%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP153^2 : 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 : n022.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 2.47s 1.49s
% Output : Refutation 2.47s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP153^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.11 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.12/0.31 % Computer : n022.cluster.edu
% 0.12/0.31 % Model : x86_64 x86_64
% 0.12/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.31 % Memory : 8042.1875MB
% 0.12/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.31 % CPULimit : 300
% 0.12/0.31 % WCLimit : 300
% 0.12/0.31 % DateTime : Tue Feb 24 08:28:06 EST 2026
% 0.12/0.31 % CPUTime :
% 0.12/0.31 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.43 ---- Original DTF file ---
% 0.15/0.43 thf(spec,logic,$$dhol).
% 0.15/0.43 %------------------------------------------------------------------------------
% 0.15/0.43 % File : ITP153^2 : TPTP v9.2.1. Released v7.5.0.
% 0.15/0.43 % Domain : Interactive Theorem Proving
% 0.15/0.43 % Problem : Sledgehammer Preferences problem prob_180__6250282_1
% 0.15/0.43 % Version : Especial.
% 0.15/0.43 % English :
% 0.15/0.43
% 0.15/0.43 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.15/0.43 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.15/0.43 % Source : [Des21]
% 0.15/0.43 % Names : Preferences/prob_180__6250282_1 [Des21]
% 0.15/0.43
% 0.15/0.43 % Status : Theorem
% 0.15/0.43 % Rating : 0.00 v7.5.0
% 0.15/0.43 % Syntax : Number of formulae : 324 ( 113 unt; 46 typ; 0 def)
% 0.15/0.43 % Number of atoms : 777 ( 217 equ; 0 cnn)
% 0.15/0.43 % Maximal formula atoms : 13 ( 2 avg)
% 0.15/0.43 % Number of connectives : 4707 ( 88 ~; 15 |; 79 &;4193 @)
% 0.15/0.43 % ( 0 <=>; 332 =>; 0 <=; 0 <~>)
% 0.15/0.43 % Maximal formula depth : 28 ( 9 avg)
% 0.15/0.43 % Number of types : 2 ( 1 usr)
% 0.15/0.43 % Number of type conns : 124 ( 124 >; 0 *; 0 +; 0 <<)
% 0.15/0.43 % Number of symbols : 46 ( 45 usr; 4 con; 0-5 aty)
% 0.15/0.43 % Number of variables : 1093 ( 47 ^; 985 !; 11 ?;1093 :)
% 0.15/0.43 % ( 50 !>; 0 ?*; 0 @-; 0 @+)
% 0.15/0.43 % SPC : TH1_THM_EQU_NAR
% 0.15/0.43
% 0.15/0.43 % Comments : This file was generated by Sledgehammer 2021-02-23 16:28:41.875
% 0.15/0.43 %------------------------------------------------------------------------------
% 0.15/0.43 % Could-be-implicit typings (3)
% 0.15/0.43 thf(ty_t_Product__Type_Oprod,type,
% 0.15/0.43 product_prod: $tType > $tType > $tType ).
% 0.15/0.43
% 0.15/0.43 thf(ty_t_Set_Oset,type,
% 0.15/0.43 set: $tType > $tType ).
% 0.15/0.43
% 0.15/0.43 thf(ty_tf_a,type,
% 0.15/0.43 a: $tType ).
% 0.15/0.43
% 0.15/0.43 % Explicit typings (43)
% 0.15/0.43 thf(sy_cl_HOL_Otype,type,
% 0.15/0.43 type:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_cl_Orderings_Obot,type,
% 0.15/0.43 bot:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_cl_Orderings_Oord,type,
% 0.15/0.43 ord:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_cl_Orderings_Oorder,type,
% 0.15/0.43 order:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_cl_Orderings_Olinorder,type,
% 0.15/0.43 linorder:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_cl_Orderings_Opreorder,type,
% 0.15/0.43 preorder:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.15/0.43 order_bot:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.15/0.43 bNF_Ca1785829860lChain:
% 0.15/0.43 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.15/0.43 uminus_uminus:
% 0.15/0.43 !>[A: $tType] : ( A > A ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Order__Relation_Olinear__order__on,type,
% 0.15/0.43 order_1409979114der_on:
% 0.15/0.43 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Order__Relation_Opreorder__on,type,
% 0.15/0.43 order_preorder_on:
% 0.15/0.43 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Order__Relation_Ounder,type,
% 0.15/0.43 order_under:
% 0.15/0.43 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.15/0.43 bot_bot:
% 0.15/0.43 !>[A: $tType] : A ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.15/0.43 ord_less_eq:
% 0.15/0.43 !>[A: $tType] : ( A > A > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 0.15/0.43 order_Greatest:
% 0.15/0.43 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oas__good__as,type,
% 0.15/0.43 prefer951318096ood_as:
% 0.15/0.43 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oat__least__as__good,type,
% 0.15/0.43 prefer310429814s_good:
% 0.15/0.43 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Ono__better__than,type,
% 0.15/0.43 prefer1532642881r_than:
% 0.15/0.43 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Opreference,type,
% 0.15/0.43 prefer199794634erence:
% 0.15/0.43 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference,type,
% 0.15/0.43 prefer1997167224erence:
% 0.15/0.43 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference__axioms,type,
% 0.15/0.43 prefer1801827867axioms:
% 0.15/0.43 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Product__Type_OPair,type,
% 0.15/0.43 product_Pair:
% 0.15/0.43 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.15/0.43 produc2004651681e_prod:
% 0.15/0.43 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.15/0.43 product_rec_prod:
% 0.15/0.43 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.15/0.43 product_fst:
% 0.15/0.43 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 0.15/0.43 product_snd:
% 0.15/0.43 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.15/0.43 product_swap:
% 0.15/0.43 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Relation_ODomain,type,
% 0.15/0.43 domain:
% 0.15/0.43 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Relation_OId__on,type,
% 0.15/0.43 id_on:
% 0.15/0.43 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Relation_ORange,type,
% 0.15/0.43 range:
% 0.15/0.43 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ B ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Relation_Orefl__on,type,
% 0.15/0.43 refl_on:
% 0.15/0.43 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Relation_Ototal__on,type,
% 0.15/0.43 total_on:
% 0.15/0.43 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Relation_Otrans,type,
% 0.15/0.43 trans:
% 0.15/0.43 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Set_OCollect,type,
% 0.15/0.43 collect:
% 0.15/0.43 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Set_Oinsert,type,
% 0.15/0.43 insert:
% 0.15/0.43 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Set_Ois__empty,type,
% 0.15/0.43 is_empty:
% 0.15/0.43 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Set_Ois__singleton,type,
% 0.15/0.43 is_singleton:
% 0.15/0.43 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Set_Othe__elem,type,
% 0.15/0.43 the_elem:
% 0.15/0.43 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_member,type,
% 0.15/0.43 member:
% 0.15/0.43 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_v_carrier,type,
% 0.15/0.43 carrier: set @ a ).
% 0.15/0.43
% 0.15/0.43 thf(sy_v_relation,type,
% 0.15/0.43 relation: set @ ( product_prod @ a @ a ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_v_y,type,
% 0.15/0.43 y: a ).
% 0.15/0.43
% 0.15/0.43 thf(sy_v_z,type,
% 0.15/0.43 z: a ).
% 0.15/0.43
% 0.15/0.43 % Relevant facts (256)
% 0.15/0.43 thf(fact_0_assms,axiom,
% 0.15/0.43 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ z @ y ) @ relation )
% 0.15/0.43 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ y @ z ) @ relation ) ) ).
% 0.15/0.43
% 0.15/0.43 % assms
% 0.15/0.43 thf(fact_1_rational__preference__axioms,axiom,
% 0.15/0.43 prefer1997167224erence @ a @ carrier @ relation ).
% 0.15/0.43
% 0.15/0.43 % rational_preference_axioms
% 0.15/0.43 thf(fact_2_preference__axioms,axiom,
% 0.15/0.43 prefer199794634erence @ a @ carrier @ relation ).
% 0.15/0.43
% 0.15/0.43 % preference_axioms
% 0.15/0.43 thf(fact_3_trans__refl,axiom,
% 0.15/0.43 order_preorder_on @ a @ carrier @ relation ).
% 0.15/0.43
% 0.15/0.43 % trans_refl
% 0.15/0.43 thf(fact_4_total,axiom,
% 0.15/0.43 total_on @ a @ carrier @ relation ).
% 0.15/0.43
% 0.15/0.43 % total
% 0.15/0.43 thf(fact_5_reflexivity,axiom,
% 0.15/0.43 refl_on @ a @ carrier @ relation ).
% 0.15/0.43
% 0.15/0.43 % reflexivity
% 0.15/0.43 thf(fact_6_compl,axiom,
% 0.15/0.43 ! [X: a] :
% 0.15/0.43 ( ( member @ a @ X @ carrier )
% 0.15/0.43 => ! [Xa: a] :
% 0.15/0.43 ( ( member @ a @ Xa @ carrier )
% 0.15/0.43 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Xa ) @ relation )
% 0.15/0.43 | ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Xa @ X ) @ relation ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % compl
% 0.15/0.43 thf(fact_7_completeD,axiom,
% 0.15/0.43 ! [X2: a,Y: a] :
% 0.15/0.43 ( ( member @ a @ X2 @ carrier )
% 0.15/0.43 => ( ( member @ a @ Y @ carrier )
% 0.15/0.43 => ( ( X2 != Y )
% 0.15/0.43 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
% 0.15/0.43 | ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X2 ) @ relation ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % completeD
% 0.15/0.43 thf(fact_8_not__outside,axiom,
% 0.15/0.43 ! [X2: a,Y: a] :
% 0.15/0.43 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
% 0.15/0.43 => ( member @ a @ X2 @ carrier ) ) ).
% 0.15/0.43
% 0.15/0.43 % not_outside
% 0.15/0.43 thf(fact_9_strict__is__neg__transitive,axiom,
% 0.15/0.43 ! [X2: a,Y: a,Z: a] :
% 0.15/0.43 ( ( ( member @ a @ X2 @ carrier )
% 0.15/0.43 & ( member @ a @ Y @ carrier )
% 0.15/0.43 & ( member @ a @ Z @ carrier ) )
% 0.15/0.43 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
% 0.15/0.43 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X2 ) @ relation ) )
% 0.15/0.43 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Z ) @ relation )
% 0.15/0.43 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X2 ) @ relation ) )
% 0.15/0.43 | ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation )
% 0.15/0.43 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % strict_is_neg_transitive
% 0.15/0.43 thf(fact_10_weak__is__transitive,axiom,
% 0.15/0.43 ! [X2: a,Y: a,Z: a] :
% 0.15/0.43 ( ( ( member @ a @ X2 @ carrier )
% 0.15/0.43 & ( member @ a @ Y @ carrier )
% 0.15/0.43 & ( member @ a @ Z @ carrier ) )
% 0.15/0.43 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
% 0.15/0.43 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
% 0.15/0.43 => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Z ) @ relation ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % weak_is_transitive
% 0.15/0.43 thf(fact_11_strict__not__refl__weak,axiom,
% 0.15/0.43 ! [X2: a,Y: a] :
% 0.15/0.43 ( ( ( member @ a @ X2 @ carrier )
% 0.15/0.43 & ( member @ a @ Y @ carrier ) )
% 0.15/0.43 => ( ( ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X2 ) @ relation ) )
% 0.15/0.43 = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
% 0.15/0.43 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X2 ) @ relation ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % strict_not_refl_weak
% 0.15/0.43 thf(fact_12_indiff__trans,axiom,
% 0.15/0.43 ! [X2: a,Y: a,Z: a] :
% 0.15/0.43 ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
% 0.15/0.43 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X2 ) @ relation ) )
% 0.15/0.43 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
% 0.15/0.43 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation ) )
% 0.15/0.43 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Z ) @ relation )
% 0.15/0.43 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X2 ) @ relation ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % indiff_trans
% 0.15/0.43 thf(fact_13_strict__trans,axiom,
% 0.15/0.43 ! [X2: a,Y: a,Z: a] :
% 0.15/0.43 ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
% 0.15/0.43 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X2 ) @ relation ) )
% 0.15/0.43 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
% 0.15/0.43 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation ) )
% 0.15/0.43 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Z ) @ relation )
% 0.15/0.43 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X2 ) @ relation ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % strict_trans
% 0.15/0.43 thf(fact_14_pref__in__at__least__as,axiom,
% 0.15/0.43 ! [X2: a,Y: a] :
% 0.15/0.43 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
% 0.15/0.43 => ( member @ a @ X2 @ ( prefer310429814s_good @ a @ Y @ carrier @ relation ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % pref_in_at_least_as
% 0.15/0.43 thf(fact_15_rational__preference_Oaxioms_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.15/0.43 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( prefer199794634erence @ A @ Carrier @ Relation ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference.axioms(1)
% 0.15/0.43 thf(fact_16_preference__def,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( prefer199794634erence @ A )
% 0.15/0.43 = ( ^ [Carrier2: set @ A,Relation2: set @ ( product_prod @ A @ A )] :
% 0.15/0.43 ( ! [X3: A,Y2: A] :
% 0.15/0.43 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ Relation2 )
% 0.15/0.43 => ( member @ A @ X3 @ Carrier2 ) )
% 0.15/0.43 & ! [X3: A,Y2: A] :
% 0.15/0.43 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ Relation2 )
% 0.15/0.43 => ( member @ A @ Y2 @ Carrier2 ) )
% 0.15/0.43 & ( order_preorder_on @ A @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % preference_def
% 0.15/0.43 thf(fact_17_preference_Ointro,axiom,
% 0.15/0.43 ! [A: $tType,Relation: set @ ( product_prod @ A @ A ),Carrier: set @ A] :
% 0.15/0.43 ( ! [X4: A,Y3: A] :
% 0.15/0.43 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ Relation )
% 0.15/0.43 => ( member @ A @ X4 @ Carrier ) )
% 0.15/0.43 => ( ! [X4: A,Y3: A] :
% 0.15/0.43 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ Relation )
% 0.15/0.43 => ( member @ A @ Y3 @ Carrier ) )
% 0.15/0.43 => ( ( order_preorder_on @ A @ Carrier @ Relation )
% 0.15/0.43 => ( prefer199794634erence @ A @ Carrier @ Relation ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % preference.intro
% 0.15/0.43 thf(fact_18_preference_Otrans__refl,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.15/0.43 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( order_preorder_on @ A @ Carrier @ Relation ) ) ).
% 0.15/0.43
% 0.15/0.43 % preference.trans_refl
% 0.15/0.43 thf(fact_19_preference_Onot__outside,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
% 0.15/0.43 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
% 0.15/0.43 => ( member @ A @ X2 @ Carrier ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % preference.not_outside
% 0.15/0.43 thf(fact_20_preference_Oreflexivity,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.15/0.43 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( refl_on @ A @ Carrier @ Relation ) ) ).
% 0.15/0.43
% 0.15/0.43 % preference.reflexivity
% 0.15/0.43 thf(fact_21_preference_Oindiff__trans,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A,Z: A] :
% 0.15/0.43 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
% 0.15/0.43 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ Relation ) )
% 0.15/0.43 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
% 0.15/0.43 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation ) )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z ) @ Relation )
% 0.15/0.43 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % preference.indiff_trans
% 0.15/0.43 thf(fact_22_rational__preference_Ocompl,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.15/0.43 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ! [X: A] :
% 0.15/0.43 ( ( member @ A @ X @ Carrier )
% 0.15/0.43 => ! [Xa: A] :
% 0.15/0.43 ( ( member @ A @ Xa @ Carrier )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Xa ) @ Relation )
% 0.15/0.43 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa @ X ) @ Relation ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference.compl
% 0.15/0.43 thf(fact_23_rational__preference_Ototal,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.15/0.43 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( total_on @ A @ Carrier @ Relation ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference.total
% 0.15/0.43 thf(fact_24_rational__preference_OcompleteD,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
% 0.15/0.43 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( ( member @ A @ X2 @ Carrier )
% 0.15/0.43 => ( ( member @ A @ Y @ Carrier )
% 0.15/0.43 => ( ( X2 != Y )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
% 0.15/0.43 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ Relation ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference.completeD
% 0.15/0.43 thf(fact_25_rational__preference_Ostrict__trans,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A,Z: A] :
% 0.15/0.43 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
% 0.15/0.43 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ Relation ) )
% 0.15/0.43 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
% 0.15/0.43 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation ) )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z ) @ Relation )
% 0.15/0.43 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference.strict_trans
% 0.15/0.43 thf(fact_26_rational__preference_Oweak__is__transitive,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A,Z: A] :
% 0.15/0.43 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( ( ( member @ A @ X2 @ Carrier )
% 0.15/0.43 & ( member @ A @ Y @ Carrier )
% 0.15/0.43 & ( member @ A @ Z @ Carrier ) )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
% 0.15/0.43 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z ) @ Relation ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference.weak_is_transitive
% 0.15/0.43 thf(fact_27_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
% 0.15/0.43 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( ( ( member @ A @ X2 @ Carrier )
% 0.15/0.43 & ( member @ A @ Y @ Carrier ) )
% 0.15/0.43 => ( ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ Relation ) )
% 0.15/0.43 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
% 0.15/0.43 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ Relation ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference.strict_not_refl_weak
% 0.15/0.43 thf(fact_28_rational__preference_Ostrict__is__neg__transitive,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A,Z: A] :
% 0.15/0.43 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( ( ( member @ A @ X2 @ Carrier )
% 0.15/0.43 & ( member @ A @ Y @ Carrier )
% 0.15/0.43 & ( member @ A @ Z @ Carrier ) )
% 0.15/0.43 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
% 0.15/0.43 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ Relation ) )
% 0.15/0.43 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z ) @ Relation )
% 0.15/0.43 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X2 ) @ Relation ) )
% 0.15/0.43 | ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation )
% 0.15/0.43 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference.strict_is_neg_transitive
% 0.15/0.43 thf(fact_29_worse__in__no__better,axiom,
% 0.15/0.43 ! [X2: a,Y: a] :
% 0.15/0.43 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
% 0.15/0.43 => ( member @ a @ Y @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % worse_in_no_better
% 0.15/0.43 thf(fact_30_as__good__as__sameIff,axiom,
% 0.15/0.43 ! [Z: a,Y: a] :
% 0.15/0.43 ( ( member @ a @ Z @ ( prefer951318096ood_as @ a @ Y @ carrier @ relation ) )
% 0.15/0.43 = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation )
% 0.15/0.43 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % as_good_as_sameIff
% 0.15/0.43 thf(fact_31_prod_Oinject,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 0.15/0.43 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.15/0.43 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 0.15/0.43 = ( ( X1 = Y1 )
% 0.15/0.43 & ( X22 = Y22 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod.inject
% 0.15/0.43 thf(fact_32_old_Oprod_Oinject,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.15/0.43 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.15/0.43 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.15/0.43 = ( ( A2 = A3 )
% 0.15/0.43 & ( B2 = B3 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % old.prod.inject
% 0.15/0.43 thf(fact_33_rational__preference_Oas__good__as__sameIff,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),Z: A,Y: A] :
% 0.15/0.43 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( ( member @ A @ Z @ ( prefer951318096ood_as @ A @ Y @ Carrier @ Relation ) )
% 0.15/0.43 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation )
% 0.15/0.43 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference.as_good_as_sameIff
% 0.15/0.43 thf(fact_34_total__onI,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A )] :
% 0.15/0.43 ( ! [X4: A,Y3: A] :
% 0.15/0.43 ( ( member @ A @ X4 @ A4 )
% 0.15/0.43 => ( ( member @ A @ Y3 @ A4 )
% 0.15/0.43 => ( ( X4 != Y3 )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R )
% 0.15/0.43 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X4 ) @ R ) ) ) ) )
% 0.15/0.43 => ( total_on @ A @ A4 @ R ) ) ).
% 0.15/0.43
% 0.15/0.43 % total_onI
% 0.15/0.43 thf(fact_35_total__on__def,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( total_on @ A )
% 0.15/0.43 = ( ^ [A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 0.15/0.43 ! [X3: A] :
% 0.15/0.43 ( ( member @ A @ X3 @ A5 )
% 0.15/0.43 => ! [Y2: A] :
% 0.15/0.43 ( ( member @ A @ Y2 @ A5 )
% 0.15/0.43 => ( ( X3 != Y2 )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
% 0.15/0.43 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X3 ) @ R2 ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % total_on_def
% 0.15/0.43 thf(fact_36_refl__onD,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A2: A] :
% 0.15/0.43 ( ( refl_on @ A @ A4 @ R )
% 0.15/0.43 => ( ( member @ A @ A2 @ A4 )
% 0.15/0.43 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % refl_onD
% 0.15/0.43 thf(fact_37_refl__onD1,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
% 0.15/0.43 ( ( refl_on @ A @ A4 @ R )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R )
% 0.15/0.43 => ( member @ A @ X2 @ A4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % refl_onD1
% 0.15/0.43 thf(fact_38_refl__onD2,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
% 0.15/0.43 ( ( refl_on @ A @ A4 @ R )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R )
% 0.15/0.43 => ( member @ A @ Y @ A4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % refl_onD2
% 0.15/0.43 thf(fact_39_refl__on__domain,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
% 0.15/0.43 ( ( refl_on @ A @ A4 @ R )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R )
% 0.15/0.43 => ( ( member @ A @ A2 @ A4 )
% 0.15/0.43 & ( member @ A @ B2 @ A4 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % refl_on_domain
% 0.15/0.43 thf(fact_40_at__least__as__goodD,axiom,
% 0.15/0.43 ! [A: $tType,Z: A,Y: A,B4: set @ A,Pr: set @ ( product_prod @ A @ A )] :
% 0.15/0.43 ( ( member @ A @ Z @ ( prefer310429814s_good @ A @ Y @ B4 @ Pr ) )
% 0.15/0.43 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Pr ) ) ).
% 0.15/0.43
% 0.15/0.43 % at_least_as_goodD
% 0.15/0.43 thf(fact_41_at__lst__asgd__ge,axiom,
% 0.15/0.43 ! [A: $tType,X2: A,Y: A,B4: set @ A,Pr: set @ ( product_prod @ A @ A )] :
% 0.15/0.43 ( ( member @ A @ X2 @ ( prefer310429814s_good @ A @ Y @ B4 @ Pr ) )
% 0.15/0.43 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Pr ) ) ).
% 0.15/0.43
% 0.15/0.43 % at_lst_asgd_ge
% 0.15/0.43 thf(fact_42_old_Oprod_Oinducts,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.15/0.43 ( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
% 0.15/0.43 => ( P @ Prod ) ) ).
% 0.15/0.43
% 0.15/0.43 % old.prod.inducts
% 0.15/0.43 thf(fact_43_old_Oprod_Oexhaust,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.15/0.43 ~ ! [A6: A,B5: B] :
% 0.15/0.43 ( Y
% 0.15/0.43 != ( product_Pair @ A @ B @ A6 @ B5 ) ) ).
% 0.15/0.43
% 0.15/0.43 % old.prod.exhaust
% 0.15/0.43 thf(fact_44_prod__induct7,axiom,
% 0.15/0.43 ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 0.15/0.43 ( ! [A6: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
% 0.15/0.43 => ( P @ X2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_induct7
% 0.15/0.43 thf(fact_45_mem__Collect__eq,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,P: A > $o] :
% 0.15/0.43 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.15/0.43 = ( P @ A2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % mem_Collect_eq
% 0.15/0.43 thf(fact_46_Collect__mem__eq,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A] :
% 0.15/0.43 ( ( collect @ A
% 0.15/0.43 @ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
% 0.15/0.43 = A4 ) ).
% 0.15/0.43
% 0.15/0.43 % Collect_mem_eq
% 0.15/0.43 thf(fact_47_Collect__cong,axiom,
% 0.15/0.43 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.15/0.43 ( ! [X4: A] :
% 0.15/0.43 ( ( P @ X4 )
% 0.15/0.43 = ( Q @ X4 ) )
% 0.15/0.43 => ( ( collect @ A @ P )
% 0.15/0.43 = ( collect @ A @ Q ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Collect_cong
% 0.15/0.43 thf(fact_48_ext,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
% 0.15/0.43 ( ! [X4: A] :
% 0.15/0.43 ( ( F3 @ X4 )
% 0.15/0.43 = ( G3 @ X4 ) )
% 0.15/0.43 => ( F3 = G3 ) ) ).
% 0.15/0.43
% 0.15/0.43 % ext
% 0.15/0.43 thf(fact_49_prod__induct6,axiom,
% 0.15/0.43 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.15/0.43 ( ! [A6: A,B5: B,C2: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
% 0.15/0.43 => ( P @ X2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_induct6
% 0.15/0.43 thf(fact_50_prod__induct5,axiom,
% 0.15/0.43 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.15/0.43 ( ! [A6: A,B5: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.15/0.43 => ( P @ X2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_induct5
% 0.15/0.43 thf(fact_51_prod__induct4,axiom,
% 0.15/0.43 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.15/0.43 ( ! [A6: A,B5: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 0.15/0.43 => ( P @ X2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_induct4
% 0.15/0.43 thf(fact_52_prod__induct3,axiom,
% 0.15/0.43 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.15/0.43 ( ! [A6: A,B5: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) )
% 0.15/0.43 => ( P @ X2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_induct3
% 0.15/0.43 thf(fact_53_prod__cases7,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 0.15/0.43 ~ ! [A6: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
% 0.15/0.43 ( Y
% 0.15/0.43 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_cases7
% 0.15/0.43 thf(fact_54_prod__cases6,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.15/0.43 ~ ! [A6: A,B5: B,C2: C,D2: D,E2: E,F2: F] :
% 0.15/0.43 ( Y
% 0.15/0.43 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_cases6
% 0.15/0.43 thf(fact_55_prod__cases5,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.15/0.43 ~ ! [A6: A,B5: B,C2: C,D2: D,E2: E] :
% 0.15/0.43 ( Y
% 0.15/0.43 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_cases5
% 0.15/0.43 thf(fact_56_prod__cases4,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.15/0.43 ~ ! [A6: A,B5: B,C2: C,D2: D] :
% 0.15/0.43 ( Y
% 0.15/0.43 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_cases4
% 0.15/0.43 thf(fact_57_prod__cases3,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.15/0.43 ~ ! [A6: A,B5: B,C2: C] :
% 0.15/0.43 ( Y
% 0.15/0.43 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_cases3
% 0.15/0.43 thf(fact_58_Pair__inject,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.15/0.43 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.15/0.43 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.15/0.43 => ~ ( ( A2 = A3 )
% 0.15/0.43 => ( B2 != B3 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Pair_inject
% 0.15/0.43 thf(fact_59_prod__cases,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.15/0.43 ( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
% 0.15/0.43 => ( P @ P2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_cases
% 0.15/0.43 thf(fact_60_surj__pair,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.15/0.43 ? [X4: A,Y3: B] :
% 0.15/0.43 ( P2
% 0.15/0.43 = ( product_Pair @ A @ B @ X4 @ Y3 ) ) ).
% 0.15/0.43
% 0.15/0.43 % surj_pair
% 0.15/0.43 thf(fact_61_rational__preference_Oworse__in__no__better,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
% 0.15/0.43 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
% 0.15/0.43 => ( member @ A @ Y @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference.worse_in_no_better
% 0.15/0.43 thf(fact_62_rational__preference_Opref__in__at__least__as,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
% 0.15/0.43 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
% 0.15/0.43 => ( member @ A @ X2 @ ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference.pref_in_at_least_as
% 0.15/0.43 thf(fact_63_at__lst__asgd__not__ge,axiom,
% 0.15/0.43 ! [X2: a,Y: a] :
% 0.15/0.43 ( ( carrier
% 0.15/0.43 != ( bot_bot @ ( set @ a ) ) )
% 0.15/0.43 => ( ( member @ a @ X2 @ carrier )
% 0.15/0.43 => ( ( member @ a @ Y @ carrier )
% 0.15/0.43 => ( ~ ( member @ a @ X2 @ ( prefer310429814s_good @ a @ Y @ carrier @ relation ) )
% 0.15/0.43 => ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % at_lst_asgd_not_ge
% 0.15/0.43 thf(fact_64_no__better__subset__pref,axiom,
% 0.15/0.43 ! [X2: a,Y: a] :
% 0.15/0.43 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
% 0.15/0.43 => ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X2 @ carrier @ relation ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % no_better_subset_pref
% 0.15/0.43 thf(fact_65_no__better__thansubset__rel,axiom,
% 0.15/0.43 ! [X2: a,Y: a] :
% 0.15/0.43 ( ( member @ a @ X2 @ carrier )
% 0.15/0.43 => ( ( member @ a @ Y @ carrier )
% 0.15/0.43 => ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X2 @ carrier @ relation ) )
% 0.15/0.43 => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % no_better_thansubset_rel
% 0.15/0.43 thf(fact_66_no__better__than__nonepty,axiom,
% 0.15/0.43 ! [X2: a] :
% 0.15/0.43 ( ( carrier
% 0.15/0.43 != ( bot_bot @ ( set @ a ) ) )
% 0.15/0.43 => ( ( member @ a @ X2 @ carrier )
% 0.15/0.43 => ( ( prefer1532642881r_than @ a @ X2 @ carrier @ relation )
% 0.15/0.43 != ( bot_bot @ ( set @ a ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % no_better_than_nonepty
% 0.15/0.43 thf(fact_67_nbt__nest,axiom,
% 0.15/0.43 ! [Y: a,X2: a] :
% 0.15/0.43 ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X2 @ carrier @ relation ) )
% 0.15/0.43 | ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ X2 @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % nbt_nest
% 0.15/0.43 thf(fact_68_old_Oprod_Orec,axiom,
% 0.15/0.43 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.15/0.43 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.15/0.43 = ( F1 @ A2 @ B2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % old.prod.rec
% 0.15/0.43 thf(fact_69_rational__preference_Ointro,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.15/0.43 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( ( prefer1801827867axioms @ A @ Carrier @ Relation )
% 0.15/0.43 => ( prefer1997167224erence @ A @ Carrier @ Relation ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference.intro
% 0.15/0.43 thf(fact_70_rational__preference__def,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( prefer1997167224erence @ A )
% 0.15/0.43 = ( ^ [Carrier2: set @ A,Relation2: set @ ( product_prod @ A @ A )] :
% 0.15/0.43 ( ( prefer199794634erence @ A @ Carrier2 @ Relation2 )
% 0.15/0.43 & ( prefer1801827867axioms @ A @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference_def
% 0.15/0.43 thf(fact_71_refl__on__empty,axiom,
% 0.15/0.43 ! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % refl_on_empty
% 0.15/0.43 thf(fact_72_preorder__on__empty,axiom,
% 0.15/0.43 ! [A: $tType] : ( order_preorder_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % preorder_on_empty
% 0.15/0.43 thf(fact_73_total__on__empty,axiom,
% 0.15/0.43 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R ) ).
% 0.15/0.43
% 0.15/0.43 % total_on_empty
% 0.15/0.43 thf(fact_74_rational__preference_Onbt__nest,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),Y: A,X2: A] :
% 0.15/0.43 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X2 @ Carrier @ Relation ) )
% 0.15/0.43 | ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ X2 @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference.nbt_nest
% 0.15/0.43 thf(fact_75_rational__preference_Ono__better__than__nonepty,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A] :
% 0.15/0.43 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( ( Carrier
% 0.15/0.43 != ( bot_bot @ ( set @ A ) ) )
% 0.15/0.43 => ( ( member @ A @ X2 @ Carrier )
% 0.15/0.43 => ( ( prefer1532642881r_than @ A @ X2 @ Carrier @ Relation )
% 0.15/0.43 != ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference.no_better_than_nonepty
% 0.15/0.43 thf(fact_76_rational__preference__axioms__def,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( prefer1801827867axioms @ A )
% 0.15/0.43 = ( total_on @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference_axioms_def
% 0.15/0.43 thf(fact_77_rational__preference__axioms_Ointro,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.15/0.43 ( ( total_on @ A @ Carrier @ Relation )
% 0.15/0.43 => ( prefer1801827867axioms @ A @ Carrier @ Relation ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference_axioms.intro
% 0.15/0.43 thf(fact_78_rational__preference_Oaxioms_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.15/0.43 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( prefer1801827867axioms @ A @ Carrier @ Relation ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference.axioms(2)
% 0.15/0.43 thf(fact_79_rational__preference_Ono__better__subset__pref,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
% 0.15/0.43 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
% 0.15/0.43 => ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X2 @ Carrier @ Relation ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference.no_better_subset_pref
% 0.15/0.43 thf(fact_80_rational__preference_Ono__better__thansubset__rel,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
% 0.15/0.43 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( ( member @ A @ X2 @ Carrier )
% 0.15/0.43 => ( ( member @ A @ Y @ Carrier )
% 0.15/0.43 => ( ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X2 @ Carrier @ Relation ) )
% 0.15/0.43 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference.no_better_thansubset_rel
% 0.15/0.43 thf(fact_81_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
% 0.15/0.43 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( ( Carrier
% 0.15/0.43 != ( bot_bot @ ( set @ A ) ) )
% 0.15/0.43 => ( ( member @ A @ X2 @ Carrier )
% 0.15/0.43 => ( ( member @ A @ Y @ Carrier )
% 0.15/0.43 => ( ~ ( member @ A @ X2 @ ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) )
% 0.15/0.43 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % rational_preference.at_lst_asgd_not_ge
% 0.15/0.43 thf(fact_82_subset__empty,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A] :
% 0.15/0.43 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 0.15/0.43 = ( A4
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_empty
% 0.15/0.43 thf(fact_83_empty__subsetI,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% 0.15/0.43
% 0.15/0.43 % empty_subsetI
% 0.15/0.43 thf(fact_84_Pair__le,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( ( ord @ B )
% 0.15/0.43 & ( ord @ A ) )
% 0.15/0.43 => ! [A2: A,B2: B,C3: A,D3: B] :
% 0.15/0.43 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C3 @ D3 ) )
% 0.15/0.43 = ( ( ord_less_eq @ A @ A2 @ C3 )
% 0.15/0.43 & ( ord_less_eq @ B @ B2 @ D3 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Pair_le
% 0.15/0.43 thf(fact_85_subsetI,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.15/0.43 ( ! [X4: A] :
% 0.15/0.43 ( ( member @ A @ X4 @ A4 )
% 0.15/0.43 => ( member @ A @ X4 @ B4 ) )
% 0.15/0.43 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.15/0.43
% 0.15/0.43 % subsetI
% 0.15/0.43 thf(fact_86_subset__antisym,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.15/0.43 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.15/0.43 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 0.15/0.43 => ( A4 = B4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_antisym
% 0.15/0.43 thf(fact_87_empty__iff,axiom,
% 0.15/0.43 ! [A: $tType,C3: A] :
% 0.15/0.43 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % empty_iff
% 0.15/0.43 thf(fact_88_all__not__in__conv,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A] :
% 0.15/0.43 ( ( ! [X3: A] :
% 0.15/0.43 ~ ( member @ A @ X3 @ A4 ) )
% 0.15/0.43 = ( A4
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % all_not_in_conv
% 0.15/0.43 thf(fact_89_empty__Collect__eq,axiom,
% 0.15/0.43 ! [A: $tType,P: A > $o] :
% 0.15/0.43 ( ( ( bot_bot @ ( set @ A ) )
% 0.15/0.43 = ( collect @ A @ P ) )
% 0.15/0.43 = ( ! [X3: A] :
% 0.15/0.43 ~ ( P @ X3 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % empty_Collect_eq
% 0.15/0.43 thf(fact_90_Collect__empty__eq,axiom,
% 0.15/0.43 ! [A: $tType,P: A > $o] :
% 0.15/0.43 ( ( ( collect @ A @ P )
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) )
% 0.15/0.43 = ( ! [X3: A] :
% 0.15/0.43 ~ ( P @ X3 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Collect_empty_eq
% 0.15/0.43 thf(fact_91_subrelI,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.15/0.43 ( ! [X4: A,Y3: B] :
% 0.15/0.43 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R )
% 0.15/0.43 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ S ) )
% 0.15/0.43 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 0.15/0.43
% 0.15/0.43 % subrelI
% 0.15/0.43 thf(fact_92_ex__in__conv,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A] :
% 0.15/0.43 ( ( ? [X3: A] : ( member @ A @ X3 @ A4 ) )
% 0.15/0.43 = ( A4
% 0.15/0.43 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ex_in_conv
% 0.15/0.43 thf(fact_93_equals0I,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A] :
% 0.15/0.43 ( ! [Y3: A] :
% 0.15/0.43 ~ ( member @ A @ Y3 @ A4 )
% 0.15/0.43 => ( A4
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % equals0I
% 0.15/0.43 thf(fact_94_equals0D,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A,A2: A] :
% 0.15/0.43 ( ( A4
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) )
% 0.15/0.43 => ~ ( member @ A @ A2 @ A4 ) ) ).
% 0.15/0.43
% 0.15/0.43 % equals0D
% 0.15/0.43 thf(fact_95_emptyE,axiom,
% 0.15/0.43 ! [A: $tType,A2: A] :
% 0.15/0.43 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % emptyE
% 0.15/0.43 thf(fact_96_Collect__mono__iff,axiom,
% 0.15/0.43 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.15/0.43 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.15/0.43 = ( ! [X3: A] :
% 0.15/0.43 ( ( P @ X3 )
% 0.15/0.43 => ( Q @ X3 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Collect_mono_iff
% 0.15/0.43 thf(fact_97_set__eq__subset,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ^ [Y4: set @ A,Z2: set @ A] : ( Y4 = Z2 ) )
% 0.15/0.43 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.15/0.43 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 0.15/0.43 & ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % set_eq_subset
% 0.15/0.43 thf(fact_98_subset__trans,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A,B4: set @ A,C4: set @ A] :
% 0.15/0.43 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.15/0.43 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
% 0.15/0.43 => ( ord_less_eq @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_trans
% 0.15/0.43 thf(fact_99_Collect__mono,axiom,
% 0.15/0.43 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.15/0.43 ( ! [X4: A] :
% 0.15/0.43 ( ( P @ X4 )
% 0.15/0.43 => ( Q @ X4 ) )
% 0.15/0.43 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Collect_mono
% 0.15/0.43 thf(fact_100_subset__refl,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% 0.15/0.43
% 0.15/0.43 % subset_refl
% 0.15/0.43 thf(fact_101_subset__iff,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ord_less_eq @ ( set @ A ) )
% 0.15/0.43 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.15/0.43 ! [T2: A] :
% 0.15/0.43 ( ( member @ A @ T2 @ A5 )
% 0.15/0.43 => ( member @ A @ T2 @ B6 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_iff
% 0.15/0.43 thf(fact_102_equalityD2,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.15/0.43 ( ( A4 = B4 )
% 0.15/0.43 => ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% 0.15/0.43
% 0.15/0.43 % equalityD2
% 0.15/0.43 thf(fact_103_equalityD1,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.15/0.43 ( ( A4 = B4 )
% 0.15/0.43 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.15/0.43
% 0.15/0.43 % equalityD1
% 0.15/0.43 thf(fact_104_subset__eq,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ord_less_eq @ ( set @ A ) )
% 0.15/0.43 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.15/0.43 ! [X3: A] :
% 0.15/0.43 ( ( member @ A @ X3 @ A5 )
% 0.15/0.43 => ( member @ A @ X3 @ B6 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_eq
% 0.15/0.43 thf(fact_105_equalityE,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.15/0.43 ( ( A4 = B4 )
% 0.15/0.43 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.15/0.43 => ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % equalityE
% 0.15/0.43 thf(fact_106_subsetD,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A,B4: set @ A,C3: A] :
% 0.15/0.43 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.15/0.43 => ( ( member @ A @ C3 @ A4 )
% 0.15/0.43 => ( member @ A @ C3 @ B4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subsetD
% 0.15/0.43 thf(fact_107_in__mono,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A,B4: set @ A,X2: A] :
% 0.15/0.43 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.15/0.43 => ( ( member @ A @ X2 @ A4 )
% 0.15/0.43 => ( member @ A @ X2 @ B4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % in_mono
% 0.15/0.43 thf(fact_108_Pair__mono,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( ( ord @ B )
% 0.15/0.43 & ( ord @ A ) )
% 0.15/0.43 => ! [X2: A,X5: A,Y: B,Y5: B] :
% 0.15/0.43 ( ( ord_less_eq @ A @ X2 @ X5 )
% 0.15/0.43 => ( ( ord_less_eq @ B @ Y @ Y5 )
% 0.15/0.43 => ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Pair_mono
% 0.15/0.43 thf(fact_109_bot__prod__def,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( ( bot @ B )
% 0.15/0.43 & ( bot @ A ) )
% 0.15/0.43 => ( ( bot_bot @ ( product_prod @ A @ B ) )
% 0.15/0.43 = ( product_Pair @ A @ B @ ( bot_bot @ A ) @ ( bot_bot @ B ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % bot_prod_def
% 0.15/0.43 thf(fact_110_bot__apply,axiom,
% 0.15/0.43 ! [C: $tType,D: $tType] :
% 0.15/0.43 ( ( bot @ C )
% 0.15/0.43 => ( ( bot_bot @ ( D > C ) )
% 0.15/0.43 = ( ^ [X3: D] : ( bot_bot @ C ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % bot_apply
% 0.15/0.43 thf(fact_111_order__refl,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( preorder @ A )
% 0.15/0.43 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % order_refl
% 0.15/0.43 thf(fact_112_subset__emptyI,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A] :
% 0.15/0.43 ( ! [X4: A] :
% 0.15/0.43 ~ ( member @ A @ X4 @ A4 )
% 0.15/0.43 => ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_emptyI
% 0.15/0.43 thf(fact_113_bot_Oextremum__uniqueI,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order_bot @ A )
% 0.15/0.43 => ! [A2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 0.15/0.43 => ( A2
% 0.15/0.43 = ( bot_bot @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % bot.extremum_uniqueI
% 0.15/0.43 thf(fact_114_bot_Oextremum__unique,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order_bot @ A )
% 0.15/0.43 => ! [A2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 0.15/0.43 = ( A2
% 0.15/0.43 = ( bot_bot @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % bot.extremum_unique
% 0.15/0.43 thf(fact_115_bot_Oextremum,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order_bot @ A )
% 0.15/0.43 => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % bot.extremum
% 0.15/0.43 thf(fact_116_bot__set__def,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( bot_bot @ ( set @ A ) )
% 0.15/0.43 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % bot_set_def
% 0.15/0.43 thf(fact_117_bot__empty__eq,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( bot_bot @ ( A > $o ) )
% 0.15/0.43 = ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % bot_empty_eq
% 0.15/0.43 thf(fact_118_le__funD,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( ord @ B )
% 0.15/0.43 => ! [F3: A > B,G3: A > B,X2: A] :
% 0.15/0.43 ( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
% 0.15/0.43 => ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G3 @ X2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % le_funD
% 0.15/0.43 thf(fact_119_le__funE,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( ord @ B )
% 0.15/0.43 => ! [F3: A > B,G3: A > B,X2: A] :
% 0.15/0.43 ( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
% 0.15/0.43 => ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G3 @ X2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % le_funE
% 0.15/0.43 thf(fact_120_le__funI,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( ord @ B )
% 0.15/0.43 => ! [F3: A > B,G3: A > B] :
% 0.15/0.43 ( ! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
% 0.15/0.43 => ( ord_less_eq @ ( A > B ) @ F3 @ G3 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % le_funI
% 0.15/0.43 thf(fact_121_le__fun__def,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( ord @ B )
% 0.15/0.43 => ( ( ord_less_eq @ ( A > B ) )
% 0.15/0.43 = ( ^ [F4: A > B,G4: A > B] :
% 0.15/0.43 ! [X3: A] : ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % le_fun_def
% 0.15/0.43 thf(fact_122_order__subst1,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( ( order @ B )
% 0.15/0.43 & ( order @ A ) )
% 0.15/0.43 => ! [A2: A,F3: B > A,B2: B,C3: B] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
% 0.15/0.43 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 0.15/0.43 => ( ! [X4: B,Y3: B] :
% 0.15/0.43 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 0.15/0.43 => ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
% 0.15/0.43 => ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % order_subst1
% 0.15/0.43 thf(fact_123_order__subst2,axiom,
% 0.15/0.43 ! [A: $tType,C: $tType] :
% 0.15/0.43 ( ( ( order @ C )
% 0.15/0.43 & ( order @ A ) )
% 0.15/0.43 => ! [A2: A,B2: A,F3: A > C,C3: C] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.43 => ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C3 )
% 0.15/0.43 => ( ! [X4: A,Y3: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.15/0.43 => ( ord_less_eq @ C @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
% 0.15/0.43 => ( ord_less_eq @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % order_subst2
% 0.15/0.43 thf(fact_124_ord__eq__le__subst,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( ( ord @ B )
% 0.15/0.43 & ( ord @ A ) )
% 0.15/0.43 => ! [A2: A,F3: B > A,B2: B,C3: B] :
% 0.15/0.43 ( ( A2
% 0.15/0.43 = ( F3 @ B2 ) )
% 0.15/0.43 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 0.15/0.43 => ( ! [X4: B,Y3: B] :
% 0.15/0.43 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 0.15/0.43 => ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
% 0.15/0.43 => ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord_eq_le_subst
% 0.15/0.43 thf(fact_125_ord__le__eq__subst,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( ( ord @ B )
% 0.15/0.43 & ( ord @ A ) )
% 0.15/0.43 => ! [A2: A,B2: A,F3: A > B,C3: B] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.43 => ( ( ( F3 @ B2 )
% 0.15/0.43 = C3 )
% 0.15/0.43 => ( ! [X4: A,Y3: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.15/0.43 => ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
% 0.15/0.43 => ( ord_less_eq @ B @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord_le_eq_subst
% 0.15/0.43 thf(fact_126_eq__iff,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.15/0.43 = ( ^ [X3: A,Y2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.15/0.43 & ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % eq_iff
% 0.15/0.43 thf(fact_127_antisym,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ! [X2: A,Y: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.15/0.43 => ( ( ord_less_eq @ A @ Y @ X2 )
% 0.15/0.43 => ( X2 = Y ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % antisym
% 0.15/0.43 thf(fact_128_linear,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [X2: A,Y: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.15/0.43 | ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % linear
% 0.15/0.43 thf(fact_129_eq__refl,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( preorder @ A )
% 0.15/0.43 => ! [X2: A,Y: A] :
% 0.15/0.43 ( ( X2 = Y )
% 0.15/0.43 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % eq_refl
% 0.15/0.43 thf(fact_130_le__cases,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [X2: A,Y: A] :
% 0.15/0.43 ( ~ ( ord_less_eq @ A @ X2 @ Y )
% 0.15/0.43 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % le_cases
% 0.15/0.43 thf(fact_131_order_Otrans,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ! [A2: A,B2: A,C3: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.43 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 0.15/0.43 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % order.trans
% 0.15/0.43 thf(fact_132_le__cases3,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [X2: A,Y: A,Z: A] :
% 0.15/0.43 ( ( ( ord_less_eq @ A @ X2 @ Y )
% 0.15/0.43 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 0.15/0.43 => ( ( ( ord_less_eq @ A @ Y @ X2 )
% 0.15/0.43 => ~ ( ord_less_eq @ A @ X2 @ Z ) )
% 0.15/0.43 => ( ( ( ord_less_eq @ A @ X2 @ Z )
% 0.15/0.43 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 0.15/0.43 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 0.15/0.43 => ~ ( ord_less_eq @ A @ Y @ X2 ) )
% 0.15/0.43 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 0.15/0.43 => ~ ( ord_less_eq @ A @ Z @ X2 ) )
% 0.15/0.43 => ~ ( ( ord_less_eq @ A @ Z @ X2 )
% 0.15/0.43 => ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % le_cases3
% 0.15/0.43 thf(fact_133_antisym__conv,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ! [Y: A,X2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ Y @ X2 )
% 0.15/0.43 => ( ( ord_less_eq @ A @ X2 @ Y )
% 0.15/0.43 = ( X2 = Y ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % antisym_conv
% 0.15/0.43 thf(fact_134_order__class_Oorder_Oeq__iff,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.15/0.43 = ( ^ [A7: A,B7: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A7 @ B7 )
% 0.15/0.43 & ( ord_less_eq @ A @ B7 @ A7 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % order_class.order.eq_iff
% 0.15/0.43 thf(fact_135_ord__eq__le__trans,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ord @ A )
% 0.15/0.43 => ! [A2: A,B2: A,C3: A] :
% 0.15/0.43 ( ( A2 = B2 )
% 0.15/0.43 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 0.15/0.43 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord_eq_le_trans
% 0.15/0.43 thf(fact_136_ord__le__eq__trans,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ord @ A )
% 0.15/0.43 => ! [A2: A,B2: A,C3: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.43 => ( ( B2 = C3 )
% 0.15/0.43 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord_le_eq_trans
% 0.15/0.43 thf(fact_137_order__class_Oorder_Oantisym,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ! [A2: A,B2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.43 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.15/0.43 => ( A2 = B2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % order_class.order.antisym
% 0.15/0.43 thf(fact_138_order__trans,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( preorder @ A )
% 0.15/0.43 => ! [X2: A,Y: A,Z: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.15/0.43 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.15/0.43 => ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % order_trans
% 0.15/0.43 thf(fact_139_dual__order_Orefl,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % dual_order.refl
% 0.15/0.43 thf(fact_140_linorder__wlog,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.15/0.43 ( ! [A6: A,B5: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A6 @ B5 )
% 0.15/0.43 => ( P @ A6 @ B5 ) )
% 0.15/0.43 => ( ! [A6: A,B5: A] :
% 0.15/0.43 ( ( P @ B5 @ A6 )
% 0.15/0.43 => ( P @ A6 @ B5 ) )
% 0.15/0.43 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % linorder_wlog
% 0.15/0.43 thf(fact_141_dual__order_Otrans,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ! [B2: A,A2: A,C3: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.15/0.43 => ( ( ord_less_eq @ A @ C3 @ B2 )
% 0.15/0.43 => ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % dual_order.trans
% 0.15/0.43 thf(fact_142_dual__order_Oeq__iff,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.15/0.43 = ( ^ [A7: A,B7: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ B7 @ A7 )
% 0.15/0.43 & ( ord_less_eq @ A @ A7 @ B7 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % dual_order.eq_iff
% 0.15/0.43 thf(fact_143_dual__order_Oantisym,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ! [B2: A,A2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.15/0.43 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.43 => ( A2 = B2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % dual_order.antisym
% 0.15/0.43 thf(fact_144_bot__fun__def,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( bot @ B )
% 0.15/0.43 => ( ( bot_bot @ ( A > B ) )
% 0.15/0.43 = ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % bot_fun_def
% 0.15/0.43 thf(fact_145_ssubst__Pair__rhs,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,R: A,S: B,R3: set @ ( product_prod @ A @ B ),S2: B] :
% 0.15/0.43 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R3 )
% 0.15/0.43 => ( ( S2 = S )
% 0.15/0.43 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R3 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ssubst_Pair_rhs
% 0.15/0.43 thf(fact_146_less__by__empty,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ ( product_prod @ A @ A ),B4: set @ ( product_prod @ A @ A )] :
% 0.15/0.43 ( ( A4
% 0.15/0.43 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.15/0.43 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B4 ) ) ).
% 0.15/0.43
% 0.15/0.43 % less_by_empty
% 0.15/0.43 thf(fact_147_transitivity,axiom,
% 0.15/0.43 trans @ a @ relation ).
% 0.15/0.43
% 0.15/0.43 % transitivity
% 0.15/0.43 thf(fact_148_Set_Ois__empty__def,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( is_empty @ A )
% 0.15/0.43 = ( ^ [A5: set @ A] :
% 0.15/0.43 ( A5
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Set.is_empty_def
% 0.15/0.43 thf(fact_149_internal__case__prod__conv,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
% 0.15/0.43 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.15/0.43 = ( C3 @ A2 @ B2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % internal_case_prod_conv
% 0.15/0.43 thf(fact_150_transD,axiom,
% 0.15/0.43 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X2: A,Y: A,Z: A] :
% 0.15/0.43 ( ( trans @ A @ R )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R )
% 0.15/0.43 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z ) @ R ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % transD
% 0.15/0.43 thf(fact_151_transE,axiom,
% 0.15/0.43 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X2: A,Y: A,Z: A] :
% 0.15/0.43 ( ( trans @ A @ R )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R )
% 0.15/0.43 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z ) @ R ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % transE
% 0.15/0.43 thf(fact_152_transI,axiom,
% 0.15/0.43 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.15/0.43 ( ! [X4: A,Y3: A,Z3: A] :
% 0.15/0.43 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R )
% 0.15/0.43 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z3 ) @ R ) ) )
% 0.15/0.43 => ( trans @ A @ R ) ) ).
% 0.15/0.43
% 0.15/0.43 % transI
% 0.15/0.43 thf(fact_153_trans__def,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( trans @ A )
% 0.15/0.43 = ( ^ [R2: set @ ( product_prod @ A @ A )] :
% 0.15/0.43 ! [X3: A,Y2: A,Z4: A] :
% 0.15/0.43 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z4 ) @ R2 )
% 0.15/0.43 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z4 ) @ R2 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % trans_def
% 0.15/0.43 thf(fact_154_trans__empty,axiom,
% 0.15/0.43 ! [A: $tType] : ( trans @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % trans_empty
% 0.15/0.43 thf(fact_155_preference_Otransitivity,axiom,
% 0.15/0.43 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 0.15/0.43 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 0.15/0.43 => ( trans @ A @ Relation ) ) ).
% 0.15/0.43
% 0.15/0.43 % preference.transitivity
% 0.15/0.43 thf(fact_156_preorder__on__def,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order_preorder_on @ A )
% 0.15/0.43 = ( ^ [A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 0.15/0.43 ( ( refl_on @ A @ A5 @ R2 )
% 0.15/0.43 & ( trans @ A @ R2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % preorder_on_def
% 0.15/0.43 thf(fact_157_Collect__empty__eq__bot,axiom,
% 0.15/0.43 ! [A: $tType,P: A > $o] :
% 0.15/0.43 ( ( ( collect @ A @ P )
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) )
% 0.15/0.43 = ( P
% 0.15/0.43 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Collect_empty_eq_bot
% 0.15/0.43 thf(fact_158_relChain__def,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( ord @ B )
% 0.15/0.43 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 0.15/0.43 = ( ^ [R2: set @ ( product_prod @ A @ A ),As: A > B] :
% 0.15/0.43 ! [I: A,J: A] :
% 0.15/0.43 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R2 )
% 0.15/0.43 => ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % relChain_def
% 0.15/0.43 thf(fact_159_Id__on__empty,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.15/0.43 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Id_on_empty
% 0.15/0.43 thf(fact_160_Id__onI,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,A4: set @ A] :
% 0.15/0.43 ( ( member @ A @ A2 @ A4 )
% 0.15/0.43 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id_on @ A @ A4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Id_onI
% 0.15/0.43 thf(fact_161_Id__on__iff,axiom,
% 0.15/0.43 ! [A: $tType,X2: A,Y: A,A4: set @ A] :
% 0.15/0.43 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( id_on @ A @ A4 ) )
% 0.15/0.43 = ( ( X2 = Y )
% 0.15/0.43 & ( member @ A @ X2 @ A4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Id_on_iff
% 0.15/0.43 thf(fact_162_Id__on__eqI,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,B2: A,A4: set @ A] :
% 0.15/0.43 ( ( A2 = B2 )
% 0.15/0.43 => ( ( member @ A @ A2 @ A4 )
% 0.15/0.43 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id_on @ A @ A4 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Id_on_eqI
% 0.15/0.43 thf(fact_163_Id__onE,axiom,
% 0.15/0.43 ! [A: $tType,C3: product_prod @ A @ A,A4: set @ A] :
% 0.15/0.43 ( ( member @ ( product_prod @ A @ A ) @ C3 @ ( id_on @ A @ A4 ) )
% 0.15/0.43 => ~ ! [X4: A] :
% 0.15/0.43 ( ( member @ A @ X4 @ A4 )
% 0.15/0.43 => ( C3
% 0.15/0.43 != ( product_Pair @ A @ A @ X4 @ X4 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Id_onE
% 0.15/0.43 thf(fact_164_trans__Id__on,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A] : ( trans @ A @ ( id_on @ A @ A4 ) ) ).
% 0.15/0.43
% 0.15/0.43 % trans_Id_on
% 0.15/0.43 thf(fact_165_refl__on__Id__on,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A] : ( refl_on @ A @ A4 @ ( id_on @ A @ A4 ) ) ).
% 0.15/0.43
% 0.15/0.43 % refl_on_Id_on
% 0.15/0.43 thf(fact_166_under__incr,axiom,
% 0.15/0.43 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
% 0.15/0.43 ( ( trans @ A @ R )
% 0.15/0.43 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R )
% 0.15/0.43 => ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R @ A2 ) @ ( order_under @ A @ R @ B2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % under_incr
% 0.15/0.43 thf(fact_167_GreatestI2__order,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ! [P: A > $o,X2: A,Q: A > $o] :
% 0.15/0.43 ( ( P @ X2 )
% 0.15/0.43 => ( ! [Y3: A] :
% 0.15/0.43 ( ( P @ Y3 )
% 0.15/0.43 => ( ord_less_eq @ A @ Y3 @ X2 ) )
% 0.15/0.43 => ( ! [X4: A] :
% 0.15/0.43 ( ( P @ X4 )
% 0.15/0.43 => ( ! [Y6: A] :
% 0.15/0.43 ( ( P @ Y6 )
% 0.15/0.43 => ( ord_less_eq @ A @ Y6 @ X4 ) )
% 0.15/0.43 => ( Q @ X4 ) ) )
% 0.15/0.43 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % GreatestI2_order
% 0.15/0.43 thf(fact_168_Greatest__equality,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ! [P: A > $o,X2: A] :
% 0.15/0.43 ( ( P @ X2 )
% 0.15/0.43 => ( ! [Y3: A] :
% 0.15/0.43 ( ( P @ Y3 )
% 0.15/0.43 => ( ord_less_eq @ A @ Y3 @ X2 ) )
% 0.15/0.43 => ( ( order_Greatest @ A @ P )
% 0.15/0.43 = X2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Greatest_equality
% 0.15/0.43 thf(fact_169_fst__bot,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( ( bot @ A )
% 0.15/0.43 & ( bot @ B ) )
% 0.15/0.43 => ( ( product_fst @ A @ B @ ( bot_bot @ ( product_prod @ A @ B ) ) )
% 0.15/0.43 = ( bot_bot @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % fst_bot
% 0.15/0.43 thf(fact_170_snd__bot,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( ( bot @ A )
% 0.15/0.43 & ( bot @ B ) )
% 0.15/0.43 => ( ( product_snd @ B @ A @ ( bot_bot @ ( product_prod @ B @ A ) ) )
% 0.15/0.43 = ( bot_bot @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % snd_bot
% 0.15/0.43 thf(fact_171_trans__singleton,axiom,
% 0.15/0.43 ! [A: $tType,A2: A] : ( trans @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % trans_singleton
% 0.15/0.43 thf(fact_172_insert__absorb2,axiom,
% 0.15/0.43 ! [A: $tType,X2: A,A4: set @ A] :
% 0.15/0.43 ( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A4 ) )
% 0.15/0.43 = ( insert @ A @ X2 @ A4 ) ) ).
% 0.15/0.43
% 0.15/0.43 % insert_absorb2
% 0.15/0.43 thf(fact_173_insert__iff,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,B2: A,A4: set @ A] :
% 0.15/0.43 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
% 0.15/0.43 = ( ( A2 = B2 )
% 0.15/0.43 | ( member @ A @ A2 @ A4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % insert_iff
% 0.15/0.43 thf(fact_174_insertCI,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,B4: set @ A,B2: A] :
% 0.15/0.43 ( ( ~ ( member @ A @ A2 @ B4 )
% 0.15/0.43 => ( A2 = B2 ) )
% 0.15/0.43 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % insertCI
% 0.15/0.43 thf(fact_175_singletonI,axiom,
% 0.15/0.43 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % singletonI
% 0.15/0.43 thf(fact_176_insert__subset,axiom,
% 0.15/0.43 ! [A: $tType,X2: A,A4: set @ A,B4: set @ A] :
% 0.15/0.43 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ A4 ) @ B4 )
% 0.15/0.43 = ( ( member @ A @ X2 @ B4 )
% 0.15/0.43 & ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % insert_subset
% 0.15/0.43 thf(fact_177_singleton__insert__inj__eq,axiom,
% 0.15/0.43 ! [A: $tType,B2: A,A2: A,A4: set @ A] :
% 0.15/0.43 ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
% 0.15/0.43 = ( insert @ A @ A2 @ A4 ) )
% 0.15/0.43 = ( ( A2 = B2 )
% 0.15/0.43 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % singleton_insert_inj_eq
% 0.15/0.43 thf(fact_178_singleton__insert__inj__eq_H,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,A4: set @ A,B2: A] :
% 0.15/0.43 ( ( ( insert @ A @ A2 @ A4 )
% 0.15/0.43 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.15/0.43 = ( ( A2 = B2 )
% 0.15/0.43 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % singleton_insert_inj_eq'
% 0.15/0.43 thf(fact_179_prod_Ocollapse,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.15/0.43 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 0.15/0.43 = Prod ) ).
% 0.15/0.43
% 0.15/0.43 % prod.collapse
% 0.15/0.43 thf(fact_180_insert__subsetI,axiom,
% 0.15/0.43 ! [A: $tType,X2: A,A4: set @ A,X6: set @ A] :
% 0.15/0.43 ( ( member @ A @ X2 @ A4 )
% 0.15/0.43 => ( ( ord_less_eq @ ( set @ A ) @ X6 @ A4 )
% 0.15/0.43 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ X6 ) @ A4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % insert_subsetI
% 0.15/0.43 thf(fact_181_less__eq__prod__def,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( ( ord @ A )
% 0.15/0.43 & ( ord @ B ) )
% 0.15/0.43 => ( ( ord_less_eq @ ( product_prod @ A @ B ) )
% 0.15/0.43 = ( ^ [X3: product_prod @ A @ B,Y2: product_prod @ A @ B] :
% 0.15/0.43 ( ( ord_less_eq @ A @ ( product_fst @ A @ B @ X3 ) @ ( product_fst @ A @ B @ Y2 ) )
% 0.15/0.43 & ( ord_less_eq @ B @ ( product_snd @ A @ B @ X3 ) @ ( product_snd @ A @ B @ Y2 ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % less_eq_prod_def
% 0.15/0.43 thf(fact_182_singleton__inject,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,B2: A] :
% 0.15/0.43 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.15/0.43 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.15/0.43 => ( A2 = B2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % singleton_inject
% 0.15/0.43 thf(fact_183_insert__not__empty,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,A4: set @ A] :
% 0.15/0.43 ( ( insert @ A @ A2 @ A4 )
% 0.15/0.43 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % insert_not_empty
% 0.15/0.43 thf(fact_184_doubleton__eq__iff,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,B2: A,C3: A,D3: A] :
% 0.15/0.43 ( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.15/0.43 = ( insert @ A @ C3 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.15/0.43 = ( ( ( A2 = C3 )
% 0.15/0.43 & ( B2 = D3 ) )
% 0.15/0.43 | ( ( A2 = D3 )
% 0.15/0.43 & ( B2 = C3 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % doubleton_eq_iff
% 0.15/0.43 thf(fact_185_singleton__iff,axiom,
% 0.15/0.43 ! [A: $tType,B2: A,A2: A] :
% 0.15/0.43 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.15/0.43 = ( B2 = A2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % singleton_iff
% 0.15/0.43 thf(fact_186_singletonD,axiom,
% 0.15/0.43 ! [A: $tType,B2: A,A2: A] :
% 0.15/0.43 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.15/0.43 => ( B2 = A2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % singletonD
% 0.15/0.43 thf(fact_187_subset__insertI2,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A,B4: set @ A,B2: A] :
% 0.15/0.43 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.15/0.43 => ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_insertI2
% 0.15/0.43 thf(fact_188_subset__insertI,axiom,
% 0.15/0.43 ! [A: $tType,B4: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( insert @ A @ A2 @ B4 ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_insertI
% 0.15/0.43 thf(fact_189_subset__insert,axiom,
% 0.15/0.43 ! [A: $tType,X2: A,A4: set @ A,B4: set @ A] :
% 0.15/0.43 ( ~ ( member @ A @ X2 @ A4 )
% 0.15/0.43 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X2 @ B4 ) )
% 0.15/0.43 = ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_insert
% 0.15/0.43 thf(fact_190_insert__mono,axiom,
% 0.15/0.43 ! [A: $tType,C4: set @ A,D4: set @ A,A2: A] :
% 0.15/0.43 ( ( ord_less_eq @ ( set @ A ) @ C4 @ D4 )
% 0.15/0.43 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C4 ) @ ( insert @ A @ A2 @ D4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % insert_mono
% 0.15/0.43 thf(fact_191_mk__disjoint__insert,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,A4: set @ A] :
% 0.15/0.43 ( ( member @ A @ A2 @ A4 )
% 0.15/0.43 => ? [B8: set @ A] :
% 0.15/0.43 ( ( A4
% 0.15/0.43 = ( insert @ A @ A2 @ B8 ) )
% 0.15/0.43 & ~ ( member @ A @ A2 @ B8 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % mk_disjoint_insert
% 0.15/0.43 thf(fact_192_insert__commute,axiom,
% 0.15/0.43 ! [A: $tType,X2: A,Y: A,A4: set @ A] :
% 0.15/0.43 ( ( insert @ A @ X2 @ ( insert @ A @ Y @ A4 ) )
% 0.15/0.43 = ( insert @ A @ Y @ ( insert @ A @ X2 @ A4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % insert_commute
% 0.15/0.43 thf(fact_193_insert__eq__iff,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,A4: set @ A,B2: A,B4: set @ A] :
% 0.15/0.43 ( ~ ( member @ A @ A2 @ A4 )
% 0.15/0.43 => ( ~ ( member @ A @ B2 @ B4 )
% 0.15/0.43 => ( ( ( insert @ A @ A2 @ A4 )
% 0.15/0.43 = ( insert @ A @ B2 @ B4 ) )
% 0.15/0.43 = ( ( ( A2 = B2 )
% 0.15/0.43 => ( A4 = B4 ) )
% 0.15/0.43 & ( ( A2 != B2 )
% 0.15/0.43 => ? [C5: set @ A] :
% 0.15/0.43 ( ( A4
% 0.15/0.43 = ( insert @ A @ B2 @ C5 ) )
% 0.15/0.43 & ~ ( member @ A @ B2 @ C5 )
% 0.15/0.43 & ( B4
% 0.15/0.43 = ( insert @ A @ A2 @ C5 ) )
% 0.15/0.43 & ~ ( member @ A @ A2 @ C5 ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % insert_eq_iff
% 0.15/0.43 thf(fact_194_insert__absorb,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,A4: set @ A] :
% 0.15/0.43 ( ( member @ A @ A2 @ A4 )
% 0.15/0.43 => ( ( insert @ A @ A2 @ A4 )
% 0.15/0.43 = A4 ) ) ).
% 0.15/0.43
% 0.15/0.43 % insert_absorb
% 0.15/0.43 thf(fact_195_insert__ident,axiom,
% 0.15/0.43 ! [A: $tType,X2: A,A4: set @ A,B4: set @ A] :
% 0.15/0.43 ( ~ ( member @ A @ X2 @ A4 )
% 0.15/0.43 => ( ~ ( member @ A @ X2 @ B4 )
% 0.15/0.43 => ( ( ( insert @ A @ X2 @ A4 )
% 0.15/0.43 = ( insert @ A @ X2 @ B4 ) )
% 0.15/0.43 = ( A4 = B4 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % insert_ident
% 0.15/0.43 thf(fact_196_Set_Oset__insert,axiom,
% 0.15/0.43 ! [A: $tType,X2: A,A4: set @ A] :
% 0.15/0.43 ( ( member @ A @ X2 @ A4 )
% 0.15/0.43 => ~ ! [B8: set @ A] :
% 0.15/0.43 ( ( A4
% 0.15/0.43 = ( insert @ A @ X2 @ B8 ) )
% 0.15/0.43 => ( member @ A @ X2 @ B8 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Set.set_insert
% 0.15/0.43 thf(fact_197_insertI2,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,B4: set @ A,B2: A] :
% 0.15/0.43 ( ( member @ A @ A2 @ B4 )
% 0.15/0.43 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % insertI2
% 0.15/0.43 thf(fact_198_insertI1,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,B4: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B4 ) ) ).
% 0.15/0.43
% 0.15/0.43 % insertI1
% 0.15/0.43 thf(fact_199_insertE,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,B2: A,A4: set @ A] :
% 0.15/0.43 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
% 0.15/0.43 => ( ( A2 != B2 )
% 0.15/0.43 => ( member @ A @ A2 @ A4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % insertE
% 0.15/0.43 thf(fact_200_prod__eq__iff,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( ^ [Y4: product_prod @ A @ B,Z2: product_prod @ A @ B] : ( Y4 = Z2 ) )
% 0.15/0.43 = ( ^ [S3: product_prod @ A @ B,T2: product_prod @ A @ B] :
% 0.15/0.43 ( ( ( product_fst @ A @ B @ S3 )
% 0.15/0.43 = ( product_fst @ A @ B @ T2 ) )
% 0.15/0.43 & ( ( product_snd @ A @ B @ S3 )
% 0.15/0.43 = ( product_snd @ A @ B @ T2 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_eq_iff
% 0.15/0.43 thf(fact_201_prod_Oexpand,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 0.15/0.43 ( ( ( ( product_fst @ A @ B @ Prod )
% 0.15/0.43 = ( product_fst @ A @ B @ Prod2 ) )
% 0.15/0.43 & ( ( product_snd @ A @ B @ Prod )
% 0.15/0.43 = ( product_snd @ A @ B @ Prod2 ) ) )
% 0.15/0.43 => ( Prod = Prod2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod.expand
% 0.15/0.43 thf(fact_202_prod__eqI,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 0.15/0.43 ( ( ( product_fst @ A @ B @ P2 )
% 0.15/0.43 = ( product_fst @ A @ B @ Q2 ) )
% 0.15/0.43 => ( ( ( product_snd @ A @ B @ P2 )
% 0.15/0.43 = ( product_snd @ A @ B @ Q2 ) )
% 0.15/0.43 => ( P2 = Q2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_eqI
% 0.15/0.43 thf(fact_203_surjective__pairing,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,T3: product_prod @ A @ B] :
% 0.15/0.43 ( T3
% 0.15/0.43 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T3 ) @ ( product_snd @ A @ B @ T3 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % surjective_pairing
% 0.15/0.43 thf(fact_204_prod_Oexhaust__sel,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.15/0.43 ( Prod
% 0.15/0.43 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod.exhaust_sel
% 0.15/0.43 thf(fact_205_fst__conv,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 0.15/0.43 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.15/0.43 = X1 ) ).
% 0.15/0.43
% 0.15/0.43 % fst_conv
% 0.15/0.43 thf(fact_206_fst__eqD,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,X2: A,Y: B,A2: A] :
% 0.15/0.43 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y ) )
% 0.15/0.43 = A2 )
% 0.15/0.43 => ( X2 = A2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % fst_eqD
% 0.15/0.43 thf(fact_207_snd__conv,axiom,
% 0.15/0.43 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 0.15/0.43 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 0.15/0.43 = X22 ) ).
% 0.15/0.43
% 0.15/0.43 % snd_conv
% 0.15/0.43 thf(fact_208_snd__eqD,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,X2: B,Y: A,A2: A] :
% 0.15/0.43 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
% 0.15/0.43 = A2 )
% 0.15/0.43 => ( Y = A2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % snd_eqD
% 0.15/0.43 thf(fact_209_refl__on__singleton,axiom,
% 0.15/0.43 ! [A: $tType,X2: A] : ( refl_on @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % refl_on_singleton
% 0.15/0.43 thf(fact_210_total__on__singleton,axiom,
% 0.15/0.43 ! [A: $tType,X2: A] : ( total_on @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % total_on_singleton
% 0.15/0.43 thf(fact_211_subset__singletonD,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A,X2: A] :
% 0.15/0.43 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.15/0.43 => ( ( A4
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) )
% 0.15/0.43 | ( A4
% 0.15/0.43 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_singletonD
% 0.15/0.43 thf(fact_212_subset__singleton__iff,axiom,
% 0.15/0.43 ! [A: $tType,X6: set @ A,A2: A] :
% 0.15/0.43 ( ( ord_less_eq @ ( set @ A ) @ X6 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.15/0.43 = ( ( X6
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) )
% 0.15/0.43 | ( X6
% 0.15/0.43 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_singleton_iff
% 0.15/0.43 thf(fact_213_snd__mono,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( ( ord @ A )
% 0.15/0.43 & ( ord @ B ) )
% 0.15/0.43 => ! [X2: product_prod @ A @ B,Y: product_prod @ A @ B] :
% 0.15/0.43 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ X2 @ Y )
% 0.15/0.43 => ( ord_less_eq @ B @ ( product_snd @ A @ B @ X2 ) @ ( product_snd @ A @ B @ Y ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % snd_mono
% 0.15/0.43 thf(fact_214_fst__mono,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( ( ord @ A )
% 0.15/0.43 & ( ord @ B ) )
% 0.15/0.43 => ! [X2: product_prod @ A @ B,Y: product_prod @ A @ B] :
% 0.15/0.43 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ X2 @ Y )
% 0.15/0.43 => ( ord_less_eq @ A @ ( product_fst @ A @ B @ X2 ) @ ( product_fst @ A @ B @ Y ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % fst_mono
% 0.15/0.43 thf(fact_215_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,P: A > B > $o,X2: A,Y: B,A2: product_prod @ A @ B] :
% 0.15/0.43 ( ( P @ X2 @ Y )
% 0.15/0.43 => ( ( A2
% 0.15/0.43 = ( product_Pair @ A @ B @ X2 @ Y ) )
% 0.15/0.43 => ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % BNF_Greatest_Fixpoint.subst_Pair
% 0.15/0.43 thf(fact_216_conjI__realizer,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q2: B] :
% 0.15/0.43 ( ( P @ P2 )
% 0.15/0.43 => ( ( Q @ Q2 )
% 0.15/0.43 => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) )
% 0.15/0.43 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % conjI_realizer
% 0.15/0.43 thf(fact_217_exI__realizer,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,P: A > B > $o,Y: A,X2: B] :
% 0.15/0.43 ( ( P @ Y @ X2 )
% 0.15/0.43 => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % exI_realizer
% 0.15/0.43 thf(fact_218_the__elem__eq,axiom,
% 0.15/0.43 ! [A: $tType,X2: A] :
% 0.15/0.43 ( ( the_elem @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.15/0.43 = X2 ) ).
% 0.15/0.43
% 0.15/0.43 % the_elem_eq
% 0.15/0.43 thf(fact_219_is__singletonI,axiom,
% 0.15/0.43 ! [A: $tType,X2: A] : ( is_singleton @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % is_singletonI
% 0.15/0.43 thf(fact_220_is__singleton__the__elem,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( is_singleton @ A )
% 0.15/0.43 = ( ^ [A5: set @ A] :
% 0.15/0.43 ( A5
% 0.15/0.43 = ( insert @ A @ ( the_elem @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % is_singleton_the_elem
% 0.15/0.43 thf(fact_221_is__singletonI_H,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A] :
% 0.15/0.43 ( ( A4
% 0.15/0.43 != ( bot_bot @ ( set @ A ) ) )
% 0.15/0.43 => ( ! [X4: A,Y3: A] :
% 0.15/0.43 ( ( member @ A @ X4 @ A4 )
% 0.15/0.43 => ( ( member @ A @ Y3 @ A4 )
% 0.15/0.43 => ( X4 = Y3 ) ) )
% 0.15/0.43 => ( is_singleton @ A @ A4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % is_singletonI'
% 0.15/0.43 thf(fact_222_is__singletonE,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A] :
% 0.15/0.43 ( ( is_singleton @ A @ A4 )
% 0.15/0.43 => ~ ! [X4: A] :
% 0.15/0.43 ( A4
% 0.15/0.43 != ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % is_singletonE
% 0.15/0.43 thf(fact_223_is__singleton__def,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( is_singleton @ A )
% 0.15/0.43 = ( ^ [A5: set @ A] :
% 0.15/0.43 ? [X3: A] :
% 0.15/0.43 ( A5
% 0.15/0.43 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % is_singleton_def
% 0.15/0.43 thf(fact_224_linear__order__on__singleton,axiom,
% 0.15/0.43 ! [A: $tType,X2: A] : ( order_1409979114der_on @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % linear_order_on_singleton
% 0.15/0.43 thf(fact_225_sndI,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,X2: product_prod @ A @ B,Y: A,Z: B] :
% 0.15/0.43 ( ( X2
% 0.15/0.43 = ( product_Pair @ A @ B @ Y @ Z ) )
% 0.15/0.43 => ( ( product_snd @ A @ B @ X2 )
% 0.15/0.43 = Z ) ) ).
% 0.15/0.43
% 0.15/0.43 % sndI
% 0.15/0.43 thf(fact_226_lnear__order__on__empty,axiom,
% 0.15/0.43 ! [A: $tType] : ( order_1409979114der_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % lnear_order_on_empty
% 0.15/0.43 thf(fact_227_fstI,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,X2: product_prod @ A @ B,Y: A,Z: B] :
% 0.15/0.43 ( ( X2
% 0.15/0.43 = ( product_Pair @ A @ B @ Y @ Z ) )
% 0.15/0.43 => ( ( product_fst @ A @ B @ X2 )
% 0.15/0.43 = Y ) ) ).
% 0.15/0.43
% 0.15/0.43 % fstI
% 0.15/0.43 thf(fact_228_eq__snd__iff,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
% 0.15/0.43 ( ( B2
% 0.15/0.43 = ( product_snd @ B @ A @ P2 ) )
% 0.15/0.43 = ( ? [A7: B] :
% 0.15/0.43 ( P2
% 0.15/0.43 = ( product_Pair @ B @ A @ A7 @ B2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % eq_snd_iff
% 0.15/0.43 thf(fact_229_eq__fst__iff,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,A2: A,P2: product_prod @ A @ B] :
% 0.15/0.43 ( ( A2
% 0.15/0.43 = ( product_fst @ A @ B @ P2 ) )
% 0.15/0.43 = ( ? [B7: B] :
% 0.15/0.43 ( P2
% 0.15/0.43 = ( product_Pair @ A @ B @ A2 @ B7 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % eq_fst_iff
% 0.15/0.43 thf(fact_230_prod_Oswap__def,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( product_swap @ A @ B )
% 0.15/0.43 = ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod.swap_def
% 0.15/0.43 thf(fact_231_Range__insert,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,A2: B,B2: A,R: set @ ( product_prod @ B @ A )] :
% 0.15/0.43 ( ( range @ B @ A @ ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A2 @ B2 ) @ R ) )
% 0.15/0.43 = ( insert @ A @ B2 @ ( range @ B @ A @ R ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Range_insert
% 0.15/0.43 thf(fact_232_swap__swap,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 0.15/0.43 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 0.15/0.43 = P2 ) ).
% 0.15/0.43
% 0.15/0.43 % swap_swap
% 0.15/0.43 thf(fact_233_swap__simp,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,X2: B,Y: A] :
% 0.15/0.43 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
% 0.15/0.43 = ( product_Pair @ A @ B @ Y @ X2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % swap_simp
% 0.15/0.43 thf(fact_234_Range__Id__on,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A] :
% 0.15/0.43 ( ( range @ A @ A @ ( id_on @ A @ A4 ) )
% 0.15/0.43 = A4 ) ).
% 0.15/0.43
% 0.15/0.43 % Range_Id_on
% 0.15/0.43 thf(fact_235_Range__empty,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( range @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Range_empty
% 0.15/0.43 thf(fact_236_fst__swap,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,X2: product_prod @ B @ A] :
% 0.15/0.43 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X2 ) )
% 0.15/0.43 = ( product_snd @ B @ A @ X2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % fst_swap
% 0.15/0.43 thf(fact_237_snd__swap,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,X2: product_prod @ A @ B] :
% 0.15/0.43 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X2 ) )
% 0.15/0.43 = ( product_fst @ A @ B @ X2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % snd_swap
% 0.15/0.43 thf(fact_238_RangeE,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A )] :
% 0.15/0.43 ( ( member @ A @ B2 @ ( range @ B @ A @ R ) )
% 0.15/0.43 => ~ ! [A6: B] :
% 0.15/0.43 ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A6 @ B2 ) @ R ) ) ).
% 0.15/0.43
% 0.15/0.43 % RangeE
% 0.15/0.43 thf(fact_239_Range__iff,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,A2: A,R: set @ ( product_prod @ B @ A )] :
% 0.15/0.43 ( ( member @ A @ A2 @ ( range @ B @ A @ R ) )
% 0.15/0.43 = ( ? [Y2: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y2 @ A2 ) @ R ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Range_iff
% 0.15/0.43 thf(fact_240_Range_Ocases,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,A2: B,R: set @ ( product_prod @ A @ B )] :
% 0.15/0.43 ( ( member @ B @ A2 @ ( range @ A @ B @ R ) )
% 0.15/0.43 => ~ ! [A6: A] :
% 0.15/0.43 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ A2 ) @ R ) ) ).
% 0.15/0.43
% 0.15/0.43 % Range.cases
% 0.15/0.43 thf(fact_241_Range_Osimps,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,A2: B,R: set @ ( product_prod @ A @ B )] :
% 0.15/0.43 ( ( member @ B @ A2 @ ( range @ A @ B @ R ) )
% 0.15/0.43 = ( ? [A7: A,B7: B] :
% 0.15/0.43 ( ( A2 = B7 )
% 0.15/0.43 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B7 ) @ R ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Range.simps
% 0.15/0.43 thf(fact_242_Range_Ointros,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,A2: A,B2: B,R: set @ ( product_prod @ A @ B )] :
% 0.15/0.43 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R )
% 0.15/0.43 => ( member @ B @ B2 @ ( range @ A @ B @ R ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Range.intros
% 0.15/0.43 thf(fact_243_Range_Oinducts,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,X2: B,R: set @ ( product_prod @ A @ B ),P: B > $o] :
% 0.15/0.43 ( ( member @ B @ X2 @ ( range @ A @ B @ R ) )
% 0.15/0.43 => ( ! [A6: A,B5: B] :
% 0.15/0.43 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ R )
% 0.15/0.43 => ( P @ B5 ) )
% 0.15/0.43 => ( P @ X2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Range.inducts
% 0.15/0.43 thf(fact_244_Range__empty__iff,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A )] :
% 0.15/0.43 ( ( ( range @ B @ A @ R )
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) )
% 0.15/0.43 = ( R
% 0.15/0.43 = ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Range_empty_iff
% 0.15/0.43 thf(fact_245_Range__mono,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.15/0.43 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S )
% 0.15/0.43 => ( ord_less_eq @ ( set @ B ) @ ( range @ A @ B @ R ) @ ( range @ A @ B @ S ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Range_mono
% 0.15/0.43 thf(fact_246_Domain__insert,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,A2: A,B2: B,R: set @ ( product_prod @ A @ B )] :
% 0.15/0.43 ( ( domain @ A @ B @ ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R ) )
% 0.15/0.43 = ( insert @ A @ A2 @ ( domain @ A @ B @ R ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Domain_insert
% 0.15/0.43 thf(fact_247_subset__Compl__singleton,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A,B2: A] :
% 0.15/0.43 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.15/0.43 = ( ~ ( member @ A @ B2 @ A4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_Compl_singleton
% 0.15/0.43 thf(fact_248_Compl__eq__Compl__iff,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.15/0.43 ( ( ( uminus_uminus @ ( set @ A ) @ A4 )
% 0.15/0.43 = ( uminus_uminus @ ( set @ A ) @ B4 ) )
% 0.15/0.43 = ( A4 = B4 ) ) ).
% 0.15/0.43
% 0.15/0.43 % Compl_eq_Compl_iff
% 0.15/0.43 thf(fact_249_Compl__iff,axiom,
% 0.15/0.43 ! [A: $tType,C3: A,A4: set @ A] :
% 0.15/0.43 ( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
% 0.15/0.43 = ( ~ ( member @ A @ C3 @ A4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Compl_iff
% 0.15/0.43 thf(fact_250_ComplI,axiom,
% 0.15/0.43 ! [A: $tType,C3: A,A4: set @ A] :
% 0.15/0.43 ( ~ ( member @ A @ C3 @ A4 )
% 0.15/0.43 => ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ComplI
% 0.15/0.43 thf(fact_251_Compl__anti__mono,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.15/0.43 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.15/0.43 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Compl_anti_mono
% 0.15/0.43 thf(fact_252_Compl__subset__Compl__iff,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.15/0.43 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) )
% 0.15/0.43 = ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% 0.15/0.43
% 0.15/0.43 % Compl_subset_Compl_iff
% 0.15/0.43 thf(fact_253_Domain__Id__on,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A] :
% 0.15/0.43 ( ( domain @ A @ A @ ( id_on @ A @ A4 ) )
% 0.15/0.43 = A4 ) ).
% 0.15/0.43
% 0.15/0.43 % Domain_Id_on
% 0.15/0.43 thf(fact_254_Domain__empty,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( domain @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Domain_empty
% 0.15/0.43 thf(fact_255_subset__Compl__self__eq,axiom,
% 0.15/0.43 ! [A: $tType,A4: set @ A] :
% 0.15/0.43 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
% 0.15/0.43 = ( A4
% 0.15/0.43 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_Compl_self_eq
% 0.15/0.43
% 0.15/0.43 % Type constructors (21)
% 0.15/0.43 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.15/0.43 ! [A8: $tType,A9: $tType] :
% 0.15/0.43 ( ( order_bot @ A9 )
% 0.15/0.43 => ( order_bot @ ( A8 > A9 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.15/0.43 ! [A8: $tType,A9: $tType] :
% 0.15/0.43 ( ( preorder @ A9 )
% 0.15/0.43 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.15/0.43 ! [A8: $tType,A9: $tType] :
% 0.15/0.43 ( ( order @ A9 )
% 0.15/0.43 => ( order @ ( A8 > A9 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_fun___Orderings_Oord,axiom,
% 0.15/0.43 ! [A8: $tType,A9: $tType] :
% 0.15/0.43 ( ( ord @ A9 )
% 0.15/0.43 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_fun___Orderings_Obot,axiom,
% 0.15/0.43 ! [A8: $tType,A9: $tType] :
% 0.15/0.43 ( ( bot @ A9 )
% 0.15/0.43 => ( bot @ ( A8 > A9 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_Set_Oset___Orderings_Oorder__bot_1,axiom,
% 0.15/0.43 ! [A8: $tType] : ( order_bot @ ( set @ A8 ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
% 0.15/0.43 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
% 0.15/0.43 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_Set_Oset___Orderings_Oord_4,axiom,
% 0.15/0.43 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_Set_Oset___Orderings_Obot_5,axiom,
% 0.15/0.43 ! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_HOL_Obool___Orderings_Oorder__bot_6,axiom,
% 0.15/0.43 order_bot @ $o ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 0.15/0.43 preorder @ $o ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.15/0.43 linorder @ $o ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_HOL_Obool___Orderings_Oorder_8,axiom,
% 0.15/0.43 order @ $o ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_HOL_Obool___Orderings_Oord_9,axiom,
% 0.15/0.43 ord @ $o ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_HOL_Obool___Orderings_Obot_10,axiom,
% 0.15/0.43 bot @ $o ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_Product__Type_Oprod___Orderings_Oorder__bot_11,axiom,
% 0.15/0.43 ! [A8: $tType,A9: $tType] :
% 0.15/0.43 ( ( ( order_bot @ A8 )
% 0.15/0.43 & ( order_bot @ A9 ) )
% 0.15/0.43 => ( order_bot @ ( product_prod @ A8 @ A9 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_12,axiom,
% 0.15/0.43 ! [A8: $tType,A9: $tType] :
% 0.15/0.43 ( ( ( preorder @ A8 )
% 0.15/0.43 & ( preorder @ A9 ) )
% 0.15/0.43 => ( preorder @ ( product_prod @ A8 @ A9 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_Product__Type_Oprod___Orderings_Oorder_13,axiom,
% 0.15/0.43 ! [A8: $tType,A9: $tType] :
% 0.15/0.43 ( ( ( order @ A8 )
% 0.15/0.43 & ( order @ A9 ) )
% 0.15/0.43 => ( order @ ( product_prod @ A8 @ A9 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_Product__Type_Oprod___Orderings_Oord_14,axiom,
% 0.15/0.43 ! [A8: $tType,A9: $tType] :
% 0.15/0.43 ( ( ( ord @ A8 )
% 0.15/0.43 & ( ord @ A9 ) )
% 0.15/0.43 => ( ord @ ( product_prod @ A8 @ A9 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_Product__Type_Oprod___Orderings_Obot_15,axiom,
% 0.15/0.43 ! [A8: $tType,A9: $tType] :
% 0.15/0.43 ( ( ( bot @ A8 )
% 0.15/0.43 & ( bot @ A9 ) )
% 0.15/0.43 => ( bot @ ( product_prod @ A8 @ A9 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Conjectures (1)
% 0.15/0.43 thf(conj_0,conjecture,
% 0.15/0.43 ( ( member @ a @ z @ carrier )
% 0.15/0.43 & ( member @ a @ y @ carrier ) ) ).
% 0.15/0.43
% 0.15/0.43 %------------------------------------------------------------------------------
% 0.15/0.43 ------------------------
% 2.30/1.36 ---- Embedded in THF ---
% 2.30/1.36 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.30/1.36 %%% Generated on Tue Feb 24 08:28:07 EST 2026
% 2.30/1.36 %%% using '$$dhol' embedding, version 1.3.0.
% 2.30/1.36 %%% Logic specification used:
% 2.30/1.36 %%% thf(spec, logic, $$dhol).
% 2.30/1.36
% 2.30/1.36 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.xCdPegwIoG/DTF2DTF29341.p
% 2.30/1.36 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.30/1.36 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.30/1.36 thf(ty_tf_a, type, a: $tType).
% 2.30/1.36 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.30/1.36 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 2.30/1.36 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.30/1.36 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.30/1.36 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.30/1.36 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.30/1.36 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: ($o))).
% 2.30/1.36 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain, type, bNF_Ca1785829860lChain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 2.30/1.36 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 2.30/1.36 thf(sy_c_Order__Relation_Olinear__order__on, type, order_1409979114der_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.30/1.36 thf(sy_c_Order__Relation_Opreorder__on, type, order_preorder_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.30/1.36 thf(sy_c_Order__Relation_Ounder, type, order_under: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (A > (set @ A)))))).
% 2.30/1.36 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.30/1.36 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.30/1.36 thf(sy_c_Orderings_Oorder__class_OGreatest, type, order_Greatest: (!> [A:$tType]: (((A > $o) > A)))).
% 2.30/1.36 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oas__good__as, type, prefer951318096ood_as: (!> [A:$tType]: ((A > ((set @ A) > ((set @ ((product_prod @ A) @ A)) > (set @ A))))))).
% 2.30/1.36 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oat__least__as__good, type, prefer310429814s_good: (!> [A:$tType]: ((A > ((set @ A) > ((set @ ((product_prod @ A) @ A)) > (set @ A))))))).
% 2.30/1.36 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Ono__better__than, type, prefer1532642881r_than: (!> [A:$tType]: ((A > ((set @ A) > ((set @ ((product_prod @ A) @ A)) > (set @ A))))))).
% 2.30/1.36 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Opreference, type, prefer199794634erence: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.30/1.36 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference, type, prefer1997167224erence: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.30/1.36 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference__axioms, type, prefer1801827867axioms: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.30/1.36 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.30/1.36 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.30/1.36 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.30/1.36 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))).
% 2.30/1.36 thf(sy_c_Product__Type_Oprod_Osnd, type, product_snd: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > B)))).
% 2.30/1.36 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 2.30/1.36 thf(sy_c_Relation_ODomain, type, domain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ A))))).
% 2.30/1.36 thf(sy_c_Relation_OId__on, type, id_on: (!> [A:$tType]: (((set @ A) > (set @ ((product_prod @ A) @ A)))))).
% 2.30/1.36 thf(sy_c_Relation_ORange, type, range: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ B))))).
% 2.30/1.36 thf(sy_c_Relation_Orefl__on, type, refl_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.30/1.36 thf(sy_c_Relation_Ototal__on, type, total_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 2.30/1.36 thf(sy_c_Relation_Otrans, type, trans: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 2.30/1.36 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.30/1.36 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.30/1.36 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.30/1.36 thf(sy_c_Set_Ois__singleton, type, is_singleton: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.30/1.36 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))).
% 2.30/1.36 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.30/1.36 thf(sy_v_carrier, type, carrier: (set @ a)).
% 2.30/1.36 thf(sy_v_relation, type, relation: (set @ ((product_prod @ a) @ a))).
% 2.30/1.36 thf(sy_v_y, type, y: a).
% 2.30/1.36 thf(sy_v_z, type, z: a).
% 2.30/1.36 thf(fact_0_assms, axiom, ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ z) @ y)) @ relation) & (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ y) @ z)) @ relation))).
% 2.30/1.36 thf(fact_1_rational__preference__axioms, axiom, (((prefer1997167224erence @ a) @ carrier) @ relation)).
% 2.30/1.36 thf(fact_2_preference__axioms, axiom, (((prefer199794634erence @ a) @ carrier) @ relation)).
% 2.30/1.36 thf(fact_3_trans__refl, axiom, (((order_preorder_on @ a) @ carrier) @ relation)).
% 2.30/1.36 thf(fact_4_total, axiom, (((total_on @ a) @ carrier) @ relation)).
% 2.30/1.36 thf(fact_5_reflexivity, axiom, (((refl_on @ a) @ carrier) @ relation)).
% 2.30/1.36 thf(fact_6_compl, axiom, (! [X:a]: (((((member @ a) @ X) @ carrier) => (! [Xa:a]: (((((member @ a) @ Xa) @ carrier) => ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X) @ Xa)) @ relation) | (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Xa) @ X)) @ relation))))))))).
% 2.30/1.36 thf(fact_7_completeD, axiom, (! [X2:a,Y:a]: (((((member @ a) @ X2) @ carrier) => ((((member @ a) @ Y) @ carrier) => ((X2 != Y) => ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X2) @ Y)) @ relation) | (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ X2)) @ relation)))))))).
% 2.30/1.36 thf(fact_8_not__outside, axiom, (! [X2:a,Y:a]: (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X2) @ Y)) @ relation) => (((member @ a) @ X2) @ carrier))))).
% 2.30/1.36 thf(fact_9_strict__is__neg__transitive, axiom, (! [X2:a,Y:a,Z:a]: ((((((member @ a) @ X2) @ carrier) & ((((member @ a) @ Y) @ carrier) & (((member @ a) @ Z) @ carrier))) => (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X2) @ Y)) @ relation) & (~ ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ X2)) @ relation)))) => (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X2) @ Z)) @ relation) & (~ ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Z) @ X2)) @ 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)))))))))).
% 2.30/1.36 thf(fact_10_weak__is__transitive, axiom, (! [X2:a,Y:a,Z:a]: ((((((member @ a) @ X2) @ carrier) & ((((member @ a) @ Y) @ carrier) & (((member @ a) @ Z) @ carrier))) => ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X2) @ Y)) @ relation) => ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ Z)) @ relation) => (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X2) @ Z)) @ relation))))))).
% 2.30/1.36 thf(fact_11_strict__not__refl__weak, axiom, (! [X2:a,Y:a]: ((((((member @ a) @ X2) @ carrier) & (((member @ a) @ Y) @ carrier)) => ((~ ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ X2)) @ relation))) = ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X2) @ Y)) @ relation) & (~ ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ X2)) @ relation))))))))).
% 2.30/1.36 thf(fact_12_indiff__trans, axiom, (! [X2:a,Y:a,Z:a]: ((((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X2) @ Y)) @ relation) & (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ X2)) @ 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) @ X2) @ Z)) @ relation) & (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Z) @ X2)) @ relation))))))).
% 2.30/1.36 thf(fact_13_strict__trans, axiom, (! [X2:a,Y:a,Z:a]: ((((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X2) @ Y)) @ relation) & (~ ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ X2)) @ 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) @ X2) @ Z)) @ relation) & (~ ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Z) @ X2)) @ relation))))))))).
% 2.30/1.36 thf(fact_14_pref__in__at__least__as, axiom, (! [X2:a,Y:a]: (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X2) @ Y)) @ relation) => (((member @ a) @ X2) @ ((((prefer310429814s_good @ a) @ Y) @ carrier) @ relation)))))).
% 2.30/1.36 thf(fact_15_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))))).
% 2.30/1.36 thf(fact_16_preference__def, axiom, (! [A:$tType]: (((prefer199794634erence @ A) = (^ [Carrier2:(set @ A),Relation2:(set @ ((product_prod @ A) @ A))]: (((! [X3:A,Y2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y2)) @ Relation2) => (((member @ A) @ X3) @ Carrier2)))) & ((! [X3:A,Y2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y2)) @ Relation2) => (((member @ A) @ Y2) @ Carrier2)))) & (((order_preorder_on @ A) @ Carrier2) @ Relation2))))))))).
% 2.30/1.36 thf(fact_17_preference_Ointro, axiom, (! [A:$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))))))).
% 2.30/1.36 thf(fact_18_preference_Otrans__refl, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer199794634erence @ A) @ Carrier) @ Relation) => (((order_preorder_on @ A) @ Carrier) @ Relation))))).
% 2.30/1.36 thf(fact_19_preference_Onot__outside, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X2:A,Y:A]: (((((prefer199794634erence @ A) @ Carrier) @ Relation) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y)) @ Relation) => (((member @ A) @ X2) @ Carrier)))))).
% 2.30/1.36 thf(fact_20_preference_Oreflexivity, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer199794634erence @ A) @ Carrier) @ Relation) => (((refl_on @ A) @ Carrier) @ Relation))))).
% 2.30/1.36 thf(fact_21_preference_Oindiff__trans, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X2:A,Y:A,Z:A]: (((((prefer199794634erence @ A) @ Carrier) @ Relation) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y)) @ Relation) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X2)) @ 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) @ X2) @ Z)) @ Relation) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ X2)) @ Relation)))))))).
% 2.30/1.36 thf(fact_22_rational__preference_Ocompl, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (! [X:A]: (((((member @ A) @ X) @ Carrier) => (! [Xa:A]: (((((member @ A) @ Xa) @ Carrier) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Xa)) @ Relation) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Xa) @ X)) @ Relation)))))))))))).
% 2.30/1.36 thf(fact_23_rational__preference_Ototal, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((total_on @ A) @ Carrier) @ Relation))))).
% 2.30/1.36 thf(fact_24_rational__preference_OcompleteD, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X2:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ A) @ X2) @ Carrier) => ((((member @ A) @ Y) @ Carrier) => ((X2 != Y) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y)) @ Relation) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X2)) @ Relation))))))))).
% 2.30/1.36 thf(fact_25_rational__preference_Ostrict__trans, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X2:A,Y:A,Z:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y)) @ Relation) & (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X2)) @ 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) @ X2) @ Z)) @ Relation) & (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ X2)) @ Relation)))))))))).
% 2.30/1.36 thf(fact_26_rational__preference_Oweak__is__transitive, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X2:A,Y:A,Z:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((((member @ A) @ X2) @ Carrier) & ((((member @ A) @ Y) @ Carrier) & (((member @ A) @ Z) @ Carrier))) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y)) @ Relation) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ Relation) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Z)) @ Relation)))))))).
% 2.30/1.36 thf(fact_27_rational__preference_Ostrict__not__refl__weak, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X2:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((((member @ A) @ X2) @ Carrier) & (((member @ A) @ Y) @ Carrier)) => ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X2)) @ Relation))) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y)) @ Relation) & (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X2)) @ Relation)))))))))).
% 2.30/1.36 thf(fact_28_rational__preference_Ostrict__is__neg__transitive, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X2:A,Y:A,Z:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((((member @ A) @ X2) @ Carrier) & ((((member @ A) @ Y) @ Carrier) & (((member @ A) @ Z) @ Carrier))) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y)) @ Relation) & (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X2)) @ Relation)))) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Z)) @ Relation) & (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ X2)) @ Relation)))) | ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ Y)) @ Relation) & (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ Relation))))))))))).
% 2.30/1.36 thf(fact_29_worse__in__no__better, axiom, (! [X2:a,Y:a]: (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X2) @ Y)) @ relation) => (((member @ a) @ Y) @ ((((prefer1532642881r_than @ a) @ Y) @ carrier) @ relation)))))).
% 2.30/1.36 thf(fact_30_as__good__as__sameIff, axiom, (! [Z:a,Y:a]: (((((member @ a) @ Z) @ ((((prefer951318096ood_as @ a) @ Y) @ carrier) @ relation)) = ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Z) @ Y)) @ relation) & (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ Y) @ Z)) @ relation)))))).
% 2.30/1.36 thf(fact_31_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y22:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y22)) = ((X1 = Y1) & (X22 = Y22)))))).
% 2.30/1.36 thf(fact_32_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A3:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A3) @ B3)) = ((A2 = A3) & (B2 = B3)))))).
% 2.30/1.36 thf(fact_33_rational__preference_Oas__good__as__sameIff, axiom, (! [A:$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))))))).
% 2.30/1.36 thf(fact_34_total__onI, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A))]: (((! [X4:A,Y3:A]: (((((member @ A) @ X4) @ A4) => ((((member @ A) @ Y3) @ A4) => ((X4 != Y3) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y3)) @ R) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X4)) @ R))))))) => (((total_on @ A) @ A4) @ R))))).
% 2.30/1.36 thf(fact_35_total__on__def, axiom, (! [A:$tType]: (((total_on @ A) = (^ [A5:(set @ A),R2:(set @ ((product_prod @ A) @ A))]: ((! [X3:A]: (((((member @ A) @ X3) @ A5) => (! [Y2:A]: (((((member @ A) @ Y2) @ A5) => ((X3 != Y2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y2)) @ R2) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ X3)) @ R2))))))))))))))).
% 2.30/1.36 thf(fact_36_refl__onD, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),A2:A]: (((((refl_on @ A) @ A4) @ R) => ((((member @ A) @ A2) @ A4) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ R)))))).
% 2.30/1.36 thf(fact_37_refl__onD1, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),X2:A,Y:A]: (((((refl_on @ A) @ A4) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y)) @ R) => (((member @ A) @ X2) @ A4)))))).
% 2.30/1.36 thf(fact_38_refl__onD2, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),X2:A,Y:A]: (((((refl_on @ A) @ A4) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y)) @ R) => (((member @ A) @ Y) @ A4)))))).
% 2.30/1.36 thf(fact_39_refl__on__domain, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),A2:A,B2:A]: (((((refl_on @ A) @ A4) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R) => ((((member @ A) @ A2) @ A4) & (((member @ A) @ B2) @ A4))))))).
% 2.30/1.36 thf(fact_40_at__least__as__goodD, axiom, (! [A:$tType,Z:A,Y:A,B4:(set @ A),Pr:(set @ ((product_prod @ A) @ A))]: (((((member @ A) @ Z) @ ((((prefer310429814s_good @ A) @ Y) @ B4) @ Pr)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z) @ Y)) @ Pr))))).
% 2.30/1.36 thf(fact_41_at__lst__asgd__ge, axiom, (! [A:$tType,X2:A,Y:A,B4:(set @ A),Pr:(set @ ((product_prod @ A) @ A))]: (((((member @ A) @ X2) @ ((((prefer310429814s_good @ A) @ Y) @ B4) @ Pr)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y)) @ Pr))))).
% 2.30/1.36 thf(fact_42_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A6:A,B5:B]: ((P @ ((((product_Pair @ A) @ B) @ A6) @ B5)))) => (P @ Prod))))).
% 2.30/1.36 thf(fact_43_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A6:A,B5:B]: ((Y != ((((product_Pair @ A) @ B) @ A6) @ B5))))))))).
% 2.30/1.36 thf(fact_44_prod__induct7, axiom, (! [G:$tType,F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))))]: (((! [A6:A,B5:B,C2:C,D2:D,E2:E,F2:F,G2:G]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G)) @ E2) @ ((((product_Pair @ F) @ G) @ F2) @ G2))))))))) => (P @ X2))))).
% 2.30/1.36 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.30/1.36 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A4)))) = A4)))).
% 2.30/1.36 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) = (Q @ X4)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.30/1.36 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F3:(A > B),G3:(A > B)]: (((! [X4:A]: (((F3 @ X4) = (G3 @ X4)))) => (F3 = G3))))).
% 2.30/1.36 thf(fact_49_prod__induct6, axiom, (! [F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: (((! [A6:A,B5:B,C2:C,D2:D,E2:E,F2:F]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2)))))))) => (P @ X2))))).
% 2.30/1.36 thf(fact_50_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A6:A,B5:B,C2:C,D2:D,E2:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P @ X2))))).
% 2.30/1.36 thf(fact_51_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A6:A,B5:B,C2:C,D2:D]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C2) @ D2)))))) => (P @ X2))))).
% 2.30/1.36 thf(fact_52_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A6:A,B5:B,C2:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A6) @ ((((product_Pair @ B) @ C) @ B5) @ C2))))) => (P @ X2))))).
% 2.30/1.36 thf(fact_53_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,G:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))))]: ((~ ((! [A6:A,B5:B,C2:C,D2:D,E2:E,F2:F,G2:G]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G)) @ E2) @ ((((product_Pair @ F) @ G) @ F2) @ G2)))))))))))))).
% 2.30/1.36 thf(fact_54_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: ((~ ((! [A6:A,B5:B,C2:C,D2:D,E2:E,F2:F]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2))))))))))))).
% 2.30/1.36 thf(fact_55_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A6:A,B5:B,C2:C,D2:D,E2:E]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 2.30/1.36 thf(fact_56_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A6:A,B5:B,C2:C,D2:D]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C2) @ D2))))))))))).
% 2.30/1.36 thf(fact_57_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A6:A,B5:B,C2:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A6) @ ((((product_Pair @ B) @ C) @ B5) @ C2)))))))))).
% 2.30/1.36 thf(fact_58_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A3:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A3) @ B3)) => (~ (((A2 = A3) => (B2 != B3)))))))).
% 2.30/1.36 thf(fact_59_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A6:A,B5:B]: ((P @ ((((product_Pair @ A) @ B) @ A6) @ B5)))) => (P @ P2))))).
% 2.30/1.36 thf(fact_60_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X4:A,Y3:B]: ((P2 = ((((product_Pair @ A) @ B) @ X4) @ Y3))))))).
% 2.30/1.36 thf(fact_61_rational__preference_Oworse__in__no__better, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X2:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y)) @ Relation) => (((member @ A) @ Y) @ ((((prefer1532642881r_than @ A) @ Y) @ Carrier) @ Relation))))))).
% 2.30/1.36 thf(fact_62_rational__preference_Opref__in__at__least__as, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X2:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y)) @ Relation) => (((member @ A) @ X2) @ ((((prefer310429814s_good @ A) @ Y) @ Carrier) @ Relation))))))).
% 2.30/1.36 thf(fact_63_at__lst__asgd__not__ge, axiom, (! [X2:a,Y:a]: (((carrier != (bot_bot @ (set @ a))) => ((((member @ a) @ X2) @ carrier) => ((((member @ a) @ Y) @ carrier) => ((~ ((((member @ a) @ X2) @ ((((prefer310429814s_good @ a) @ Y) @ carrier) @ relation)))) => (~ ((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X2) @ Y)) @ relation)))))))))).
% 2.30/1.36 thf(fact_64_no__better__subset__pref, axiom, (! [X2:a,Y:a]: (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X2) @ Y)) @ relation) => (((ord_less_eq @ (set @ a)) @ ((((prefer1532642881r_than @ a) @ Y) @ carrier) @ relation)) @ ((((prefer1532642881r_than @ a) @ X2) @ carrier) @ relation)))))).
% 2.30/1.36 thf(fact_65_no__better__thansubset__rel, axiom, (! [X2:a,Y:a]: (((((member @ a) @ X2) @ carrier) => ((((member @ a) @ Y) @ carrier) => ((((ord_less_eq @ (set @ a)) @ ((((prefer1532642881r_than @ a) @ Y) @ carrier) @ relation)) @ ((((prefer1532642881r_than @ a) @ X2) @ carrier) @ relation)) => (((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ X2) @ Y)) @ relation))))))).
% 2.30/1.36 thf(fact_66_no__better__than__nonepty, axiom, (! [X2:a]: (((carrier != (bot_bot @ (set @ a))) => ((((member @ a) @ X2) @ carrier) => (((((prefer1532642881r_than @ a) @ X2) @ carrier) @ relation) != (bot_bot @ (set @ a)))))))).
% 2.30/1.36 thf(fact_67_nbt__nest, axiom, (! [Y:a,X2:a]: (((((ord_less_eq @ (set @ a)) @ ((((prefer1532642881r_than @ a) @ Y) @ carrier) @ relation)) @ ((((prefer1532642881r_than @ a) @ X2) @ carrier) @ relation)) | (((ord_less_eq @ (set @ a)) @ ((((prefer1532642881r_than @ a) @ X2) @ carrier) @ relation)) @ ((((prefer1532642881r_than @ a) @ Y) @ carrier) @ relation)))))).
% 2.30/1.36 thf(fact_68_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))).
% 2.30/1.36 thf(fact_69_rational__preference_Ointro, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer199794634erence @ A) @ Carrier) @ Relation) => ((((prefer1801827867axioms @ A) @ Carrier) @ Relation) => (((prefer1997167224erence @ A) @ Carrier) @ Relation)))))).
% 2.30/1.36 thf(fact_70_rational__preference__def, axiom, (! [A:$tType]: (((prefer1997167224erence @ A) = (^ [Carrier2:(set @ A),Relation2:(set @ ((product_prod @ A) @ A))]: (((((prefer199794634erence @ A) @ Carrier2) @ Relation2) & (((prefer1801827867axioms @ A) @ Carrier2) @ Relation2)))))))).
% 2.30/1.36 thf(fact_71_refl__on__empty, axiom, (! [A:$tType]: ((((refl_on @ A) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 2.30/1.36 thf(fact_72_preorder__on__empty, axiom, (! [A:$tType]: ((((order_preorder_on @ A) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 2.30/1.36 thf(fact_73_total__on__empty, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((total_on @ A) @ (bot_bot @ (set @ A))) @ R)))).
% 2.30/1.36 thf(fact_74_rational__preference_Onbt__nest, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),Y:A,X2:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((ord_less_eq @ (set @ A)) @ ((((prefer1532642881r_than @ A) @ Y) @ Carrier) @ Relation)) @ ((((prefer1532642881r_than @ A) @ X2) @ Carrier) @ Relation)) | (((ord_less_eq @ (set @ A)) @ ((((prefer1532642881r_than @ A) @ X2) @ Carrier) @ Relation)) @ ((((prefer1532642881r_than @ A) @ Y) @ Carrier) @ Relation))))))).
% 2.30/1.36 thf(fact_75_rational__preference_Ono__better__than__nonepty, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X2:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((Carrier != (bot_bot @ (set @ A))) => ((((member @ A) @ X2) @ Carrier) => (((((prefer1532642881r_than @ A) @ X2) @ Carrier) @ Relation) != (bot_bot @ (set @ A))))))))).
% 2.30/1.36 thf(fact_76_rational__preference__axioms__def, axiom, (! [A:$tType]: (((prefer1801827867axioms @ A) = (total_on @ A))))).
% 2.30/1.36 thf(fact_77_rational__preference__axioms_Ointro, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((total_on @ A) @ Carrier) @ Relation) => (((prefer1801827867axioms @ A) @ Carrier) @ Relation))))).
% 2.30/1.36 thf(fact_78_rational__preference_Oaxioms_I2_J, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => (((prefer1801827867axioms @ A) @ Carrier) @ Relation))))).
% 2.30/1.36 thf(fact_79_rational__preference_Ono__better__subset__pref, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X2:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y)) @ Relation) => (((ord_less_eq @ (set @ A)) @ ((((prefer1532642881r_than @ A) @ Y) @ Carrier) @ Relation)) @ ((((prefer1532642881r_than @ A) @ X2) @ Carrier) @ Relation))))))).
% 2.30/1.36 thf(fact_80_rational__preference_Ono__better__thansubset__rel, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X2:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((((member @ A) @ X2) @ Carrier) => ((((member @ A) @ Y) @ Carrier) => ((((ord_less_eq @ (set @ A)) @ ((((prefer1532642881r_than @ A) @ Y) @ Carrier) @ Relation)) @ ((((prefer1532642881r_than @ A) @ X2) @ Carrier) @ Relation)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y)) @ Relation)))))))).
% 2.30/1.36 thf(fact_81_rational__preference_Oat__lst__asgd__not__ge, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A)),X2:A,Y:A]: (((((prefer1997167224erence @ A) @ Carrier) @ Relation) => ((Carrier != (bot_bot @ (set @ A))) => ((((member @ A) @ X2) @ Carrier) => ((((member @ A) @ Y) @ Carrier) => ((~ ((((member @ A) @ X2) @ ((((prefer310429814s_good @ A) @ Y) @ Carrier) @ Relation)))) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y)) @ Relation))))))))))).
% 2.30/1.36 thf(fact_82_subset__empty, axiom, (! [A:$tType,A4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ (bot_bot @ (set @ A))) = (A4 = (bot_bot @ (set @ A))))))).
% 2.30/1.36 thf(fact_83_empty__subsetI, axiom, (! [A:$tType,A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A4)))).
% 2.30/1.36 thf(fact_84_Pair__le, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:B,C3:A,D3:B]: (((((ord_less_eq @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Pair @ A) @ B) @ C3) @ D3)) = ((((ord_less_eq @ A) @ A2) @ C3) & (((ord_less_eq @ B) @ B2) @ D3))))))))).
% 2.30/1.36 thf(fact_85_subsetI, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((! [X4:A]: (((((member @ A) @ X4) @ A4) => (((member @ A) @ X4) @ B4)))) => (((ord_less_eq @ (set @ A)) @ A4) @ B4))))).
% 2.30/1.36 thf(fact_86_subset__antisym, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ A4) => (A4 = B4)))))).
% 2.30/1.36 thf(fact_87_empty__iff, axiom, (! [A:$tType,C3:A]: ((~ ((((member @ A) @ C3) @ (bot_bot @ (set @ A)))))))).
% 2.30/1.36 thf(fact_88_all__not__in__conv, axiom, (! [A:$tType,A4:(set @ A)]: (((! [X3:A]: ((~ ((((member @ A) @ X3) @ A4))))) = (A4 = (bot_bot @ (set @ A))))))).
% 2.30/1.36 thf(fact_89_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X3:A]: ((~ ((P @ X3))))))))).
% 2.30/1.36 thf(fact_90_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X3:A]: ((~ ((P @ X3))))))))).
% 2.30/1.36 thf(fact_91_subrelI, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((! [X4:A,Y3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y3)) @ R) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y3)) @ S)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S))))).
% 2.30/1.36 thf(fact_92_ex__in__conv, axiom, (! [A:$tType,A4:(set @ A)]: (((? [X3:A]: ((((member @ A) @ X3) @ A4))) = (A4 != (bot_bot @ (set @ A))))))).
% 2.30/1.36 thf(fact_93_equals0I, axiom, (! [A:$tType,A4:(set @ A)]: (((! [Y3:A]: ((~ ((((member @ A) @ Y3) @ A4))))) => (A4 = (bot_bot @ (set @ A))))))).
% 2.30/1.36 thf(fact_94_equals0D, axiom, (! [A:$tType,A4:(set @ A),A2:A]: (((A4 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A4))))))).
% 2.30/1.36 thf(fact_95_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.30/1.36 thf(fact_96_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X3:A]: (((P @ X3) => (Q @ X3)))))))).
% 2.30/1.36 thf(fact_97_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y4:(set @ A),Z2:(set @ A)]: ((Y4 = Z2))) = (^ [A5:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B6) & (((ord_less_eq @ (set @ A)) @ B6) @ A5)))))))).
% 2.30/1.36 thf(fact_98_subset__trans, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ C4) => (((ord_less_eq @ (set @ A)) @ A4) @ C4)))))).
% 2.30/1.36 thf(fact_99_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) => (Q @ X4)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.30/1.36 thf(fact_100_subset__refl, axiom, (! [A:$tType,A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A4) @ A4)))).
% 2.30/1.36 thf(fact_101_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B6:(set @ A)]: ((! [T2:A]: (((((member @ A) @ T2) @ A5) => (((member @ A) @ T2) @ B6)))))))))).
% 2.30/1.36 thf(fact_102_equalityD2, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((A4 = B4) => (((ord_less_eq @ (set @ A)) @ B4) @ A4))))).
% 2.30/1.36 thf(fact_103_equalityD1, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((A4 = B4) => (((ord_less_eq @ (set @ A)) @ A4) @ B4))))).
% 2.30/1.36 thf(fact_104_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B6:(set @ A)]: ((! [X3:A]: (((((member @ A) @ X3) @ A5) => (((member @ A) @ X3) @ B6)))))))))).
% 2.30/1.36 thf(fact_105_equalityE, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((A4 = B4) => (~ (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (~ ((((ord_less_eq @ (set @ A)) @ B4) @ A4)))))))))).
% 2.30/1.36 thf(fact_106_subsetD, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((member @ A) @ C3) @ A4) => (((member @ A) @ C3) @ B4)))))).
% 2.30/1.36 thf(fact_107_in__mono, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),X2:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((member @ A) @ X2) @ A4) => (((member @ A) @ X2) @ B4)))))).
% 2.30/1.36 thf(fact_108_Pair__mono, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [X2:A,X5:A,Y:B,Y5:B]: (((((ord_less_eq @ A) @ X2) @ X5) => ((((ord_less_eq @ B) @ Y) @ Y5) => (((ord_less_eq @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) @ ((((product_Pair @ A) @ B) @ X5) @ Y5)))))))))).
% 2.30/1.36 thf(fact_109_bot__prod__def, axiom, (! [A:$tType,B:$tType]: ((((bot @ B) & (bot @ A)) => ((bot_bot @ ((product_prod @ A) @ B)) = ((((product_Pair @ A) @ B) @ (bot_bot @ A)) @ (bot_bot @ B))))))).
% 2.30/1.36 thf(fact_110_bot__apply, axiom, (! [C:$tType,D:$tType]: (((bot @ C) => ((bot_bot @ (D > C)) = (^ [X3:D]: ((bot_bot @ C)))))))).
% 2.30/1.36 thf(fact_111_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((((ord_less_eq @ A) @ X2) @ X2))))))).
% 2.30/1.36 thf(fact_112_subset__emptyI, axiom, (! [A:$tType,A4:(set @ A)]: (((! [X4:A]: ((~ ((((member @ A) @ X4) @ A4))))) => (((ord_less_eq @ (set @ A)) @ A4) @ (bot_bot @ (set @ A))))))).
% 2.30/1.36 thf(fact_113_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (bot_bot @ A)) => (A2 = (bot_bot @ A))))))))).
% 2.30/1.36 thf(fact_114_bot_Oextremum__unique, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (bot_bot @ A)) = (A2 = (bot_bot @ A))))))))).
% 2.30/1.36 thf(fact_115_bot_Oextremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A2))))))).
% 2.30/1.36 thf(fact_116_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 2.30/1.36 thf(fact_117_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X3:A]: ((((member @ A) @ X3) @ (bot_bot @ (set @ A))))))))).
% 2.30/1.36 thf(fact_118_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G3:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F3) @ G3) => (((ord_less_eq @ B) @ (F3 @ X2)) @ (G3 @ X2))))))))).
% 2.30/1.36 thf(fact_119_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G3:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F3) @ G3) => (((ord_less_eq @ B) @ (F3 @ X2)) @ (G3 @ X2))))))))).
% 2.30/1.36 thf(fact_120_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G3:(A > B)]: (((! [X4:A]: ((((ord_less_eq @ B) @ (F3 @ X4)) @ (G3 @ X4)))) => (((ord_less_eq @ (A > B)) @ F3) @ G3)))))))).
% 2.30/1.36 thf(fact_121_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F4:(A > B),G4:(A > B)]: ((! [X3:A]: ((((ord_less_eq @ B) @ (F4 @ X3)) @ (G4 @ X3))))))))))).
% 2.30/1.36 thf(fact_122_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F3:(B > A),B2:B,C3:B]: (((((ord_less_eq @ A) @ A2) @ (F3 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C3) => ((! [X4:B,Y3:B]: (((((ord_less_eq @ B) @ X4) @ Y3) => (((ord_less_eq @ A) @ (F3 @ X4)) @ (F3 @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F3 @ C3))))))))))).
% 2.30/1.36 thf(fact_123_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F3:(A > C),C3:C]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C) @ (F3 @ B2)) @ C3) => ((! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => (((ord_less_eq @ C) @ (F3 @ X4)) @ (F3 @ Y3))))) => (((ord_less_eq @ C) @ (F3 @ A2)) @ C3)))))))))).
% 2.30/1.36 thf(fact_124_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F3:(B > A),B2:B,C3:B]: (((A2 = (F3 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C3) => ((! [X4:B,Y3:B]: (((((ord_less_eq @ B) @ X4) @ Y3) => (((ord_less_eq @ A) @ (F3 @ X4)) @ (F3 @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F3 @ C3))))))))))).
% 2.30/1.36 thf(fact_125_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:A,F3:(A > B),C3:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F3 @ B2) = C3) => ((! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => (((ord_less_eq @ B) @ (F3 @ X4)) @ (F3 @ Y3))))) => (((ord_less_eq @ B) @ (F3 @ A2)) @ C3)))))))))).
% 2.30/1.36 thf(fact_126_eq__iff, axiom, (! [A:$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))))))))).
% 2.30/1.36 thf(fact_127_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ X2) => (X2 = Y))))))))).
% 2.30/1.36 thf(fact_128_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) | (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.30/1.36 thf(fact_129_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((X2 = Y) => (((ord_less_eq @ A) @ X2) @ Y)))))))).
% 2.30/1.36 thf(fact_130_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Y))) => (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.30/1.36 thf(fact_131_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C3:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 2.30/1.36 thf(fact_132_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A,Z:A]: ((((((ord_less_eq @ A) @ X2) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z)))) => (((((ord_less_eq @ A) @ Y) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Z)))) => (((((ord_less_eq @ A) @ X2) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y)))) => (((((ord_less_eq @ A) @ Z) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X2)))) => (((((ord_less_eq @ A) @ Y) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X2)))) => (~ (((((ord_less_eq @ A) @ Z) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Y))))))))))))))))).
% 2.30/1.36 thf(fact_133_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ Y) @ X2) => ((((ord_less_eq @ A) @ X2) @ Y) = (X2 = Y))))))))).
% 2.30/1.36 thf(fact_134_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [A7:A,B7:A]: (((((ord_less_eq @ A) @ A7) @ B7) & (((ord_less_eq @ A) @ B7) @ A7))))))))).
% 2.30/1.36 thf(fact_135_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C3:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 2.30/1.36 thf(fact_136_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C3:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 2.30/1.36 thf(fact_137_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.30/1.36 thf(fact_138_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less_eq @ A) @ X2) @ Z))))))))).
% 2.30/1.36 thf(fact_139_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.30/1.36 thf(fact_140_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A6:A,B5:A]: (((((ord_less_eq @ A) @ A6) @ B5) => ((P @ A6) @ B5)))) => ((! [A6:A,B5:A]: ((((P @ B5) @ A6) => ((P @ A6) @ B5)))) => ((P @ A2) @ B2))))))))).
% 2.30/1.36 thf(fact_141_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C3:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C3) @ B2) => (((ord_less_eq @ A) @ C3) @ A2))))))))).
% 2.30/1.36 thf(fact_142_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [A7:A,B7:A]: (((((ord_less_eq @ A) @ B7) @ A7) & (((ord_less_eq @ A) @ A7) @ B7))))))))).
% 2.30/1.36 thf(fact_143_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.30/1.36 thf(fact_144_bot__fun__def, axiom, (! [B:$tType,A:$tType]: (((bot @ B) => ((bot_bot @ (A > B)) = (^ [X3:A]: ((bot_bot @ B)))))))).
% 2.30/1.36 thf(fact_145_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R:A,S:B,R3:(set @ ((product_prod @ A) @ B)),S2:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S)) @ R3) => ((S2 = S) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S2)) @ R3)))))).
% 2.30/1.36 thf(fact_146_less__by__empty, axiom, (! [A:$tType,A4:(set @ ((product_prod @ A) @ A)),B4:(set @ ((product_prod @ A) @ A))]: (((A4 = (bot_bot @ (set @ ((product_prod @ A) @ A)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ A4) @ B4))))).
% 2.30/1.36 thf(fact_147_transitivity, axiom, ((trans @ a) @ relation)).
% 2.30/1.36 thf(fact_148_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A5:(set @ A)]: ((A5 = (bot_bot @ (set @ A))))))))).
% 2.30/1.36 thf(fact_149_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C3:(B > (C > A)),A2:B,B2:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C3) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((C3 @ A2) @ B2))))).
% 2.30/1.36 thf(fact_150_transD, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),X2:A,Y:A,Z:A]: ((((trans @ A) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ R) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Z)) @ R))))))).
% 2.30/1.36 thf(fact_151_transE, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),X2:A,Y:A,Z:A]: ((((trans @ A) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z)) @ R) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Z)) @ R))))))).
% 2.30/1.36 thf(fact_152_transI, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((! [X4:A,Y3:A,Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y3)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ Z3)) @ R) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Z3)) @ R))))) => ((trans @ A) @ R))))).
% 2.30/1.36 thf(fact_153_trans__def, axiom, (! [A:$tType]: (((trans @ A) = (^ [R2:(set @ ((product_prod @ A) @ A))]: ((! [X3:A,Y2:A,Z4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y2)) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ Z4)) @ R2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Z4)) @ R2))))))))))).
% 2.30/1.36 thf(fact_154_trans__empty, axiom, (! [A:$tType]: (((trans @ A) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 2.30/1.36 thf(fact_155_preference_Otransitivity, axiom, (! [A:$tType,Carrier:(set @ A),Relation:(set @ ((product_prod @ A) @ A))]: (((((prefer199794634erence @ A) @ Carrier) @ Relation) => ((trans @ A) @ Relation))))).
% 2.30/1.36 thf(fact_156_preorder__on__def, axiom, (! [A:$tType]: (((order_preorder_on @ A) = (^ [A5:(set @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((refl_on @ A) @ A5) @ R2) & ((trans @ A) @ R2)))))))).
% 2.30/1.36 thf(fact_157_Collect__empty__eq__bot, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (P = (bot_bot @ (A > $o))))))).
% 2.30/1.36 thf(fact_158_relChain__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (((bNF_Ca1785829860lChain @ A) @ B) = (^ [R2:(set @ ((product_prod @ A) @ A)),As:(A > B)]: ((! [I:A,J:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I) @ J)) @ R2) => (((ord_less_eq @ B) @ (As @ I)) @ (As @ J)))))))))))).
% 2.30/1.36 thf(fact_159_Id__on__empty, axiom, (! [A:$tType]: ((((id_on @ A) @ (bot_bot @ (set @ A))) = (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 2.30/1.36 thf(fact_160_Id__onI, axiom, (! [A:$tType,A2:A,A4:(set @ A)]: (((((member @ A) @ A2) @ A4) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ ((id_on @ A) @ A4)))))).
% 2.30/1.36 thf(fact_161_Id__on__iff, axiom, (! [A:$tType,X2:A,Y:A,A4:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y)) @ ((id_on @ A) @ A4)) = ((X2 = Y) & (((member @ A) @ X2) @ A4)))))).
% 2.30/1.36 thf(fact_162_Id__on__eqI, axiom, (! [A:$tType,A2:A,B2:A,A4:(set @ A)]: (((A2 = B2) => ((((member @ A) @ A2) @ A4) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((id_on @ A) @ A4))))))).
% 2.30/1.36 thf(fact_163_Id__onE, axiom, (! [A:$tType,C3:((product_prod @ A) @ A),A4:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ C3) @ ((id_on @ A) @ A4)) => (~ ((! [X4:A]: (((((member @ A) @ X4) @ A4) => (C3 != ((((product_Pair @ A) @ A) @ X4) @ X4))))))))))).
% 2.30/1.36 thf(fact_164_trans__Id__on, axiom, (! [A:$tType,A4:(set @ A)]: (((trans @ A) @ ((id_on @ A) @ A4))))).
% 2.30/1.36 thf(fact_165_refl__on__Id__on, axiom, (! [A:$tType,A4:(set @ A)]: ((((refl_on @ A) @ A4) @ ((id_on @ A) @ A4))))).
% 2.30/1.36 thf(fact_166_under__incr, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A2:A,B2:A]: ((((trans @ A) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R) => (((ord_less_eq @ (set @ A)) @ (((order_under @ A) @ R) @ A2)) @ (((order_under @ A) @ R) @ B2))))))).
% 2.30/1.36 thf(fact_167_GreatestI2__order, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X2:A,Q:(A > $o)]: (((P @ X2) => ((! [Y3:A]: (((P @ Y3) => (((ord_less_eq @ A) @ Y3) @ X2)))) => ((! [X4:A]: (((P @ X4) => ((! [Y6:A]: (((P @ Y6) => (((ord_less_eq @ A) @ Y6) @ X4)))) => (Q @ X4))))) => (Q @ ((order_Greatest @ A) @ P))))))))))).
% 2.30/1.36 thf(fact_168_Greatest__equality, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X2:A]: (((P @ X2) => ((! [Y3:A]: (((P @ Y3) => (((ord_less_eq @ A) @ Y3) @ X2)))) => (((order_Greatest @ A) @ P) = X2))))))))).
% 2.30/1.36 thf(fact_169_fst__bot, axiom, (! [B:$tType,A:$tType]: ((((bot @ A) & (bot @ B)) => ((((product_fst @ A) @ B) @ (bot_bot @ ((product_prod @ A) @ B))) = (bot_bot @ A)))))).
% 2.30/1.36 thf(fact_170_snd__bot, axiom, (! [B:$tType,A:$tType]: ((((bot @ A) & (bot @ B)) => ((((product_snd @ B) @ A) @ (bot_bot @ ((product_prod @ B) @ A))) = (bot_bot @ A)))))).
% 2.30/1.36 thf(fact_171_trans__singleton, axiom, (! [A:$tType,A2:A]: (((trans @ A) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 2.30/1.36 thf(fact_172_insert__absorb2, axiom, (! [A:$tType,X2:A,A4:(set @ A)]: (((((insert @ A) @ X2) @ (((insert @ A) @ X2) @ A4)) = (((insert @ A) @ X2) @ A4))))).
% 2.30/1.36 thf(fact_173_insert__iff, axiom, (! [A:$tType,A2:A,B2:A,A4:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B2) @ A4)) = ((A2 = B2) | (((member @ A) @ A2) @ A4)))))).
% 2.30/1.36 thf(fact_174_insertCI, axiom, (! [A:$tType,A2:A,B4:(set @ A),B2:A]: ((((~ ((((member @ A) @ A2) @ B4))) => (A2 = B2)) => (((member @ A) @ A2) @ (((insert @ A) @ B2) @ B4)))))).
% 2.30/1.36 thf(fact_175_singletonI, axiom, (! [A:$tType,A2:A]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.30/1.36 thf(fact_176_insert__subset, axiom, (! [A:$tType,X2:A,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X2) @ A4)) @ B4) = ((((member @ A) @ X2) @ B4) & (((ord_less_eq @ (set @ A)) @ A4) @ B4)))))).
% 2.30/1.36 thf(fact_177_singleton__insert__inj__eq, axiom, (! [A:$tType,B2:A,A2:A,A4:(set @ A)]: ((((((insert @ A) @ B2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ A2) @ A4)) = ((A2 = B2) & (((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))))))).
% 2.30/1.36 thf(fact_178_singleton__insert__inj__eq_H, axiom, (! [A:$tType,A2:A,A4:(set @ A),B2:A]: ((((((insert @ A) @ A2) @ A4) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = ((A2 = B2) & (((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))))))).
% 2.30/1.36 thf(fact_179_prod_Ocollapse, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)) = Prod)))).
% 2.30/1.36 thf(fact_180_insert__subsetI, axiom, (! [A:$tType,X2:A,A4:(set @ A),X6:(set @ A)]: (((((member @ A) @ X2) @ A4) => ((((ord_less_eq @ (set @ A)) @ X6) @ A4) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X2) @ X6)) @ A4)))))).
% 2.30/1.36 thf(fact_181_less__eq__prod__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => ((ord_less_eq @ ((product_prod @ A) @ B)) = (^ [X3:((product_prod @ A) @ B),Y2:((product_prod @ A) @ B)]: (((((ord_less_eq @ A) @ (((product_fst @ A) @ B) @ X3)) @ (((product_fst @ A) @ B) @ Y2)) & (((ord_less_eq @ B) @ (((product_snd @ A) @ B) @ X3)) @ (((product_snd @ A) @ B) @ Y2)))))))))).
% 2.30/1.36 thf(fact_182_singleton__inject, axiom, (! [A:$tType,A2:A,B2:A]: ((((((insert @ A) @ A2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) => (A2 = B2))))).
% 2.30/1.36 thf(fact_183_insert__not__empty, axiom, (! [A:$tType,A2:A,A4:(set @ A)]: (((((insert @ A) @ A2) @ A4) != (bot_bot @ (set @ A)))))).
% 2.30/1.36 thf(fact_184_doubleton__eq__iff, axiom, (! [A:$tType,A2:A,B2:A,C3:A,D3:A]: ((((((insert @ A) @ A2) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C3) @ (((insert @ A) @ D3) @ (bot_bot @ (set @ A))))) = (((A2 = C3) & (B2 = D3)) | ((A2 = D3) & (B2 = C3))))))).
% 2.30/1.36 thf(fact_185_singleton__iff, axiom, (! [A:$tType,B2:A,A2:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = (B2 = A2))))).
% 2.30/1.36 thf(fact_186_singletonD, axiom, (! [A:$tType,B2:A,A2:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) => (B2 = A2))))).
% 2.30/1.36 thf(fact_187_subset__insertI2, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),B2:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ B2) @ B4)))))).
% 2.30/1.36 thf(fact_188_subset__insertI, axiom, (! [A:$tType,B4:(set @ A),A2:A]: ((((ord_less_eq @ (set @ A)) @ B4) @ (((insert @ A) @ A2) @ B4))))).
% 2.30/1.36 thf(fact_189_subset__insert, axiom, (! [A:$tType,X2:A,A4:(set @ A),B4:(set @ A)]: (((~ ((((member @ A) @ X2) @ A4))) => ((((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ X2) @ B4)) = (((ord_less_eq @ (set @ A)) @ A4) @ B4)))))).
% 2.30/1.36 thf(fact_190_insert__mono, axiom, (! [A:$tType,C4:(set @ A),D4:(set @ A),A2:A]: (((((ord_less_eq @ (set @ A)) @ C4) @ D4) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ A2) @ C4)) @ (((insert @ A) @ A2) @ D4)))))).
% 2.30/1.36 thf(fact_191_mk__disjoint__insert, axiom, (! [A:$tType,A2:A,A4:(set @ A)]: (((((member @ A) @ A2) @ A4) => (? [B8:(set @ A)]: (((A4 = (((insert @ A) @ A2) @ B8)) & (~ ((((member @ A) @ A2) @ B8)))))))))).
% 2.30/1.36 thf(fact_192_insert__commute, axiom, (! [A:$tType,X2:A,Y:A,A4:(set @ A)]: (((((insert @ A) @ X2) @ (((insert @ A) @ Y) @ A4)) = (((insert @ A) @ Y) @ (((insert @ A) @ X2) @ A4)))))).
% 2.30/1.36 thf(fact_193_insert__eq__iff, axiom, (! [A:$tType,A2:A,A4:(set @ A),B2:A,B4:(set @ A)]: (((~ ((((member @ A) @ A2) @ A4))) => ((~ ((((member @ A) @ B2) @ B4))) => (((((insert @ A) @ A2) @ A4) = (((insert @ A) @ B2) @ B4)) = (((A2 = B2) => (A4 = B4)) & ((A2 != B2) => (? [C5:(set @ A)]: (((A4 = (((insert @ A) @ B2) @ C5)) & ((~ ((((member @ A) @ B2) @ C5))) & ((B4 = (((insert @ A) @ A2) @ C5)) & (~ ((((member @ A) @ A2) @ C5)))))))))))))))).
% 2.30/1.36 thf(fact_194_insert__absorb, axiom, (! [A:$tType,A2:A,A4:(set @ A)]: (((((member @ A) @ A2) @ A4) => ((((insert @ A) @ A2) @ A4) = A4))))).
% 2.30/1.36 thf(fact_195_insert__ident, axiom, (! [A:$tType,X2:A,A4:(set @ A),B4:(set @ A)]: (((~ ((((member @ A) @ X2) @ A4))) => ((~ ((((member @ A) @ X2) @ B4))) => (((((insert @ A) @ X2) @ A4) = (((insert @ A) @ X2) @ B4)) = (A4 = B4))))))).
% 2.30/1.36 thf(fact_196_Set_Oset__insert, axiom, (! [A:$tType,X2:A,A4:(set @ A)]: (((((member @ A) @ X2) @ A4) => (~ ((! [B8:(set @ A)]: (((A4 = (((insert @ A) @ X2) @ B8)) => (((member @ A) @ X2) @ B8)))))))))).
% 2.30/1.36 thf(fact_197_insertI2, axiom, (! [A:$tType,A2:A,B4:(set @ A),B2:A]: (((((member @ A) @ A2) @ B4) => (((member @ A) @ A2) @ (((insert @ A) @ B2) @ B4)))))).
% 2.30/1.36 thf(fact_198_insertI1, axiom, (! [A:$tType,A2:A,B4:(set @ A)]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ B4))))).
% 2.30/1.36 thf(fact_199_insertE, axiom, (! [A:$tType,A2:A,B2:A,A4:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B2) @ A4)) => ((A2 != B2) => (((member @ A) @ A2) @ A4)))))).
% 2.30/1.36 thf(fact_200_prod__eq__iff, axiom, (! [B:$tType,A:$tType]: (((^ [Y4:((product_prod @ A) @ B),Z2:((product_prod @ A) @ B)]: ((Y4 = Z2))) = (^ [S3:((product_prod @ A) @ B),T2:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ S3) = (((product_fst @ A) @ B) @ T2)) & ((((product_snd @ A) @ B) @ S3) = (((product_snd @ A) @ B) @ T2))))))))).
% 2.30/1.36 thf(fact_201_prod_Oexpand, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B),Prod2:((product_prod @ A) @ B)]: (((((((product_fst @ A) @ B) @ Prod) = (((product_fst @ A) @ B) @ Prod2)) & ((((product_snd @ A) @ B) @ Prod) = (((product_snd @ A) @ B) @ Prod2))) => (Prod = Prod2))))).
% 2.30/1.36 thf(fact_202_prod__eqI, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B),Q2:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ P2) = (((product_fst @ A) @ B) @ Q2)) => (((((product_snd @ A) @ B) @ P2) = (((product_snd @ A) @ B) @ Q2)) => (P2 = Q2)))))).
% 2.30/1.36 thf(fact_203_surjective__pairing, axiom, (! [B:$tType,A:$tType,T3:((product_prod @ A) @ B)]: ((T3 = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ T3)) @ (((product_snd @ A) @ B) @ T3)))))).
% 2.30/1.36 thf(fact_204_prod_Oexhaust__sel, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))).
% 2.30/1.36 thf(fact_205_fst__conv, axiom, (! [B:$tType,A:$tType,X1:A,X22:B]: (((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X1) @ X22)) = X1)))).
% 2.30/1.36 thf(fact_206_fst__eqD, axiom, (! [B:$tType,A:$tType,X2:A,Y:B,A2:A]: ((((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) = A2) => (X2 = A2))))).
% 2.30/1.36 thf(fact_207_snd__conv, axiom, (! [Aa:$tType,A:$tType,X1:Aa,X22:A]: (((((product_snd @ Aa) @ A) @ ((((product_Pair @ Aa) @ A) @ X1) @ X22)) = X22)))).
% 2.30/1.36 thf(fact_208_snd__eqD, axiom, (! [B:$tType,A:$tType,X2:B,Y:A,A2:A]: ((((((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X2) @ Y)) = A2) => (Y = A2))))).
% 2.30/1.36 thf(fact_209_refl__on__singleton, axiom, (! [A:$tType,X2:A]: ((((refl_on @ A) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ X2)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 2.30/1.36 thf(fact_210_total__on__singleton, axiom, (! [A:$tType,X2:A]: ((((total_on @ A) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ X2)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 2.30/1.36 thf(fact_211_subset__singletonD, axiom, (! [A:$tType,A4:(set @ A),X2:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) => ((A4 = (bot_bot @ (set @ A))) | (A4 = (((insert @ A) @ X2) @ (bot_bot @ (set @ A))))))))).
% 2.30/1.36 thf(fact_212_subset__singleton__iff, axiom, (! [A:$tType,X6:(set @ A),A2:A]: (((((ord_less_eq @ (set @ A)) @ X6) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = ((X6 = (bot_bot @ (set @ A))) | (X6 = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))))).
% 2.30/1.36 thf(fact_213_snd__mono, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => (! [X2:((product_prod @ A) @ B),Y:((product_prod @ A) @ B)]: (((((ord_less_eq @ ((product_prod @ A) @ B)) @ X2) @ Y) => (((ord_less_eq @ B) @ (((product_snd @ A) @ B) @ X2)) @ (((product_snd @ A) @ B) @ Y))))))))).
% 2.30/1.36 thf(fact_214_fst__mono, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => (! [X2:((product_prod @ A) @ B),Y:((product_prod @ A) @ B)]: (((((ord_less_eq @ ((product_prod @ A) @ B)) @ X2) @ Y) => (((ord_less_eq @ A) @ (((product_fst @ A) @ B) @ X2)) @ (((product_fst @ A) @ B) @ Y))))))))).
% 2.30/1.36 thf(fact_215_BNF__Greatest__Fixpoint_Osubst__Pair, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),X2:A,Y:B,A2:((product_prod @ A) @ B)]: ((((P @ X2) @ Y) => ((A2 = ((((product_Pair @ A) @ B) @ X2) @ Y)) => ((P @ (((product_fst @ A) @ B) @ A2)) @ (((product_snd @ A) @ B) @ A2))))))).
% 2.30/1.36 thf(fact_216_conjI__realizer, axiom, (! [A:$tType,B:$tType,P:(A > $o),P2:A,Q:(B > $o),Q2:B]: (((P @ P2) => ((Q @ Q2) => ((P @ (((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ P2) @ Q2))) & (Q @ (((product_snd @ A) @ B) @ ((((product_Pair @ A) @ B) @ P2) @ Q2))))))))).
% 2.30/1.36 thf(fact_217_exI__realizer, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Y:A,X2:B]: ((((P @ Y) @ X2) => ((P @ (((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X2) @ Y))) @ (((product_fst @ B) @ A) @ ((((product_Pair @ B) @ A) @ X2) @ Y))))))).
% 2.30/1.36 thf(fact_218_the__elem__eq, axiom, (! [A:$tType,X2:A]: ((((the_elem @ A) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) = X2)))).
% 2.30/1.36 thf(fact_219_is__singletonI, axiom, (! [A:$tType,X2:A]: (((is_singleton @ A) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A))))))).
% 2.30/1.36 thf(fact_220_is__singleton__the__elem, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A5:(set @ A)]: ((A5 = (((insert @ A) @ ((the_elem @ A) @ A5)) @ (bot_bot @ (set @ A)))))))))).
% 2.30/1.36 thf(fact_221_is__singletonI_H, axiom, (! [A:$tType,A4:(set @ A)]: (((A4 != (bot_bot @ (set @ A))) => ((! [X4:A,Y3:A]: (((((member @ A) @ X4) @ A4) => ((((member @ A) @ Y3) @ A4) => (X4 = Y3))))) => ((is_singleton @ A) @ A4)))))).
% 2.30/1.36 thf(fact_222_is__singletonE, axiom, (! [A:$tType,A4:(set @ A)]: ((((is_singleton @ A) @ A4) => (~ ((! [X4:A]: ((A4 != (((insert @ A) @ X4) @ (bot_bot @ (set @ A)))))))))))).
% 2.30/1.36 thf(fact_223_is__singleton__def, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A5:(set @ A)]: ((? [X3:A]: ((A5 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))))))).
% 2.30/1.36 thf(fact_224_linear__order__on__singleton, axiom, (! [A:$tType,X2:A]: ((((order_1409979114der_on @ A) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ X2)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 2.30/1.36 thf(fact_225_sndI, axiom, (! [A:$tType,B:$tType,X2:((product_prod @ A) @ B),Y:A,Z:B]: (((X2 = ((((product_Pair @ A) @ B) @ Y) @ Z)) => ((((product_snd @ A) @ B) @ X2) = Z))))).
% 2.30/1.36 thf(fact_226_lnear__order__on__empty, axiom, (! [A:$tType]: ((((order_1409979114der_on @ A) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 2.30/1.36 thf(fact_227_fstI, axiom, (! [B:$tType,A:$tType,X2:((product_prod @ A) @ B),Y:A,Z:B]: (((X2 = ((((product_Pair @ A) @ B) @ Y) @ Z)) => ((((product_fst @ A) @ B) @ X2) = Y))))).
% 2.30/1.36 thf(fact_228_eq__snd__iff, axiom, (! [A:$tType,B:$tType,B2:A,P2:((product_prod @ B) @ A)]: (((B2 = (((product_snd @ B) @ A) @ P2)) = (? [A7:B]: ((P2 = ((((product_Pair @ B) @ A) @ A7) @ B2)))))))).
% 2.30/1.36 thf(fact_229_eq__fst__iff, axiom, (! [A:$tType,B:$tType,A2:A,P2:((product_prod @ A) @ B)]: (((A2 = (((product_fst @ A) @ B) @ P2)) = (? [B7:B]: ((P2 = ((((product_Pair @ A) @ B) @ A2) @ B7)))))))).
% 2.30/1.36 thf(fact_230_prod_Oswap__def, axiom, (! [B:$tType,A:$tType]: ((((product_swap @ A) @ B) = (^ [P3:((product_prod @ A) @ B)]: (((((product_Pair @ B) @ A) @ (((product_snd @ A) @ B) @ P3)) @ (((product_fst @ A) @ B) @ P3)))))))).
% 2.30/1.36 thf(fact_231_Range__insert, axiom, (! [A:$tType,B:$tType,A2:B,B2:A,R:(set @ ((product_prod @ B) @ A))]: (((((range @ B) @ A) @ (((insert @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A2) @ B2)) @ R)) = (((insert @ A) @ B2) @ (((range @ B) @ A) @ R)))))).
% 2.30/1.36 thf(fact_232_swap__swap, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P2)) = P2)))).
% 2.30/1.36 thf(fact_233_swap__simp, axiom, (! [A:$tType,B:$tType,X2:B,Y:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X2) @ Y)) = ((((product_Pair @ A) @ B) @ Y) @ X2))))).
% 2.30/1.36 thf(fact_234_Range__Id__on, axiom, (! [A:$tType,A4:(set @ A)]: (((((range @ A) @ A) @ ((id_on @ A) @ A4)) = A4)))).
% 2.30/1.36 thf(fact_235_Range__empty, axiom, (! [B:$tType,A:$tType]: (((((range @ B) @ A) @ (bot_bot @ (set @ ((product_prod @ B) @ A)))) = (bot_bot @ (set @ A)))))).
% 2.30/1.36 thf(fact_236_fst__swap, axiom, (! [A:$tType,B:$tType,X2:((product_prod @ B) @ A)]: (((((product_fst @ A) @ B) @ (((product_swap @ B) @ A) @ X2)) = (((product_snd @ B) @ A) @ X2))))).
% 2.30/1.36 thf(fact_237_snd__swap, axiom, (! [B:$tType,A:$tType,X2:((product_prod @ A) @ B)]: (((((product_snd @ B) @ A) @ (((product_swap @ A) @ B) @ X2)) = (((product_fst @ A) @ B) @ X2))))).
% 2.30/1.36 thf(fact_238_RangeE, axiom, (! [A:$tType,B:$tType,B2:A,R:(set @ ((product_prod @ B) @ A))]: (((((member @ A) @ B2) @ (((range @ B) @ A) @ R)) => (~ ((! [A6:B]: ((~ ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A6) @ B2)) @ R))))))))))).
% 2.30/1.36 thf(fact_239_Range__iff, axiom, (! [A:$tType,B:$tType,A2:A,R:(set @ ((product_prod @ B) @ A))]: (((((member @ A) @ A2) @ (((range @ B) @ A) @ R)) = (? [Y2:B]: ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ Y2) @ A2)) @ R))))))).
% 2.30/1.36 thf(fact_240_Range_Ocases, axiom, (! [B:$tType,A:$tType,A2:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ B) @ A2) @ (((range @ A) @ B) @ R)) => (~ ((! [A6:A]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A6) @ A2)) @ R))))))))))).
% 2.30/1.36 thf(fact_241_Range_Osimps, axiom, (! [B:$tType,A:$tType,A2:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ B) @ A2) @ (((range @ A) @ B) @ R)) = (? [A7:A,B7:B]: (((A2 = B7) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A7) @ B7)) @ R)))))))).
% 2.30/1.36 thf(fact_242_Range_Ointros, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ R) => (((member @ B) @ B2) @ (((range @ A) @ B) @ R)))))).
% 2.30/1.36 thf(fact_243_Range_Oinducts, axiom, (! [A:$tType,B:$tType,X2:B,R:(set @ ((product_prod @ A) @ B)),P:(B > $o)]: (((((member @ B) @ X2) @ (((range @ A) @ B) @ R)) => ((! [A6:A,B5:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A6) @ B5)) @ R) => (P @ B5)))) => (P @ X2)))))).
% 2.30/1.36 thf(fact_244_Range__empty__iff, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ B) @ A))]: ((((((range @ B) @ A) @ R) = (bot_bot @ (set @ A))) = (R = (bot_bot @ (set @ ((product_prod @ B) @ A)))))))).
% 2.30/1.36 thf(fact_245_Range__mono, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S) => (((ord_less_eq @ (set @ B)) @ (((range @ A) @ B) @ R)) @ (((range @ A) @ B) @ S)))))).
% 2.30/1.36 thf(fact_246_Domain__insert, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,R:(set @ ((product_prod @ A) @ B))]: (((((domain @ A) @ B) @ (((insert @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ R)) = (((insert @ A) @ A2) @ (((domain @ A) @ B) @ R)))))).
% 2.30/1.36 thf(fact_247_subset__Compl__singleton, axiom, (! [A:$tType,A4:(set @ A),B2:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ ((uminus_uminus @ (set @ A)) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))) = (~ ((((member @ A) @ B2) @ A4))))))).
% 2.30/1.36 thf(fact_248_Compl__eq__Compl__iff, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((uminus_uminus @ (set @ A)) @ A4) = ((uminus_uminus @ (set @ A)) @ B4)) = (A4 = B4))))).
% 2.30/1.36 thf(fact_249_Compl__iff, axiom, (! [A:$tType,C3:A,A4:(set @ A)]: (((((member @ A) @ C3) @ ((uminus_uminus @ (set @ A)) @ A4)) = (~ ((((member @ A) @ C3) @ A4))))))).
% 2.30/1.36 thf(fact_250_ComplI, axiom, (! [A:$tType,C3:A,A4:(set @ A)]: (((~ ((((member @ A) @ C3) @ A4))) => (((member @ A) @ C3) @ ((uminus_uminus @ (set @ A)) @ A4)))))).
% 2.30/1.36 thf(fact_251_Compl__anti__mono, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ B4)) @ ((uminus_uminus @ (set @ A)) @ A4)))))).
% 2.30/1.36 thf(fact_252_Compl__subset__Compl__iff, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A4)) @ ((uminus_uminus @ (set @ A)) @ B4)) = (((ord_less_eq @ (set @ A)) @ B4) @ A4))))).
% 2.30/1.36 thf(fact_253_Domain__Id__on, axiom, (! [A:$tType,A4:(set @ A)]: (((((domain @ A) @ A) @ ((id_on @ A) @ A4)) = A4)))).
% 2.30/1.36 thf(fact_254_Domain__empty, axiom, (! [B:$tType,A:$tType]: (((((domain @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ B)))) = (bot_bot @ (set @ A)))))).
% 2.30/1.36 thf(fact_255_subset__Compl__self__eq, axiom, (! [A:$tType,A4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ ((uminus_uminus @ (set @ A)) @ A4)) = (A4 = (bot_bot @ (set @ A))))))).
% 2.30/1.36 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A8:$tType,A9:$tType]: (((order_bot @ A9) => (order_bot @ (A8 > A9)))))).
% 2.30/1.36 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A8 > A9)))))).
% 2.30/1.36 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: (((order @ A9) => (order @ (A8 > A9)))))).
% 2.30/1.36 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: (((ord @ A9) => (ord @ (A8 > A9)))))).
% 2.30/1.36 thf(tcon_fun___Orderings_Obot, axiom, (! [A8:$tType,A9:$tType]: (((bot @ A9) => (bot @ (A8 > A9)))))).
% 2.30/1.36 thf(tcon_Set_Oset___Orderings_Oorder__bot_1, axiom, (! [A8:$tType]: ((order_bot @ (set @ A8))))).
% 2.30/1.36 thf(tcon_Set_Oset___Orderings_Opreorder_2, axiom, (! [A8:$tType]: ((preorder @ (set @ A8))))).
% 2.30/1.36 thf(tcon_Set_Oset___Orderings_Oorder_3, axiom, (! [A8:$tType]: ((order @ (set @ A8))))).
% 2.30/1.36 thf(tcon_Set_Oset___Orderings_Oord_4, axiom, (! [A8:$tType]: ((ord @ (set @ A8))))).
% 2.30/1.36 thf(tcon_Set_Oset___Orderings_Obot_5, axiom, (! [A8:$tType]: ((bot @ (set @ A8))))).
% 2.30/1.36 thf(tcon_HOL_Obool___Orderings_Oorder__bot_6, axiom, (order_bot @ $o)).
% 2.30/1.36 thf(tcon_HOL_Obool___Orderings_Opreorder_7, axiom, (preorder @ $o)).
% 2.30/1.36 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 2.30/1.36 thf(tcon_HOL_Obool___Orderings_Oorder_8, axiom, (order @ $o)).
% 2.30/1.36 thf(tcon_HOL_Obool___Orderings_Oord_9, axiom, (ord @ $o)).
% 2.30/1.36 thf(tcon_HOL_Obool___Orderings_Obot_10, axiom, (bot @ $o)).
% 2.30/1.36 thf(tcon_Product__Type_Oprod___Orderings_Oorder__bot_11, axiom, (! [A8:$tType,A9:$tType]: ((((order_bot @ A8) & (order_bot @ A9)) => (order_bot @ ((product_prod @ A8) @ A9)))))).
% 2.30/1.36 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_12, axiom, (! [A8:$tType,A9:$tType]: ((((preorder @ A8) & (preorder @ A9)) => (preorder @ ((product_prod @ A8) @ A9)))))).
% 2.30/1.36 thf(tcon_Product__Type_Oprod___Orderings_Oorder_13, axiom, (! [A8:$tType,A9:$tType]: ((((order @ A8) & (order @ A9)) => (order @ ((product_prod @ A8) @ A9)))))).
% 2.30/1.36 thf(tcon_Product__Type_Oprod___Orderings_Oord_14, axiom, (! [A8:$tType,A9:$tType]: ((((ord @ A8) & (ord @ A9)) => (ord @ ((product_prod @ A8) @ A9)))))).
% 2.30/1.36 thf(tcon_Product__Type_Oprod___Orderings_Obot_15, axiom, (! [A8:$tType,A9:$tType]: ((((bot @ A8) & (bot @ A9)) => (bot @ ((product_prod @ A8) @ A9)))))).
% 2.30/1.36 thf(conj_0, conjecture, ((((member @ a) @ z) @ carrier) & (((member @ a) @ y) @ carrier))).
% 2.30/1.36 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.xCdPegwIoG/DTF2DTF29341.p
% 2.30/1.36 ------------------------
% 2.47/1.40 ---- Cleaned THF ---
% 2.47/1.40 thf(ty_t_Product__Type_Oprod,type,
% 2.47/1.40 product_prod: $tType > $tType > $tType ).
% 2.47/1.40
% 2.47/1.40 thf(ty_t_Set_Oset,type,
% 2.47/1.40 set: $tType > $tType ).
% 2.47/1.40
% 2.47/1.40 thf(ty_tf_a,type,
% 2.47/1.40 a: $tType ).
% 2.47/1.40
% 2.47/1.40 thf(sy_cl_HOL_Otype,type,
% 2.47/1.40 type:
% 2.47/1.40 !>[A: $tType] : $o ).
% 2.47/1.40
% 2.47/1.40 thf(sy_cl_Orderings_Obot,type,
% 2.47/1.40 bot:
% 2.47/1.40 !>[A: $tType] : $o ).
% 2.47/1.40
% 2.47/1.40 thf(sy_cl_Orderings_Oord,type,
% 2.47/1.40 ord:
% 2.47/1.40 !>[A: $tType] : $o ).
% 2.47/1.40
% 2.47/1.40 thf(sy_cl_Orderings_Oorder,type,
% 2.47/1.40 order:
% 2.47/1.40 !>[A: $tType] : $o ).
% 2.47/1.40
% 2.47/1.40 thf(sy_cl_Orderings_Olinorder,type,
% 2.47/1.40 linorder:
% 2.47/1.40 !>[A: $tType] : $o ).
% 2.47/1.40
% 2.47/1.40 thf(sy_cl_Orderings_Opreorder,type,
% 2.47/1.40 preorder:
% 2.47/1.40 !>[A: $tType] : $o ).
% 2.47/1.40
% 2.47/1.40 thf(sy_cl_Orderings_Oorder__bot,type,
% 2.47/1.40 order_bot:
% 2.47/1.40 !>[A: $tType] : $o ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 2.47/1.40 bNF_Ca1785829860lChain:
% 2.47/1.40 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 2.47/1.40 uminus_uminus:
% 2.47/1.40 !>[A: $tType] : ( A > A ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Order__Relation_Olinear__order__on,type,
% 2.47/1.40 order_1409979114der_on:
% 2.47/1.40 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Order__Relation_Opreorder__on,type,
% 2.47/1.40 order_preorder_on:
% 2.47/1.40 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Order__Relation_Ounder,type,
% 2.47/1.40 order_under:
% 2.47/1.40 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.47/1.40 bot_bot:
% 2.47/1.40 !>[A: $tType] : A ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.47/1.40 ord_less_eq:
% 2.47/1.40 !>[A: $tType] : ( A > A > $o ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 2.47/1.40 order_Greatest:
% 2.47/1.40 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oas__good__as,type,
% 2.47/1.40 prefer951318096ood_as:
% 2.47/1.40 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oat__least__as__good,type,
% 2.47/1.40 prefer310429814s_good:
% 2.47/1.40 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Ono__better__than,type,
% 2.47/1.40 prefer1532642881r_than:
% 2.47/1.40 !>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Opreference,type,
% 2.47/1.40 prefer199794634erence:
% 2.47/1.40 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference,type,
% 2.47/1.40 prefer1997167224erence:
% 2.47/1.40 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference__axioms,type,
% 2.47/1.40 prefer1801827867axioms:
% 2.47/1.40 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Product__Type_OPair,type,
% 2.47/1.40 product_Pair:
% 2.47/1.40 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 2.47/1.40 produc2004651681e_prod:
% 2.47/1.40 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.47/1.40 product_rec_prod:
% 2.47/1.40 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 2.47/1.40 product_fst:
% 2.47/1.40 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 2.47/1.40 product_snd:
% 2.47/1.40 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 2.47/1.40 product_swap:
% 2.47/1.40 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Relation_ODomain,type,
% 2.47/1.40 domain:
% 2.47/1.40 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Relation_OId__on,type,
% 2.47/1.40 id_on:
% 2.47/1.40 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Relation_ORange,type,
% 2.47/1.40 range:
% 2.47/1.40 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ B ) ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Relation_Orefl__on,type,
% 2.47/1.40 refl_on:
% 2.47/1.40 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Relation_Ototal__on,type,
% 2.47/1.40 total_on:
% 2.47/1.40 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Relation_Otrans,type,
% 2.47/1.40 trans:
% 2.47/1.40 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Set_OCollect,type,
% 2.47/1.40 collect:
% 2.47/1.40 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Set_Oinsert,type,
% 2.47/1.40 insert:
% 2.47/1.40 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Set_Ois__empty,type,
% 2.47/1.40 is_empty:
% 2.47/1.40 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Set_Ois__singleton,type,
% 2.47/1.40 is_singleton:
% 2.47/1.40 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_Set_Othe__elem,type,
% 2.47/1.40 the_elem:
% 2.47/1.40 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_c_member,type,
% 2.47/1.40 member:
% 2.47/1.40 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_v_carrier,type,
% 2.47/1.40 carrier: set @ a ).
% 2.47/1.40
% 2.47/1.40 thf(sy_v_relation,type,
% 2.47/1.40 relation: set @ ( product_prod @ a @ a ) ).
% 2.47/1.40
% 2.47/1.40 thf(sy_v_y,type,
% 2.47/1.40 y: a ).
% 2.47/1.40
% 2.47/1.40 thf(sy_v_z,type,
% 2.47/1.40 z: a ).
% 2.47/1.40
% 2.47/1.40 thf(fact_0_assms,axiom,
% 2.47/1.40 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ z @ y ) @ relation )
% 2.47/1.40 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ y @ z ) @ relation ) ) ).
% 2.47/1.40
% 2.47/1.40 thf(fact_1_rational__preference__axioms,axiom,
% 2.47/1.40 prefer1997167224erence @ a @ carrier @ relation ).
% 2.47/1.40
% 2.47/1.40 thf(fact_2_preference__axioms,axiom,
% 2.47/1.40 prefer199794634erence @ a @ carrier @ relation ).
% 2.47/1.40
% 2.47/1.40 thf(fact_3_trans__refl,axiom,
% 2.47/1.40 order_preorder_on @ a @ carrier @ relation ).
% 2.47/1.40
% 2.47/1.40 thf(fact_4_total,axiom,
% 2.47/1.40 total_on @ a @ carrier @ relation ).
% 2.47/1.40
% 2.47/1.40 thf(fact_5_reflexivity,axiom,
% 2.47/1.40 refl_on @ a @ carrier @ relation ).
% 2.47/1.40
% 2.47/1.40 thf(fact_6_compl,axiom,
% 2.47/1.40 ! [X: a] :
% 2.47/1.40 ( ( member @ a @ X @ carrier )
% 2.47/1.40 => ! [Xa: a] :
% 2.47/1.40 ( ( member @ a @ Xa @ carrier )
% 2.47/1.40 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Xa ) @ relation )
% 2.47/1.40 | ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Xa @ X ) @ relation ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_7_completeD,axiom,
% 2.47/1.41 ! [X2: a,Y: a] :
% 2.47/1.41 ( ( member @ a @ X2 @ carrier )
% 2.47/1.41 => ( ( member @ a @ Y @ carrier )
% 2.47/1.41 => ( ( X2 != Y )
% 2.47/1.41 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
% 2.47/1.41 | ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X2 ) @ relation ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_8_not__outside,axiom,
% 2.47/1.41 ! [X2: a,Y: a] :
% 2.47/1.41 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
% 2.47/1.41 => ( member @ a @ X2 @ carrier ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_9_strict__is__neg__transitive,axiom,
% 2.47/1.41 ! [X2: a,Y: a,Z: a] :
% 2.47/1.41 ( ( ( member @ a @ X2 @ carrier )
% 2.47/1.41 & ( member @ a @ Y @ carrier )
% 2.47/1.41 & ( member @ a @ Z @ carrier ) )
% 2.47/1.41 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
% 2.47/1.41 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X2 ) @ relation ) )
% 2.47/1.41 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Z ) @ relation )
% 2.47/1.41 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X2 ) @ relation ) )
% 2.47/1.41 | ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation )
% 2.47/1.41 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_10_weak__is__transitive,axiom,
% 2.47/1.41 ! [X2: a,Y: a,Z: a] :
% 2.47/1.41 ( ( ( member @ a @ X2 @ carrier )
% 2.47/1.41 & ( member @ a @ Y @ carrier )
% 2.47/1.41 & ( member @ a @ Z @ carrier ) )
% 2.47/1.41 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
% 2.47/1.41 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
% 2.47/1.41 => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Z ) @ relation ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_11_strict__not__refl__weak,axiom,
% 2.47/1.41 ! [X2: a,Y: a] :
% 2.47/1.41 ( ( ( member @ a @ X2 @ carrier )
% 2.47/1.41 & ( member @ a @ Y @ carrier ) )
% 2.47/1.41 => ( ( ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X2 ) @ relation ) )
% 2.47/1.41 = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
% 2.47/1.41 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X2 ) @ relation ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_12_indiff__trans,axiom,
% 2.47/1.41 ! [X2: a,Y: a,Z: a] :
% 2.47/1.41 ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
% 2.47/1.41 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X2 ) @ relation ) )
% 2.47/1.41 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
% 2.47/1.41 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation ) )
% 2.47/1.41 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Z ) @ relation )
% 2.47/1.41 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X2 ) @ relation ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_13_strict__trans,axiom,
% 2.47/1.41 ! [X2: a,Y: a,Z: a] :
% 2.47/1.41 ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
% 2.47/1.41 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X2 ) @ relation ) )
% 2.47/1.41 => ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
% 2.47/1.41 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation ) )
% 2.47/1.41 => ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Z ) @ relation )
% 2.47/1.41 & ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X2 ) @ relation ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_14_pref__in__at__least__as,axiom,
% 2.47/1.41 ! [X2: a,Y: a] :
% 2.47/1.41 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
% 2.47/1.41 => ( member @ a @ X2 @ ( prefer310429814s_good @ a @ Y @ carrier @ relation ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_15_rational__preference_Oaxioms_I1_J,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 2.47/1.41 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( prefer199794634erence @ A @ Carrier @ Relation ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_16_preference__def,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( prefer199794634erence @ A )
% 2.47/1.41 = ( ^ [Carrier2: set @ A,Relation2: set @ ( product_prod @ A @ A )] :
% 2.47/1.41 ( ! [X3: A,Y2: A] :
% 2.47/1.41 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ Relation2 )
% 2.47/1.41 => ( member @ A @ X3 @ Carrier2 ) )
% 2.47/1.41 & ! [X3: A,Y2: A] :
% 2.47/1.41 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ Relation2 )
% 2.47/1.41 => ( member @ A @ Y2 @ Carrier2 ) )
% 2.47/1.41 & ( order_preorder_on @ A @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_17_preference_Ointro,axiom,
% 2.47/1.41 ! [A: $tType,Relation: set @ ( product_prod @ A @ A ),Carrier: set @ A] :
% 2.47/1.41 ( ! [X4: A,Y3: A] :
% 2.47/1.41 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ Relation )
% 2.47/1.41 => ( member @ A @ X4 @ Carrier ) )
% 2.47/1.41 => ( ! [X4: A,Y3: A] :
% 2.47/1.41 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ Relation )
% 2.47/1.41 => ( member @ A @ Y3 @ Carrier ) )
% 2.47/1.41 => ( ( order_preorder_on @ A @ Carrier @ Relation )
% 2.47/1.41 => ( prefer199794634erence @ A @ Carrier @ Relation ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_18_preference_Otrans__refl,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 2.47/1.41 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( order_preorder_on @ A @ Carrier @ Relation ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_19_preference_Onot__outside,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
% 2.47/1.41 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
% 2.47/1.41 => ( member @ A @ X2 @ Carrier ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_20_preference_Oreflexivity,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 2.47/1.41 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( refl_on @ A @ Carrier @ Relation ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_21_preference_Oindiff__trans,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A,Z: A] :
% 2.47/1.41 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
% 2.47/1.41 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ Relation ) )
% 2.47/1.41 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
% 2.47/1.41 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation ) )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z ) @ Relation )
% 2.47/1.41 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_22_rational__preference_Ocompl,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 2.47/1.41 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ! [X: A] :
% 2.47/1.41 ( ( member @ A @ X @ Carrier )
% 2.47/1.41 => ! [Xa: A] :
% 2.47/1.41 ( ( member @ A @ Xa @ Carrier )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Xa ) @ Relation )
% 2.47/1.41 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa @ X ) @ Relation ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_23_rational__preference_Ototal,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 2.47/1.41 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( total_on @ A @ Carrier @ Relation ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_24_rational__preference_OcompleteD,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
% 2.47/1.41 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( ( member @ A @ X2 @ Carrier )
% 2.47/1.41 => ( ( member @ A @ Y @ Carrier )
% 2.47/1.41 => ( ( X2 != Y )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
% 2.47/1.41 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ Relation ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_25_rational__preference_Ostrict__trans,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A,Z: A] :
% 2.47/1.41 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
% 2.47/1.41 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ Relation ) )
% 2.47/1.41 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
% 2.47/1.41 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation ) )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z ) @ Relation )
% 2.47/1.41 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X2 ) @ Relation ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_26_rational__preference_Oweak__is__transitive,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A,Z: A] :
% 2.47/1.41 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( ( ( member @ A @ X2 @ Carrier )
% 2.47/1.41 & ( member @ A @ Y @ Carrier )
% 2.47/1.41 & ( member @ A @ Z @ Carrier ) )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
% 2.47/1.41 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z ) @ Relation ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_27_rational__preference_Ostrict__not__refl__weak,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
% 2.47/1.41 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( ( ( member @ A @ X2 @ Carrier )
% 2.47/1.41 & ( member @ A @ Y @ Carrier ) )
% 2.47/1.41 => ( ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ Relation ) )
% 2.47/1.41 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
% 2.47/1.41 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ Relation ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_28_rational__preference_Ostrict__is__neg__transitive,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A,Z: A] :
% 2.47/1.41 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( ( ( member @ A @ X2 @ Carrier )
% 2.47/1.41 & ( member @ A @ Y @ Carrier )
% 2.47/1.41 & ( member @ A @ Z @ Carrier ) )
% 2.47/1.41 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
% 2.47/1.41 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ Relation ) )
% 2.47/1.41 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z ) @ Relation )
% 2.47/1.41 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X2 ) @ Relation ) )
% 2.47/1.41 | ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation )
% 2.47/1.41 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_29_worse__in__no__better,axiom,
% 2.47/1.41 ! [X2: a,Y: a] :
% 2.47/1.41 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
% 2.47/1.41 => ( member @ a @ Y @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_30_as__good__as__sameIff,axiom,
% 2.47/1.41 ! [Z: a,Y: a] :
% 2.47/1.41 ( ( member @ a @ Z @ ( prefer951318096ood_as @ a @ Y @ carrier @ relation ) )
% 2.47/1.41 = ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation )
% 2.47/1.41 & ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_31_prod_Oinject,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 2.47/1.41 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 2.47/1.41 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 2.47/1.41 = ( ( X1 = Y1 )
% 2.47/1.41 & ( X22 = Y22 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_32_old_Oprod_Oinject,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 2.47/1.41 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.47/1.41 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.47/1.41 = ( ( A2 = A3 )
% 2.47/1.41 & ( B2 = B3 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_33_rational__preference_Oas__good__as__sameIff,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),Z: A,Y: A] :
% 2.47/1.41 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( ( member @ A @ Z @ ( prefer951318096ood_as @ A @ Y @ Carrier @ Relation ) )
% 2.47/1.41 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation )
% 2.47/1.41 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_34_total__onI,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A )] :
% 2.47/1.41 ( ! [X4: A,Y3: A] :
% 2.47/1.41 ( ( member @ A @ X4 @ A4 )
% 2.47/1.41 => ( ( member @ A @ Y3 @ A4 )
% 2.47/1.41 => ( ( X4 != Y3 )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R )
% 2.47/1.41 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X4 ) @ R ) ) ) ) )
% 2.47/1.41 => ( total_on @ A @ A4 @ R ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_35_total__on__def,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( total_on @ A )
% 2.47/1.41 = ( ^ [A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 2.47/1.41 ! [X3: A] :
% 2.47/1.41 ( ( member @ A @ X3 @ A5 )
% 2.47/1.41 => ! [Y2: A] :
% 2.47/1.41 ( ( member @ A @ Y2 @ A5 )
% 2.47/1.41 => ( ( X3 != Y2 )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
% 2.47/1.41 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X3 ) @ R2 ) ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_36_refl__onD,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A2: A] :
% 2.47/1.41 ( ( refl_on @ A @ A4 @ R )
% 2.47/1.41 => ( ( member @ A @ A2 @ A4 )
% 2.47/1.41 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_37_refl__onD1,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
% 2.47/1.41 ( ( refl_on @ A @ A4 @ R )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R )
% 2.47/1.41 => ( member @ A @ X2 @ A4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_38_refl__onD2,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
% 2.47/1.41 ( ( refl_on @ A @ A4 @ R )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R )
% 2.47/1.41 => ( member @ A @ Y @ A4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_39_refl__on__domain,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
% 2.47/1.41 ( ( refl_on @ A @ A4 @ R )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R )
% 2.47/1.41 => ( ( member @ A @ A2 @ A4 )
% 2.47/1.41 & ( member @ A @ B2 @ A4 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_40_at__least__as__goodD,axiom,
% 2.47/1.41 ! [A: $tType,Z: A,Y: A,B4: set @ A,Pr: set @ ( product_prod @ A @ A )] :
% 2.47/1.41 ( ( member @ A @ Z @ ( prefer310429814s_good @ A @ Y @ B4 @ Pr ) )
% 2.47/1.41 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Pr ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_41_at__lst__asgd__ge,axiom,
% 2.47/1.41 ! [A: $tType,X2: A,Y: A,B4: set @ A,Pr: set @ ( product_prod @ A @ A )] :
% 2.47/1.41 ( ( member @ A @ X2 @ ( prefer310429814s_good @ A @ Y @ B4 @ Pr ) )
% 2.47/1.41 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Pr ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_42_old_Oprod_Oinducts,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.47/1.41 ( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
% 2.47/1.41 => ( P @ Prod ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_43_old_Oprod_Oexhaust,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 2.47/1.41 ~ ! [A6: A,B5: B] :
% 2.47/1.41 ( Y
% 2.47/1.41 != ( product_Pair @ A @ B @ A6 @ B5 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_44_prod__induct7,axiom,
% 2.47/1.41 ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 2.47/1.41 ( ! [A6: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
% 2.47/1.41 => ( P @ X2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_45_mem__Collect__eq,axiom,
% 2.47/1.41 ! [A: $tType,A2: A,P: A > $o] :
% 2.47/1.41 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.47/1.41 = ( P @ A2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_46_Collect__mem__eq,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A] :
% 2.47/1.41 ( ( collect @ A
% 2.47/1.41 @ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
% 2.47/1.41 = A4 ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_47_Collect__cong,axiom,
% 2.47/1.41 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.47/1.41 ( ! [X4: A] :
% 2.47/1.41 ( ( P @ X4 )
% 2.47/1.41 = ( Q @ X4 ) )
% 2.47/1.41 => ( ( collect @ A @ P )
% 2.47/1.41 = ( collect @ A @ Q ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_48_ext,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
% 2.47/1.41 ( ! [X4: A] :
% 2.47/1.41 ( ( F3 @ X4 )
% 2.47/1.41 = ( G3 @ X4 ) )
% 2.47/1.41 => ( F3 = G3 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_49_prod__induct6,axiom,
% 2.47/1.41 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 2.47/1.41 ( ! [A6: A,B5: B,C2: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
% 2.47/1.41 => ( P @ X2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_50_prod__induct5,axiom,
% 2.47/1.41 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.47/1.41 ( ! [A6: A,B5: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 2.47/1.41 => ( P @ X2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_51_prod__induct4,axiom,
% 2.47/1.41 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.47/1.41 ( ! [A6: A,B5: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 2.47/1.41 => ( P @ X2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_52_prod__induct3,axiom,
% 2.47/1.41 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.47/1.41 ( ! [A6: A,B5: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) )
% 2.47/1.41 => ( P @ X2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_53_prod__cases7,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 2.47/1.41 ~ ! [A6: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
% 2.47/1.41 ( Y
% 2.47/1.41 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_54_prod__cases6,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 2.47/1.41 ~ ! [A6: A,B5: B,C2: C,D2: D,E2: E,F2: F] :
% 2.47/1.41 ( Y
% 2.47/1.41 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_55_prod__cases5,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.47/1.41 ~ ! [A6: A,B5: B,C2: C,D2: D,E2: E] :
% 2.47/1.41 ( Y
% 2.47/1.41 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_56_prod__cases4,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.47/1.41 ~ ! [A6: A,B5: B,C2: C,D2: D] :
% 2.47/1.41 ( Y
% 2.47/1.41 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_57_prod__cases3,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.47/1.41 ~ ! [A6: A,B5: B,C2: C] :
% 2.47/1.41 ( Y
% 2.47/1.41 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_58_Pair__inject,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 2.47/1.41 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.47/1.41 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.47/1.41 => ~ ( ( A2 = A3 )
% 2.47/1.41 => ( B2 != B3 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_59_prod__cases,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 2.47/1.41 ( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
% 2.47/1.41 => ( P @ P2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_60_surj__pair,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 2.47/1.41 ? [X4: A,Y3: B] :
% 2.47/1.41 ( P2
% 2.47/1.41 = ( product_Pair @ A @ B @ X4 @ Y3 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_61_rational__preference_Oworse__in__no__better,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
% 2.47/1.41 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
% 2.47/1.41 => ( member @ A @ Y @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_62_rational__preference_Opref__in__at__least__as,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
% 2.47/1.41 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
% 2.47/1.41 => ( member @ A @ X2 @ ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_63_at__lst__asgd__not__ge,axiom,
% 2.47/1.41 ! [X2: a,Y: a] :
% 2.47/1.41 ( ( carrier
% 2.47/1.41 != ( bot_bot @ ( set @ a ) ) )
% 2.47/1.41 => ( ( member @ a @ X2 @ carrier )
% 2.47/1.41 => ( ( member @ a @ Y @ carrier )
% 2.47/1.41 => ( ~ ( member @ a @ X2 @ ( prefer310429814s_good @ a @ Y @ carrier @ relation ) )
% 2.47/1.41 => ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_64_no__better__subset__pref,axiom,
% 2.47/1.41 ! [X2: a,Y: a] :
% 2.47/1.41 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation )
% 2.47/1.41 => ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X2 @ carrier @ relation ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_65_no__better__thansubset__rel,axiom,
% 2.47/1.41 ! [X2: a,Y: a] :
% 2.47/1.41 ( ( member @ a @ X2 @ carrier )
% 2.47/1.41 => ( ( member @ a @ Y @ carrier )
% 2.47/1.41 => ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X2 @ carrier @ relation ) )
% 2.47/1.41 => ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Y ) @ relation ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_66_no__better__than__nonepty,axiom,
% 2.47/1.41 ! [X2: a] :
% 2.47/1.41 ( ( carrier
% 2.47/1.41 != ( bot_bot @ ( set @ a ) ) )
% 2.47/1.41 => ( ( member @ a @ X2 @ carrier )
% 2.47/1.41 => ( ( prefer1532642881r_than @ a @ X2 @ carrier @ relation )
% 2.47/1.41 != ( bot_bot @ ( set @ a ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_67_nbt__nest,axiom,
% 2.47/1.41 ! [Y: a,X2: a] :
% 2.47/1.41 ( ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ X2 @ carrier @ relation ) )
% 2.47/1.41 | ( ord_less_eq @ ( set @ a ) @ ( prefer1532642881r_than @ a @ X2 @ carrier @ relation ) @ ( prefer1532642881r_than @ a @ Y @ carrier @ relation ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_68_old_Oprod_Orec,axiom,
% 2.47/1.41 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 2.47/1.41 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.47/1.41 = ( F1 @ A2 @ B2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_69_rational__preference_Ointro,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 2.47/1.41 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( ( prefer1801827867axioms @ A @ Carrier @ Relation )
% 2.47/1.41 => ( prefer1997167224erence @ A @ Carrier @ Relation ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_70_rational__preference__def,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( prefer1997167224erence @ A )
% 2.47/1.41 = ( ^ [Carrier2: set @ A,Relation2: set @ ( product_prod @ A @ A )] :
% 2.47/1.41 ( ( prefer199794634erence @ A @ Carrier2 @ Relation2 )
% 2.47/1.41 & ( prefer1801827867axioms @ A @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_71_refl__on__empty,axiom,
% 2.47/1.41 ! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_72_preorder__on__empty,axiom,
% 2.47/1.41 ! [A: $tType] : ( order_preorder_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_73_total__on__empty,axiom,
% 2.47/1.41 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_74_rational__preference_Onbt__nest,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),Y: A,X2: A] :
% 2.47/1.41 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X2 @ Carrier @ Relation ) )
% 2.47/1.41 | ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ X2 @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_75_rational__preference_Ono__better__than__nonepty,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A] :
% 2.47/1.41 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( ( Carrier
% 2.47/1.41 != ( bot_bot @ ( set @ A ) ) )
% 2.47/1.41 => ( ( member @ A @ X2 @ Carrier )
% 2.47/1.41 => ( ( prefer1532642881r_than @ A @ X2 @ Carrier @ Relation )
% 2.47/1.41 != ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_76_rational__preference__axioms__def,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( prefer1801827867axioms @ A )
% 2.47/1.41 = ( total_on @ A ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_77_rational__preference__axioms_Ointro,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 2.47/1.41 ( ( total_on @ A @ Carrier @ Relation )
% 2.47/1.41 => ( prefer1801827867axioms @ A @ Carrier @ Relation ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_78_rational__preference_Oaxioms_I2_J,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 2.47/1.41 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( prefer1801827867axioms @ A @ Carrier @ Relation ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_79_rational__preference_Ono__better__subset__pref,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
% 2.47/1.41 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation )
% 2.47/1.41 => ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X2 @ Carrier @ Relation ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_80_rational__preference_Ono__better__thansubset__rel,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
% 2.47/1.41 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( ( member @ A @ X2 @ Carrier )
% 2.47/1.41 => ( ( member @ A @ Y @ Carrier )
% 2.47/1.41 => ( ( ord_less_eq @ ( set @ A ) @ ( prefer1532642881r_than @ A @ Y @ Carrier @ Relation ) @ ( prefer1532642881r_than @ A @ X2 @ Carrier @ Relation ) )
% 2.47/1.41 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_81_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X2: A,Y: A] :
% 2.47/1.41 ( ( prefer1997167224erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( ( Carrier
% 2.47/1.41 != ( bot_bot @ ( set @ A ) ) )
% 2.47/1.41 => ( ( member @ A @ X2 @ Carrier )
% 2.47/1.41 => ( ( member @ A @ Y @ Carrier )
% 2.47/1.41 => ( ~ ( member @ A @ X2 @ ( prefer310429814s_good @ A @ Y @ Carrier @ Relation ) )
% 2.47/1.41 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ Relation ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_82_subset__empty,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A] :
% 2.47/1.41 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 2.47/1.41 = ( A4
% 2.47/1.41 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_83_empty__subsetI,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_84_Pair__le,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType] :
% 2.47/1.41 ( ( ( ord @ B )
% 2.47/1.41 & ( ord @ A ) )
% 2.47/1.41 => ! [A2: A,B2: B,C3: A,D3: B] :
% 2.47/1.41 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C3 @ D3 ) )
% 2.47/1.41 = ( ( ord_less_eq @ A @ A2 @ C3 )
% 2.47/1.41 & ( ord_less_eq @ B @ B2 @ D3 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_85_subsetI,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.47/1.41 ( ! [X4: A] :
% 2.47/1.41 ( ( member @ A @ X4 @ A4 )
% 2.47/1.41 => ( member @ A @ X4 @ B4 ) )
% 2.47/1.41 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_86_subset__antisym,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.47/1.41 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.47/1.41 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 2.47/1.41 => ( A4 = B4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_87_empty__iff,axiom,
% 2.47/1.41 ! [A: $tType,C3: A] :
% 2.47/1.41 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_88_all__not__in__conv,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A] :
% 2.47/1.41 ( ( ! [X3: A] :
% 2.47/1.41 ~ ( member @ A @ X3 @ A4 ) )
% 2.47/1.41 = ( A4
% 2.47/1.41 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_89_empty__Collect__eq,axiom,
% 2.47/1.41 ! [A: $tType,P: A > $o] :
% 2.47/1.41 ( ( ( bot_bot @ ( set @ A ) )
% 2.47/1.41 = ( collect @ A @ P ) )
% 2.47/1.41 = ( ! [X3: A] :
% 2.47/1.41 ~ ( P @ X3 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_90_Collect__empty__eq,axiom,
% 2.47/1.41 ! [A: $tType,P: A > $o] :
% 2.47/1.41 ( ( ( collect @ A @ P )
% 2.47/1.41 = ( bot_bot @ ( set @ A ) ) )
% 2.47/1.41 = ( ! [X3: A] :
% 2.47/1.41 ~ ( P @ X3 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_91_subrelI,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 2.47/1.41 ( ! [X4: A,Y3: B] :
% 2.47/1.41 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R )
% 2.47/1.41 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ S ) )
% 2.47/1.41 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_92_ex__in__conv,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A] :
% 2.47/1.41 ( ( ? [X3: A] : ( member @ A @ X3 @ A4 ) )
% 2.47/1.41 = ( A4
% 2.47/1.41 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_93_equals0I,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A] :
% 2.47/1.41 ( ! [Y3: A] :
% 2.47/1.41 ~ ( member @ A @ Y3 @ A4 )
% 2.47/1.41 => ( A4
% 2.47/1.41 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_94_equals0D,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A,A2: A] :
% 2.47/1.41 ( ( A4
% 2.47/1.41 = ( bot_bot @ ( set @ A ) ) )
% 2.47/1.41 => ~ ( member @ A @ A2 @ A4 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_95_emptyE,axiom,
% 2.47/1.41 ! [A: $tType,A2: A] :
% 2.47/1.41 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_96_Collect__mono__iff,axiom,
% 2.47/1.41 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.47/1.41 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.47/1.41 = ( ! [X3: A] :
% 2.47/1.41 ( ( P @ X3 )
% 2.47/1.41 => ( Q @ X3 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_97_set__eq__subset,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( ^ [Y4: set @ A,Z2: set @ A] : ( Y4 = Z2 ) )
% 2.47/1.41 = ( ^ [A5: set @ A,B6: set @ A] :
% 2.47/1.41 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
% 2.47/1.41 & ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_98_subset__trans,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A,B4: set @ A,C4: set @ A] :
% 2.47/1.41 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.47/1.41 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
% 2.47/1.41 => ( ord_less_eq @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_99_Collect__mono,axiom,
% 2.47/1.41 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.47/1.41 ( ! [X4: A] :
% 2.47/1.41 ( ( P @ X4 )
% 2.47/1.41 => ( Q @ X4 ) )
% 2.47/1.41 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_100_subset__refl,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_101_subset__iff,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( ord_less_eq @ ( set @ A ) )
% 2.47/1.41 = ( ^ [A5: set @ A,B6: set @ A] :
% 2.47/1.41 ! [T2: A] :
% 2.47/1.41 ( ( member @ A @ T2 @ A5 )
% 2.47/1.41 => ( member @ A @ T2 @ B6 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_102_equalityD2,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.47/1.41 ( ( A4 = B4 )
% 2.47/1.41 => ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_103_equalityD1,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.47/1.41 ( ( A4 = B4 )
% 2.47/1.41 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_104_subset__eq,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( ord_less_eq @ ( set @ A ) )
% 2.47/1.41 = ( ^ [A5: set @ A,B6: set @ A] :
% 2.47/1.41 ! [X3: A] :
% 2.47/1.41 ( ( member @ A @ X3 @ A5 )
% 2.47/1.41 => ( member @ A @ X3 @ B6 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_105_equalityE,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.47/1.41 ( ( A4 = B4 )
% 2.47/1.41 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.47/1.41 => ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_106_subsetD,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A,B4: set @ A,C3: A] :
% 2.47/1.41 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.47/1.41 => ( ( member @ A @ C3 @ A4 )
% 2.47/1.41 => ( member @ A @ C3 @ B4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_107_in__mono,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A,B4: set @ A,X2: A] :
% 2.47/1.41 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.47/1.41 => ( ( member @ A @ X2 @ A4 )
% 2.47/1.41 => ( member @ A @ X2 @ B4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_108_Pair__mono,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType] :
% 2.47/1.41 ( ( ( ord @ B )
% 2.47/1.41 & ( ord @ A ) )
% 2.47/1.41 => ! [X2: A,X5: A,Y: B,Y5: B] :
% 2.47/1.41 ( ( ord_less_eq @ A @ X2 @ X5 )
% 2.47/1.41 => ( ( ord_less_eq @ B @ Y @ Y5 )
% 2.47/1.41 => ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_109_bot__prod__def,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType] :
% 2.47/1.41 ( ( ( bot @ B )
% 2.47/1.41 & ( bot @ A ) )
% 2.47/1.41 => ( ( bot_bot @ ( product_prod @ A @ B ) )
% 2.47/1.41 = ( product_Pair @ A @ B @ ( bot_bot @ A ) @ ( bot_bot @ B ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_110_bot__apply,axiom,
% 2.47/1.41 ! [C: $tType,D: $tType] :
% 2.47/1.41 ( ( bot @ C )
% 2.47/1.41 => ( ( bot_bot @ ( D > C ) )
% 2.47/1.41 = ( ^ [X3: D] : ( bot_bot @ C ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_111_order__refl,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( preorder @ A )
% 2.47/1.41 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_112_subset__emptyI,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A] :
% 2.47/1.41 ( ! [X4: A] :
% 2.47/1.41 ~ ( member @ A @ X4 @ A4 )
% 2.47/1.41 => ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_113_bot_Oextremum__uniqueI,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( order_bot @ A )
% 2.47/1.41 => ! [A2: A] :
% 2.47/1.41 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 2.47/1.41 => ( A2
% 2.47/1.41 = ( bot_bot @ A ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_114_bot_Oextremum__unique,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( order_bot @ A )
% 2.47/1.41 => ! [A2: A] :
% 2.47/1.41 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 2.47/1.41 = ( A2
% 2.47/1.41 = ( bot_bot @ A ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_115_bot_Oextremum,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( order_bot @ A )
% 2.47/1.41 => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_116_bot__set__def,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( bot_bot @ ( set @ A ) )
% 2.47/1.41 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_117_bot__empty__eq,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( bot_bot @ ( A > $o ) )
% 2.47/1.41 = ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_118_le__funD,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType] :
% 2.47/1.41 ( ( ord @ B )
% 2.47/1.41 => ! [F3: A > B,G3: A > B,X2: A] :
% 2.47/1.41 ( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
% 2.47/1.41 => ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G3 @ X2 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_119_le__funE,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType] :
% 2.47/1.41 ( ( ord @ B )
% 2.47/1.41 => ! [F3: A > B,G3: A > B,X2: A] :
% 2.47/1.41 ( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
% 2.47/1.41 => ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G3 @ X2 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_120_le__funI,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType] :
% 2.47/1.41 ( ( ord @ B )
% 2.47/1.41 => ! [F3: A > B,G3: A > B] :
% 2.47/1.41 ( ! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
% 2.47/1.41 => ( ord_less_eq @ ( A > B ) @ F3 @ G3 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_121_le__fun__def,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType] :
% 2.47/1.41 ( ( ord @ B )
% 2.47/1.41 => ( ( ord_less_eq @ ( A > B ) )
% 2.47/1.41 = ( ^ [F4: A > B,G4: A > B] :
% 2.47/1.41 ! [X3: A] : ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_122_order__subst1,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType] :
% 2.47/1.41 ( ( ( order @ B )
% 2.47/1.41 & ( order @ A ) )
% 2.47/1.41 => ! [A2: A,F3: B > A,B2: B,C3: B] :
% 2.47/1.41 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
% 2.47/1.41 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 2.47/1.41 => ( ! [X4: B,Y3: B] :
% 2.47/1.41 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 2.47/1.41 => ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
% 2.47/1.41 => ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_123_order__subst2,axiom,
% 2.47/1.41 ! [A: $tType,C: $tType] :
% 2.47/1.41 ( ( ( order @ C )
% 2.47/1.41 & ( order @ A ) )
% 2.47/1.41 => ! [A2: A,B2: A,F3: A > C,C3: C] :
% 2.47/1.41 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.47/1.41 => ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C3 )
% 2.47/1.41 => ( ! [X4: A,Y3: A] :
% 2.47/1.41 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.47/1.41 => ( ord_less_eq @ C @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
% 2.47/1.41 => ( ord_less_eq @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_124_ord__eq__le__subst,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType] :
% 2.47/1.41 ( ( ( ord @ B )
% 2.47/1.41 & ( ord @ A ) )
% 2.47/1.41 => ! [A2: A,F3: B > A,B2: B,C3: B] :
% 2.47/1.41 ( ( A2
% 2.47/1.41 = ( F3 @ B2 ) )
% 2.47/1.41 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 2.47/1.41 => ( ! [X4: B,Y3: B] :
% 2.47/1.41 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 2.47/1.41 => ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
% 2.47/1.41 => ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_125_ord__le__eq__subst,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType] :
% 2.47/1.41 ( ( ( ord @ B )
% 2.47/1.41 & ( ord @ A ) )
% 2.47/1.41 => ! [A2: A,B2: A,F3: A > B,C3: B] :
% 2.47/1.41 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.47/1.41 => ( ( ( F3 @ B2 )
% 2.47/1.41 = C3 )
% 2.47/1.41 => ( ! [X4: A,Y3: A] :
% 2.47/1.41 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.47/1.41 => ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
% 2.47/1.41 => ( ord_less_eq @ B @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_126_eq__iff,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( order @ A )
% 2.47/1.41 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 2.47/1.41 = ( ^ [X3: A,Y2: A] :
% 2.47/1.41 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.47/1.41 & ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_127_antisym,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( order @ A )
% 2.47/1.41 => ! [X2: A,Y: A] :
% 2.47/1.41 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.47/1.41 => ( ( ord_less_eq @ A @ Y @ X2 )
% 2.47/1.41 => ( X2 = Y ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_128_linear,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( linorder @ A )
% 2.47/1.41 => ! [X2: A,Y: A] :
% 2.47/1.41 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.47/1.41 | ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_129_eq__refl,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( preorder @ A )
% 2.47/1.41 => ! [X2: A,Y: A] :
% 2.47/1.41 ( ( X2 = Y )
% 2.47/1.41 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_130_le__cases,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( linorder @ A )
% 2.47/1.41 => ! [X2: A,Y: A] :
% 2.47/1.41 ( ~ ( ord_less_eq @ A @ X2 @ Y )
% 2.47/1.41 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_131_order_Otrans,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( order @ A )
% 2.47/1.41 => ! [A2: A,B2: A,C3: A] :
% 2.47/1.41 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.47/1.41 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 2.47/1.41 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_132_le__cases3,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( linorder @ A )
% 2.47/1.41 => ! [X2: A,Y: A,Z: A] :
% 2.47/1.41 ( ( ( ord_less_eq @ A @ X2 @ Y )
% 2.47/1.41 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 2.47/1.41 => ( ( ( ord_less_eq @ A @ Y @ X2 )
% 2.47/1.41 => ~ ( ord_less_eq @ A @ X2 @ Z ) )
% 2.47/1.41 => ( ( ( ord_less_eq @ A @ X2 @ Z )
% 2.47/1.41 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 2.47/1.41 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 2.47/1.41 => ~ ( ord_less_eq @ A @ Y @ X2 ) )
% 2.47/1.41 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 2.47/1.41 => ~ ( ord_less_eq @ A @ Z @ X2 ) )
% 2.47/1.41 => ~ ( ( ord_less_eq @ A @ Z @ X2 )
% 2.47/1.41 => ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_133_antisym__conv,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( order @ A )
% 2.47/1.41 => ! [Y: A,X2: A] :
% 2.47/1.41 ( ( ord_less_eq @ A @ Y @ X2 )
% 2.47/1.41 => ( ( ord_less_eq @ A @ X2 @ Y )
% 2.47/1.41 = ( X2 = Y ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_134_order__class_Oorder_Oeq__iff,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( order @ A )
% 2.47/1.41 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 2.47/1.41 = ( ^ [A7: A,B7: A] :
% 2.47/1.41 ( ( ord_less_eq @ A @ A7 @ B7 )
% 2.47/1.41 & ( ord_less_eq @ A @ B7 @ A7 ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_135_ord__eq__le__trans,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( ord @ A )
% 2.47/1.41 => ! [A2: A,B2: A,C3: A] :
% 2.47/1.41 ( ( A2 = B2 )
% 2.47/1.41 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 2.47/1.41 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_136_ord__le__eq__trans,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( ord @ A )
% 2.47/1.41 => ! [A2: A,B2: A,C3: A] :
% 2.47/1.41 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.47/1.41 => ( ( B2 = C3 )
% 2.47/1.41 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_137_order__class_Oorder_Oantisym,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( order @ A )
% 2.47/1.41 => ! [A2: A,B2: A] :
% 2.47/1.41 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.47/1.41 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.47/1.41 => ( A2 = B2 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_138_order__trans,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( preorder @ A )
% 2.47/1.41 => ! [X2: A,Y: A,Z: A] :
% 2.47/1.41 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.47/1.41 => ( ( ord_less_eq @ A @ Y @ Z )
% 2.47/1.41 => ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_139_dual__order_Orefl,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( order @ A )
% 2.47/1.41 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_140_linorder__wlog,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( linorder @ A )
% 2.47/1.41 => ! [P: A > A > $o,A2: A,B2: A] :
% 2.47/1.41 ( ! [A6: A,B5: A] :
% 2.47/1.41 ( ( ord_less_eq @ A @ A6 @ B5 )
% 2.47/1.41 => ( P @ A6 @ B5 ) )
% 2.47/1.41 => ( ! [A6: A,B5: A] :
% 2.47/1.41 ( ( P @ B5 @ A6 )
% 2.47/1.41 => ( P @ A6 @ B5 ) )
% 2.47/1.41 => ( P @ A2 @ B2 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_141_dual__order_Otrans,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( order @ A )
% 2.47/1.41 => ! [B2: A,A2: A,C3: A] :
% 2.47/1.41 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.47/1.41 => ( ( ord_less_eq @ A @ C3 @ B2 )
% 2.47/1.41 => ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_142_dual__order_Oeq__iff,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( order @ A )
% 2.47/1.41 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 2.47/1.41 = ( ^ [A7: A,B7: A] :
% 2.47/1.41 ( ( ord_less_eq @ A @ B7 @ A7 )
% 2.47/1.41 & ( ord_less_eq @ A @ A7 @ B7 ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_143_dual__order_Oantisym,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( order @ A )
% 2.47/1.41 => ! [B2: A,A2: A] :
% 2.47/1.41 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.47/1.41 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.47/1.41 => ( A2 = B2 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_144_bot__fun__def,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType] :
% 2.47/1.41 ( ( bot @ B )
% 2.47/1.41 => ( ( bot_bot @ ( A > B ) )
% 2.47/1.41 = ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_145_ssubst__Pair__rhs,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,R: A,S: B,R3: set @ ( product_prod @ A @ B ),S2: B] :
% 2.47/1.41 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R3 )
% 2.47/1.41 => ( ( S2 = S )
% 2.47/1.41 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R3 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_146_less__by__empty,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ ( product_prod @ A @ A ),B4: set @ ( product_prod @ A @ A )] :
% 2.47/1.41 ( ( A4
% 2.47/1.41 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 2.47/1.41 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B4 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_147_transitivity,axiom,
% 2.47/1.41 trans @ a @ relation ).
% 2.47/1.41
% 2.47/1.41 thf(fact_148_Set_Ois__empty__def,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( is_empty @ A )
% 2.47/1.41 = ( ^ [A5: set @ A] :
% 2.47/1.41 ( A5
% 2.47/1.41 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_149_internal__case__prod__conv,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
% 2.47/1.41 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 2.47/1.41 = ( C3 @ A2 @ B2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_150_transD,axiom,
% 2.47/1.41 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X2: A,Y: A,Z: A] :
% 2.47/1.41 ( ( trans @ A @ R )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R )
% 2.47/1.41 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z ) @ R ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_151_transE,axiom,
% 2.47/1.41 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X2: A,Y: A,Z: A] :
% 2.47/1.41 ( ( trans @ A @ R )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R )
% 2.47/1.41 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z ) @ R ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_152_transI,axiom,
% 2.47/1.41 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 2.47/1.41 ( ! [X4: A,Y3: A,Z3: A] :
% 2.47/1.41 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R )
% 2.47/1.41 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z3 ) @ R ) ) )
% 2.47/1.41 => ( trans @ A @ R ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_153_trans__def,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( trans @ A )
% 2.47/1.41 = ( ^ [R2: set @ ( product_prod @ A @ A )] :
% 2.47/1.41 ! [X3: A,Y2: A,Z4: A] :
% 2.47/1.41 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z4 ) @ R2 )
% 2.47/1.41 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z4 ) @ R2 ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_154_trans__empty,axiom,
% 2.47/1.41 ! [A: $tType] : ( trans @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_155_preference_Otransitivity,axiom,
% 2.47/1.41 ! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
% 2.47/1.41 ( ( prefer199794634erence @ A @ Carrier @ Relation )
% 2.47/1.41 => ( trans @ A @ Relation ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_156_preorder__on__def,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( order_preorder_on @ A )
% 2.47/1.41 = ( ^ [A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
% 2.47/1.41 ( ( refl_on @ A @ A5 @ R2 )
% 2.47/1.41 & ( trans @ A @ R2 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_157_Collect__empty__eq__bot,axiom,
% 2.47/1.41 ! [A: $tType,P: A > $o] :
% 2.47/1.41 ( ( ( collect @ A @ P )
% 2.47/1.41 = ( bot_bot @ ( set @ A ) ) )
% 2.47/1.41 = ( P
% 2.47/1.41 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_158_relChain__def,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType] :
% 2.47/1.41 ( ( ord @ B )
% 2.47/1.41 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 2.47/1.41 = ( ^ [R2: set @ ( product_prod @ A @ A ),As: A > B] :
% 2.47/1.41 ! [I: A,J: A] :
% 2.47/1.41 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R2 )
% 2.47/1.41 => ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_159_Id__on__empty,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
% 2.47/1.41 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_160_Id__onI,axiom,
% 2.47/1.41 ! [A: $tType,A2: A,A4: set @ A] :
% 2.47/1.41 ( ( member @ A @ A2 @ A4 )
% 2.47/1.41 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id_on @ A @ A4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_161_Id__on__iff,axiom,
% 2.47/1.41 ! [A: $tType,X2: A,Y: A,A4: set @ A] :
% 2.47/1.41 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( id_on @ A @ A4 ) )
% 2.47/1.41 = ( ( X2 = Y )
% 2.47/1.41 & ( member @ A @ X2 @ A4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_162_Id__on__eqI,axiom,
% 2.47/1.41 ! [A: $tType,A2: A,B2: A,A4: set @ A] :
% 2.47/1.41 ( ( A2 = B2 )
% 2.47/1.41 => ( ( member @ A @ A2 @ A4 )
% 2.47/1.41 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id_on @ A @ A4 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_163_Id__onE,axiom,
% 2.47/1.41 ! [A: $tType,C3: product_prod @ A @ A,A4: set @ A] :
% 2.47/1.41 ( ( member @ ( product_prod @ A @ A ) @ C3 @ ( id_on @ A @ A4 ) )
% 2.47/1.41 => ~ ! [X4: A] :
% 2.47/1.41 ( ( member @ A @ X4 @ A4 )
% 2.47/1.41 => ( C3
% 2.47/1.41 != ( product_Pair @ A @ A @ X4 @ X4 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_164_trans__Id__on,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A] : ( trans @ A @ ( id_on @ A @ A4 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_165_refl__on__Id__on,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A] : ( refl_on @ A @ A4 @ ( id_on @ A @ A4 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_166_under__incr,axiom,
% 2.47/1.41 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
% 2.47/1.41 ( ( trans @ A @ R )
% 2.47/1.41 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R )
% 2.47/1.41 => ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R @ A2 ) @ ( order_under @ A @ R @ B2 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_167_GreatestI2__order,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( order @ A )
% 2.47/1.41 => ! [P: A > $o,X2: A,Q: A > $o] :
% 2.47/1.41 ( ( P @ X2 )
% 2.47/1.41 => ( ! [Y3: A] :
% 2.47/1.41 ( ( P @ Y3 )
% 2.47/1.41 => ( ord_less_eq @ A @ Y3 @ X2 ) )
% 2.47/1.41 => ( ! [X4: A] :
% 2.47/1.41 ( ( P @ X4 )
% 2.47/1.41 => ( ! [Y6: A] :
% 2.47/1.41 ( ( P @ Y6 )
% 2.47/1.41 => ( ord_less_eq @ A @ Y6 @ X4 ) )
% 2.47/1.41 => ( Q @ X4 ) ) )
% 2.47/1.41 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_168_Greatest__equality,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( order @ A )
% 2.47/1.41 => ! [P: A > $o,X2: A] :
% 2.47/1.41 ( ( P @ X2 )
% 2.47/1.41 => ( ! [Y3: A] :
% 2.47/1.41 ( ( P @ Y3 )
% 2.47/1.41 => ( ord_less_eq @ A @ Y3 @ X2 ) )
% 2.47/1.41 => ( ( order_Greatest @ A @ P )
% 2.47/1.41 = X2 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_169_fst__bot,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType] :
% 2.47/1.41 ( ( ( bot @ A )
% 2.47/1.41 & ( bot @ B ) )
% 2.47/1.41 => ( ( product_fst @ A @ B @ ( bot_bot @ ( product_prod @ A @ B ) ) )
% 2.47/1.41 = ( bot_bot @ A ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_170_snd__bot,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType] :
% 2.47/1.41 ( ( ( bot @ A )
% 2.47/1.41 & ( bot @ B ) )
% 2.47/1.41 => ( ( product_snd @ B @ A @ ( bot_bot @ ( product_prod @ B @ A ) ) )
% 2.47/1.41 = ( bot_bot @ A ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_171_trans__singleton,axiom,
% 2.47/1.41 ! [A: $tType,A2: A] : ( trans @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_172_insert__absorb2,axiom,
% 2.47/1.41 ! [A: $tType,X2: A,A4: set @ A] :
% 2.47/1.41 ( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A4 ) )
% 2.47/1.41 = ( insert @ A @ X2 @ A4 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_173_insert__iff,axiom,
% 2.47/1.41 ! [A: $tType,A2: A,B2: A,A4: set @ A] :
% 2.47/1.41 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
% 2.47/1.41 = ( ( A2 = B2 )
% 2.47/1.41 | ( member @ A @ A2 @ A4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_174_insertCI,axiom,
% 2.47/1.41 ! [A: $tType,A2: A,B4: set @ A,B2: A] :
% 2.47/1.41 ( ( ~ ( member @ A @ A2 @ B4 )
% 2.47/1.41 => ( A2 = B2 ) )
% 2.47/1.41 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_175_singletonI,axiom,
% 2.47/1.41 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_176_insert__subset,axiom,
% 2.47/1.41 ! [A: $tType,X2: A,A4: set @ A,B4: set @ A] :
% 2.47/1.41 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ A4 ) @ B4 )
% 2.47/1.41 = ( ( member @ A @ X2 @ B4 )
% 2.47/1.41 & ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_177_singleton__insert__inj__eq,axiom,
% 2.47/1.41 ! [A: $tType,B2: A,A2: A,A4: set @ A] :
% 2.47/1.41 ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
% 2.47/1.41 = ( insert @ A @ A2 @ A4 ) )
% 2.47/1.41 = ( ( A2 = B2 )
% 2.47/1.41 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_178_singleton__insert__inj__eq_H,axiom,
% 2.47/1.41 ! [A: $tType,A2: A,A4: set @ A,B2: A] :
% 2.47/1.41 ( ( ( insert @ A @ A2 @ A4 )
% 2.47/1.41 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.47/1.41 = ( ( A2 = B2 )
% 2.47/1.41 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_179_prod_Ocollapse,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.47/1.41 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 2.47/1.41 = Prod ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_180_insert__subsetI,axiom,
% 2.47/1.41 ! [A: $tType,X2: A,A4: set @ A,X6: set @ A] :
% 2.47/1.41 ( ( member @ A @ X2 @ A4 )
% 2.47/1.41 => ( ( ord_less_eq @ ( set @ A ) @ X6 @ A4 )
% 2.47/1.41 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ X6 ) @ A4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_181_less__eq__prod__def,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType] :
% 2.47/1.41 ( ( ( ord @ A )
% 2.47/1.41 & ( ord @ B ) )
% 2.47/1.41 => ( ( ord_less_eq @ ( product_prod @ A @ B ) )
% 2.47/1.41 = ( ^ [X3: product_prod @ A @ B,Y2: product_prod @ A @ B] :
% 2.47/1.41 ( ( ord_less_eq @ A @ ( product_fst @ A @ B @ X3 ) @ ( product_fst @ A @ B @ Y2 ) )
% 2.47/1.41 & ( ord_less_eq @ B @ ( product_snd @ A @ B @ X3 ) @ ( product_snd @ A @ B @ Y2 ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_182_singleton__inject,axiom,
% 2.47/1.41 ! [A: $tType,A2: A,B2: A] :
% 2.47/1.41 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 2.47/1.41 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.47/1.41 => ( A2 = B2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_183_insert__not__empty,axiom,
% 2.47/1.41 ! [A: $tType,A2: A,A4: set @ A] :
% 2.47/1.41 ( ( insert @ A @ A2 @ A4 )
% 2.47/1.41 != ( bot_bot @ ( set @ A ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_184_doubleton__eq__iff,axiom,
% 2.47/1.41 ! [A: $tType,A2: A,B2: A,C3: A,D3: A] :
% 2.47/1.41 ( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.47/1.41 = ( insert @ A @ C3 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.47/1.41 = ( ( ( A2 = C3 )
% 2.47/1.41 & ( B2 = D3 ) )
% 2.47/1.41 | ( ( A2 = D3 )
% 2.47/1.41 & ( B2 = C3 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_185_singleton__iff,axiom,
% 2.47/1.41 ! [A: $tType,B2: A,A2: A] :
% 2.47/1.41 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.47/1.41 = ( B2 = A2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_186_singletonD,axiom,
% 2.47/1.41 ! [A: $tType,B2: A,A2: A] :
% 2.47/1.41 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.47/1.41 => ( B2 = A2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_187_subset__insertI2,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A,B4: set @ A,B2: A] :
% 2.47/1.41 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.47/1.41 => ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_188_subset__insertI,axiom,
% 2.47/1.41 ! [A: $tType,B4: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( insert @ A @ A2 @ B4 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_189_subset__insert,axiom,
% 2.47/1.41 ! [A: $tType,X2: A,A4: set @ A,B4: set @ A] :
% 2.47/1.41 ( ~ ( member @ A @ X2 @ A4 )
% 2.47/1.41 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X2 @ B4 ) )
% 2.47/1.41 = ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_190_insert__mono,axiom,
% 2.47/1.41 ! [A: $tType,C4: set @ A,D4: set @ A,A2: A] :
% 2.47/1.41 ( ( ord_less_eq @ ( set @ A ) @ C4 @ D4 )
% 2.47/1.41 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C4 ) @ ( insert @ A @ A2 @ D4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_191_mk__disjoint__insert,axiom,
% 2.47/1.41 ! [A: $tType,A2: A,A4: set @ A] :
% 2.47/1.41 ( ( member @ A @ A2 @ A4 )
% 2.47/1.41 => ? [B8: set @ A] :
% 2.47/1.41 ( ( A4
% 2.47/1.41 = ( insert @ A @ A2 @ B8 ) )
% 2.47/1.41 & ~ ( member @ A @ A2 @ B8 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_192_insert__commute,axiom,
% 2.47/1.41 ! [A: $tType,X2: A,Y: A,A4: set @ A] :
% 2.47/1.41 ( ( insert @ A @ X2 @ ( insert @ A @ Y @ A4 ) )
% 2.47/1.41 = ( insert @ A @ Y @ ( insert @ A @ X2 @ A4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_193_insert__eq__iff,axiom,
% 2.47/1.41 ! [A: $tType,A2: A,A4: set @ A,B2: A,B4: set @ A] :
% 2.47/1.41 ( ~ ( member @ A @ A2 @ A4 )
% 2.47/1.41 => ( ~ ( member @ A @ B2 @ B4 )
% 2.47/1.41 => ( ( ( insert @ A @ A2 @ A4 )
% 2.47/1.41 = ( insert @ A @ B2 @ B4 ) )
% 2.47/1.41 = ( ( ( A2 = B2 )
% 2.47/1.41 => ( A4 = B4 ) )
% 2.47/1.41 & ( ( A2 != B2 )
% 2.47/1.41 => ? [C5: set @ A] :
% 2.47/1.41 ( ( A4
% 2.47/1.41 = ( insert @ A @ B2 @ C5 ) )
% 2.47/1.41 & ~ ( member @ A @ B2 @ C5 )
% 2.47/1.41 & ( B4
% 2.47/1.41 = ( insert @ A @ A2 @ C5 ) )
% 2.47/1.41 & ~ ( member @ A @ A2 @ C5 ) ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_194_insert__absorb,axiom,
% 2.47/1.41 ! [A: $tType,A2: A,A4: set @ A] :
% 2.47/1.41 ( ( member @ A @ A2 @ A4 )
% 2.47/1.41 => ( ( insert @ A @ A2 @ A4 )
% 2.47/1.41 = A4 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_195_insert__ident,axiom,
% 2.47/1.41 ! [A: $tType,X2: A,A4: set @ A,B4: set @ A] :
% 2.47/1.41 ( ~ ( member @ A @ X2 @ A4 )
% 2.47/1.41 => ( ~ ( member @ A @ X2 @ B4 )
% 2.47/1.41 => ( ( ( insert @ A @ X2 @ A4 )
% 2.47/1.41 = ( insert @ A @ X2 @ B4 ) )
% 2.47/1.41 = ( A4 = B4 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_196_Set_Oset__insert,axiom,
% 2.47/1.41 ! [A: $tType,X2: A,A4: set @ A] :
% 2.47/1.41 ( ( member @ A @ X2 @ A4 )
% 2.47/1.41 => ~ ! [B8: set @ A] :
% 2.47/1.41 ( ( A4
% 2.47/1.41 = ( insert @ A @ X2 @ B8 ) )
% 2.47/1.41 => ( member @ A @ X2 @ B8 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_197_insertI2,axiom,
% 2.47/1.41 ! [A: $tType,A2: A,B4: set @ A,B2: A] :
% 2.47/1.41 ( ( member @ A @ A2 @ B4 )
% 2.47/1.41 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_198_insertI1,axiom,
% 2.47/1.41 ! [A: $tType,A2: A,B4: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B4 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_199_insertE,axiom,
% 2.47/1.41 ! [A: $tType,A2: A,B2: A,A4: set @ A] :
% 2.47/1.41 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
% 2.47/1.41 => ( ( A2 != B2 )
% 2.47/1.41 => ( member @ A @ A2 @ A4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_200_prod__eq__iff,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType] :
% 2.47/1.41 ( ( ^ [Y4: product_prod @ A @ B,Z2: product_prod @ A @ B] : ( Y4 = Z2 ) )
% 2.47/1.41 = ( ^ [S3: product_prod @ A @ B,T2: product_prod @ A @ B] :
% 2.47/1.41 ( ( ( product_fst @ A @ B @ S3 )
% 2.47/1.41 = ( product_fst @ A @ B @ T2 ) )
% 2.47/1.41 & ( ( product_snd @ A @ B @ S3 )
% 2.47/1.41 = ( product_snd @ A @ B @ T2 ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_201_prod_Oexpand,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 2.47/1.41 ( ( ( ( product_fst @ A @ B @ Prod )
% 2.47/1.41 = ( product_fst @ A @ B @ Prod2 ) )
% 2.47/1.41 & ( ( product_snd @ A @ B @ Prod )
% 2.47/1.41 = ( product_snd @ A @ B @ Prod2 ) ) )
% 2.47/1.41 => ( Prod = Prod2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_202_prod__eqI,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 2.47/1.41 ( ( ( product_fst @ A @ B @ P2 )
% 2.47/1.41 = ( product_fst @ A @ B @ Q2 ) )
% 2.47/1.41 => ( ( ( product_snd @ A @ B @ P2 )
% 2.47/1.41 = ( product_snd @ A @ B @ Q2 ) )
% 2.47/1.41 => ( P2 = Q2 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_203_surjective__pairing,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,T3: product_prod @ A @ B] :
% 2.47/1.41 ( T3
% 2.47/1.41 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T3 ) @ ( product_snd @ A @ B @ T3 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_204_prod_Oexhaust__sel,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.47/1.41 ( Prod
% 2.47/1.41 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_205_fst__conv,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 2.47/1.41 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 2.47/1.41 = X1 ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_206_fst__eqD,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,X2: A,Y: B,A2: A] :
% 2.47/1.41 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y ) )
% 2.47/1.41 = A2 )
% 2.47/1.41 => ( X2 = A2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_207_snd__conv,axiom,
% 2.47/1.41 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 2.47/1.41 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 2.47/1.41 = X22 ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_208_snd__eqD,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,X2: B,Y: A,A2: A] :
% 2.47/1.41 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
% 2.47/1.41 = A2 )
% 2.47/1.41 => ( Y = A2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_209_refl__on__singleton,axiom,
% 2.47/1.41 ! [A: $tType,X2: A] : ( refl_on @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_210_total__on__singleton,axiom,
% 2.47/1.41 ! [A: $tType,X2: A] : ( total_on @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_211_subset__singletonD,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A,X2: A] :
% 2.47/1.41 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.47/1.41 => ( ( A4
% 2.47/1.41 = ( bot_bot @ ( set @ A ) ) )
% 2.47/1.41 | ( A4
% 2.47/1.41 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_212_subset__singleton__iff,axiom,
% 2.47/1.41 ! [A: $tType,X6: set @ A,A2: A] :
% 2.47/1.41 ( ( ord_less_eq @ ( set @ A ) @ X6 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.47/1.41 = ( ( X6
% 2.47/1.41 = ( bot_bot @ ( set @ A ) ) )
% 2.47/1.41 | ( X6
% 2.47/1.41 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_213_snd__mono,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType] :
% 2.47/1.41 ( ( ( ord @ A )
% 2.47/1.41 & ( ord @ B ) )
% 2.47/1.41 => ! [X2: product_prod @ A @ B,Y: product_prod @ A @ B] :
% 2.47/1.41 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ X2 @ Y )
% 2.47/1.41 => ( ord_less_eq @ B @ ( product_snd @ A @ B @ X2 ) @ ( product_snd @ A @ B @ Y ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_214_fst__mono,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType] :
% 2.47/1.41 ( ( ( ord @ A )
% 2.47/1.41 & ( ord @ B ) )
% 2.47/1.41 => ! [X2: product_prod @ A @ B,Y: product_prod @ A @ B] :
% 2.47/1.41 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ X2 @ Y )
% 2.47/1.41 => ( ord_less_eq @ A @ ( product_fst @ A @ B @ X2 ) @ ( product_fst @ A @ B @ Y ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_215_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,P: A > B > $o,X2: A,Y: B,A2: product_prod @ A @ B] :
% 2.47/1.41 ( ( P @ X2 @ Y )
% 2.47/1.41 => ( ( A2
% 2.47/1.41 = ( product_Pair @ A @ B @ X2 @ Y ) )
% 2.47/1.41 => ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_216_conjI__realizer,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q2: B] :
% 2.47/1.41 ( ( P @ P2 )
% 2.47/1.41 => ( ( Q @ Q2 )
% 2.47/1.41 => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) )
% 2.47/1.41 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_217_exI__realizer,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,P: A > B > $o,Y: A,X2: B] :
% 2.47/1.41 ( ( P @ Y @ X2 )
% 2.47/1.41 => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_218_the__elem__eq,axiom,
% 2.47/1.41 ! [A: $tType,X2: A] :
% 2.47/1.41 ( ( the_elem @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.47/1.41 = X2 ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_219_is__singletonI,axiom,
% 2.47/1.41 ! [A: $tType,X2: A] : ( is_singleton @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_220_is__singleton__the__elem,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( is_singleton @ A )
% 2.47/1.41 = ( ^ [A5: set @ A] :
% 2.47/1.41 ( A5
% 2.47/1.41 = ( insert @ A @ ( the_elem @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_221_is__singletonI_H,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A] :
% 2.47/1.41 ( ( A4
% 2.47/1.41 != ( bot_bot @ ( set @ A ) ) )
% 2.47/1.41 => ( ! [X4: A,Y3: A] :
% 2.47/1.41 ( ( member @ A @ X4 @ A4 )
% 2.47/1.41 => ( ( member @ A @ Y3 @ A4 )
% 2.47/1.41 => ( X4 = Y3 ) ) )
% 2.47/1.41 => ( is_singleton @ A @ A4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_222_is__singletonE,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A] :
% 2.47/1.41 ( ( is_singleton @ A @ A4 )
% 2.47/1.41 => ~ ! [X4: A] :
% 2.47/1.41 ( A4
% 2.47/1.41 != ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_223_is__singleton__def,axiom,
% 2.47/1.41 ! [A: $tType] :
% 2.47/1.41 ( ( is_singleton @ A )
% 2.47/1.41 = ( ^ [A5: set @ A] :
% 2.47/1.41 ? [X3: A] :
% 2.47/1.41 ( A5
% 2.47/1.41 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_224_linear__order__on__singleton,axiom,
% 2.47/1.41 ! [A: $tType,X2: A] : ( order_1409979114der_on @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_225_sndI,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,X2: product_prod @ A @ B,Y: A,Z: B] :
% 2.47/1.41 ( ( X2
% 2.47/1.41 = ( product_Pair @ A @ B @ Y @ Z ) )
% 2.47/1.41 => ( ( product_snd @ A @ B @ X2 )
% 2.47/1.41 = Z ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_226_lnear__order__on__empty,axiom,
% 2.47/1.41 ! [A: $tType] : ( order_1409979114der_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_227_fstI,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,X2: product_prod @ A @ B,Y: A,Z: B] :
% 2.47/1.41 ( ( X2
% 2.47/1.41 = ( product_Pair @ A @ B @ Y @ Z ) )
% 2.47/1.41 => ( ( product_fst @ A @ B @ X2 )
% 2.47/1.41 = Y ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_228_eq__snd__iff,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
% 2.47/1.41 ( ( B2
% 2.47/1.41 = ( product_snd @ B @ A @ P2 ) )
% 2.47/1.41 = ( ? [A7: B] :
% 2.47/1.41 ( P2
% 2.47/1.41 = ( product_Pair @ B @ A @ A7 @ B2 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_229_eq__fst__iff,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,A2: A,P2: product_prod @ A @ B] :
% 2.47/1.41 ( ( A2
% 2.47/1.41 = ( product_fst @ A @ B @ P2 ) )
% 2.47/1.41 = ( ? [B7: B] :
% 2.47/1.41 ( P2
% 2.47/1.41 = ( product_Pair @ A @ B @ A2 @ B7 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_230_prod_Oswap__def,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType] :
% 2.47/1.41 ( ( product_swap @ A @ B )
% 2.47/1.41 = ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_231_Range__insert,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,A2: B,B2: A,R: set @ ( product_prod @ B @ A )] :
% 2.47/1.41 ( ( range @ B @ A @ ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A2 @ B2 ) @ R ) )
% 2.47/1.41 = ( insert @ A @ B2 @ ( range @ B @ A @ R ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_232_swap__swap,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 2.47/1.41 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 2.47/1.41 = P2 ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_233_swap__simp,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,X2: B,Y: A] :
% 2.47/1.41 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
% 2.47/1.41 = ( product_Pair @ A @ B @ Y @ X2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_234_Range__Id__on,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A] :
% 2.47/1.41 ( ( range @ A @ A @ ( id_on @ A @ A4 ) )
% 2.47/1.41 = A4 ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_235_Range__empty,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType] :
% 2.47/1.41 ( ( range @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.47/1.41 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_236_fst__swap,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,X2: product_prod @ B @ A] :
% 2.47/1.41 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X2 ) )
% 2.47/1.41 = ( product_snd @ B @ A @ X2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_237_snd__swap,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,X2: product_prod @ A @ B] :
% 2.47/1.41 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X2 ) )
% 2.47/1.41 = ( product_fst @ A @ B @ X2 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_238_RangeE,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A )] :
% 2.47/1.41 ( ( member @ A @ B2 @ ( range @ B @ A @ R ) )
% 2.47/1.41 => ~ ! [A6: B] :
% 2.47/1.41 ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A6 @ B2 ) @ R ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_239_Range__iff,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,A2: A,R: set @ ( product_prod @ B @ A )] :
% 2.47/1.41 ( ( member @ A @ A2 @ ( range @ B @ A @ R ) )
% 2.47/1.41 = ( ? [Y2: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y2 @ A2 ) @ R ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_240_Range_Ocases,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,A2: B,R: set @ ( product_prod @ A @ B )] :
% 2.47/1.41 ( ( member @ B @ A2 @ ( range @ A @ B @ R ) )
% 2.47/1.41 => ~ ! [A6: A] :
% 2.47/1.41 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ A2 ) @ R ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_241_Range_Osimps,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,A2: B,R: set @ ( product_prod @ A @ B )] :
% 2.47/1.41 ( ( member @ B @ A2 @ ( range @ A @ B @ R ) )
% 2.47/1.41 = ( ? [A7: A,B7: B] :
% 2.47/1.41 ( ( A2 = B7 )
% 2.47/1.41 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B7 ) @ R ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_242_Range_Ointros,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,A2: A,B2: B,R: set @ ( product_prod @ A @ B )] :
% 2.47/1.41 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R )
% 2.47/1.41 => ( member @ B @ B2 @ ( range @ A @ B @ R ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_243_Range_Oinducts,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,X2: B,R: set @ ( product_prod @ A @ B ),P: B > $o] :
% 2.47/1.41 ( ( member @ B @ X2 @ ( range @ A @ B @ R ) )
% 2.47/1.41 => ( ! [A6: A,B5: B] :
% 2.47/1.41 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ R )
% 2.47/1.41 => ( P @ B5 ) )
% 2.47/1.41 => ( P @ X2 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_244_Range__empty__iff,axiom,
% 2.47/1.41 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A )] :
% 2.47/1.41 ( ( ( range @ B @ A @ R )
% 2.47/1.41 = ( bot_bot @ ( set @ A ) ) )
% 2.47/1.41 = ( R
% 2.47/1.41 = ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_245_Range__mono,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 2.47/1.41 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S )
% 2.47/1.41 => ( ord_less_eq @ ( set @ B ) @ ( range @ A @ B @ R ) @ ( range @ A @ B @ S ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_246_Domain__insert,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType,A2: A,B2: B,R: set @ ( product_prod @ A @ B )] :
% 2.47/1.41 ( ( domain @ A @ B @ ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R ) )
% 2.47/1.41 = ( insert @ A @ A2 @ ( domain @ A @ B @ R ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_247_subset__Compl__singleton,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A,B2: A] :
% 2.47/1.41 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.47/1.41 = ( ~ ( member @ A @ B2 @ A4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_248_Compl__eq__Compl__iff,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.47/1.41 ( ( ( uminus_uminus @ ( set @ A ) @ A4 )
% 2.47/1.41 = ( uminus_uminus @ ( set @ A ) @ B4 ) )
% 2.47/1.41 = ( A4 = B4 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_249_Compl__iff,axiom,
% 2.47/1.41 ! [A: $tType,C3: A,A4: set @ A] :
% 2.47/1.41 ( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
% 2.47/1.41 = ( ~ ( member @ A @ C3 @ A4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_250_ComplI,axiom,
% 2.47/1.41 ! [A: $tType,C3: A,A4: set @ A] :
% 2.47/1.41 ( ~ ( member @ A @ C3 @ A4 )
% 2.47/1.41 => ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_251_Compl__anti__mono,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.47/1.41 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.47/1.41 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_252_Compl__subset__Compl__iff,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.47/1.41 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) )
% 2.47/1.41 = ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_253_Domain__Id__on,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A] :
% 2.47/1.41 ( ( domain @ A @ A @ ( id_on @ A @ A4 ) )
% 2.47/1.41 = A4 ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_254_Domain__empty,axiom,
% 2.47/1.41 ! [B: $tType,A: $tType] :
% 2.47/1.41 ( ( domain @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.47/1.41 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(fact_255_subset__Compl__self__eq,axiom,
% 2.47/1.41 ! [A: $tType,A4: set @ A] :
% 2.47/1.41 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
% 2.47/1.41 = ( A4
% 2.47/1.41 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 2.47/1.41 ! [A8: $tType,A9: $tType] :
% 2.47/1.41 ( ( order_bot @ A9 )
% 2.47/1.41 => ( order_bot @ ( A8 > A9 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.47/1.41 ! [A8: $tType,A9: $tType] :
% 2.47/1.41 ( ( preorder @ A9 )
% 2.47/1.41 => ( preorder @ ( A8 > A9 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.47/1.41 ! [A8: $tType,A9: $tType] :
% 2.47/1.41 ( ( order @ A9 )
% 2.47/1.41 => ( order @ ( A8 > A9 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_fun___Orderings_Oord,axiom,
% 2.47/1.41 ! [A8: $tType,A9: $tType] :
% 2.47/1.41 ( ( ord @ A9 )
% 2.47/1.41 => ( ord @ ( A8 > A9 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_fun___Orderings_Obot,axiom,
% 2.47/1.41 ! [A8: $tType,A9: $tType] :
% 2.47/1.41 ( ( bot @ A9 )
% 2.47/1.41 => ( bot @ ( A8 > A9 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_Set_Oset___Orderings_Oorder__bot_1,axiom,
% 2.47/1.41 ! [A8: $tType] : ( order_bot @ ( set @ A8 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
% 2.47/1.41 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
% 2.47/1.41 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_Set_Oset___Orderings_Oord_4,axiom,
% 2.47/1.41 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_Set_Oset___Orderings_Obot_5,axiom,
% 2.47/1.41 ! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_HOL_Obool___Orderings_Oorder__bot_6,axiom,
% 2.47/1.41 order_bot @ $o ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 2.47/1.41 preorder @ $o ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.47/1.41 linorder @ $o ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_HOL_Obool___Orderings_Oorder_8,axiom,
% 2.47/1.41 order @ $o ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_HOL_Obool___Orderings_Oord_9,axiom,
% 2.47/1.41 ord @ $o ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_HOL_Obool___Orderings_Obot_10,axiom,
% 2.47/1.41 bot @ $o ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_Product__Type_Oprod___Orderings_Oorder__bot_11,axiom,
% 2.47/1.41 ! [A8: $tType,A9: $tType] :
% 2.47/1.41 ( ( ( order_bot @ A8 )
% 2.47/1.41 & ( order_bot @ A9 ) )
% 2.47/1.41 => ( order_bot @ ( product_prod @ A8 @ A9 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_12,axiom,
% 2.47/1.41 ! [A8: $tType,A9: $tType] :
% 2.47/1.41 ( ( ( preorder @ A8 )
% 2.47/1.41 & ( preorder @ A9 ) )
% 2.47/1.41 => ( preorder @ ( product_prod @ A8 @ A9 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_Product__Type_Oprod___Orderings_Oorder_13,axiom,
% 2.47/1.41 ! [A8: $tType,A9: $tType] :
% 2.47/1.41 ( ( ( order @ A8 )
% 2.47/1.41 & ( order @ A9 ) )
% 2.47/1.41 => ( order @ ( product_prod @ A8 @ A9 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_Product__Type_Oprod___Orderings_Oord_14,axiom,
% 2.47/1.41 ! [A8: $tType,A9: $tType] :
% 2.47/1.41 ( ( ( ord @ A8 )
% 2.47/1.41 & ( ord @ A9 ) )
% 2.47/1.41 => ( ord @ ( product_prod @ A8 @ A9 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(tcon_Product__Type_Oprod___Orderings_Obot_15,axiom,
% 2.47/1.41 ! [A8: $tType,A9: $tType] :
% 2.47/1.41 ( ( ( bot @ A8 )
% 2.47/1.41 & ( bot @ A9 ) )
% 2.47/1.41 => ( bot @ ( product_prod @ A8 @ A9 ) ) ) ).
% 2.47/1.41
% 2.47/1.41 thf(conj_0,conjecture,
% 2.47/1.41 ( ( member @ a @ z @ carrier )
% 2.47/1.41 & ( member @ a @ y @ carrier ) ) ).
% 2.47/1.41 ------------------------
% 2.47/1.43 % (29552)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_29341 for (2999ds/27Mi)
% 2.47/1.43 % (29550)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_29341 for (2999ds/183Mi)
% 2.47/1.43 % (29556)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_29341 for (2999ds/18Mi)
% 2.47/1.43 % (29554)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_29341 for (2999ds/2Mi)
% 2.47/1.43 % (29551)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_29341 for (2999ds/4Mi)
% 2.47/1.43 % (29555)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_29341 for (2999ds/275Mi)
% 2.47/1.43 % (29554)Instruction limit reached!
% 2.47/1.43 % (29554)------------------------------
% 2.47/1.43 % (29554)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.43 % (29554)Termination reason: Unknown
% 2.47/1.43 % (29554)Termination phase: shuffling
% 2.47/1.43
% 2.47/1.43 % (29554)Memory used [KB]: 1407
% 2.47/1.43 % (29554)Time elapsed: 0.003 s
% 2.47/1.43 % (29554)Instructions burned: 4 (million)
% 2.47/1.43 % (29554)------------------------------
% 2.47/1.43 % (29554)------------------------------
% 2.47/1.43 % (29551)Instruction limit reached!
% 2.47/1.43 % (29551)------------------------------
% 2.47/1.43 % (29551)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.43 % (29551)Termination reason: Unknown
% 2.47/1.43 % (29551)Termination phase: shuffling
% 2.47/1.43
% 2.47/1.43 % (29551)Memory used [KB]: 1407
% 2.47/1.43 % (29551)Time elapsed: 0.004 s
% 2.47/1.43 % (29551)Instructions burned: 7 (million)
% 2.47/1.43 % (29551)------------------------------
% 2.47/1.43 % (29551)------------------------------
% 2.47/1.43 % (29556)Instruction limit reached!
% 2.47/1.43 % (29556)------------------------------
% 2.47/1.43 % (29556)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.43 % (29556)Termination reason: Unknown
% 2.47/1.43 % (29556)Termination phase: shuffling
% 2.47/1.43
% 2.47/1.43 % (29556)Memory used [KB]: 1663
% 2.47/1.43 % (29556)Time elapsed: 0.008 s
% 2.47/1.43 % (29556)Instructions burned: 18 (million)
% 2.47/1.43 % (29556)------------------------------
% 2.47/1.43 % (29556)------------------------------
% 2.47/1.43 % (29553)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_29341 for (2999ds/2Mi)
% 2.47/1.44 % (29552)Instruction limit reached!
% 2.47/1.44 % (29552)------------------------------
% 2.47/1.44 % (29552)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.44 % (29552)Termination reason: Unknown
% 2.47/1.44 % (29552)Termination phase: Property scanning
% 2.47/1.44
% 2.47/1.44 % (29552)Memory used [KB]: 1791
% 2.47/1.44 % (29552)Time elapsed: 0.011 s
% 2.47/1.44 % (29552)Instructions burned: 27 (million)
% 2.47/1.44 % (29552)------------------------------
% 2.47/1.44 % (29552)------------------------------
% 2.47/1.44 % (29553)Instruction limit reached!
% 2.47/1.44 % (29553)------------------------------
% 2.47/1.44 % (29553)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.44 % (29553)Termination reason: Unknown
% 2.47/1.44 % (29553)Termination phase: shuffling
% 2.47/1.44
% 2.47/1.44 % (29553)Memory used [KB]: 1407
% 2.47/1.44 % (29553)Time elapsed: 0.004 s
% 2.47/1.44 % (29553)Instructions burned: 3 (million)
% 2.47/1.44 % (29553)------------------------------
% 2.47/1.44 % (29553)------------------------------
% 2.47/1.44 % (29564)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_29341 for (2999ds/3Mi)
% 2.47/1.44 % (29565)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_29341 for (2999ds/37Mi)
% 2.47/1.44 % (29564)Instruction limit reached!
% 2.47/1.44 % (29564)------------------------------
% 2.47/1.44 % (29564)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.44 % (29564)Termination reason: Unknown
% 2.47/1.44 % (29564)Termination phase: shuffling
% 2.47/1.44
% 2.47/1.44 % (29564)Memory used [KB]: 1407
% 2.47/1.44 % (29564)Time elapsed: 0.002 s
% 2.47/1.44 % (29564)Instructions burned: 3 (million)
% 2.47/1.44 % (29564)------------------------------
% 2.47/1.44 % (29564)------------------------------
% 2.47/1.44 % (29568)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_29341 for (2999ds/15Mi)
% 2.47/1.45 % (29569)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_29341 for (2999ds/3Mi)
% 2.47/1.45 % (29569)Instruction limit reached!
% 2.47/1.45 % (29569)------------------------------
% 2.47/1.45 % (29569)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.45 % (29569)Termination reason: Unknown
% 2.47/1.45 % (29569)Termination phase: shuffling
% 2.47/1.45
% 2.47/1.45 % (29569)Memory used [KB]: 1407
% 2.47/1.45 % (29569)Time elapsed: 0.002 s
% 2.47/1.45 % (29569)Instructions burned: 3 (million)
% 2.47/1.45 % (29569)------------------------------
% 2.47/1.45 % (29569)------------------------------
% 2.47/1.45 % (29555)Refutation not found, incomplete strategy
% 2.47/1.45 % (29555)------------------------------
% 2.47/1.45 % (29555)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.45 % (29555)Termination reason: Refutation not found, incomplete strategy
% 2.47/1.45
% 2.47/1.45
% 2.47/1.45 % (29555)Memory used [KB]: 6652
% 2.47/1.45 % (29555)Time elapsed: 0.022 s
% 2.47/1.45 % (29555)Instructions burned: 66 (million)
% 2.47/1.45 % (29555)------------------------------
% 2.47/1.45 % (29555)------------------------------
% 2.47/1.45 % (29568)Instruction limit reached!
% 2.47/1.45 % (29568)------------------------------
% 2.47/1.45 % (29568)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.45 % (29568)Termination reason: Unknown
% 2.47/1.45 % (29568)Termination phase: shuffling
% 2.47/1.45
% 2.47/1.45 % (29568)Memory used [KB]: 1535
% 2.47/1.45 % (29568)Time elapsed: 0.006 s
% 2.47/1.45 % (29568)Instructions burned: 15 (million)
% 2.47/1.45 % (29568)------------------------------
% 2.47/1.45 % (29568)------------------------------
% 2.47/1.45 % (29571)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_29341 for (2999ds/7Mi)
% 2.47/1.45 % (29565)Instruction limit reached!
% 2.47/1.45 % (29565)------------------------------
% 2.47/1.45 % (29565)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.45 % (29565)Termination reason: Unknown
% 2.47/1.45 % (29565)Termination phase: shuffling
% 2.47/1.45
% 2.47/1.45 % (29565)Memory used [KB]: 2174
% 2.47/1.45 % (29565)Time elapsed: 0.012 s
% 2.47/1.45 % (29565)Instructions burned: 39 (million)
% 2.47/1.45 % (29565)------------------------------
% 2.47/1.45 % (29565)------------------------------
% 2.47/1.45 % (29571)Instruction limit reached!
% 2.47/1.45 % (29571)------------------------------
% 2.47/1.45 % (29571)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.45 % (29571)Termination reason: Unknown
% 2.47/1.45 % (29571)Termination phase: shuffling
% 2.47/1.45
% 2.47/1.45 % (29571)Memory used [KB]: 1535
% 2.47/1.45 % (29571)Time elapsed: 0.004 s
% 2.47/1.45 % (29571)Instructions burned: 9 (million)
% 2.47/1.45 % (29571)------------------------------
% 2.47/1.45 % (29571)------------------------------
% 2.47/1.45 % (29570)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_29341 for (2999ds/1041Mi)
% 2.47/1.46 % (29572)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_29341 for (2999ds/16Mi)
% 2.47/1.46 % (29573)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_29341 for (2999ds/3Mi)
% 2.47/1.46 % (29574)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_29341 for (2999ds/3Mi)
% 2.47/1.46 % (29573)Instruction limit reached!
% 2.47/1.46 % (29573)------------------------------
% 2.47/1.46 % (29573)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.46 % (29573)Termination reason: Unknown
% 2.47/1.46 % (29573)Termination phase: shuffling
% 2.47/1.46
% 2.47/1.46 % (29573)Memory used [KB]: 1407
% 2.47/1.46 % (29573)Time elapsed: 0.002 s
% 2.47/1.46 % (29573)Instructions burned: 3 (million)
% 2.47/1.46 % (29573)------------------------------
% 2.47/1.46 % (29573)------------------------------
% 2.47/1.46 % (29574)Instruction limit reached!
% 2.47/1.46 % (29574)------------------------------
% 2.47/1.46 % (29574)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.46 % (29574)Termination reason: Unknown
% 2.47/1.46 % (29574)Termination phase: shuffling
% 2.47/1.46
% 2.47/1.46 % (29574)Memory used [KB]: 1407
% 2.47/1.46 % (29574)Time elapsed: 0.002 s
% 2.47/1.46 % (29574)Instructions burned: 3 (million)
% 2.47/1.46 % (29574)------------------------------
% 2.47/1.46 % (29574)------------------------------
% 2.47/1.46 % (29575)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_29341 for (2999ds/7Mi)
% 2.47/1.46 % (29578)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_29341 for (2999ds/3Mi)
% 2.47/1.46 % (29578)Instruction limit reached!
% 2.47/1.46 % (29578)------------------------------
% 2.47/1.46 % (29578)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.46 % (29578)Termination reason: Unknown
% 2.47/1.46 % (29578)Termination phase: shuffling
% 2.47/1.46
% 2.47/1.46 % (29578)Memory used [KB]: 1407
% 2.47/1.46 % (29578)Time elapsed: 0.002 s
% 2.47/1.46 % (29578)Instructions burned: 3 (million)
% 2.47/1.46 % (29578)------------------------------
% 2.47/1.46 % (29578)------------------------------
% 2.47/1.46 % (29575)Instruction limit reached!
% 2.47/1.46 % (29575)------------------------------
% 2.47/1.46 % (29575)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.46 % (29575)Termination reason: Unknown
% 2.47/1.46 % (29575)Termination phase: shuffling
% 2.47/1.46
% 2.47/1.46 % (29575)Memory used [KB]: 1535
% 2.47/1.46 % (29575)Time elapsed: 0.004 s
% 2.47/1.46 % (29575)Instructions burned: 9 (million)
% 2.47/1.46 % (29575)------------------------------
% 2.47/1.46 % (29575)------------------------------
% 2.47/1.46 % (29572)Instruction limit reached!
% 2.47/1.46 % (29572)------------------------------
% 2.47/1.46 % (29572)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.46 % (29572)Termination reason: Unknown
% 2.47/1.46 % (29572)Termination phase: shuffling
% 2.47/1.46
% 2.47/1.46 % (29572)Memory used [KB]: 1663
% 2.47/1.46 % (29572)Time elapsed: 0.007 s
% 2.47/1.46 % (29572)Instructions burned: 18 (million)
% 2.47/1.46 % (29572)------------------------------
% 2.47/1.46 % (29572)------------------------------
% 2.47/1.47 % (29587)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_29341 for (2999ds/4Mi)
% 2.47/1.47 % (29588)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_29341 for (2999ds/18Mi)
% 2.47/1.47 % (29587)Instruction limit reached!
% 2.47/1.47 % (29587)------------------------------
% 2.47/1.47 % (29587)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.47 % (29587)Termination reason: Unknown
% 2.47/1.47 % (29587)Termination phase: shuffling
% 2.47/1.47
% 2.47/1.47 % (29587)Memory used [KB]: 1407
% 2.47/1.47 % (29587)Time elapsed: 0.003 s
% 2.47/1.47 % (29587)Instructions burned: 6 (million)
% 2.47/1.47 % (29587)------------------------------
% 2.47/1.47 % (29587)------------------------------
% 2.47/1.47 % (29590)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_29341 for (2999ds/710Mi)
% 2.47/1.47 % (29591)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_29341 for (2999ds/6Mi)
% 2.47/1.47 % (29592)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_29341 for (2999ds/902Mi)
% 2.47/1.47 % (29591)Instruction limit reached!
% 2.47/1.47 % (29591)------------------------------
% 2.47/1.47 % (29591)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.47 % (29591)Termination reason: Unknown
% 2.47/1.47 % (29591)Termination phase: shuffling
% 2.47/1.47
% 2.47/1.47 % (29591)Memory used [KB]: 1407
% 2.47/1.47 % (29591)Time elapsed: 0.003 s
% 2.47/1.47 % (29591)Instructions burned: 6 (million)
% 2.47/1.47 % (29591)------------------------------
% 2.47/1.47 % (29591)------------------------------
% 2.47/1.47 % (29588)Instruction limit reached!
% 2.47/1.47 % (29588)------------------------------
% 2.47/1.47 % (29588)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.47 % (29588)Termination reason: Unknown
% 2.47/1.47 % (29588)Termination phase: shuffling
% 2.47/1.47
% 2.47/1.47 % (29588)Memory used [KB]: 1663
% 2.47/1.47 % (29588)Time elapsed: 0.007 s
% 2.47/1.47 % (29588)Instructions burned: 18 (million)
% 2.47/1.47 % (29588)------------------------------
% 2.47/1.47 % (29588)------------------------------
% 2.47/1.48 % (29597)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_29341 for (2999ds/21Mi)
% 2.47/1.48 % (29550)Instruction limit reached!
% 2.47/1.48 % (29550)------------------------------
% 2.47/1.48 % (29550)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.48 % (29550)Termination reason: Unknown
% 2.47/1.48 % (29550)Termination phase: Saturation
% 2.47/1.48
% 2.47/1.48 % (29550)Memory used [KB]: 7164
% 2.47/1.48 % (29550)Time elapsed: 0.054 s
% 2.47/1.48 % (29550)Instructions burned: 186 (million)
% 2.47/1.48 % (29550)------------------------------
% 2.47/1.48 % (29550)------------------------------
% 2.47/1.48 % (29598)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_29341 for (2999ds/5Mi)
% 2.47/1.48 % (29599)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_29341 for (2999ds/6Mi)
% 2.47/1.48 % (29598)Instruction limit reached!
% 2.47/1.48 % (29598)------------------------------
% 2.47/1.48 % (29598)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.48 % (29598)Termination reason: Unknown
% 2.47/1.48 % (29598)Termination phase: shuffling
% 2.47/1.48
% 2.47/1.48 % (29598)Memory used [KB]: 1407
% 2.47/1.48 % (29598)Time elapsed: 0.003 s
% 2.47/1.48 % (29598)Instructions burned: 6 (million)
% 2.47/1.48 % (29598)------------------------------
% 2.47/1.48 % (29598)------------------------------
% 2.47/1.48 % (29599)Instruction limit reached!
% 2.47/1.48 % (29599)------------------------------
% 2.47/1.48 % (29599)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.48 % (29599)Termination reason: Unknown
% 2.47/1.48 % (29599)Termination phase: shuffling
% 2.47/1.48
% 2.47/1.48 % (29599)Memory used [KB]: 1407
% 2.47/1.48 % (29599)Time elapsed: 0.003 s
% 2.47/1.48 % (29599)Instructions burned: 6 (million)
% 2.47/1.48 % (29599)------------------------------
% 2.47/1.48 % (29599)------------------------------
% 2.47/1.49 % (29597)Instruction limit reached!
% 2.47/1.49 % (29597)------------------------------
% 2.47/1.49 % (29597)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.49 % (29597)Termination reason: Unknown
% 2.47/1.49 % (29597)Termination phase: shuffling
% 2.47/1.49
% 2.47/1.49 % (29597)Memory used [KB]: 1791
% 2.47/1.49 % (29597)Time elapsed: 0.009 s
% 2.47/1.49 % (29597)Instructions burned: 23 (million)
% 2.47/1.49 % (29597)------------------------------
% 2.47/1.49 % (29597)------------------------------
% 2.47/1.49 % (29590)First to succeed.
% 2.47/1.49 % (29590)Refutation found. Thanks to Tanya!
% 2.47/1.49 % SZS status Theorem for DTF2THF_29341
% 2.47/1.49 % SZS output start Proof for DTF2THF_29341
% 2.47/1.49 thf(type_def_5, type, product_prod: ($tType * $tType) > $tType).
% 2.47/1.49 thf(type_def_6, type, set: ($tType) > $tType).
% 2.47/1.49 thf(type_def_8, type, a: $tType).
% 2.47/1.49 thf(func_def_0, type, product_prod: ($tType * $tType) > $tType).
% 2.47/1.49 thf(func_def_1, type, set: ($tType) > $tType).
% 2.47/1.49 thf(func_def_2, type, a: $tType).
% 2.47/1.49 thf(func_def_3, type, type: !>[X0: $tType]:($o)).
% 2.47/1.49 thf(func_def_4, type, bot: !>[X0: $tType]:($o)).
% 2.47/1.49 thf(func_def_5, type, ord: !>[X0: $tType]:($o)).
% 2.47/1.49 thf(func_def_6, type, order: !>[X0: $tType]:($o)).
% 2.47/1.49 thf(func_def_7, type, linorder: !>[X0: $tType]:($o)).
% 2.47/1.49 thf(func_def_8, type, preorder: !>[X0: $tType]:($o)).
% 2.47/1.49 thf(func_def_9, type, order_bot: !>[X0: $tType]:($o)).
% 2.47/1.49 thf(func_def_10, type, bNF_Ca1785829860lChain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 2.47/1.49 thf(func_def_11, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 2.47/1.49 thf(func_def_12, type, order_1409979114der_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 2.47/1.49 thf(func_def_13, type, order_preorder_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 2.47/1.49 thf(func_def_14, type, order_under: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0 > set @ X0)).
% 2.47/1.49 thf(func_def_15, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.47/1.49 thf(func_def_16, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.47/1.49 thf(func_def_17, type, order_Greatest: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.47/1.49 thf(func_def_18, type, prefer951318096ood_as: !>[X0: $tType]:(X0 > set @ X0 > set @ product_prod @ X0 @ X0 > set @ X0)).
% 2.47/1.49 thf(func_def_19, type, prefer310429814s_good: !>[X0: $tType]:(X0 > set @ X0 > set @ product_prod @ X0 @ X0 > set @ X0)).
% 2.47/1.49 thf(func_def_20, type, prefer1532642881r_than: !>[X0: $tType]:(X0 > set @ X0 > set @ product_prod @ X0 @ X0 > set @ X0)).
% 2.47/1.49 thf(func_def_21, type, prefer199794634erence: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 2.47/1.49 thf(func_def_22, type, prefer1997167224erence: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 2.47/1.49 thf(func_def_23, type, prefer1801827867axioms: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 2.47/1.49 thf(func_def_24, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 2.47/1.49 thf(func_def_25, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.47/1.49 thf(func_def_26, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.47/1.49 thf(func_def_27, type, product_fst: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.47/1.49 thf(func_def_28, type, product_snd: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.47/1.49 thf(func_def_29, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 2.47/1.49 thf(func_def_30, type, domain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X0)).
% 2.47/1.49 thf(func_def_31, type, id_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0)).
% 2.47/1.49 thf(func_def_32, type, range: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X1)).
% 2.47/1.49 thf(func_def_33, type, refl_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 2.47/1.49 thf(func_def_34, type, total_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 2.47/1.49 thf(func_def_35, type, trans: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 2.47/1.49 thf(func_def_36, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.47/1.49 thf(func_def_37, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 2.47/1.49 thf(func_def_38, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 2.47/1.49 thf(func_def_39, type, is_singleton: !>[X0: $tType]:(set @ X0 > $o)).
% 2.47/1.49 thf(func_def_40, type, the_elem: !>[X0: $tType]:(set @ X0 > X0)).
% 2.47/1.49 thf(func_def_41, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.47/1.49 thf(func_def_42, type, carrier: set @ a).
% 2.47/1.49 thf(func_def_43, type, relation: set @ product_prod @ a @ a).
% 2.47/1.49 thf(func_def_44, type, y: a).
% 2.47/1.49 thf(func_def_45, type, z: a).
% 2.47/1.49 thf(func_def_61, type, sK0: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.47/1.49 thf(f810,plain,(
% 2.47/1.49 $false),
% 2.47/1.49 inference(avatar_sat_refutation,[status(thm)],[f801,f808,f809])).
% 2.47/1.49 thf(f809,plain,(
% 2.47/1.49 spl1_2),
% 2.47/1.49 inference(avatar_split_clause,[status(thm)],[f805,f798])).
% 2.47/1.49 thf(f798,definition,(
% 2.47/1.49 spl1_2 <=> ((member @ a @ y @ carrier) = $true)),
% 2.47/1.49 introduced(definition,[new_symbols(naming,[spl1_2])],[avatar_definition])).
% 2.47/1.49 thf(f805,plain,(
% 2.47/1.49 ((member @ a @ y @ carrier) = $true)),
% 2.47/1.49 inference(trivial_inequality_removal,[status(thm)],[f804])).
% 2.47/1.49 thf(f804,plain,(
% 2.47/1.49 ((member @ a @ y @ carrier) = $true) | ($true != $true)),
% 2.47/1.49 inference(superposition,[status(thm)],[f788,f779])).
% 2.47/1.49 thf(f779,plain,(
% 2.47/1.49 ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ y @ z) @ relation) = $true)),
% 2.47/1.49 inference(cnf_transformation,[status(thm)],[f442])).
% 2.47/1.49 thf(f442,plain,(
% 2.47/1.49 ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ y @ z) @ relation) = $true) & ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ z @ y) @ relation) = $true)),
% 2.47/1.49 inference(fool_elimination,[status(thm)],[f441])).
% 2.47/1.49 thf(f441,plain,(
% 2.47/1.49 (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ z @ y) @ relation) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ y @ z) @ relation)),
% 2.47/1.49 inference(rectify,[status(thm)],[f1])).
% 2.47/1.49 thf(f1,axiom,(
% 2.47/1.49 (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ z @ y) @ relation) & (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ y @ z) @ relation)),
% 2.47/1.49 file('/export/starexec/sandbox/tmp/tmp.xCdPegwIoG/DTF2THF_29341.p',fact_0_assms)).
% 2.47/1.49 thf(f788,plain,(
% 2.47/1.49 ( ! [X0 : a,X1 : a] : (($true != (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ X0 @ X1) @ relation)) | ($true = (member @ a @ X0 @ carrier))) )),
% 2.47/1.49 inference(cnf_transformation,[status(thm)],[f763])).
% 2.47/1.49 thf(f763,plain,(
% 2.47/1.49 ! [X0 : a,X1 : a] : (($true = (member @ a @ X0 @ carrier)) | ($true != (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ X0 @ X1) @ relation)))),
% 2.47/1.49 inference(ennf_transformation,[status(thm)],[f538])).
% 2.47/1.49 thf(f538,plain,(
% 2.47/1.49 ! [X0 : a,X1 : a] : (($true = (member @ product_prod @ a @ a @ (product_Pair @ a @ a @ X0 @ X1) @ relation)) => ($true = (member @ a @ X0 @ carrier)))),
% 2.47/1.49 inference(fool_elimination,[status(thm)],[f537])).
% 2.47/1.49 thf(f537,plain,(
% 2.47/1.49 ! [X0 : a,X1 : a] : ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ X0 @ X1) @ relation) => (member @ a @ X0 @ carrier))),
% 2.47/1.49 inference(rectify,[status(thm)],[f9])).
% 2.47/1.49 thf(f9,axiom,(
% 2.47/1.49 ! [X6 : a,X7 : a] : ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ X6 @ X7) @ relation) => (member @ a @ X6 @ carrier))),
% 2.47/1.49 file('/export/starexec/sandbox/tmp/tmp.xCdPegwIoG/DTF2THF_29341.p',fact_8_not__outside)).
% 2.47/1.49 thf(f808,plain,(
% 2.47/1.49 spl1_1),
% 2.47/1.49 inference(avatar_contradiction_clause,[status(thm)],[f807])).
% 2.47/1.49 thf(f807,plain,(
% 2.47/1.49 $false | spl1_1),
% 2.47/1.49 inference(subsumption_resolution,[status(thm)],[f806,f796])).
% 2.47/1.49 thf(f796,plain,(
% 2.47/1.49 ((member @ a @ z @ carrier) != $true) | spl1_1),
% 2.47/1.49 inference(avatar_component_clause,[status(thm)],[f794])).
% 2.47/1.49 thf(f794,definition,(
% 2.47/1.49 spl1_1 <=> ((member @ a @ z @ carrier) = $true)),
% 2.47/1.49 introduced(definition,[new_symbols(naming,[spl1_1])],[avatar_definition])).
% 2.47/1.49 thf(f806,plain,(
% 2.47/1.49 ((member @ a @ z @ carrier) = $true)),
% 2.47/1.49 inference(trivial_inequality_removal,[status(thm)],[f803])).
% 2.47/1.49 thf(f803,plain,(
% 2.47/1.49 ((member @ a @ z @ carrier) = $true) | ($true != $true)),
% 2.47/1.49 inference(superposition,[status(thm)],[f788,f778])).
% 2.47/1.49 thf(f778,plain,(
% 2.47/1.49 ((member @ product_prod @ a @ a @ (product_Pair @ a @ a @ z @ y) @ relation) = $true)),
% 2.47/1.49 inference(cnf_transformation,[status(thm)],[f442])).
% 2.47/1.49 thf(f801,plain,(
% 2.47/1.49 ~spl1_1 | ~spl1_2),
% 2.47/1.49 inference(avatar_split_clause,[status(thm)],[f786,f798,f794])).
% 2.47/1.49 thf(f786,plain,(
% 2.47/1.49 ((member @ a @ y @ carrier) != $true) | ((member @ a @ z @ carrier) != $true)),
% 2.47/1.49 inference(cnf_transformation,[status(thm)],[f758])).
% 2.47/1.49 thf(f758,plain,(
% 2.47/1.49 ((member @ a @ y @ carrier) != $true) | ((member @ a @ z @ carrier) != $true)),
% 2.47/1.49 inference(ennf_transformation,[status(thm)],[f514])).
% 2.47/1.49 thf(f514,plain,(
% 2.47/1.49 ~(((member @ a @ y @ carrier) = $true) & ((member @ a @ z @ carrier) = $true))),
% 2.47/1.49 inference(fool_elimination,[status(thm)],[f513])).
% 2.47/1.49 thf(f513,plain,(
% 2.47/1.49 ~((member @ a @ y @ carrier) & (member @ a @ z @ carrier))),
% 2.47/1.49 inference(rectify,[status(thm)],[f279])).
% 2.47/1.49 thf(f279,negated_conjecture,(
% 2.47/1.49 ~((member @ a @ y @ carrier) & (member @ a @ z @ carrier))),
% 2.47/1.49 inference(negated_conjecture,[status(cth)],[f278])).
% 2.47/1.49 thf(f278,conjecture,(
% 2.47/1.49 (member @ a @ y @ carrier) & (member @ a @ z @ carrier)),
% 2.47/1.49 file('/export/starexec/sandbox/tmp/tmp.xCdPegwIoG/DTF2THF_29341.p',conj_0)).
% 2.47/1.49 % SZS output end Proof for DTF2THF_29341
% 2.47/1.49 % (29590)------------------------------
% 2.47/1.49 % (29590)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.49 % (29590)Termination reason: Refutation
% 2.47/1.49
% 2.47/1.49 % (29590)Memory used [KB]: 6396
% 2.47/1.49 % (29590)Time elapsed: 0.017 s
% 2.47/1.49 % (29590)Instructions burned: 50 (million)
% 2.47/1.49 % (29590)------------------------------
% 2.47/1.49 % (29590)------------------------------
% 2.47/1.49 % (29542)Success in time 0.068 s
%------------------------------------------------------------------------------