.47 = ( ^ [A4: set @ A,B4: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.27/0.47 & ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % set_eq_subset
% 0.27/0.47 thf(fact_78_Collect__mono__iff,axiom,
% 0.27/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.27/0.47 = ( ! [X3: A] :
% 0.27/0.47 ( ( P @ X3 )
% 0.27/0.47 => ( Q @ X3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Collect_mono_iff
% 0.27/0.47 thf(fact_79_finite,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( finite_finite @ A )
% 0.27/0.47 => ! [A2: set @ A] : ( finite_finite2 @ A @ A2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % finite
% 0.27/0.47 thf(fact_80_finite__set__choice,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,A2: set @ A,P: A > B > $o] :
% 0.27/0.47 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.47 => ( ! [X4: A] :
% 0.27/0.47 ( ( member @ A @ X4 @ A2 )
% 0.27/0.47 => ? [X_1: B] : ( P @ X4 @ X_1 ) )
% 0.27/0.47 => ? [F2: A > B] :
% 0.27/0.47 ! [X2: A] :
% 0.27/0.47 ( ( member @ A @ X2 @ A2 )
% 0.27/0.47 => ( P @ X2 @ ( F2 @ X2 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % finite_set_choice
% 0.27/0.47 thf(fact_81_rational__preference_Osame__at__least__as__equal,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),Z: A,Y: A] :
% 0.27/0.47 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation )
% 0.27/0.47 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation ) )
% 0.27/0.47 => ( ( prefer310429814s_good @ A @ Z @ Carrier @ Relation )
% 0.27/0.47 = ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference.same_at_least_as_equal
% 0.27/0.47 thf(fact_82_rational__preference_Opref__in__at__least__as,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.27/0.47 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.27/0.47 => ( member @ A @ X @ ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference.pref_in_at_least_as
% 0.27/0.47 thf(fact_83_rational__preference_Oas__good__as__sameIff,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),Z: A,Y: A] :
% 0.27/0.47 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( ( member @ A @ Z @ ( prefer951318096ood_as @ A @ Y @ Carrier @ Relation ) )
% 0.27/0.47 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation )
% 0.27/0.47 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference.as_good_as_sameIff
% 0.27/0.47 thf(fact_84_rational__preference_Oas__good__asIff,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.27/0.47 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( ( member @ A @ X @ ( prefer951318096ood_as @ A @ Y @ Carrier @ Relation ) )
% 0.27/0.47 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 0.27/0.47 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference.as_good_asIff
% 0.27/0.47 thf(fact_85_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.27/0.47 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( ( Carrier
% 0.27/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 => ( ( member @ A @ X @ Carrier )
% 0.27/0.47 => ( ( member @ A @ Y @ Carrier )
% 0.27/0.47 => ( ~ ( member @ A @ X @ ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) )
% 0.27/0.47 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference.at_lst_asgd_not_ge
% 0.27/0.47 thf(fact_86_Pair__mono,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType] :
% 0.27/0.47 ( ( ( ord @ B )
% 0.27/0.47 & ( ord @ A ) )
% 0.27/0.47 => ! [X: A,X5: A,Y: B,Y5: B] :
% 0.27/0.47 ( ( ord_less_eq @ A @ X @ X5 )
% 0.27/0.47 => ( ( ord_less_eq @ B @ Y @ Y5 )
% 0.27/0.47 => ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Pair_mono
% 0.27/0.47 thf(fact_87_finite__has__maximal2,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [A2: set @ A,A3: A] :
% 0.27/0.47 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.47 => ( ( member @ A @ A3 @ A2 )
% 0.27/0.47 => ? [X4: A] :
% 0.27/0.47 ( ( member @ A @ X4 @ A2 )
% 0.27/0.47 & ( ord_less_eq @ A @ A3 @ X4 )
% 0.27/0.47 & ! [Xa: A] :
% 0.27/0.47 ( ( member @ A @ Xa @ A2 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ X4 @ Xa )
% 0.27/0.47 => ( X4 = Xa ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % finite_has_maximal2
% 0.27/0.47 thf(fact_88_finite__has__minimal2,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [A2: set @ A,A3: A] :
% 0.27/0.47 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.47 => ( ( member @ A @ A3 @ A2 )
% 0.27/0.47 => ? [X4: A] :
% 0.27/0.47 ( ( member @ A @ X4 @ A2 )
% 0.27/0.47 & ( ord_less_eq @ A @ X4 @ A3 )
% 0.27/0.47 & ! [Xa: A] :
% 0.27/0.47 ( ( member @ A @ Xa @ A2 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ Xa @ X4 )
% 0.27/0.47 => ( X4 = Xa ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % finite_has_minimal2
% 0.27/0.47 thf(fact_89_bot__prod__def,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType] :
% 0.27/0.47 ( ( ( bot @ B )
% 0.27/0.47 & ( bot @ A ) )
% 0.27/0.47 => ( ( bot_bot @ ( product_prod @ A @ B ) )
% 0.27/0.47 = ( product_Pair @ A @ B @ ( bot_bot @ A ) @ ( bot_bot @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % bot_prod_def
% 0.27/0.47 thf(fact_90_infinite__imp__nonempty,axiom,
% 0.27/0.47 ! [A: $tType,S: set @ A] :
% 0.27/0.47 ( ~ ( finite_finite2 @ A @ S )
% 0.27/0.47 => ( S
% 0.27/0.47 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % infinite_imp_nonempty
% 0.27/0.47 thf(fact_91_finite_OemptyI,axiom,
% 0.27/0.47 ! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % finite.emptyI
% 0.27/0.47 thf(fact_92_finite__subset,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.27/0.47 => ( ( finite_finite2 @ A @ B3 )
% 0.27/0.47 => ( finite_finite2 @ A @ A2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % finite_subset
% 0.27/0.47 thf(fact_93_infinite__super,axiom,
% 0.27/0.47 ! [A: $tType,S: set @ A,T3: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ S @ T3 )
% 0.27/0.47 => ( ~ ( finite_finite2 @ A @ S )
% 0.27/0.47 => ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % infinite_super
% 0.27/0.47 thf(fact_94_rev__finite__subset,axiom,
% 0.27/0.47 ! [A: $tType,B3: set @ A,A2: set @ A] :
% 0.27/0.47 ( ( finite_finite2 @ A @ B3 )
% 0.27/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.27/0.47 => ( finite_finite2 @ A @ A2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rev_finite_subset
% 0.27/0.47 thf(fact_95_old_Oprod_Oinject,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,A3: A,B2: B,A5: A,B5: B] :
% 0.27/0.47 ( ( ( product_Pair @ A @ B @ A3 @ B2 )
% 0.27/0.47 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.27/0.47 = ( ( A3 = A5 )
% 0.27/0.47 & ( B2 = B5 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % old.prod.inject
% 0.27/0.47 thf(fact_96_prod_Oinject,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 0.27/0.47 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.27/0.47 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 0.27/0.47 = ( ( X1 = Y1 )
% 0.27/0.47 & ( X22 = Y22 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod.inject
% 0.27/0.47 thf(fact_97_bot__apply,axiom,
% 0.27/0.47 ! [C: $tType,D2: $tType] :
% 0.27/0.47 ( ( bot @ C )
% 0.27/0.47 => ( ( bot_bot @ ( D2 > C ) )
% 0.27/0.47 = ( ^ [X3: D2] : ( bot_bot @ C ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % bot_apply
% 0.27/0.47 thf(fact_98_order__refl,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( preorder @ A )
% 0.27/0.47 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_refl
% 0.27/0.47 thf(fact_99_preorder__on__empty,axiom,
% 0.27/0.47 ! [A: $tType] : ( order_preorder_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % preorder_on_empty
% 0.27/0.47 thf(fact_100_total__on__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( total_on @ A )
% 0.27/0.47 = ( ^ [A4: set @ A,R: set @ ( product_prod @ A @ A )] :
% 0.27/0.47 ! [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ A4 )
% 0.27/0.47 => ! [Y2: A] :
% 0.27/0.47 ( ( member @ A @ Y2 @ A4 )
% 0.27/0.47 => ( ( X3 != Y2 )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R )
% 0.27/0.47 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X3 ) @ R ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % total_on_def
% 0.27/0.47 thf(fact_101_total__onI,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 0.27/0.47 ( ! [X4: A,Y3: A] :
% 0.27/0.47 ( ( member @ A @ X4 @ A2 )
% 0.27/0.47 => ( ( member @ A @ Y3 @ A2 )
% 0.27/0.47 => ( ( X4 != Y3 )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R2 )
% 0.27/0.47 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X4 ) @ R2 ) ) ) ) )
% 0.27/0.47 => ( total_on @ A @ A2 @ R2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % total_onI
% 0.27/0.47 thf(fact_102_total__on__empty,axiom,
% 0.27/0.47 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R2 ) ).
% 0.27/0.47
% 0.27/0.47 % total_on_empty
% 0.27/0.47 thf(fact_103_bot__empty__eq,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( bot_bot @ ( A > $o ) )
% 0.27/0.47 = ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % bot_empty_eq
% 0.27/0.47 thf(fact_104_bot__set__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( bot_bot @ ( set @ A ) )
% 0.27/0.47 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % bot_set_def
% 0.27/0.47 thf(fact_105_dual__order_Oantisym,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [B2: A,A3: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.27/0.47 => ( A3 = B2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % dual_order.antisym
% 0.27/0.47 thf(fact_106_dual__order_Oeq__iff,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.27/0.47 = ( ^ [A6: A,B6: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ B6 @ A6 )
% 0.27/0.47 & ( ord_less_eq @ A @ A6 @ B6 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % dual_order.eq_iff
% 0.27/0.47 thf(fact_107_dual__order_Otrans,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [B2: A,A3: A,C2: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.27/0.47 => ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % dual_order.trans
% 0.27/0.47 thf(fact_108_linorder__wlog,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linorder @ A )
% 0.27/0.47 => ! [P: A > A > $o,A3: A,B2: A] :
% 0.27/0.47 ( ! [A7: A,B7: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A7 @ B7 )
% 0.27/0.47 => ( P @ A7 @ B7 ) )
% 0.27/0.47 => ( ! [A7: A,B7: A] :
% 0.27/0.47 ( ( P @ B7 @ A7 )
% 0.27/0.47 => ( P @ A7 @ B7 ) )
% 0.27/0.47 => ( P @ A3 @ B2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % linorder_wlog
% 0.27/0.47 thf(fact_109_dual__order_Orefl,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.27/0.47
% 0.27/0.47 % dual_order.refl
% 0.27/0.47 thf(fact_110_order__trans,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( preorder @ A )
% 0.27/0.47 => ! [X: A,Y: A,Z: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.27/0.47 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.27/0.47 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_trans
% 0.27/0.47 thf(fact_111_order__class_Oorder_Oantisym,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [A3: A,B2: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.27/0.47 => ( A3 = B2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_class.order.antisym
% 0.27/0.47 thf(fact_112_ord__le__eq__trans,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ord @ A )
% 0.27/0.47 => ! [A3: A,B2: A,C2: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.27/0.47 => ( ( B2 = C2 )
% 0.27/0.47 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_le_eq_trans
% 0.27/0.47 thf(fact_113_ord__eq__le__trans,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ord @ A )
% 0.27/0.47 => ! [A3: A,B2: A,C2: A] :
% 0.27/0.47 ( ( A3 = B2 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.27/0.47 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_eq_le_trans
% 0.27/0.47 thf(fact_114_order__class_Oorder_Oeq__iff,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.27/0.47 = ( ^ [A6: A,B6: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A6 @ B6 )
% 0.27/0.47 & ( ord_less_eq @ A @ B6 @ A6 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_class.order.eq_iff
% 0.27/0.47 thf(fact_115_antisym__conv,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [Y: A,X: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ Y @ X )
% 0.27/0.47 => ( ( ord_less_eq @ A @ X @ Y )
% 0.27/0.47 = ( X = Y ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % antisym_conv
% 0.27/0.47 thf(fact_116_le__cases3,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linorder @ A )
% 0.27/0.47 => ! [X: A,Y: A,Z: A] :
% 0.27/0.47 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.27/0.47 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 0.27/0.47 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.27/0.47 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 0.27/0.47 => ( ( ( ord_less_eq @ A @ X @ Z )
% 0.27/0.47 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 0.27/0.47 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 0.27/0.47 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.27/0.47 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 0.27/0.47 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 0.27/0.47 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 0.27/0.47 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_cases3
% 0.27/0.47 thf(fact_117_order_Otrans,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [A3: A,B2: A,C2: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.27/0.47 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order.trans
% 0.27/0.47 thf(fact_118_le__cases,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linorder @ A )
% 0.27/0.47 => ! [X: A,Y: A] :
% 0.27/0.47 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.27/0.47 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_cases
% 0.27/0.47 thf(fact_119_eq__refl,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( preorder @ A )
% 0.27/0.47 => ! [X: A,Y: A] :
% 0.27/0.47 ( ( X = Y )
% 0.27/0.47 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % eq_refl
% 0.27/0.47 thf(fact_120_linear,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linorder @ A )
% 0.27/0.47 => ! [X: A,Y: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.27/0.47 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % linear
% 0.27/0.47 thf(fact_121_antisym,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [X: A,Y: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.27/0.47 => ( ( ord_less_eq @ A @ Y @ X )
% 0.27/0.47 => ( X = Y ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % antisym
% 0.27/0.47 thf(fact_122_eq__iff,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.27/0.47 = ( ^ [X3: A,Y2: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.27/0.47 & ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % eq_iff
% 0.27/0.47 thf(fact_123_ord__le__eq__subst,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType] :
% 0.27/0.47 ( ( ( ord @ B )
% 0.27/0.47 & ( ord @ A ) )
% 0.27/0.47 => ! [A3: A,B2: A,F: A > B,C2: B] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.27/0.47 => ( ( ( F @ B2 )
% 0.27/0.47 = C2 )
% 0.27/0.47 => ( ! [X4: A,Y3: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.27/0.47 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.27/0.47 => ( ord_less_eq @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_le_eq_subst
% 0.27/0.47 thf(fact_124_ord__eq__le__subst,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType] :
% 0.27/0.47 ( ( ( ord @ B )
% 0.27/0.47 & ( ord @ A ) )
% 0.27/0.47 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 0.27/0.47 ( ( A3
% 0.27/0.47 = ( F @ B2 ) )
% 0.27/0.47 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.27/0.47 => ( ! [X4: B,Y3: B] :
% 0.27/0.47 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 0.27/0.47 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.27/0.47 => ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_eq_le_subst
% 0.27/0.47 thf(fact_125_order__subst2,axiom,
% 0.27/0.47 ! [A: $tType,C: $tType] :
% 0.27/0.47 ( ( ( order @ C )
% 0.27/0.47 & ( order @ A ) )
% 0.27/0.47 => ! [A3: A,B2: A,F: A > C,C2: C] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.27/0.47 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 0.27/0.47 => ( ! [X4: A,Y3: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.27/0.47 => ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.27/0.47 => ( ord_less_eq @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_subst2
% 0.27/0.47 thf(fact_126_order__subst1,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType] :
% 0.27/0.47 ( ( ( order @ B )
% 0.27/0.47 & ( order @ A ) )
% 0.27/0.47 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
% 0.27/0.47 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.27/0.47 => ( ! [X4: B,Y3: B] :
% 0.27/0.47 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 0.27/0.47 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.27/0.47 => ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_subst1
% 0.27/0.47 thf(fact_127_le__fun__def,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( ord @ B )
% 0.27/0.47 => ( ( ord_less_eq @ ( A > B ) )
% 0.27/0.47 = ( ^ [F3: A > B,G2: A > B] :
% 0.27/0.47 ! [X3: A] : ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_fun_def
% 0.27/0.47 thf(fact_128_le__funI,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( ord @ B )
% 0.27/0.47 => ! [F: A > B,G: A > B] :
% 0.27/0.47 ( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.27/0.47 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_funI
% 0.27/0.47 thf(fact_129_le__funE,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( ord @ B )
% 0.27/0.47 => ! [F: A > B,G: A > B,X: A] :
% 0.27/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.27/0.47 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_funE
% 0.27/0.47 thf(fact_130_le__funD,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( ord @ B )
% 0.27/0.47 => ! [F: A > B,G: A > B,X: A] :
% 0.27/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.27/0.47 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_funD
% 0.27/0.47 thf(fact_131_bot__fun__def,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( bot @ B )
% 0.27/0.47 => ( ( bot_bot @ ( A > B ) )
% 0.27/0.47 = ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % bot_fun_def
% 0.27/0.47 thf(fact_132_surj__pair,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.27/0.47 ? [X4: A,Y3: B] :
% 0.27/0.47 ( P2
% 0.27/0.47 = ( product_Pair @ A @ B @ X4 @ Y3 ) ) ).
% 0.27/0.47
% 0.27/0.47 % surj_pair
% 0.27/0.47 thf(fact_133_prod__cases,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.27/0.47 ( ! [A7: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A7 @ B7 ) )
% 0.27/0.47 => ( P @ P2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_cases
% 0.27/0.47 thf(fact_134_Pair__inject,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,A3: A,B2: B,A5: A,B5: B] :
% 0.27/0.47 ( ( ( product_Pair @ A @ B @ A3 @ B2 )
% 0.27/0.47 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.27/0.47 => ~ ( ( A3 = A5 )
% 0.27/0.47 => ( B2 != B5 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Pair_inject
% 0.27/0.47 thf(fact_135_prod__cases3,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.27/0.47 ~ ! [A7: A,B7: B,C4: C] :
% 0.27/0.47 ( Y
% 0.27/0.47 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A7 @ ( product_Pair @ B @ C @ B7 @ C4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_cases3
% 0.27/0.47 thf(fact_136_prod__cases4,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,C: $tType,D2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 0.27/0.47 ~ ! [A7: A,B7: B,C4: C,D3: D2] :
% 0.27/0.47 ( Y
% 0.27/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B7 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_cases4
% 0.27/0.47 thf(fact_137_prod__cases5,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 0.27/0.47 ~ ! [A7: A,B7: B,C4: C,D3: D2,E2: E] :
% 0.27/0.47 ( Y
% 0.27/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C4 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_cases5
% 0.27/0.47 thf(fact_138_prod__cases6,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) )] :
% 0.27/0.47 ~ ! [A7: A,B7: B,C4: C,D3: D2,E2: E,F2: F4] :
% 0.27/0.47 ( Y
% 0.27/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ F4 ) @ D3 @ ( product_Pair @ E @ F4 @ E2 @ F2 ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_cases6
% 0.27/0.47 thf(fact_139_prod__cases7,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F4: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
% 0.27/0.47 ~ ! [A7: A,B7: B,C4: C,D3: D2,E2: E,F2: F4,G4: G3] :
% 0.27/0.47 ( Y
% 0.27/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F4 @ G3 ) @ E2 @ ( product_Pair @ F4 @ G3 @ F2 @ G4 ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_cases7
% 0.27/0.47 thf(fact_140_prod__induct3,axiom,
% 0.27/0.47 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.27/0.47 ( ! [A7: A,B7: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A7 @ ( product_Pair @ B @ C @ B7 @ C4 ) ) )
% 0.27/0.47 => ( P @ X ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_induct3
% 0.27/0.47 thf(fact_141_prod__induct4,axiom,
% 0.27/0.47 ! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 0.27/0.47 ( ! [A7: A,B7: B,C4: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B7 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) )
% 0.27/0.47 => ( P @ X ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_induct4
% 0.27/0.47 thf(fact_142_prod__induct5,axiom,
% 0.27/0.47 ! [E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 0.27/0.47 ( ! [A7: A,B7: B,C4: C,D3: D2,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C4 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) )
% 0.27/0.47 => ( P @ X ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_induct5
% 0.27/0.47 thf(fact_143_prod__induct6,axiom,
% 0.27/0.47 ! [F4: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) )] :
% 0.27/0.47 ( ! [A7: A,B7: B,C4: C,D3: D2,E2: E,F2: F4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ F4 ) @ D3 @ ( product_Pair @ E @ F4 @ E2 @ F2 ) ) ) ) ) )
% 0.27/0.47 => ( P @ X ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_induct6
% 0.27/0.47 thf(fact_144_prod__induct7,axiom,
% 0.27/0.47 ! [G3: $tType,F4: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
% 0.27/0.47 ( ! [A7: A,B7: B,C4: C,D3: D2,E2: E,F2: F4,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F4 @ G3 ) @ E2 @ ( product_Pair @ F4 @ G3 @ F2 @ G4 ) ) ) ) ) ) )
% 0.27/0.47 => ( P @ X ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_induct7
% 0.27/0.47 thf(fact_145_old_Oprod_Oexhaust,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.27/0.47 ~ ! [A7: A,B7: B] :
% 0.27/0.47 ( Y
% 0.27/0.47 != ( product_Pair @ A @ B @ A7 @ B7 ) ) ).
% 0.27/0.47
% 0.27/0.47 % old.prod.exhaust
% 0.27/0.47 thf(fact_146_old_Oprod_Oinducts,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.27/0.47 ( ! [A7: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A7 @ B7 ) )
% 0.27/0.47 => ( P @ Prod ) ) ).
% 0.27/0.47
% 0.27/0.47 % old.prod.inducts
% 0.27/0.47 thf(fact_147_subrelI,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 0.27/0.47 ( ! [X4: A,Y3: B] :
% 0.27/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R2 )
% 0.27/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ S2 ) )
% 0.27/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % subrelI
% 0.27/0.47 thf(fact_148_bot_Oextremum__uniqueI,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order_bot @ A )
% 0.27/0.47 => ! [A3: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.27/0.47 => ( A3
% 0.27/0.47 = ( bot_bot @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % bot.extremum_uniqueI
% 0.27/0.47 thf(fact_149_bot_Oextremum__unique,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order_bot @ A )
% 0.27/0.47 => ! [A3: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.27/0.47 = ( A3
% 0.27/0.47 = ( bot_bot @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % bot.extremum_unique
% 0.27/0.47 thf(fact_150_bot_Oextremum,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order_bot @ A )
% 0.27/0.47 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 0.27/0.47
% 0.27/0.47 % bot.extremum
% 0.27/0.47 thf(fact_151_refl__on__empty,axiom,
% 0.27/0.47 ! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % refl_on_empty
% 0.27/0.47 thf(fact_152_refl__onD,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A] :
% 0.27/0.47 ( ( refl_on @ A @ A2 @ R2 )
% 0.27/0.47 => ( ( member @ A @ A3 @ A2 )
% 0.27/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % refl_onD
% 0.27/0.47 thf(fact_153_refl__onD1,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.27/0.47 ( ( refl_on @ A @ A2 @ R2 )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
% 0.27/0.47 => ( member @ A @ X @ A2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % refl_onD1
% 0.27/0.47 thf(fact_154_refl__onD2,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.27/0.47 ( ( refl_on @ A @ A2 @ R2 )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
% 0.27/0.47 => ( member @ A @ Y @ A2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % refl_onD2
% 0.27/0.47 thf(fact_155_refl__on__domain,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 0.27/0.47 ( ( refl_on @ A @ A2 @ R2 )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
% 0.27/0.47 => ( ( member @ A @ A3 @ A2 )
% 0.27/0.47 & ( member @ A @ B2 @ A2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % refl_on_domain
% 0.27/0.47 thf(fact_156_old_Oprod_Orec,axiom,
% 0.27/0.47 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B2: B] :
% 0.27/0.47 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
% 0.27/0.47 = ( F1 @ A3 @ B2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % old.prod.rec
% 0.27/0.47 thf(fact_157_rational__preference__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( prefer1997167224erence @ A )
% 0.27/0.47 = ( ^ [Carrier2: set @ A,Relation2: set @ ( product_prod @ A @ A )] :
% 0.27/0.47 ( ( prefer199794634erence @ A @ Carrier2 @ Relation2 )
% 0.27/0.47 & ( prefer1801827867axioms @ A @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference_def
% 0.27/0.47 thf(fact_158_rational__preference_Ointro,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.27/0.47 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( ( prefer1801827867axioms @ A @ Carrier @ Relation )
% 0.27/0.47 => ( prefer1997167224erence @ A @ Carrier @ Relation ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference.intro
% 0.27/0.47 thf(fact_159_finite__transitivity__chain,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,R3: A > A > $o] :
% 0.27/0.47 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.47 => ( ! [X4: A] :
% 0.27/0.47 ~ ( R3 @ X4 @ X4 )
% 0.27/0.47 => ( ! [X4: A,Y3: A,Z3: A] :
% 0.27/0.47 ( ( R3 @ X4 @ Y3 )
% 0.27/0.47 => ( ( R3 @ Y3 @ Z3 )
% 0.27/0.47 => ( R3 @ X4 @ Z3 ) ) )
% 0.27/0.47 => ( ! [X4: A] :
% 0.27/0.47 ( ( member @ A @ X4 @ A2 )
% 0.27/0.47 => ? [Y6: A] :
% 0.27/0.47 ( ( member @ A @ Y6 @ A2 )
% 0.27/0.47 & ( R3 @ X4 @ Y6 ) ) )
% 0.27/0.47 => ( A2
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % finite_transitivity_chain
% 0.27/0.47 thf(fact_160_subset__emptyI,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A] :
% 0.27/0.47 ( ! [X4: A] :
% 0.27/0.47 ~ ( member @ A @ X4 @ A2 )
% 0.27/0.47 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_emptyI
% 0.27/0.47 thf(fact_161_ssubst__Pair__rhs,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,R2: A,S2: B,R3: set @ ( product_prod @ A @ B ),S3: B] :
% 0.27/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S2 ) @ R3 )
% 0.27/0.47 => ( ( S3 = S2 )
% 0.27/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S3 ) @ R3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ssubst_Pair_rhs
% 0.27/0.47 thf(fact_162_rational__preference__axioms_Ointro,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.27/0.47 ( ( total_on @ A @ Carrier @ Relation )
% 0.27/0.47 => ( prefer1801827867axioms @ A @ Carrier @ Relation ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference_axioms.intro
% 0.27/0.47 thf(fact_163_rational__preference__axioms__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( prefer1801827867axioms @ A )
% 0.27/0.47 = ( total_on @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference_axioms_def
% 0.27/0.47 thf(fact_164_rational__preference_Oaxioms_I2_J,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.27/0.47 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( prefer1801827867axioms @ A @ Carrier @ Relation ) ) ).
% 0.27/0.47
% 0.27/0.47 % rational_preference.axioms(2)
% 0.27/0.47 thf(fact_165_Collect__empty__eq__bot,axiom,
% 0.27/0.47 ! [A: $tType,P: A > $o] :
% 0.27/0.47 ( ( ( collect @ A @ P )
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 = ( P
% 0.27/0.47 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Collect_empty_eq_bot
% 0.27/0.47 thf(fact_166_less__by__empty,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A )] :
% 0.27/0.47 ( ( A2
% 0.27/0.47 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.27/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A2 @ B3 ) ) ).
% 0.27/0.47
% 0.27/0.47 % less_by_empty
% 0.27/0.47 thf(fact_167_finite__indexed__bound,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( linorder @ A )
% 0.27/0.47 => ! [A2: set @ B,P: B > A > $o] :
% 0.27/0.47 ( ( finite_finite2 @ B @ A2 )
% 0.27/0.47 => ( ! [X4: B] :
% 0.27/0.47 ( ( member @ B @ X4 @ A2 )
% 0.27/0.47 => ? [X_1: A] : ( P @ X4 @ X_1 ) )
% 0.27/0.47 => ? [M: A] :
% 0.27/0.47 ! [X2: B] :
% 0.27/0.47 ( ( member @ B @ X2 @ A2 )
% 0.27/0.47 => ? [K: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ K @ M )
% 0.27/0.47 & ( P @ X2 @ K ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % finite_indexed_bound
% 0.27/0.47 thf(fact_168_transitivity,axiom,
% 0.27/0.47 trans @ a @ relation ).
% 0.27/0.47
% 0.27/0.47 % transitivity
% 0.27/0.47 thf(fact_169_Set_Ois__empty__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( is_empty @ A )
% 0.27/0.47 = ( ^ [A4: set @ A] :
% 0.27/0.47 ( A4
% 0.27/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Set.is_empty_def
% 0.27/0.47 thf(fact_170_internal__case__prod__conv,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B2: C] :
% 0.27/0.47 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B2 ) )
% 0.27/0.47 = ( C2 @ A3 @ B2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % internal_case_prod_conv
% 0.27/0.47 thf(fact_171_preference_Otransitivity,axiom,
% 0.27/0.47 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.27/0.47 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.27/0.47 => ( trans @ A @ Relation ) ) ).
% 0.27/0.47
% 0.27/0.47 % preference.transitivity
% 0.27/0.47 thf(fact_172_trans__empty,axiom,
% 0.27/0.47 ! [A: $tType] : ( trans @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % trans_empty
% 0.27/0.47 thf(fact_173_transD,axiom,
% 0.27/0.47 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 0.27/0.47 ( ( trans @ A @ R2 )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R2 )
% 0.27/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % transD
% 0.27/0.47 thf(fact_174_transE,axiom,
% 0.27/0.47 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 0.27/0.47 ( ( trans @ A @ R2 )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R2 )
% 0.27/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % transE
% 0.27/0.47 thf(fact_175_transI,axiom,
% 0.27/0.47 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 0.27/0.47 ( ! [X4: A,Y3: A,Z3: A] :
% 0.27/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R2 )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
% 0.27/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z3 ) @ R2 ) ) )
% 0.27/0.47 => ( trans @ A @ R2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % transI
% 0.27/0.47 thf(fact_176_trans__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( trans @ A )
% 0.27/0.47 = ( ^ [R: set @ ( product_prod @ A @ A )] :
% 0.27/0.47 ! [X3: A,Y2: A,Z4: A] :
% 0.27/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z4 ) @ R )
% 0.27/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z4 ) @ R ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % trans_def
% 0.27/0.47 thf(fact_177_preorder__on__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order_preorder_on @ A )
% 0.27/0.47 = ( ^ [A4: set @ A,R: set @ ( product_prod @ A @ A )] :
% 0.27/0.47 ( ( refl_on @ A @ A4 @ R )
% 0.27/0.47 & ( trans @ A @ R ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % preorder_on_def
% 0.27/0.47 thf(fact_178_relChain__def,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( ord @ B )
% 0.27/0.47 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 0.27/0.47 = ( ^ [R: set @ ( product_prod @ A @ A ),As: A > B] :
% 0.27/0.47 ! [I: A,J: A] :
% 0.27/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
% 0.27/0.47 => ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % relChain_def
% 0.27/0.47 thf(fact_179_Id__on__empty,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Id_on_empty
% 0.27/0.47 thf(fact_180_under__incr,axiom,
% 0.27/0.47 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 0.27/0.47 ( ( trans @ A @ R2 )
% 0.27/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
% 0.27/0.47 => ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R2 @ A3 ) @ ( order_under @ A @ R2 @ B2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % under_incr
% 0.27/0.47 thf(fact_181_Id__onI,axiom,
% 0.27/0.47 ! [A: $tType,A3: A,A2: set @ A] :
% 0.27/0.47 ( ( member @ A @ A3 @ A2 )
% 0.27/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id_on @ A @ A2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Id_onI
% 0.27/0.47 thf(fact_182_trans__Id__on,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A] : ( trans @ A @ ( id_on @ A @ A2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % trans_Id_on
% 0.27/0.47 thf(fact_183_Id__on__iff,axiom,
% 0.27/0.47 ! [A: $tType,X: A,Y: A,A2: set @ A] :
% 0.27/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( id_on @ A @ A2 ) )
% 0.27/0.47 = ( ( X = Y )
% 0.27/0.47 & ( member @ A @ X @ A2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Id_on_iff
% 0.27/0.47 thf(fact_184_Id__on__eqI,axiom,
% 0.27/0.47 ! [A: $tType,A3: A,B2: A,A2: set @ A] :
% 0.27/0.47 ( ( A3 = B2 )
% 0.27/0.47 => ( ( member @ A @ A3 @ A2 )
% 0.27/0.47 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id_on @ A @ A2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Id_on_eqI
% 0.27/0.47 thf(fact_185_Id__onE,axiom,
% 0.27/0.47 ! [A: $tType,C2: product_prod @ A @ A,A2: set @ A] :
% 0.27/0.47 ( ( member @ ( product_prod @ A @ A ) @ C2 @ ( id_on @ A @ A2 ) )
% 0.27/0.47 => ~ ! [X4: A] :
% 0.27/0.47 ( ( member @ A @ X4 @ A2 )
% 0.27/0.47 => ( C2
% 0.27/0.47 != ( product_Pair @ A @ A @ X4 @ X4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Id_onE
% 0.27/0.47 thf(fact_186_refl__on__Id__on,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A] : ( refl_on @ A @ A2 @ ( id_on @ A @ A2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % refl_on_Id_on
% 0.27/0.47 thf(fact_187_arg__min__least,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( linorder @ B )
% 0.27/0.47 => ! [S: set @ A,Y: A,F: A > B] :
% 0.27/0.47 ( ( finite_finite2 @ A @ S )
% 0.27/0.47 => ( ( S
% 0.27/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 => ( ( member @ A @ Y @ S )
% 0.27/0.47 => ( ord_less_eq @ B @ ( F @ ( lattic1704895705min_on @ A @ B @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % arg_min_least
% 0.27/0.47 thf(fact_188_Greatest__equality,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [P: A > $o,X: A] :
% 0.27/0.47 ( ( P @ X )
% 0.27/0.47 => ( ! [Y3: A] :
% 0.27/0.47 ( ( P @ Y3 )
% 0.27/0.47 => ( ord_less_eq @ A @ Y3 @ X ) )
% 0.27/0.47 => ( ( order_Greatest @ A @ P )
% 0.27/0.47 = X ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Greatest_equality
% 0.27/0.47 thf(fact_189_GreatestI2__order,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [P: A > $o,X: A,Q: A > $o] :
% 0.27/0.47 ( ( P @ X )
% 0.27/0.47 => ( ! [Y3: A] :
% 0.27/0.47 ( ( P @ Y3 )
% 0.27/0.47 => ( ord_less_eq @ A @ Y3 @ X ) )
% 0.27/0.47 => ( ! [X4: A] :
% 0.27/0.47 ( ( P @ X4 )
% 0.27/0.47 => ( ! [Y6: A] :
% 0.27/0.47 ( ( P @ Y6 )
% 0.27/0.47 => ( ord_less_eq @ A @ Y6 @ X4 ) )
% 0.27/0.47 => ( Q @ X4 ) ) )
% 0.27/0.47 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % GreatestI2_order
% 0.27/0.47 thf(fact_190_arg__min__if__finite_I1_J,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( order @ B )
% 0.27/0.47 => ! [S: set @ A,F: A > B] :
% 0.27/0.47 ( ( finite_finite2 @ A @ S )
% 0.27/0.47 => ( ( S
% 0.27/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 => ( member @ A @ ( lattic1704895705min_on @ A @ B @ F @ S ) @ S ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % arg_min_if_finite(1)
% 0.27/0.47 thf(fact_191_Inf__fin_Osubset__imp,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_inf @ A )
% 0.27/0.47 => ! [A2: set @ A,B3: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.27/0.47 => ( ( A2
% 0.27/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 => ( ( finite_finite2 @ A @ B3 )
% 0.27/0.47 => ( ord_less_eq @ A @ ( lattic2065218050nf_fin @ A @ B3 ) @ ( lattic2065218050nf_fin @ A @ A2 ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Inf_fin.subset_imp
% 0.27/0.47 thf(fact_192_Sup__fin_Osubset__imp,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [A2: set @ A,B3: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.27/0.47 => ( ( A2
% 0.27/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 => ( ( finite_finite2 @ A @ B3 )
% 0.27/0.47 => ( ord_less_eq @ A @ ( lattic477160up_fin @ A @ A2 ) @ ( lattic477160up_fin @ A @ B3 ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Sup_fin.subset_imp
% 0.27/0.47 thf(fact_193_Inf__fin__le__Sup__fin,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( lattice @ A )
% 0.27/0.47 => ! [A2: set @ A] :
% 0.27/0.47 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.47 => ( ( A2
% 0.27/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 => ( ord_less_eq @ A @ ( lattic2065218050nf_fin @ A @ A2 ) @ ( lattic477160up_fin @ A @ A2 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Inf_fin_le_Sup_fin
% 0.27/0.47 thf(fact_194_Sup__fin_OcoboundedI,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [A2: set @ A,A3: A] :
% 0.27/0.47 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.47 => ( ( member @ A @ A3 @ A2 )
% 0.27/0.47 => ( ord_less_eq @ A @ A3 @ ( lattic477160up_fin @ A @ A2 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Sup_fin.coboundedI
% 0.27/0.47 thf(fact_195_Inf__fin_OcoboundedI,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_inf @ A )
% 0.27/0.47 => ! [A2: set @ A,A3: A] :
% 0.27/0.47 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.47 => ( ( member @ A @ A3 @ A2 )
% 0.27/0.47 => ( ord_less_eq @ A @ ( lattic2065218050nf_fin @ A @ A2 ) @ A3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Inf_fin.coboundedI
% 0.27/0.47 thf(fact_196_Inf__fin_OboundedE,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_inf @ A )
% 0.27/0.47 => ! [A2: set @ A,X: A] :
% 0.27/0.47 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.47 => ( ( A2
% 0.27/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.47 => ( ( ord_less_eq @ A @ X @ ( lattic2065218050nf_fin @ A @ A2 ) )
% 0.27/0.47 => ! [A8: A] :
% 0.27/0.47 ( ( member @ A @ A8 @ A2 )
% 0.27/0.47 => ( ord_less_eq @ A @ X @ A8 ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Inf_fin.boundedE
% 0.27/0.48 thf(fact_197_Inf__fin_OboundedI,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( semilattice_inf @ A )
% 0.27/0.48 => ! [A2: set @ A,X: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ! [A7: A] :
% 0.27/0.48 ( ( member @ A @ A7 @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ X @ A7 ) )
% 0.27/0.48 => ( ord_less_eq @ A @ X @ ( lattic2065218050nf_fin @ A @ A2 ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Inf_fin.boundedI
% 0.27/0.48 thf(fact_198_Sup__fin_OboundedE,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( semilattice_sup @ A )
% 0.27/0.48 => ! [A2: set @ A,X: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( ord_less_eq @ A @ ( lattic477160up_fin @ A @ A2 ) @ X )
% 0.27/0.48 => ! [A8: A] :
% 0.27/0.48 ( ( member @ A @ A8 @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ A8 @ X ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Sup_fin.boundedE
% 0.27/0.48 thf(fact_199_Sup__fin_OboundedI,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( semilattice_sup @ A )
% 0.27/0.48 => ! [A2: set @ A,X: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ! [A7: A] :
% 0.27/0.48 ( ( member @ A @ A7 @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ A7 @ X ) )
% 0.27/0.48 => ( ord_less_eq @ A @ ( lattic477160up_fin @ A @ A2 ) @ X ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Sup_fin.boundedI
% 0.27/0.48 thf(fact_200_Inf__fin_Obounded__iff,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( semilattice_inf @ A )
% 0.27/0.48 => ! [A2: set @ A,X: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( ord_less_eq @ A @ X @ ( lattic2065218050nf_fin @ A @ A2 ) )
% 0.27/0.48 = ( ! [X3: A] :
% 0.27/0.48 ( ( member @ A @ X3 @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Inf_fin.bounded_iff
% 0.27/0.48 thf(fact_201_Sup__fin_Obounded__iff,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( semilattice_sup @ A )
% 0.27/0.48 => ! [A2: set @ A,X: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( ord_less_eq @ A @ ( lattic477160up_fin @ A @ A2 ) @ X )
% 0.27/0.48 = ( ! [X3: A] :
% 0.27/0.48 ( ( member @ A @ X3 @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Sup_fin.bounded_iff
% 0.27/0.48 thf(fact_202_Max_Osubset__imp,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,B3: set @ A] :
% 0.27/0.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( finite_finite2 @ A @ B3 )
% 0.27/0.48 => ( ord_less_eq @ A @ ( lattic929149872er_Max @ A @ A2 ) @ ( lattic929149872er_Max @ A @ B3 ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Max.subset_imp
% 0.27/0.48 thf(fact_203_Max__mono,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [M2: set @ A,N: set @ A] :
% 0.27/0.48 ( ( ord_less_eq @ ( set @ A ) @ M2 @ N )
% 0.27/0.48 => ( ( M2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( finite_finite2 @ A @ N )
% 0.27/0.48 => ( ord_less_eq @ A @ ( lattic929149872er_Max @ A @ M2 ) @ ( lattic929149872er_Max @ A @ N ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Max_mono
% 0.27/0.48 thf(fact_204_Max_Obounded__iff,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,X: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( ord_less_eq @ A @ ( lattic929149872er_Max @ A @ A2 ) @ X )
% 0.27/0.48 = ( ! [X3: A] :
% 0.27/0.48 ( ( member @ A @ X3 @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Max.bounded_iff
% 0.27/0.48 thf(fact_205_Max_OcoboundedI,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,A3: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( member @ A @ A3 @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ A3 @ ( lattic929149872er_Max @ A @ A2 ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Max.coboundedI
% 0.27/0.48 thf(fact_206_Max__eq__if,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,B3: set @ A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( finite_finite2 @ A @ B3 )
% 0.27/0.48 => ( ! [X4: A] :
% 0.27/0.48 ( ( member @ A @ X4 @ A2 )
% 0.27/0.48 => ? [Xa: A] :
% 0.27/0.48 ( ( member @ A @ Xa @ B3 )
% 0.27/0.48 & ( ord_less_eq @ A @ X4 @ Xa ) ) )
% 0.27/0.48 => ( ! [X4: A] :
% 0.27/0.48 ( ( member @ A @ X4 @ B3 )
% 0.27/0.48 => ? [Xa: A] :
% 0.27/0.48 ( ( member @ A @ Xa @ A2 )
% 0.27/0.48 & ( ord_less_eq @ A @ X4 @ Xa ) ) )
% 0.27/0.48 => ( ( lattic929149872er_Max @ A @ A2 )
% 0.27/0.48 = ( lattic929149872er_Max @ A @ B3 ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Max_eq_if
% 0.27/0.48 thf(fact_207_Max__eqI,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,X: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ! [Y3: A] :
% 0.27/0.48 ( ( member @ A @ Y3 @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ Y3 @ X ) )
% 0.27/0.48 => ( ( member @ A @ X @ A2 )
% 0.27/0.48 => ( ( lattic929149872er_Max @ A @ A2 )
% 0.27/0.48 = X ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Max_eqI
% 0.27/0.48 thf(fact_208_Max__ge,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,X: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( member @ A @ X @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ X @ ( lattic929149872er_Max @ A @ A2 ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Max_ge
% 0.27/0.48 thf(fact_209_Max__in,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( member @ A @ ( lattic929149872er_Max @ A @ A2 ) @ A2 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Max_in
% 0.27/0.48 thf(fact_210_Max__eq__iff,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,M3: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( ( lattic929149872er_Max @ A @ A2 )
% 0.27/0.48 = M3 )
% 0.27/0.48 = ( ( member @ A @ M3 @ A2 )
% 0.27/0.48 & ! [X3: A] :
% 0.27/0.48 ( ( member @ A @ X3 @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ X3 @ M3 ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Max_eq_iff
% 0.27/0.48 thf(fact_211_Max__ge__iff,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,X: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( ord_less_eq @ A @ X @ ( lattic929149872er_Max @ A @ A2 ) )
% 0.27/0.48 = ( ? [X3: A] :
% 0.27/0.48 ( ( member @ A @ X3 @ A2 )
% 0.27/0.48 & ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Max_ge_iff
% 0.27/0.48 thf(fact_212_eq__Max__iff,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,M3: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( M3
% 0.27/0.48 = ( lattic929149872er_Max @ A @ A2 ) )
% 0.27/0.48 = ( ( member @ A @ M3 @ A2 )
% 0.27/0.48 & ! [X3: A] :
% 0.27/0.48 ( ( member @ A @ X3 @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ X3 @ M3 ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % eq_Max_iff
% 0.27/0.48 thf(fact_213_Max_OboundedE,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,X: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( ord_less_eq @ A @ ( lattic929149872er_Max @ A @ A2 ) @ X )
% 0.27/0.48 => ! [A8: A] :
% 0.27/0.48 ( ( member @ A @ A8 @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ A8 @ X ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Max.boundedE
% 0.27/0.48 thf(fact_214_Max_OboundedI,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,X: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ! [A7: A] :
% 0.27/0.48 ( ( member @ A @ A7 @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ A7 @ X ) )
% 0.27/0.48 => ( ord_less_eq @ A @ ( lattic929149872er_Max @ A @ A2 ) @ X ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Max.boundedI
% 0.27/0.48 thf(fact_215_Min__antimono,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [M2: set @ A,N: set @ A] :
% 0.27/0.48 ( ( ord_less_eq @ ( set @ A ) @ M2 @ N )
% 0.27/0.48 => ( ( M2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( finite_finite2 @ A @ N )
% 0.27/0.48 => ( ord_less_eq @ A @ ( lattic929674654er_Min @ A @ N ) @ ( lattic929674654er_Min @ A @ M2 ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Min_antimono
% 0.27/0.48 thf(fact_216_Min_Osubset__imp,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,B3: set @ A] :
% 0.27/0.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( finite_finite2 @ A @ B3 )
% 0.27/0.48 => ( ord_less_eq @ A @ ( lattic929674654er_Min @ A @ B3 ) @ ( lattic929674654er_Min @ A @ A2 ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Min.subset_imp
% 0.27/0.48 thf(fact_217_Min_Obounded__iff,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,X: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( ord_less_eq @ A @ X @ ( lattic929674654er_Min @ A @ A2 ) )
% 0.27/0.48 = ( ! [X3: A] :
% 0.27/0.48 ( ( member @ A @ X3 @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Min.bounded_iff
% 0.27/0.48 thf(fact_218_Min__in,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( member @ A @ ( lattic929674654er_Min @ A @ A2 ) @ A2 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Min_in
% 0.27/0.48 thf(fact_219_Min__le,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,X: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( member @ A @ X @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ ( lattic929674654er_Min @ A @ A2 ) @ X ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Min_le
% 0.27/0.48 thf(fact_220_Min__eqI,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,X: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ! [Y3: A] :
% 0.27/0.48 ( ( member @ A @ Y3 @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ X @ Y3 ) )
% 0.27/0.48 => ( ( member @ A @ X @ A2 )
% 0.27/0.48 => ( ( lattic929674654er_Min @ A @ A2 )
% 0.27/0.48 = X ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Min_eqI
% 0.27/0.48 thf(fact_221_Min_OcoboundedI,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,A3: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( member @ A @ A3 @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ ( lattic929674654er_Min @ A @ A2 ) @ A3 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Min.coboundedI
% 0.27/0.48 thf(fact_222_Min_OboundedI,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,X: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ! [A7: A] :
% 0.27/0.48 ( ( member @ A @ A7 @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ X @ A7 ) )
% 0.27/0.48 => ( ord_less_eq @ A @ X @ ( lattic929674654er_Min @ A @ A2 ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Min.boundedI
% 0.27/0.48 thf(fact_223_Min_OboundedE,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,X: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( ord_less_eq @ A @ X @ ( lattic929674654er_Min @ A @ A2 ) )
% 0.27/0.48 => ! [A8: A] :
% 0.27/0.48 ( ( member @ A @ A8 @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ X @ A8 ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Min.boundedE
% 0.27/0.48 thf(fact_224_eq__Min__iff,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,M3: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( M3
% 0.27/0.48 = ( lattic929674654er_Min @ A @ A2 ) )
% 0.27/0.48 = ( ( member @ A @ M3 @ A2 )
% 0.27/0.48 & ! [X3: A] :
% 0.27/0.48 ( ( member @ A @ X3 @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ M3 @ X3 ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % eq_Min_iff
% 0.27/0.48 thf(fact_225_Min__le__iff,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,X: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( ord_less_eq @ A @ ( lattic929674654er_Min @ A @ A2 ) @ X )
% 0.27/0.48 = ( ? [X3: A] :
% 0.27/0.48 ( ( member @ A @ X3 @ A2 )
% 0.27/0.48 & ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Min_le_iff
% 0.27/0.48 thf(fact_226_Min__eq__iff,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( linorder @ A )
% 0.27/0.48 => ! [A2: set @ A,M3: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( ( lattic929674654er_Min @ A @ A2 )
% 0.27/0.48 = M3 )
% 0.27/0.48 = ( ( member @ A @ M3 @ A2 )
% 0.27/0.48 & ! [X3: A] :
% 0.27/0.48 ( ( member @ A @ X3 @ A2 )
% 0.27/0.48 => ( ord_less_eq @ A @ M3 @ X3 ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Min_eq_iff
% 0.27/0.48 thf(fact_227_Inf__fin_Osubset,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( semilattice_inf @ A )
% 0.27/0.48 => ! [A2: set @ A,B3: set @ A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( B3
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
% 0.27/0.48 => ( ( inf_inf @ A @ ( lattic2065218050nf_fin @ A @ B3 ) @ ( lattic2065218050nf_fin @ A @ A2 ) )
% 0.27/0.48 = ( lattic2065218050nf_fin @ A @ A2 ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Inf_fin.subset
% 0.27/0.48 thf(fact_228_Sup__fin_Osubset,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( semilattice_sup @ A )
% 0.27/0.48 => ! [A2: set @ A,B3: set @ A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( B3
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
% 0.27/0.48 => ( ( sup_sup @ A @ ( lattic477160up_fin @ A @ B3 ) @ ( lattic477160up_fin @ A @ A2 ) )
% 0.27/0.48 = ( lattic477160up_fin @ A @ A2 ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Sup_fin.subset
% 0.27/0.48 thf(fact_229_Un__empty,axiom,
% 0.27/0.48 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.27/0.48 ( ( ( sup_sup @ ( set @ A ) @ A2 @ B3 )
% 0.27/0.48 = ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 = ( ( A2
% 0.27/0.48 = ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 & ( B3
% 0.27/0.48 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Un_empty
% 0.27/0.48 thf(fact_230_Int__subset__iff,axiom,
% 0.27/0.48 ! [A: $tType,C3: set @ A,A2: set @ A,B3: set @ A] :
% 0.27/0.48 ( ( ord_less_eq @ ( set @ A ) @ C3 @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) )
% 0.27/0.48 = ( ( ord_less_eq @ ( set @ A ) @ C3 @ A2 )
% 0.27/0.48 & ( ord_less_eq @ ( set @ A ) @ C3 @ B3 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Int_subset_iff
% 0.27/0.48 thf(fact_231_Un__subset__iff,axiom,
% 0.27/0.48 ! [A: $tType,A2: set @ A,B3: set @ A,C3: set @ A] :
% 0.27/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) @ C3 )
% 0.27/0.48 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 0.27/0.48 & ( ord_less_eq @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Un_subset_iff
% 0.27/0.48 thf(fact_232_finite__Int,axiom,
% 0.27/0.48 ! [A: $tType,F5: set @ A,G5: set @ A] :
% 0.27/0.48 ( ( ( finite_finite2 @ A @ F5 )
% 0.27/0.48 | ( finite_finite2 @ A @ G5 ) )
% 0.27/0.48 => ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F5 @ G5 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % finite_Int
% 0.27/0.48 thf(fact_233_finite__Un,axiom,
% 0.27/0.48 ! [A: $tType,F5: set @ A,G5: set @ A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F5 @ G5 ) )
% 0.27/0.48 = ( ( finite_finite2 @ A @ F5 )
% 0.27/0.48 & ( finite_finite2 @ A @ G5 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % finite_Un
% 0.27/0.48 thf(fact_234_inf__Pair__Pair,axiom,
% 0.27/0.48 ! [A: $tType,B: $tType] :
% 0.27/0.48 ( ( ( inf @ B )
% 0.27/0.48 & ( inf @ A ) )
% 0.27/0.48 => ! [A3: A,B2: B,C2: A,D: B] :
% 0.27/0.48 ( ( inf_inf @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D ) )
% 0.27/0.48 = ( product_Pair @ A @ B @ ( inf_inf @ A @ A3 @ C2 ) @ ( inf_inf @ B @ B2 @ D ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % inf_Pair_Pair
% 0.27/0.48 thf(fact_235_sup__Pair__Pair,axiom,
% 0.27/0.48 ! [A: $tType,B: $tType] :
% 0.27/0.48 ( ( ( sup @ B )
% 0.27/0.48 & ( sup @ A ) )
% 0.27/0.48 => ! [A3: A,B2: B,C2: A,D: B] :
% 0.27/0.48 ( ( sup_sup @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D ) )
% 0.27/0.48 = ( product_Pair @ A @ B @ ( sup_sup @ A @ A3 @ C2 ) @ ( sup_sup @ B @ B2 @ D ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % sup_Pair_Pair
% 0.27/0.48 thf(fact_236_inf__Sup__absorb,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( lattice @ A )
% 0.27/0.48 => ! [A2: set @ A,A3: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( member @ A @ A3 @ A2 )
% 0.27/0.48 => ( ( inf_inf @ A @ A3 @ ( lattic477160up_fin @ A @ A2 ) )
% 0.27/0.48 = A3 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % inf_Sup_absorb
% 0.27/0.48 thf(fact_237_sup__Inf__absorb,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( lattice @ A )
% 0.27/0.48 => ! [A2: set @ A,A3: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( member @ A @ A3 @ A2 )
% 0.27/0.48 => ( ( sup_sup @ A @ ( lattic2065218050nf_fin @ A @ A2 ) @ A3 )
% 0.27/0.48 = A3 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % sup_Inf_absorb
% 0.27/0.48 thf(fact_238_Sup__fin_Oin__idem,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( semilattice_sup @ A )
% 0.27/0.48 => ! [A2: set @ A,X: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( member @ A @ X @ A2 )
% 0.27/0.48 => ( ( sup_sup @ A @ X @ ( lattic477160up_fin @ A @ A2 ) )
% 0.27/0.48 = ( lattic477160up_fin @ A @ A2 ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Sup_fin.in_idem
% 0.27/0.48 thf(fact_239_Inf__fin_Oin__idem,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( semilattice_inf @ A )
% 0.27/0.48 => ! [A2: set @ A,X: A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( member @ A @ X @ A2 )
% 0.27/0.48 => ( ( inf_inf @ A @ X @ ( lattic2065218050nf_fin @ A @ A2 ) )
% 0.27/0.48 = ( lattic2065218050nf_fin @ A @ A2 ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Inf_fin.in_idem
% 0.27/0.48 thf(fact_240_Sup__fin_Ounion,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( semilattice_sup @ A )
% 0.27/0.48 => ! [A2: set @ A,B3: set @ A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( finite_finite2 @ A @ B3 )
% 0.27/0.48 => ( ( B3
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( lattic477160up_fin @ A @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) )
% 0.27/0.48 = ( sup_sup @ A @ ( lattic477160up_fin @ A @ A2 ) @ ( lattic477160up_fin @ A @ B3 ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Sup_fin.union
% 0.27/0.48 thf(fact_241_Inf__fin_Ounion,axiom,
% 0.27/0.48 ! [A: $tType] :
% 0.27/0.48 ( ( semilattice_inf @ A )
% 0.27/0.48 => ! [A2: set @ A,B3: set @ A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ A2 )
% 0.27/0.48 => ( ( A2
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( finite_finite2 @ A @ B3 )
% 0.27/0.48 => ( ( B3
% 0.27/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.27/0.48 => ( ( lattic2065218050nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) )
% 0.27/0.48 = ( inf_inf @ A @ ( lattic2065218050nf_fin @ A @ A2 ) @ ( lattic2065218050nf_fin @ A @ B3 ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Inf_fin.union
% 0.27/0.48 thf(fact_242_trans__Int,axiom,
% 0.27/0.48 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
% 0.27/0.48 ( ( trans @ A @ R2 )
% 0.27/0.48 => ( ( trans @ A @ S2 )
% 0.27/0.48 => ( trans @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S2 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % trans_Int
% 0.27/0.48 thf(fact_243_refl__on__Un,axiom,
% 0.27/0.48 ! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A ),B3: set @ A,S2: set @ ( product_prod @ A @ A )] :
% 0.27/0.48 ( ( refl_on @ A @ A2 @ R2 )
% 0.27/0.48 => ( ( refl_on @ A @ B3 @ S2 )
% 0.27/0.48 => ( refl_on @ A @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S2 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % refl_on_Un
% 0.27/0.48 thf(fact_244_refl__on__Int,axiom,
% 0.27/0.48 ! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A ),B3: set @ A,S2: set @ ( product_prod @ A @ A )] :
% 0.27/0.48 ( ( refl_on @ A @ A2 @ R2 )
% 0.27/0.48 => ( ( refl_on @ A @ B3 @ S2 )
% 0.27/0.48 => ( refl_on @ A @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S2 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % refl_on_Int
% 0.27/0.48 thf(fact_245_finite__UnI,axiom,
% 0.27/0.48 ! [A: $tType,F5: set @ A,G5: set @ A] :
% 0.27/0.48 ( ( finite_finite2 @ A @ F5 )
% 0.27/0.48 => ( ( finite_finite2 @ A @ G5 )
% 0.27/0.48 => ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F5 @ G5 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % finite_UnI
% 0.27/0.48 thf(fact_246_Un__infinite,axiom,
% 0.27/0.48 ! [A: $tType,S: set @ A,T3: set @ A] :
% 0.27/0.48 ( ~ ( finite_finite2 @ A @ S )
% 0.27/0.48 => ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T3 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Un_infinite
% 0.27/0.48 thf(fact_247_infinite__Un,axiom,
% 0.27/0.48 ! [A: $tType,S: set @ A,T3: set @ A] :
% 0.27/0.48 ( ( ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T3 ) ) )
% 0.27/0.48 = ( ~ ( finite_finite2 @ A @ S )
% 0.27/0.48 | ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % infinite_Un
% 0.27/0.48 thf(fact_248_Un__Int__assoc__eq,axiom,
% 0.27/0.48 ! [A: $tType,A2: set @ A,B3: set @ A,C3: set @ A] :
% 0.27/0.48 ( ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) @ C3 )
% 0.27/0.48 = ( inf_inf @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) ) )
% 0.27/0.48 = ( ord_less_eq @ ( set @ A ) @ C3 @ A2 ) ) ).
% 0.27/0.48
% 0.27/0.48 % Un_Int_assoc_eq
% 0.27/0.48 thf(fact_249_Int__mono,axiom,
% 0.27/0.48 ! [A: $tType,A2: set @ A,C3: set @ A,B3: set @ A,D4: set @ A] :
% 0.27/0.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 0.27/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ D4 )
% 0.27/0.48 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) @ ( inf_inf @ ( set @ A ) @ C3 @ D4 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Int_mono
% 0.27/0.48 thf(fact_250_Int__lower1,axiom,
% 0.27/0.48 ! [A: $tType,A2: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) @ A2 ) ).
% 0.27/0.48
% 0.27/0.48 % Int_lower1
% 0.27/0.48 thf(fact_251_Int__lower2,axiom,
% 0.27/0.48 ! [A: $tType,A2: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) @ B3 ) ).
% 0.27/0.48
% 0.27/0.48 % Int_lower2
% 0.27/0.48 thf(fact_252_Int__absorb1,axiom,
% 0.27/0.48 ! [A: $tType,B3: set @ A,A2: set @ A] :
% 0.27/0.48 ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
% 0.27/0.48 => ( ( inf_inf @ ( set @ A ) @ A2 @ B3 )
% 0.27/0.48 = B3 ) ) ).
% 0.27/0.48
% 0.27/0.48 % Int_absorb1
% 0.27/0.48 thf(fact_253_Int__absorb2,axiom,
% 0.27/0.48 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.27/0.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.27/0.48 => ( ( inf_inf @ ( set @ A ) @ A2 @ B3 )
% 0.27/0.48 = A2 ) ) ).
% 0.27/0.48
% 0.27/0.48 % Int_absorb2
% 0.27/0.48 thf(fact_254_Int__greatest,axiom,
% 0.27/0.48 ! [A: $tType,C3: set @ A,A2: set @ A,B3: set @ A] :
% 0.27/0.48 ( ( ord_less_eq @ ( set @ A ) @ C3 @ A2 )
% 0.27/0.48 => ( ( ord_less_eq @ ( set @ A ) @ C3 @ B3 )
% 0.27/0.48 => ( ord_less_eq @ ( set @ A ) @ C3 @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Int_greatest
% 0.27/0.48 thf(fact_255_Int__Collect__mono,axiom,
% 0.27/0.48 ! [A: $tType,A2: set @ A,B3: set @ A,P: A > $o,Q: A > $o] :
% 0.27/0.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.27/0.48 => ( ! [X4: A] :
% 0.27/0.48 ( ( member @ A @ X4 @ A2 )
% 0.27/0.48 => ( ( P @ X4 )
% 0.27/0.48 => ( Q @ X4 ) ) )
% 0.27/0.48 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B3 @ ( collect @ A @ Q ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Int_Collect_mono
% 0.27/0.48
% 0.27/0.48 % Type constructors (45)
% 0.27/0.48 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( semilattice_sup @ A10 )
% 0.27/0.48 => ( semilattice_sup @ ( A9 > A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( semilattice_inf @ A10 )
% 0.27/0.48 => ( semilattice_inf @ ( A9 > A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( order_bot @ A10 )
% 0.27/0.48 => ( order_bot @ ( A9 > A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( preorder @ A10 )
% 0.27/0.48 => ( preorder @ ( A9 > A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( ( finite_finite @ A9 )
% 0.27/0.48 & ( finite_finite @ A10 ) )
% 0.27/0.48 => ( finite_finite @ ( A9 > A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( lattice @ A10 )
% 0.27/0.48 => ( lattice @ ( A9 > A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( order @ A10 )
% 0.27/0.48 => ( order @ ( A9 > A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Orderings_Oord,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( ord @ A10 )
% 0.27/0.48 => ( ord @ ( A9 > A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Orderings_Obot,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( bot @ A10 )
% 0.27/0.48 => ( bot @ ( A9 > A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Lattices_Osup,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( semilattice_sup @ A10 )
% 0.27/0.48 => ( sup @ ( A9 > A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_fun___Lattices_Oinf,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( semilattice_inf @ A10 )
% 0.27/0.48 => ( inf @ ( A9 > A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1,axiom,
% 0.27/0.48 ! [A9: $tType] : ( semilattice_sup @ ( set @ A9 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_2,axiom,
% 0.27/0.48 ! [A9: $tType] : ( semilattice_inf @ ( set @ A9 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Orderings_Oorder__bot_3,axiom,
% 0.27/0.48 ! [A9: $tType] : ( order_bot @ ( set @ A9 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 0.27/0.48 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
% 0.27/0.48 ! [A9: $tType] :
% 0.27/0.48 ( ( finite_finite @ A9 )
% 0.27/0.48 => ( finite_finite @ ( set @ A9 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Lattices_Olattice_6,axiom,
% 0.27/0.48 ! [A9: $tType] : ( lattice @ ( set @ A9 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Orderings_Oorder_7,axiom,
% 0.27/0.48 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
% 0.27/0.48 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Orderings_Obot_9,axiom,
% 0.27/0.48 ! [A9: $tType] : ( bot @ ( set @ A9 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Lattices_Osup_10,axiom,
% 0.27/0.48 ! [A9: $tType] : ( sup @ ( set @ A9 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Set_Oset___Lattices_Oinf_11,axiom,
% 0.27/0.48 ! [A9: $tType] : ( inf @ ( set @ A9 ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_12,axiom,
% 0.27/0.48 semilattice_sup @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_13,axiom,
% 0.27/0.48 semilattice_inf @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Orderings_Oorder__bot_14,axiom,
% 0.27/0.48 order_bot @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Orderings_Opreorder_15,axiom,
% 0.27/0.48 preorder @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.27/0.48 linorder @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Finite__Set_Ofinite_16,axiom,
% 0.27/0.48 finite_finite @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Lattices_Olattice_17,axiom,
% 0.27/0.48 lattice @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Orderings_Oorder_18,axiom,
% 0.27/0.48 order @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Orderings_Oord_19,axiom,
% 0.27/0.48 ord @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Orderings_Obot_20,axiom,
% 0.27/0.48 bot @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Lattices_Osup_21,axiom,
% 0.27/0.48 sup @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_HOL_Obool___Lattices_Oinf_22,axiom,
% 0.27/0.48 inf @ $o ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Product__Type_Oprod___Lattices_Osemilattice__sup_23,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( ( semilattice_sup @ A9 )
% 0.27/0.48 & ( semilattice_sup @ A10 ) )
% 0.27/0.48 => ( semilattice_sup @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Product__Type_Oprod___Lattices_Osemilattice__inf_24,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( ( semilattice_inf @ A9 )
% 0.27/0.48 & ( semilattice_inf @ A10 ) )
% 0.27/0.48 => ( semilattice_inf @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Product__Type_Oprod___Orderings_Oorder__bot_25,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( ( order_bot @ A9 )
% 0.27/0.48 & ( order_bot @ A10 ) )
% 0.27/0.48 => ( order_bot @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_26,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( ( preorder @ A9 )
% 0.27/0.48 & ( preorder @ A10 ) )
% 0.27/0.48 => ( preorder @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_27,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( ( finite_finite @ A9 )
% 0.27/0.48 & ( finite_finite @ A10 ) )
% 0.27/0.48 => ( finite_finite @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Product__Type_Oprod___Lattices_Olattice_28,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( ( lattice @ A9 )
% 0.27/0.48 & ( lattice @ A10 ) )
% 0.27/0.48 => ( lattice @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Product__Type_Oprod___Orderings_Oorder_29,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( ( order @ A9 )
% 0.27/0.48 & ( order @ A10 ) )
% 0.27/0.48 => ( order @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Product__Type_Oprod___Orderings_Oord_30,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( ( ord @ A9 )
% 0.27/0.48 & ( ord @ A10 ) )
% 0.27/0.48 => ( ord @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Product__Type_Oprod___Orderings_Obot_31,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( ( bot @ A9 )
% 0.27/0.48 & ( bot @ A10 ) )
% 0.27/0.48 => ( bot @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Product__Type_Oprod___Lattices_Osup_32,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( ( sup @ A9 )
% 0.27/0.48 & ( sup @ A10 ) )
% 0.27/0.48 => ( sup @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(tcon_Product__Type_Oprod___Lattices_Oinf_33,axiom,
% 0.27/0.48 ! [A9: $tType,A10: $tType] :
% 0.27/0.48 ( ( ( inf @ A9 )
% 0.27/0.48 & ( inf @ A10 ) )
% 0.27/0.48 => ( inf @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % Conjectures (1)
% 0.27/0.48 thf(conj_0,conjecture,
% 0.27/0.48 ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ x @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ x @ carrier @ relation ) )
% 0.27/0.48 | ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ x @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ x @ carrier @ relation ) ) ) ).
% 0.27/0.48
% 0.27/0.48 %------------------------------------------------------------------------------
% 0.27/0.48 ------------------------
% 2.73/1.57 ---- Embedded in THF ---
% 2.73/1.57 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.73/1.57 %%% Generated on Tue Feb 24 08:28:50 EST 2026
% 2.73/1.57 %%% using '$$dhol' embedding, version 1.3.0.
% 2.73/1.57 %%% Logic specification used:
% 2.73/1.57 %%% thf(spec, logic, $$dhol).
% 2.73/1.57
% 2.73/1.57 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.bwuTtVduWu/DTF2DTF22979.p
% 2.73/1.57 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.73/1.57 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.73/1.57 thf(ty_tf_a, type, a: $tType).
% 2.73/1.57 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.73/1.57 thf(sy_cl_Lattices_Oinf, type, inf: (!> [A:$tType]: ($o))).
% 2.73/1.57 thf(sy_cl_Lattices_Osup, type, sup: (!> [A:$tType]: ($o))).
% 2.73/1.57 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 2.73/1.57 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.73/1.57 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.73/1.57 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: ($o))).
% 2.73/1.57 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 2.73/1.57 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.73/1.57 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.73/1.57 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: ($o))).
% 2.73/1.57 thf(sy_cl_Lattices_Osemilattice__inf, type, semilattice_inf: (!> [A:$tType]: ($o))).
% 2.73/1.57 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: ($o))).
% 2.73/1.57 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain, type, bNF_Ca1785829860lChain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 2.73/1.57 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.73/1.57 thf(sy_c_Lattices_Oinf__class_Oinf, type, inf_inf: (!> [A:$tType]: ((A > (A > A))))).
% 2.73/1.57 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 2.73/1.57 thf(sy_c_Lattices__Big_Olinorder__class_OMax, type, lattic929149872er_Max: (!> [A:$tType]: (((set @ A) > A)))).
% 2.73/1.57 thf(sy_c_Lattices__Big_Olinorder__class_OMin, type, lattic929674654er_Min: (!> [A:$tType]: (((set @ A) > A)))).
% 2.73/1.57 thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on, type, lattic1704895705min_on: (!> [B:$tType,A:$tType]: (((B > A) > ((set @ B) > B))))).
% 2.73/1.57 thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin, type, lattic2065218050nf_fin: (!> [A:$tType]: (((set @ A) > A)))).
% 2.73/1.57 thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin, type, lattic477160up_fin: (!> [A:$tType]: (((set @ A) > A)))).
% 2.73/1.57 thf(sy_c_Order__Relation_Opreorder__on, type, order_preorder_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.73/1.57 thf(sy_c_Order__Relation_Ounder, type, order_under: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (A > (set @ A)))))).
% 2.73/1.57 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.73/1.57 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.73/1.57 thf(sy_c_Orderings_Oorder__class_OGreatest, type, order_Greatest: (!> [A:$tType]: (((A > $o) > A)))).
% 2.73/1.57 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oas__good__as, type, prefer951318096ood_as: (!> [A:$tType]: ((A > ((set @ A) > ((set @ ((product_prod @ A) @ A)) > (set @ A))))))).
% 2.73/1.57 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oat__least__as__good, type, prefer310429814s_good: (!> [A:$tType]: ((A > ((set @ A) > ((set @ ((product_prod @ A) @ A)) > (set @ A))))))).
% 2.73/1.57 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Ono__better__than, type, prefer1532642881r_than: (!> [A:$tType]: ((A > ((set @ A) > ((set @ ((product_prod @ A) @ A)) > (set @ A))))))).
% 2.73/1.57 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Opreference, type, prefer199794634erence: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.73/1.57 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference, type, prefer1997167224erence: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.73/1.57 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference__axioms, type, prefer1801827867axioms: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.73/1.57 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.73/1.57 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.73/1.57 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.73/1.57 thf(sy_c_Relation_OId__on, type, id_on: (!> [A:$tType]: (((set @ A) > (set @ ((product_prod @ A) @ A)))))).
% 2.73/1.57 thf(sy_c_Relation_Orefl__on, type, refl_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.73/1.57 thf(sy_c_Relation_Ototal__on, type, total_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.73/1.57 thf(sy_c_Relation_Otrans, type, trans: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 2.73/1.57 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.73/1.57 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.73/1.57 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.73/1.57 thf(sy_v_carrier, type, carrier: (set @ a)).
% 2.73/1.57 thf(sy_v_relation, type, relation: (set @ ((product_prod @ a) @ a))).
% 2.73/1.57 thf(sy_v_x, type, x: a).
% 2.73/1.57 thf(sy_v_y, type, y: a).
% 2.73/1.57 thf(fact_0_assms_I3_J, axiom, (((member @ a) @ y) @ carrier)).
% 2.73/1.57 thf(fact_1_assms_I2_J, axiom, (((member @ a) @ x) @ carrier)).
% 2.73/1.57 thf(fact_2_rational__preference__axioms, axiom, (((prefer1997167224erence @ a) @ carrier) @ relation)).
% 2.73/1.57 thf(fact_3_nbt__nest, axiom, (! [Y:a,X:a]: (((((ord_less_eq @ (set @ a)) @ ((((prefer1532642881r_than @ a) @ Y) @ carrier) @ relation)) @ ((((prefer1532642881r_than @ a) @ X) @ carrier) @ relation)) | (((ord_less_eq @ (set @ a)) @ ((((prefer1532642881r_than @ a) @ X) @ carrier) @ relation)) @ ((((prefer1532642881r_than @ a) @ Y) @ carrier) @ relation)))))).
% 2.73/1.57 thf(fact_4_assms_I1_J, axiom, ((finite_finite2 @ a) @ carrier)).
% 2.73/1.57 thf(fact_5_preference__axioms, axiom, (((prefer199794634erence @ a) @ carrier) @ relation)).
% 2.73/1.57 thf(fact_6_trans__refl, axiom, (((order_preorder_on @ a) @ carrier) @ relation)).
% 2.73/1.57 thf(fact_7_no__better__subset__pref, axiom, (! [X:a,Y:a]: (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) => (((ord_less_eq @ (set @ a)) @ ((((prefer1532642881r_than @ a) @ Y) @ carrier) @ relation)) @ ((((prefer1532642881r_than @ a) @ X) @ carrier) @ relation)))))).
% 2.73/1.57 thf(fact_8_no__better__thansubset__rel, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ carrier) => ((((member @ a) @ Y) @ carrier) => ((((ord_less_eq @ (set @ a)) @ ((((prefer1532642881r_than @ a) @ Y) @ carrier) @ relation)) @ ((((prefer1532642881r_than @ a) @ X) @ carrier) @ relation)) => (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation))))))).
% 2.73/1.57 thf(fact_9_no__better__than__nonepty, axiom, (! [X:a]: (((carrier != (bot_bot @ (set @ a))) => ((((member @ a) @ X) @ carrier) => (((((prefer1532642881r_than @ a) @ X) @ carrier) @ relation) != (bot_bot @ (set @ a)))))))).
% 2.73/1.57 thf(fact_10_total, axiom, (((total_on @ a) @ carrier) @ relation)).
% 2.73/1.57 thf(fact_11_worse__in__no__better, axiom, (! [X:a,Y:a]: (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) => (((member @ a) @ Y) @ ((((prefer1532642881r_than @ a) @ Y) @ carrier) @ relation)))))).
% 2.73/1.57 thf(fact_12_reflexivity, axiom, (((refl_on @ a) @ carrier) @ relation)).
% 2.73/1.57 thf(fact_13_compl, axiom, (! [X2:a]: (((((member @ a) @ X2) @ carrier) => (! [Xa:a]: (((((member @ a) @ Xa) @ carrier) => ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X2) @ Xa)) @ relation) | (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Xa) @ X2)) @ relation))))))))).
% 2.73/1.57 thf(fact_14_completeD, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ carrier) => ((((member @ a) @ Y) @ carrier) => ((X != Y) => ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) | (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ X)) @ relation)))))))).
% 2.73/1.57 thf(fact_15_not__outside, axiom, (! [X:a,Y:a]: (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) => (((member @ a) @ X) @ carrier))))).
% 2.73/1.57 thf(fact_16_strict__is__neg__transitive, axiom, (! [X:a,Y:a,Z:a]: ((((((member @ a) @ X) @ carrier) & ((((member @ a) @ Y) @ carrier) & (((member @ a) @ Z) @ carrier))) => (((((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) @ X) @ Z)) @ relation) & (~ ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Z) @ X)) @ relation)))) | ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Z) @ Y)) @ relation) & (~ ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ Z)) @ relation)))))))))).
% 2.73/1.57 thf(fact_17_weak__is__transitive, axiom, (! [X:a,Y:a,Z:a]: ((((((member @ a) @ X) @ carrier) & ((((member @ a) @ Y) @ carrier) & (((member @ a) @ Z) @ carrier))) => ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) => ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ Z)) @ relation) => (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Z)) @ relation))))))).
% 2.73/1.57 thf(fact_18_strict__trans, axiom, (! [X:a,Y:a,Z:a]: ((((((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))))))))).
% 2.73/1.57 thf(fact_19_indiff__trans, axiom, (! [X:a,Y:a,Z:a]: ((((((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))))))).
% 2.73/1.57 thf(fact_20_strict__not__refl__weak, axiom, (! [X:a,Y:a]: ((((((member @ a) @ X) @ carrier) & (((member @ a) @ Y) @ carrier)) => ((~ ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ X)) @ 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))))))))).
% 2.73/1.57 thf(fact_21_rational__preference_Oaxioms_I1_J, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((prefer199794634erence @ A) @ Carrier) @ Relation))))).
% 2.73/1.57 thf(fact_22_preference__def, axiom, (! [A:$tType]: (((prefer199794634erence @ A) = (^ [Carrier2:(set @ A),Relation2:(set @ ((product_prod @ A) @ A))]: (((! [X3:A,Y2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y2)) @ Relation2) => (((member @ A) @ X3) @ Carrier2)))) & ((! [X3:A,Y2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y2)) @ Relation2) => (((member @ A) @ Y2) @ Carrier2)))) & (((order_preorder_on @ A) @ Carrier2) @ Relation2))))))))).
% 2.73/1.57 thf(fact_23_preference_Ointro, axiom, (! [A:$tType,Relation:(set @ ((product_prod @ A) @ A)),Carrier:(set @ A)]: (((! [X4:A,Y3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y3)) @ Relation) => (((member @ A) @ X4) @ Carrier)))) => ((! [X4:A,Y3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y3)) @ Relation) => (((member @ A) @ Y3) @ Carrier)))) => ((((order_preorder_on @ A) @ Carrier) @ Relation) => (((prefer199794634erence @ A) @ Carrier) @ Relation))))))).
% 2.73/1.57 thf(fact_24_preference_Otrans__refl, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer199794634erence @ A) @ Carrier) @ Relation) => (((order_preorder_on @ A) @ Carrier) @ Relation))))).
% 2.73/1.57 thf(fact_25_preference_Onot__outside, axiom, (! [A:$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)))))).
% 2.73/1.57 thf(fact_26_preference_Oreflexivity, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer199794634erence @ A) @ Carrier) @ Relation) => (((refl_on @ A) @ Carrier) @ Relation))))).
% 2.73/1.57 thf(fact_27_preference_Oindiff__trans, axiom, (! [A:$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)))))))).
% 2.73/1.57 thf(fact_28_rational__preference_Ocompl, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (! [X2:A]: (((((member @ A) @ X2) @ Carrier) => (! [Xa:A]: (((((member @ A) @ Xa) @ Carrier) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Xa)) @ Relation) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Xa) @ X2)) @ Relation)))))))))))).
% 2.73/1.57 thf(fact_29_rational__preference_Ototal, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((total_on @ A) @ Carrier) @ Relation))))).
% 2.73/1.57 thf(fact_30_rational__preference_OcompleteD, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ A) @ X) @ Carrier) => ((((member @ A) @ Y) @ Carrier) => ((X != Y) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ Relation))))))))).
% 2.73/1.57 thf(fact_31_rational__preference_Ostrict__trans, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A,Z:A]: (((((prefer1997167224erence @ 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)))))))))).
% 2.73/1.57 thf(fact_32_rational__preference_Oweak__is__transitive, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A,Z:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((((member @ A) @ X) @ Carrier) & ((((member @ A) @ Y) @ Carrier) & (((member @ A) @ Z) @ Carrier))) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ Relation) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z)) @ Relation)))))))).
% 2.73/1.57 thf(fact_33_rational__preference_Ostrict__not__refl__weak, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((((member @ A) @ X) @ Carrier) & (((member @ A) @ Y) @ Carrier)) => ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ 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)))))))))).
% 2.73/1.57 thf(fact_34_rational__preference_Ostrict__is__neg__transitive, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A,Z:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((((member @ A) @ X) @ Carrier) & ((((member @ A) @ Y) @ Carrier) & (((member @ A) @ Z) @ Carrier))) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) & (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ Relation)))) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z)) @ Relation) & (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ X)) @ Relation)))) | ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ Y)) @ Relation) & (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ Relation))))))))))).
% 2.73/1.57 thf(fact_35_rational__preference_Oworse__in__no__better, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) => (((member @ A) @ Y) @ ((((prefer1532642881r_than @ A) @ Y) @ Carrier) @ Relation))))))).
% 2.73/1.57 thf(fact_36_rational__preference_Ono__better__than__nonepty, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((Carrier != (bot_bot @ (set @ A))) => ((((member @ A) @ X) @ Carrier) => (((((prefer1532642881r_than @ A) @ X) @ Carrier) @ Relation) != (bot_bot @ (set @ A))))))))).
% 2.73/1.57 thf(fact_37_rational__preference_Ono__better__thansubset__rel, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ A) @ X) @ Carrier) => ((((member @ A) @ Y) @ Carrier) => ((((ord_less_eq @ (set @ A)) @ ((((prefer1532642881r_than @ A) @ Y) @ Carrier) @ Relation)) @ ((((prefer1532642881r_than @ A) @ X) @ Carrier) @ Relation)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation)))))))).
% 2.73/1.57 thf(fact_38_rational__preference_Ono__better__subset__pref, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) => (((ord_less_eq @ (set @ A)) @ ((((prefer1532642881r_than @ A) @ Y) @ Carrier) @ Relation)) @ ((((prefer1532642881r_than @ A) @ X) @ Carrier) @ Relation))))))).
% 2.73/1.57 thf(fact_39_rational__preference_Onbt__nest, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),Y:A,X:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((ord_less_eq @ (set @ A)) @ ((((prefer1532642881r_than @ A) @ Y) @ Carrier) @ Relation)) @ ((((prefer1532642881r_than @ A) @ X) @ Carrier) @ Relation)) | (((ord_less_eq @ (set @ A)) @ ((((prefer1532642881r_than @ A) @ X) @ Carrier) @ Relation)) @ ((((prefer1532642881r_than @ A) @ Y) @ Carrier) @ Relation))))))).
% 2.73/1.57 thf(fact_40_at__lst__asgd__not__ge, axiom, (! [X:a,Y:a]: (((carrier != (bot_bot @ (set @ a))) => ((((member @ a) @ X) @ carrier) => ((((member @ a) @ Y) @ carrier) => ((~ ((((member @ a) @ X) @ ((((prefer310429814s_good @ a) @ Y) @ carrier) @ relation)))) => (~ ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation)))))))))).
% 2.73/1.57 thf(fact_41_same__at__least__as__equal, axiom, (! [Z:a,Y:a]: ((((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Z) @ Y)) @ relation) & (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ Z)) @ relation)) => (((((prefer310429814s_good @ a) @ Z) @ carrier) @ relation) = ((((prefer310429814s_good @ a) @ Y) @ carrier) @ relation)))))).
% 2.73/1.57 thf(fact_42_pref__in__at__least__as, axiom, (! [X:a,Y:a]: (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) => (((member @ a) @ X) @ ((((prefer310429814s_good @ a) @ Y) @ carrier) @ relation)))))).
% 2.73/1.57 thf(fact_43_empty__subsetI, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A2)))).
% 2.73/1.57 thf(fact_44_subset__empty, axiom, (! [A:$tType,A2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ (bot_bot @ (set @ A))) = (A2 = (bot_bot @ (set @ A))))))).
% 2.73/1.57 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 2.73/1.57 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A2)))) = A2)))).
% 2.73/1.57 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) = (Q @ X4)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.73/1.57 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X4:A]: (((F @ X4) = (G @ X4)))) => (F = G))))).
% 2.73/1.57 thf(fact_49_Pair__le, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,B2:B,C2:A,D:B]: (((((ord_less_eq @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ ((((product_Pair @ A) @ B) @ C2) @ D)) = ((((ord_less_eq @ A) @ A3) @ C2) & (((ord_less_eq @ B) @ B2) @ D))))))))).
% 2.73/1.57 thf(fact_50_as__good__as__sameIff, axiom, (! [Z:a,Y:a]: (((((member @ a) @ Z) @ ((((prefer951318096ood_as @ a) @ Y) @ carrier) @ relation)) = ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Z) @ Y)) @ relation) & (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ Z)) @ relation)))))).
% 2.73/1.57 thf(fact_51_as__good__asIff, axiom, (! [X:a,Y:a]: (((((member @ a) @ X) @ ((((prefer951318096ood_as @ a) @ Y) @ carrier) @ relation)) = ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Y)) @ relation) & (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ X)) @ relation)))))).
% 2.73/1.57 thf(fact_52_finite__has__maximal, axiom, (! [A:$tType]: (((order @ A) => (! [A2:(set @ A)]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => (? [X4:A]: (((((member @ A) @ X4) @ A2) & (! [Xa:A]: (((((member @ A) @ Xa) @ A2) => ((((ord_less_eq @ A) @ X4) @ Xa) => (X4 = Xa)))))))))))))))).
% 2.73/1.57 thf(fact_53_finite__has__minimal, axiom, (! [A:$tType]: (((order @ A) => (! [A2:(set @ A)]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => (? [X4:A]: (((((member @ A) @ X4) @ A2) & (! [Xa:A]: (((((member @ A) @ Xa) @ A2) => ((((ord_less_eq @ A) @ Xa) @ X4) => (X4 = Xa)))))))))))))))).
% 2.73/1.57 thf(fact_54_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X3:A]: ((~ ((P @ X3))))))))).
% 2.73/1.57 thf(fact_55_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X3:A]: ((~ ((P @ X3))))))))).
% 2.73/1.57 thf(fact_56_all__not__in__conv, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X3:A]: ((~ ((((member @ A) @ X3) @ A2))))) = (A2 = (bot_bot @ (set @ A))))))).
% 2.73/1.57 thf(fact_57_empty__iff, axiom, (! [A:$tType,C2:A]: ((~ ((((member @ A) @ C2) @ (bot_bot @ (set @ A)))))))).
% 2.73/1.57 thf(fact_58_subsetI, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: (((! [X4:A]: (((((member @ A) @ X4) @ A2) => (((member @ A) @ X4) @ B3)))) => (((ord_less_eq @ (set @ A)) @ A2) @ B3))))).
% 2.73/1.57 thf(fact_59_subset__antisym, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ A2) => (A2 = B3)))))).
% 2.73/1.57 thf(fact_60_finite__code, axiom, (! [A:$tType]: (((finite_finite @ A) => ((finite_finite2 @ A) = (^ [A4:(set @ A)]: ($true))))))).
% 2.73/1.57 thf(fact_61_at__least__as__goodD, axiom, (! [A:$tType,Z:A,Y:A,B3:(set @ A),Pr:(set @ ((product_prod @ A) @ A))]: (((((member @ A) @ Z) @ ((((prefer310429814s_good @ A) @ Y) @ B3) @ Pr)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ Y)) @ Pr))))).
% 2.73/1.57 thf(fact_62_at__lst__asgd__ge, axiom, (! [A:$tType,X:A,Y:A,B3:(set @ A),Pr:(set @ ((product_prod @ A) @ A))]: (((((member @ A) @ X) @ ((((prefer310429814s_good @ A) @ Y) @ B3) @ Pr)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Pr))))).
% 2.73/1.57 thf(fact_63_ex__in__conv, axiom, (! [A:$tType,A2:(set @ A)]: (((? [X3:A]: ((((member @ A) @ X3) @ A2))) = (A2 != (bot_bot @ (set @ A))))))).
% 2.73/1.57 thf(fact_64_equals0I, axiom, (! [A:$tType,A2:(set @ A)]: (((! [Y3:A]: ((~ ((((member @ A) @ Y3) @ A2))))) => (A2 = (bot_bot @ (set @ A))))))).
% 2.73/1.57 thf(fact_65_equals0D, axiom, (! [A:$tType,A2:(set @ A),A3:A]: (((A2 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A3) @ A2))))))).
% 2.73/1.57 thf(fact_66_emptyE, axiom, (! [A:$tType,A3:A]: ((~ ((((member @ A) @ A3) @ (bot_bot @ (set @ A)))))))).
% 2.73/1.57 thf(fact_67_in__mono, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((((member @ A) @ X) @ A2) => (((member @ A) @ X) @ B3)))))).
% 2.73/1.57 thf(fact_68_subsetD, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((((member @ A) @ C2) @ A2) => (((member @ A) @ C2) @ B3)))))).
% 2.73/1.57 thf(fact_69_equalityE, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: (((A2 = B3) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => (~ ((((ord_less_eq @ (set @ A)) @ B3) @ A2)))))))))).
% 2.73/1.57 thf(fact_70_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A4:(set @ A),B4:(set @ A)]: ((! [X3:A]: (((((member @ A) @ X3) @ A4) => (((member @ A) @ X3) @ B4)))))))))).
% 2.73/1.57 thf(fact_71_equalityD1, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: (((A2 = B3) => (((ord_less_eq @ (set @ A)) @ A2) @ B3))))).
% 2.73/1.57 thf(fact_72_equalityD2, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: (((A2 = B3) => (((ord_less_eq @ (set @ A)) @ B3) @ A2))))).
% 2.73/1.57 thf(fact_73_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A4:(set @ A),B4:(set @ A)]: ((! [T2:A]: (((((member @ A) @ T2) @ A4) => (((member @ A) @ T2) @ B4)))))))))).
% 2.73/1.57 thf(fact_74_subset__refl, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ A2)))).
% 2.73/1.57 thf(fact_75_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) => (Q @ X4)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.73/1.57 thf(fact_76_subset__trans, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ C3) => (((ord_less_eq @ (set @ A)) @ A2) @ C3)))))).
% 2.73/1.57 thf(fact_77_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y4:(set @ A),Z2:(set @ A)]: ((Y4 = Z2))) = (^ [A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) & (((ord_less_eq @ (set @ A)) @ B4) @ A4)))))))).
% 2.73/1.57 thf(fact_78_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X3:A]: (((P @ X3) => (Q @ X3)))))))).
% 2.73/1.57 thf(fact_79_finite, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A2:(set @ A)]: (((finite_finite2 @ A) @ A2))))))).
% 2.73/1.57 thf(fact_80_finite__set__choice, axiom, (! [B:$tType,A:$tType,A2:(set @ A),P:(A > (B > $o))]: ((((finite_finite2 @ A) @ A2) => ((! [X4:A]: (((((member @ A) @ X4) @ A2) => (? [X_1:B]: (((P @ X4) @ X_1)))))) => (? [F2:(A > B)]: ((! [X2:A]: (((((member @ A) @ X2) @ A2) => ((P @ X2) @ (F2 @ X2)))))))))))).
% 2.73/1.57 thf(fact_81_rational__preference_Osame__at__least__as__equal, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),Z:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ Y)) @ Relation) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ Relation)) => (((((prefer310429814s_good @ A) @ Z) @ Carrier) @ Relation) = ((((prefer310429814s_good @ A) @ Y) @ Carrier) @ Relation))))))).
% 2.73/1.57 thf(fact_82_rational__preference_Opref__in__at__least__as, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) => (((member @ A) @ X) @ ((((prefer310429814s_good @ A) @ Y) @ Carrier) @ Relation))))))).
% 2.73/1.57 thf(fact_83_rational__preference_Oas__good__as__sameIff, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),Z:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ A) @ Z) @ ((((prefer951318096ood_as @ A) @ Y) @ Carrier) @ Relation)) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ Y)) @ Relation) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ Relation))))))).
% 2.73/1.57 thf(fact_84_rational__preference_Oas__good__asIff, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ A) @ X) @ ((((prefer951318096ood_as @ A) @ Y) @ Carrier) @ Relation)) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ Relation))))))).
% 2.73/1.57 thf(fact_85_rational__preference_Oat__lst__asgd__not__ge, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((Carrier != (bot_bot @ (set @ A))) => ((((member @ A) @ X) @ Carrier) => ((((member @ A) @ Y) @ Carrier) => ((~ ((((member @ A) @ X) @ ((((prefer310429814s_good @ A) @ Y) @ Carrier) @ Relation)))) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ Relation))))))))))).
% 2.73/1.57 thf(fact_86_Pair__mono, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [X:A,X5:A,Y:B,Y5:B]: (((((ord_less_eq @ A) @ X) @ X5) => ((((ord_less_eq @ B) @ Y) @ Y5) => (((ord_less_eq @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ ((((product_Pair @ A) @ B) @ X5) @ Y5)))))))))).
% 2.73/1.57 thf(fact_87_finite__has__maximal2, axiom, (! [A:$tType]: (((order @ A) => (! [A2:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A2) => ((((member @ A) @ A3) @ A2) => (? [X4:A]: (((((member @ A) @ X4) @ A2) & ((((ord_less_eq @ A) @ A3) @ X4) & (! [Xa:A]: (((((member @ A) @ Xa) @ A2) => ((((ord_less_eq @ A) @ X4) @ Xa) => (X4 = Xa))))))))))))))))).
% 2.73/1.57 thf(fact_88_finite__has__minimal2, axiom, (! [A:$tType]: (((order @ A) => (! [A2:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A2) => ((((member @ A) @ A3) @ A2) => (? [X4:A]: (((((member @ A) @ X4) @ A2) & ((((ord_less_eq @ A) @ X4) @ A3) & (! [Xa:A]: (((((member @ A) @ Xa) @ A2) => ((((ord_less_eq @ A) @ Xa) @ X4) => (X4 = Xa))))))))))))))))).
% 2.73/1.57 thf(fact_89_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))))))).
% 2.73/1.57 thf(fact_90_infinite__imp__nonempty, axiom, (! [A:$tType,S:(set @ A)]: (((~ (((finite_finite2 @ A) @ S))) => (S != (bot_bot @ (set @ A))))))).
% 2.73/1.57 thf(fact_91_finite_OemptyI, axiom, (! [A:$tType]: (((finite_finite2 @ A) @ (bot_bot @ (set @ A)))))).
% 2.73/1.57 thf(fact_92_finite__subset, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => (((finite_finite2 @ A) @ B3) => ((finite_finite2 @ A) @ A2)))))).
% 2.73/1.57 thf(fact_93_infinite__super, axiom, (! [A:$tType,S:(set @ A),T3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S) @ T3) => ((~ (((finite_finite2 @ A) @ S))) => (~ (((finite_finite2 @ A) @ T3)))))))).
% 2.73/1.57 thf(fact_94_rev__finite__subset, axiom, (! [A:$tType,B3:(set @ A),A2:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((finite_finite2 @ A) @ A2)))))).
% 2.73/1.57 thf(fact_95_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A3:A,B2:B,A5:A,B5:B]: (((((((product_Pair @ A) @ B) @ A3) @ B2) = ((((product_Pair @ A) @ B) @ A5) @ B5)) = ((A3 = A5) & (B2 = B5)))))).
% 2.73/1.57 thf(fact_96_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y22:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y22)) = ((X1 = Y1) & (X22 = Y22)))))).
% 2.73/1.57 thf(fact_97_bot__apply, axiom, (! [C:$tType,D2:$tType]: (((bot @ C) => ((bot_bot @ (D2 > C)) = (^ [X3:D2]: ((bot_bot @ C)))))))).
% 2.73/1.57 thf(fact_98_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.73/1.57 thf(fact_99_preorder__on__empty, axiom, (! [A:$tType]: ((((order_preorder_on @ A) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 2.73/1.57 thf(fact_100_total__on__def, axiom, (! [A:$tType]: (((total_on @ A) = (^ [A4:(set @ A),R:(set @ ((product_prod @ A) @ A))]: ((! [X3:A]: (((((member @ A) @ X3) @ A4) => (! [Y2:A]: (((((member @ A) @ Y2) @ A4) => ((X3 != Y2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y2)) @ R) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ X3)) @ R))))))))))))))).
% 2.73/1.57 thf(fact_101_total__onI, axiom, (! [A:$tType,A2:(set @ A),R2:(set @ ((product_prod @ A) @ A))]: (((! [X4:A,Y3:A]: (((((member @ A) @ X4) @ A2) => ((((member @ A) @ Y3) @ A2) => ((X4 != Y3) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y3)) @ R2) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X4)) @ R2))))))) => (((total_on @ A) @ A2) @ R2))))).
% 2.73/1.57 thf(fact_102_total__on__empty, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: ((((total_on @ A) @ (bot_bot @ (set @ A))) @ R2)))).
% 2.73/1.57 thf(fact_103_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X3:A]: ((((member @ A) @ X3) @ (bot_bot @ (set @ A))))))))).
% 2.73/1.57 thf(fact_104_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 2.73/1.57 thf(fact_105_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_less_eq @ A) @ A3) @ B2) => (A3 = B2))))))))).
% 2.73/1.57 thf(fact_106_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [A6:A,B6:A]: (((((ord_less_eq @ A) @ B6) @ A6) & (((ord_less_eq @ A) @ A6) @ B6))))))))).
% 2.73/1.57 thf(fact_107_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ C2) @ A3))))))))).
% 2.73/1.57 thf(fact_108_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A3:A,B2:A]: (((! [A7:A,B7:A]: (((((ord_less_eq @ A) @ A7) @ B7) => ((P @ A7) @ B7)))) => ((! [A7:A,B7:A]: ((((P @ B7) @ A7) => ((P @ A7) @ B7)))) => ((P @ A3) @ B2))))))))).
% 2.73/1.57 thf(fact_109_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 2.73/1.57 thf(fact_110_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less_eq @ A) @ X) @ Z))))))))).
% 2.73/1.57 thf(fact_111_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ B2) @ A3) => (A3 = B2))))))))).
% 2.73/1.57 thf(fact_112_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((B2 = C2) => (((ord_less_eq @ A) @ A3) @ C2))))))))).
% 2.73/1.57 thf(fact_113_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B2:A,C2:A]: (((A3 = B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A3) @ C2))))))))).
% 2.73/1.57 thf(fact_114_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [A6:A,B6:A]: (((((ord_less_eq @ A) @ A6) @ B6) & (((ord_less_eq @ A) @ B6) @ A6))))))))).
% 2.73/1.57 thf(fact_115_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.73/1.57 thf(fact_116_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A,Z:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z)))) => (((((ord_less_eq @ A) @ X) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y)))) => (((((ord_less_eq @ A) @ Z) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X)))) => (~ (((((ord_less_eq @ A) @ Z) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 2.73/1.57 thf(fact_117_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A3) @ C2))))))))).
% 2.73/1.57 thf(fact_118_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.73/1.57 thf(fact_119_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.73/1.57 thf(fact_120_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.73/1.57 thf(fact_121_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 2.73/1.57 thf(fact_122_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) & (((ord_less_eq @ A) @ Y2) @ X3))))))))).
% 2.73/1.57 thf(fact_123_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,B2:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A3) @ B2) => (((F @ B2) = C2) => ((! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => (((ord_less_eq @ B) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less_eq @ B) @ (F @ A3)) @ C2)))))))))).
% 2.73/1.57 thf(fact_124_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,F:(B > A),B2:B,C2:B]: (((A3 = (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X4:B,Y3:B]: (((((ord_less_eq @ B) @ X4) @ Y3) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A3) @ (F @ C2))))))))))).
% 2.73/1.57 thf(fact_125_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B2:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ C) @ (F @ B2)) @ C2) => ((! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => (((ord_less_eq @ C) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less_eq @ C) @ (F @ A3)) @ C2)))))))))).
% 2.73/1.57 thf(fact_126_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B2:B,C2:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X4:B,Y3:B]: (((((ord_less_eq @ B) @ X4) @ Y3) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A3) @ (F @ C2))))))))))).
% 2.73/1.57 thf(fact_127_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F3:(A > B),G2:(A > B)]: ((! [X3:A]: ((((ord_less_eq @ B) @ (F3 @ X3)) @ (G2 @ X3))))))))))).
% 2.73/1.57 thf(fact_128_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X4:A]: ((((ord_less_eq @ B) @ (F @ X4)) @ (G @ X4)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.73/1.57 thf(fact_129_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.73/1.57 thf(fact_130_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.73/1.57 thf(fact_131_bot__fun__def, axiom, (! [B:$tType,A:$tType]: (((bot @ B) => ((bot_bot @ (A > B)) = (^ [X3:A]: ((bot_bot @ B)))))))).
% 2.73/1.57 thf(fact_132_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X4:A,Y3:B]: ((P2 = ((((product_Pair @ A) @ B) @ X4) @ Y3))))))).
% 2.73/1.57 thf(fact_133_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A7:A,B7:B]: ((P @ ((((product_Pair @ A) @ B) @ A7) @ B7)))) => (P @ P2))))).
% 2.73/1.57 thf(fact_134_Pair__inject, axiom, (! [A:$tType,B:$tType,A3:A,B2:B,A5:A,B5:B]: (((((((product_Pair @ A) @ B) @ A3) @ B2) = ((((product_Pair @ A) @ B) @ A5) @ B5)) => (~ (((A3 = A5) => (B2 != B5)))))))).
% 2.73/1.57 thf(fact_135_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A7:A,B7:B,C4:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A7) @ ((((product_Pair @ B) @ C) @ B7) @ C4)))))))))).
% 2.73/1.57 thf(fact_136_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2)))]: ((~ ((! [A7:A,B7:B,C4:C,D3:D2]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) @ A7) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D2)) @ B7) @ ((((product_Pair @ C) @ D2) @ C4) @ D3))))))))))).
% 2.73/1.57 thf(fact_137_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))))]: ((~ ((! [A7:A,B7:B,C4:C,D3:D2,E2:E]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E)))) @ A7) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ E)) @ C4) @ ((((product_Pair @ D2) @ E) @ D3) @ E2)))))))))))).
% 2.73/1.57 thf(fact_138_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E:$tType,F4:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F4)))))]: ((~ ((! [A7:A,B7:B,C4:C,D3:D2,E2:E,F2:F4]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F4))))) @ A7) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F4)))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F4))) @ C4) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ F4)) @ D3) @ ((((product_Pair @ E) @ F4) @ E2) @ F2))))))))))))).
% 2.73/1.57 thf(fact_139_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E:$tType,F4:$tType,G3:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F4) @ G3))))))]: ((~ ((! [A7:A,B7:B,C4:C,D3:D2,E2:E,F2:F4,G4:G3]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F4) @ G3)))))) @ A7) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F4) @ G3))))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F4) @ G3)))) @ C4) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ ((product_prod @ F4) @ G3))) @ D3) @ ((((product_Pair @ E) @ ((product_prod @ F4) @ G3)) @ E2) @ ((((product_Pair @ F4) @ G3) @ F2) @ G4)))))))))))))).
% 2.73/1.57 thf(fact_140_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A7:A,B7:B,C4:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A7) @ ((((product_Pair @ B) @ C) @ B7) @ C4))))) => (P @ X))))).
% 2.73/1.57 thf(fact_141_prod__induct4, axiom, (! [D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2)))]: (((! [A7:A,B7:B,C4:C,D3:D2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) @ A7) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D2)) @ B7) @ ((((product_Pair @ C) @ D2) @ C4) @ D3)))))) => (P @ X))))).
% 2.73/1.57 thf(fact_142_prod__induct5, axiom, (! [E:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))))]: (((! [A7:A,B7:B,C4:C,D3:D2,E2:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E)))) @ A7) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ E)) @ C4) @ ((((product_Pair @ D2) @ E) @ D3) @ E2))))))) => (P @ X))))).
% 2.73/1.57 thf(fact_143_prod__induct6, axiom, (! [F4:$tType,E:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F4))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F4)))))]: (((! [A7:A,B7:B,C4:C,D3:D2,E2:E,F2:F4]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F4))))) @ A7) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F4)))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F4))) @ C4) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ F4)) @ D3) @ ((((product_Pair @ E) @ F4) @ E2) @ F2)))))))) => (P @ X))))).
% 2.73/1.57 thf(fact_144_prod__induct7, axiom, (! [G3:$tType,F4:$tType,E:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F4) @ G3)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F4) @ G3))))))]: (((! [A7:A,B7:B,C4:C,D3:D2,E2:E,F2:F4,G4:G3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F4) @ G3)))))) @ A7) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F4) @ G3))))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F4) @ G3)))) @ C4) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ ((product_prod @ F4) @ G3))) @ D3) @ ((((product_Pair @ E) @ ((product_prod @ F4) @ G3)) @ E2) @ ((((product_Pair @ F4) @ G3) @ F2) @ G4))))))))) => (P @ X))))).
% 2.73/1.57 thf(fact_145_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A7:A,B7:B]: ((Y != ((((product_Pair @ A) @ B) @ A7) @ B7))))))))).
% 2.73/1.57 thf(fact_146_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A7:A,B7:B]: ((P @ ((((product_Pair @ A) @ B) @ A7) @ B7)))) => (P @ Prod))))).
% 2.73/1.57 thf(fact_147_subrelI, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B)),S2:(set @ ((product_prod @ A) @ B))]: (((! [X4:A,Y3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y3)) @ R2) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y3)) @ S2)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R2) @ S2))))).
% 2.73/1.57 thf(fact_148_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) => (A3 = (bot_bot @ A))))))))).
% 2.73/1.57 thf(fact_149_bot_Oextremum__unique, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) = (A3 = (bot_bot @ A))))))))).
% 2.73/1.57 thf(fact_150_bot_Oextremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A3))))))).
% 2.73/1.57 thf(fact_151_refl__on__empty, axiom, (! [A:$tType]: ((((refl_on @ A) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 2.73/1.57 thf(fact_152_refl__onD, axiom, (! [A:$tType,A2:(set @ A),R2:(set @ ((product_prod @ A) @ A)),A3:A]: (((((refl_on @ A) @ A2) @ R2) => ((((member @ A) @ A3) @ A2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ A3)) @ R2)))))).
% 2.73/1.57 thf(fact_153_refl__onD1, axiom, (! [A:$tType,A2:(set @ A),R2:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((refl_on @ A) @ A2) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R2) => (((member @ A) @ X) @ A2)))))).
% 2.73/1.57 thf(fact_154_refl__onD2, axiom, (! [A:$tType,A2:(set @ A),R2:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((refl_on @ A) @ A2) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R2) => (((member @ A) @ Y) @ A2)))))).
% 2.73/1.57 thf(fact_155_refl__on__domain, axiom, (! [A:$tType,A2:(set @ A),R2:(set @ ((product_prod @ A) @ A)),A3:A,B2:A]: (((((refl_on @ A) @ A2) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R2) => ((((member @ A) @ A3) @ A2) & (((member @ A) @ B2) @ A2))))))).
% 2.73/1.57 thf(fact_156_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A3:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) = ((F1 @ A3) @ B2))))).
% 2.73/1.57 thf(fact_157_rational__preference__def, axiom, (! [A:$tType]: (((prefer1997167224erence @ A) = (^ [Carrier2:(set @ A),Relation2:(set @ ((product_prod @ A) @ A))]: (((((prefer199794634erence @ A) @ Carrier2) @ Relation2) & (((prefer1801827867axioms @ A) @ Carrier2) @ Relation2)))))))).
% 2.73/1.57 thf(fact_158_rational__preference_Ointro, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer199794634erence @ A) @ Carrier) @ Relation) => ((((prefer1801827867axioms @ A) @ Carrier) @ Relation) => (((prefer1997167224erence @ A) @ Carrier) @ Relation)))))).
% 2.73/1.57 thf(fact_159_finite__transitivity__chain, axiom, (! [A:$tType,A2:(set @ A),R3:(A > (A > $o))]: ((((finite_finite2 @ A) @ A2) => ((! [X4:A]: ((~ (((R3 @ X4) @ X4))))) => ((! [X4:A,Y3:A,Z3:A]: ((((R3 @ X4) @ Y3) => (((R3 @ Y3) @ Z3) => ((R3 @ X4) @ Z3))))) => ((! [X4:A]: (((((member @ A) @ X4) @ A2) => (? [Y6:A]: (((((member @ A) @ Y6) @ A2) & ((R3 @ X4) @ Y6))))))) => (A2 = (bot_bot @ (set @ A)))))))))).
% 2.73/1.57 thf(fact_160_subset__emptyI, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X4:A]: ((~ ((((member @ A) @ X4) @ A2))))) => (((ord_less_eq @ (set @ A)) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.73/1.57 thf(fact_161_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R2:A,S2:B,R3:(set @ ((product_prod @ A) @ B)),S3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R2) @ S2)) @ R3) => ((S3 = S2) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R2) @ S3)) @ R3)))))).
% 2.73/1.57 thf(fact_162_rational__preference__axioms_Ointro, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((total_on @ A) @ Carrier) @ Relation) => (((prefer1801827867axioms @ A) @ Carrier) @ Relation))))).
% 2.73/1.57 thf(fact_163_rational__preference__axioms__def, axiom, (! [A:$tType]: (((prefer1801827867axioms @ A) = (total_on @ A))))).
% 2.73/1.57 thf(fact_164_rational__preference_Oaxioms_I2_J, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((prefer1801827867axioms @ A) @ Carrier) @ Relation))))).
% 2.73/1.57 thf(fact_165_Collect__empty__eq__bot, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (P = (bot_bot @ (A > $o))))))).
% 2.73/1.57 thf(fact_166_less__by__empty, axiom, (! [A:$tType,A2:(set @ ((product_prod @ A) @ A)),B3:(set @ ((product_prod @ A) @ A))]: (((A2 = (bot_bot @ (set @ ((product_prod @ A) @ A)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ A2) @ B3))))).
% 2.73/1.57 thf(fact_167_finite__indexed__bound, axiom, (! [B:$tType,A:$tType]: (((linorder @ A) => (! [A2:(set @ B),P:(B > (A > $o))]: ((((finite_finite2 @ B) @ A2) => ((! [X4:B]: (((((member @ B) @ X4) @ A2) => (? [X_1:A]: (((P @ X4) @ X_1)))))) => (? [M:A]: ((! [X2:B]: (((((member @ B) @ X2) @ A2) => (? [K:A]: (((((ord_less_eq @ A) @ K) @ M) & ((P @ X2) @ K))))))))))))))))).
% 2.73/1.57 thf(fact_168_transitivity, axiom, ((trans @ a) @ relation)).
% 2.73/1.57 thf(fact_169_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A4:(set @ A)]: ((A4 = (bot_bot @ (set @ A))))))))).
% 2.73/1.57 thf(fact_170_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C2:(B > (C > A)),A3:B,B2:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C2) @ ((((product_Pair @ B) @ C) @ A3) @ B2)) = ((C2 @ A3) @ B2))))).
% 2.73/1.57 thf(fact_171_preference_Otransitivity, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer199794634erence @ A) @ Carrier) @ Relation) => ((trans @ A) @ Relation))))).
% 2.73/1.57 thf(fact_172_trans__empty, axiom, (! [A:$tType]: (((trans @ A) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 2.73/1.57 thf(fact_173_transD, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),X:A,Y:A,Z:A]: ((((trans @ A) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ R2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z)) @ R2))))))).
% 2.73/1.57 thf(fact_174_transE, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),X:A,Y:A,Z:A]: ((((trans @ A) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ R2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z)) @ R2))))))).
% 2.73/1.57 thf(fact_175_transI, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((! [X4:A,Y3:A,Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y3)) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ Z3)) @ R2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Z3)) @ R2))))) => ((trans @ A) @ R2))))).
% 2.73/1.57 thf(fact_176_trans__def, axiom, (! [A:$tType]: (((trans @ A) = (^ [R:(set @ ((product_prod @ A) @ A))]: ((! [X3:A,Y2:A,Z4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y2)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ Z4)) @ R) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Z4)) @ R))))))))))).
% 2.73/1.57 thf(fact_177_preorder__on__def, axiom, (! [A:$tType]: (((order_preorder_on @ A) = (^ [A4:(set @ A),R:(set @ ((product_prod @ A) @ A))]: (((((refl_on @ A) @ A4) @ R) & ((trans @ A) @ R)))))))).
% 2.73/1.57 thf(fact_178_relChain__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (((bNF_Ca1785829860lChain @ A) @ B) = (^ [R:(set @ ((product_prod @ A) @ A)),As:(A > B)]: ((! [I:A,J:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I) @ J)) @ R) => (((ord_less_eq @ B) @ (As @ I)) @ (As @ J)))))))))))).
% 2.73/1.57 thf(fact_179_Id__on__empty, axiom, (! [A:$tType]: ((((id_on @ A) @ (bot_bot @ (set @ A))) = (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 2.73/1.57 thf(fact_180_under__incr, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),A3:A,B2:A]: ((((trans @ A) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R2) => (((ord_less_eq @ (set @ A)) @ (((order_under @ A) @ R2) @ A3)) @ (((order_under @ A) @ R2) @ B2))))))).
% 2.73/1.57 thf(fact_181_Id__onI, axiom, (! [A:$tType,A3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ A2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ A3)) @ ((id_on @ A) @ A2)))))).
% 2.73/1.57 thf(fact_182_trans__Id__on, axiom, (! [A:$tType,A2:(set @ A)]: (((trans @ A) @ ((id_on @ A) @ A2))))).
% 2.73/1.57 thf(fact_183_Id__on__iff, axiom, (! [A:$tType,X:A,Y:A,A2:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((id_on @ A) @ A2)) = ((X = Y) & (((member @ A) @ X) @ A2)))))).
% 2.73/1.57 thf(fact_184_Id__on__eqI, axiom, (! [A:$tType,A3:A,B2:A,A2:(set @ A)]: (((A3 = B2) => ((((member @ A) @ A3) @ A2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((id_on @ A) @ A2))))))).
% 2.73/1.57 thf(fact_185_Id__onE, axiom, (! [A:$tType,C2:((product_prod @ A) @ A),A2:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ C2) @ ((id_on @ A) @ A2)) => (~ ((! [X4:A]: (((((member @ A) @ X4) @ A2) => (C2 != ((((product_Pair @ A) @ A) @ X4) @ X4))))))))))).
% 2.73/1.57 thf(fact_186_refl__on__Id__on, axiom, (! [A:$tType,A2:(set @ A)]: ((((refl_on @ A) @ A2) @ ((id_on @ A) @ A2))))).
% 2.73/1.57 thf(fact_187_arg__min__least, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (! [S:(set @ A),Y:A,F:(A > B)]: ((((finite_finite2 @ A) @ S) => ((S != (bot_bot @ (set @ A))) => ((((member @ A) @ Y) @ S) => (((ord_less_eq @ B) @ (F @ ((((lattic1704895705min_on @ A) @ B) @ F) @ S))) @ (F @ Y))))))))))).
% 2.73/1.57 thf(fact_188_Greatest__equality, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X:A]: (((P @ X) => ((! [Y3:A]: (((P @ Y3) => (((ord_less_eq @ A) @ Y3) @ X)))) => (((order_Greatest @ A) @ P) = X))))))))).
% 2.73/1.57 thf(fact_189_GreatestI2__order, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X:A,Q:(A > $o)]: (((P @ X) => ((! [Y3:A]: (((P @ Y3) => (((ord_less_eq @ A) @ Y3) @ X)))) => ((! [X4:A]: (((P @ X4) => ((! [Y6:A]: (((P @ Y6) => (((ord_less_eq @ A) @ Y6) @ X4)))) => (Q @ X4))))) => (Q @ ((order_Greatest @ A) @ P))))))))))).
% 2.73/1.57 thf(fact_190_arg__min__if__finite_I1_J, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (! [S:(set @ A),F:(A > B)]: ((((finite_finite2 @ A) @ S) => ((S != (bot_bot @ (set @ A))) => (((member @ A) @ ((((lattic1704895705min_on @ A) @ B) @ F) @ S)) @ S))))))))).
% 2.73/1.57 thf(fact_191_Inf__fin_Osubset__imp, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((A2 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B3) => (((ord_less_eq @ A) @ ((lattic2065218050nf_fin @ A) @ B3)) @ ((lattic2065218050nf_fin @ A) @ A2))))))))))).
% 2.73/1.57 thf(fact_192_Sup__fin_Osubset__imp, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((A2 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B3) => (((ord_less_eq @ A) @ ((lattic477160up_fin @ A) @ A2)) @ ((lattic477160up_fin @ A) @ B3))))))))))).
% 2.73/1.57 thf(fact_193_Inf__fin__le__Sup__fin, axiom, (! [A:$tType]: (((lattice @ A) => (! [A2:(set @ A)]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => (((ord_less_eq @ A) @ ((lattic2065218050nf_fin @ A) @ A2)) @ ((lattic477160up_fin @ A) @ A2)))))))))).
% 2.73/1.57 thf(fact_194_Sup__fin_OcoboundedI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A2) => ((((member @ A) @ A3) @ A2) => (((ord_less_eq @ A) @ A3) @ ((lattic477160up_fin @ A) @ A2)))))))))).
% 2.73/1.57 thf(fact_195_Inf__fin_OcoboundedI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A2) => ((((member @ A) @ A3) @ A2) => (((ord_less_eq @ A) @ ((lattic2065218050nf_fin @ A) @ A2)) @ A3))))))))).
% 2.73/1.57 thf(fact_196_Inf__fin_OboundedE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:(set @ A),X:A]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ X) @ ((lattic2065218050nf_fin @ A) @ A2)) => (! [A8:A]: (((((member @ A) @ A8) @ A2) => (((ord_less_eq @ A) @ X) @ A8))))))))))))).
% 2.73/1.57 thf(fact_197_Inf__fin_OboundedI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:(set @ A),X:A]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => ((! [A7:A]: (((((member @ A) @ A7) @ A2) => (((ord_less_eq @ A) @ X) @ A7)))) => (((ord_less_eq @ A) @ X) @ ((lattic2065218050nf_fin @ A) @ A2))))))))))).
% 2.73/1.57 thf(fact_198_Sup__fin_OboundedE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:(set @ A),X:A]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ ((lattic477160up_fin @ A) @ A2)) @ X) => (! [A8:A]: (((((member @ A) @ A8) @ A2) => (((ord_less_eq @ A) @ A8) @ X))))))))))))).
% 2.73/1.57 thf(fact_199_Sup__fin_OboundedI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:(set @ A),X:A]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => ((! [A7:A]: (((((member @ A) @ A7) @ A2) => (((ord_less_eq @ A) @ A7) @ X)))) => (((ord_less_eq @ A) @ ((lattic477160up_fin @ A) @ A2)) @ X)))))))))).
% 2.73/1.57 thf(fact_200_Inf__fin_Obounded__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:(set @ A),X:A]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ X) @ ((lattic2065218050nf_fin @ A) @ A2)) = (! [X3:A]: (((((member @ A) @ X3) @ A2) => (((ord_less_eq @ A) @ X) @ X3))))))))))))).
% 2.73/1.57 thf(fact_201_Sup__fin_Obounded__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:(set @ A),X:A]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ ((lattic477160up_fin @ A) @ A2)) @ X) = (! [X3:A]: (((((member @ A) @ X3) @ A2) => (((ord_less_eq @ A) @ X3) @ X))))))))))))).
% 2.73/1.57 thf(fact_202_Max_Osubset__imp, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((A2 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B3) => (((ord_less_eq @ A) @ ((lattic929149872er_Max @ A) @ A2)) @ ((lattic929149872er_Max @ A) @ B3))))))))))).
% 2.73/1.57 thf(fact_203_Max__mono, axiom, (! [A:$tType]: (((linorder @ A) => (! [M2:(set @ A),N:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ M2) @ N) => ((M2 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ N) => (((ord_less_eq @ A) @ ((lattic929149872er_Max @ A) @ M2)) @ ((lattic929149872er_Max @ A) @ N))))))))))).
% 2.73/1.57 thf(fact_204_Max_Obounded__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),X:A]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ ((lattic929149872er_Max @ A) @ A2)) @ X) = (! [X3:A]: (((((member @ A) @ X3) @ A2) => (((ord_less_eq @ A) @ X3) @ X))))))))))))).
% 2.73/1.57 thf(fact_205_Max_OcoboundedI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A2) => ((((member @ A) @ A3) @ A2) => (((ord_less_eq @ A) @ A3) @ ((lattic929149872er_Max @ A) @ A2)))))))))).
% 2.73/1.57 thf(fact_206_Max__eq__if, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A2) => (((finite_finite2 @ A) @ B3) => ((! [X4:A]: (((((member @ A) @ X4) @ A2) => (? [Xa:A]: (((((member @ A) @ Xa) @ B3) & (((ord_less_eq @ A) @ X4) @ Xa))))))) => ((! [X4:A]: (((((member @ A) @ X4) @ B3) => (? [Xa:A]: (((((member @ A) @ Xa) @ A2) & (((ord_less_eq @ A) @ X4) @ Xa))))))) => (((lattic929149872er_Max @ A) @ A2) = ((lattic929149872er_Max @ A) @ B3)))))))))))).
% 2.73/1.57 thf(fact_207_Max__eqI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),X:A]: ((((finite_finite2 @ A) @ A2) => ((! [Y3:A]: (((((member @ A) @ Y3) @ A2) => (((ord_less_eq @ A) @ Y3) @ X)))) => ((((member @ A) @ X) @ A2) => (((lattic929149872er_Max @ A) @ A2) = X)))))))))).
% 2.73/1.57 thf(fact_208_Max__ge, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),X:A]: ((((finite_finite2 @ A) @ A2) => ((((member @ A) @ X) @ A2) => (((ord_less_eq @ A) @ X) @ ((lattic929149872er_Max @ A) @ A2)))))))))).
% 2.73/1.57 thf(fact_209_Max__in, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A)]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => (((member @ A) @ ((lattic929149872er_Max @ A) @ A2)) @ A2))))))))).
% 2.73/1.57 thf(fact_210_Max__eq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),M3:A]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => ((((lattic929149872er_Max @ A) @ A2) = M3) = ((((member @ A) @ M3) @ A2) & (! [X3:A]: (((((member @ A) @ X3) @ A2) => (((ord_less_eq @ A) @ X3) @ M3)))))))))))))).
% 2.73/1.57 thf(fact_211_Max__ge__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),X:A]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ X) @ ((lattic929149872er_Max @ A) @ A2)) = (? [X3:A]: (((((member @ A) @ X3) @ A2) & (((ord_less_eq @ A) @ X) @ X3))))))))))))).
% 2.73/1.57 thf(fact_212_eq__Max__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),M3:A]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => ((M3 = ((lattic929149872er_Max @ A) @ A2)) = ((((member @ A) @ M3) @ A2) & (! [X3:A]: (((((member @ A) @ X3) @ A2) => (((ord_less_eq @ A) @ X3) @ M3)))))))))))))).
% 2.73/1.57 thf(fact_213_Max_OboundedE, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),X:A]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ ((lattic929149872er_Max @ A) @ A2)) @ X) => (! [A8:A]: (((((member @ A) @ A8) @ A2) => (((ord_less_eq @ A) @ A8) @ X))))))))))))).
% 2.73/1.57 thf(fact_214_Max_OboundedI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),X:A]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => ((! [A7:A]: (((((member @ A) @ A7) @ A2) => (((ord_less_eq @ A) @ A7) @ X)))) => (((ord_less_eq @ A) @ ((lattic929149872er_Max @ A) @ A2)) @ X)))))))))).
% 2.73/1.57 thf(fact_215_Min__antimono, axiom, (! [A:$tType]: (((linorder @ A) => (! [M2:(set @ A),N:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ M2) @ N) => ((M2 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ N) => (((ord_less_eq @ A) @ ((lattic929674654er_Min @ A) @ N)) @ ((lattic929674654er_Min @ A) @ M2))))))))))).
% 2.73/1.57 thf(fact_216_Min_Osubset__imp, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((A2 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B3) => (((ord_less_eq @ A) @ ((lattic929674654er_Min @ A) @ B3)) @ ((lattic929674654er_Min @ A) @ A2))))))))))).
% 2.73/1.57 thf(fact_217_Min_Obounded__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),X:A]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ X) @ ((lattic929674654er_Min @ A) @ A2)) = (! [X3:A]: (((((member @ A) @ X3) @ A2) => (((ord_less_eq @ A) @ X) @ X3))))))))))))).
% 2.73/1.57 thf(fact_218_Min__in, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A)]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => (((member @ A) @ ((lattic929674654er_Min @ A) @ A2)) @ A2))))))))).
% 2.73/1.57 thf(fact_219_Min__le, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),X:A]: ((((finite_finite2 @ A) @ A2) => ((((member @ A) @ X) @ A2) => (((ord_less_eq @ A) @ ((lattic929674654er_Min @ A) @ A2)) @ X))))))))).
% 2.73/1.57 thf(fact_220_Min__eqI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),X:A]: ((((finite_finite2 @ A) @ A2) => ((! [Y3:A]: (((((member @ A) @ Y3) @ A2) => (((ord_less_eq @ A) @ X) @ Y3)))) => ((((member @ A) @ X) @ A2) => (((lattic929674654er_Min @ A) @ A2) = X)))))))))).
% 2.73/1.57 thf(fact_221_Min_OcoboundedI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A2) => ((((member @ A) @ A3) @ A2) => (((ord_less_eq @ A) @ ((lattic929674654er_Min @ A) @ A2)) @ A3))))))))).
% 2.73/1.57 thf(fact_222_Min_OboundedI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),X:A]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => ((! [A7:A]: (((((member @ A) @ A7) @ A2) => (((ord_less_eq @ A) @ X) @ A7)))) => (((ord_less_eq @ A) @ X) @ ((lattic929674654er_Min @ A) @ A2))))))))))).
% 2.73/1.57 thf(fact_223_Min_OboundedE, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),X:A]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ X) @ ((lattic929674654er_Min @ A) @ A2)) => (! [A8:A]: (((((member @ A) @ A8) @ A2) => (((ord_less_eq @ A) @ X) @ A8))))))))))))).
% 2.73/1.57 thf(fact_224_eq__Min__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),M3:A]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => ((M3 = ((lattic929674654er_Min @ A) @ A2)) = ((((member @ A) @ M3) @ A2) & (! [X3:A]: (((((member @ A) @ X3) @ A2) => (((ord_less_eq @ A) @ M3) @ X3)))))))))))))).
% 2.73/1.57 thf(fact_225_Min__le__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),X:A]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ ((lattic929674654er_Min @ A) @ A2)) @ X) = (? [X3:A]: (((((member @ A) @ X3) @ A2) & (((ord_less_eq @ A) @ X3) @ X))))))))))))).
% 2.73/1.57 thf(fact_226_Min__eq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:(set @ A),M3:A]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => ((((lattic929674654er_Min @ A) @ A2) = M3) = ((((member @ A) @ M3) @ A2) & (! [X3:A]: (((((member @ A) @ X3) @ A2) => (((ord_less_eq @ A) @ M3) @ X3)))))))))))))).
% 2.73/1.57 thf(fact_227_Inf__fin_Osubset, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A2) => ((B3 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ (set @ A)) @ B3) @ A2) => ((((inf_inf @ A) @ ((lattic2065218050nf_fin @ A) @ B3)) @ ((lattic2065218050nf_fin @ A) @ A2)) = ((lattic2065218050nf_fin @ A) @ A2))))))))))).
% 2.73/1.57 thf(fact_228_Sup__fin_Osubset, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A2) => ((B3 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ (set @ A)) @ B3) @ A2) => ((((sup_sup @ A) @ ((lattic477160up_fin @ A) @ B3)) @ ((lattic477160up_fin @ A) @ A2)) = ((lattic477160up_fin @ A) @ A2))))))))))).
% 2.73/1.57 thf(fact_229_Un__empty, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: ((((((sup_sup @ (set @ A)) @ A2) @ B3) = (bot_bot @ (set @ A))) = ((A2 = (bot_bot @ (set @ A))) & (B3 = (bot_bot @ (set @ A)))))))).
% 2.73/1.57 thf(fact_230_Int__subset__iff, axiom, (! [A:$tType,C3:(set @ A),A2:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ C3) @ (((inf_inf @ (set @ A)) @ A2) @ B3)) = ((((ord_less_eq @ (set @ A)) @ C3) @ A2) & (((ord_less_eq @ (set @ A)) @ C3) @ B3)))))).
% 2.73/1.57 thf(fact_231_Un__subset__iff, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B3)) @ C3) = ((((ord_less_eq @ (set @ A)) @ A2) @ C3) & (((ord_less_eq @ (set @ A)) @ B3) @ C3)))))).
% 2.73/1.57 thf(fact_232_finite__Int, axiom, (! [A:$tType,F5:(set @ A),G5:(set @ A)]: (((((finite_finite2 @ A) @ F5) | ((finite_finite2 @ A) @ G5)) => ((finite_finite2 @ A) @ (((inf_inf @ (set @ A)) @ F5) @ G5)))))).
% 2.73/1.57 thf(fact_233_finite__Un, axiom, (! [A:$tType,F5:(set @ A),G5:(set @ A)]: ((((finite_finite2 @ A) @ (((sup_sup @ (set @ A)) @ F5) @ G5)) = (((finite_finite2 @ A) @ F5) & ((finite_finite2 @ A) @ G5)))))).
% 2.73/1.57 thf(fact_234_inf__Pair__Pair, axiom, (! [A:$tType,B:$tType]: ((((inf @ B) & (inf @ A)) => (! [A3:A,B2:B,C2:A,D:B]: (((((inf_inf @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ ((((product_Pair @ A) @ B) @ C2) @ D)) = ((((product_Pair @ A) @ B) @ (((inf_inf @ A) @ A3) @ C2)) @ (((inf_inf @ B) @ B2) @ D))))))))).
% 2.73/1.57 thf(fact_235_sup__Pair__Pair, axiom, (! [A:$tType,B:$tType]: ((((sup @ B) & (sup @ A)) => (! [A3:A,B2:B,C2:A,D:B]: (((((sup_sup @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ ((((product_Pair @ A) @ B) @ C2) @ D)) = ((((product_Pair @ A) @ B) @ (((sup_sup @ A) @ A3) @ C2)) @ (((sup_sup @ B) @ B2) @ D))))))))).
% 2.73/1.57 thf(fact_236_inf__Sup__absorb, axiom, (! [A:$tType]: (((lattice @ A) => (! [A2:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A2) => ((((member @ A) @ A3) @ A2) => ((((inf_inf @ A) @ A3) @ ((lattic477160up_fin @ A) @ A2)) = A3))))))))).
% 2.73/1.57 thf(fact_237_sup__Inf__absorb, axiom, (! [A:$tType]: (((lattice @ A) => (! [A2:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A2) => ((((member @ A) @ A3) @ A2) => ((((sup_sup @ A) @ ((lattic2065218050nf_fin @ A) @ A2)) @ A3) = A3))))))))).
% 2.73/1.57 thf(fact_238_Sup__fin_Oin__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:(set @ A),X:A]: ((((finite_finite2 @ A) @ A2) => ((((member @ A) @ X) @ A2) => ((((sup_sup @ A) @ X) @ ((lattic477160up_fin @ A) @ A2)) = ((lattic477160up_fin @ A) @ A2)))))))))).
% 2.73/1.57 thf(fact_239_Inf__fin_Oin__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:(set @ A),X:A]: ((((finite_finite2 @ A) @ A2) => ((((member @ A) @ X) @ A2) => ((((inf_inf @ A) @ X) @ ((lattic2065218050nf_fin @ A) @ A2)) = ((lattic2065218050nf_fin @ A) @ A2)))))))))).
% 2.73/1.57 thf(fact_240_Sup__fin_Ounion, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B3) => ((B3 != (bot_bot @ (set @ A))) => (((lattic477160up_fin @ A) @ (((sup_sup @ (set @ A)) @ A2) @ B3)) = (((sup_sup @ A) @ ((lattic477160up_fin @ A) @ A2)) @ ((lattic477160up_fin @ A) @ B3))))))))))))).
% 2.73/1.57 thf(fact_241_Inf__fin_Ounion, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B3) => ((B3 != (bot_bot @ (set @ A))) => (((lattic2065218050nf_fin @ A) @ (((sup_sup @ (set @ A)) @ A2) @ B3)) = (((inf_inf @ A) @ ((lattic2065218050nf_fin @ A) @ A2)) @ ((lattic2065218050nf_fin @ A) @ B3))))))))))))).
% 2.73/1.57 thf(fact_242_trans__Int, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),S2:(set @ ((product_prod @ A) @ A))]: ((((trans @ A) @ R2) => (((trans @ A) @ S2) => ((trans @ A) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R2) @ S2))))))).
% 2.73/1.57 thf(fact_243_refl__on__Un, axiom, (! [A:$tType,A2:(set @ A),R2:(set @ ((product_prod @ A) @ A)),B3:(set @ A),S2:(set @ ((product_prod @ A) @ A))]: (((((refl_on @ A) @ A2) @ R2) => ((((refl_on @ A) @ B3) @ S2) => (((refl_on @ A) @ (((sup_sup @ (set @ A)) @ A2) @ B3)) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ R2) @ S2))))))).
% 2.73/1.57 thf(fact_244_refl__on__Int, axiom, (! [A:$tType,A2:(set @ A),R2:(set @ ((product_prod @ A) @ A)),B3:(set @ A),S2:(set @ ((product_prod @ A) @ A))]: (((((refl_on @ A) @ A2) @ R2) => ((((refl_on @ A) @ B3) @ S2) => (((refl_on @ A) @ (((inf_inf @ (set @ A)) @ A2) @ B3)) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R2) @ S2))))))).
% 2.73/1.57 thf(fact_245_finite__UnI, axiom, (! [A:$tType,F5:(set @ A),G5:(set @ A)]: ((((finite_finite2 @ A) @ F5) => (((finite_finite2 @ A) @ G5) => ((finite_finite2 @ A) @ (((sup_sup @ (set @ A)) @ F5) @ G5))))))).
% 2.73/1.57 thf(fact_246_Un__infinite, axiom, (! [A:$tType,S:(set @ A),T3:(set @ A)]: (((~ (((finite_finite2 @ A) @ S))) => (~ (((finite_finite2 @ A) @ (((sup_sup @ (set @ A)) @ S) @ T3)))))))).
% 2.73/1.57 thf(fact_247_infinite__Un, axiom, (! [A:$tType,S:(set @ A),T3:(set @ A)]: (((~ (((finite_finite2 @ A) @ (((sup_sup @ (set @ A)) @ S) @ T3)))) = ((~ (((finite_finite2 @ A) @ S))) | (~ (((finite_finite2 @ A) @ T3)))))))).
% 2.73/1.57 thf(fact_248_Un__Int__assoc__eq, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A),C3:(set @ A)]: ((((((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A2) @ B3)) @ C3) = (((inf_inf @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ B3) @ C3))) = (((ord_less_eq @ (set @ A)) @ C3) @ A2))))).
% 2.73/1.57 thf(fact_249_Int__mono, axiom, (! [A:$tType,A2:(set @ A),C3:(set @ A),B3:(set @ A),D4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ C3) => ((((ord_less_eq @ (set @ A)) @ B3) @ D4) => (((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A2) @ B3)) @ (((inf_inf @ (set @ A)) @ C3) @ D4))))))).
% 2.73/1.57 thf(fact_250_Int__lower1, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A2) @ B3)) @ A2)))).
% 2.73/1.57 thf(fact_251_Int__lower2, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A2) @ B3)) @ B3)))).
% 2.73/1.57 thf(fact_252_Int__absorb1, axiom, (! [A:$tType,B3:(set @ A),A2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B3) @ A2) => ((((inf_inf @ (set @ A)) @ A2) @ B3) = B3))))).
% 2.73/1.57 thf(fact_253_Int__absorb2, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((((inf_inf @ (set @ A)) @ A2) @ B3) = A2))))).
% 2.73/1.57 thf(fact_254_Int__greatest, axiom, (! [A:$tType,C3:(set @ A),A2:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ C3) @ A2) => ((((ord_less_eq @ (set @ A)) @ C3) @ B3) => (((ord_less_eq @ (set @ A)) @ C3) @ (((inf_inf @ (set @ A)) @ A2) @ B3))))))).
% 2.73/1.57 thf(fact_255_Int__Collect__mono, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A),P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((! [X4:A]: (((((member @ A) @ X4) @ A2) => ((P @ X4) => (Q @ X4))))) => (((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A2) @ ((collect @ A) @ P))) @ (((inf_inf @ (set @ A)) @ B3) @ ((collect @ A) @ Q)))))))).
% 2.73/1.57 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A9:$tType,A10:$tType]: (((semilattice_sup @ A10) => (semilattice_sup @ (A9 > A10)))))).
% 2.73/1.57 thf(tcon_fun___Lattices_Osemilattice__inf, axiom, (! [A9:$tType,A10:$tType]: (((semilattice_inf @ A10) => (semilattice_inf @ (A9 > A10)))))).
% 2.73/1.57 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A9:$tType,A10:$tType]: (((order_bot @ A10) => (order_bot @ (A9 > A10)))))).
% 2.73/1.57 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A9:$tType,A10:$tType]: (((preorder @ A10) => (preorder @ (A9 > A10)))))).
% 2.73/1.57 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A9:$tType,A10:$tType]: ((((finite_finite @ A9) & (finite_finite @ A10)) => (finite_finite @ (A9 > A10)))))).
% 2.73/1.57 thf(tcon_fun___Lattices_Olattice, axiom, (! [A9:$tType,A10:$tType]: (((lattice @ A10) => (lattice @ (A9 > A10)))))).
% 2.73/1.57 thf(tcon_fun___Orderings_Oorder, axiom, (! [A9:$tType,A10:$tType]: (((order @ A10) => (order @ (A9 > A10)))))).
% 2.73/1.57 thf(tcon_fun___Orderings_Oord, axiom, (! [A9:$tType,A10:$tType]: (((ord @ A10) => (ord @ (A9 > A10)))))).
% 2.73/1.57 thf(tcon_fun___Orderings_Obot, axiom, (! [A9:$tType,A10:$tType]: (((bot @ A10) => (bot @ (A9 > A10)))))).
% 2.73/1.57 thf(tcon_fun___Lattices_Osup, axiom, (! [A9:$tType,A10:$tType]: (((semilattice_sup @ A10) => (sup @ (A9 > A10)))))).
% 2.73/1.57 thf(tcon_fun___Lattices_Oinf, axiom, (! [A9:$tType,A10:$tType]: (((semilattice_inf @ A10) => (inf @ (A9 > A10)))))).
% 2.73/1.57 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1, axiom, (! [A9:$tType]: ((semilattice_sup @ (set @ A9))))).
% 2.73/1.57 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_2, axiom, (! [A9:$tType]: ((semilattice_inf @ (set @ A9))))).
% 2.73/1.57 thf(tcon_Set_Oset___Orderings_Oorder__bot_3, axiom, (! [A9:$tType]: ((order_bot @ (set @ A9))))).
% 2.73/1.57 thf(tcon_Set_Oset___Orderings_Opreorder_4, axiom, (! [A9:$tType]: ((preorder @ (set @ A9))))).
% 2.73/1.57 thf(tcon_Set_Oset___Finite__Set_Ofinite_5, axiom, (! [A9:$tType]: (((finite_finite @ A9) => (finite_finite @ (set @ A9)))))).
% 2.73/1.57 thf(tcon_Set_Oset___Lattices_Olattice_6, axiom, (! [A9:$tType]: ((lattice @ (set @ A9))))).
% 2.73/1.57 thf(tcon_Set_Oset___Orderings_Oorder_7, axiom, (! [A9:$tType]: ((order @ (set @ A9))))).
% 2.73/1.57 thf(tcon_Set_Oset___Orderings_Oord_8, axiom, (! [A9:$tType]: ((ord @ (set @ A9))))).
% 2.73/1.57 thf(tcon_Set_Oset___Orderings_Obot_9, axiom, (! [A9:$tType]: ((bot @ (set @ A9))))).
% 2.73/1.57 thf(tcon_Set_Oset___Lattices_Osup_10, axiom, (! [A9:$tType]: ((sup @ (set @ A9))))).
% 2.73/1.57 thf(tcon_Set_Oset___Lattices_Oinf_11, axiom, (! [A9:$tType]: ((inf @ (set @ A9))))).
% 2.73/1.57 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_12, axiom, (semilattice_sup @ $o)).
% 2.73/1.57 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_13, axiom, (semilattice_inf @ $o)).
% 2.73/1.57 thf(tcon_HOL_Obool___Orderings_Oorder__bot_14, axiom, (order_bot @ $o)).
% 2.73/1.57 thf(tcon_HOL_Obool___Orderings_Opreorder_15, axiom, (preorder @ $o)).
% 2.73/1.57 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 2.73/1.57 thf(tcon_HOL_Obool___Finite__Set_Ofinite_16, axiom, (finite_finite @ $o)).
% 2.73/1.57 thf(tcon_HOL_Obool___Lattices_Olattice_17, axiom, (lattice @ $o)).
% 2.73/1.57 thf(tcon_HOL_Obool___Orderings_Oorder_18, axiom, (order @ $o)).
% 2.73/1.57 thf(tcon_HOL_Obool___Orderings_Oord_19, axiom, (ord @ $o)).
% 2.73/1.57 thf(tcon_HOL_Obool___Orderings_Obot_20, axiom, (bot @ $o)).
% 2.73/1.57 thf(tcon_HOL_Obool___Lattices_Osup_21, axiom, (sup @ $o)).
% 2.73/1.57 thf(tcon_HOL_Obool___Lattices_Oinf_22, axiom, (inf @ $o)).
% 2.73/1.57 thf(tcon_Product__Type_Oprod___Lattices_Osemilattice__sup_23, axiom, (! [A9:$tType,A10:$tType]: ((((semilattice_sup @ A9) & (semilattice_sup @ A10)) => (semilattice_sup @ ((product_prod @ A9) @ A10)))))).
% 2.73/1.57 thf(tcon_Product__Type_Oprod___Lattices_Osemilattice__inf_24, axiom, (! [A9:$tType,A10:$tType]: ((((semilattice_inf @ A9) & (semilattice_inf @ A10)) => (semilattice_inf @ ((product_prod @ A9) @ A10)))))).
% 2.73/1.57 thf(tcon_Product__Type_Oprod___Orderings_Oorder__bot_25, axiom, (! [A9:$tType,A10:$tType]: ((((order_bot @ A9) & (order_bot @ A10)) => (order_bot @ ((product_prod @ A9) @ A10)))))).
% 2.73/1.57 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_26, axiom, (! [A9:$tType,A10:$tType]: ((((preorder @ A9) & (preorder @ A10)) => (preorder @ ((product_prod @ A9) @ A10)))))).
% 2.73/1.57 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_27, axiom, (! [A9:$tType,A10:$tType]: ((((finite_finite @ A9) & (finite_finite @ A10)) => (finite_finite @ ((product_prod @ A9) @ A10)))))).
% 2.73/1.57 thf(tcon_Product__Type_Oprod___Lattices_Olattice_28, axiom, (! [A9:$tType,A10:$tType]: ((((lattice @ A9) & (lattice @ A10)) => (lattice @ ((product_prod @ A9) @ A10)))))).
% 2.73/1.57 thf(tcon_Product__Type_Oprod___Orderings_Oorder_29, axiom, (! [A9:$tType,A10:$tType]: ((((order @ A9) & (order @ A10)) => (order @ ((product_prod @ A9) @ A10)))))).
% 2.73/1.57 thf(tcon_Product__Type_Oprod___Orderings_Oord_30, axiom, (! [A9:$tType,A10:$tType]: ((((ord @ A9) & (ord @ A10)) => (ord @ ((product_prod @ A9) @ A10)))))).
% 2.73/1.57 thf(tcon_Product__Type_Oprod___Orderings_Obot_31, axiom, (! [A9:$tType,A10:$tType]: ((((bot @ A9) & (bot @ A10)) => (bot @ ((product_prod @ A9) @ A10)))))).
% 2.73/1.57 thf(tcon_Product__Type_Oprod___Lattices_Osup_32, axiom, (! [A9:$tType,A10:$tType]: ((((sup @ A9) & (sup @ A10)) => (sup @ ((product_prod @ A9) @ A10)))))).
% 2.73/1.57 thf(tcon_Product__Type_Oprod___Lattices_Oinf_33, axiom, (! [A9:$tType,A10:$tType]: ((((inf @ A9) & (inf @ A10)) => (inf @ ((product_prod @ A9) @ A10)))))).
% 2.73/1.57 thf(conj_0, conjecture, ((((ord_less_eq @ (set @ a)) @ ((((prefer1532642881r_than @ a) @ x) @ carrier) @ relation)) @ ((((prefer1532642881r_than @ a) @ x) @ carrier) @ relation)) | (((ord_less_eq @ (set @ a)) @ ((((prefer1532642881r_than @ a) @ x) @ carrier) @ relation)) @ ((((prefer1532642881r_than @ a) @ x) @ carrier) @ relation)))).
% 2.73/1.57 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.bwuTtVduWu/DTF2DTF22979.p
% 2.73/1.57 ------------------------
% 2.90/1.63 ---- Cleaned THF ---
% 2.90/1.63 thf(ty_t_Product__Type_Oprod,type,
% 2.90/1.63 product_prod: $tType > $tType > $tType ).
% 2.90/1.63
% 2.90/1.63 thf(ty_t_Set_Oset,type,
% 2.90/1.63 set: $tType > $tType ).
% 2.90/1.63
% 2.90/1.63 thf(ty_tf_a,type,
% 2.90/1.63 a: $tType ).
% 2.90/1.63
% 2.90/1.63 thf(sy_cl_HOL_Otype,type,
% 2.90/1.63 type:
% 2.90/1.63 !>[A: $tType] : $o ).
% 2.90/1.63
% 2.90/1.63 thf(sy_cl_Lattices_Oinf,type,
% 2.90/1.63 inf:
% 2.90/1.63 !>[A: $tType] : $o ).
% 2.90/1.63
% 2.90/1.63 thf(sy_cl_Lattices_Osup,type,
% 2.90/1.63 sup:
% 2.90/1.63 !>[A: $tType] : $o ).
% 2.90/1.63
% 2.90/1.63 thf(sy_cl_Orderings_Obot,type,
% 2.90/1.63 bot:
% 2.90/1.63 !>[A: $tType] : $o ).
% 2.90/1.63
% 2.90/1.63 thf(sy_cl_Orderings_Oord,type,
% 2.90/1.63 ord:
% 2.90/1.63 !>[A: $tType] : $o ).
% 2.90/1.63
% 2.90/1.63 thf(sy_cl_Orderings_Oorder,type,
% 2.90/1.63 order:
% 2.90/1.63 !>[A: $tType] : $o ).
% 2.90/1.63
% 2.90/1.63 thf(sy_cl_Lattices_Olattice,type,
% 2.90/1.63 lattice:
% 2.90/1.63 !>[A: $tType] : $o ).
% 2.90/1.63
% 2.90/1.63 thf(sy_cl_Finite__Set_Ofinite,type,
% 2.90/1.63 finite_finite:
% 2.90/1.63 !>[A: $tType] : $o ).
% 2.90/1.63
% 2.90/1.63 thf(sy_cl_Orderings_Olinorder,type,
% 2.90/1.63 linorder:
% 2.90/1.63 !>[A: $tType] : $o ).
% 2.90/1.63
% 2.90/1.63 thf(sy_cl_Orderings_Opreorder,type,
% 2.90/1.63 preorder:
% 2.90/1.63 !>[A: $tType] : $o ).
% 2.90/1.63
% 2.90/1.63 thf(sy_cl_Orderings_Oorder__bot,type,
% 2.90/1.63 order_bot:
% 2.90/1.63 !>[A: $tType] : $o ).
% 2.90/1.63
% 2.90/1.63 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 2.90/1.63 semilattice_inf:
% 2.90/1.63 !>[A: $tType] : $o ).
% 2.90/1.63
% 2.90/1.63 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 2.90/1.63 semilattice_sup:
% 2.90/1.63 !>[A: $tType] : $o ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 2.90/1.63 bNF_Ca1785829860lChain:
% 2.90/1.63 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Finite__Set_Ofinite,type,
% 2.90/1.63 finite_finite2:
% 2.90/1.63 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 2.90/1.63 inf_inf:
% 2.90/1.63 !>[A: $tType] : ( A > A > A ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Lattices_Osup__class_Osup,type,
% 2.90/1.63 sup_sup:
% 2.90/1.63 !>[A: $tType] : ( A > A > A ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
% 2.90/1.63 lattic929149872er_Max:
% 2.90/1.63 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Lattices__Big_Olinorder__class_OMin,type,
% 2.90/1.63 lattic929674654er_Min:
% 2.90/1.63 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
% 2.90/1.63 lattic1704895705min_on:
% 2.90/1.63 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
% 2.90/1.63 lattic2065218050nf_fin:
% 2.90/1.63 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
% 2.90/1.63 lattic477160up_fin:
% 2.90/1.63 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Order__Relation_Opreorder__on,type,
% 2.90/1.63 order_preorder_on:
% 2.90/1.63 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Order__Relation_Ounder,type,
% 2.90/1.63 order_under:
% 2.90/1.63 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.90/1.63 bot_bot:
% 2.90/1.63 !>[A: $tType] : A ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.90/1.63 ord_less_eq:
% 2.90/1.63 !>[A: $tType] : ( A > A > $o ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 2.90/1.63 order_Greatest:
% 2.90/1.63 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oas__good__as,type,
% 2.90/1.63 prefer951318096ood_as:
% 2.90/1.63 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oat__least__as__good,type,
% 2.90/1.63 prefer310429814s_good:
% 2.90/1.63 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Ono__better__than,type,
% 2.90/1.63 prefer1532642881r_than:
% 2.90/1.63 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Opreference,type,
% 2.90/1.63 prefer199794634erence:
% 2.90/1.63 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference,type,
% 2.90/1.63 prefer1997167224erence:
% 2.90/1.63 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference__axioms,type,
% 2.90/1.63 prefer1801827867axioms:
% 2.90/1.63 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Product__Type_OPair,type,
% 2.90/1.63 product_Pair:
% 2.90/1.63 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 2.90/1.63 produc2004651681e_prod:
% 2.90/1.63 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.90/1.63 product_rec_prod:
% 2.90/1.63 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Relation_OId__on,type,
% 2.90/1.63 id_on:
% 2.90/1.63 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Relation_Orefl__on,type,
% 2.90/1.63 refl_on:
% 2.90/1.63 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Relation_Ototal__on,type,
% 2.90/1.63 total_on:
% 2.90/1.63 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Relation_Otrans,type,
% 2.90/1.63 trans:
% 2.90/1.63 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Set_OCollect,type,
% 2.90/1.63 collect:
% 2.90/1.63 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_Set_Ois__empty,type,
% 2.90/1.63 is_empty:
% 2.90/1.63 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_c_member,type,
% 2.90/1.63 member:
% 2.90/1.63 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_v_carrier,type,
% 2.90/1.63 carrier: set @ a ).
% 2.90/1.63
% 2.90/1.63 thf(sy_v_relation,type,
% 2.90/1.63 relation: set @ ( product_prod @ a @ a ) ).
% 2.90/1.63
% 2.90/1.63 thf(sy_v_x,type,
% 2.90/1.63 x: a ).
% 2.90/1.63
% 2.90/1.63 thf(sy_v_y,type,
% 2.90/1.63 y: a ).
% 2.90/1.63
% 2.90/1.63 thf(fact_0_assms_I3_J,axiom,
% 2.90/1.63 member @ a @ y @ carrier ).
% 2.90/1.63
% 2.90/1.63 thf(fact_1_assms_I2_J,axiom,
% 2.90/1.63 member @ a @ x @ carrier ).
% 2.90/1.63
% 2.90/1.63 thf(fact_2_rational__preference__axioms,axiom,
% 2.90/1.63 prefer1997167224erence @ a @ carrier @ relation ).
% 2.90/1.63
% 2.90/1.63 thf(fact_3_nbt__nest,axiom,
% 2.90/1.63 ! [Y: a,X: a] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) )
% 2.90/1.63 | ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_4_assms_I1_J,axiom,
% 2.90/1.63 finite_finite2 @ a @ carrier ).
% 2.90/1.63
% 2.90/1.63 thf(fact_5_preference__axioms,axiom,
% 2.90/1.63 prefer199794634erence @ a @ carrier @ relation ).
% 2.90/1.63
% 2.90/1.63 thf(fact_6_trans__refl,axiom,
% 2.90/1.63 order_preorder_on @ a @ carrier @ relation ).
% 2.90/1.63
% 2.90/1.63 thf(fact_7_no__better__subset__pref,axiom,
% 2.90/1.63 ! [X: a,Y: a] :
% 2.90/1.63 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 2.90/1.63 => ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_8_no__better__thansubset__rel,axiom,
% 2.90/1.63 ! [X: a,Y: a] :
% 2.90/1.63 ( ( member @ a @ X @ carrier )
% 2.90/1.63 => ( ( member @ a @ Y @ carrier )
% 2.90/1.63 => ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X @ carrier @ relation ) )
% 2.90/1.63 => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_9_no__better__than__nonepty,axiom,
% 2.90/1.63 ! [X: a] :
% 2.90/1.63 ( ( carrier
% 2.90/1.63 != ( bot_bot @ ( set @ a ) ) )
% 2.90/1.63 => ( ( member @ a @ X @ carrier )
% 2.90/1.63 => ( ( prefer1532642881r_than @ a @ X @ carrier @ relation )
% 2.90/1.63 != ( bot_bot @ ( set @ a ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_10_total,axiom,
% 2.90/1.63 total_on @ a @ carrier @ relation ).
% 2.90/1.63
% 2.90/1.63 thf(fact_11_worse__in__no__better,axiom,
% 2.90/1.63 ! [X: a,Y: a] :
% 2.90/1.63 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 2.90/1.63 => ( member @ a @ Y @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_12_reflexivity,axiom,
% 2.90/1.63 refl_on @ a @ carrier @ relation ).
% 2.90/1.63
% 2.90/1.63 thf(fact_13_compl,axiom,
% 2.90/1.63 ! [X2: a] :
% 2.90/1.63 ( ( member @ a @ X2 @ carrier )
% 2.90/1.63 => ! [Xa: a] :
% 2.90/1.63 ( ( member @ a @ Xa @ carrier )
% 2.90/1.63 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Xa ) @ relation )
% 2.90/1.63 | ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Xa @ X2 ) @ relation ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_14_completeD,axiom,
% 2.90/1.63 ! [X: a,Y: a] :
% 2.90/1.63 ( ( member @ a @ X @ carrier )
% 2.90/1.63 => ( ( member @ a @ Y @ carrier )
% 2.90/1.63 => ( ( X != Y )
% 2.90/1.63 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 2.90/1.63 | ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_15_not__outside,axiom,
% 2.90/1.63 ! [X: a,Y: a] :
% 2.90/1.63 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 2.90/1.63 => ( member @ a @ X @ carrier ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_16_strict__is__neg__transitive,axiom,
% 2.90/1.63 ! [X: a,Y: a,Z: a] :
% 2.90/1.63 ( ( ( member @ a @ X @ carrier )
% 2.90/1.63 & ( member @ a @ Y @ carrier )
% 2.90/1.63 & ( member @ a @ Z @ carrier ) )
% 2.90/1.63 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 2.90/1.63 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
% 2.90/1.63 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation )
% 2.90/1.63 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X ) @ relation ) )
% 2.90/1.63 | ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation )
% 2.90/1.63 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_17_weak__is__transitive,axiom,
% 2.90/1.63 ! [X: a,Y: a,Z: a] :
% 2.90/1.63 ( ( ( member @ a @ X @ carrier )
% 2.90/1.63 & ( member @ a @ Y @ carrier )
% 2.90/1.63 & ( member @ a @ Z @ carrier ) )
% 2.90/1.63 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 2.90/1.63 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
% 2.90/1.63 => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_18_strict__trans,axiom,
% 2.90/1.63 ! [X: a,Y: a,Z: a] :
% 2.90/1.63 ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 2.90/1.63 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
% 2.90/1.63 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
% 2.90/1.63 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation ) )
% 2.90/1.63 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation )
% 2.90/1.63 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X ) @ relation ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_19_indiff__trans,axiom,
% 2.90/1.63 ! [X: a,Y: a,Z: a] :
% 2.90/1.63 ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 2.90/1.63 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
% 2.90/1.63 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
% 2.90/1.63 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation ) )
% 2.90/1.63 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation )
% 2.90/1.63 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X ) @ relation ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_20_strict__not__refl__weak,axiom,
% 2.90/1.63 ! [X: a,Y: a] :
% 2.90/1.63 ( ( ( member @ a @ X @ carrier )
% 2.90/1.63 & ( member @ a @ Y @ carrier ) )
% 2.90/1.63 => ( ( ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
% 2.90/1.63 = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 2.90/1.63 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_21_rational__preference_Oaxioms_I1_J,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( prefer199794634erence @ A @ Carrier @ Relation ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_22_preference__def,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( prefer199794634erence @ A )
% 2.90/1.63 = ( ^ [Carrier2: set @ A,Relation2: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ( ! [X3: A,Y2: A] :
% 2.90/1.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ Relation2 )
% 2.90/1.63 => ( member @ A @ X3 @ Carrier2 ) )
% 2.90/1.63 & ! [X3: A,Y2: A] :
% 2.90/1.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ Relation2 )
% 2.90/1.63 => ( member @ A @ Y2 @ Carrier2 ) )
% 2.90/1.63 & ( order_preorder_on @ A @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_23_preference_Ointro,axiom,
% 2.90/1.63 ! [A: $tType,Relation: set @ ( product_prod @ A @ A ),Carrier: set @ A] :
% 2.90/1.63 ( ! [X4: A,Y3: A] :
% 2.90/1.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ Relation )
% 2.90/1.63 => ( member @ A @ X4 @ Carrier ) )
% 2.90/1.63 => ( ! [X4: A,Y3: A] :
% 2.90/1.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ Relation )
% 2.90/1.63 => ( member @ A @ Y3 @ Carrier ) )
% 2.90/1.63 => ( ( order_preorder_on @ A @ Carrier @ Relation )
% 2.90/1.63 => ( prefer199794634erence @ A @ Carrier @ Relation ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_24_preference_Otrans__refl,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( order_preorder_on @ A @ Carrier @ Relation ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_25_preference_Onot__outside,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 2.90/1.63 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 2.90/1.63 => ( member @ A @ X @ Carrier ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_26_preference_Oreflexivity,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( refl_on @ A @ Carrier @ Relation ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_27_preference_Oindiff__trans,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 2.90/1.63 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 2.90/1.63 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 2.90/1.63 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
% 2.90/1.63 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation ) )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation )
% 2.90/1.63 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X ) @ Relation ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_28_rational__preference_Ocompl,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ! [X2: A] :
% 2.90/1.63 ( ( member @ A @ X2 @ Carrier )
% 2.90/1.63 => ! [Xa: A] :
% 2.90/1.63 ( ( member @ A @ Xa @ Carrier )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa ) @ Relation )
% 2.90/1.63 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_29_rational__preference_Ototal,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( total_on @ A @ Carrier @ Relation ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_30_rational__preference_OcompleteD,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 2.90/1.63 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( ( member @ A @ X @ Carrier )
% 2.90/1.63 => ( ( member @ A @ Y @ Carrier )
% 2.90/1.63 => ( ( X != Y )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 2.90/1.63 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_31_rational__preference_Ostrict__trans,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 2.90/1.63 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 2.90/1.63 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 2.90/1.63 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
% 2.90/1.63 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation ) )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation )
% 2.90/1.63 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X ) @ Relation ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_32_rational__preference_Oweak__is__transitive,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 2.90/1.63 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( ( ( member @ A @ X @ Carrier )
% 2.90/1.63 & ( member @ A @ Y @ Carrier )
% 2.90/1.63 & ( member @ A @ Z @ Carrier ) )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
% 2.90/1.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_33_rational__preference_Ostrict__not__refl__weak,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 2.90/1.63 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( ( ( member @ A @ X @ Carrier )
% 2.90/1.63 & ( member @ A @ Y @ Carrier ) )
% 2.90/1.63 => ( ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 2.90/1.63 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 2.90/1.63 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_34_rational__preference_Ostrict__is__neg__transitive,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 2.90/1.63 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( ( ( member @ A @ X @ Carrier )
% 2.90/1.63 & ( member @ A @ Y @ Carrier )
% 2.90/1.63 & ( member @ A @ Z @ Carrier ) )
% 2.90/1.63 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 2.90/1.63 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
% 2.90/1.63 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation )
% 2.90/1.63 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X ) @ Relation ) )
% 2.90/1.63 | ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation )
% 2.90/1.63 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_35_rational__preference_Oworse__in__no__better,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 2.90/1.63 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 2.90/1.63 => ( member @ A @ Y @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_36_rational__preference_Ono__better__than__nonepty,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A] :
% 2.90/1.63 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( ( Carrier
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( member @ A @ X @ Carrier )
% 2.90/1.63 => ( ( prefer1532642881r_than @ A @ X @ Carrier @ Relation )
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_37_rational__preference_Ono__better__thansubset__rel,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 2.90/1.63 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( ( member @ A @ X @ Carrier )
% 2.90/1.63 => ( ( member @ A @ Y @ Carrier )
% 2.90/1.63 => ( ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) )
% 2.90/1.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_38_rational__preference_Ono__better__subset__pref,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 2.90/1.63 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 2.90/1.63 => ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_39_rational__preference_Onbt__nest,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),Y: A,X: A] :
% 2.90/1.63 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) )
% 2.90/1.63 | ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ X @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_40_at__lst__asgd__not__ge,axiom,
% 2.90/1.63 ! [X: a,Y: a] :
% 2.90/1.63 ( ( carrier
% 2.90/1.63 != ( bot_bot @ ( set @ a ) ) )
% 2.90/1.63 => ( ( member @ a @ X @ carrier )
% 2.90/1.63 => ( ( member @ a @ Y @ carrier )
% 2.90/1.63 => ( ~ ( member @ a @ X @ ( prefer310429814s_good @ a @ Y @ carrier @ relation ) )
% 2.90/1.63 => ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_41_same__at__least__as__equal,axiom,
% 2.90/1.63 ! [Z: a,Y: a] :
% 2.90/1.63 ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation )
% 2.90/1.63 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation ) )
% 2.90/1.63 => ( ( prefer310429814s_good @ a @ Z @ carrier @ relation )
% 2.90/1.63 = ( prefer310429814s_good @ a @ Y @ carrier @ relation ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_42_pref__in__at__least__as,axiom,
% 2.90/1.63 ! [X: a,Y: a] :
% 2.90/1.63 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 2.90/1.63 => ( member @ a @ X @ ( prefer310429814s_good @ a @ Y @ carrier @ relation ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_43_empty__subsetI,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_44_subset__empty,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 = ( A2
% 2.90/1.63 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_45_mem__Collect__eq,axiom,
% 2.90/1.63 ! [A: $tType,A3: A,P: A > $o] :
% 2.90/1.63 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 2.90/1.63 = ( P @ A3 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_46_Collect__mem__eq,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A] :
% 2.90/1.63 ( ( collect @ A
% 2.90/1.63 @ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
% 2.90/1.63 = A2 ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_47_Collect__cong,axiom,
% 2.90/1.63 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.90/1.63 ( ! [X4: A] :
% 2.90/1.63 ( ( P @ X4 )
% 2.90/1.63 = ( Q @ X4 ) )
% 2.90/1.63 => ( ( collect @ A @ P )
% 2.90/1.63 = ( collect @ A @ Q ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_48_ext,axiom,
% 2.90/1.63 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.90/1.63 ( ! [X4: A] :
% 2.90/1.63 ( ( F @ X4 )
% 2.90/1.63 = ( G @ X4 ) )
% 2.90/1.63 => ( F = G ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_49_Pair__le,axiom,
% 2.90/1.63 ! [A: $tType,B: $tType] :
% 2.90/1.63 ( ( ( ord @ B )
% 2.90/1.63 & ( ord @ A ) )
% 2.90/1.63 => ! [A3: A,B2: B,C2: A,D: B] :
% 2.90/1.63 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D ) )
% 2.90/1.63 = ( ( ord_less_eq @ A @ A3 @ C2 )
% 2.90/1.63 & ( ord_less_eq @ B @ B2 @ D ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_50_as__good__as__sameIff,axiom,
% 2.90/1.63 ! [Z: a,Y: a] :
% 2.90/1.63 ( ( member @ a @ Z @ ( prefer951318096ood_as @ a @ Y @ carrier @ relation ) )
% 2.90/1.63 = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation )
% 2.90/1.63 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_51_as__good__asIff,axiom,
% 2.90/1.63 ! [X: a,Y: a] :
% 2.90/1.63 ( ( member @ a @ X @ ( prefer951318096ood_as @ a @ Y @ carrier @ relation ) )
% 2.90/1.63 = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
% 2.90/1.63 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_52_finite__has__maximal,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( order @ A )
% 2.90/1.63 => ! [A2: set @ A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ? [X4: A] :
% 2.90/1.63 ( ( member @ A @ X4 @ A2 )
% 2.90/1.63 & ! [Xa: A] :
% 2.90/1.63 ( ( member @ A @ Xa @ A2 )
% 2.90/1.63 => ( ( ord_less_eq @ A @ X4 @ Xa )
% 2.90/1.63 => ( X4 = Xa ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_53_finite__has__minimal,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( order @ A )
% 2.90/1.63 => ! [A2: set @ A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ? [X4: A] :
% 2.90/1.63 ( ( member @ A @ X4 @ A2 )
% 2.90/1.63 & ! [Xa: A] :
% 2.90/1.63 ( ( member @ A @ Xa @ A2 )
% 2.90/1.63 => ( ( ord_less_eq @ A @ Xa @ X4 )
% 2.90/1.63 => ( X4 = Xa ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_54_empty__Collect__eq,axiom,
% 2.90/1.63 ! [A: $tType,P: A > $o] :
% 2.90/1.63 ( ( ( bot_bot @ ( set @ A ) )
% 2.90/1.63 = ( collect @ A @ P ) )
% 2.90/1.63 = ( ! [X3: A] :
% 2.90/1.63 ~ ( P @ X3 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_55_Collect__empty__eq,axiom,
% 2.90/1.63 ! [A: $tType,P: A > $o] :
% 2.90/1.63 ( ( ( collect @ A @ P )
% 2.90/1.63 = ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 = ( ! [X3: A] :
% 2.90/1.63 ~ ( P @ X3 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_56_all__not__in__conv,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A] :
% 2.90/1.63 ( ( ! [X3: A] :
% 2.90/1.63 ~ ( member @ A @ X3 @ A2 ) )
% 2.90/1.63 = ( A2
% 2.90/1.63 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_57_empty__iff,axiom,
% 2.90/1.63 ! [A: $tType,C2: A] :
% 2.90/1.63 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_58_subsetI,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 2.90/1.63 ( ! [X4: A] :
% 2.90/1.63 ( ( member @ A @ X4 @ A2 )
% 2.90/1.63 => ( member @ A @ X4 @ B3 ) )
% 2.90/1.63 => ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_59_subset__antisym,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 2.90/1.63 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
% 2.90/1.63 => ( A2 = B3 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_60_finite__code,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( finite_finite @ A )
% 2.90/1.63 => ( ( finite_finite2 @ A )
% 2.90/1.63 = ( ^ [A4: set @ A] : $true ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_61_at__least__as__goodD,axiom,
% 2.90/1.63 ! [A: $tType,Z: A,Y: A,B3: set @ A,Pr: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ( ( member @ A @ Z @ ( prefer310429814s_good @ A @ Y @ B3 @ Pr ) )
% 2.90/1.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Pr ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_62_at__lst__asgd__ge,axiom,
% 2.90/1.63 ! [A: $tType,X: A,Y: A,B3: set @ A,Pr: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ( ( member @ A @ X @ ( prefer310429814s_good @ A @ Y @ B3 @ Pr ) )
% 2.90/1.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Pr ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_63_ex__in__conv,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A] :
% 2.90/1.63 ( ( ? [X3: A] : ( member @ A @ X3 @ A2 ) )
% 2.90/1.63 = ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_64_equals0I,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A] :
% 2.90/1.63 ( ! [Y3: A] :
% 2.90/1.63 ~ ( member @ A @ Y3 @ A2 )
% 2.90/1.63 => ( A2
% 2.90/1.63 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_65_equals0D,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,A3: A] :
% 2.90/1.63 ( ( A2
% 2.90/1.63 = ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ~ ( member @ A @ A3 @ A2 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_66_emptyE,axiom,
% 2.90/1.63 ! [A: $tType,A3: A] :
% 2.90/1.63 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_67_in__mono,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,B3: set @ A,X: A] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 2.90/1.63 => ( ( member @ A @ X @ A2 )
% 2.90/1.63 => ( member @ A @ X @ B3 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_68_subsetD,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,B3: set @ A,C2: A] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 2.90/1.63 => ( ( member @ A @ C2 @ A2 )
% 2.90/1.63 => ( member @ A @ C2 @ B3 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_69_equalityE,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 2.90/1.63 ( ( A2 = B3 )
% 2.90/1.63 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 2.90/1.63 => ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_70_subset__eq,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) )
% 2.90/1.63 = ( ^ [A4: set @ A,B4: set @ A] :
% 2.90/1.63 ! [X3: A] :
% 2.90/1.63 ( ( member @ A @ X3 @ A4 )
% 2.90/1.63 => ( member @ A @ X3 @ B4 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_71_equalityD1,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 2.90/1.63 ( ( A2 = B3 )
% 2.90/1.63 => ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_72_equalityD2,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 2.90/1.63 ( ( A2 = B3 )
% 2.90/1.63 => ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_73_subset__iff,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) )
% 2.90/1.63 = ( ^ [A4: set @ A,B4: set @ A] :
% 2.90/1.63 ! [T2: A] :
% 2.90/1.63 ( ( member @ A @ T2 @ A4 )
% 2.90/1.63 => ( member @ A @ T2 @ B4 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_74_subset__refl,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_75_Collect__mono,axiom,
% 2.90/1.63 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.90/1.63 ( ! [X4: A] :
% 2.90/1.63 ( ( P @ X4 )
% 2.90/1.63 => ( Q @ X4 ) )
% 2.90/1.63 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_76_subset__trans,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,B3: set @ A,C3: set @ A] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 2.90/1.63 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
% 2.90/1.63 => ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_77_set__eq__subset,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( ^ [Y4: set @ A,Z2: set @ A] : ( Y4 = Z2 ) )
% 2.90/1.63 = ( ^ [A4: set @ A,B4: set @ A] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.90/1.63 & ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_78_Collect__mono__iff,axiom,
% 2.90/1.63 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.90/1.63 = ( ! [X3: A] :
% 2.90/1.63 ( ( P @ X3 )
% 2.90/1.63 => ( Q @ X3 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_79_finite,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( finite_finite @ A )
% 2.90/1.63 => ! [A2: set @ A] : ( finite_finite2 @ A @ A2 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_80_finite__set__choice,axiom,
% 2.90/1.63 ! [B: $tType,A: $tType,A2: set @ A,P: A > B > $o] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ! [X4: A] :
% 2.90/1.63 ( ( member @ A @ X4 @ A2 )
% 2.90/1.63 => ? [X_1: B] : ( P @ X4 @ X_1 ) )
% 2.90/1.63 => ? [F2: A > B] :
% 2.90/1.63 ! [X2: A] :
% 2.90/1.63 ( ( member @ A @ X2 @ A2 )
% 2.90/1.63 => ( P @ X2 @ ( F2 @ X2 ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_81_rational__preference_Osame__at__least__as__equal,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),Z: A,Y: A] :
% 2.90/1.63 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation )
% 2.90/1.63 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation ) )
% 2.90/1.63 => ( ( prefer310429814s_good @ A @ Z @ Carrier @ Relation )
% 2.90/1.63 = ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_82_rational__preference_Opref__in__at__least__as,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 2.90/1.63 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 2.90/1.63 => ( member @ A @ X @ ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_83_rational__preference_Oas__good__as__sameIff,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),Z: A,Y: A] :
% 2.90/1.63 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( ( member @ A @ Z @ ( prefer951318096ood_as @ A @ Y @ Carrier @ Relation ) )
% 2.90/1.63 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation )
% 2.90/1.63 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_84_rational__preference_Oas__good__asIff,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 2.90/1.63 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( ( member @ A @ X @ ( prefer951318096ood_as @ A @ Y @ Carrier @ Relation ) )
% 2.90/1.63 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
% 2.90/1.63 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_85_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 2.90/1.63 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( ( Carrier
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( member @ A @ X @ Carrier )
% 2.90/1.63 => ( ( member @ A @ Y @ Carrier )
% 2.90/1.63 => ( ~ ( member @ A @ X @ ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) )
% 2.90/1.63 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_86_Pair__mono,axiom,
% 2.90/1.63 ! [A: $tType,B: $tType] :
% 2.90/1.63 ( ( ( ord @ B )
% 2.90/1.63 & ( ord @ A ) )
% 2.90/1.63 => ! [X: A,X5: A,Y: B,Y5: B] :
% 2.90/1.63 ( ( ord_less_eq @ A @ X @ X5 )
% 2.90/1.63 => ( ( ord_less_eq @ B @ Y @ Y5 )
% 2.90/1.63 => ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_87_finite__has__maximal2,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( order @ A )
% 2.90/1.63 => ! [A2: set @ A,A3: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( member @ A @ A3 @ A2 )
% 2.90/1.63 => ? [X4: A] :
% 2.90/1.63 ( ( member @ A @ X4 @ A2 )
% 2.90/1.63 & ( ord_less_eq @ A @ A3 @ X4 )
% 2.90/1.63 & ! [Xa: A] :
% 2.90/1.63 ( ( member @ A @ Xa @ A2 )
% 2.90/1.63 => ( ( ord_less_eq @ A @ X4 @ Xa )
% 2.90/1.63 => ( X4 = Xa ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_88_finite__has__minimal2,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( order @ A )
% 2.90/1.63 => ! [A2: set @ A,A3: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( member @ A @ A3 @ A2 )
% 2.90/1.63 => ? [X4: A] :
% 2.90/1.63 ( ( member @ A @ X4 @ A2 )
% 2.90/1.63 & ( ord_less_eq @ A @ X4 @ A3 )
% 2.90/1.63 & ! [Xa: A] :
% 2.90/1.63 ( ( member @ A @ Xa @ A2 )
% 2.90/1.63 => ( ( ord_less_eq @ A @ Xa @ X4 )
% 2.90/1.63 => ( X4 = Xa ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_89_bot__prod__def,axiom,
% 2.90/1.63 ! [A: $tType,B: $tType] :
% 2.90/1.63 ( ( ( bot @ B )
% 2.90/1.63 & ( bot @ A ) )
% 2.90/1.63 => ( ( bot_bot @ ( product_prod @ A @ B ) )
% 2.90/1.63 = ( product_Pair @ A @ B @ ( bot_bot @ A ) @ ( bot_bot @ B ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_90_infinite__imp__nonempty,axiom,
% 2.90/1.63 ! [A: $tType,S: set @ A] :
% 2.90/1.63 ( ~ ( finite_finite2 @ A @ S )
% 2.90/1.63 => ( S
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_91_finite_OemptyI,axiom,
% 2.90/1.63 ! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_92_finite__subset,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 2.90/1.63 => ( ( finite_finite2 @ A @ B3 )
% 2.90/1.63 => ( finite_finite2 @ A @ A2 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_93_infinite__super,axiom,
% 2.90/1.63 ! [A: $tType,S: set @ A,T3: set @ A] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ S @ T3 )
% 2.90/1.63 => ( ~ ( finite_finite2 @ A @ S )
% 2.90/1.63 => ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_94_rev__finite__subset,axiom,
% 2.90/1.63 ! [A: $tType,B3: set @ A,A2: set @ A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ B3 )
% 2.90/1.63 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 2.90/1.63 => ( finite_finite2 @ A @ A2 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_95_old_Oprod_Oinject,axiom,
% 2.90/1.63 ! [A: $tType,B: $tType,A3: A,B2: B,A5: A,B5: B] :
% 2.90/1.63 ( ( ( product_Pair @ A @ B @ A3 @ B2 )
% 2.90/1.63 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 2.90/1.63 = ( ( A3 = A5 )
% 2.90/1.63 & ( B2 = B5 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_96_prod_Oinject,axiom,
% 2.90/1.63 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 2.90/1.63 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 2.90/1.63 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 2.90/1.63 = ( ( X1 = Y1 )
% 2.90/1.63 & ( X22 = Y22 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_97_bot__apply,axiom,
% 2.90/1.63 ! [C: $tType,D2: $tType] :
% 2.90/1.63 ( ( bot @ C )
% 2.90/1.63 => ( ( bot_bot @ ( D2 > C ) )
% 2.90/1.63 = ( ^ [X3: D2] : ( bot_bot @ C ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_98_order__refl,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( preorder @ A )
% 2.90/1.63 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_99_preorder__on__empty,axiom,
% 2.90/1.63 ! [A: $tType] : ( order_preorder_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_100_total__on__def,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( total_on @ A )
% 2.90/1.63 = ( ^ [A4: set @ A,R: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ! [X3: A] :
% 2.90/1.63 ( ( member @ A @ X3 @ A4 )
% 2.90/1.63 => ! [Y2: A] :
% 2.90/1.63 ( ( member @ A @ Y2 @ A4 )
% 2.90/1.63 => ( ( X3 != Y2 )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R )
% 2.90/1.63 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X3 ) @ R ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_101_total__onI,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ( ! [X4: A,Y3: A] :
% 2.90/1.63 ( ( member @ A @ X4 @ A2 )
% 2.90/1.63 => ( ( member @ A @ Y3 @ A2 )
% 2.90/1.63 => ( ( X4 != Y3 )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R2 )
% 2.90/1.63 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X4 ) @ R2 ) ) ) ) )
% 2.90/1.63 => ( total_on @ A @ A2 @ R2 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_102_total__on__empty,axiom,
% 2.90/1.63 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R2 ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_103_bot__empty__eq,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( bot_bot @ ( A > $o ) )
% 2.90/1.63 = ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_104_bot__set__def,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( bot_bot @ ( set @ A ) )
% 2.90/1.63 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_105_dual__order_Oantisym,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( order @ A )
% 2.90/1.63 => ! [B2: A,A3: A] :
% 2.90/1.63 ( ( ord_less_eq @ A @ B2 @ A3 )
% 2.90/1.63 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 2.90/1.63 => ( A3 = B2 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_106_dual__order_Oeq__iff,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( order @ A )
% 2.90/1.63 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 2.90/1.63 = ( ^ [A6: A,B6: A] :
% 2.90/1.63 ( ( ord_less_eq @ A @ B6 @ A6 )
% 2.90/1.63 & ( ord_less_eq @ A @ A6 @ B6 ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_107_dual__order_Otrans,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( order @ A )
% 2.90/1.63 => ! [B2: A,A3: A,C2: A] :
% 2.90/1.63 ( ( ord_less_eq @ A @ B2 @ A3 )
% 2.90/1.63 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 2.90/1.63 => ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_108_linorder__wlog,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [P: A > A > $o,A3: A,B2: A] :
% 2.90/1.63 ( ! [A7: A,B7: A] :
% 2.90/1.63 ( ( ord_less_eq @ A @ A7 @ B7 )
% 2.90/1.63 => ( P @ A7 @ B7 ) )
% 2.90/1.63 => ( ! [A7: A,B7: A] :
% 2.90/1.63 ( ( P @ B7 @ A7 )
% 2.90/1.63 => ( P @ A7 @ B7 ) )
% 2.90/1.63 => ( P @ A3 @ B2 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_109_dual__order_Orefl,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( order @ A )
% 2.90/1.63 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_110_order__trans,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( preorder @ A )
% 2.90/1.63 => ! [X: A,Y: A,Z: A] :
% 2.90/1.63 ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.63 => ( ( ord_less_eq @ A @ Y @ Z )
% 2.90/1.63 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_111_order__class_Oorder_Oantisym,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( order @ A )
% 2.90/1.63 => ! [A3: A,B2: A] :
% 2.90/1.63 ( ( ord_less_eq @ A @ A3 @ B2 )
% 2.90/1.63 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 2.90/1.63 => ( A3 = B2 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_112_ord__le__eq__trans,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( ord @ A )
% 2.90/1.63 => ! [A3: A,B2: A,C2: A] :
% 2.90/1.63 ( ( ord_less_eq @ A @ A3 @ B2 )
% 2.90/1.63 => ( ( B2 = C2 )
% 2.90/1.63 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_113_ord__eq__le__trans,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( ord @ A )
% 2.90/1.63 => ! [A3: A,B2: A,C2: A] :
% 2.90/1.63 ( ( A3 = B2 )
% 2.90/1.63 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 2.90/1.63 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_114_order__class_Oorder_Oeq__iff,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( order @ A )
% 2.90/1.63 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 2.90/1.63 = ( ^ [A6: A,B6: A] :
% 2.90/1.63 ( ( ord_less_eq @ A @ A6 @ B6 )
% 2.90/1.63 & ( ord_less_eq @ A @ B6 @ A6 ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_115_antisym__conv,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( order @ A )
% 2.90/1.63 => ! [Y: A,X: A] :
% 2.90/1.63 ( ( ord_less_eq @ A @ Y @ X )
% 2.90/1.63 => ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.63 = ( X = Y ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_116_le__cases3,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [X: A,Y: A,Z: A] :
% 2.90/1.63 ( ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.63 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 2.90/1.63 => ( ( ( ord_less_eq @ A @ Y @ X )
% 2.90/1.63 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 2.90/1.63 => ( ( ( ord_less_eq @ A @ X @ Z )
% 2.90/1.63 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 2.90/1.63 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 2.90/1.63 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 2.90/1.63 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 2.90/1.63 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 2.90/1.63 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 2.90/1.63 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_117_order_Otrans,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( order @ A )
% 2.90/1.63 => ! [A3: A,B2: A,C2: A] :
% 2.90/1.63 ( ( ord_less_eq @ A @ A3 @ B2 )
% 2.90/1.63 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 2.90/1.63 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_118_le__cases,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [X: A,Y: A] :
% 2.90/1.63 ( ~ ( ord_less_eq @ A @ X @ Y )
% 2.90/1.63 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_119_eq__refl,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( preorder @ A )
% 2.90/1.63 => ! [X: A,Y: A] :
% 2.90/1.63 ( ( X = Y )
% 2.90/1.63 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_120_linear,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [X: A,Y: A] :
% 2.90/1.63 ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.63 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_121_antisym,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( order @ A )
% 2.90/1.63 => ! [X: A,Y: A] :
% 2.90/1.63 ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.63 => ( ( ord_less_eq @ A @ Y @ X )
% 2.90/1.63 => ( X = Y ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_122_eq__iff,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( order @ A )
% 2.90/1.63 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 2.90/1.63 = ( ^ [X3: A,Y2: A] :
% 2.90/1.63 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.90/1.63 & ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_123_ord__le__eq__subst,axiom,
% 2.90/1.63 ! [A: $tType,B: $tType] :
% 2.90/1.63 ( ( ( ord @ B )
% 2.90/1.63 & ( ord @ A ) )
% 2.90/1.63 => ! [A3: A,B2: A,F: A > B,C2: B] :
% 2.90/1.63 ( ( ord_less_eq @ A @ A3 @ B2 )
% 2.90/1.63 => ( ( ( F @ B2 )
% 2.90/1.63 = C2 )
% 2.90/1.63 => ( ! [X4: A,Y3: A] :
% 2.90/1.63 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.90/1.63 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.90/1.63 => ( ord_less_eq @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_124_ord__eq__le__subst,axiom,
% 2.90/1.63 ! [A: $tType,B: $tType] :
% 2.90/1.63 ( ( ( ord @ B )
% 2.90/1.63 & ( ord @ A ) )
% 2.90/1.63 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 2.90/1.63 ( ( A3
% 2.90/1.63 = ( F @ B2 ) )
% 2.90/1.63 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 2.90/1.63 => ( ! [X4: B,Y3: B] :
% 2.90/1.63 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 2.90/1.63 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.90/1.63 => ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_125_order__subst2,axiom,
% 2.90/1.63 ! [A: $tType,C: $tType] :
% 2.90/1.63 ( ( ( order @ C )
% 2.90/1.63 & ( order @ A ) )
% 2.90/1.63 => ! [A3: A,B2: A,F: A > C,C2: C] :
% 2.90/1.63 ( ( ord_less_eq @ A @ A3 @ B2 )
% 2.90/1.63 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 2.90/1.63 => ( ! [X4: A,Y3: A] :
% 2.90/1.63 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.90/1.63 => ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.90/1.63 => ( ord_less_eq @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_126_order__subst1,axiom,
% 2.90/1.63 ! [A: $tType,B: $tType] :
% 2.90/1.63 ( ( ( order @ B )
% 2.90/1.63 & ( order @ A ) )
% 2.90/1.63 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 2.90/1.63 ( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
% 2.90/1.63 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 2.90/1.63 => ( ! [X4: B,Y3: B] :
% 2.90/1.63 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 2.90/1.63 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 2.90/1.63 => ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_127_le__fun__def,axiom,
% 2.90/1.63 ! [B: $tType,A: $tType] :
% 2.90/1.63 ( ( ord @ B )
% 2.90/1.63 => ( ( ord_less_eq @ ( A > B ) )
% 2.90/1.63 = ( ^ [F3: A > B,G2: A > B] :
% 2.90/1.63 ! [X3: A] : ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_128_le__funI,axiom,
% 2.90/1.63 ! [B: $tType,A: $tType] :
% 2.90/1.63 ( ( ord @ B )
% 2.90/1.63 => ! [F: A > B,G: A > B] :
% 2.90/1.63 ( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
% 2.90/1.63 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_129_le__funE,axiom,
% 2.90/1.63 ! [B: $tType,A: $tType] :
% 2.90/1.63 ( ( ord @ B )
% 2.90/1.63 => ! [F: A > B,G: A > B,X: A] :
% 2.90/1.63 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.90/1.63 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_130_le__funD,axiom,
% 2.90/1.63 ! [B: $tType,A: $tType] :
% 2.90/1.63 ( ( ord @ B )
% 2.90/1.63 => ! [F: A > B,G: A > B,X: A] :
% 2.90/1.63 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.90/1.63 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_131_bot__fun__def,axiom,
% 2.90/1.63 ! [B: $tType,A: $tType] :
% 2.90/1.63 ( ( bot @ B )
% 2.90/1.63 => ( ( bot_bot @ ( A > B ) )
% 2.90/1.63 = ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_132_surj__pair,axiom,
% 2.90/1.63 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 2.90/1.63 ? [X4: A,Y3: B] :
% 2.90/1.63 ( P2
% 2.90/1.63 = ( product_Pair @ A @ B @ X4 @ Y3 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_133_prod__cases,axiom,
% 2.90/1.63 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 2.90/1.63 ( ! [A7: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A7 @ B7 ) )
% 2.90/1.63 => ( P @ P2 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_134_Pair__inject,axiom,
% 2.90/1.63 ! [A: $tType,B: $tType,A3: A,B2: B,A5: A,B5: B] :
% 2.90/1.63 ( ( ( product_Pair @ A @ B @ A3 @ B2 )
% 2.90/1.63 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 2.90/1.63 => ~ ( ( A3 = A5 )
% 2.90/1.63 => ( B2 != B5 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_135_prod__cases3,axiom,
% 2.90/1.63 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.90/1.63 ~ ! [A7: A,B7: B,C4: C] :
% 2.90/1.63 ( Y
% 2.90/1.63 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A7 @ ( product_Pair @ B @ C @ B7 @ C4 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_136_prod__cases4,axiom,
% 2.90/1.63 ! [A: $tType,B: $tType,C: $tType,D2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 2.90/1.63 ~ ! [A7: A,B7: B,C4: C,D3: D2] :
% 2.90/1.63 ( Y
% 2.90/1.63 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B7 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_137_prod__cases5,axiom,
% 2.90/1.63 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 2.90/1.63 ~ ! [A7: A,B7: B,C4: C,D3: D2,E2: E] :
% 2.90/1.63 ( Y
% 2.90/1.63 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C4 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_138_prod__cases6,axiom,
% 2.90/1.63 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) )] :
% 2.90/1.63 ~ ! [A7: A,B7: B,C4: C,D3: D2,E2: E,F2: F4] :
% 2.90/1.63 ( Y
% 2.90/1.63 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ F4 ) @ D3 @ ( product_Pair @ E @ F4 @ E2 @ F2 ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_139_prod__cases7,axiom,
% 2.90/1.63 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F4: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
% 2.90/1.63 ~ ! [A7: A,B7: B,C4: C,D3: D2,E2: E,F2: F4,G4: G3] :
% 2.90/1.63 ( Y
% 2.90/1.63 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F4 @ G3 ) @ E2 @ ( product_Pair @ F4 @ G3 @ F2 @ G4 ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_140_prod__induct3,axiom,
% 2.90/1.63 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.90/1.63 ( ! [A7: A,B7: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A7 @ ( product_Pair @ B @ C @ B7 @ C4 ) ) )
% 2.90/1.63 => ( P @ X ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_141_prod__induct4,axiom,
% 2.90/1.63 ! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 2.90/1.63 ( ! [A7: A,B7: B,C4: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B7 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) )
% 2.90/1.63 => ( P @ X ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_142_prod__induct5,axiom,
% 2.90/1.63 ! [E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 2.90/1.63 ( ! [A7: A,B7: B,C4: C,D3: D2,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C4 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) )
% 2.90/1.63 => ( P @ X ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_143_prod__induct6,axiom,
% 2.90/1.63 ! [F4: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) )] :
% 2.90/1.63 ( ! [A7: A,B7: B,C4: C,D3: D2,E2: E,F2: F4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ F4 ) @ D3 @ ( product_Pair @ E @ F4 @ E2 @ F2 ) ) ) ) ) )
% 2.90/1.63 => ( P @ X ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_144_prod__induct7,axiom,
% 2.90/1.63 ! [G3: $tType,F4: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
% 2.90/1.63 ( ! [A7: A,B7: B,C4: C,D3: D2,E2: E,F2: F4,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F4 @ G3 ) @ E2 @ ( product_Pair @ F4 @ G3 @ F2 @ G4 ) ) ) ) ) ) )
% 2.90/1.63 => ( P @ X ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_145_old_Oprod_Oexhaust,axiom,
% 2.90/1.63 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 2.90/1.63 ~ ! [A7: A,B7: B] :
% 2.90/1.63 ( Y
% 2.90/1.63 != ( product_Pair @ A @ B @ A7 @ B7 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_146_old_Oprod_Oinducts,axiom,
% 2.90/1.63 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.90/1.63 ( ! [A7: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A7 @ B7 ) )
% 2.90/1.63 => ( P @ Prod ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_147_subrelI,axiom,
% 2.90/1.63 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 2.90/1.63 ( ! [X4: A,Y3: B] :
% 2.90/1.63 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R2 )
% 2.90/1.63 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ S2 ) )
% 2.90/1.63 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_148_bot_Oextremum__uniqueI,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( order_bot @ A )
% 2.90/1.63 => ! [A3: A] :
% 2.90/1.63 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 2.90/1.63 => ( A3
% 2.90/1.63 = ( bot_bot @ A ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_149_bot_Oextremum__unique,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( order_bot @ A )
% 2.90/1.63 => ! [A3: A] :
% 2.90/1.63 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 2.90/1.63 = ( A3
% 2.90/1.63 = ( bot_bot @ A ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_150_bot_Oextremum,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( order_bot @ A )
% 2.90/1.63 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_151_refl__on__empty,axiom,
% 2.90/1.63 ! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_152_refl__onD,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A] :
% 2.90/1.63 ( ( refl_on @ A @ A2 @ R2 )
% 2.90/1.63 => ( ( member @ A @ A3 @ A2 )
% 2.90/1.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R2 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_153_refl__onD1,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 2.90/1.63 ( ( refl_on @ A @ A2 @ R2 )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
% 2.90/1.63 => ( member @ A @ X @ A2 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_154_refl__onD2,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 2.90/1.63 ( ( refl_on @ A @ A2 @ R2 )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
% 2.90/1.63 => ( member @ A @ Y @ A2 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_155_refl__on__domain,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 2.90/1.63 ( ( refl_on @ A @ A2 @ R2 )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
% 2.90/1.63 => ( ( member @ A @ A3 @ A2 )
% 2.90/1.63 & ( member @ A @ B2 @ A2 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_156_old_Oprod_Orec,axiom,
% 2.90/1.63 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B2: B] :
% 2.90/1.63 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
% 2.90/1.63 = ( F1 @ A3 @ B2 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_157_rational__preference__def,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( prefer1997167224erence @ A )
% 2.90/1.63 = ( ^ [Carrier2: set @ A,Relation2: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ( ( prefer199794634erence @ A @ Carrier2 @ Relation2 )
% 2.90/1.63 & ( prefer1801827867axioms @ A @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_158_rational__preference_Ointro,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( ( prefer1801827867axioms @ A @ Carrier @ Relation )
% 2.90/1.63 => ( prefer1997167224erence @ A @ Carrier @ Relation ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_159_finite__transitivity__chain,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,R3: A > A > $o] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ! [X4: A] :
% 2.90/1.63 ~ ( R3 @ X4 @ X4 )
% 2.90/1.63 => ( ! [X4: A,Y3: A,Z3: A] :
% 2.90/1.63 ( ( R3 @ X4 @ Y3 )
% 2.90/1.63 => ( ( R3 @ Y3 @ Z3 )
% 2.90/1.63 => ( R3 @ X4 @ Z3 ) ) )
% 2.90/1.63 => ( ! [X4: A] :
% 2.90/1.63 ( ( member @ A @ X4 @ A2 )
% 2.90/1.63 => ? [Y6: A] :
% 2.90/1.63 ( ( member @ A @ Y6 @ A2 )
% 2.90/1.63 & ( R3 @ X4 @ Y6 ) ) )
% 2.90/1.63 => ( A2
% 2.90/1.63 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_160_subset__emptyI,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A] :
% 2.90/1.63 ( ! [X4: A] :
% 2.90/1.63 ~ ( member @ A @ X4 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_161_ssubst__Pair__rhs,axiom,
% 2.90/1.63 ! [B: $tType,A: $tType,R2: A,S2: B,R3: set @ ( product_prod @ A @ B ),S3: B] :
% 2.90/1.63 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S2 ) @ R3 )
% 2.90/1.63 => ( ( S3 = S2 )
% 2.90/1.63 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S3 ) @ R3 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_162_rational__preference__axioms_Ointro,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ( ( total_on @ A @ Carrier @ Relation )
% 2.90/1.63 => ( prefer1801827867axioms @ A @ Carrier @ Relation ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_163_rational__preference__axioms__def,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( prefer1801827867axioms @ A )
% 2.90/1.63 = ( total_on @ A ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_164_rational__preference_Oaxioms_I2_J,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( prefer1801827867axioms @ A @ Carrier @ Relation ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_165_Collect__empty__eq__bot,axiom,
% 2.90/1.63 ! [A: $tType,P: A > $o] :
% 2.90/1.63 ( ( ( collect @ A @ P )
% 2.90/1.63 = ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 = ( P
% 2.90/1.63 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_166_less__by__empty,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ( ( A2
% 2.90/1.63 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 2.90/1.63 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A2 @ B3 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_167_finite__indexed__bound,axiom,
% 2.90/1.63 ! [B: $tType,A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ B,P: B > A > $o] :
% 2.90/1.63 ( ( finite_finite2 @ B @ A2 )
% 2.90/1.63 => ( ! [X4: B] :
% 2.90/1.63 ( ( member @ B @ X4 @ A2 )
% 2.90/1.63 => ? [X_1: A] : ( P @ X4 @ X_1 ) )
% 2.90/1.63 => ? [M: A] :
% 2.90/1.63 ! [X2: B] :
% 2.90/1.63 ( ( member @ B @ X2 @ A2 )
% 2.90/1.63 => ? [K: A] :
% 2.90/1.63 ( ( ord_less_eq @ A @ K @ M )
% 2.90/1.63 & ( P @ X2 @ K ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_168_transitivity,axiom,
% 2.90/1.63 trans @ a @ relation ).
% 2.90/1.63
% 2.90/1.63 thf(fact_169_Set_Ois__empty__def,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( is_empty @ A )
% 2.90/1.63 = ( ^ [A4: set @ A] :
% 2.90/1.63 ( A4
% 2.90/1.63 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_170_internal__case__prod__conv,axiom,
% 2.90/1.63 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B2: C] :
% 2.90/1.63 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B2 ) )
% 2.90/1.63 = ( C2 @ A3 @ B2 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_171_preference_Otransitivity,axiom,
% 2.90/1.63 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 2.90/1.63 => ( trans @ A @ Relation ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_172_trans__empty,axiom,
% 2.90/1.63 ! [A: $tType] : ( trans @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_173_transD,axiom,
% 2.90/1.63 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 2.90/1.63 ( ( trans @ A @ R2 )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R2 )
% 2.90/1.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R2 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_174_transE,axiom,
% 2.90/1.63 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
% 2.90/1.63 ( ( trans @ A @ R2 )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R2 )
% 2.90/1.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R2 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_175_transI,axiom,
% 2.90/1.63 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ( ! [X4: A,Y3: A,Z3: A] :
% 2.90/1.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R2 )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
% 2.90/1.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z3 ) @ R2 ) ) )
% 2.90/1.63 => ( trans @ A @ R2 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_176_trans__def,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( trans @ A )
% 2.90/1.63 = ( ^ [R: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ! [X3: A,Y2: A,Z4: A] :
% 2.90/1.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z4 ) @ R )
% 2.90/1.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z4 ) @ R ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_177_preorder__on__def,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( order_preorder_on @ A )
% 2.90/1.63 = ( ^ [A4: set @ A,R: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ( ( refl_on @ A @ A4 @ R )
% 2.90/1.63 & ( trans @ A @ R ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_178_relChain__def,axiom,
% 2.90/1.63 ! [B: $tType,A: $tType] :
% 2.90/1.63 ( ( ord @ B )
% 2.90/1.63 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 2.90/1.63 = ( ^ [R: set @ ( product_prod @ A @ A ),As: A > B] :
% 2.90/1.63 ! [I: A,J: A] :
% 2.90/1.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
% 2.90/1.63 => ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_179_Id__on__empty,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_180_under__incr,axiom,
% 2.90/1.63 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 2.90/1.63 ( ( trans @ A @ R2 )
% 2.90/1.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
% 2.90/1.63 => ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R2 @ A3 ) @ ( order_under @ A @ R2 @ B2 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_181_Id__onI,axiom,
% 2.90/1.63 ! [A: $tType,A3: A,A2: set @ A] :
% 2.90/1.63 ( ( member @ A @ A3 @ A2 )
% 2.90/1.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id_on @ A @ A2 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_182_trans__Id__on,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A] : ( trans @ A @ ( id_on @ A @ A2 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_183_Id__on__iff,axiom,
% 2.90/1.63 ! [A: $tType,X: A,Y: A,A2: set @ A] :
% 2.90/1.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( id_on @ A @ A2 ) )
% 2.90/1.63 = ( ( X = Y )
% 2.90/1.63 & ( member @ A @ X @ A2 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_184_Id__on__eqI,axiom,
% 2.90/1.63 ! [A: $tType,A3: A,B2: A,A2: set @ A] :
% 2.90/1.63 ( ( A3 = B2 )
% 2.90/1.63 => ( ( member @ A @ A3 @ A2 )
% 2.90/1.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id_on @ A @ A2 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_185_Id__onE,axiom,
% 2.90/1.63 ! [A: $tType,C2: product_prod @ A @ A,A2: set @ A] :
% 2.90/1.63 ( ( member @ ( product_prod @ A @ A ) @ C2 @ ( id_on @ A @ A2 ) )
% 2.90/1.63 => ~ ! [X4: A] :
% 2.90/1.63 ( ( member @ A @ X4 @ A2 )
% 2.90/1.63 => ( C2
% 2.90/1.63 != ( product_Pair @ A @ A @ X4 @ X4 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_186_refl__on__Id__on,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A] : ( refl_on @ A @ A2 @ ( id_on @ A @ A2 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_187_arg__min__least,axiom,
% 2.90/1.63 ! [B: $tType,A: $tType] :
% 2.90/1.63 ( ( linorder @ B )
% 2.90/1.63 => ! [S: set @ A,Y: A,F: A > B] :
% 2.90/1.63 ( ( finite_finite2 @ A @ S )
% 2.90/1.63 => ( ( S
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( member @ A @ Y @ S )
% 2.90/1.63 => ( ord_less_eq @ B @ ( F @ ( lattic1704895705min_on @ A @ B @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_188_Greatest__equality,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( order @ A )
% 2.90/1.63 => ! [P: A > $o,X: A] :
% 2.90/1.63 ( ( P @ X )
% 2.90/1.63 => ( ! [Y3: A] :
% 2.90/1.63 ( ( P @ Y3 )
% 2.90/1.63 => ( ord_less_eq @ A @ Y3 @ X ) )
% 2.90/1.63 => ( ( order_Greatest @ A @ P )
% 2.90/1.63 = X ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_189_GreatestI2__order,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( order @ A )
% 2.90/1.63 => ! [P: A > $o,X: A,Q: A > $o] :
% 2.90/1.63 ( ( P @ X )
% 2.90/1.63 => ( ! [Y3: A] :
% 2.90/1.63 ( ( P @ Y3 )
% 2.90/1.63 => ( ord_less_eq @ A @ Y3 @ X ) )
% 2.90/1.63 => ( ! [X4: A] :
% 2.90/1.63 ( ( P @ X4 )
% 2.90/1.63 => ( ! [Y6: A] :
% 2.90/1.63 ( ( P @ Y6 )
% 2.90/1.63 => ( ord_less_eq @ A @ Y6 @ X4 ) )
% 2.90/1.63 => ( Q @ X4 ) ) )
% 2.90/1.63 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_190_arg__min__if__finite_I1_J,axiom,
% 2.90/1.63 ! [B: $tType,A: $tType] :
% 2.90/1.63 ( ( order @ B )
% 2.90/1.63 => ! [S: set @ A,F: A > B] :
% 2.90/1.63 ( ( finite_finite2 @ A @ S )
% 2.90/1.63 => ( ( S
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( member @ A @ ( lattic1704895705min_on @ A @ B @ F @ S ) @ S ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_191_Inf__fin_Osubset__imp,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( semilattice_inf @ A )
% 2.90/1.63 => ! [A2: set @ A,B3: set @ A] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( finite_finite2 @ A @ B3 )
% 2.90/1.63 => ( ord_less_eq @ A @ ( lattic2065218050nf_fin @ A @ B3 ) @ ( lattic2065218050nf_fin @ A @ A2 ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_192_Sup__fin_Osubset__imp,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( semilattice_sup @ A )
% 2.90/1.63 => ! [A2: set @ A,B3: set @ A] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( finite_finite2 @ A @ B3 )
% 2.90/1.63 => ( ord_less_eq @ A @ ( lattic477160up_fin @ A @ A2 ) @ ( lattic477160up_fin @ A @ B3 ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_193_Inf__fin__le__Sup__fin,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( lattice @ A )
% 2.90/1.63 => ! [A2: set @ A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ord_less_eq @ A @ ( lattic2065218050nf_fin @ A @ A2 ) @ ( lattic477160up_fin @ A @ A2 ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_194_Sup__fin_OcoboundedI,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( semilattice_sup @ A )
% 2.90/1.63 => ! [A2: set @ A,A3: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( member @ A @ A3 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ A3 @ ( lattic477160up_fin @ A @ A2 ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_195_Inf__fin_OcoboundedI,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( semilattice_inf @ A )
% 2.90/1.63 => ! [A2: set @ A,A3: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( member @ A @ A3 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ ( lattic2065218050nf_fin @ A @ A2 ) @ A3 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_196_Inf__fin_OboundedE,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( semilattice_inf @ A )
% 2.90/1.63 => ! [A2: set @ A,X: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( ord_less_eq @ A @ X @ ( lattic2065218050nf_fin @ A @ A2 ) )
% 2.90/1.63 => ! [A8: A] :
% 2.90/1.63 ( ( member @ A @ A8 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ X @ A8 ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_197_Inf__fin_OboundedI,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( semilattice_inf @ A )
% 2.90/1.63 => ! [A2: set @ A,X: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ! [A7: A] :
% 2.90/1.63 ( ( member @ A @ A7 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ X @ A7 ) )
% 2.90/1.63 => ( ord_less_eq @ A @ X @ ( lattic2065218050nf_fin @ A @ A2 ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_198_Sup__fin_OboundedE,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( semilattice_sup @ A )
% 2.90/1.63 => ! [A2: set @ A,X: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( ord_less_eq @ A @ ( lattic477160up_fin @ A @ A2 ) @ X )
% 2.90/1.63 => ! [A8: A] :
% 2.90/1.63 ( ( member @ A @ A8 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ A8 @ X ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_199_Sup__fin_OboundedI,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( semilattice_sup @ A )
% 2.90/1.63 => ! [A2: set @ A,X: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ! [A7: A] :
% 2.90/1.63 ( ( member @ A @ A7 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ A7 @ X ) )
% 2.90/1.63 => ( ord_less_eq @ A @ ( lattic477160up_fin @ A @ A2 ) @ X ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_200_Inf__fin_Obounded__iff,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( semilattice_inf @ A )
% 2.90/1.63 => ! [A2: set @ A,X: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( ord_less_eq @ A @ X @ ( lattic2065218050nf_fin @ A @ A2 ) )
% 2.90/1.63 = ( ! [X3: A] :
% 2.90/1.63 ( ( member @ A @ X3 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_201_Sup__fin_Obounded__iff,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( semilattice_sup @ A )
% 2.90/1.63 => ! [A2: set @ A,X: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( ord_less_eq @ A @ ( lattic477160up_fin @ A @ A2 ) @ X )
% 2.90/1.63 = ( ! [X3: A] :
% 2.90/1.63 ( ( member @ A @ X3 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_202_Max_Osubset__imp,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,B3: set @ A] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( finite_finite2 @ A @ B3 )
% 2.90/1.63 => ( ord_less_eq @ A @ ( lattic929149872er_Max @ A @ A2 ) @ ( lattic929149872er_Max @ A @ B3 ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_203_Max__mono,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [M2: set @ A,N: set @ A] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ M2 @ N )
% 2.90/1.63 => ( ( M2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( finite_finite2 @ A @ N )
% 2.90/1.63 => ( ord_less_eq @ A @ ( lattic929149872er_Max @ A @ M2 ) @ ( lattic929149872er_Max @ A @ N ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_204_Max_Obounded__iff,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,X: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( ord_less_eq @ A @ ( lattic929149872er_Max @ A @ A2 ) @ X )
% 2.90/1.63 = ( ! [X3: A] :
% 2.90/1.63 ( ( member @ A @ X3 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_205_Max_OcoboundedI,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,A3: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( member @ A @ A3 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ A3 @ ( lattic929149872er_Max @ A @ A2 ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_206_Max__eq__if,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,B3: set @ A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( finite_finite2 @ A @ B3 )
% 2.90/1.63 => ( ! [X4: A] :
% 2.90/1.63 ( ( member @ A @ X4 @ A2 )
% 2.90/1.63 => ? [Xa: A] :
% 2.90/1.63 ( ( member @ A @ Xa @ B3 )
% 2.90/1.63 & ( ord_less_eq @ A @ X4 @ Xa ) ) )
% 2.90/1.63 => ( ! [X4: A] :
% 2.90/1.63 ( ( member @ A @ X4 @ B3 )
% 2.90/1.63 => ? [Xa: A] :
% 2.90/1.63 ( ( member @ A @ Xa @ A2 )
% 2.90/1.63 & ( ord_less_eq @ A @ X4 @ Xa ) ) )
% 2.90/1.63 => ( ( lattic929149872er_Max @ A @ A2 )
% 2.90/1.63 = ( lattic929149872er_Max @ A @ B3 ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_207_Max__eqI,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,X: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ! [Y3: A] :
% 2.90/1.63 ( ( member @ A @ Y3 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ Y3 @ X ) )
% 2.90/1.63 => ( ( member @ A @ X @ A2 )
% 2.90/1.63 => ( ( lattic929149872er_Max @ A @ A2 )
% 2.90/1.63 = X ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_208_Max__ge,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,X: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( member @ A @ X @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ X @ ( lattic929149872er_Max @ A @ A2 ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_209_Max__in,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( member @ A @ ( lattic929149872er_Max @ A @ A2 ) @ A2 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_210_Max__eq__iff,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,M3: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( ( lattic929149872er_Max @ A @ A2 )
% 2.90/1.63 = M3 )
% 2.90/1.63 = ( ( member @ A @ M3 @ A2 )
% 2.90/1.63 & ! [X3: A] :
% 2.90/1.63 ( ( member @ A @ X3 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ X3 @ M3 ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_211_Max__ge__iff,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,X: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( ord_less_eq @ A @ X @ ( lattic929149872er_Max @ A @ A2 ) )
% 2.90/1.63 = ( ? [X3: A] :
% 2.90/1.63 ( ( member @ A @ X3 @ A2 )
% 2.90/1.63 & ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_212_eq__Max__iff,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,M3: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( M3
% 2.90/1.63 = ( lattic929149872er_Max @ A @ A2 ) )
% 2.90/1.63 = ( ( member @ A @ M3 @ A2 )
% 2.90/1.63 & ! [X3: A] :
% 2.90/1.63 ( ( member @ A @ X3 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ X3 @ M3 ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_213_Max_OboundedE,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,X: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( ord_less_eq @ A @ ( lattic929149872er_Max @ A @ A2 ) @ X )
% 2.90/1.63 => ! [A8: A] :
% 2.90/1.63 ( ( member @ A @ A8 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ A8 @ X ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_214_Max_OboundedI,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,X: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ! [A7: A] :
% 2.90/1.63 ( ( member @ A @ A7 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ A7 @ X ) )
% 2.90/1.63 => ( ord_less_eq @ A @ ( lattic929149872er_Max @ A @ A2 ) @ X ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_215_Min__antimono,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [M2: set @ A,N: set @ A] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ M2 @ N )
% 2.90/1.63 => ( ( M2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( finite_finite2 @ A @ N )
% 2.90/1.63 => ( ord_less_eq @ A @ ( lattic929674654er_Min @ A @ N ) @ ( lattic929674654er_Min @ A @ M2 ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_216_Min_Osubset__imp,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,B3: set @ A] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( finite_finite2 @ A @ B3 )
% 2.90/1.63 => ( ord_less_eq @ A @ ( lattic929674654er_Min @ A @ B3 ) @ ( lattic929674654er_Min @ A @ A2 ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_217_Min_Obounded__iff,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,X: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( ord_less_eq @ A @ X @ ( lattic929674654er_Min @ A @ A2 ) )
% 2.90/1.63 = ( ! [X3: A] :
% 2.90/1.63 ( ( member @ A @ X3 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_218_Min__in,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( member @ A @ ( lattic929674654er_Min @ A @ A2 ) @ A2 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_219_Min__le,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,X: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( member @ A @ X @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ ( lattic929674654er_Min @ A @ A2 ) @ X ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_220_Min__eqI,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,X: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ! [Y3: A] :
% 2.90/1.63 ( ( member @ A @ Y3 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ X @ Y3 ) )
% 2.90/1.63 => ( ( member @ A @ X @ A2 )
% 2.90/1.63 => ( ( lattic929674654er_Min @ A @ A2 )
% 2.90/1.63 = X ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_221_Min_OcoboundedI,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,A3: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( member @ A @ A3 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ ( lattic929674654er_Min @ A @ A2 ) @ A3 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_222_Min_OboundedI,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,X: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ! [A7: A] :
% 2.90/1.63 ( ( member @ A @ A7 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ X @ A7 ) )
% 2.90/1.63 => ( ord_less_eq @ A @ X @ ( lattic929674654er_Min @ A @ A2 ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_223_Min_OboundedE,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,X: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( ord_less_eq @ A @ X @ ( lattic929674654er_Min @ A @ A2 ) )
% 2.90/1.63 => ! [A8: A] :
% 2.90/1.63 ( ( member @ A @ A8 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ X @ A8 ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_224_eq__Min__iff,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,M3: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( M3
% 2.90/1.63 = ( lattic929674654er_Min @ A @ A2 ) )
% 2.90/1.63 = ( ( member @ A @ M3 @ A2 )
% 2.90/1.63 & ! [X3: A] :
% 2.90/1.63 ( ( member @ A @ X3 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ M3 @ X3 ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_225_Min__le__iff,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,X: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( ord_less_eq @ A @ ( lattic929674654er_Min @ A @ A2 ) @ X )
% 2.90/1.63 = ( ? [X3: A] :
% 2.90/1.63 ( ( member @ A @ X3 @ A2 )
% 2.90/1.63 & ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_226_Min__eq__iff,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( linorder @ A )
% 2.90/1.63 => ! [A2: set @ A,M3: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( ( lattic929674654er_Min @ A @ A2 )
% 2.90/1.63 = M3 )
% 2.90/1.63 = ( ( member @ A @ M3 @ A2 )
% 2.90/1.63 & ! [X3: A] :
% 2.90/1.63 ( ( member @ A @ X3 @ A2 )
% 2.90/1.63 => ( ord_less_eq @ A @ M3 @ X3 ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_227_Inf__fin_Osubset,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( semilattice_inf @ A )
% 2.90/1.63 => ! [A2: set @ A,B3: set @ A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( B3
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
% 2.90/1.63 => ( ( inf_inf @ A @ ( lattic2065218050nf_fin @ A @ B3 ) @ ( lattic2065218050nf_fin @ A @ A2 ) )
% 2.90/1.63 = ( lattic2065218050nf_fin @ A @ A2 ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_228_Sup__fin_Osubset,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( semilattice_sup @ A )
% 2.90/1.63 => ! [A2: set @ A,B3: set @ A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( B3
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
% 2.90/1.63 => ( ( sup_sup @ A @ ( lattic477160up_fin @ A @ B3 ) @ ( lattic477160up_fin @ A @ A2 ) )
% 2.90/1.63 = ( lattic477160up_fin @ A @ A2 ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_229_Un__empty,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 2.90/1.63 ( ( ( sup_sup @ ( set @ A ) @ A2 @ B3 )
% 2.90/1.63 = ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 = ( ( A2
% 2.90/1.63 = ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 & ( B3
% 2.90/1.63 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_230_Int__subset__iff,axiom,
% 2.90/1.63 ! [A: $tType,C3: set @ A,A2: set @ A,B3: set @ A] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ C3 @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) )
% 2.90/1.63 = ( ( ord_less_eq @ ( set @ A ) @ C3 @ A2 )
% 2.90/1.63 & ( ord_less_eq @ ( set @ A ) @ C3 @ B3 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_231_Un__subset__iff,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,B3: set @ A,C3: set @ A] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) @ C3 )
% 2.90/1.63 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 2.90/1.63 & ( ord_less_eq @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_232_finite__Int,axiom,
% 2.90/1.63 ! [A: $tType,F5: set @ A,G5: set @ A] :
% 2.90/1.63 ( ( ( finite_finite2 @ A @ F5 )
% 2.90/1.63 | ( finite_finite2 @ A @ G5 ) )
% 2.90/1.63 => ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F5 @ G5 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_233_finite__Un,axiom,
% 2.90/1.63 ! [A: $tType,F5: set @ A,G5: set @ A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F5 @ G5 ) )
% 2.90/1.63 = ( ( finite_finite2 @ A @ F5 )
% 2.90/1.63 & ( finite_finite2 @ A @ G5 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_234_inf__Pair__Pair,axiom,
% 2.90/1.63 ! [A: $tType,B: $tType] :
% 2.90/1.63 ( ( ( inf @ B )
% 2.90/1.63 & ( inf @ A ) )
% 2.90/1.63 => ! [A3: A,B2: B,C2: A,D: B] :
% 2.90/1.63 ( ( inf_inf @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D ) )
% 2.90/1.63 = ( product_Pair @ A @ B @ ( inf_inf @ A @ A3 @ C2 ) @ ( inf_inf @ B @ B2 @ D ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_235_sup__Pair__Pair,axiom,
% 2.90/1.63 ! [A: $tType,B: $tType] :
% 2.90/1.63 ( ( ( sup @ B )
% 2.90/1.63 & ( sup @ A ) )
% 2.90/1.63 => ! [A3: A,B2: B,C2: A,D: B] :
% 2.90/1.63 ( ( sup_sup @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D ) )
% 2.90/1.63 = ( product_Pair @ A @ B @ ( sup_sup @ A @ A3 @ C2 ) @ ( sup_sup @ B @ B2 @ D ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_236_inf__Sup__absorb,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( lattice @ A )
% 2.90/1.63 => ! [A2: set @ A,A3: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( member @ A @ A3 @ A2 )
% 2.90/1.63 => ( ( inf_inf @ A @ A3 @ ( lattic477160up_fin @ A @ A2 ) )
% 2.90/1.63 = A3 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_237_sup__Inf__absorb,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( lattice @ A )
% 2.90/1.63 => ! [A2: set @ A,A3: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( member @ A @ A3 @ A2 )
% 2.90/1.63 => ( ( sup_sup @ A @ ( lattic2065218050nf_fin @ A @ A2 ) @ A3 )
% 2.90/1.63 = A3 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_238_Sup__fin_Oin__idem,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( semilattice_sup @ A )
% 2.90/1.63 => ! [A2: set @ A,X: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( member @ A @ X @ A2 )
% 2.90/1.63 => ( ( sup_sup @ A @ X @ ( lattic477160up_fin @ A @ A2 ) )
% 2.90/1.63 = ( lattic477160up_fin @ A @ A2 ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_239_Inf__fin_Oin__idem,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( semilattice_inf @ A )
% 2.90/1.63 => ! [A2: set @ A,X: A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( member @ A @ X @ A2 )
% 2.90/1.63 => ( ( inf_inf @ A @ X @ ( lattic2065218050nf_fin @ A @ A2 ) )
% 2.90/1.63 = ( lattic2065218050nf_fin @ A @ A2 ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_240_Sup__fin_Ounion,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( semilattice_sup @ A )
% 2.90/1.63 => ! [A2: set @ A,B3: set @ A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( finite_finite2 @ A @ B3 )
% 2.90/1.63 => ( ( B3
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( lattic477160up_fin @ A @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) )
% 2.90/1.63 = ( sup_sup @ A @ ( lattic477160up_fin @ A @ A2 ) @ ( lattic477160up_fin @ A @ B3 ) ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_241_Inf__fin_Ounion,axiom,
% 2.90/1.63 ! [A: $tType] :
% 2.90/1.63 ( ( semilattice_inf @ A )
% 2.90/1.63 => ! [A2: set @ A,B3: set @ A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ A2 )
% 2.90/1.63 => ( ( A2
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( finite_finite2 @ A @ B3 )
% 2.90/1.63 => ( ( B3
% 2.90/1.63 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.63 => ( ( lattic2065218050nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) )
% 2.90/1.63 = ( inf_inf @ A @ ( lattic2065218050nf_fin @ A @ A2 ) @ ( lattic2065218050nf_fin @ A @ B3 ) ) ) ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_242_trans__Int,axiom,
% 2.90/1.63 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ( ( trans @ A @ R2 )
% 2.90/1.63 => ( ( trans @ A @ S2 )
% 2.90/1.63 => ( trans @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S2 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_243_refl__on__Un,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A ),B3: set @ A,S2: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ( ( refl_on @ A @ A2 @ R2 )
% 2.90/1.63 => ( ( refl_on @ A @ B3 @ S2 )
% 2.90/1.63 => ( refl_on @ A @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S2 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_244_refl__on__Int,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,R2: set @ ( product_prod @ A @ A ),B3: set @ A,S2: set @ ( product_prod @ A @ A )] :
% 2.90/1.63 ( ( refl_on @ A @ A2 @ R2 )
% 2.90/1.63 => ( ( refl_on @ A @ B3 @ S2 )
% 2.90/1.63 => ( refl_on @ A @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S2 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_245_finite__UnI,axiom,
% 2.90/1.63 ! [A: $tType,F5: set @ A,G5: set @ A] :
% 2.90/1.63 ( ( finite_finite2 @ A @ F5 )
% 2.90/1.63 => ( ( finite_finite2 @ A @ G5 )
% 2.90/1.63 => ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F5 @ G5 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_246_Un__infinite,axiom,
% 2.90/1.63 ! [A: $tType,S: set @ A,T3: set @ A] :
% 2.90/1.63 ( ~ ( finite_finite2 @ A @ S )
% 2.90/1.63 => ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T3 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_247_infinite__Un,axiom,
% 2.90/1.63 ! [A: $tType,S: set @ A,T3: set @ A] :
% 2.90/1.63 ( ( ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T3 ) ) )
% 2.90/1.63 = ( ~ ( finite_finite2 @ A @ S )
% 2.90/1.63 | ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_248_Un__Int__assoc__eq,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,B3: set @ A,C3: set @ A] :
% 2.90/1.63 ( ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) @ C3 )
% 2.90/1.63 = ( inf_inf @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) ) )
% 2.90/1.63 = ( ord_less_eq @ ( set @ A ) @ C3 @ A2 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_249_Int__mono,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,C3: set @ A,B3: set @ A,D4: set @ A] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 2.90/1.63 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ D4 )
% 2.90/1.63 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) @ ( inf_inf @ ( set @ A ) @ C3 @ D4 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_250_Int__lower1,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) @ A2 ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_251_Int__lower2,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) @ B3 ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_252_Int__absorb1,axiom,
% 2.90/1.63 ! [A: $tType,B3: set @ A,A2: set @ A] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
% 2.90/1.63 => ( ( inf_inf @ ( set @ A ) @ A2 @ B3 )
% 2.90/1.63 = B3 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_253_Int__absorb2,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 2.90/1.63 => ( ( inf_inf @ ( set @ A ) @ A2 @ B3 )
% 2.90/1.63 = A2 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_254_Int__greatest,axiom,
% 2.90/1.63 ! [A: $tType,C3: set @ A,A2: set @ A,B3: set @ A] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ C3 @ A2 )
% 2.90/1.63 => ( ( ord_less_eq @ ( set @ A ) @ C3 @ B3 )
% 2.90/1.63 => ( ord_less_eq @ ( set @ A ) @ C3 @ ( inf_inf @ ( set @ A ) @ A2 @ B3 ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(fact_255_Int__Collect__mono,axiom,
% 2.90/1.63 ! [A: $tType,A2: set @ A,B3: set @ A,P: A > $o,Q: A > $o] :
% 2.90/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 2.90/1.63 => ( ! [X4: A] :
% 2.90/1.63 ( ( member @ A @ X4 @ A2 )
% 2.90/1.63 => ( ( P @ X4 )
% 2.90/1.63 => ( Q @ X4 ) ) )
% 2.90/1.63 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B3 @ ( collect @ A @ Q ) ) ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( semilattice_sup @ A10 )
% 2.90/1.63 => ( semilattice_sup @ ( A9 > A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( semilattice_inf @ A10 )
% 2.90/1.63 => ( semilattice_inf @ ( A9 > A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( order_bot @ A10 )
% 2.90/1.63 => ( order_bot @ ( A9 > A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( preorder @ A10 )
% 2.90/1.63 => ( preorder @ ( A9 > A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( ( finite_finite @ A9 )
% 2.90/1.63 & ( finite_finite @ A10 ) )
% 2.90/1.63 => ( finite_finite @ ( A9 > A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_fun___Lattices_Olattice,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( lattice @ A10 )
% 2.90/1.63 => ( lattice @ ( A9 > A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( order @ A10 )
% 2.90/1.63 => ( order @ ( A9 > A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_fun___Orderings_Oord,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( ord @ A10 )
% 2.90/1.63 => ( ord @ ( A9 > A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_fun___Orderings_Obot,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( bot @ A10 )
% 2.90/1.63 => ( bot @ ( A9 > A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_fun___Lattices_Osup,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( semilattice_sup @ A10 )
% 2.90/1.63 => ( sup @ ( A9 > A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_fun___Lattices_Oinf,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( semilattice_inf @ A10 )
% 2.90/1.63 => ( inf @ ( A9 > A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1,axiom,
% 2.90/1.63 ! [A9: $tType] : ( semilattice_sup @ ( set @ A9 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_2,axiom,
% 2.90/1.63 ! [A9: $tType] : ( semilattice_inf @ ( set @ A9 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Set_Oset___Orderings_Oorder__bot_3,axiom,
% 2.90/1.63 ! [A9: $tType] : ( order_bot @ ( set @ A9 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 2.90/1.63 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
% 2.90/1.63 ! [A9: $tType] :
% 2.90/1.63 ( ( finite_finite @ A9 )
% 2.90/1.63 => ( finite_finite @ ( set @ A9 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Set_Oset___Lattices_Olattice_6,axiom,
% 2.90/1.63 ! [A9: $tType] : ( lattice @ ( set @ A9 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Set_Oset___Orderings_Oorder_7,axiom,
% 2.90/1.63 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
% 2.90/1.63 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Set_Oset___Orderings_Obot_9,axiom,
% 2.90/1.63 ! [A9: $tType] : ( bot @ ( set @ A9 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Set_Oset___Lattices_Osup_10,axiom,
% 2.90/1.63 ! [A9: $tType] : ( sup @ ( set @ A9 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Set_Oset___Lattices_Oinf_11,axiom,
% 2.90/1.63 ! [A9: $tType] : ( inf @ ( set @ A9 ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_12,axiom,
% 2.90/1.63 semilattice_sup @ $o ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_13,axiom,
% 2.90/1.63 semilattice_inf @ $o ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_HOL_Obool___Orderings_Oorder__bot_14,axiom,
% 2.90/1.63 order_bot @ $o ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_HOL_Obool___Orderings_Opreorder_15,axiom,
% 2.90/1.63 preorder @ $o ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.90/1.63 linorder @ $o ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_HOL_Obool___Finite__Set_Ofinite_16,axiom,
% 2.90/1.63 finite_finite @ $o ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_HOL_Obool___Lattices_Olattice_17,axiom,
% 2.90/1.63 lattice @ $o ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_HOL_Obool___Orderings_Oorder_18,axiom,
% 2.90/1.63 order @ $o ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_HOL_Obool___Orderings_Oord_19,axiom,
% 2.90/1.63 ord @ $o ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_HOL_Obool___Orderings_Obot_20,axiom,
% 2.90/1.63 bot @ $o ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_HOL_Obool___Lattices_Osup_21,axiom,
% 2.90/1.63 sup @ $o ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_HOL_Obool___Lattices_Oinf_22,axiom,
% 2.90/1.63 inf @ $o ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Product__Type_Oprod___Lattices_Osemilattice__sup_23,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( ( semilattice_sup @ A9 )
% 2.90/1.63 & ( semilattice_sup @ A10 ) )
% 2.90/1.63 => ( semilattice_sup @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Product__Type_Oprod___Lattices_Osemilattice__inf_24,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( ( semilattice_inf @ A9 )
% 2.90/1.63 & ( semilattice_inf @ A10 ) )
% 2.90/1.63 => ( semilattice_inf @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Product__Type_Oprod___Orderings_Oorder__bot_25,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( ( order_bot @ A9 )
% 2.90/1.63 & ( order_bot @ A10 ) )
% 2.90/1.63 => ( order_bot @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_26,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( ( preorder @ A9 )
% 2.90/1.63 & ( preorder @ A10 ) )
% 2.90/1.63 => ( preorder @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_27,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( ( finite_finite @ A9 )
% 2.90/1.63 & ( finite_finite @ A10 ) )
% 2.90/1.63 => ( finite_finite @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Product__Type_Oprod___Lattices_Olattice_28,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( ( lattice @ A9 )
% 2.90/1.63 & ( lattice @ A10 ) )
% 2.90/1.63 => ( lattice @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Product__Type_Oprod___Orderings_Oorder_29,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( ( order @ A9 )
% 2.90/1.63 & ( order @ A10 ) )
% 2.90/1.63 => ( order @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Product__Type_Oprod___Orderings_Oord_30,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( ( ord @ A9 )
% 2.90/1.63 & ( ord @ A10 ) )
% 2.90/1.63 => ( ord @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Product__Type_Oprod___Orderings_Obot_31,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( ( bot @ A9 )
% 2.90/1.63 & ( bot @ A10 ) )
% 2.90/1.63 => ( bot @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Product__Type_Oprod___Lattices_Osup_32,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( ( sup @ A9 )
% 2.90/1.63 & ( sup @ A10 ) )
% 2.90/1.63 => ( sup @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(tcon_Product__Type_Oprod___Lattices_Oinf_33,axiom,
% 2.90/1.63 ! [A9: $tType,A10: $tType] :
% 2.90/1.63 ( ( ( inf @ A9 )
% 2.90/1.63 & ( inf @ A10 ) )
% 2.90/1.63 => ( inf @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.90/1.63
% 2.90/1.63 thf(conj_0,conjecture,
% 2.90/1.63 ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ x @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ x @ carrier @ relation ) )
% 2.90/1.63 | ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ x @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ x @ carrier @ relation ) ) ) ).
% 2.90/1.63 ------------------------
% 2.90/1.67 % (23261)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_22979 for (2999ds/2Mi)
% 2.90/1.67 % (23260)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22979 for (2999ds/2Mi)
% 2.90/1.67 % (23261)Instruction limit reached!
% 2.90/1.67 % (23261)------------------------------
% 2.90/1.67 % (23261)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.67 % (23261)Termination reason: Unknown
% 2.90/1.67 % (23261)Termination phase: shuffling
% 2.90/1.67
% 2.90/1.67 % (23261)Memory used [KB]: 1407
% 2.90/1.67 % (23261)Time elapsed: 0.003 s
% 2.90/1.67 % (23261)Instructions burned: 4 (million)
% 2.90/1.67 % (23261)------------------------------
% 2.90/1.67 % (23261)------------------------------
% 2.90/1.67 % (23260)Instruction limit reached!
% 2.90/1.67 % (23260)------------------------------
% 2.90/1.67 % (23260)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.67 % (23260)Termination reason: Unknown
% 2.90/1.67 % (23260)Termination phase: shuffling
% 2.90/1.67
% 2.90/1.67 % (23260)Memory used [KB]: 1407
% 2.90/1.67 % (23260)Time elapsed: 0.003 s
% 2.90/1.67 % (23260)Instructions burned: 4 (million)
% 2.90/1.67 % (23260)------------------------------
% 2.90/1.67 % (23260)------------------------------
% 2.90/1.67 % (23262)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_22979 for (2999ds/275Mi)
% 2.90/1.67 % (23257)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22979 for (2999ds/183Mi)
% 2.90/1.67 % (23258)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_22979 for (2999ds/4Mi)
% 2.90/1.67 % (23259)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_22979 for (2999ds/27Mi)
% 2.90/1.67 % (23263)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_22979 for (2999ds/18Mi)
% 2.90/1.68 % (23258)Instruction limit reached!
% 2.90/1.68 % (23258)------------------------------
% 2.90/1.68 % (23258)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.68 % (23258)Termination reason: Unknown
% 2.90/1.68 % (23258)Termination phase: shuffling
% 2.90/1.68
% 2.90/1.68 % (23258)Memory used [KB]: 1407
% 2.90/1.68 % (23258)Time elapsed: 0.005 s
% 2.90/1.68 % (23258)Instructions burned: 5 (million)
% 2.90/1.68 % (23258)------------------------------
% 2.90/1.68 % (23258)------------------------------
% 2.90/1.68 % (23264)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22979 for (2999ds/3Mi)
% 2.90/1.68 % (23265)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_22979 for (2999ds/37Mi)
% 2.90/1.68 % (23264)Instruction limit reached!
% 2.90/1.68 % (23264)------------------------------
% 2.90/1.68 % (23264)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.68 % (23264)Termination reason: Unknown
% 2.90/1.68 % (23264)Termination phase: shuffling
% 2.90/1.68
% 2.90/1.68 % (23264)Memory used [KB]: 1407
% 2.90/1.68 % (23264)Time elapsed: 0.003 s
% 2.90/1.68 % (23264)Instructions burned: 5 (million)
% 2.90/1.68 % (23264)------------------------------
% 2.90/1.68 % (23264)------------------------------
% 2.90/1.69 % (23263)Instruction limit reached!
% 2.90/1.69 % (23263)------------------------------
% 2.90/1.69 % (23263)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.69 % (23263)Termination reason: Unknown
% 2.90/1.69 % (23263)Termination phase: shuffling
% 2.90/1.69
% 2.90/1.69 % (23263)Memory used [KB]: 1791
% 2.90/1.69 % (23263)Time elapsed: 0.014 s
% 2.90/1.69 % (23263)Instructions burned: 18 (million)
% 2.90/1.69 % (23263)------------------------------
% 2.90/1.69 % (23263)------------------------------
% 2.90/1.69 % (23267)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_22979 for (2999ds/3Mi)
% 2.90/1.69 % (23265)Instruction limit reached!
% 2.90/1.69 % (23265)------------------------------
% 2.90/1.69 % (23265)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.69 % (23265)Termination reason: Unknown
% 2.90/1.69 % (23265)Termination phase: shuffling
% 2.90/1.69
% 2.90/1.69 % (23265)Memory used [KB]: 2174
% 2.90/1.69 % (23265)Time elapsed: 0.012 s
% 2.90/1.69 % (23265)Instructions burned: 38 (million)
% 2.90/1.69 % (23265)------------------------------
% 2.90/1.69 % (23265)------------------------------
% 2.90/1.69 % (23267)Instruction limit reached!
% 2.90/1.69 % (23267)------------------------------
% 2.90/1.69 % (23267)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.69 % (23267)Termination reason: Unknown
% 2.90/1.69 % (23267)Termination phase: shuffling
% 2.90/1.69
% 2.90/1.69 % (23267)Memory used [KB]: 1407
% 2.90/1.69 % (23267)Time elapsed: 0.003 s
% 2.90/1.69 % (23267)Instructions burned: 5 (million)
% 2.90/1.69 % (23267)------------------------------
% 2.90/1.69 % (23267)------------------------------
% 2.90/1.69 % (23259)Instruction limit reached!
% 2.90/1.69 % (23259)------------------------------
% 2.90/1.69 % (23259)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.69 % (23259)Termination reason: Unknown
% 2.90/1.69 % (23259)Termination phase: shuffling
% 2.90/1.69
% 2.90/1.69 % (23259)Memory used [KB]: 1918
% 2.90/1.69 % (23259)Time elapsed: 0.020 s
% 2.90/1.69 % (23259)Instructions burned: 27 (million)
% 2.90/1.69 % (23259)------------------------------
% 2.90/1.69 % (23259)------------------------------
% 2.90/1.69 % (23266)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_22979 for (2999ds/15Mi)
% 2.96/1.70 % (23269)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_22979 for (2999ds/7Mi)
% 2.96/1.70 % (23270)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_22979 for (2999ds/16Mi)
% 2.96/1.70 % (23269)Instruction limit reached!
% 2.96/1.70 % (23269)------------------------------
% 2.96/1.70 % (23269)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.70 % (23269)Termination reason: Unknown
% 2.96/1.70 % (23269)Termination phase: shuffling
% 2.96/1.70
% 2.96/1.70 % (23269)Memory used [KB]: 1535
% 2.96/1.70 % (23269)Time elapsed: 0.004 s
% 2.96/1.70 % (23269)Instructions burned: 8 (million)
% 2.96/1.70 % (23269)------------------------------
% 2.96/1.70 % (23269)------------------------------
% 2.96/1.70 % (23262)First to succeed.
% 2.96/1.70 % (23266)Instruction limit reached!
% 2.96/1.70 % (23266)------------------------------
% 2.96/1.70 % (23266)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.70 % (23266)Termination reason: Unknown
% 2.96/1.70 % (23266)Termination phase: shuffling
% 2.96/1.70 % (23268)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_22979 for (2999ds/1041Mi)
% 2.96/1.70
% 2.96/1.70 % (23266)Memory used [KB]: 1663
% 2.96/1.70 % (23266)Time elapsed: 0.012 s
% 2.96/1.70 % (23266)Instructions burned: 15 (million)
% 2.96/1.70 % (23266)------------------------------
% 2.96/1.70 % (23266)------------------------------
% 2.96/1.70 % (23262)Refutation found. Thanks to Tanya!
% 2.96/1.70 % SZS status Theorem for DTF2THF_22979
% 2.96/1.70 % SZS output start Proof for DTF2THF_22979
% 2.96/1.70 thf(type_def_5, type, product_prod: ($tType * $tType) > $tType).
% 2.96/1.70 thf(type_def_6, type, set: ($tType) > $tType).
% 2.96/1.70 thf(type_def_8, type, a: $tType).
% 2.96/1.70 thf(func_def_0, type, product_prod: ($tType * $tType) > $tType).
% 2.96/1.70 thf(func_def_1, type, set: ($tType) > $tType).
% 2.96/1.70 thf(func_def_2, type, a: $tType).
% 2.96/1.70 thf(func_def_3, type, type: !>[X0: $tType]:($o)).
% 2.96/1.70 thf(func_def_4, type, inf: !>[X0: $tType]:($o)).
% 2.96/1.70 thf(func_def_5, type, sup: !>[X0: $tType]:($o)).
% 2.96/1.70 thf(func_def_6, type, bot: !>[X0: $tType]:($o)).
% 2.96/1.70 thf(func_def_7, type, ord: !>[X0: $tType]:($o)).
% 2.96/1.70 thf(func_def_8, type, order: !>[X0: $tType]:($o)).
% 2.96/1.70 thf(func_def_9, type, lattice: !>[X0: $tType]:($o)).
% 2.96/1.70 thf(func_def_10, type, finite_finite: !>[X0: $tType]:($o)).
% 2.96/1.70 thf(func_def_11, type, linorder: !>[X0: $tType]:($o)).
% 2.96/1.70 thf(func_def_12, type, preorder: !>[X0: $tType]:($o)).
% 2.96/1.70 thf(func_def_13, type, order_bot: !>[X0: $tType]:($o)).
% 2.96/1.70 thf(func_def_14, type, semilattice_inf: !>[X0: $tType]:($o)).
% 2.96/1.70 thf(func_def_15, type, semilattice_sup: !>[X0: $tType]:($o)).
% 2.96/1.70 thf(func_def_16, type, bNF_Ca1785829860lChain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 2.96/1.70 thf(func_def_17, type, finite_finite2: !>[X0: $tType]:(set @ X0 > $o)).
% 2.96/1.70 thf(func_def_18, type, inf_inf: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.96/1.70 thf(func_def_19, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.96/1.70 thf(func_def_20, type, lattic929149872er_Max: !>[X0: $tType]:(set @ X0 > X0)).
% 2.96/1.70 thf(func_def_21, type, lattic929674654er_Min: !>[X0: $tType]:(set @ X0 > X0)).
% 2.96/1.70 thf(func_def_22, type, lattic1704895705min_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X0)).
% 2.96/1.70 thf(func_def_23, type, lattic2065218050nf_fin: !>[X0: $tType]:(set @ X0 > X0)).
% 2.96/1.70 thf(func_def_24, type, lattic477160up_fin: !>[X0: $tType]:(set @ X0 > X0)).
% 2.96/1.70 thf(func_def_25, type, order_preorder_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 2.96/1.70 thf(func_def_26, type, order_under: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0 > set @ X0)).
% 2.96/1.70 thf(func_def_27, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.96/1.70 thf(func_def_28, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.96/1.70 thf(func_def_29, type, order_Greatest: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.96/1.70 thf(func_def_30, type, prefer951318096ood_as: !>[X0: $tType]:(X0 > set @ X0 > set @ product_prod @ X0 @ X0 > set @ X0)).
% 2.96/1.70 thf(func_def_31, type, prefer310429814s_good: !>[X0: $tType]:(X0 > set @ X0 > set @ product_prod @ X0 @ X0 > set @ X0)).
% 2.96/1.70 thf(func_def_32, type, prefer1532642881r_than: !>[X0: $tType]:(X0 > set @ X0 > set @ product_prod @ X0 @ X0 > set @ X0)).
% 2.96/1.70 thf(func_def_33, type, prefer199794634erence: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 2.96/1.70 thf(func_def_34, type, prefer1997167224erence: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 2.96/1.70 thf(func_def_35, type, prefer1801827867axioms: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 2.96/1.70 thf(func_def_36, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 2.96/1.70 thf(func_def_37, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.96/1.70 thf(func_def_38, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.96/1.70 thf(func_def_39, type, id_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0)).
% 2.96/1.70 thf(func_def_40, type, refl_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 2.96/1.70 thf(func_def_41, type, total_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 2.96/1.70 thf(func_def_42, type, trans: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 2.96/1.70 thf(func_def_43, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.96/1.70 thf(func_def_44, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 2.96/1.70 thf(func_def_45, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.96/1.70 thf(func_def_46, type, carrier: set @ a).
% 2.96/1.70 thf(func_def_47, type, relation: set @ product_prod @ a @ a).
% 2.96/1.70 thf(func_def_48, type, x: a).
% 2.96/1.70 thf(func_def_49, type, y: a).
% 2.96/1.70 thf(func_def_64, type, sK0: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.96/1.70 thf(func_def_65, type, sK1: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.96/1.70 thf(func_def_66, type, sK2: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.96/1.70 thf(func_def_67, type, sK3: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.96/1.70 thf(func_def_68, type, sK4: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.96/1.70 thf(func_def_69, type, sK5: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.96/1.70 thf(func_def_70, type, sK6: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 2.96/1.70 thf(func_def_71, type, sK7: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 2.96/1.70 thf(func_def_72, type, sK8: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 2.96/1.71 thf(func_def_73, type, sK9: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 2.96/1.71 thf(func_def_74, type, sK10: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 2.96/1.71 thf(func_def_75, type, sK11: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.96/1.71 thf(func_def_76, type, sK12: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.96/1.71 thf(func_def_77, type, sK13: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.96/1.71 thf(func_def_78, type, sK14: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.96/1.71 thf(func_def_79, type, sK15: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.96/1.71 thf(func_def_80, type, sK16: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 2.96/1.71 thf(func_def_81, type, sK17: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 2.96/1.71 thf(func_def_82, type, sK18: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 2.96/1.71 thf(func_def_83, type, sK19: !>[X0: $tType]:(set @ X0 > X0)).
% 2.96/1.71 thf(func_def_84, type, sK20: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 2.96/1.71 thf(func_def_85, type, sK21: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 2.96/1.71 thf(func_def_86, type, sK22: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 2.96/1.71 thf(func_def_87, type, sK23: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 2.96/1.71 thf(func_def_88, type, sK24: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.96/1.71 thf(func_def_89, type, sK25: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.96/1.71 thf(func_def_90, type, sK26: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.96/1.71 thf(func_def_91, type, sK27: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.96/1.71 thf(func_def_92, type, sK28: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.96/1.71 thf(func_def_93, type, sK29: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.96/1.71 thf(func_def_94, type, sK30: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.96/1.71 thf(func_def_95, type, sK31: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.96/1.71 thf(func_def_96, type, sK32: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.96/1.71 thf(func_def_97, type, sK33: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.96/1.71 thf(func_def_98, type, sK34: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.96/1.71 thf(func_def_99, type, sK35: !>[X0: $tType]:(set @ X0 > X0)).
% 2.96/1.71 thf(func_def_100, type, sK36: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X0)).
% 2.96/1.71 thf(func_def_101, type, sK37: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X1)).
% 2.96/1.71 thf(func_def_102, type, sK38: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X2)).
% 2.96/1.71 thf(func_def_103, type, sK39: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X3)).
% 2.96/1.71 thf(func_def_104, type, sK40: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X4)).
% 2.96/1.71 thf(func_def_105, type, sK41: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X5)).
% 2.96/1.71 thf(func_def_106, type, sK42: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X6)).
% 2.96/1.71 thf(func_def_107, type, sK43: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 2.96/1.71 thf(func_def_108, type, sK44: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.96/1.71 thf(func_def_109, type, sK45: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.96/1.71 thf(func_def_110, type, sK46: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.96/1.71 thf(func_def_111, type, sK47: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.96/1.71 thf(func_def_112, type, sK48: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.96/1.71 thf(func_def_113, type, sK49: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1)).
% 2.96/1.71 thf(func_def_114, type, sK50: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1 > X0)).
% 2.96/1.71 thf(func_def_115, type, sK51: !>[X0: $tType]:(set @ X0 > X0)).
% 2.96/1.71 thf(func_def_116, type, sK52: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > X0)).
% 2.96/1.71 thf(func_def_117, type, sK53: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.96/1.71 thf(func_def_118, type, sK54: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.96/1.71 thf(func_def_119, type, sK55: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.96/1.71 thf(func_def_120, type, sK56: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.96/1.71 thf(func_def_121, type, sK57: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 2.96/1.71 thf(func_def_122, type, sK58: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.96/1.71 thf(func_def_123, type, sK59: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.96/1.71 thf(func_def_124, type, sK60: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.96/1.71 thf(func_def_125, type, sK61: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.96/1.71 thf(func_def_126, type, sK62: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.96/1.71 thf(func_def_127, type, sK63: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X1)).
% 2.96/1.71 thf(func_def_128, type, sK64: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X0)).
% 2.96/1.71 thf(func_def_129, type, sK65: !>[X0: $tType]:(set @ X0 > X0)).
% 2.96/1.71 thf(func_def_130, type, sK66: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X0)).
% 2.96/1.71 thf(func_def_131, type, sK67: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X1)).
% 2.96/1.71 thf(func_def_132, type, sK68: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X2)).
% 2.96/1.71 thf(func_def_133, type, sK69: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X3)).
% 2.96/1.71 thf(func_def_134, type, sK70: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X4)).
% 2.96/1.71 thf(func_def_135, type, sK71: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X5)).
% 2.96/1.71 thf(func_def_136, type, sK72: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X6)).
% 2.96/1.71 thf(func_def_138, type, ph74: !>[X0: $tType]:(X0)).
% 2.96/1.71 thf(f1174,plain,(
% 2.96/1.71 $false),
% 2.96/1.71 inference(subsumption_resolution,[status(thm)],[f1148,f1173])).
% 2.96/1.71 thf(f1173,plain,(
% 2.96/1.71 ( ! [X0 : $tType,X2 : set @ X0] : (((ord_less_eq @ set @ X0 @ X2 @ X2) = $true)) )),
% 2.96/1.71 inference(equality_resolution,[status(thm)],[f1136])).
% 2.96/1.71 thf(f1136,plain,(
% 2.96/1.71 ( ! [X0 : $tType,X2 : set @ X0,X1 : set @ X0] : (($true = (ord_less_eq @ set @ X0 @ X1 @ X2)) | (X1 != X2)) )),
% 2.96/1.71 inference(cnf_transformation,[status(thm)],[f907])).
% 2.96/1.71 thf(f907,plain,(
% 2.96/1.71 ! [X0 : $tType,X1 : set @ X0,X2 : set @ X0] : ((((ord_less_eq @ set @ X0 @ X2 @ X1) = $true) & ($true = (ord_less_eq @ set @ X0 @ X1 @ X2))) | (X1 != X2))),
% 2.96/1.71 inference(ennf_transformation,[status(thm)],[f877])).
% 2.96/1.71 thf(f877,plain,(
% 2.96/1.71 ! [X0 : $tType,X1 : set @ X0,X2 : set @ X0] : ((X1 = X2) => ~(($true = (ord_less_eq @ set @ X0 @ X1 @ X2)) => ((ord_less_eq @ set @ X0 @ X2 @ X1) != $true)))),
% 2.96/1.71 inference(flattening,[status(thm)],[f574])).
% 2.96/1.71 thf(f574,plain,(
% 2.96/1.71 ! [X0 : $tType,X1 : set @ X0,X2 : set @ X0] : ((X1 = X2) => ~(($true = (ord_less_eq @ set @ X0 @ X1 @ X2)) => ~((ord_less_eq @ set @ X0 @ X2 @ X1) = $true)))),
% 2.96/1.71 inference(fool_elimination,[status(thm)],[f573])).
% 2.96/1.71 thf(f573,plain,(
% 2.96/1.71 ! [X0 : $tType,X1 : set @ X0,X2 : set @ X0] : ((X1 = X2) => ~((ord_less_eq @ set @ X0 @ X1 @ X2) => ~(ord_less_eq @ set @ X0 @ X2 @ X1)))),
% 2.96/1.71 inference(rectify,[status(thm)],[f70])).
% 2.96/1.71 thf(f70,axiom,(
% 2.96/1.71 ! [X0 : $tType,X17 : set @ X0,X26 : set @ X0] : ((X17 = X26) => ~((ord_less_eq @ set @ X0 @ X17 @ X26) => ~(ord_less_eq @ set @ X0 @ X26 @ X17)))),
% 2.96/1.71 file('/export/starexec/sandbox/tmp/tmp.bwuTtVduWu/DTF2THF_22979.p',fact_69_equalityE)).
% 2.96/1.71 thf(f1148,plain,(
% 2.96/1.71 ((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation) @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) != $true)),
% 2.96/1.71 inference(cnf_transformation,[status(thm)],[f945])).
% 2.96/1.71 thf(f945,plain,(
% 2.96/1.71 ((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation) @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) != $true) & ((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation) @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) != $true)),
% 2.96/1.71 inference(ennf_transformation,[status(thm)],[f488])).
% 2.96/1.71 thf(f488,plain,(
% 2.96/1.71 ~(((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation) @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) = $true) | ((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation) @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) = $true))),
% 2.96/1.71 inference(fool_elimination,[status(thm)],[f487])).
% 2.96/1.71 thf(f487,plain,(
% 2.96/1.71 ~((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation) @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) | (ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation) @ (prefer1532642881r_than @ a @ x @ carrier @ relation)))),
% 2.96/1.71 inference(rectify,[status(thm)],[f303])).
% 2.96/1.71 thf(f303,negated_conjecture,(
% 2.96/1.71 ~((ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation) @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) | (ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation) @ (prefer1532642881r_than @ a @ x @ carrier @ relation)))),
% 2.96/1.71 inference(negated_conjecture,[status(cth)],[f302])).
% 2.96/1.71 thf(f302,conjecture,(
% 2.96/1.71 (ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation) @ (prefer1532642881r_than @ a @ x @ carrier @ relation)) | (ord_less_eq @ set @ a @ (prefer1532642881r_than @ a @ x @ carrier @ relation) @ (prefer1532642881r_than @ a @ x @ carrier @ relation))),
% 2.96/1.71 file('/export/starexec/sandbox/tmp/tmp.bwuTtVduWu/DTF2THF_22979.p',conj_0)).
% 2.96/1.71 % SZS output end Proof for DTF2THF_22979
% 2.96/1.71 % (23262)------------------------------
% 2.96/1.71 % (23262)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.71 % (23262)Termination reason: Refutation
% 2.96/1.71
% 2.96/1.71 % (23262)Memory used [KB]: 6908
% 2.96/1.71 % (23262)Time elapsed: 0.036 s
% 2.96/1.71 % (23262)Instructions burned: 81 (million)
% 2.96/1.71 % (23262)------------------------------
% 2.96/1.71 % (23262)------------------------------
% 2.96/1.71 % (23256)Success in time 0.064 s
%------------------------------------------------------------------------------