%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP031^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n012.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 08:54:59 AM UTC 2026
% Result : Theorem 2.35s 1.39s
% Output : Refutation 2.35s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.07 % Problem : ITP031^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.08 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.08/0.27 % Computer : n012.cluster.edu
% 0.08/0.27 % Model : x86_64 x86_64
% 0.08/0.27 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.27 % Memory : 8042.1875MB
% 0.08/0.27 % OS : Linux 3.10.0-693.el7.x86_64
% 0.08/0.27 % CPULimit : 300
% 0.08/0.27 % WCLimit : 300
% 0.08/0.27 % DateTime : Tue Feb 24 08:23:45 EST 2026
% 0.08/0.28 % CPUTime :
% 0.08/0.28 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.14/0.36 ---- Original DTF file ---
% 0.14/0.36 thf(spec,logic,$$dhol).
% 0.14/0.36 %------------------------------------------------------------------------------
% 0.14/0.36 % File : ITP031^2 : TPTP v9.2.1. Released v7.5.0.
% 0.14/0.36 % Domain : Interactive Theorem Proving
% 0.14/0.36 % Problem : Sledgehammer BinaryTree problem prob_301__3253080_1
% 0.14/0.36 % Version : Especial.
% 0.14/0.36 % English :
% 0.14/0.36
% 0.14/0.36 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.14/0.36 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.14/0.36 % Source : [Des21]
% 0.14/0.36 % Names : BinaryTree/prob_301__3253080_1 [Des21]
% 0.14/0.36
% 0.14/0.36 % Status : Theorem
% 0.14/0.36 % Rating : 0.00 v7.5.0
% 0.14/0.36 % Syntax : Number of formulae : 355 ( 139 unt; 52 typ; 0 def)
% 0.14/0.36 % Number of atoms : 778 ( 300 equ; 0 cnn)
% 0.14/0.36 % Maximal formula atoms : 7 ( 2 avg)
% 0.14/0.36 % Number of connectives : 3436 ( 90 ~; 17 |; 66 &;2943 @)
% 0.14/0.36 % ( 0 <=>; 320 =>; 0 <=; 0 <~>)
% 0.14/0.36 % Maximal formula depth : 17 ( 7 avg)
% 0.14/0.36 % Number of types : 3 ( 2 usr)
% 0.14/0.36 % Number of type conns : 147 ( 147 >; 0 *; 0 +; 0 <<)
% 0.14/0.36 % Number of symbols : 53 ( 50 usr; 6 con; 0-5 aty)
% 0.14/0.36 % Number of variables : 1009 ( 92 ^; 856 !; 18 ?;1009 :)
% 0.14/0.36 % ( 43 !>; 0 ?*; 0 @-; 0 @+)
% 0.14/0.36 % SPC : TH1_THM_EQU_NAR
% 0.14/0.36
% 0.14/0.36 % Comments : This file was generated by Sledgehammer 2021-02-23 16:16:01.044
% 0.14/0.36 %------------------------------------------------------------------------------
% 0.14/0.36 % Could-be-implicit typings (4)
% 0.14/0.36 thf(ty_t_BinaryTree__Mirabelle__pchhvghoao_OTree,type,
% 0.14/0.36 binary1291135688e_Tree: $tType > $tType ).
% 0.14/0.36
% 0.14/0.36 thf(ty_t_Set_Oset,type,
% 0.14/0.36 set: $tType > $tType ).
% 0.14/0.36
% 0.14/0.36 thf(ty_t_Int_Oint,type,
% 0.14/0.36 int: $tType ).
% 0.14/0.36
% 0.14/0.36 thf(ty_tf_a,type,
% 0.14/0.36 a: $tType ).
% 0.14/0.36
% 0.14/0.36 % Explicit typings (48)
% 0.14/0.36 thf(sy_cl_Lattices_Obounded__lattice,type,
% 0.14/0.36 bounded_lattice:
% 0.14/0.36 !>[A: $tType] : $o ).
% 0.14/0.36
% 0.14/0.36 thf(sy_cl_HOL_Otype,type,
% 0.14/0.36 type:
% 0.14/0.36 !>[A: $tType] : $o ).
% 0.14/0.36
% 0.14/0.36 thf(sy_cl_Groups_Ominus,type,
% 0.14/0.36 minus:
% 0.14/0.36 !>[A: $tType] : $o ).
% 0.14/0.36
% 0.14/0.36 thf(sy_cl_Orderings_Obot,type,
% 0.14/0.36 bot:
% 0.14/0.36 !>[A: $tType] : $o ).
% 0.14/0.36
% 0.14/0.36 thf(sy_cl_Orderings_Oord,type,
% 0.14/0.36 ord:
% 0.14/0.36 !>[A: $tType] : $o ).
% 0.14/0.36
% 0.14/0.36 thf(sy_cl_Orderings_Oorder,type,
% 0.14/0.36 order:
% 0.14/0.36 !>[A: $tType] : $o ).
% 0.14/0.36
% 0.14/0.36 thf(sy_cl_Groups_Ogroup__add,type,
% 0.14/0.36 group_add:
% 0.14/0.36 !>[A: $tType] : $o ).
% 0.14/0.36
% 0.14/0.36 thf(sy_cl_Lattices_Olattice,type,
% 0.14/0.36 lattice:
% 0.14/0.36 !>[A: $tType] : $o ).
% 0.14/0.36
% 0.14/0.36 thf(sy_cl_Orderings_Ono__bot,type,
% 0.14/0.36 no_bot:
% 0.14/0.36 !>[A: $tType] : $o ).
% 0.14/0.36
% 0.14/0.36 thf(sy_cl_Orderings_Ono__top,type,
% 0.14/0.36 no_top:
% 0.14/0.36 !>[A: $tType] : $o ).
% 0.14/0.36
% 0.14/0.36 thf(sy_cl_Orderings_Olinorder,type,
% 0.14/0.36 linorder:
% 0.14/0.36 !>[A: $tType] : $o ).
% 0.14/0.36
% 0.14/0.36 thf(sy_cl_Orderings_Opreorder,type,
% 0.14/0.36 preorder:
% 0.14/0.36 !>[A: $tType] : $o ).
% 0.14/0.36
% 0.14/0.36 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.14/0.36 order_bot:
% 0.14/0.36 !>[A: $tType] : $o ).
% 0.14/0.36
% 0.14/0.36 thf(sy_cl_Orderings_Owellorder,type,
% 0.14/0.36 wellorder:
% 0.14/0.36 !>[A: $tType] : $o ).
% 0.14/0.36
% 0.14/0.36 thf(sy_cl_Orderings_Odense__order,type,
% 0.14/0.36 dense_order:
% 0.14/0.36 !>[A: $tType] : $o ).
% 0.14/0.36
% 0.14/0.36 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.14/0.36 semilattice_sup:
% 0.14/0.36 !>[A: $tType] : $o ).
% 0.14/0.36
% 0.14/0.36 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.14/0.36 ordered_ab_group_add:
% 0.14/0.36 !>[A: $tType] : $o ).
% 0.14/0.36
% 0.14/0.36 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.14/0.36 cancel146912293up_add:
% 0.14/0.36 !>[A: $tType] : $o ).
% 0.14/0.36
% 0.14/0.36 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
% 0.14/0.36 bounde1808546759up_bot:
% 0.14/0.36 !>[A: $tType] : $o ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OT,type,
% 0.14/0.36 binary210054475elle_T:
% 0.14/0.36 !>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OTip,type,
% 0.14/0.36 binary1746293266le_Tip:
% 0.14/0.36 !>[A: $tType] : ( binary1291135688e_Tree @ A ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_Oset__Tree,type,
% 0.14/0.36 binary2130109271t_Tree:
% 0.14/0.36 !>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > ( set @ A ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Obinsert,type,
% 0.14/0.36 binary1830089824insert:
% 0.14/0.36 !>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Oeqs,type,
% 0.14/0.36 binary64540844le_eqs:
% 0.14/0.36 !>[A: $tType] : ( ( A > int ) > A > ( set @ A ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Omemb,type,
% 0.14/0.36 binary827270440e_memb:
% 0.14/0.36 !>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > $o ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsetOf,type,
% 0.14/0.36 binary1653327646_setOf:
% 0.14/0.36 !>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > ( set @ A ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsortedTree,type,
% 0.14/0.36 binary1610619414edTree:
% 0.14/0.36 !>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > $o ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Osorted__distinct__pred,type,
% 0.14/0.36 binary231205461t_pred:
% 0.14/0.36 !>[A: $tType] : ( ( A > int ) > A > A > ( binary1291135688e_Tree @ A ) > $o ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.14/0.36 minus_minus:
% 0.14/0.36 !>[A: $tType] : ( A > A > A ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_HOL_OThe,type,
% 0.14/0.36 the:
% 0.14/0.36 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_If,type,
% 0.14/0.36 if:
% 0.14/0.36 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.14/0.36 sup_sup:
% 0.14/0.36 !>[A: $tType] : ( A > A > A ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.14/0.36 bot_bot:
% 0.14/0.36 !>[A: $tType] : A ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.14/0.36 ord_less:
% 0.14/0.36 !>[A: $tType] : ( A > A > $o ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.14/0.36 ord_less_eq:
% 0.14/0.36 !>[A: $tType] : ( A > A > $o ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_Set_OCollect,type,
% 0.14/0.36 collect:
% 0.14/0.36 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_Set_Oinsert,type,
% 0.14/0.36 insert:
% 0.14/0.36 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_Set_Ois__empty,type,
% 0.14/0.36 is_empty:
% 0.14/0.36 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_Set_Ois__singleton,type,
% 0.14/0.36 is_singleton:
% 0.14/0.36 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_Set_Opairwise,type,
% 0.14/0.36 pairwise:
% 0.14/0.36 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_Set_Oremove,type,
% 0.14/0.36 remove:
% 0.14/0.36 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_Set_Othe__elem,type,
% 0.14/0.36 the_elem:
% 0.14/0.36 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_c_member,type,
% 0.14/0.36 member:
% 0.14/0.36 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.14/0.36
% 0.14/0.36 thf(sy_v_e,type,
% 0.14/0.36 e: a ).
% 0.14/0.36
% 0.14/0.36 thf(sy_v_h,type,
% 0.14/0.36 h: a > int ).
% 0.14/0.36
% 0.14/0.36 thf(sy_v_t1____,type,
% 0.14/0.36 t1: binary1291135688e_Tree @ a ).
% 0.14/0.36
% 0.14/0.36 thf(sy_v_t2____,type,
% 0.14/0.36 t2: binary1291135688e_Tree @ a ).
% 0.14/0.36
% 0.14/0.36 thf(sy_v_x____,type,
% 0.14/0.36 x: a ).
% 0.14/0.36
% 0.14/0.36 % Relevant facts (256)
% 0.14/0.36 thf(fact_0_eqsLessX,axiom,
% 0.14/0.36 ! [X: a] :
% 0.14/0.36 ( ( member @ a @ X @ ( binary64540844le_eqs @ a @ h @ e ) )
% 0.14/0.36 => ( ord_less @ int @ ( h @ X ) @ ( h @ x ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % eqsLessX
% 0.14/0.36 thf(fact_1_eLess,axiom,
% 0.14/0.36 ord_less @ int @ ( h @ e ) @ ( h @ x ) ).
% 0.14/0.36
% 0.14/0.36 % eLess
% 0.14/0.36 thf(fact_2_res,axiom,
% 0.14/0.36 ( ( binary1830089824insert @ a @ h @ e @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) )
% 0.14/0.36 = ( binary210054475elle_T @ a @ ( binary1830089824insert @ a @ h @ e @ t1 ) @ x @ t2 ) ) ).
% 0.14/0.36
% 0.14/0.36 % res
% 0.14/0.36 thf(fact_3_s1,axiom,
% 0.14/0.36 binary1610619414edTree @ a @ h @ t1 ).
% 0.14/0.36
% 0.14/0.36 % s1
% 0.14/0.36 thf(fact_4_s2,axiom,
% 0.14/0.36 binary1610619414edTree @ a @ h @ t2 ).
% 0.14/0.36
% 0.14/0.36 % s2
% 0.14/0.36 thf(fact_5_s,axiom,
% 0.14/0.36 binary1610619414edTree @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) ).
% 0.14/0.36
% 0.14/0.36 % s
% 0.14/0.36 thf(fact_6_sorted__distinct,axiom,
% 0.14/0.36 ! [A: $tType,H: A > int,A2: A,B: A,T: binary1291135688e_Tree @ A] : ( binary231205461t_pred @ A @ H @ A2 @ B @ T ) ).
% 0.14/0.36
% 0.14/0.36 % sorted_distinct
% 0.14/0.36 thf(fact_7_c1,axiom,
% 0.14/0.36 ( ( binary1653327646_setOf @ a @ ( binary1830089824insert @ a @ h @ e @ t1 ) )
% 0.14/0.36 = ( sup_sup @ ( set @ a ) @ ( minus_minus @ ( set @ a ) @ ( binary1653327646_setOf @ a @ t1 ) @ ( binary64540844le_eqs @ a @ h @ e ) ) @ ( insert @ a @ e @ ( bot_bot @ ( set @ a ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % c1
% 0.14/0.36 thf(fact_8_c2,axiom,
% 0.14/0.36 ( ( binary1653327646_setOf @ a @ ( binary1830089824insert @ a @ h @ e @ t2 ) )
% 0.14/0.36 = ( sup_sup @ ( set @ a ) @ ( minus_minus @ ( set @ a ) @ ( binary1653327646_setOf @ a @ t2 ) @ ( binary64540844le_eqs @ a @ h @ e ) ) @ ( insert @ a @ e @ ( bot_bot @ ( set @ a ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % c2
% 0.14/0.36 thf(fact_9_eqs__def,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( binary64540844le_eqs @ A )
% 0.14/0.36 = ( ^ [H2: A > int,X2: A] :
% 0.14/0.36 ( collect @ A
% 0.14/0.36 @ ^ [Y: A] :
% 0.14/0.36 ( ( H2 @ Y )
% 0.14/0.36 = ( H2 @ X2 ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % eqs_def
% 0.14/0.36 thf(fact_10_h1,axiom,
% 0.14/0.36 ( ( binary1610619414edTree @ a @ h @ t1 )
% 0.14/0.36 => ( ( binary1653327646_setOf @ a @ ( binary1830089824insert @ a @ h @ e @ t1 ) )
% 0.14/0.36 = ( sup_sup @ ( set @ a ) @ ( minus_minus @ ( set @ a ) @ ( binary1653327646_setOf @ a @ t1 ) @ ( binary64540844le_eqs @ a @ h @ e ) ) @ ( insert @ a @ e @ ( bot_bot @ ( set @ a ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % h1
% 0.14/0.36 thf(fact_11_h2,axiom,
% 0.14/0.36 ( ( binary1610619414edTree @ a @ h @ t2 )
% 0.14/0.36 => ( ( binary1653327646_setOf @ a @ ( binary1830089824insert @ a @ h @ e @ t2 ) )
% 0.14/0.36 = ( sup_sup @ ( set @ a ) @ ( minus_minus @ ( set @ a ) @ ( binary1653327646_setOf @ a @ t2 ) @ ( binary64540844le_eqs @ a @ h @ e ) ) @ ( insert @ a @ e @ ( bot_bot @ ( set @ a ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % h2
% 0.14/0.36 thf(fact_12__092_060open_062sortedTree_Ah_ATip_A_092_060longrightarrow_062_AsetOf_A_Ibinsert_Ah_Ae_ATip_J_A_061_AsetOf_ATip_A_N_Aeqs_Ah_Ae_A_092_060union_062_A_123e_125_092_060close_062,axiom,
% 0.14/0.36 ( ( binary1610619414edTree @ a @ h @ ( binary1746293266le_Tip @ a ) )
% 0.14/0.36 => ( ( binary1653327646_setOf @ a @ ( binary1830089824insert @ a @ h @ e @ ( binary1746293266le_Tip @ a ) ) )
% 0.14/0.36 = ( sup_sup @ ( set @ a ) @ ( minus_minus @ ( set @ a ) @ ( binary1653327646_setOf @ a @ ( binary1746293266le_Tip @ a ) ) @ ( binary64540844le_eqs @ a @ h @ e ) ) @ ( insert @ a @ e @ ( bot_bot @ ( set @ a ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % \<open>sortedTree h Tip \<longrightarrow> setOf (binsert h e Tip) = setOf Tip - eqs h e \<union> {e}\<close>
% 0.14/0.36 thf(fact_13_Tree_Oinject,axiom,
% 0.14/0.36 ! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A,Y21: binary1291135688e_Tree @ A,Y22: A,Y23: binary1291135688e_Tree @ A] :
% 0.14/0.36 ( ( ( binary210054475elle_T @ A @ X21 @ X22 @ X23 )
% 0.14/0.36 = ( binary210054475elle_T @ A @ Y21 @ Y22 @ Y23 ) )
% 0.14/0.36 = ( ( X21 = Y21 )
% 0.14/0.36 & ( X22 = Y22 )
% 0.14/0.36 & ( X23 = Y23 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Tree.inject
% 0.14/0.36 thf(fact_14_setOf_Osimps_I2_J,axiom,
% 0.14/0.36 ! [A: $tType,T1: binary1291135688e_Tree @ A,X3: A,T2: binary1291135688e_Tree @ A] :
% 0.14/0.36 ( ( binary1653327646_setOf @ A @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
% 0.14/0.36 = ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( binary1653327646_setOf @ A @ T1 ) @ ( binary1653327646_setOf @ A @ T2 ) ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % setOf.simps(2)
% 0.14/0.36 thf(fact_15_setOf_Osimps_I1_J,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( binary1653327646_setOf @ A @ ( binary1746293266le_Tip @ A ) )
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % setOf.simps(1)
% 0.14/0.36 thf(fact_16_Tree_Odistinct_I1_J,axiom,
% 0.14/0.36 ! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
% 0.14/0.36 ( ( binary1746293266le_Tip @ A )
% 0.14/0.36 != ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ).
% 0.14/0.36
% 0.14/0.36 % Tree.distinct(1)
% 0.14/0.36 thf(fact_17_binsert_Osimps_I2_J,axiom,
% 0.14/0.36 ! [A: $tType,H: A > int,E: A,X3: A,T1: binary1291135688e_Tree @ A,T2: binary1291135688e_Tree @ A] :
% 0.14/0.36 ( ( ( ord_less @ int @ ( H @ E ) @ ( H @ X3 ) )
% 0.14/0.36 => ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
% 0.14/0.36 = ( binary210054475elle_T @ A @ ( binary1830089824insert @ A @ H @ E @ T1 ) @ X3 @ T2 ) ) )
% 0.14/0.36 & ( ~ ( ord_less @ int @ ( H @ E ) @ ( H @ X3 ) )
% 0.14/0.36 => ( ( ( ord_less @ int @ ( H @ X3 ) @ ( H @ E ) )
% 0.14/0.36 => ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
% 0.14/0.36 = ( binary210054475elle_T @ A @ T1 @ X3 @ ( binary1830089824insert @ A @ H @ E @ T2 ) ) ) )
% 0.14/0.36 & ( ~ ( ord_less @ int @ ( H @ X3 ) @ ( H @ E ) )
% 0.14/0.36 => ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
% 0.14/0.36 = ( binary210054475elle_T @ A @ T1 @ E @ T2 ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % binsert.simps(2)
% 0.14/0.36 thf(fact_18_binsert_Osimps_I1_J,axiom,
% 0.14/0.36 ! [A: $tType,H: A > int,E: A] :
% 0.14/0.36 ( ( binary1830089824insert @ A @ H @ E @ ( binary1746293266le_Tip @ A ) )
% 0.14/0.36 = ( binary210054475elle_T @ A @ ( binary1746293266le_Tip @ A ) @ E @ ( binary1746293266le_Tip @ A ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % binsert.simps(1)
% 0.14/0.36 thf(fact_19_sortedTree_Osimps_I2_J,axiom,
% 0.14/0.36 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X3: A,T2: binary1291135688e_Tree @ A] :
% 0.14/0.36 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
% 0.14/0.36 = ( ( binary1610619414edTree @ A @ H @ T1 )
% 0.14/0.36 & ! [X2: A] :
% 0.14/0.36 ( ( member @ A @ X2 @ ( binary1653327646_setOf @ A @ T1 ) )
% 0.14/0.36 => ( ord_less @ int @ ( H @ X2 ) @ ( H @ X3 ) ) )
% 0.14/0.36 & ! [X2: A] :
% 0.14/0.36 ( ( member @ A @ X2 @ ( binary1653327646_setOf @ A @ T2 ) )
% 0.14/0.36 => ( ord_less @ int @ ( H @ X3 ) @ ( H @ X2 ) ) )
% 0.14/0.36 & ( binary1610619414edTree @ A @ H @ T2 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sortedTree.simps(2)
% 0.14/0.36 thf(fact_20_sortedTree_Osimps_I1_J,axiom,
% 0.14/0.36 ! [A: $tType,H: A > int] : ( binary1610619414edTree @ A @ H @ ( binary1746293266le_Tip @ A ) ) ).
% 0.14/0.36
% 0.14/0.36 % sortedTree.simps(1)
% 0.14/0.36 thf(fact_21_sortLemmaL,axiom,
% 0.14/0.36 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X3: A,T2: binary1291135688e_Tree @ A] :
% 0.14/0.36 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
% 0.14/0.36 => ( binary1610619414edTree @ A @ H @ T1 ) ) ).
% 0.14/0.36
% 0.14/0.36 % sortLemmaL
% 0.14/0.36 thf(fact_22_sortLemmaR,axiom,
% 0.14/0.36 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X3: A,T2: binary1291135688e_Tree @ A] :
% 0.14/0.36 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
% 0.14/0.36 => ( binary1610619414edTree @ A @ H @ T2 ) ) ).
% 0.14/0.36
% 0.14/0.36 % sortLemmaR
% 0.14/0.36 thf(fact_23_Tree_Oinduct,axiom,
% 0.14/0.36 ! [A: $tType,P: ( binary1291135688e_Tree @ A ) > $o,Tree: binary1291135688e_Tree @ A] :
% 0.14/0.36 ( ( P @ ( binary1746293266le_Tip @ A ) )
% 0.14/0.36 => ( ! [X1: binary1291135688e_Tree @ A,X24: A,X32: binary1291135688e_Tree @ A] :
% 0.14/0.36 ( ( P @ X1 )
% 0.14/0.36 => ( ( P @ X32 )
% 0.14/0.36 => ( P @ ( binary210054475elle_T @ A @ X1 @ X24 @ X32 ) ) ) )
% 0.14/0.36 => ( P @ Tree ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Tree.induct
% 0.14/0.36 thf(fact_24_Tree_Oexhaust,axiom,
% 0.14/0.36 ! [A: $tType,Y2: binary1291135688e_Tree @ A] :
% 0.14/0.36 ( ( Y2
% 0.14/0.36 != ( binary1746293266le_Tip @ A ) )
% 0.14/0.36 => ~ ! [X212: binary1291135688e_Tree @ A,X222: A,X232: binary1291135688e_Tree @ A] :
% 0.14/0.36 ( Y2
% 0.14/0.36 != ( binary210054475elle_T @ A @ X212 @ X222 @ X232 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Tree.exhaust
% 0.14/0.36 thf(fact_25_sorted__distinct__pred__def,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( binary231205461t_pred @ A )
% 0.14/0.36 = ( ^ [H2: A > int,A3: A,B2: A,T3: binary1291135688e_Tree @ A] :
% 0.14/0.36 ( ( ( binary1610619414edTree @ A @ H2 @ T3 )
% 0.14/0.36 & ( member @ A @ A3 @ ( binary1653327646_setOf @ A @ T3 ) )
% 0.14/0.36 & ( member @ A @ B2 @ ( binary1653327646_setOf @ A @ T3 ) )
% 0.14/0.36 & ( ( H2 @ A3 )
% 0.14/0.36 = ( H2 @ B2 ) ) )
% 0.14/0.36 => ( A3 = B2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sorted_distinct_pred_def
% 0.14/0.36 thf(fact_26_insert__Diff__single,axiom,
% 0.14/0.36 ! [A: $tType,A2: A,A4: set @ A] :
% 0.14/0.36 ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.14/0.36 = ( insert @ A @ A2 @ A4 ) ) ).
% 0.14/0.36
% 0.14/0.36 % insert_Diff_single
% 0.14/0.36 thf(fact_27_singleton__conv,axiom,
% 0.14/0.36 ! [A: $tType,A2: A] :
% 0.14/0.36 ( ( collect @ A
% 0.14/0.36 @ ^ [X2: A] : ( X2 = A2 ) )
% 0.14/0.36 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % singleton_conv
% 0.14/0.36 thf(fact_28_singleton__conv2,axiom,
% 0.14/0.36 ! [A: $tType,A2: A] :
% 0.14/0.36 ( ( collect @ A
% 0.14/0.36 @ ( ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.14/0.36 @ A2 ) )
% 0.14/0.36 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % singleton_conv2
% 0.14/0.36 thf(fact_29_Un__Diff__cancel,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ( sup_sup @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) )
% 0.14/0.36 = ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ).
% 0.14/0.36
% 0.14/0.36 % Un_Diff_cancel
% 0.14/0.36 thf(fact_30_Un__Diff__cancel2,axiom,
% 0.14/0.36 ! [A: $tType,B3: set @ A,A4: set @ A] :
% 0.14/0.36 ( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) @ A4 )
% 0.14/0.36 = ( sup_sup @ ( set @ A ) @ B3 @ A4 ) ) ).
% 0.14/0.36
% 0.14/0.36 % Un_Diff_cancel2
% 0.14/0.36 thf(fact_31_Diff__insert0,axiom,
% 0.14/0.36 ! [A: $tType,X3: A,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ~ ( member @ A @ X3 @ A4 )
% 0.14/0.36 => ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ B3 ) )
% 0.14/0.36 = ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Diff_insert0
% 0.14/0.36 thf(fact_32_insert__Diff1,axiom,
% 0.14/0.36 ! [A: $tType,X3: A,B3: set @ A,A4: set @ A] :
% 0.14/0.36 ( ( member @ A @ X3 @ B3 )
% 0.14/0.36 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A4 ) @ B3 )
% 0.14/0.36 = ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % insert_Diff1
% 0.14/0.36 thf(fact_33_Un__insert__left,axiom,
% 0.14/0.36 ! [A: $tType,A2: A,B3: set @ A,C: set @ A] :
% 0.14/0.36 ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C )
% 0.14/0.36 = ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B3 @ C ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Un_insert_left
% 0.14/0.36 thf(fact_34_Un__insert__right,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,A2: A,B3: set @ A] :
% 0.14/0.36 ( ( sup_sup @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B3 ) )
% 0.14/0.36 = ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Un_insert_right
% 0.14/0.36 thf(fact_35_Diff__empty,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A] :
% 0.14/0.36 ( ( minus_minus @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 0.14/0.36 = A4 ) ).
% 0.14/0.36
% 0.14/0.36 % Diff_empty
% 0.14/0.36 thf(fact_36_empty__Diff,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A] :
% 0.14/0.36 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 )
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % empty_Diff
% 0.14/0.36 thf(fact_37_empty__Collect__eq,axiom,
% 0.14/0.36 ! [A: $tType,P: A > $o] :
% 0.14/0.36 ( ( ( bot_bot @ ( set @ A ) )
% 0.14/0.36 = ( collect @ A @ P ) )
% 0.14/0.36 = ( ! [X2: A] :
% 0.14/0.36 ~ ( P @ X2 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % empty_Collect_eq
% 0.14/0.36 thf(fact_38_Collect__empty__eq,axiom,
% 0.14/0.36 ! [A: $tType,P: A > $o] :
% 0.14/0.36 ( ( ( collect @ A @ P )
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) )
% 0.14/0.36 = ( ! [X2: A] :
% 0.14/0.36 ~ ( P @ X2 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Collect_empty_eq
% 0.14/0.36 thf(fact_39_all__not__in__conv,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A] :
% 0.14/0.36 ( ( ! [X2: A] :
% 0.14/0.36 ~ ( member @ A @ X2 @ A4 ) )
% 0.14/0.36 = ( A4
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % all_not_in_conv
% 0.14/0.36 thf(fact_40_empty__iff,axiom,
% 0.14/0.36 ! [A: $tType,C2: A] :
% 0.14/0.36 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % empty_iff
% 0.14/0.36 thf(fact_41_insert__absorb2,axiom,
% 0.14/0.36 ! [A: $tType,X3: A,A4: set @ A] :
% 0.14/0.36 ( ( insert @ A @ X3 @ ( insert @ A @ X3 @ A4 ) )
% 0.14/0.36 = ( insert @ A @ X3 @ A4 ) ) ).
% 0.14/0.36
% 0.14/0.36 % insert_absorb2
% 0.14/0.36 thf(fact_42_insert__iff,axiom,
% 0.14/0.36 ! [A: $tType,A2: A,B: A,A4: set @ A] :
% 0.14/0.36 ( ( member @ A @ A2 @ ( insert @ A @ B @ A4 ) )
% 0.14/0.36 = ( ( A2 = B )
% 0.14/0.36 | ( member @ A @ A2 @ A4 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % insert_iff
% 0.14/0.36 thf(fact_43_insertCI,axiom,
% 0.14/0.36 ! [A: $tType,A2: A,B3: set @ A,B: A] :
% 0.14/0.36 ( ( ~ ( member @ A @ A2 @ B3 )
% 0.14/0.36 => ( A2 = B ) )
% 0.14/0.36 => ( member @ A @ A2 @ ( insert @ A @ B @ B3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % insertCI
% 0.14/0.36 thf(fact_44_Un__iff,axiom,
% 0.14/0.36 ! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 0.14/0.36 = ( ( member @ A @ C2 @ A4 )
% 0.14/0.36 | ( member @ A @ C2 @ B3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Un_iff
% 0.14/0.36 thf(fact_45_mem__Collect__eq,axiom,
% 0.14/0.36 ! [A: $tType,A2: A,P: A > $o] :
% 0.14/0.36 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.14/0.36 = ( P @ A2 ) ) ).
% 0.14/0.36
% 0.14/0.36 % mem_Collect_eq
% 0.14/0.36 thf(fact_46_Collect__mem__eq,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A] :
% 0.14/0.36 ( ( collect @ A
% 0.14/0.36 @ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
% 0.14/0.36 = A4 ) ).
% 0.14/0.36
% 0.14/0.36 % Collect_mem_eq
% 0.14/0.36 thf(fact_47_Collect__cong,axiom,
% 0.14/0.36 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.14/0.36 ( ! [X4: A] :
% 0.14/0.36 ( ( P @ X4 )
% 0.14/0.36 = ( Q @ X4 ) )
% 0.14/0.36 => ( ( collect @ A @ P )
% 0.14/0.36 = ( collect @ A @ Q ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Collect_cong
% 0.14/0.36 thf(fact_48_ext,axiom,
% 0.14/0.36 ! [B4: $tType,A: $tType,F: A > B4,G: A > B4] :
% 0.14/0.36 ( ! [X4: A] :
% 0.14/0.36 ( ( F @ X4 )
% 0.14/0.36 = ( G @ X4 ) )
% 0.14/0.36 => ( F = G ) ) ).
% 0.14/0.36
% 0.14/0.36 % ext
% 0.14/0.36 thf(fact_49_UnCI,axiom,
% 0.14/0.36 ! [A: $tType,C2: A,B3: set @ A,A4: set @ A] :
% 0.14/0.36 ( ( ~ ( member @ A @ C2 @ B3 )
% 0.14/0.36 => ( member @ A @ C2 @ A4 ) )
% 0.14/0.36 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % UnCI
% 0.14/0.36 thf(fact_50_Diff__idemp,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ B3 )
% 0.14/0.36 = ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ).
% 0.14/0.36
% 0.14/0.36 % Diff_idemp
% 0.14/0.36 thf(fact_51_Diff__iff,axiom,
% 0.14/0.36 ! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 0.14/0.36 = ( ( member @ A @ C2 @ A4 )
% 0.14/0.36 & ~ ( member @ A @ C2 @ B3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Diff_iff
% 0.14/0.36 thf(fact_52_DiffI,axiom,
% 0.14/0.36 ! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ( member @ A @ C2 @ A4 )
% 0.14/0.36 => ( ~ ( member @ A @ C2 @ B3 )
% 0.14/0.36 => ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % DiffI
% 0.14/0.36 thf(fact_53_singletonI,axiom,
% 0.14/0.36 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % singletonI
% 0.14/0.36 thf(fact_54_Un__empty,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ( ( sup_sup @ ( set @ A ) @ A4 @ B3 )
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) )
% 0.14/0.36 = ( ( A4
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) )
% 0.14/0.36 & ( B3
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Un_empty
% 0.14/0.36 thf(fact_55_Diff__cancel,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A] :
% 0.14/0.36 ( ( minus_minus @ ( set @ A ) @ A4 @ A4 )
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Diff_cancel
% 0.14/0.36 thf(fact_56_ex__in__conv,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A] :
% 0.14/0.36 ( ( ? [X2: A] : ( member @ A @ X2 @ A4 ) )
% 0.14/0.36 = ( A4
% 0.14/0.36 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % ex_in_conv
% 0.14/0.36 thf(fact_57_equals0I,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A] :
% 0.14/0.36 ( ! [Y4: A] :
% 0.14/0.36 ~ ( member @ A @ Y4 @ A4 )
% 0.14/0.36 => ( A4
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % equals0I
% 0.14/0.36 thf(fact_58_equals0D,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,A2: A] :
% 0.14/0.36 ( ( A4
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) )
% 0.14/0.36 => ~ ( member @ A @ A2 @ A4 ) ) ).
% 0.14/0.36
% 0.14/0.36 % equals0D
% 0.14/0.36 thf(fact_59_emptyE,axiom,
% 0.14/0.36 ! [A: $tType,A2: A] :
% 0.14/0.36 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % emptyE
% 0.14/0.36 thf(fact_60_mk__disjoint__insert,axiom,
% 0.14/0.36 ! [A: $tType,A2: A,A4: set @ A] :
% 0.14/0.36 ( ( member @ A @ A2 @ A4 )
% 0.14/0.36 => ? [B5: set @ A] :
% 0.14/0.36 ( ( A4
% 0.14/0.36 = ( insert @ A @ A2 @ B5 ) )
% 0.14/0.36 & ~ ( member @ A @ A2 @ B5 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % mk_disjoint_insert
% 0.14/0.36 thf(fact_61_insert__commute,axiom,
% 0.14/0.36 ! [A: $tType,X3: A,Y2: A,A4: set @ A] :
% 0.14/0.36 ( ( insert @ A @ X3 @ ( insert @ A @ Y2 @ A4 ) )
% 0.14/0.36 = ( insert @ A @ Y2 @ ( insert @ A @ X3 @ A4 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % insert_commute
% 0.14/0.36 thf(fact_62_insert__eq__iff,axiom,
% 0.14/0.36 ! [A: $tType,A2: A,A4: set @ A,B: A,B3: set @ A] :
% 0.14/0.36 ( ~ ( member @ A @ A2 @ A4 )
% 0.14/0.36 => ( ~ ( member @ A @ B @ B3 )
% 0.14/0.36 => ( ( ( insert @ A @ A2 @ A4 )
% 0.14/0.36 = ( insert @ A @ B @ B3 ) )
% 0.14/0.36 = ( ( ( A2 = B )
% 0.14/0.36 => ( A4 = B3 ) )
% 0.14/0.36 & ( ( A2 != B )
% 0.14/0.36 => ? [C3: set @ A] :
% 0.14/0.36 ( ( A4
% 0.14/0.36 = ( insert @ A @ B @ C3 ) )
% 0.14/0.36 & ~ ( member @ A @ B @ C3 )
% 0.14/0.36 & ( B3
% 0.14/0.36 = ( insert @ A @ A2 @ C3 ) )
% 0.14/0.36 & ~ ( member @ A @ A2 @ C3 ) ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % insert_eq_iff
% 0.14/0.36 thf(fact_63_insert__absorb,axiom,
% 0.14/0.36 ! [A: $tType,A2: A,A4: set @ A] :
% 0.14/0.36 ( ( member @ A @ A2 @ A4 )
% 0.14/0.36 => ( ( insert @ A @ A2 @ A4 )
% 0.14/0.36 = A4 ) ) ).
% 0.14/0.36
% 0.14/0.36 % insert_absorb
% 0.14/0.36 thf(fact_64_insert__ident,axiom,
% 0.14/0.36 ! [A: $tType,X3: A,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ~ ( member @ A @ X3 @ A4 )
% 0.14/0.36 => ( ~ ( member @ A @ X3 @ B3 )
% 0.14/0.36 => ( ( ( insert @ A @ X3 @ A4 )
% 0.14/0.36 = ( insert @ A @ X3 @ B3 ) )
% 0.14/0.36 = ( A4 = B3 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % insert_ident
% 0.14/0.36 thf(fact_65_Set_Oset__insert,axiom,
% 0.14/0.36 ! [A: $tType,X3: A,A4: set @ A] :
% 0.14/0.36 ( ( member @ A @ X3 @ A4 )
% 0.14/0.36 => ~ ! [B5: set @ A] :
% 0.14/0.36 ( ( A4
% 0.14/0.36 = ( insert @ A @ X3 @ B5 ) )
% 0.14/0.36 => ( member @ A @ X3 @ B5 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Set.set_insert
% 0.14/0.36 thf(fact_66_insertI2,axiom,
% 0.14/0.36 ! [A: $tType,A2: A,B3: set @ A,B: A] :
% 0.14/0.36 ( ( member @ A @ A2 @ B3 )
% 0.14/0.36 => ( member @ A @ A2 @ ( insert @ A @ B @ B3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % insertI2
% 0.14/0.36 thf(fact_67_insertI1,axiom,
% 0.14/0.36 ! [A: $tType,A2: A,B3: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B3 ) ) ).
% 0.14/0.36
% 0.14/0.36 % insertI1
% 0.14/0.36 thf(fact_68_insertE,axiom,
% 0.14/0.36 ! [A: $tType,A2: A,B: A,A4: set @ A] :
% 0.14/0.36 ( ( member @ A @ A2 @ ( insert @ A @ B @ A4 ) )
% 0.14/0.36 => ( ( A2 != B )
% 0.14/0.36 => ( member @ A @ A2 @ A4 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % insertE
% 0.14/0.36 thf(fact_69_Un__left__commute,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,B3: set @ A,C: set @ A] :
% 0.14/0.36 ( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B3 @ C ) )
% 0.14/0.36 = ( sup_sup @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A4 @ C ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Un_left_commute
% 0.14/0.36 thf(fact_70_Un__left__absorb,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 0.14/0.36 = ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ).
% 0.14/0.36
% 0.14/0.36 % Un_left_absorb
% 0.14/0.36 thf(fact_71_Un__commute,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( sup_sup @ ( set @ A ) )
% 0.14/0.36 = ( ^ [A5: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Un_commute
% 0.14/0.36 thf(fact_72_Un__absorb,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A] :
% 0.14/0.36 ( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
% 0.14/0.36 = A4 ) ).
% 0.14/0.36
% 0.14/0.36 % Un_absorb
% 0.14/0.36 thf(fact_73_Un__assoc,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,B3: set @ A,C: set @ A] :
% 0.14/0.36 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C )
% 0.14/0.36 = ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B3 @ C ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Un_assoc
% 0.14/0.36 thf(fact_74_ball__Un,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,B3: set @ A,P: A > $o] :
% 0.14/0.36 ( ( ! [X2: A] :
% 0.14/0.36 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 0.14/0.36 => ( P @ X2 ) ) )
% 0.14/0.36 = ( ! [X2: A] :
% 0.14/0.36 ( ( member @ A @ X2 @ A4 )
% 0.14/0.36 => ( P @ X2 ) )
% 0.14/0.36 & ! [X2: A] :
% 0.14/0.36 ( ( member @ A @ X2 @ B3 )
% 0.14/0.36 => ( P @ X2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % ball_Un
% 0.14/0.36 thf(fact_75_bex__Un,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,B3: set @ A,P: A > $o] :
% 0.14/0.36 ( ( ? [X2: A] :
% 0.14/0.36 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 0.14/0.36 & ( P @ X2 ) ) )
% 0.14/0.36 = ( ? [X2: A] :
% 0.14/0.36 ( ( member @ A @ X2 @ A4 )
% 0.14/0.36 & ( P @ X2 ) )
% 0.14/0.36 | ? [X2: A] :
% 0.14/0.36 ( ( member @ A @ X2 @ B3 )
% 0.14/0.36 & ( P @ X2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % bex_Un
% 0.14/0.36 thf(fact_76_UnI2,axiom,
% 0.14/0.36 ! [A: $tType,C2: A,B3: set @ A,A4: set @ A] :
% 0.14/0.36 ( ( member @ A @ C2 @ B3 )
% 0.14/0.36 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % UnI2
% 0.14/0.36 thf(fact_77_UnI1,axiom,
% 0.14/0.36 ! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ( member @ A @ C2 @ A4 )
% 0.14/0.36 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % UnI1
% 0.14/0.36 thf(fact_78_UnE,axiom,
% 0.14/0.36 ! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 0.14/0.36 => ( ~ ( member @ A @ C2 @ A4 )
% 0.14/0.36 => ( member @ A @ C2 @ B3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % UnE
% 0.14/0.36 thf(fact_79_DiffD2,axiom,
% 0.14/0.36 ! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 0.14/0.36 => ~ ( member @ A @ C2 @ B3 ) ) ).
% 0.14/0.36
% 0.14/0.36 % DiffD2
% 0.14/0.36 thf(fact_80_DiffD1,axiom,
% 0.14/0.36 ! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 0.14/0.36 => ( member @ A @ C2 @ A4 ) ) ).
% 0.14/0.36
% 0.14/0.36 % DiffD1
% 0.14/0.36 thf(fact_81_DiffE,axiom,
% 0.14/0.36 ! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 0.14/0.36 => ~ ( ( member @ A @ C2 @ A4 )
% 0.14/0.36 => ( member @ A @ C2 @ B3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % DiffE
% 0.14/0.36 thf(fact_82_empty__def,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( bot_bot @ ( set @ A ) )
% 0.14/0.36 = ( collect @ A
% 0.14/0.36 @ ^ [X2: A] : $false ) ) ).
% 0.14/0.36
% 0.14/0.36 % empty_def
% 0.14/0.36 thf(fact_83_insert__Collect,axiom,
% 0.14/0.36 ! [A: $tType,A2: A,P: A > $o] :
% 0.14/0.36 ( ( insert @ A @ A2 @ ( collect @ A @ P ) )
% 0.14/0.36 = ( collect @ A
% 0.14/0.36 @ ^ [U: A] :
% 0.14/0.36 ( ( U != A2 )
% 0.14/0.36 => ( P @ U ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % insert_Collect
% 0.14/0.36 thf(fact_84_insert__compr,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( insert @ A )
% 0.14/0.36 = ( ^ [A3: A,B6: set @ A] :
% 0.14/0.36 ( collect @ A
% 0.14/0.36 @ ^ [X2: A] :
% 0.14/0.36 ( ( X2 = A3 )
% 0.14/0.36 | ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % insert_compr
% 0.14/0.36 thf(fact_85_Collect__disj__eq,axiom,
% 0.14/0.36 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.14/0.36 ( ( collect @ A
% 0.14/0.36 @ ^ [X2: A] :
% 0.14/0.36 ( ( P @ X2 )
% 0.14/0.36 | ( Q @ X2 ) ) )
% 0.14/0.36 = ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Collect_disj_eq
% 0.14/0.36 thf(fact_86_Un__def,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( sup_sup @ ( set @ A ) )
% 0.14/0.36 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.14/0.36 ( collect @ A
% 0.14/0.36 @ ^ [X2: A] :
% 0.14/0.36 ( ( member @ A @ X2 @ A5 )
% 0.14/0.36 | ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Un_def
% 0.14/0.36 thf(fact_87_set__diff__eq,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( minus_minus @ ( set @ A ) )
% 0.14/0.36 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.14/0.36 ( collect @ A
% 0.14/0.36 @ ^ [X2: A] :
% 0.14/0.36 ( ( member @ A @ X2 @ A5 )
% 0.14/0.36 & ~ ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % set_diff_eq
% 0.14/0.36 thf(fact_88_singleton__inject,axiom,
% 0.14/0.36 ! [A: $tType,A2: A,B: A] :
% 0.14/0.36 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.14/0.36 = ( insert @ A @ B @ ( bot_bot @ ( set @ A ) ) ) )
% 0.14/0.36 => ( A2 = B ) ) ).
% 0.14/0.36
% 0.14/0.36 % singleton_inject
% 0.14/0.36 thf(fact_89_insert__not__empty,axiom,
% 0.14/0.36 ! [A: $tType,A2: A,A4: set @ A] :
% 0.14/0.36 ( ( insert @ A @ A2 @ A4 )
% 0.14/0.36 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % insert_not_empty
% 0.14/0.36 thf(fact_90_doubleton__eq__iff,axiom,
% 0.14/0.36 ! [A: $tType,A2: A,B: A,C2: A,D: A] :
% 0.14/0.36 ( ( ( insert @ A @ A2 @ ( insert @ A @ B @ ( bot_bot @ ( set @ A ) ) ) )
% 0.14/0.36 = ( insert @ A @ C2 @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.14/0.36 = ( ( ( A2 = C2 )
% 0.14/0.36 & ( B = D ) )
% 0.14/0.36 | ( ( A2 = D )
% 0.14/0.36 & ( B = C2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % doubleton_eq_iff
% 0.14/0.36 thf(fact_91_singleton__iff,axiom,
% 0.14/0.36 ! [A: $tType,B: A,A2: A] :
% 0.14/0.36 ( ( member @ A @ B @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.14/0.36 = ( B = A2 ) ) ).
% 0.14/0.36
% 0.14/0.36 % singleton_iff
% 0.14/0.36 thf(fact_92_singletonD,axiom,
% 0.14/0.36 ! [A: $tType,B: A,A2: A] :
% 0.14/0.36 ( ( member @ A @ B @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.14/0.36 => ( B = A2 ) ) ).
% 0.14/0.36
% 0.14/0.36 % singletonD
% 0.14/0.36 thf(fact_93_Un__empty__right,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A] :
% 0.14/0.36 ( ( sup_sup @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 0.14/0.36 = A4 ) ).
% 0.14/0.36
% 0.14/0.36 % Un_empty_right
% 0.14/0.36 thf(fact_94_Un__empty__left,axiom,
% 0.14/0.36 ! [A: $tType,B3: set @ A] :
% 0.14/0.36 ( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B3 )
% 0.14/0.36 = B3 ) ).
% 0.14/0.36
% 0.14/0.36 % Un_empty_left
% 0.14/0.36 thf(fact_95_insert__Diff__if,axiom,
% 0.14/0.36 ! [A: $tType,X3: A,B3: set @ A,A4: set @ A] :
% 0.14/0.36 ( ( ( member @ A @ X3 @ B3 )
% 0.14/0.36 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A4 ) @ B3 )
% 0.14/0.36 = ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) )
% 0.14/0.36 & ( ~ ( member @ A @ X3 @ B3 )
% 0.14/0.36 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A4 ) @ B3 )
% 0.14/0.36 = ( insert @ A @ X3 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % insert_Diff_if
% 0.14/0.36 thf(fact_96_Un__Diff,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,B3: set @ A,C: set @ A] :
% 0.14/0.36 ( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C )
% 0.14/0.36 = ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ C ) @ ( minus_minus @ ( set @ A ) @ B3 @ C ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Un_Diff
% 0.14/0.36 thf(fact_97_Collect__conv__if2,axiom,
% 0.14/0.36 ! [A: $tType,P: A > $o,A2: A] :
% 0.14/0.36 ( ( ( P @ A2 )
% 0.14/0.36 => ( ( collect @ A
% 0.14/0.36 @ ^ [X2: A] :
% 0.14/0.36 ( ( A2 = X2 )
% 0.14/0.36 & ( P @ X2 ) ) )
% 0.14/0.36 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.14/0.36 & ( ~ ( P @ A2 )
% 0.14/0.36 => ( ( collect @ A
% 0.14/0.36 @ ^ [X2: A] :
% 0.14/0.36 ( ( A2 = X2 )
% 0.14/0.36 & ( P @ X2 ) ) )
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Collect_conv_if2
% 0.14/0.36 thf(fact_98_Collect__conv__if,axiom,
% 0.14/0.36 ! [A: $tType,P: A > $o,A2: A] :
% 0.14/0.36 ( ( ( P @ A2 )
% 0.14/0.36 => ( ( collect @ A
% 0.14/0.36 @ ^ [X2: A] :
% 0.14/0.36 ( ( X2 = A2 )
% 0.14/0.36 & ( P @ X2 ) ) )
% 0.14/0.36 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.14/0.36 & ( ~ ( P @ A2 )
% 0.14/0.36 => ( ( collect @ A
% 0.14/0.36 @ ^ [X2: A] :
% 0.14/0.36 ( ( X2 = A2 )
% 0.14/0.36 & ( P @ X2 ) ) )
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Collect_conv_if
% 0.14/0.36 thf(fact_99_insert__def,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( insert @ A )
% 0.14/0.36 = ( ^ [A3: A] :
% 0.14/0.36 ( sup_sup @ ( set @ A )
% 0.14/0.36 @ ( collect @ A
% 0.14/0.36 @ ^ [X2: A] : ( X2 = A3 ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % insert_def
% 0.14/0.36 thf(fact_100_singleton__Un__iff,axiom,
% 0.14/0.36 ! [A: $tType,X3: A,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ( ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
% 0.14/0.36 = ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 0.14/0.36 = ( ( ( A4
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) )
% 0.14/0.36 & ( B3
% 0.14/0.36 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.14/0.36 | ( ( A4
% 0.14/0.36 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.14/0.36 & ( B3
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) ) )
% 0.14/0.36 | ( ( A4
% 0.14/0.36 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.14/0.36 & ( B3
% 0.14/0.36 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % singleton_Un_iff
% 0.14/0.36 thf(fact_101_Un__singleton__iff,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,B3: set @ A,X3: A] :
% 0.14/0.36 ( ( ( sup_sup @ ( set @ A ) @ A4 @ B3 )
% 0.14/0.36 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.14/0.36 = ( ( ( A4
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) )
% 0.14/0.36 & ( B3
% 0.14/0.36 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.14/0.36 | ( ( A4
% 0.14/0.36 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.14/0.36 & ( B3
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) ) )
% 0.14/0.36 | ( ( A4
% 0.14/0.36 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.14/0.36 & ( B3
% 0.14/0.36 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Un_singleton_iff
% 0.14/0.36 thf(fact_102_insert__is__Un,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( insert @ A )
% 0.14/0.36 = ( ^ [A3: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % insert_is_Un
% 0.14/0.36 thf(fact_103_Diff__insert__absorb,axiom,
% 0.14/0.36 ! [A: $tType,X3: A,A4: set @ A] :
% 0.14/0.36 ( ~ ( member @ A @ X3 @ A4 )
% 0.14/0.36 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A4 ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.14/0.36 = A4 ) ) ).
% 0.14/0.36
% 0.14/0.36 % Diff_insert_absorb
% 0.14/0.36 thf(fact_104_Diff__insert2,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,A2: A,B3: set @ A] :
% 0.14/0.36 ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B3 ) )
% 0.14/0.36 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ).
% 0.14/0.36
% 0.14/0.36 % Diff_insert2
% 0.14/0.36 thf(fact_105_insert__Diff,axiom,
% 0.14/0.36 ! [A: $tType,A2: A,A4: set @ A] :
% 0.14/0.36 ( ( member @ A @ A2 @ A4 )
% 0.14/0.36 => ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.14/0.36 = A4 ) ) ).
% 0.14/0.36
% 0.14/0.36 % insert_Diff
% 0.14/0.36 thf(fact_106_Diff__insert,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,A2: A,B3: set @ A] :
% 0.14/0.36 ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B3 ) )
% 0.14/0.36 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Diff_insert
% 0.14/0.36 thf(fact_107_sup__bot_Oright__neutral,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( bounde1808546759up_bot @ A )
% 0.14/0.36 => ! [A2: A] :
% 0.14/0.36 ( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
% 0.14/0.36 = A2 ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup_bot.right_neutral
% 0.14/0.36 thf(fact_108_sup__bot_Oneutr__eq__iff,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( bounde1808546759up_bot @ A )
% 0.14/0.36 => ! [A2: A,B: A] :
% 0.14/0.36 ( ( ( bot_bot @ A )
% 0.14/0.36 = ( sup_sup @ A @ A2 @ B ) )
% 0.14/0.36 = ( ( A2
% 0.14/0.36 = ( bot_bot @ A ) )
% 0.14/0.36 & ( B
% 0.14/0.36 = ( bot_bot @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup_bot.neutr_eq_iff
% 0.14/0.36 thf(fact_109_sup__bot_Oleft__neutral,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( bounde1808546759up_bot @ A )
% 0.14/0.36 => ! [A2: A] :
% 0.14/0.36 ( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
% 0.14/0.36 = A2 ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup_bot.left_neutral
% 0.14/0.36 thf(fact_110_sup__bot_Oeq__neutr__iff,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( bounde1808546759up_bot @ A )
% 0.14/0.36 => ! [A2: A,B: A] :
% 0.14/0.36 ( ( ( sup_sup @ A @ A2 @ B )
% 0.14/0.36 = ( bot_bot @ A ) )
% 0.14/0.36 = ( ( A2
% 0.14/0.36 = ( bot_bot @ A ) )
% 0.14/0.36 & ( B
% 0.14/0.36 = ( bot_bot @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup_bot.eq_neutr_iff
% 0.14/0.36 thf(fact_111_sup__eq__bot__iff,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( bounde1808546759up_bot @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ( ( sup_sup @ A @ X3 @ Y2 )
% 0.14/0.36 = ( bot_bot @ A ) )
% 0.14/0.36 = ( ( X3
% 0.14/0.36 = ( bot_bot @ A ) )
% 0.14/0.36 & ( Y2
% 0.14/0.36 = ( bot_bot @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup_eq_bot_iff
% 0.14/0.36 thf(fact_112_bot__eq__sup__iff,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( bounde1808546759up_bot @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ( ( bot_bot @ A )
% 0.14/0.36 = ( sup_sup @ A @ X3 @ Y2 ) )
% 0.14/0.36 = ( ( X3
% 0.14/0.36 = ( bot_bot @ A ) )
% 0.14/0.36 & ( Y2
% 0.14/0.36 = ( bot_bot @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % bot_eq_sup_iff
% 0.14/0.36 thf(fact_113_sup__bot__right,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( bounde1808546759up_bot @ A )
% 0.14/0.36 => ! [X3: A] :
% 0.14/0.36 ( ( sup_sup @ A @ X3 @ ( bot_bot @ A ) )
% 0.14/0.36 = X3 ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup_bot_right
% 0.14/0.36 thf(fact_114_sup__bot__left,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( bounde1808546759up_bot @ A )
% 0.14/0.36 => ! [X3: A] :
% 0.14/0.36 ( ( sup_sup @ A @ ( bot_bot @ A ) @ X3 )
% 0.14/0.36 = X3 ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup_bot_left
% 0.14/0.36 thf(fact_115_memb__spec,axiom,
% 0.14/0.36 ! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,X3: A] :
% 0.14/0.36 ( ( binary1610619414edTree @ A @ H @ T )
% 0.14/0.36 => ( ( binary827270440e_memb @ A @ H @ X3 @ T )
% 0.14/0.36 = ( member @ A @ X3 @ ( binary1653327646_setOf @ A @ T ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % memb_spec
% 0.14/0.36 thf(fact_116_minus__apply,axiom,
% 0.14/0.36 ! [B4: $tType,A: $tType] :
% 0.14/0.36 ( ( minus @ B4 )
% 0.14/0.36 => ( ( minus_minus @ ( A > B4 ) )
% 0.14/0.36 = ( ^ [A5: A > B4,B6: A > B4,X2: A] : ( minus_minus @ B4 @ ( A5 @ X2 ) @ ( B6 @ X2 ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % minus_apply
% 0.14/0.36 thf(fact_117_sup__apply,axiom,
% 0.14/0.36 ! [B4: $tType,A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ B4 )
% 0.14/0.36 => ( ( sup_sup @ ( A > B4 ) )
% 0.14/0.36 = ( ^ [F2: A > B4,G2: A > B4,X2: A] : ( sup_sup @ B4 @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup_apply
% 0.14/0.36 thf(fact_118_sup_Oidem,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ! [A2: A] :
% 0.14/0.36 ( ( sup_sup @ A @ A2 @ A2 )
% 0.14/0.36 = A2 ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup.idem
% 0.14/0.36 thf(fact_119_sup__idem,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ! [X3: A] :
% 0.14/0.36 ( ( sup_sup @ A @ X3 @ X3 )
% 0.14/0.36 = X3 ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup_idem
% 0.14/0.36 thf(fact_120_sup_Oleft__idem,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ! [A2: A,B: A] :
% 0.14/0.36 ( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B ) )
% 0.14/0.36 = ( sup_sup @ A @ A2 @ B ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup.left_idem
% 0.14/0.36 thf(fact_121_sup__left__idem,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) )
% 0.14/0.36 = ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup_left_idem
% 0.14/0.36 thf(fact_122_sup_Oright__idem,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ! [A2: A,B: A] :
% 0.14/0.36 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B ) @ B )
% 0.14/0.36 = ( sup_sup @ A @ A2 @ B ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup.right_idem
% 0.14/0.36 thf(fact_123_bot__set__def,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( bot_bot @ ( set @ A ) )
% 0.14/0.36 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % bot_set_def
% 0.14/0.36 thf(fact_124_sup__set__def,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( sup_sup @ ( set @ A ) )
% 0.14/0.36 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.14/0.36 ( collect @ A
% 0.14/0.36 @ ( sup_sup @ ( A > $o )
% 0.14/0.36 @ ^ [X2: A] : ( member @ A @ X2 @ A5 )
% 0.14/0.36 @ ^ [X2: A] : ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup_set_def
% 0.14/0.36 thf(fact_125_fun__diff__def,axiom,
% 0.14/0.36 ! [B4: $tType,A: $tType] :
% 0.14/0.36 ( ( minus @ B4 )
% 0.14/0.36 => ( ( minus_minus @ ( A > B4 ) )
% 0.14/0.36 = ( ^ [A5: A > B4,B6: A > B4,X2: A] : ( minus_minus @ B4 @ ( A5 @ X2 ) @ ( B6 @ X2 ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % fun_diff_def
% 0.14/0.36 thf(fact_126_inf__sup__aci_I8_J,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( lattice @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) )
% 0.14/0.36 = ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % inf_sup_aci(8)
% 0.14/0.36 thf(fact_127_inf__sup__aci_I7_J,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( lattice @ A )
% 0.14/0.36 => ! [X3: A,Y2: A,Z2: A] :
% 0.14/0.36 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z2 ) )
% 0.14/0.36 = ( sup_sup @ A @ Y2 @ ( sup_sup @ A @ X3 @ Z2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % inf_sup_aci(7)
% 0.14/0.36 thf(fact_128_inf__sup__aci_I6_J,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( lattice @ A )
% 0.14/0.36 => ! [X3: A,Y2: A,Z2: A] :
% 0.14/0.36 ( ( sup_sup @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z2 )
% 0.14/0.36 = ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % inf_sup_aci(6)
% 0.14/0.36 thf(fact_129_inf__sup__aci_I5_J,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( lattice @ A )
% 0.14/0.36 => ( ( sup_sup @ A )
% 0.14/0.36 = ( ^ [X2: A,Y: A] : ( sup_sup @ A @ Y @ X2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % inf_sup_aci(5)
% 0.14/0.36 thf(fact_130_sup__fun__def,axiom,
% 0.14/0.36 ! [B4: $tType,A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ B4 )
% 0.14/0.36 => ( ( sup_sup @ ( A > B4 ) )
% 0.14/0.36 = ( ^ [F2: A > B4,G2: A > B4,X2: A] : ( sup_sup @ B4 @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup_fun_def
% 0.14/0.36 thf(fact_131_boolean__algebra__cancel_Osup1,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ! [A4: A,K: A,A2: A,B: A] :
% 0.14/0.36 ( ( A4
% 0.14/0.36 = ( sup_sup @ A @ K @ A2 ) )
% 0.14/0.36 => ( ( sup_sup @ A @ A4 @ B )
% 0.14/0.36 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % boolean_algebra_cancel.sup1
% 0.14/0.36 thf(fact_132_boolean__algebra__cancel_Osup2,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ! [B3: A,K: A,B: A,A2: A] :
% 0.14/0.36 ( ( B3
% 0.14/0.36 = ( sup_sup @ A @ K @ B ) )
% 0.14/0.36 => ( ( sup_sup @ A @ A2 @ B3 )
% 0.14/0.36 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % boolean_algebra_cancel.sup2
% 0.14/0.36 thf(fact_133_sup_Oassoc,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ! [A2: A,B: A,C2: A] :
% 0.14/0.36 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B ) @ C2 )
% 0.14/0.36 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B @ C2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup.assoc
% 0.14/0.36 thf(fact_134_sup__assoc,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ! [X3: A,Y2: A,Z2: A] :
% 0.14/0.36 ( ( sup_sup @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z2 )
% 0.14/0.36 = ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup_assoc
% 0.14/0.36 thf(fact_135_sup_Ocommute,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ( ( sup_sup @ A )
% 0.14/0.36 = ( ^ [A3: A,B2: A] : ( sup_sup @ A @ B2 @ A3 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup.commute
% 0.14/0.36 thf(fact_136_sup__commute,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ( ( sup_sup @ A )
% 0.14/0.36 = ( ^ [X2: A,Y: A] : ( sup_sup @ A @ Y @ X2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup_commute
% 0.14/0.36 thf(fact_137_sup_Oleft__commute,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ! [B: A,A2: A,C2: A] :
% 0.14/0.36 ( ( sup_sup @ A @ B @ ( sup_sup @ A @ A2 @ C2 ) )
% 0.14/0.36 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B @ C2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup.left_commute
% 0.14/0.36 thf(fact_138_sup__left__commute,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ! [X3: A,Y2: A,Z2: A] :
% 0.14/0.36 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z2 ) )
% 0.14/0.36 = ( sup_sup @ A @ Y2 @ ( sup_sup @ A @ X3 @ Z2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup_left_commute
% 0.14/0.36 thf(fact_139_less__supI1,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ! [X3: A,A2: A,B: A] :
% 0.14/0.36 ( ( ord_less @ A @ X3 @ A2 )
% 0.14/0.36 => ( ord_less @ A @ X3 @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % less_supI1
% 0.14/0.36 thf(fact_140_less__supI2,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ! [X3: A,B: A,A2: A] :
% 0.14/0.36 ( ( ord_less @ A @ X3 @ B )
% 0.14/0.36 => ( ord_less @ A @ X3 @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % less_supI2
% 0.14/0.36 thf(fact_141_sup_Ostrict__boundedE,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ! [B: A,C2: A,A2: A] :
% 0.14/0.36 ( ( ord_less @ A @ ( sup_sup @ A @ B @ C2 ) @ A2 )
% 0.14/0.36 => ~ ( ( ord_less @ A @ B @ A2 )
% 0.14/0.36 => ~ ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup.strict_boundedE
% 0.14/0.36 thf(fact_142_sup_Ostrict__order__iff,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ( ( ord_less @ A )
% 0.14/0.36 = ( ^ [B2: A,A3: A] :
% 0.14/0.36 ( ( A3
% 0.14/0.36 = ( sup_sup @ A @ A3 @ B2 ) )
% 0.14/0.36 & ( A3 != B2 ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup.strict_order_iff
% 0.14/0.36 thf(fact_143_sup_Ostrict__coboundedI1,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ! [C2: A,A2: A,B: A] :
% 0.14/0.36 ( ( ord_less @ A @ C2 @ A2 )
% 0.14/0.36 => ( ord_less @ A @ C2 @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup.strict_coboundedI1
% 0.14/0.36 thf(fact_144_sup_Ostrict__coboundedI2,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ! [C2: A,B: A,A2: A] :
% 0.14/0.36 ( ( ord_less @ A @ C2 @ B )
% 0.14/0.36 => ( ord_less @ A @ C2 @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup.strict_coboundedI2
% 0.14/0.36 thf(fact_145_the__elem__eq,axiom,
% 0.14/0.36 ! [A: $tType,X3: A] :
% 0.14/0.36 ( ( the_elem @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.14/0.36 = X3 ) ).
% 0.14/0.36
% 0.14/0.36 % the_elem_eq
% 0.14/0.36 thf(fact_146_bot__apply,axiom,
% 0.14/0.36 ! [C4: $tType,D2: $tType] :
% 0.14/0.36 ( ( bot @ C4 )
% 0.14/0.36 => ( ( bot_bot @ ( D2 > C4 ) )
% 0.14/0.36 = ( ^ [X2: D2] : ( bot_bot @ C4 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % bot_apply
% 0.14/0.36 thf(fact_147_is__singletonI,axiom,
% 0.14/0.36 ! [A: $tType,X3: A] : ( is_singleton @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % is_singletonI
% 0.14/0.36 thf(fact_148_diff__strict__mono,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( ordered_ab_group_add @ A )
% 0.14/0.36 => ! [A2: A,B: A,D: A,C2: A] :
% 0.14/0.36 ( ( ord_less @ A @ A2 @ B )
% 0.14/0.36 => ( ( ord_less @ A @ D @ C2 )
% 0.14/0.36 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % diff_strict_mono
% 0.14/0.36 thf(fact_149_sup1CI,axiom,
% 0.14/0.36 ! [A: $tType,B3: A > $o,X3: A,A4: A > $o] :
% 0.14/0.36 ( ( ~ ( B3 @ X3 )
% 0.14/0.36 => ( A4 @ X3 ) )
% 0.14/0.36 => ( sup_sup @ ( A > $o ) @ A4 @ B3 @ X3 ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup1CI
% 0.14/0.36 thf(fact_150_sup1E,axiom,
% 0.14/0.36 ! [A: $tType,A4: A > $o,B3: A > $o,X3: A] :
% 0.14/0.36 ( ( sup_sup @ ( A > $o ) @ A4 @ B3 @ X3 )
% 0.14/0.36 => ( ~ ( A4 @ X3 )
% 0.14/0.36 => ( B3 @ X3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup1E
% 0.14/0.36 thf(fact_151_sup1I1,axiom,
% 0.14/0.36 ! [A: $tType,A4: A > $o,X3: A,B3: A > $o] :
% 0.14/0.36 ( ( A4 @ X3 )
% 0.14/0.36 => ( sup_sup @ ( A > $o ) @ A4 @ B3 @ X3 ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup1I1
% 0.14/0.36 thf(fact_152_sup1I2,axiom,
% 0.14/0.36 ! [A: $tType,B3: A > $o,X3: A,A4: A > $o] :
% 0.14/0.36 ( ( B3 @ X3 )
% 0.14/0.36 => ( sup_sup @ ( A > $o ) @ A4 @ B3 @ X3 ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup1I2
% 0.14/0.36 thf(fact_153_not__psubset__empty,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A] :
% 0.14/0.36 ~ ( ord_less @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % not_psubset_empty
% 0.14/0.36 thf(fact_154_psubset__imp__ex__mem,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
% 0.14/0.36 => ? [B7: A] : ( member @ A @ B7 @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % psubset_imp_ex_mem
% 0.14/0.36 thf(fact_155_minus__set__def,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( minus_minus @ ( set @ A ) )
% 0.14/0.36 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.14/0.36 ( collect @ A
% 0.14/0.36 @ ( minus_minus @ ( A > $o )
% 0.14/0.36 @ ^ [X2: A] : ( member @ A @ X2 @ A5 )
% 0.14/0.36 @ ^ [X2: A] : ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % minus_set_def
% 0.14/0.36 thf(fact_156_is__singleton__the__elem,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( is_singleton @ A )
% 0.14/0.36 = ( ^ [A5: set @ A] :
% 0.14/0.36 ( A5
% 0.14/0.36 = ( insert @ A @ ( the_elem @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % is_singleton_the_elem
% 0.14/0.36 thf(fact_157_is__singletonI_H,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A] :
% 0.14/0.36 ( ( A4
% 0.14/0.36 != ( bot_bot @ ( set @ A ) ) )
% 0.14/0.36 => ( ! [X4: A,Y4: A] :
% 0.14/0.36 ( ( member @ A @ X4 @ A4 )
% 0.14/0.36 => ( ( member @ A @ Y4 @ A4 )
% 0.14/0.36 => ( X4 = Y4 ) ) )
% 0.14/0.36 => ( is_singleton @ A @ A4 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % is_singletonI'
% 0.14/0.36 thf(fact_158_dual__order_Ostrict__implies__not__eq,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( order @ A )
% 0.14/0.36 => ! [B: A,A2: A] :
% 0.14/0.36 ( ( ord_less @ A @ B @ A2 )
% 0.14/0.36 => ( A2 != B ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % dual_order.strict_implies_not_eq
% 0.14/0.36 thf(fact_159_order_Ostrict__implies__not__eq,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( order @ A )
% 0.14/0.36 => ! [A2: A,B: A] :
% 0.14/0.36 ( ( ord_less @ A @ A2 @ B )
% 0.14/0.36 => ( A2 != B ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % order.strict_implies_not_eq
% 0.14/0.36 thf(fact_160_not__less__iff__gr__or__eq,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( linorder @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 0.14/0.36 = ( ( ord_less @ A @ Y2 @ X3 )
% 0.14/0.36 | ( X3 = Y2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % not_less_iff_gr_or_eq
% 0.14/0.36 thf(fact_161_dual__order_Ostrict__trans,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( order @ A )
% 0.14/0.36 => ! [B: A,A2: A,C2: A] :
% 0.14/0.36 ( ( ord_less @ A @ B @ A2 )
% 0.14/0.36 => ( ( ord_less @ A @ C2 @ B )
% 0.14/0.36 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % dual_order.strict_trans
% 0.14/0.36 thf(fact_162_linorder__less__wlog,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( linorder @ A )
% 0.14/0.36 => ! [P: A > A > $o,A2: A,B: A] :
% 0.14/0.36 ( ! [A6: A,B7: A] :
% 0.14/0.36 ( ( ord_less @ A @ A6 @ B7 )
% 0.14/0.36 => ( P @ A6 @ B7 ) )
% 0.14/0.36 => ( ! [A6: A] : ( P @ A6 @ A6 )
% 0.14/0.36 => ( ! [A6: A,B7: A] :
% 0.14/0.36 ( ( P @ B7 @ A6 )
% 0.14/0.36 => ( P @ A6 @ B7 ) )
% 0.14/0.36 => ( P @ A2 @ B ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % linorder_less_wlog
% 0.14/0.36 thf(fact_163_exists__least__iff,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( wellorder @ A )
% 0.14/0.36 => ( ( ^ [P2: A > $o] :
% 0.14/0.36 ? [X5: A] : ( P2 @ X5 ) )
% 0.14/0.36 = ( ^ [P3: A > $o] :
% 0.14/0.36 ? [N: A] :
% 0.14/0.36 ( ( P3 @ N )
% 0.14/0.36 & ! [M: A] :
% 0.14/0.36 ( ( ord_less @ A @ M @ N )
% 0.14/0.36 => ~ ( P3 @ M ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % exists_least_iff
% 0.14/0.36 thf(fact_164_less__imp__not__less,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( preorder @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ( ord_less @ A @ X3 @ Y2 )
% 0.14/0.36 => ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % less_imp_not_less
% 0.14/0.36 thf(fact_165_order_Ostrict__trans,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( order @ A )
% 0.14/0.36 => ! [A2: A,B: A,C2: A] :
% 0.14/0.36 ( ( ord_less @ A @ A2 @ B )
% 0.14/0.36 => ( ( ord_less @ A @ B @ C2 )
% 0.14/0.36 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % order.strict_trans
% 0.14/0.36 thf(fact_166_dual__order_Oirrefl,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( order @ A )
% 0.14/0.36 => ! [A2: A] :
% 0.14/0.36 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.14/0.36
% 0.14/0.36 % dual_order.irrefl
% 0.14/0.36 thf(fact_167_linorder__cases,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( linorder @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ~ ( ord_less @ A @ X3 @ Y2 )
% 0.14/0.36 => ( ( X3 != Y2 )
% 0.14/0.36 => ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % linorder_cases
% 0.14/0.36 thf(fact_168_less__imp__triv,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( preorder @ A )
% 0.14/0.36 => ! [X3: A,Y2: A,P: $o] :
% 0.14/0.36 ( ( ord_less @ A @ X3 @ Y2 )
% 0.14/0.36 => ( ( ord_less @ A @ Y2 @ X3 )
% 0.14/0.36 => P ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % less_imp_triv
% 0.14/0.36 thf(fact_169_less__imp__not__eq2,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( order @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ( ord_less @ A @ X3 @ Y2 )
% 0.14/0.36 => ( Y2 != X3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % less_imp_not_eq2
% 0.14/0.36 thf(fact_170_antisym__conv3,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( linorder @ A )
% 0.14/0.36 => ! [Y2: A,X3: A] :
% 0.14/0.36 ( ~ ( ord_less @ A @ Y2 @ X3 )
% 0.14/0.36 => ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 0.14/0.36 = ( X3 = Y2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % antisym_conv3
% 0.14/0.36 thf(fact_171_less__induct,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( wellorder @ A )
% 0.14/0.36 => ! [P: A > $o,A2: A] :
% 0.14/0.36 ( ! [X4: A] :
% 0.14/0.36 ( ! [Y5: A] :
% 0.14/0.36 ( ( ord_less @ A @ Y5 @ X4 )
% 0.14/0.36 => ( P @ Y5 ) )
% 0.14/0.36 => ( P @ X4 ) )
% 0.14/0.36 => ( P @ A2 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % less_induct
% 0.14/0.36 thf(fact_172_less__not__sym,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( preorder @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ( ord_less @ A @ X3 @ Y2 )
% 0.14/0.36 => ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % less_not_sym
% 0.14/0.36 thf(fact_173_less__imp__not__eq,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( order @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ( ord_less @ A @ X3 @ Y2 )
% 0.14/0.36 => ( X3 != Y2 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % less_imp_not_eq
% 0.14/0.36 thf(fact_174_dual__order_Oasym,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( order @ A )
% 0.14/0.36 => ! [B: A,A2: A] :
% 0.14/0.36 ( ( ord_less @ A @ B @ A2 )
% 0.14/0.36 => ~ ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % dual_order.asym
% 0.14/0.36 thf(fact_175_ord__less__eq__trans,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( ord @ A )
% 0.14/0.36 => ! [A2: A,B: A,C2: A] :
% 0.14/0.36 ( ( ord_less @ A @ A2 @ B )
% 0.14/0.36 => ( ( B = C2 )
% 0.14/0.36 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % ord_less_eq_trans
% 0.14/0.36 thf(fact_176_ord__eq__less__trans,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( ord @ A )
% 0.14/0.36 => ! [A2: A,B: A,C2: A] :
% 0.14/0.36 ( ( A2 = B )
% 0.14/0.36 => ( ( ord_less @ A @ B @ C2 )
% 0.14/0.36 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % ord_eq_less_trans
% 0.14/0.36 thf(fact_177_less__irrefl,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( preorder @ A )
% 0.14/0.36 => ! [X3: A] :
% 0.14/0.36 ~ ( ord_less @ A @ X3 @ X3 ) ) ).
% 0.14/0.36
% 0.14/0.36 % less_irrefl
% 0.14/0.36 thf(fact_178_less__linear,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( linorder @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ( ord_less @ A @ X3 @ Y2 )
% 0.14/0.36 | ( X3 = Y2 )
% 0.14/0.36 | ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % less_linear
% 0.14/0.36 thf(fact_179_less__trans,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( preorder @ A )
% 0.14/0.36 => ! [X3: A,Y2: A,Z2: A] :
% 0.14/0.36 ( ( ord_less @ A @ X3 @ Y2 )
% 0.14/0.36 => ( ( ord_less @ A @ Y2 @ Z2 )
% 0.14/0.36 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % less_trans
% 0.14/0.36 thf(fact_180_less__asym_H,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( preorder @ A )
% 0.14/0.36 => ! [A2: A,B: A] :
% 0.14/0.36 ( ( ord_less @ A @ A2 @ B )
% 0.14/0.36 => ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % less_asym'
% 0.14/0.36 thf(fact_181_less__asym,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( preorder @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ( ord_less @ A @ X3 @ Y2 )
% 0.14/0.36 => ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % less_asym
% 0.14/0.36 thf(fact_182_less__imp__neq,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( order @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ( ord_less @ A @ X3 @ Y2 )
% 0.14/0.36 => ( X3 != Y2 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % less_imp_neq
% 0.14/0.36 thf(fact_183_dense,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( dense_order @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ( ord_less @ A @ X3 @ Y2 )
% 0.14/0.36 => ? [Z3: A] :
% 0.14/0.36 ( ( ord_less @ A @ X3 @ Z3 )
% 0.14/0.36 & ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % dense
% 0.14/0.36 thf(fact_184_order_Oasym,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( order @ A )
% 0.14/0.36 => ! [A2: A,B: A] :
% 0.14/0.36 ( ( ord_less @ A @ A2 @ B )
% 0.14/0.36 => ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % order.asym
% 0.14/0.36 thf(fact_185_neq__iff,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( linorder @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ( X3 != Y2 )
% 0.14/0.36 = ( ( ord_less @ A @ X3 @ Y2 )
% 0.14/0.36 | ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % neq_iff
% 0.14/0.36 thf(fact_186_neqE,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( linorder @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ( X3 != Y2 )
% 0.14/0.36 => ( ~ ( ord_less @ A @ X3 @ Y2 )
% 0.14/0.36 => ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % neqE
% 0.14/0.36 thf(fact_187_gt__ex,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( no_top @ A )
% 0.14/0.36 => ! [X3: A] :
% 0.14/0.36 ? [X_1: A] : ( ord_less @ A @ X3 @ X_1 ) ) ).
% 0.14/0.36
% 0.14/0.36 % gt_ex
% 0.14/0.36 thf(fact_188_lt__ex,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( no_bot @ A )
% 0.14/0.36 => ! [X3: A] :
% 0.14/0.36 ? [Y4: A] : ( ord_less @ A @ Y4 @ X3 ) ) ).
% 0.14/0.36
% 0.14/0.36 % lt_ex
% 0.14/0.36 thf(fact_189_order__less__subst2,axiom,
% 0.14/0.36 ! [A: $tType,C4: $tType] :
% 0.14/0.36 ( ( ( order @ C4 )
% 0.14/0.36 & ( order @ A ) )
% 0.14/0.36 => ! [A2: A,B: A,F: A > C4,C2: C4] :
% 0.14/0.36 ( ( ord_less @ A @ A2 @ B )
% 0.14/0.36 => ( ( ord_less @ C4 @ ( F @ B ) @ C2 )
% 0.14/0.36 => ( ! [X4: A,Y4: A] :
% 0.14/0.36 ( ( ord_less @ A @ X4 @ Y4 )
% 0.14/0.36 => ( ord_less @ C4 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.14/0.36 => ( ord_less @ C4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % order_less_subst2
% 0.14/0.36 thf(fact_190_order__less__subst1,axiom,
% 0.14/0.36 ! [A: $tType,B4: $tType] :
% 0.14/0.36 ( ( ( order @ B4 )
% 0.14/0.36 & ( order @ A ) )
% 0.14/0.36 => ! [A2: A,F: B4 > A,B: B4,C2: B4] :
% 0.14/0.36 ( ( ord_less @ A @ A2 @ ( F @ B ) )
% 0.14/0.36 => ( ( ord_less @ B4 @ B @ C2 )
% 0.14/0.36 => ( ! [X4: B4,Y4: B4] :
% 0.14/0.36 ( ( ord_less @ B4 @ X4 @ Y4 )
% 0.14/0.36 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.14/0.36 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % order_less_subst1
% 0.14/0.36 thf(fact_191_ord__less__eq__subst,axiom,
% 0.14/0.36 ! [A: $tType,B4: $tType] :
% 0.14/0.36 ( ( ( ord @ B4 )
% 0.14/0.36 & ( ord @ A ) )
% 0.14/0.36 => ! [A2: A,B: A,F: A > B4,C2: B4] :
% 0.14/0.36 ( ( ord_less @ A @ A2 @ B )
% 0.14/0.36 => ( ( ( F @ B )
% 0.14/0.36 = C2 )
% 0.14/0.36 => ( ! [X4: A,Y4: A] :
% 0.14/0.36 ( ( ord_less @ A @ X4 @ Y4 )
% 0.14/0.36 => ( ord_less @ B4 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.14/0.36 => ( ord_less @ B4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % ord_less_eq_subst
% 0.14/0.36 thf(fact_192_ord__eq__less__subst,axiom,
% 0.14/0.36 ! [A: $tType,B4: $tType] :
% 0.14/0.36 ( ( ( ord @ B4 )
% 0.14/0.36 & ( ord @ A ) )
% 0.14/0.36 => ! [A2: A,F: B4 > A,B: B4,C2: B4] :
% 0.14/0.36 ( ( A2
% 0.14/0.36 = ( F @ B ) )
% 0.14/0.36 => ( ( ord_less @ B4 @ B @ C2 )
% 0.14/0.36 => ( ! [X4: B4,Y4: B4] :
% 0.14/0.36 ( ( ord_less @ B4 @ X4 @ Y4 )
% 0.14/0.36 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.14/0.36 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % ord_eq_less_subst
% 0.14/0.36 thf(fact_193_bot__fun__def,axiom,
% 0.14/0.36 ! [B4: $tType,A: $tType] :
% 0.14/0.36 ( ( bot @ B4 )
% 0.14/0.36 => ( ( bot_bot @ ( A > B4 ) )
% 0.14/0.36 = ( ^ [X2: A] : ( bot_bot @ B4 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % bot_fun_def
% 0.14/0.36 thf(fact_194_diff__right__commute,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( cancel146912293up_add @ A )
% 0.14/0.36 => ! [A2: A,C2: A,B: A] :
% 0.14/0.36 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B )
% 0.14/0.36 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C2 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % diff_right_commute
% 0.14/0.36 thf(fact_195_diff__eq__diff__eq,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( group_add @ A )
% 0.14/0.36 => ! [A2: A,B: A,C2: A,D: A] :
% 0.14/0.36 ( ( ( minus_minus @ A @ A2 @ B )
% 0.14/0.36 = ( minus_minus @ A @ C2 @ D ) )
% 0.14/0.36 => ( ( A2 = B )
% 0.14/0.36 = ( C2 = D ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % diff_eq_diff_eq
% 0.14/0.36 thf(fact_196_is__singleton__def,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( is_singleton @ A )
% 0.14/0.36 = ( ^ [A5: set @ A] :
% 0.14/0.36 ? [X2: A] :
% 0.14/0.36 ( A5
% 0.14/0.36 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % is_singleton_def
% 0.14/0.36 thf(fact_197_is__singletonE,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A] :
% 0.14/0.36 ( ( is_singleton @ A @ A4 )
% 0.14/0.36 => ~ ! [X4: A] :
% 0.14/0.36 ( A4
% 0.14/0.36 != ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % is_singletonE
% 0.14/0.36 thf(fact_198_bot_Onot__eq__extremum,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( order_bot @ A )
% 0.14/0.36 => ! [A2: A] :
% 0.14/0.36 ( ( A2
% 0.14/0.36 != ( bot_bot @ A ) )
% 0.14/0.36 = ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % bot.not_eq_extremum
% 0.14/0.36 thf(fact_199_bot_Oextremum__strict,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( order_bot @ A )
% 0.14/0.36 => ! [A2: A] :
% 0.14/0.36 ~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % bot.extremum_strict
% 0.14/0.36 thf(fact_200_diff__strict__right__mono,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( ordered_ab_group_add @ A )
% 0.14/0.36 => ! [A2: A,B: A,C2: A] :
% 0.14/0.36 ( ( ord_less @ A @ A2 @ B )
% 0.14/0.36 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B @ C2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % diff_strict_right_mono
% 0.14/0.36 thf(fact_201_diff__strict__left__mono,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( ordered_ab_group_add @ A )
% 0.14/0.36 => ! [B: A,A2: A,C2: A] :
% 0.14/0.36 ( ( ord_less @ A @ B @ A2 )
% 0.14/0.36 => ( ord_less @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % diff_strict_left_mono
% 0.14/0.36 thf(fact_202_diff__eq__diff__less,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( ordered_ab_group_add @ A )
% 0.14/0.36 => ! [A2: A,B: A,C2: A,D: A] :
% 0.14/0.36 ( ( ( minus_minus @ A @ A2 @ B )
% 0.14/0.36 = ( minus_minus @ A @ C2 @ D ) )
% 0.14/0.36 => ( ( ord_less @ A @ A2 @ B )
% 0.14/0.36 = ( ord_less @ A @ C2 @ D ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % diff_eq_diff_less
% 0.14/0.36 thf(fact_203_sup__Un__eq,axiom,
% 0.14/0.36 ! [A: $tType,R: set @ A,S: set @ A] :
% 0.14/0.36 ( ( sup_sup @ ( A > $o )
% 0.14/0.36 @ ^ [X2: A] : ( member @ A @ X2 @ R )
% 0.14/0.36 @ ^ [X2: A] : ( member @ A @ X2 @ S ) )
% 0.14/0.36 = ( ^ [X2: A] : ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ R @ S ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup_Un_eq
% 0.14/0.36 thf(fact_204_bot__empty__eq,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( bot_bot @ ( A > $o ) )
% 0.14/0.36 = ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % bot_empty_eq
% 0.14/0.36 thf(fact_205_Collect__empty__eq__bot,axiom,
% 0.14/0.36 ! [A: $tType,P: A > $o] :
% 0.14/0.36 ( ( ( collect @ A @ P )
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) )
% 0.14/0.36 = ( P
% 0.14/0.36 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Collect_empty_eq_bot
% 0.14/0.36 thf(fact_206_psubset__trans,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,B3: set @ A,C: set @ A] :
% 0.14/0.36 ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
% 0.14/0.36 => ( ( ord_less @ ( set @ A ) @ B3 @ C )
% 0.14/0.36 => ( ord_less @ ( set @ A ) @ A4 @ C ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % psubset_trans
% 0.14/0.36 thf(fact_207_less__set__def,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( ord_less @ ( set @ A ) )
% 0.14/0.36 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.14/0.36 ( ord_less @ ( A > $o )
% 0.14/0.36 @ ^ [X2: A] : ( member @ A @ X2 @ A5 )
% 0.14/0.36 @ ^ [X2: A] : ( member @ A @ X2 @ B6 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % less_set_def
% 0.14/0.36 thf(fact_208_psubsetD,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,B3: set @ A,C2: A] :
% 0.14/0.36 ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
% 0.14/0.36 => ( ( member @ A @ C2 @ A4 )
% 0.14/0.36 => ( member @ A @ C2 @ B3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % psubsetD
% 0.14/0.36 thf(fact_209_remove__def,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( remove @ A )
% 0.14/0.36 = ( ^ [X2: A,A5: set @ A] : ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % remove_def
% 0.14/0.36 thf(fact_210_Tree_Osimps_I15_J,axiom,
% 0.14/0.36 ! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
% 0.14/0.36 ( ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) )
% 0.14/0.36 = ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( binary2130109271t_Tree @ A @ X21 ) @ ( insert @ A @ X22 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( binary2130109271t_Tree @ A @ X23 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Tree.simps(15)
% 0.14/0.36 thf(fact_211_member__remove,axiom,
% 0.14/0.36 ! [A: $tType,X3: A,Y2: A,A4: set @ A] :
% 0.14/0.36 ( ( member @ A @ X3 @ ( remove @ A @ Y2 @ A4 ) )
% 0.14/0.36 = ( ( member @ A @ X3 @ A4 )
% 0.14/0.36 & ( X3 != Y2 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % member_remove
% 0.14/0.36 thf(fact_212_Tree_Oset__intros_I3_J,axiom,
% 0.14/0.36 ! [A: $tType,Ya: A,X23: binary1291135688e_Tree @ A,X21: binary1291135688e_Tree @ A,X22: A] :
% 0.14/0.36 ( ( member @ A @ Ya @ ( binary2130109271t_Tree @ A @ X23 ) )
% 0.14/0.36 => ( member @ A @ Ya @ ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Tree.set_intros(3)
% 0.14/0.36 thf(fact_213_Tree_Oset__intros_I2_J,axiom,
% 0.14/0.36 ! [A: $tType,X22: A,X21: binary1291135688e_Tree @ A,X23: binary1291135688e_Tree @ A] : ( member @ A @ X22 @ ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Tree.set_intros(2)
% 0.14/0.36 thf(fact_214_Tree_Oset__intros_I1_J,axiom,
% 0.14/0.36 ! [A: $tType,Y2: A,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
% 0.14/0.36 ( ( member @ A @ Y2 @ ( binary2130109271t_Tree @ A @ X21 ) )
% 0.14/0.36 => ( member @ A @ Y2 @ ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Tree.set_intros(1)
% 0.14/0.36 thf(fact_215_Tree_Oset__cases,axiom,
% 0.14/0.36 ! [A: $tType,E: A,A2: binary1291135688e_Tree @ A] :
% 0.14/0.36 ( ( member @ A @ E @ ( binary2130109271t_Tree @ A @ A2 ) )
% 0.14/0.36 => ( ! [Z1: binary1291135688e_Tree @ A] :
% 0.14/0.36 ( ? [Z22: A,Z32: binary1291135688e_Tree @ A] :
% 0.14/0.36 ( A2
% 0.14/0.36 = ( binary210054475elle_T @ A @ Z1 @ Z22 @ Z32 ) )
% 0.14/0.36 => ~ ( member @ A @ E @ ( binary2130109271t_Tree @ A @ Z1 ) ) )
% 0.14/0.36 => ( ! [Z1: binary1291135688e_Tree @ A,Z32: binary1291135688e_Tree @ A] :
% 0.14/0.36 ( A2
% 0.14/0.36 != ( binary210054475elle_T @ A @ Z1 @ E @ Z32 ) )
% 0.14/0.36 => ~ ! [Z1: binary1291135688e_Tree @ A,Z22: A,Z32: binary1291135688e_Tree @ A] :
% 0.14/0.36 ( ( A2
% 0.14/0.36 = ( binary210054475elle_T @ A @ Z1 @ Z22 @ Z32 ) )
% 0.14/0.36 => ~ ( member @ A @ E @ ( binary2130109271t_Tree @ A @ Z32 ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Tree.set_cases
% 0.14/0.36 thf(fact_216_Tree_Osimps_I14_J,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( binary2130109271t_Tree @ A @ ( binary1746293266le_Tip @ A ) )
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Tree.simps(14)
% 0.14/0.36 thf(fact_217_Set_Ois__empty__def,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( is_empty @ A )
% 0.14/0.36 = ( ^ [A5: set @ A] :
% 0.14/0.36 ( A5
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Set.is_empty_def
% 0.14/0.36 thf(fact_218_the__elem__def,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( the_elem @ A )
% 0.14/0.36 = ( ^ [X6: set @ A] :
% 0.14/0.36 ( the @ A
% 0.14/0.36 @ ^ [X2: A] :
% 0.14/0.36 ( X6
% 0.14/0.36 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % the_elem_def
% 0.14/0.36 thf(fact_219_the__sym__eq__trivial,axiom,
% 0.14/0.36 ! [A: $tType,X3: A] :
% 0.14/0.36 ( ( the @ A
% 0.14/0.36 @ ( ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.14/0.36 @ X3 ) )
% 0.14/0.36 = X3 ) ).
% 0.14/0.36
% 0.14/0.36 % the_sym_eq_trivial
% 0.14/0.36 thf(fact_220_the__eq__trivial,axiom,
% 0.14/0.36 ! [A: $tType,A2: A] :
% 0.14/0.36 ( ( the @ A
% 0.14/0.36 @ ^ [X2: A] : ( X2 = A2 ) )
% 0.14/0.36 = A2 ) ).
% 0.14/0.36
% 0.14/0.36 % the_eq_trivial
% 0.14/0.36 thf(fact_221_the__equality,axiom,
% 0.14/0.36 ! [A: $tType,P: A > $o,A2: A] :
% 0.14/0.36 ( ( P @ A2 )
% 0.14/0.36 => ( ! [X4: A] :
% 0.14/0.36 ( ( P @ X4 )
% 0.14/0.36 => ( X4 = A2 ) )
% 0.14/0.36 => ( ( the @ A @ P )
% 0.14/0.36 = A2 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % the_equality
% 0.14/0.36 thf(fact_222_theI,axiom,
% 0.14/0.36 ! [A: $tType,P: A > $o,A2: A] :
% 0.14/0.36 ( ( P @ A2 )
% 0.14/0.36 => ( ! [X4: A] :
% 0.14/0.36 ( ( P @ X4 )
% 0.14/0.36 => ( X4 = A2 ) )
% 0.14/0.36 => ( P @ ( the @ A @ P ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % theI
% 0.14/0.36 thf(fact_223_theI_H,axiom,
% 0.14/0.36 ! [A: $tType,P: A > $o] :
% 0.14/0.36 ( ? [X: A] :
% 0.14/0.36 ( ( P @ X )
% 0.14/0.36 & ! [Y4: A] :
% 0.14/0.36 ( ( P @ Y4 )
% 0.14/0.36 => ( Y4 = X ) ) )
% 0.14/0.36 => ( P @ ( the @ A @ P ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % theI'
% 0.14/0.36 thf(fact_224_theI2,axiom,
% 0.14/0.36 ! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
% 0.14/0.36 ( ( P @ A2 )
% 0.14/0.36 => ( ! [X4: A] :
% 0.14/0.36 ( ( P @ X4 )
% 0.14/0.36 => ( X4 = A2 ) )
% 0.14/0.36 => ( ! [X4: A] :
% 0.14/0.36 ( ( P @ X4 )
% 0.14/0.36 => ( Q @ X4 ) )
% 0.14/0.36 => ( Q @ ( the @ A @ P ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % theI2
% 0.14/0.36 thf(fact_225_If__def,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( if @ A )
% 0.14/0.36 = ( ^ [P3: $o,X2: A,Y: A] :
% 0.14/0.36 ( the @ A
% 0.14/0.36 @ ^ [Z4: A] :
% 0.14/0.36 ( ( P3
% 0.14/0.36 => ( Z4 = X2 ) )
% 0.14/0.36 & ( ~ P3
% 0.14/0.36 => ( Z4 = Y ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % If_def
% 0.14/0.36 thf(fact_226_the1I2,axiom,
% 0.14/0.36 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.14/0.36 ( ? [X: A] :
% 0.14/0.36 ( ( P @ X )
% 0.14/0.36 & ! [Y4: A] :
% 0.14/0.36 ( ( P @ Y4 )
% 0.14/0.36 => ( Y4 = X ) ) )
% 0.14/0.36 => ( ! [X4: A] :
% 0.14/0.36 ( ( P @ X4 )
% 0.14/0.36 => ( Q @ X4 ) )
% 0.14/0.36 => ( Q @ ( the @ A @ P ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % the1I2
% 0.14/0.36 thf(fact_227_the1__equality,axiom,
% 0.14/0.36 ! [A: $tType,P: A > $o,A2: A] :
% 0.14/0.36 ( ? [X: A] :
% 0.14/0.36 ( ( P @ X )
% 0.14/0.36 & ! [Y4: A] :
% 0.14/0.36 ( ( P @ Y4 )
% 0.14/0.36 => ( Y4 = X ) ) )
% 0.14/0.36 => ( ( P @ A2 )
% 0.14/0.36 => ( ( the @ A @ P )
% 0.14/0.36 = A2 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % the1_equality
% 0.14/0.36 thf(fact_228_psubset__insert__iff,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,X3: A,B3: set @ A] :
% 0.14/0.36 ( ( ord_less @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ B3 ) )
% 0.14/0.36 = ( ( ( member @ A @ X3 @ B3 )
% 0.14/0.36 => ( ord_less @ ( set @ A ) @ A4 @ B3 ) )
% 0.14/0.36 & ( ~ ( member @ A @ X3 @ B3 )
% 0.14/0.36 => ( ( ( member @ A @ X3 @ A4 )
% 0.14/0.36 => ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) )
% 0.14/0.36 & ( ~ ( member @ A @ X3 @ A4 )
% 0.14/0.36 => ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % psubset_insert_iff
% 0.14/0.36 thf(fact_229_pairwise__alt,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( pairwise @ A )
% 0.14/0.36 = ( ^ [R2: A > A > $o,S2: set @ A] :
% 0.14/0.36 ! [X2: A] :
% 0.14/0.36 ( ( member @ A @ X2 @ S2 )
% 0.14/0.36 => ! [Y: A] :
% 0.14/0.36 ( ( member @ A @ Y @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.14/0.36 => ( R2 @ X2 @ Y ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % pairwise_alt
% 0.14/0.36 thf(fact_230_order__refl,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( preorder @ A )
% 0.14/0.36 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 0.14/0.36
% 0.14/0.36 % order_refl
% 0.14/0.36 thf(fact_231_subset__antisym,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 0.14/0.36 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
% 0.14/0.36 => ( A4 = B3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % subset_antisym
% 0.14/0.36 thf(fact_232_subsetI,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ! [X4: A] :
% 0.14/0.36 ( ( member @ A @ X4 @ A4 )
% 0.14/0.36 => ( member @ A @ X4 @ B3 ) )
% 0.14/0.36 => ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ).
% 0.14/0.36
% 0.14/0.36 % subsetI
% 0.14/0.36 thf(fact_233_sup_Obounded__iff,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ! [B: A,C2: A,A2: A] :
% 0.14/0.36 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B @ C2 ) @ A2 )
% 0.14/0.36 = ( ( ord_less_eq @ A @ B @ A2 )
% 0.14/0.36 & ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % sup.bounded_iff
% 0.14/0.36 thf(fact_234_le__sup__iff,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A )
% 0.14/0.36 => ! [X3: A,Y2: A,Z2: A] :
% 0.14/0.36 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z2 )
% 0.14/0.36 = ( ( ord_less_eq @ A @ X3 @ Z2 )
% 0.14/0.36 & ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % le_sup_iff
% 0.14/0.36 thf(fact_235_empty__subsetI,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% 0.14/0.36
% 0.14/0.36 % empty_subsetI
% 0.14/0.36 thf(fact_236_subset__empty,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A] :
% 0.14/0.36 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 0.14/0.36 = ( A4
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % subset_empty
% 0.14/0.36 thf(fact_237_insert__subset,axiom,
% 0.14/0.36 ! [A: $tType,X3: A,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A4 ) @ B3 )
% 0.14/0.36 = ( ( member @ A @ X3 @ B3 )
% 0.14/0.36 & ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % insert_subset
% 0.14/0.36 thf(fact_238_Un__subset__iff,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,B3: set @ A,C: set @ A] :
% 0.14/0.36 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C )
% 0.14/0.36 = ( ( ord_less_eq @ ( set @ A ) @ A4 @ C )
% 0.14/0.36 & ( ord_less_eq @ ( set @ A ) @ B3 @ C ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % Un_subset_iff
% 0.14/0.36 thf(fact_239_psubsetI,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 0.14/0.36 => ( ( A4 != B3 )
% 0.14/0.36 => ( ord_less @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % psubsetI
% 0.14/0.36 thf(fact_240_singleton__insert__inj__eq_H,axiom,
% 0.14/0.36 ! [A: $tType,A2: A,A4: set @ A,B: A] :
% 0.14/0.36 ( ( ( insert @ A @ A2 @ A4 )
% 0.14/0.36 = ( insert @ A @ B @ ( bot_bot @ ( set @ A ) ) ) )
% 0.14/0.36 = ( ( A2 = B )
% 0.14/0.36 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % singleton_insert_inj_eq'
% 0.14/0.36 thf(fact_241_singleton__insert__inj__eq,axiom,
% 0.14/0.36 ! [A: $tType,B: A,A2: A,A4: set @ A] :
% 0.14/0.36 ( ( ( insert @ A @ B @ ( bot_bot @ ( set @ A ) ) )
% 0.14/0.36 = ( insert @ A @ A2 @ A4 ) )
% 0.14/0.36 = ( ( A2 = B )
% 0.14/0.36 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % singleton_insert_inj_eq
% 0.14/0.36 thf(fact_242_Diff__eq__empty__iff,axiom,
% 0.14/0.36 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 0.14/0.36 ( ( ( minus_minus @ ( set @ A ) @ A4 @ B3 )
% 0.14/0.36 = ( bot_bot @ ( set @ A ) ) )
% 0.14/0.36 = ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ).
% 0.14/0.36
% 0.14/0.36 % Diff_eq_empty_iff
% 0.14/0.36 thf(fact_243_leD,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( order @ A )
% 0.14/0.36 => ! [Y2: A,X3: A] :
% 0.14/0.36 ( ( ord_less_eq @ A @ Y2 @ X3 )
% 0.14/0.36 => ~ ( ord_less @ A @ X3 @ Y2 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % leD
% 0.14/0.36 thf(fact_244_leI,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( linorder @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ~ ( ord_less @ A @ X3 @ Y2 )
% 0.14/0.36 => ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % leI
% 0.14/0.36 thf(fact_245_le__less,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( order @ A )
% 0.14/0.36 => ( ( ord_less_eq @ A )
% 0.14/0.36 = ( ^ [X2: A,Y: A] :
% 0.14/0.36 ( ( ord_less @ A @ X2 @ Y )
% 0.14/0.36 | ( X2 = Y ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % le_less
% 0.14/0.36 thf(fact_246_less__le,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( order @ A )
% 0.14/0.36 => ( ( ord_less @ A )
% 0.14/0.36 = ( ^ [X2: A,Y: A] :
% 0.14/0.36 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.14/0.36 & ( X2 != Y ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % less_le
% 0.14/0.36 thf(fact_247_order__le__less__subst1,axiom,
% 0.14/0.36 ! [A: $tType,B4: $tType] :
% 0.14/0.36 ( ( ( order @ B4 )
% 0.14/0.36 & ( order @ A ) )
% 0.14/0.36 => ! [A2: A,F: B4 > A,B: B4,C2: B4] :
% 0.14/0.36 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 0.14/0.36 => ( ( ord_less @ B4 @ B @ C2 )
% 0.14/0.36 => ( ! [X4: B4,Y4: B4] :
% 0.14/0.36 ( ( ord_less @ B4 @ X4 @ Y4 )
% 0.14/0.36 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.14/0.36 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % order_le_less_subst1
% 0.14/0.36 thf(fact_248_order__le__less__subst2,axiom,
% 0.14/0.36 ! [A: $tType,C4: $tType] :
% 0.14/0.36 ( ( ( order @ C4 )
% 0.14/0.36 & ( order @ A ) )
% 0.14/0.36 => ! [A2: A,B: A,F: A > C4,C2: C4] :
% 0.14/0.36 ( ( ord_less_eq @ A @ A2 @ B )
% 0.14/0.36 => ( ( ord_less @ C4 @ ( F @ B ) @ C2 )
% 0.14/0.36 => ( ! [X4: A,Y4: A] :
% 0.14/0.36 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.14/0.36 => ( ord_less_eq @ C4 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.14/0.36 => ( ord_less @ C4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % order_le_less_subst2
% 0.14/0.36 thf(fact_249_order__less__le__subst1,axiom,
% 0.14/0.36 ! [A: $tType,B4: $tType] :
% 0.14/0.36 ( ( ( order @ B4 )
% 0.14/0.36 & ( order @ A ) )
% 0.14/0.36 => ! [A2: A,F: B4 > A,B: B4,C2: B4] :
% 0.14/0.36 ( ( ord_less @ A @ A2 @ ( F @ B ) )
% 0.14/0.36 => ( ( ord_less_eq @ B4 @ B @ C2 )
% 0.14/0.36 => ( ! [X4: B4,Y4: B4] :
% 0.14/0.36 ( ( ord_less_eq @ B4 @ X4 @ Y4 )
% 0.14/0.36 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.14/0.36 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % order_less_le_subst1
% 0.14/0.36 thf(fact_250_order__less__le__subst2,axiom,
% 0.14/0.36 ! [A: $tType,C4: $tType] :
% 0.14/0.36 ( ( ( order @ C4 )
% 0.14/0.36 & ( order @ A ) )
% 0.14/0.36 => ! [A2: A,B: A,F: A > C4,C2: C4] :
% 0.14/0.36 ( ( ord_less @ A @ A2 @ B )
% 0.14/0.36 => ( ( ord_less_eq @ C4 @ ( F @ B ) @ C2 )
% 0.14/0.36 => ( ! [X4: A,Y4: A] :
% 0.14/0.36 ( ( ord_less @ A @ X4 @ Y4 )
% 0.14/0.36 => ( ord_less @ C4 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.14/0.36 => ( ord_less @ C4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % order_less_le_subst2
% 0.14/0.36 thf(fact_251_not__le,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( linorder @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ( ~ ( ord_less_eq @ A @ X3 @ Y2 ) )
% 0.14/0.36 = ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % not_le
% 0.14/0.36 thf(fact_252_not__less,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( linorder @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 0.14/0.36 = ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % not_less
% 0.14/0.36 thf(fact_253_le__neq__trans,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( order @ A )
% 0.14/0.36 => ! [A2: A,B: A] :
% 0.14/0.36 ( ( ord_less_eq @ A @ A2 @ B )
% 0.14/0.36 => ( ( A2 != B )
% 0.14/0.36 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % le_neq_trans
% 0.14/0.36 thf(fact_254_antisym__conv1,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( order @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ~ ( ord_less @ A @ X3 @ Y2 )
% 0.14/0.36 => ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.14/0.36 = ( X3 = Y2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % antisym_conv1
% 0.14/0.36 thf(fact_255_antisym__conv2,axiom,
% 0.14/0.36 ! [A: $tType] :
% 0.14/0.36 ( ( order @ A )
% 0.14/0.36 => ! [X3: A,Y2: A] :
% 0.14/0.36 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.14/0.36 => ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 0.14/0.36 = ( X3 = Y2 ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 % antisym_conv2
% 0.14/0.36
% 0.14/0.36 % Type constructors (43)
% 0.14/0.36 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 0.14/0.36 bounded_lattice @ $o ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 0.14/0.36 ! [A7: $tType] : ( bounded_lattice @ ( set @ A7 ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 0.14/0.36 ! [A7: $tType,A8: $tType] :
% 0.14/0.36 ( ( bounded_lattice @ A8 )
% 0.14/0.36 => ( bounded_lattice @ ( A7 > A8 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
% 0.14/0.36 ! [A7: $tType,A8: $tType] :
% 0.14/0.36 ( ( bounded_lattice @ A8 )
% 0.14/0.36 => ( bounde1808546759up_bot @ ( A7 > A8 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.14/0.36 ! [A7: $tType,A8: $tType] :
% 0.14/0.36 ( ( semilattice_sup @ A8 )
% 0.14/0.36 => ( semilattice_sup @ ( A7 > A8 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.14/0.36 ! [A7: $tType,A8: $tType] :
% 0.14/0.36 ( ( order_bot @ A8 )
% 0.14/0.36 => ( order_bot @ ( A7 > A8 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.14/0.36 ! [A7: $tType,A8: $tType] :
% 0.14/0.36 ( ( preorder @ A8 )
% 0.14/0.36 => ( preorder @ ( A7 > A8 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.14/0.36 ! [A7: $tType,A8: $tType] :
% 0.14/0.36 ( ( lattice @ A8 )
% 0.14/0.36 => ( lattice @ ( A7 > A8 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.14/0.36 ! [A7: $tType,A8: $tType] :
% 0.14/0.36 ( ( order @ A8 )
% 0.14/0.36 => ( order @ ( A7 > A8 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_fun___Orderings_Oord,axiom,
% 0.14/0.36 ! [A7: $tType,A8: $tType] :
% 0.14/0.36 ( ( ord @ A8 )
% 0.14/0.36 => ( ord @ ( A7 > A8 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_fun___Orderings_Obot,axiom,
% 0.14/0.36 ! [A7: $tType,A8: $tType] :
% 0.14/0.36 ( ( bot @ A8 )
% 0.14/0.36 => ( bot @ ( A7 > A8 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_fun___Groups_Ominus,axiom,
% 0.14/0.36 ! [A7: $tType,A8: $tType] :
% 0.14/0.36 ( ( minus @ A8 )
% 0.14/0.36 => ( minus @ ( A7 > A8 ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.14/0.36 cancel146912293up_add @ int ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 0.14/0.36 ordered_ab_group_add @ int ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Int_Oint___Lattices_Osemilattice__sup_3,axiom,
% 0.14/0.36 semilattice_sup @ int ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Int_Oint___Orderings_Opreorder_4,axiom,
% 0.14/0.36 preorder @ int ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 0.14/0.36 linorder @ int ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
% 0.14/0.36 no_top @ int ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
% 0.14/0.36 no_bot @ int ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Int_Oint___Lattices_Olattice_5,axiom,
% 0.14/0.36 lattice @ int ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 0.14/0.36 group_add @ int ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Int_Oint___Orderings_Oorder_6,axiom,
% 0.14/0.36 order @ int ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Int_Oint___Orderings_Oord_7,axiom,
% 0.14/0.36 ord @ int ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Int_Oint___Groups_Ominus_8,axiom,
% 0.14/0.36 minus @ int ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_9,axiom,
% 0.14/0.36 ! [A7: $tType] : ( bounde1808546759up_bot @ ( set @ A7 ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_10,axiom,
% 0.14/0.36 ! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Set_Oset___Orderings_Oorder__bot_11,axiom,
% 0.14/0.36 ! [A7: $tType] : ( order_bot @ ( set @ A7 ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Set_Oset___Orderings_Opreorder_12,axiom,
% 0.14/0.36 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Set_Oset___Lattices_Olattice_13,axiom,
% 0.14/0.36 ! [A7: $tType] : ( lattice @ ( set @ A7 ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Set_Oset___Orderings_Oorder_14,axiom,
% 0.14/0.36 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Set_Oset___Orderings_Oord_15,axiom,
% 0.14/0.36 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Set_Oset___Orderings_Obot_16,axiom,
% 0.14/0.36 ! [A7: $tType] : ( bot @ ( set @ A7 ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_Set_Oset___Groups_Ominus_17,axiom,
% 0.14/0.36 ! [A7: $tType] : ( minus @ ( set @ A7 ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_18,axiom,
% 0.14/0.36 bounde1808546759up_bot @ $o ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_19,axiom,
% 0.14/0.36 semilattice_sup @ $o ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_HOL_Obool___Orderings_Oorder__bot_20,axiom,
% 0.14/0.36 order_bot @ $o ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_HOL_Obool___Orderings_Opreorder_21,axiom,
% 0.14/0.36 preorder @ $o ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_HOL_Obool___Orderings_Olinorder_22,axiom,
% 0.14/0.36 linorder @ $o ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_HOL_Obool___Lattices_Olattice_23,axiom,
% 0.14/0.36 lattice @ $o ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_HOL_Obool___Orderings_Oorder_24,axiom,
% 0.14/0.36 order @ $o ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_HOL_Obool___Orderings_Oord_25,axiom,
% 0.14/0.36 ord @ $o ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_HOL_Obool___Orderings_Obot_26,axiom,
% 0.14/0.36 bot @ $o ).
% 0.14/0.36
% 0.14/0.36 thf(tcon_HOL_Obool___Groups_Ominus_27,axiom,
% 0.14/0.36 minus @ $o ).
% 0.14/0.36
% 0.14/0.36 % Helper facts (3)
% 0.14/0.36 thf(help_If_3_1_T,axiom,
% 0.14/0.36 ! [P: $o] :
% 0.14/0.36 ( ( P = $true )
% 0.14/0.36 | ( P = $false ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(help_If_2_1_T,axiom,
% 0.14/0.36 ! [A: $tType,X3: A,Y2: A] :
% 0.14/0.36 ( ( if @ A @ $false @ X3 @ Y2 )
% 0.14/0.36 = Y2 ) ).
% 0.14/0.36
% 0.14/0.36 thf(help_If_1_1_T,axiom,
% 0.14/0.36 ! [A: $tType,X3: A,Y2: A] :
% 0.14/0.36 ( ( if @ A @ $true @ X3 @ Y2 )
% 0.14/0.36 = X3 ) ).
% 0.14/0.36
% 0.14/0.36 % Conjectures (1)
% 0.14/0.36 thf(conj_0,conjecture,
% 0.14/0.36 ! [X4: a] :
% 0.14/0.36 ( ( member @ a @ X4 @ ( binary64540844le_eqs @ a @ h @ e ) )
% 0.14/0.36 => ( ( h @ X4 )
% 0.14/0.36 != ( h @ x ) ) ) ).
% 0.14/0.36
% 0.14/0.36 %------------------------------------------------------------------------------
% 0.14/0.36 ------------------------
% 2.22/1.25 ---- Embedded in THF ---
% 2.22/1.25 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.22/1.25 %%% Generated on Tue Feb 24 08:23:46 EST 2026
% 2.22/1.25 %%% using '$$dhol' embedding, version 1.3.0.
% 2.22/1.25 %%% Logic specification used:
% 2.22/1.25 %%% thf(spec, logic, $$dhol).
% 2.22/1.25
% 2.22/1.25 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.dsmPofI642/DTF2DTF4426.p
% 2.22/1.25 thf(ty_t_BinaryTree__Mirabelle__pchhvghoao_OTree, type, binary1291135688e_Tree: ($tType > $tType)).
% 2.22/1.25 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.22/1.25 thf(ty_t_Int_Oint, type, int: $tType).
% 2.22/1.25 thf(ty_tf_a, type, a: $tType).
% 2.22/1.25 thf(sy_cl_Lattices_Obounded__lattice, type, bounded_lattice: (!> [A:$tType]: ($o))).
% 2.22/1.25 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.22/1.25 thf(sy_cl_Groups_Ominus, type, minus: (!> [A:$tType]: ($o))).
% 2.22/1.25 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 2.22/1.25 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.22/1.25 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.22/1.25 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.22/1.25 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: ($o))).
% 2.22/1.25 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 2.22/1.25 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: ($o))).
% 2.22/1.25 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.22/1.25 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.22/1.25 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: ($o))).
% 2.22/1.25 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 2.22/1.25 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: ($o))).
% 2.22/1.25 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: ($o))).
% 2.22/1.25 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 2.22/1.25 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: ($o))).
% 2.22/1.25 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot, type, bounde1808546759up_bot: (!> [A:$tType]: ($o))).
% 2.22/1.25 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OT, type, binary210054475elle_T: (!> [A:$tType]: (((binary1291135688e_Tree @ A) > (A > ((binary1291135688e_Tree @ A) > (binary1291135688e_Tree @ A))))))).
% 2.22/1.25 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OTip, type, binary1746293266le_Tip: (!> [A:$tType]: ((binary1291135688e_Tree @ A)))).
% 2.22/1.25 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_Oset__Tree, type, binary2130109271t_Tree: (!> [A:$tType]: (((binary1291135688e_Tree @ A) > (set @ A))))).
% 2.22/1.25 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Obinsert, type, binary1830089824insert: (!> [A:$tType]: (((A > int) > (A > ((binary1291135688e_Tree @ A) > (binary1291135688e_Tree @ A))))))).
% 2.22/1.25 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Oeqs, type, binary64540844le_eqs: (!> [A:$tType]: (((A > int) > (A > (set @ A)))))).
% 2.22/1.25 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Omemb, type, binary827270440e_memb: (!> [A:$tType]: (((A > int) > (A > ((binary1291135688e_Tree @ A) > $o)))))).
% 2.22/1.25 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsetOf, type, binary1653327646_setOf: (!> [A:$tType]: (((binary1291135688e_Tree @ A) > (set @ A))))).
% 2.22/1.25 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsortedTree, type, binary1610619414edTree: (!> [A:$tType]: (((A > int) > ((binary1291135688e_Tree @ A) > $o))))).
% 2.22/1.25 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Osorted__distinct__pred, type, binary231205461t_pred: (!> [A:$tType]: (((A > int) > (A > (A > ((binary1291135688e_Tree @ A) > $o))))))).
% 2.22/1.25 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.22/1.25 thf(sy_c_HOL_OThe, type, the: (!> [A:$tType]: (((A > $o) > A)))).
% 2.22/1.25 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.22/1.25 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 2.22/1.25 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.22/1.25 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.22/1.25 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.22/1.25 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.22/1.25 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.22/1.25 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.22/1.25 thf(sy_c_Set_Ois__singleton, type, is_singleton: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.22/1.25 thf(sy_c_Set_Opairwise, type, pairwise: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > $o))))).
% 2.22/1.25 thf(sy_c_Set_Oremove, type, remove: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.22/1.25 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))).
% 2.22/1.25 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.22/1.25 thf(sy_v_e, type, e: a).
% 2.22/1.25 thf(sy_v_h, type, h: (a > int)).
% 2.22/1.25 thf(sy_v_t1____, type, t1: (binary1291135688e_Tree @ a)).
% 2.22/1.25 thf(sy_v_t2____, type, t2: (binary1291135688e_Tree @ a)).
% 2.22/1.25 thf(sy_v_x____, type, x: a).
% 2.22/1.25 thf(fact_0_eqsLessX, axiom, (! [X:a]: (((((member @ a) @ X) @ (((binary64540844le_eqs @ a) @ h) @ e)) => (((ord_less @ int) @ (h @ X)) @ (h @ x)))))).
% 2.22/1.25 thf(fact_1_eLess, axiom, (((ord_less @ int) @ (h @ e)) @ (h @ x))).
% 2.22/1.25 thf(fact_2_res, axiom, (((((binary1830089824insert @ a) @ h) @ e) @ ((((binary210054475elle_T @ a) @ t1) @ x) @ t2)) = ((((binary210054475elle_T @ a) @ ((((binary1830089824insert @ a) @ h) @ e) @ t1)) @ x) @ t2))).
% 2.22/1.25 thf(fact_3_s1, axiom, (((binary1610619414edTree @ a) @ h) @ t1)).
% 2.22/1.25 thf(fact_4_s2, axiom, (((binary1610619414edTree @ a) @ h) @ t2)).
% 2.22/1.25 thf(fact_5_s, axiom, (((binary1610619414edTree @ a) @ h) @ ((((binary210054475elle_T @ a) @ t1) @ x) @ t2))).
% 2.22/1.25 thf(fact_6_sorted__distinct, axiom, (! [A:$tType,H:(A > int),A2:A,B:A,T:(binary1291135688e_Tree @ A)]: ((((((binary231205461t_pred @ A) @ H) @ A2) @ B) @ T)))).
% 2.22/1.25 thf(fact_7_c1, axiom, (((binary1653327646_setOf @ a) @ ((((binary1830089824insert @ a) @ h) @ e) @ t1)) = (((sup_sup @ (set @ a)) @ (((minus_minus @ (set @ a)) @ ((binary1653327646_setOf @ a) @ t1)) @ (((binary64540844le_eqs @ a) @ h) @ e))) @ (((insert @ a) @ e) @ (bot_bot @ (set @ a)))))).
% 2.22/1.25 thf(fact_8_c2, axiom, (((binary1653327646_setOf @ a) @ ((((binary1830089824insert @ a) @ h) @ e) @ t2)) = (((sup_sup @ (set @ a)) @ (((minus_minus @ (set @ a)) @ ((binary1653327646_setOf @ a) @ t2)) @ (((binary64540844le_eqs @ a) @ h) @ e))) @ (((insert @ a) @ e) @ (bot_bot @ (set @ a)))))).
% 2.22/1.25 thf(fact_9_eqs__def, axiom, (! [A:$tType]: (((binary64540844le_eqs @ A) = (^ [H2:(A > int),X2:A]: (((collect @ A) @ (^ [Y:A]: (((H2 @ Y) = (H2 @ X2))))))))))).
% 2.22/1.25 thf(fact_10_h1, axiom, ((((binary1610619414edTree @ a) @ h) @ t1) => (((binary1653327646_setOf @ a) @ ((((binary1830089824insert @ a) @ h) @ e) @ t1)) = (((sup_sup @ (set @ a)) @ (((minus_minus @ (set @ a)) @ ((binary1653327646_setOf @ a) @ t1)) @ (((binary64540844le_eqs @ a) @ h) @ e))) @ (((insert @ a) @ e) @ (bot_bot @ (set @ a))))))).
% 2.22/1.25 thf(fact_11_h2, axiom, ((((binary1610619414edTree @ a) @ h) @ t2) => (((binary1653327646_setOf @ a) @ ((((binary1830089824insert @ a) @ h) @ e) @ t2)) = (((sup_sup @ (set @ a)) @ (((minus_minus @ (set @ a)) @ ((binary1653327646_setOf @ a) @ t2)) @ (((binary64540844le_eqs @ a) @ h) @ e))) @ (((insert @ a) @ e) @ (bot_bot @ (set @ a))))))).
% 2.22/1.25 thf(fact_12__092_060open_062sortedTree_Ah_ATip_A_092_060longrightarrow_062_AsetOf_A_Ibinsert_Ah_Ae_ATip_J_A_061_AsetOf_ATip_A_N_Aeqs_Ah_Ae_A_092_060union_062_A_123e_125_092_060close_062, axiom, ((((binary1610619414edTree @ a) @ h) @ (binary1746293266le_Tip @ a)) => (((binary1653327646_setOf @ a) @ ((((binary1830089824insert @ a) @ h) @ e) @ (binary1746293266le_Tip @ a))) = (((sup_sup @ (set @ a)) @ (((minus_minus @ (set @ a)) @ ((binary1653327646_setOf @ a) @ (binary1746293266le_Tip @ a))) @ (((binary64540844le_eqs @ a) @ h) @ e))) @ (((insert @ a) @ e) @ (bot_bot @ (set @ a))))))).
% 2.22/1.25 thf(fact_13_Tree_Oinject, axiom, (! [A:$tType,X21:(binary1291135688e_Tree @ A),X22:A,X23:(binary1291135688e_Tree @ A),Y21:(binary1291135688e_Tree @ A),Y22:A,Y23:(binary1291135688e_Tree @ A)]: (((((((binary210054475elle_T @ A) @ X21) @ X22) @ X23) = ((((binary210054475elle_T @ A) @ Y21) @ Y22) @ Y23)) = ((X21 = Y21) & ((X22 = Y22) & (X23 = Y23))))))).
% 2.22/1.25 thf(fact_14_setOf_Osimps_I2_J, axiom, (! [A:$tType,T1:(binary1291135688e_Tree @ A),X3:A,T2:(binary1291135688e_Tree @ A)]: ((((binary1653327646_setOf @ A) @ ((((binary210054475elle_T @ A) @ T1) @ X3) @ T2)) = (((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ ((binary1653327646_setOf @ A) @ T1)) @ ((binary1653327646_setOf @ A) @ T2))) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))).
% 2.22/1.25 thf(fact_15_setOf_Osimps_I1_J, axiom, (! [A:$tType]: ((((binary1653327646_setOf @ A) @ (binary1746293266le_Tip @ A)) = (bot_bot @ (set @ A)))))).
% 2.22/1.25 thf(fact_16_Tree_Odistinct_I1_J, axiom, (! [A:$tType,X21:(binary1291135688e_Tree @ A),X22:A,X23:(binary1291135688e_Tree @ A)]: (((binary1746293266le_Tip @ A) != ((((binary210054475elle_T @ A) @ X21) @ X22) @ X23))))).
% 2.22/1.25 thf(fact_17_binsert_Osimps_I2_J, axiom, (! [A:$tType,H:(A > int),E:A,X3:A,T1:(binary1291135688e_Tree @ A),T2:(binary1291135688e_Tree @ A)]: ((((((ord_less @ int) @ (H @ E)) @ (H @ X3)) => (((((binary1830089824insert @ A) @ H) @ E) @ ((((binary210054475elle_T @ A) @ T1) @ X3) @ T2)) = ((((binary210054475elle_T @ A) @ ((((binary1830089824insert @ A) @ H) @ E) @ T1)) @ X3) @ T2))) & ((~ ((((ord_less @ int) @ (H @ E)) @ (H @ X3)))) => (((((ord_less @ int) @ (H @ X3)) @ (H @ E)) => (((((binary1830089824insert @ A) @ H) @ E) @ ((((binary210054475elle_T @ A) @ T1) @ X3) @ T2)) = ((((binary210054475elle_T @ A) @ T1) @ X3) @ ((((binary1830089824insert @ A) @ H) @ E) @ T2)))) & ((~ ((((ord_less @ int) @ (H @ X3)) @ (H @ E)))) => (((((binary1830089824insert @ A) @ H) @ E) @ ((((binary210054475elle_T @ A) @ T1) @ X3) @ T2)) = ((((binary210054475elle_T @ A) @ T1) @ E) @ T2))))))))).
% 2.22/1.25 thf(fact_18_binsert_Osimps_I1_J, axiom, (! [A:$tType,H:(A > int),E:A]: ((((((binary1830089824insert @ A) @ H) @ E) @ (binary1746293266le_Tip @ A)) = ((((binary210054475elle_T @ A) @ (binary1746293266le_Tip @ A)) @ E) @ (binary1746293266le_Tip @ A)))))).
% 2.22/1.25 thf(fact_19_sortedTree_Osimps_I2_J, axiom, (! [A:$tType,H:(A > int),T1:(binary1291135688e_Tree @ A),X3:A,T2:(binary1291135688e_Tree @ A)]: (((((binary1610619414edTree @ A) @ H) @ ((((binary210054475elle_T @ A) @ T1) @ X3) @ T2)) = ((((binary1610619414edTree @ A) @ H) @ T1) & ((! [X2:A]: (((((member @ A) @ X2) @ ((binary1653327646_setOf @ A) @ T1)) => (((ord_less @ int) @ (H @ X2)) @ (H @ X3))))) & ((! [X2:A]: (((((member @ A) @ X2) @ ((binary1653327646_setOf @ A) @ T2)) => (((ord_less @ int) @ (H @ X3)) @ (H @ X2))))) & (((binary1610619414edTree @ A) @ H) @ T2)))))))).
% 2.22/1.25 thf(fact_20_sortedTree_Osimps_I1_J, axiom, (! [A:$tType,H:(A > int)]: ((((binary1610619414edTree @ A) @ H) @ (binary1746293266le_Tip @ A))))).
% 2.22/1.25 thf(fact_21_sortLemmaL, axiom, (! [A:$tType,H:(A > int),T1:(binary1291135688e_Tree @ A),X3:A,T2:(binary1291135688e_Tree @ A)]: (((((binary1610619414edTree @ A) @ H) @ ((((binary210054475elle_T @ A) @ T1) @ X3) @ T2)) => (((binary1610619414edTree @ A) @ H) @ T1))))).
% 2.22/1.25 thf(fact_22_sortLemmaR, axiom, (! [A:$tType,H:(A > int),T1:(binary1291135688e_Tree @ A),X3:A,T2:(binary1291135688e_Tree @ A)]: (((((binary1610619414edTree @ A) @ H) @ ((((binary210054475elle_T @ A) @ T1) @ X3) @ T2)) => (((binary1610619414edTree @ A) @ H) @ T2))))).
% 2.22/1.25 thf(fact_23_Tree_Oinduct, axiom, (! [A:$tType,P:((binary1291135688e_Tree @ A) > $o),Tree:(binary1291135688e_Tree @ A)]: (((P @ (binary1746293266le_Tip @ A)) => ((! [X1:(binary1291135688e_Tree @ A),X24:A,X32:(binary1291135688e_Tree @ A)]: (((P @ X1) => ((P @ X32) => (P @ ((((binary210054475elle_T @ A) @ X1) @ X24) @ X32)))))) => (P @ Tree)))))).
% 2.22/1.25 thf(fact_24_Tree_Oexhaust, axiom, (! [A:$tType,Y2:(binary1291135688e_Tree @ A)]: (((Y2 != (binary1746293266le_Tip @ A)) => (~ ((! [X212:(binary1291135688e_Tree @ A),X222:A,X232:(binary1291135688e_Tree @ A)]: ((Y2 != ((((binary210054475elle_T @ A) @ X212) @ X222) @ X232)))))))))).
% 2.22/1.25 thf(fact_25_sorted__distinct__pred__def, axiom, (! [A:$tType]: (((binary231205461t_pred @ A) = (^ [H2:(A > int),A3:A,B2:A,T3:(binary1291135688e_Tree @ A)]: ((((((binary1610619414edTree @ A) @ H2) @ T3) & ((((member @ A) @ A3) @ ((binary1653327646_setOf @ A) @ T3)) & ((((member @ A) @ B2) @ ((binary1653327646_setOf @ A) @ T3)) & ((H2 @ A3) = (H2 @ B2))))) => (A3 = B2)))))))).
% 2.22/1.25 thf(fact_26_insert__Diff__single, axiom, (! [A:$tType,A2:A,A4:(set @ A)]: (((((insert @ A) @ A2) @ (((minus_minus @ (set @ A)) @ A4) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = (((insert @ A) @ A2) @ A4))))).
% 2.22/1.25 thf(fact_27_singleton__conv, axiom, (! [A:$tType,A2:A]: ((((collect @ A) @ (^ [X2:A]: ((X2 = A2)))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.22/1.25 thf(fact_28_singleton__conv2, axiom, (! [A:$tType,A2:A]: ((((collect @ A) @ ((^ [Y3:A,Z:A]: ((Y3 = Z))) @ A2)) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.22/1.25 thf(fact_29_Un__Diff__cancel, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ (((minus_minus @ (set @ A)) @ B3) @ A4)) = (((sup_sup @ (set @ A)) @ A4) @ B3))))).
% 2.22/1.25 thf(fact_30_Un__Diff__cancel2, axiom, (! [A:$tType,B3:(set @ A),A4:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((minus_minus @ (set @ A)) @ B3) @ A4)) @ A4) = (((sup_sup @ (set @ A)) @ B3) @ A4))))).
% 2.22/1.25 thf(fact_31_Diff__insert0, axiom, (! [A:$tType,X3:A,A4:(set @ A),B3:(set @ A)]: (((~ ((((member @ A) @ X3) @ A4))) => ((((minus_minus @ (set @ A)) @ A4) @ (((insert @ A) @ X3) @ B3)) = (((minus_minus @ (set @ A)) @ A4) @ B3)))))).
% 2.22/1.25 thf(fact_32_insert__Diff1, axiom, (! [A:$tType,X3:A,B3:(set @ A),A4:(set @ A)]: (((((member @ A) @ X3) @ B3) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X3) @ A4)) @ B3) = (((minus_minus @ (set @ A)) @ A4) @ B3)))))).
% 2.22/1.25 thf(fact_33_Un__insert__left, axiom, (! [A:$tType,A2:A,B3:(set @ A),C:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((insert @ A) @ A2) @ B3)) @ C) = (((insert @ A) @ A2) @ (((sup_sup @ (set @ A)) @ B3) @ C)))))).
% 2.22/1.25 thf(fact_34_Un__insert__right, axiom, (! [A:$tType,A4:(set @ A),A2:A,B3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ (((insert @ A) @ A2) @ B3)) = (((insert @ A) @ A2) @ (((sup_sup @ (set @ A)) @ A4) @ B3)))))).
% 2.22/1.25 thf(fact_35_Diff__empty, axiom, (! [A:$tType,A4:(set @ A)]: (((((minus_minus @ (set @ A)) @ A4) @ (bot_bot @ (set @ A))) = A4)))).
% 2.22/1.25 thf(fact_36_empty__Diff, axiom, (! [A:$tType,A4:(set @ A)]: (((((minus_minus @ (set @ A)) @ (bot_bot @ (set @ A))) @ A4) = (bot_bot @ (set @ A)))))).
% 2.22/1.25 thf(fact_37_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X2:A]: ((~ ((P @ X2))))))))).
% 2.22/1.25 thf(fact_38_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X2:A]: ((~ ((P @ X2))))))))).
% 2.22/1.25 thf(fact_39_all__not__in__conv, axiom, (! [A:$tType,A4:(set @ A)]: (((! [X2:A]: ((~ ((((member @ A) @ X2) @ A4))))) = (A4 = (bot_bot @ (set @ A))))))).
% 2.22/1.25 thf(fact_40_empty__iff, axiom, (! [A:$tType,C2:A]: ((~ ((((member @ A) @ C2) @ (bot_bot @ (set @ A)))))))).
% 2.22/1.25 thf(fact_41_insert__absorb2, axiom, (! [A:$tType,X3:A,A4:(set @ A)]: (((((insert @ A) @ X3) @ (((insert @ A) @ X3) @ A4)) = (((insert @ A) @ X3) @ A4))))).
% 2.22/1.25 thf(fact_42_insert__iff, axiom, (! [A:$tType,A2:A,B:A,A4:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B) @ A4)) = ((A2 = B) | (((member @ A) @ A2) @ A4)))))).
% 2.22/1.25 thf(fact_43_insertCI, axiom, (! [A:$tType,A2:A,B3:(set @ A),B:A]: ((((~ ((((member @ A) @ A2) @ B3))) => (A2 = B)) => (((member @ A) @ A2) @ (((insert @ A) @ B) @ B3)))))).
% 2.22/1.25 thf(fact_44_Un__iff, axiom, (! [A:$tType,C2:A,A4:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A4) @ B3)) = ((((member @ A) @ C2) @ A4) | (((member @ A) @ C2) @ B3)))))).
% 2.22/1.25 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.22/1.25 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A4)))) = A4)))).
% 2.22/1.25 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) = (Q @ X4)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.22/1.25 thf(fact_48_ext, axiom, (! [B4:$tType,A:$tType,F:(A > B4),G:(A > B4)]: (((! [X4:A]: (((F @ X4) = (G @ X4)))) => (F = G))))).
% 2.22/1.25 thf(fact_49_UnCI, axiom, (! [A:$tType,C2:A,B3:(set @ A),A4:(set @ A)]: ((((~ ((((member @ A) @ C2) @ B3))) => (((member @ A) @ C2) @ A4)) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A4) @ B3)))))).
% 2.22/1.25 thf(fact_50_Diff__idemp, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A4) @ B3)) @ B3) = (((minus_minus @ (set @ A)) @ A4) @ B3))))).
% 2.22/1.25 thf(fact_51_Diff__iff, axiom, (! [A:$tType,C2:A,A4:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A4) @ B3)) = ((((member @ A) @ C2) @ A4) & (~ ((((member @ A) @ C2) @ B3)))))))).
% 2.22/1.25 thf(fact_52_DiffI, axiom, (! [A:$tType,C2:A,A4:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ A4) => ((~ ((((member @ A) @ C2) @ B3))) => (((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A4) @ B3))))))).
% 2.22/1.25 thf(fact_53_singletonI, axiom, (! [A:$tType,A2:A]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.22/1.25 thf(fact_54_Un__empty, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: ((((((sup_sup @ (set @ A)) @ A4) @ B3) = (bot_bot @ (set @ A))) = ((A4 = (bot_bot @ (set @ A))) & (B3 = (bot_bot @ (set @ A)))))))).
% 2.22/1.25 thf(fact_55_Diff__cancel, axiom, (! [A:$tType,A4:(set @ A)]: (((((minus_minus @ (set @ A)) @ A4) @ A4) = (bot_bot @ (set @ A)))))).
% 2.22/1.25 thf(fact_56_ex__in__conv, axiom, (! [A:$tType,A4:(set @ A)]: (((? [X2:A]: ((((member @ A) @ X2) @ A4))) = (A4 != (bot_bot @ (set @ A))))))).
% 2.22/1.25 thf(fact_57_equals0I, axiom, (! [A:$tType,A4:(set @ A)]: (((! [Y4:A]: ((~ ((((member @ A) @ Y4) @ A4))))) => (A4 = (bot_bot @ (set @ A))))))).
% 2.22/1.25 thf(fact_58_equals0D, axiom, (! [A:$tType,A4:(set @ A),A2:A]: (((A4 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A4))))))).
% 2.22/1.25 thf(fact_59_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.22/1.25 thf(fact_60_mk__disjoint__insert, axiom, (! [A:$tType,A2:A,A4:(set @ A)]: (((((member @ A) @ A2) @ A4) => (? [B5:(set @ A)]: (((A4 = (((insert @ A) @ A2) @ B5)) & (~ ((((member @ A) @ A2) @ B5)))))))))).
% 2.22/1.25 thf(fact_61_insert__commute, axiom, (! [A:$tType,X3:A,Y2:A,A4:(set @ A)]: (((((insert @ A) @ X3) @ (((insert @ A) @ Y2) @ A4)) = (((insert @ A) @ Y2) @ (((insert @ A) @ X3) @ A4)))))).
% 2.22/1.25 thf(fact_62_insert__eq__iff, axiom, (! [A:$tType,A2:A,A4:(set @ A),B:A,B3:(set @ A)]: (((~ ((((member @ A) @ A2) @ A4))) => ((~ ((((member @ A) @ B) @ B3))) => (((((insert @ A) @ A2) @ A4) = (((insert @ A) @ B) @ B3)) = (((A2 = B) => (A4 = B3)) & ((A2 != B) => (? [C3:(set @ A)]: (((A4 = (((insert @ A) @ B) @ C3)) & ((~ ((((member @ A) @ B) @ C3))) & ((B3 = (((insert @ A) @ A2) @ C3)) & (~ ((((member @ A) @ A2) @ C3)))))))))))))))).
% 2.22/1.25 thf(fact_63_insert__absorb, axiom, (! [A:$tType,A2:A,A4:(set @ A)]: (((((member @ A) @ A2) @ A4) => ((((insert @ A) @ A2) @ A4) = A4))))).
% 2.22/1.25 thf(fact_64_insert__ident, axiom, (! [A:$tType,X3:A,A4:(set @ A),B3:(set @ A)]: (((~ ((((member @ A) @ X3) @ A4))) => ((~ ((((member @ A) @ X3) @ B3))) => (((((insert @ A) @ X3) @ A4) = (((insert @ A) @ X3) @ B3)) = (A4 = B3))))))).
% 2.22/1.25 thf(fact_65_Set_Oset__insert, axiom, (! [A:$tType,X3:A,A4:(set @ A)]: (((((member @ A) @ X3) @ A4) => (~ ((! [B5:(set @ A)]: (((A4 = (((insert @ A) @ X3) @ B5)) => (((member @ A) @ X3) @ B5)))))))))).
% 2.22/1.25 thf(fact_66_insertI2, axiom, (! [A:$tType,A2:A,B3:(set @ A),B:A]: (((((member @ A) @ A2) @ B3) => (((member @ A) @ A2) @ (((insert @ A) @ B) @ B3)))))).
% 2.22/1.25 thf(fact_67_insertI1, axiom, (! [A:$tType,A2:A,B3:(set @ A)]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ B3))))).
% 2.22/1.25 thf(fact_68_insertE, axiom, (! [A:$tType,A2:A,B:A,A4:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B) @ A4)) => ((A2 != B) => (((member @ A) @ A2) @ A4)))))).
% 2.22/1.25 thf(fact_69_Un__left__commute, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ (((sup_sup @ (set @ A)) @ B3) @ C)) = (((sup_sup @ (set @ A)) @ B3) @ (((sup_sup @ (set @ A)) @ A4) @ C)))))).
% 2.22/1.25 thf(fact_70_Un__left__absorb, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ (((sup_sup @ (set @ A)) @ A4) @ B3)) = (((sup_sup @ (set @ A)) @ A4) @ B3))))).
% 2.22/1.25 thf(fact_71_Un__commute, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A5:(set @ A),B6:(set @ A)]: ((((sup_sup @ (set @ A)) @ B6) @ A5))))))).
% 2.22/1.25 thf(fact_72_Un__absorb, axiom, (! [A:$tType,A4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ A4) = A4)))).
% 2.22/1.25 thf(fact_73_Un__assoc, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A4) @ B3)) @ C) = (((sup_sup @ (set @ A)) @ A4) @ (((sup_sup @ (set @ A)) @ B3) @ C)))))).
% 2.22/1.25 thf(fact_74_ball__Un, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),P:(A > $o)]: (((! [X2:A]: (((((member @ A) @ X2) @ (((sup_sup @ (set @ A)) @ A4) @ B3)) => (P @ X2)))) = ((! [X2:A]: (((((member @ A) @ X2) @ A4) => (P @ X2)))) & (! [X2:A]: (((((member @ A) @ X2) @ B3) => (P @ X2))))))))).
% 2.22/1.25 thf(fact_75_bex__Un, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),P:(A > $o)]: (((? [X2:A]: (((((member @ A) @ X2) @ (((sup_sup @ (set @ A)) @ A4) @ B3)) & (P @ X2)))) = ((? [X2:A]: (((((member @ A) @ X2) @ A4) & (P @ X2)))) | (? [X2:A]: (((((member @ A) @ X2) @ B3) & (P @ X2))))))))).
% 2.22/1.25 thf(fact_76_UnI2, axiom, (! [A:$tType,C2:A,B3:(set @ A),A4:(set @ A)]: (((((member @ A) @ C2) @ B3) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A4) @ B3)))))).
% 2.22/1.25 thf(fact_77_UnI1, axiom, (! [A:$tType,C2:A,A4:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ A4) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A4) @ B3)))))).
% 2.22/1.25 thf(fact_78_UnE, axiom, (! [A:$tType,C2:A,A4:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A4) @ B3)) => ((~ ((((member @ A) @ C2) @ A4))) => (((member @ A) @ C2) @ B3)))))).
% 2.22/1.25 thf(fact_79_DiffD2, axiom, (! [A:$tType,C2:A,A4:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A4) @ B3)) => (~ ((((member @ A) @ C2) @ B3))))))).
% 2.22/1.25 thf(fact_80_DiffD1, axiom, (! [A:$tType,C2:A,A4:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A4) @ B3)) => (((member @ A) @ C2) @ A4))))).
% 2.22/1.25 thf(fact_81_DiffE, axiom, (! [A:$tType,C2:A,A4:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A4) @ B3)) => (~ (((((member @ A) @ C2) @ A4) => (((member @ A) @ C2) @ B3)))))))).
% 2.22/1.25 thf(fact_82_empty__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (^ [X2:A]: ($false))))))).
% 2.22/1.25 thf(fact_83_insert__Collect, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((insert @ A) @ A2) @ ((collect @ A) @ P)) = ((collect @ A) @ (^ [U:A]: (((U != A2) => (P @ U))))))))).
% 2.22/1.25 thf(fact_84_insert__compr, axiom, (! [A:$tType]: (((insert @ A) = (^ [A3:A,B6:(set @ A)]: (((collect @ A) @ (^ [X2:A]: (((X2 = A3) | (((member @ A) @ X2) @ B6))))))))))).
% 2.22/1.25 thf(fact_85_Collect__disj__eq, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: ((((collect @ A) @ (^ [X2:A]: (((P @ X2) | (Q @ X2))))) = (((sup_sup @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.22/1.25 thf(fact_86_Un__def, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A5:(set @ A),B6:(set @ A)]: (((collect @ A) @ (^ [X2:A]: (((((member @ A) @ X2) @ A5) | (((member @ A) @ X2) @ B6))))))))))).
% 2.22/1.25 thf(fact_87_set__diff__eq, axiom, (! [A:$tType]: (((minus_minus @ (set @ A)) = (^ [A5:(set @ A),B6:(set @ A)]: (((collect @ A) @ (^ [X2:A]: (((((member @ A) @ X2) @ A5) & (~ ((((member @ A) @ X2) @ B6))))))))))))).
% 2.22/1.25 thf(fact_88_singleton__inject, axiom, (! [A:$tType,A2:A,B:A]: ((((((insert @ A) @ A2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B) @ (bot_bot @ (set @ A)))) => (A2 = B))))).
% 2.22/1.25 thf(fact_89_insert__not__empty, axiom, (! [A:$tType,A2:A,A4:(set @ A)]: (((((insert @ A) @ A2) @ A4) != (bot_bot @ (set @ A)))))).
% 2.22/1.25 thf(fact_90_doubleton__eq__iff, axiom, (! [A:$tType,A2:A,B:A,C2:A,D:A]: ((((((insert @ A) @ A2) @ (((insert @ A) @ B) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C2) @ (((insert @ A) @ D) @ (bot_bot @ (set @ A))))) = (((A2 = C2) & (B = D)) | ((A2 = D) & (B = C2))))))).
% 2.22/1.25 thf(fact_91_singleton__iff, axiom, (! [A:$tType,B:A,A2:A]: (((((member @ A) @ B) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = (B = A2))))).
% 2.22/1.25 thf(fact_92_singletonD, axiom, (! [A:$tType,B:A,A2:A]: (((((member @ A) @ B) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) => (B = A2))))).
% 2.22/1.25 thf(fact_93_Un__empty__right, axiom, (! [A:$tType,A4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ (bot_bot @ (set @ A))) = A4)))).
% 2.22/1.25 thf(fact_94_Un__empty__left, axiom, (! [A:$tType,B3:(set @ A)]: (((((sup_sup @ (set @ A)) @ (bot_bot @ (set @ A))) @ B3) = B3)))).
% 2.22/1.25 thf(fact_95_insert__Diff__if, axiom, (! [A:$tType,X3:A,B3:(set @ A),A4:(set @ A)]: ((((((member @ A) @ X3) @ B3) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X3) @ A4)) @ B3) = (((minus_minus @ (set @ A)) @ A4) @ B3))) & ((~ ((((member @ A) @ X3) @ B3))) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X3) @ A4)) @ B3) = (((insert @ A) @ X3) @ (((minus_minus @ (set @ A)) @ A4) @ B3)))))))).
% 2.22/1.25 thf(fact_96_Un__Diff, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A4) @ B3)) @ C) = (((sup_sup @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A4) @ C)) @ (((minus_minus @ (set @ A)) @ B3) @ C)))))).
% 2.22/1.25 thf(fact_97_Collect__conv__if2, axiom, (! [A:$tType,P:(A > $o),A2:A]: ((((P @ A2) => (((collect @ A) @ (^ [X2:A]: (((A2 = X2) & (P @ X2))))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) & ((~ ((P @ A2))) => (((collect @ A) @ (^ [X2:A]: (((A2 = X2) & (P @ X2))))) = (bot_bot @ (set @ A)))))))).
% 2.22/1.25 thf(fact_98_Collect__conv__if, axiom, (! [A:$tType,P:(A > $o),A2:A]: ((((P @ A2) => (((collect @ A) @ (^ [X2:A]: (((X2 = A2) & (P @ X2))))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) & ((~ ((P @ A2))) => (((collect @ A) @ (^ [X2:A]: (((X2 = A2) & (P @ X2))))) = (bot_bot @ (set @ A)))))))).
% 2.22/1.25 thf(fact_99_insert__def, axiom, (! [A:$tType]: (((insert @ A) = (^ [A3:A]: (((sup_sup @ (set @ A)) @ ((collect @ A) @ (^ [X2:A]: ((X2 = A3))))))))))).
% 2.22/1.25 thf(fact_100_singleton__Un__iff, axiom, (! [A:$tType,X3:A,A4:(set @ A),B3:(set @ A)]: ((((((insert @ A) @ X3) @ (bot_bot @ (set @ A))) = (((sup_sup @ (set @ A)) @ A4) @ B3)) = (((A4 = (bot_bot @ (set @ A))) & (B3 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) | (((A4 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) & (B3 = (bot_bot @ (set @ A)))) | ((A4 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) & (B3 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))))).
% 2.22/1.25 thf(fact_101_Un__singleton__iff, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),X3:A]: ((((((sup_sup @ (set @ A)) @ A4) @ B3) = (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = (((A4 = (bot_bot @ (set @ A))) & (B3 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) | (((A4 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) & (B3 = (bot_bot @ (set @ A)))) | ((A4 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) & (B3 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))))).
% 2.22/1.25 thf(fact_102_insert__is__Un, axiom, (! [A:$tType]: (((insert @ A) = (^ [A3:A]: (((sup_sup @ (set @ A)) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))))))))).
% 2.22/1.25 thf(fact_103_Diff__insert__absorb, axiom, (! [A:$tType,X3:A,A4:(set @ A)]: (((~ ((((member @ A) @ X3) @ A4))) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X3) @ A4)) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = A4))))).
% 2.22/1.25 thf(fact_104_Diff__insert2, axiom, (! [A:$tType,A4:(set @ A),A2:A,B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A4) @ (((insert @ A) @ A2) @ B3)) = (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A4) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) @ B3))))).
% 2.22/1.25 thf(fact_105_insert__Diff, axiom, (! [A:$tType,A2:A,A4:(set @ A)]: (((((member @ A) @ A2) @ A4) => ((((insert @ A) @ A2) @ (((minus_minus @ (set @ A)) @ A4) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = A4))))).
% 2.22/1.25 thf(fact_106_Diff__insert, axiom, (! [A:$tType,A4:(set @ A),A2:A,B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A4) @ (((insert @ A) @ A2) @ B3)) = (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A4) @ B3)) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.22/1.25 thf(fact_107_sup__bot_Oright__neutral, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [A2:A]: (((((sup_sup @ A) @ A2) @ (bot_bot @ A)) = A2))))))).
% 2.22/1.25 thf(fact_108_sup__bot_Oneutr__eq__iff, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [A2:A,B:A]: ((((bot_bot @ A) = (((sup_sup @ A) @ A2) @ B)) = ((A2 = (bot_bot @ A)) & (B = (bot_bot @ A)))))))))).
% 2.22/1.25 thf(fact_109_sup__bot_Oleft__neutral, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [A2:A]: (((((sup_sup @ A) @ (bot_bot @ A)) @ A2) = A2))))))).
% 2.22/1.25 thf(fact_110_sup__bot_Oeq__neutr__iff, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [A2:A,B:A]: ((((((sup_sup @ A) @ A2) @ B) = (bot_bot @ A)) = ((A2 = (bot_bot @ A)) & (B = (bot_bot @ A)))))))))).
% 2.22/1.25 thf(fact_111_sup__eq__bot__iff, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [X3:A,Y2:A]: ((((((sup_sup @ A) @ X3) @ Y2) = (bot_bot @ A)) = ((X3 = (bot_bot @ A)) & (Y2 = (bot_bot @ A)))))))))).
% 2.22/1.25 thf(fact_112_bot__eq__sup__iff, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [X3:A,Y2:A]: ((((bot_bot @ A) = (((sup_sup @ A) @ X3) @ Y2)) = ((X3 = (bot_bot @ A)) & (Y2 = (bot_bot @ A)))))))))).
% 2.22/1.25 thf(fact_113_sup__bot__right, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [X3:A]: (((((sup_sup @ A) @ X3) @ (bot_bot @ A)) = X3))))))).
% 2.22/1.25 thf(fact_114_sup__bot__left, axiom, (! [A:$tType]: (((bounde1808546759up_bot @ A) => (! [X3:A]: (((((sup_sup @ A) @ (bot_bot @ A)) @ X3) = X3))))))).
% 2.22/1.25 thf(fact_115_memb__spec, axiom, (! [A:$tType,H:(A > int),T:(binary1291135688e_Tree @ A),X3:A]: (((((binary1610619414edTree @ A) @ H) @ T) => (((((binary827270440e_memb @ A) @ H) @ X3) @ T) = (((member @ A) @ X3) @ ((binary1653327646_setOf @ A) @ T))))))).
% 2.22/1.25 thf(fact_116_minus__apply, axiom, (! [B4:$tType,A:$tType]: (((minus @ B4) => ((minus_minus @ (A > B4)) = (^ [A5:(A > B4),B6:(A > B4),X2:A]: ((((minus_minus @ B4) @ (A5 @ X2)) @ (B6 @ X2))))))))).
% 2.22/1.25 thf(fact_117_sup__apply, axiom, (! [B4:$tType,A:$tType]: (((semilattice_sup @ B4) => ((sup_sup @ (A > B4)) = (^ [F2:(A > B4),G2:(A > B4),X2:A]: ((((sup_sup @ B4) @ (F2 @ X2)) @ (G2 @ X2))))))))).
% 2.22/1.25 thf(fact_118_sup_Oidem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A]: (((((sup_sup @ A) @ A2) @ A2) = A2))))))).
% 2.22/1.25 thf(fact_119_sup__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A]: (((((sup_sup @ A) @ X3) @ X3) = X3))))))).
% 2.22/1.25 thf(fact_120_sup_Oleft__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B:A]: (((((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ A2) @ B)) = (((sup_sup @ A) @ A2) @ B)))))))).
% 2.22/1.25 thf(fact_121_sup__left__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,Y2:A]: (((((sup_sup @ A) @ X3) @ (((sup_sup @ A) @ X3) @ Y2)) = (((sup_sup @ A) @ X3) @ Y2)))))))).
% 2.22/1.25 thf(fact_122_sup_Oright__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A2) @ B)) @ B) = (((sup_sup @ A) @ A2) @ B)))))))).
% 2.22/1.25 thf(fact_123_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 2.22/1.25 thf(fact_124_sup__set__def, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A5:(set @ A),B6:(set @ A)]: (((collect @ A) @ (((sup_sup @ (A > $o)) @ (^ [X2:A]: ((((member @ A) @ X2) @ A5)))) @ (^ [X2:A]: ((((member @ A) @ X2) @ B6))))))))))).
% 2.22/1.25 thf(fact_125_fun__diff__def, axiom, (! [B4:$tType,A:$tType]: (((minus @ B4) => ((minus_minus @ (A > B4)) = (^ [A5:(A > B4),B6:(A > B4),X2:A]: ((((minus_minus @ B4) @ (A5 @ X2)) @ (B6 @ X2))))))))).
% 2.22/1.25 thf(fact_126_inf__sup__aci_I8_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X3:A,Y2:A]: (((((sup_sup @ A) @ X3) @ (((sup_sup @ A) @ X3) @ Y2)) = (((sup_sup @ A) @ X3) @ Y2)))))))).
% 2.22/1.25 thf(fact_127_inf__sup__aci_I7_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X3:A,Y2:A,Z2:A]: (((((sup_sup @ A) @ X3) @ (((sup_sup @ A) @ Y2) @ Z2)) = (((sup_sup @ A) @ Y2) @ (((sup_sup @ A) @ X3) @ Z2))))))))).
% 2.22/1.25 thf(fact_128_inf__sup__aci_I6_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X3:A,Y2:A,Z2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X3) @ Y2)) @ Z2) = (((sup_sup @ A) @ X3) @ (((sup_sup @ A) @ Y2) @ Z2))))))))).
% 2.22/1.25 thf(fact_129_inf__sup__aci_I5_J, axiom, (! [A:$tType]: (((lattice @ A) => ((sup_sup @ A) = (^ [X2:A,Y:A]: ((((sup_sup @ A) @ Y) @ X2)))))))).
% 2.22/1.25 thf(fact_130_sup__fun__def, axiom, (! [B4:$tType,A:$tType]: (((semilattice_sup @ B4) => ((sup_sup @ (A > B4)) = (^ [F2:(A > B4),G2:(A > B4),X2:A]: ((((sup_sup @ B4) @ (F2 @ X2)) @ (G2 @ X2))))))))).
% 2.22/1.25 thf(fact_131_boolean__algebra__cancel_Osup1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A4:A,K:A,A2:A,B:A]: (((A4 = (((sup_sup @ A) @ K) @ A2)) => ((((sup_sup @ A) @ A4) @ B) = (((sup_sup @ A) @ K) @ (((sup_sup @ A) @ A2) @ B)))))))))).
% 2.22/1.25 thf(fact_132_boolean__algebra__cancel_Osup2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B3:A,K:A,B:A,A2:A]: (((B3 = (((sup_sup @ A) @ K) @ B)) => ((((sup_sup @ A) @ A2) @ B3) = (((sup_sup @ A) @ K) @ (((sup_sup @ A) @ A2) @ B)))))))))).
% 2.22/1.25 thf(fact_133_sup_Oassoc, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B:A,C2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A2) @ B)) @ C2) = (((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ B) @ C2))))))))).
% 2.22/1.25 thf(fact_134_sup__assoc, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,Y2:A,Z2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X3) @ Y2)) @ Z2) = (((sup_sup @ A) @ X3) @ (((sup_sup @ A) @ Y2) @ Z2))))))))).
% 2.22/1.25 thf(fact_135_sup_Ocommute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((sup_sup @ A) = (^ [A3:A,B2:A]: ((((sup_sup @ A) @ B2) @ A3)))))))).
% 2.22/1.25 thf(fact_136_sup__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((sup_sup @ A) = (^ [X2:A,Y:A]: ((((sup_sup @ A) @ Y) @ X2)))))))).
% 2.22/1.25 thf(fact_137_sup_Oleft__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B:A,A2:A,C2:A]: (((((sup_sup @ A) @ B) @ (((sup_sup @ A) @ A2) @ C2)) = (((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ B) @ C2))))))))).
% 2.22/1.25 thf(fact_138_sup__left__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,Y2:A,Z2:A]: (((((sup_sup @ A) @ X3) @ (((sup_sup @ A) @ Y2) @ Z2)) = (((sup_sup @ A) @ Y2) @ (((sup_sup @ A) @ X3) @ Z2))))))))).
% 2.22/1.25 thf(fact_139_less__supI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,A2:A,B:A]: (((((ord_less @ A) @ X3) @ A2) => (((ord_less @ A) @ X3) @ (((sup_sup @ A) @ A2) @ B))))))))).
% 2.22/1.25 thf(fact_140_less__supI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,B:A,A2:A]: (((((ord_less @ A) @ X3) @ B) => (((ord_less @ A) @ X3) @ (((sup_sup @ A) @ A2) @ B))))))))).
% 2.22/1.25 thf(fact_141_sup_Ostrict__boundedE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B:A,C2:A,A2:A]: (((((ord_less @ A) @ (((sup_sup @ A) @ B) @ C2)) @ A2) => (~ (((((ord_less @ A) @ B) @ A2) => (~ ((((ord_less @ A) @ C2) @ A2))))))))))))).
% 2.22/1.25 thf(fact_142_sup_Ostrict__order__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less @ A) = (^ [B2:A,A3:A]: (((A3 = (((sup_sup @ A) @ A3) @ B2)) & (A3 != B2))))))))).
% 2.22/1.25 thf(fact_143_sup_Ostrict__coboundedI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,A2:A,B:A]: (((((ord_less @ A) @ C2) @ A2) => (((ord_less @ A) @ C2) @ (((sup_sup @ A) @ A2) @ B))))))))).
% 2.22/1.25 thf(fact_144_sup_Ostrict__coboundedI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,B:A,A2:A]: (((((ord_less @ A) @ C2) @ B) => (((ord_less @ A) @ C2) @ (((sup_sup @ A) @ A2) @ B))))))))).
% 2.22/1.25 thf(fact_145_the__elem__eq, axiom, (! [A:$tType,X3:A]: ((((the_elem @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = X3)))).
% 2.22/1.25 thf(fact_146_bot__apply, axiom, (! [C4:$tType,D2:$tType]: (((bot @ C4) => ((bot_bot @ (D2 > C4)) = (^ [X2:D2]: ((bot_bot @ C4)))))))).
% 2.22/1.25 thf(fact_147_is__singletonI, axiom, (! [A:$tType,X3:A]: (((is_singleton @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))).
% 2.22/1.25 thf(fact_148_diff__strict__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,D:A,C2:A]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less @ A) @ D) @ C2) => (((ord_less @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ (((minus_minus @ A) @ B) @ D)))))))))).
% 2.22/1.25 thf(fact_149_sup1CI, axiom, (! [A:$tType,B3:(A > $o),X3:A,A4:(A > $o)]: ((((~ ((B3 @ X3))) => (A4 @ X3)) => ((((sup_sup @ (A > $o)) @ A4) @ B3) @ X3))))).
% 2.22/1.25 thf(fact_150_sup1E, axiom, (! [A:$tType,A4:(A > $o),B3:(A > $o),X3:A]: ((((((sup_sup @ (A > $o)) @ A4) @ B3) @ X3) => ((~ ((A4 @ X3))) => (B3 @ X3)))))).
% 2.22/1.25 thf(fact_151_sup1I1, axiom, (! [A:$tType,A4:(A > $o),X3:A,B3:(A > $o)]: (((A4 @ X3) => ((((sup_sup @ (A > $o)) @ A4) @ B3) @ X3))))).
% 2.22/1.25 thf(fact_152_sup1I2, axiom, (! [A:$tType,B3:(A > $o),X3:A,A4:(A > $o)]: (((B3 @ X3) => ((((sup_sup @ (A > $o)) @ A4) @ B3) @ X3))))).
% 2.22/1.25 thf(fact_153_not__psubset__empty, axiom, (! [A:$tType,A4:(set @ A)]: ((~ ((((ord_less @ (set @ A)) @ A4) @ (bot_bot @ (set @ A)))))))).
% 2.22/1.25 thf(fact_154_psubset__imp__ex__mem, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B3) => (? [B7:A]: ((((member @ A) @ B7) @ (((minus_minus @ (set @ A)) @ B3) @ A4)))))))).
% 2.22/1.25 thf(fact_155_minus__set__def, axiom, (! [A:$tType]: (((minus_minus @ (set @ A)) = (^ [A5:(set @ A),B6:(set @ A)]: (((collect @ A) @ (((minus_minus @ (A > $o)) @ (^ [X2:A]: ((((member @ A) @ X2) @ A5)))) @ (^ [X2:A]: ((((member @ A) @ X2) @ B6))))))))))).
% 2.22/1.25 thf(fact_156_is__singleton__the__elem, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A5:(set @ A)]: ((A5 = (((insert @ A) @ ((the_elem @ A) @ A5)) @ (bot_bot @ (set @ A)))))))))).
% 2.22/1.25 thf(fact_157_is__singletonI_H, axiom, (! [A:$tType,A4:(set @ A)]: (((A4 != (bot_bot @ (set @ A))) => ((! [X4:A,Y4:A]: (((((member @ A) @ X4) @ A4) => ((((member @ A) @ Y4) @ A4) => (X4 = Y4))))) => ((is_singleton @ A) @ A4)))))).
% 2.22/1.25 thf(fact_158_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A]: (((((ord_less @ A) @ B) @ A2) => (A2 != B)))))))).
% 2.22/1.25 thf(fact_159_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (A2 != B)))))))).
% 2.22/1.25 thf(fact_160_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((~ ((((ord_less @ A) @ X3) @ Y2))) = ((((ord_less @ A) @ Y2) @ X3) | (X3 = Y2))))))))).
% 2.22/1.25 thf(fact_161_dual__order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A,C2:A]: (((((ord_less @ A) @ B) @ A2) => ((((ord_less @ A) @ C2) @ B) => (((ord_less @ A) @ C2) @ A2))))))))).
% 2.22/1.25 thf(fact_162_linorder__less__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B:A]: (((! [A6:A,B7:A]: (((((ord_less @ A) @ A6) @ B7) => ((P @ A6) @ B7)))) => ((! [A6:A]: (((P @ A6) @ A6))) => ((! [A6:A,B7:A]: ((((P @ B7) @ A6) => ((P @ A6) @ B7)))) => ((P @ A2) @ B)))))))))).
% 2.22/1.25 thf(fact_163_exists__least__iff, axiom, (! [A:$tType]: (((wellorder @ A) => ((^ [P2:(A > $o)]: ((? [X5:A]: ((P2 @ X5))))) = (^ [P3:(A > $o)]: ((? [N:A]: (((P3 @ N) & (! [M:A]: (((((ord_less @ A) @ M) @ N) => (~ ((P3 @ M)))))))))))))))).
% 2.22/1.25 thf(fact_164_less__imp__not__less, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X3)))))))))).
% 2.22/1.25 thf(fact_165_order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A,C2:A]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less @ A) @ B) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.22/1.25 thf(fact_166_dual__order_Oirrefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 2.22/1.25 thf(fact_167_linorder__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((~ ((((ord_less @ A) @ X3) @ Y2))) => ((X3 != Y2) => (((ord_less @ A) @ Y2) @ X3))))))))).
% 2.22/1.25 thf(fact_168_less__imp__triv, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y2:A,P:$o]: (((((ord_less @ A) @ X3) @ Y2) => ((((ord_less @ A) @ Y2) @ X3) => P)))))))).
% 2.22/1.25 thf(fact_169_less__imp__not__eq2, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (Y2 != X3)))))))).
% 2.22/1.25 thf(fact_170_antisym__conv3, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y2:A,X3:A]: (((~ ((((ord_less @ A) @ Y2) @ X3))) => ((~ ((((ord_less @ A) @ X3) @ Y2))) = (X3 = Y2))))))))).
% 2.22/1.25 thf(fact_171_less__induct, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),A2:A]: (((! [X4:A]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X4) => (P @ Y5)))) => (P @ X4)))) => (P @ A2)))))))).
% 2.22/1.25 thf(fact_172_less__not__sym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X3)))))))))).
% 2.22/1.25 thf(fact_173_less__imp__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (X3 != Y2)))))))).
% 2.22/1.25 thf(fact_174_dual__order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A]: (((((ord_less @ A) @ B) @ A2) => (~ ((((ord_less @ A) @ A2) @ B)))))))))).
% 2.22/1.25 thf(fact_175_ord__less__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B:A,C2:A]: (((((ord_less @ A) @ A2) @ B) => ((B = C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.22/1.25 thf(fact_176_ord__eq__less__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B:A,C2:A]: (((A2 = B) => ((((ord_less @ A) @ B) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.22/1.25 thf(fact_177_less__irrefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A]: ((~ ((((ord_less @ A) @ X3) @ X3))))))))).
% 2.22/1.25 thf(fact_178_less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) | ((X3 = Y2) | (((ord_less @ A) @ Y2) @ X3))))))))).
% 2.22/1.25 thf(fact_179_less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X3) @ Y2) => ((((ord_less @ A) @ Y2) @ Z2) => (((ord_less @ A) @ X3) @ Z2))))))))).
% 2.22/1.25 thf(fact_180_less__asym_H, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (~ ((((ord_less @ A) @ B) @ A2)))))))))).
% 2.22/1.25 thf(fact_181_less__asym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X3)))))))))).
% 2.22/1.25 thf(fact_182_less__imp__neq, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (X3 != Y2)))))))).
% 2.22/1.25 thf(fact_183_dense, axiom, (! [A:$tType]: (((dense_order @ A) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (? [Z3:A]: (((((ord_less @ A) @ X3) @ Z3) & (((ord_less @ A) @ Z3) @ Y2))))))))))).
% 2.22/1.25 thf(fact_184_order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (~ ((((ord_less @ A) @ B) @ A2)))))))))).
% 2.22/1.25 thf(fact_185_neq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((X3 != Y2) = ((((ord_less @ A) @ X3) @ Y2) | (((ord_less @ A) @ Y2) @ X3))))))))).
% 2.22/1.25 thf(fact_186_neqE, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((X3 != Y2) => ((~ ((((ord_less @ A) @ X3) @ Y2))) => (((ord_less @ A) @ Y2) @ X3))))))))).
% 2.22/1.25 thf(fact_187_gt__ex, axiom, (! [A:$tType]: (((no_top @ A) => (! [X3:A]: ((? [X_1:A]: ((((ord_less @ A) @ X3) @ X_1))))))))).
% 2.22/1.25 thf(fact_188_lt__ex, axiom, (! [A:$tType]: (((no_bot @ A) => (! [X3:A]: ((? [Y4:A]: ((((ord_less @ A) @ Y4) @ X3))))))))).
% 2.22/1.25 thf(fact_189_order__less__subst2, axiom, (! [A:$tType,C4:$tType]: ((((order @ C4) & (order @ A)) => (! [A2:A,B:A,F:(A > C4),C2:C4]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less @ C4) @ (F @ B)) @ C2) => ((! [X4:A,Y4:A]: (((((ord_less @ A) @ X4) @ Y4) => (((ord_less @ C4) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ C4) @ (F @ A2)) @ C2)))))))))).
% 2.22/1.25 thf(fact_190_order__less__subst1, axiom, (! [A:$tType,B4:$tType]: ((((order @ B4) & (order @ A)) => (! [A2:A,F:(B4 > A),B:B4,C2:B4]: (((((ord_less @ A) @ A2) @ (F @ B)) => ((((ord_less @ B4) @ B) @ C2) => ((! [X4:B4,Y4:B4]: (((((ord_less @ B4) @ X4) @ Y4) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C2))))))))))).
% 2.22/1.25 thf(fact_191_ord__less__eq__subst, axiom, (! [A:$tType,B4:$tType]: ((((ord @ B4) & (ord @ A)) => (! [A2:A,B:A,F:(A > B4),C2:B4]: (((((ord_less @ A) @ A2) @ B) => (((F @ B) = C2) => ((! [X4:A,Y4:A]: (((((ord_less @ A) @ X4) @ Y4) => (((ord_less @ B4) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ B4) @ (F @ A2)) @ C2)))))))))).
% 2.22/1.25 thf(fact_192_ord__eq__less__subst, axiom, (! [A:$tType,B4:$tType]: ((((ord @ B4) & (ord @ A)) => (! [A2:A,F:(B4 > A),B:B4,C2:B4]: (((A2 = (F @ B)) => ((((ord_less @ B4) @ B) @ C2) => ((! [X4:B4,Y4:B4]: (((((ord_less @ B4) @ X4) @ Y4) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C2))))))))))).
% 2.22/1.25 thf(fact_193_bot__fun__def, axiom, (! [B4:$tType,A:$tType]: (((bot @ B4) => ((bot_bot @ (A > B4)) = (^ [X2:A]: ((bot_bot @ B4)))))))).
% 2.22/1.25 thf(fact_194_diff__right__commute, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,C2:A,B:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ B) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B)) @ C2)))))))).
% 2.22/1.25 thf(fact_195_diff__eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B:A,C2:A,D:A]: ((((((minus_minus @ A) @ A2) @ B) = (((minus_minus @ A) @ C2) @ D)) => ((A2 = B) = (C2 = D))))))))).
% 2.22/1.25 thf(fact_196_is__singleton__def, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A5:(set @ A)]: ((? [X2:A]: ((A5 = (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))))))))))).
% 2.22/1.25 thf(fact_197_is__singletonE, axiom, (! [A:$tType,A4:(set @ A)]: ((((is_singleton @ A) @ A4) => (~ ((! [X4:A]: ((A4 != (((insert @ A) @ X4) @ (bot_bot @ (set @ A)))))))))))).
% 2.22/1.25 thf(fact_198_bot_Onot__eq__extremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: (((A2 != (bot_bot @ A)) = (((ord_less @ A) @ (bot_bot @ A)) @ A2)))))))).
% 2.22/1.25 thf(fact_199_bot_Oextremum__strict, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ (bot_bot @ A)))))))))).
% 2.22/1.25 thf(fact_200_diff__strict__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,C2:A]: (((((ord_less @ A) @ A2) @ B) => (((ord_less @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ (((minus_minus @ A) @ B) @ C2))))))))).
% 2.22/1.25 thf(fact_201_diff__strict__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B:A,A2:A,C2:A]: (((((ord_less @ A) @ B) @ A2) => (((ord_less @ A) @ (((minus_minus @ A) @ C2) @ A2)) @ (((minus_minus @ A) @ C2) @ B))))))))).
% 2.22/1.25 thf(fact_202_diff__eq__diff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,C2:A,D:A]: ((((((minus_minus @ A) @ A2) @ B) = (((minus_minus @ A) @ C2) @ D)) => ((((ord_less @ A) @ A2) @ B) = (((ord_less @ A) @ C2) @ D))))))))).
% 2.22/1.25 thf(fact_203_sup__Un__eq, axiom, (! [A:$tType,R:(set @ A),S:(set @ A)]: (((((sup_sup @ (A > $o)) @ (^ [X2:A]: ((((member @ A) @ X2) @ R)))) @ (^ [X2:A]: ((((member @ A) @ X2) @ S)))) = (^ [X2:A]: ((((member @ A) @ X2) @ (((sup_sup @ (set @ A)) @ R) @ S)))))))).
% 2.22/1.25 thf(fact_204_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X2:A]: ((((member @ A) @ X2) @ (bot_bot @ (set @ A))))))))).
% 2.22/1.25 thf(fact_205_Collect__empty__eq__bot, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (P = (bot_bot @ (A > $o))))))).
% 2.22/1.25 thf(fact_206_psubset__trans, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B3) => ((((ord_less @ (set @ A)) @ B3) @ C) => (((ord_less @ (set @ A)) @ A4) @ C)))))).
% 2.22/1.25 thf(fact_207_less__set__def, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A5:(set @ A),B6:(set @ A)]: ((((ord_less @ (A > $o)) @ (^ [X2:A]: ((((member @ A) @ X2) @ A5)))) @ (^ [X2:A]: ((((member @ A) @ X2) @ B6)))))))))).
% 2.22/1.25 thf(fact_208_psubsetD, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C2:A]: (((((ord_less @ (set @ A)) @ A4) @ B3) => ((((member @ A) @ C2) @ A4) => (((member @ A) @ C2) @ B3)))))).
% 2.22/1.25 thf(fact_209_remove__def, axiom, (! [A:$tType]: (((remove @ A) = (^ [X2:A,A5:(set @ A)]: ((((minus_minus @ (set @ A)) @ A5) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))))))))).
% 2.22/1.25 thf(fact_210_Tree_Osimps_I15_J, axiom, (! [A:$tType,X21:(binary1291135688e_Tree @ A),X22:A,X23:(binary1291135688e_Tree @ A)]: ((((binary2130109271t_Tree @ A) @ ((((binary210054475elle_T @ A) @ X21) @ X22) @ X23)) = (((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ ((binary2130109271t_Tree @ A) @ X21)) @ (((insert @ A) @ X22) @ (bot_bot @ (set @ A))))) @ ((binary2130109271t_Tree @ A) @ X23)))))).
% 2.22/1.25 thf(fact_211_member__remove, axiom, (! [A:$tType,X3:A,Y2:A,A4:(set @ A)]: (((((member @ A) @ X3) @ (((remove @ A) @ Y2) @ A4)) = ((((member @ A) @ X3) @ A4) & (X3 != Y2)))))).
% 2.22/1.25 thf(fact_212_Tree_Oset__intros_I3_J, axiom, (! [A:$tType,Ya:A,X23:(binary1291135688e_Tree @ A),X21:(binary1291135688e_Tree @ A),X22:A]: (((((member @ A) @ Ya) @ ((binary2130109271t_Tree @ A) @ X23)) => (((member @ A) @ Ya) @ ((binary2130109271t_Tree @ A) @ ((((binary210054475elle_T @ A) @ X21) @ X22) @ X23))))))).
% 2.22/1.25 thf(fact_213_Tree_Oset__intros_I2_J, axiom, (! [A:$tType,X22:A,X21:(binary1291135688e_Tree @ A),X23:(binary1291135688e_Tree @ A)]: ((((member @ A) @ X22) @ ((binary2130109271t_Tree @ A) @ ((((binary210054475elle_T @ A) @ X21) @ X22) @ X23)))))).
% 2.22/1.25 thf(fact_214_Tree_Oset__intros_I1_J, axiom, (! [A:$tType,Y2:A,X21:(binary1291135688e_Tree @ A),X22:A,X23:(binary1291135688e_Tree @ A)]: (((((member @ A) @ Y2) @ ((binary2130109271t_Tree @ A) @ X21)) => (((member @ A) @ Y2) @ ((binary2130109271t_Tree @ A) @ ((((binary210054475elle_T @ A) @ X21) @ X22) @ X23))))))).
% 2.22/1.25 thf(fact_215_Tree_Oset__cases, axiom, (! [A:$tType,E:A,A2:(binary1291135688e_Tree @ A)]: (((((member @ A) @ E) @ ((binary2130109271t_Tree @ A) @ A2)) => ((! [Z1:(binary1291135688e_Tree @ A)]: (((? [Z22:A,Z32:(binary1291135688e_Tree @ A)]: ((A2 = ((((binary210054475elle_T @ A) @ Z1) @ Z22) @ Z32)))) => (~ ((((member @ A) @ E) @ ((binary2130109271t_Tree @ A) @ Z1))))))) => ((! [Z1:(binary1291135688e_Tree @ A),Z32:(binary1291135688e_Tree @ A)]: ((A2 != ((((binary210054475elle_T @ A) @ Z1) @ E) @ Z32)))) => (~ ((! [Z1:(binary1291135688e_Tree @ A),Z22:A,Z32:(binary1291135688e_Tree @ A)]: (((A2 = ((((binary210054475elle_T @ A) @ Z1) @ Z22) @ Z32)) => (~ ((((member @ A) @ E) @ ((binary2130109271t_Tree @ A) @ Z32))))))))))))))).
% 2.22/1.25 thf(fact_216_Tree_Osimps_I14_J, axiom, (! [A:$tType]: ((((binary2130109271t_Tree @ A) @ (binary1746293266le_Tip @ A)) = (bot_bot @ (set @ A)))))).
% 2.22/1.25 thf(fact_217_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A5:(set @ A)]: ((A5 = (bot_bot @ (set @ A))))))))).
% 2.22/1.25 thf(fact_218_the__elem__def, axiom, (! [A:$tType]: (((the_elem @ A) = (^ [X6:(set @ A)]: (((the @ A) @ (^ [X2:A]: ((X6 = (((insert @ A) @ X2) @ (bot_bot @ (set @ A))))))))))))).
% 2.22/1.25 thf(fact_219_the__sym__eq__trivial, axiom, (! [A:$tType,X3:A]: ((((the @ A) @ ((^ [Y3:A,Z:A]: ((Y3 = Z))) @ X3)) = X3)))).
% 2.22/1.25 thf(fact_220_the__eq__trivial, axiom, (! [A:$tType,A2:A]: ((((the @ A) @ (^ [X2:A]: ((X2 = A2)))) = A2)))).
% 2.22/1.25 thf(fact_221_the__equality, axiom, (! [A:$tType,P:(A > $o),A2:A]: (((P @ A2) => ((! [X4:A]: (((P @ X4) => (X4 = A2)))) => (((the @ A) @ P) = A2)))))).
% 2.22/1.25 thf(fact_222_theI, axiom, (! [A:$tType,P:(A > $o),A2:A]: (((P @ A2) => ((! [X4:A]: (((P @ X4) => (X4 = A2)))) => (P @ ((the @ A) @ P))))))).
% 2.22/1.25 thf(fact_223_theI_H, axiom, (! [A:$tType,P:(A > $o)]: (((? [X:A]: (((P @ X) & (! [Y4:A]: (((P @ Y4) => (Y4 = X))))))) => (P @ ((the @ A) @ P)))))).
% 2.22/1.25 thf(fact_224_theI2, axiom, (! [A:$tType,P:(A > $o),A2:A,Q:(A > $o)]: (((P @ A2) => ((! [X4:A]: (((P @ X4) => (X4 = A2)))) => ((! [X4:A]: (((P @ X4) => (Q @ X4)))) => (Q @ ((the @ A) @ P)))))))).
% 2.22/1.25 thf(fact_225_If__def, axiom, (! [A:$tType]: (((if @ A) = (^ [P3:$o,X2:A,Y:A]: (((the @ A) @ (^ [Z4:A]: (((P3 => (Z4 = X2)) & ((~ (P3)) => (Z4 = Y)))))))))))).
% 2.22/1.25 thf(fact_226_the1I2, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((? [X:A]: (((P @ X) & (! [Y4:A]: (((P @ Y4) => (Y4 = X))))))) => ((! [X4:A]: (((P @ X4) => (Q @ X4)))) => (Q @ ((the @ A) @ P))))))).
% 2.22/1.25 thf(fact_227_the1__equality, axiom, (! [A:$tType,P:(A > $o),A2:A]: (((? [X:A]: (((P @ X) & (! [Y4:A]: (((P @ Y4) => (Y4 = X))))))) => ((P @ A2) => (((the @ A) @ P) = A2)))))).
% 2.22/1.25 thf(fact_228_psubset__insert__iff, axiom, (! [A:$tType,A4:(set @ A),X3:A,B3:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ (((insert @ A) @ X3) @ B3)) = (((((member @ A) @ X3) @ B3) => (((ord_less @ (set @ A)) @ A4) @ B3)) & ((~ ((((member @ A) @ X3) @ B3))) => (((((member @ A) @ X3) @ A4) => (((ord_less @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A4) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) @ B3)) & ((~ ((((member @ A) @ X3) @ A4))) => (((ord_less_eq @ (set @ A)) @ A4) @ B3))))))))).
% 2.22/1.25 thf(fact_229_pairwise__alt, axiom, (! [A:$tType]: (((pairwise @ A) = (^ [R2:(A > (A > $o)),S2:(set @ A)]: ((! [X2:A]: (((((member @ A) @ X2) @ S2) => (! [Y:A]: (((((member @ A) @ Y) @ (((minus_minus @ (set @ A)) @ S2) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A))))) => ((R2 @ X2) @ Y))))))))))))).
% 2.22/1.25 thf(fact_230_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A]: ((((ord_less_eq @ A) @ X3) @ X3))))))).
% 2.22/1.25 thf(fact_231_subset__antisym, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ A4) => (A4 = B3)))))).
% 2.22/1.25 thf(fact_232_subsetI, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((! [X4:A]: (((((member @ A) @ X4) @ A4) => (((member @ A) @ X4) @ B3)))) => (((ord_less_eq @ (set @ A)) @ A4) @ B3))))).
% 2.22/1.25 thf(fact_233_sup_Obounded__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B:A,C2:A,A2:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B) @ C2)) @ A2) = ((((ord_less_eq @ A) @ B) @ A2) & (((ord_less_eq @ A) @ C2) @ A2))))))))).
% 2.22/1.25 thf(fact_234_le__sup__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,Y2:A,Z2:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ X3) @ Y2)) @ Z2) = ((((ord_less_eq @ A) @ X3) @ Z2) & (((ord_less_eq @ A) @ Y2) @ Z2))))))))).
% 2.22/1.25 thf(fact_235_empty__subsetI, axiom, (! [A:$tType,A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A4)))).
% 2.22/1.25 thf(fact_236_subset__empty, axiom, (! [A:$tType,A4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ (bot_bot @ (set @ A))) = (A4 = (bot_bot @ (set @ A))))))).
% 2.22/1.25 thf(fact_237_insert__subset, axiom, (! [A:$tType,X3:A,A4:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X3) @ A4)) @ B3) = ((((member @ A) @ X3) @ B3) & (((ord_less_eq @ (set @ A)) @ A4) @ B3)))))).
% 2.22/1.25 thf(fact_238_Un__subset__iff, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A),C:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A4) @ B3)) @ C) = ((((ord_less_eq @ (set @ A)) @ A4) @ C) & (((ord_less_eq @ (set @ A)) @ B3) @ C)))))).
% 2.22/1.25 thf(fact_239_psubsetI, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B3) => ((A4 != B3) => (((ord_less @ (set @ A)) @ A4) @ B3)))))).
% 2.22/1.25 thf(fact_240_singleton__insert__inj__eq_H, axiom, (! [A:$tType,A2:A,A4:(set @ A),B:A]: ((((((insert @ A) @ A2) @ A4) = (((insert @ A) @ B) @ (bot_bot @ (set @ A)))) = ((A2 = B) & (((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ B) @ (bot_bot @ (set @ A))))))))).
% 2.22/1.25 thf(fact_241_singleton__insert__inj__eq, axiom, (! [A:$tType,B:A,A2:A,A4:(set @ A)]: ((((((insert @ A) @ B) @ (bot_bot @ (set @ A))) = (((insert @ A) @ A2) @ A4)) = ((A2 = B) & (((ord_less_eq @ (set @ A)) @ A4) @ (((insert @ A) @ B) @ (bot_bot @ (set @ A))))))))).
% 2.22/1.25 thf(fact_242_Diff__eq__empty__iff, axiom, (! [A:$tType,A4:(set @ A),B3:(set @ A)]: ((((((minus_minus @ (set @ A)) @ A4) @ B3) = (bot_bot @ (set @ A))) = (((ord_less_eq @ (set @ A)) @ A4) @ B3))))).
% 2.22/1.25 thf(fact_243_leD, axiom, (! [A:$tType]: (((order @ A) => (! [Y2:A,X3:A]: (((((ord_less_eq @ A) @ Y2) @ X3) => (~ ((((ord_less @ A) @ X3) @ Y2)))))))))).
% 2.22/1.25 thf(fact_244_leI, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((~ ((((ord_less @ A) @ X3) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X3)))))))).
% 2.22/1.25 thf(fact_245_le__less, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) | (X2 = Y))))))))).
% 2.22/1.25 thf(fact_246_less__le, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) & (X2 != Y))))))))).
% 2.22/1.25 thf(fact_247_order__le__less__subst1, axiom, (! [A:$tType,B4:$tType]: ((((order @ B4) & (order @ A)) => (! [A2:A,F:(B4 > A),B:B4,C2:B4]: (((((ord_less_eq @ A) @ A2) @ (F @ B)) => ((((ord_less @ B4) @ B) @ C2) => ((! [X4:B4,Y4:B4]: (((((ord_less @ B4) @ X4) @ Y4) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C2))))))))))).
% 2.22/1.25 thf(fact_248_order__le__less__subst2, axiom, (! [A:$tType,C4:$tType]: ((((order @ C4) & (order @ A)) => (! [A2:A,B:A,F:(A > C4),C2:C4]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less @ C4) @ (F @ B)) @ C2) => ((! [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) => (((ord_less_eq @ C4) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ C4) @ (F @ A2)) @ C2)))))))))).
% 2.22/1.25 thf(fact_249_order__less__le__subst1, axiom, (! [A:$tType,B4:$tType]: ((((order @ B4) & (order @ A)) => (! [A2:A,F:(B4 > A),B:B4,C2:B4]: (((((ord_less @ A) @ A2) @ (F @ B)) => ((((ord_less_eq @ B4) @ B) @ C2) => ((! [X4:B4,Y4:B4]: (((((ord_less_eq @ B4) @ X4) @ Y4) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C2))))))))))).
% 2.22/1.25 thf(fact_250_order__less__le__subst2, axiom, (! [A:$tType,C4:$tType]: ((((order @ C4) & (order @ A)) => (! [A2:A,B:A,F:(A > C4),C2:C4]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less_eq @ C4) @ (F @ B)) @ C2) => ((! [X4:A,Y4:A]: (((((ord_less @ A) @ X4) @ Y4) => (((ord_less @ C4) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ C4) @ (F @ A2)) @ C2)))))))))).
% 2.22/1.25 thf(fact_251_not__le, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X3) @ Y2))) = (((ord_less @ A) @ Y2) @ X3)))))))).
% 2.22/1.25 thf(fact_252_not__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((~ ((((ord_less @ A) @ X3) @ Y2))) = (((ord_less_eq @ A) @ Y2) @ X3)))))))).
% 2.22/1.25 thf(fact_253_le__neq__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((A2 != B) => (((ord_less @ A) @ A2) @ B))))))))).
% 2.22/1.25 thf(fact_254_antisym__conv1, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y2:A]: (((~ ((((ord_less @ A) @ X3) @ Y2))) => ((((ord_less_eq @ A) @ X3) @ Y2) = (X3 = Y2))))))))).
% 2.22/1.25 thf(fact_255_antisym__conv2, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => ((~ ((((ord_less @ A) @ X3) @ Y2))) = (X3 = Y2))))))))).
% 2.22/1.25 thf(tcon_HOL_Obool___Lattices_Obounded__lattice, axiom, (bounded_lattice @ $o)).
% 2.22/1.25 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1, axiom, (! [A7:$tType]: ((bounded_lattice @ (set @ A7))))).
% 2.22/1.25 thf(tcon_fun___Lattices_Obounded__lattice_2, axiom, (! [A7:$tType,A8:$tType]: (((bounded_lattice @ A8) => (bounded_lattice @ (A7 > A8)))))).
% 2.22/1.25 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot, axiom, (! [A7:$tType,A8:$tType]: (((bounded_lattice @ A8) => (bounde1808546759up_bot @ (A7 > A8)))))).
% 2.22/1.25 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A7:$tType,A8:$tType]: (((semilattice_sup @ A8) => (semilattice_sup @ (A7 > A8)))))).
% 2.22/1.25 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A7:$tType,A8:$tType]: (((order_bot @ A8) => (order_bot @ (A7 > A8)))))).
% 2.22/1.25 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: (((preorder @ A8) => (preorder @ (A7 > A8)))))).
% 2.22/1.25 thf(tcon_fun___Lattices_Olattice, axiom, (! [A7:$tType,A8:$tType]: (((lattice @ A8) => (lattice @ (A7 > A8)))))).
% 2.22/1.25 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: (((order @ A8) => (order @ (A7 > A8)))))).
% 2.22/1.25 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: (((ord @ A8) => (ord @ (A7 > A8)))))).
% 2.22/1.25 thf(tcon_fun___Orderings_Obot, axiom, (! [A7:$tType,A8:$tType]: (((bot @ A8) => (bot @ (A7 > A8)))))).
% 2.22/1.25 thf(tcon_fun___Groups_Ominus, axiom, (! [A7:$tType,A8:$tType]: (((minus @ A8) => (minus @ (A7 > A8)))))).
% 2.22/1.25 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add, axiom, (cancel146912293up_add @ int)).
% 2.22/1.25 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add, axiom, (ordered_ab_group_add @ int)).
% 2.22/1.25 thf(tcon_Int_Oint___Lattices_Osemilattice__sup_3, axiom, (semilattice_sup @ int)).
% 2.22/1.25 thf(tcon_Int_Oint___Orderings_Opreorder_4, axiom, (preorder @ int)).
% 2.22/1.25 thf(tcon_Int_Oint___Orderings_Olinorder, axiom, (linorder @ int)).
% 2.22/1.25 thf(tcon_Int_Oint___Orderings_Ono__top, axiom, (no_top @ int)).
% 2.22/1.25 thf(tcon_Int_Oint___Orderings_Ono__bot, axiom, (no_bot @ int)).
% 2.22/1.25 thf(tcon_Int_Oint___Lattices_Olattice_5, axiom, (lattice @ int)).
% 2.22/1.25 thf(tcon_Int_Oint___Groups_Ogroup__add, axiom, (group_add @ int)).
% 2.22/1.25 thf(tcon_Int_Oint___Orderings_Oorder_6, axiom, (order @ int)).
% 2.22/1.25 thf(tcon_Int_Oint___Orderings_Oord_7, axiom, (ord @ int)).
% 2.22/1.25 thf(tcon_Int_Oint___Groups_Ominus_8, axiom, (minus @ int)).
% 2.22/1.25 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_9, axiom, (! [A7:$tType]: ((bounde1808546759up_bot @ (set @ A7))))).
% 2.22/1.25 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_10, axiom, (! [A7:$tType]: ((semilattice_sup @ (set @ A7))))).
% 2.22/1.25 thf(tcon_Set_Oset___Orderings_Oorder__bot_11, axiom, (! [A7:$tType]: ((order_bot @ (set @ A7))))).
% 2.22/1.25 thf(tcon_Set_Oset___Orderings_Opreorder_12, axiom, (! [A7:$tType]: ((preorder @ (set @ A7))))).
% 2.22/1.25 thf(tcon_Set_Oset___Lattices_Olattice_13, axiom, (! [A7:$tType]: ((lattice @ (set @ A7))))).
% 2.22/1.25 thf(tcon_Set_Oset___Orderings_Oorder_14, axiom, (! [A7:$tType]: ((order @ (set @ A7))))).
% 2.22/1.25 thf(tcon_Set_Oset___Orderings_Oord_15, axiom, (! [A7:$tType]: ((ord @ (set @ A7))))).
% 2.22/1.25 thf(tcon_Set_Oset___Orderings_Obot_16, axiom, (! [A7:$tType]: ((bot @ (set @ A7))))).
% 2.22/1.25 thf(tcon_Set_Oset___Groups_Ominus_17, axiom, (! [A7:$tType]: ((minus @ (set @ A7))))).
% 2.22/1.25 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_18, axiom, (bounde1808546759up_bot @ $o)).
% 2.22/1.25 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_19, axiom, (semilattice_sup @ $o)).
% 2.22/1.25 thf(tcon_HOL_Obool___Orderings_Oorder__bot_20, axiom, (order_bot @ $o)).
% 2.22/1.25 thf(tcon_HOL_Obool___Orderings_Opreorder_21, axiom, (preorder @ $o)).
% 2.22/1.25 thf(tcon_HOL_Obool___Orderings_Olinorder_22, axiom, (linorder @ $o)).
% 2.22/1.25 thf(tcon_HOL_Obool___Lattices_Olattice_23, axiom, (lattice @ $o)).
% 2.22/1.25 thf(tcon_HOL_Obool___Orderings_Oorder_24, axiom, (order @ $o)).
% 2.22/1.25 thf(tcon_HOL_Obool___Orderings_Oord_25, axiom, (ord @ $o)).
% 2.22/1.25 thf(tcon_HOL_Obool___Orderings_Obot_26, axiom, (bot @ $o)).
% 2.22/1.25 thf(tcon_HOL_Obool___Groups_Ominus_27, axiom, (minus @ $o)).
% 2.22/1.25 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.22/1.25 thf(help_If_2_1_T, axiom, (! [A:$tType,X3:A,Y2:A]: ((((((if @ A) @ $false) @ X3) @ Y2) = Y2)))).
% 2.22/1.25 thf(help_If_1_1_T, axiom, (! [A:$tType,X3:A,Y2:A]: ((((((if @ A) @ $true) @ X3) @ Y2) = X3)))).
% 2.22/1.25 thf(conj_0, conjecture, (! [X4:a]: (((((member @ a) @ X4) @ (((binary64540844le_eqs @ a) @ h) @ e)) => ((h @ X4) != (h @ x)))))).
% 2.22/1.25 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.dsmPofI642/DTF2DTF4426.p
% 2.22/1.25 ------------------------
% 2.22/1.27 ---- Cleaned THF ---
% 2.22/1.27 thf(ty_t_BinaryTree__Mirabelle__pchhvghoao_OTree,type,
% 2.22/1.27 binary1291135688e_Tree: $tType > $tType ).
% 2.22/1.27
% 2.22/1.27 thf(ty_t_Set_Oset,type,
% 2.22/1.27 set: $tType > $tType ).
% 2.22/1.27
% 2.22/1.27 thf(ty_t_Int_Oint,type,
% 2.22/1.27 int: $tType ).
% 2.22/1.27
% 2.22/1.27 thf(ty_tf_a,type,
% 2.22/1.27 a: $tType ).
% 2.22/1.27
% 2.22/1.27 thf(sy_cl_Lattices_Obounded__lattice,type,
% 2.22/1.27 bounded_lattice:
% 2.22/1.27 !>[A: $tType] : $o ).
% 2.22/1.27
% 2.22/1.27 thf(sy_cl_HOL_Otype,type,
% 2.22/1.27 type:
% 2.22/1.27 !>[A: $tType] : $o ).
% 2.22/1.27
% 2.22/1.27 thf(sy_cl_Groups_Ominus,type,
% 2.22/1.27 minus:
% 2.22/1.27 !>[A: $tType] : $o ).
% 2.22/1.27
% 2.22/1.27 thf(sy_cl_Orderings_Obot,type,
% 2.22/1.27 bot:
% 2.22/1.27 !>[A: $tType] : $o ).
% 2.22/1.27
% 2.22/1.27 thf(sy_cl_Orderings_Oord,type,
% 2.22/1.27 ord:
% 2.22/1.27 !>[A: $tType] : $o ).
% 2.22/1.27
% 2.22/1.27 thf(sy_cl_Orderings_Oorder,type,
% 2.22/1.27 order:
% 2.22/1.27 !>[A: $tType] : $o ).
% 2.22/1.27
% 2.22/1.27 thf(sy_cl_Groups_Ogroup__add,type,
% 2.22/1.27 group_add:
% 2.22/1.27 !>[A: $tType] : $o ).
% 2.22/1.27
% 2.22/1.27 thf(sy_cl_Lattices_Olattice,type,
% 2.22/1.27 lattice:
% 2.22/1.27 !>[A: $tType] : $o ).
% 2.22/1.27
% 2.22/1.27 thf(sy_cl_Orderings_Ono__bot,type,
% 2.22/1.27 no_bot:
% 2.22/1.27 !>[A: $tType] : $o ).
% 2.22/1.27
% 2.22/1.27 thf(sy_cl_Orderings_Ono__top,type,
% 2.22/1.27 no_top:
% 2.22/1.27 !>[A: $tType] : $o ).
% 2.22/1.27
% 2.22/1.27 thf(sy_cl_Orderings_Olinorder,type,
% 2.22/1.27 linorder:
% 2.22/1.27 !>[A: $tType] : $o ).
% 2.22/1.27
% 2.22/1.27 thf(sy_cl_Orderings_Opreorder,type,
% 2.22/1.27 preorder:
% 2.22/1.27 !>[A: $tType] : $o ).
% 2.22/1.27
% 2.22/1.27 thf(sy_cl_Orderings_Oorder__bot,type,
% 2.22/1.27 order_bot:
% 2.22/1.27 !>[A: $tType] : $o ).
% 2.22/1.27
% 2.22/1.27 thf(sy_cl_Orderings_Owellorder,type,
% 2.22/1.27 wellorder:
% 2.22/1.27 !>[A: $tType] : $o ).
% 2.22/1.27
% 2.22/1.27 thf(sy_cl_Orderings_Odense__order,type,
% 2.22/1.27 dense_order:
% 2.22/1.27 !>[A: $tType] : $o ).
% 2.22/1.27
% 2.22/1.27 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 2.22/1.27 semilattice_sup:
% 2.22/1.27 !>[A: $tType] : $o ).
% 2.22/1.27
% 2.22/1.27 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.22/1.27 ordered_ab_group_add:
% 2.22/1.27 !>[A: $tType] : $o ).
% 2.22/1.27
% 2.22/1.27 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.22/1.27 cancel146912293up_add:
% 2.22/1.27 !>[A: $tType] : $o ).
% 2.22/1.27
% 2.22/1.27 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
% 2.22/1.27 bounde1808546759up_bot:
% 2.22/1.27 !>[A: $tType] : $o ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OT,type,
% 2.22/1.27 binary210054475elle_T:
% 2.22/1.27 !>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OTip,type,
% 2.22/1.27 binary1746293266le_Tip:
% 2.22/1.27 !>[A: $tType] : ( binary1291135688e_Tree @ A ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_Oset__Tree,type,
% 2.22/1.27 binary2130109271t_Tree:
% 2.22/1.27 !>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > ( set @ A ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Obinsert,type,
% 2.22/1.27 binary1830089824insert:
% 2.22/1.27 !>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Oeqs,type,
% 2.22/1.27 binary64540844le_eqs:
% 2.22/1.27 !>[A: $tType] : ( ( A > int ) > A > ( set @ A ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Omemb,type,
% 2.22/1.27 binary827270440e_memb:
% 2.22/1.27 !>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > $o ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsetOf,type,
% 2.22/1.27 binary1653327646_setOf:
% 2.22/1.27 !>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > ( set @ A ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsortedTree,type,
% 2.22/1.27 binary1610619414edTree:
% 2.22/1.27 !>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > $o ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Osorted__distinct__pred,type,
% 2.22/1.27 binary231205461t_pred:
% 2.22/1.27 !>[A: $tType] : ( ( A > int ) > A > A > ( binary1291135688e_Tree @ A ) > $o ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.22/1.27 minus_minus:
% 2.22/1.27 !>[A: $tType] : ( A > A > A ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_HOL_OThe,type,
% 2.22/1.27 the:
% 2.22/1.27 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_If,type,
% 2.22/1.27 if:
% 2.22/1.27 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_Lattices_Osup__class_Osup,type,
% 2.22/1.27 sup_sup:
% 2.22/1.27 !>[A: $tType] : ( A > A > A ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.22/1.27 bot_bot:
% 2.22/1.27 !>[A: $tType] : A ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.22/1.27 ord_less:
% 2.22/1.27 !>[A: $tType] : ( A > A > $o ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.22/1.27 ord_less_eq:
% 2.22/1.27 !>[A: $tType] : ( A > A > $o ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_Set_OCollect,type,
% 2.22/1.27 collect:
% 2.22/1.27 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_Set_Oinsert,type,
% 2.22/1.27 insert:
% 2.22/1.27 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_Set_Ois__empty,type,
% 2.22/1.27 is_empty:
% 2.22/1.27 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_Set_Ois__singleton,type,
% 2.22/1.27 is_singleton:
% 2.22/1.27 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_Set_Opairwise,type,
% 2.22/1.27 pairwise:
% 2.22/1.27 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_Set_Oremove,type,
% 2.22/1.27 remove:
% 2.22/1.27 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_Set_Othe__elem,type,
% 2.22/1.27 the_elem:
% 2.22/1.27 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_c_member,type,
% 2.22/1.27 member:
% 2.22/1.27 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.22/1.27
% 2.22/1.27 thf(sy_v_e,type,
% 2.22/1.27 e: a ).
% 2.22/1.27
% 2.22/1.27 thf(sy_v_h,type,
% 2.22/1.27 h: a > int ).
% 2.22/1.27
% 2.22/1.27 thf(sy_v_t1____,type,
% 2.22/1.27 t1: binary1291135688e_Tree @ a ).
% 2.22/1.27
% 2.22/1.27 thf(sy_v_t2____,type,
% 2.22/1.27 t2: binary1291135688e_Tree @ a ).
% 2.22/1.27
% 2.22/1.27 thf(sy_v_x____,type,
% 2.22/1.27 x: a ).
% 2.22/1.27
% 2.22/1.27 thf(fact_0_eqsLessX,axiom,
% 2.22/1.27 ! [X: a] :
% 2.22/1.27 ( ( member @ a @ X @ ( binary64540844le_eqs @ a @ h @ e ) )
% 2.22/1.27 => ( ord_less @ int @ ( h @ X ) @ ( h @ x ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_1_eLess,axiom,
% 2.22/1.27 ord_less @ int @ ( h @ e ) @ ( h @ x ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_2_res,axiom,
% 2.22/1.27 ( ( binary1830089824insert @ a @ h @ e @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) )
% 2.22/1.27 = ( binary210054475elle_T @ a @ ( binary1830089824insert @ a @ h @ e @ t1 ) @ x @ t2 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_3_s1,axiom,
% 2.22/1.27 binary1610619414edTree @ a @ h @ t1 ).
% 2.22/1.27
% 2.22/1.27 thf(fact_4_s2,axiom,
% 2.22/1.27 binary1610619414edTree @ a @ h @ t2 ).
% 2.22/1.27
% 2.22/1.27 thf(fact_5_s,axiom,
% 2.22/1.27 binary1610619414edTree @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_6_sorted__distinct,axiom,
% 2.22/1.27 ! [A: $tType,H: A > int,A2: A,B: A,T: binary1291135688e_Tree @ A] : ( binary231205461t_pred @ A @ H @ A2 @ B @ T ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_7_c1,axiom,
% 2.22/1.27 ( ( binary1653327646_setOf @ a @ ( binary1830089824insert @ a @ h @ e @ t1 ) )
% 2.22/1.27 = ( sup_sup @ ( set @ a ) @ ( minus_minus @ ( set @ a ) @ ( binary1653327646_setOf @ a @ t1 ) @ ( binary64540844le_eqs @ a @ h @ e ) ) @ ( insert @ a @ e @ ( bot_bot @ ( set @ a ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_8_c2,axiom,
% 2.22/1.27 ( ( binary1653327646_setOf @ a @ ( binary1830089824insert @ a @ h @ e @ t2 ) )
% 2.22/1.27 = ( sup_sup @ ( set @ a ) @ ( minus_minus @ ( set @ a ) @ ( binary1653327646_setOf @ a @ t2 ) @ ( binary64540844le_eqs @ a @ h @ e ) ) @ ( insert @ a @ e @ ( bot_bot @ ( set @ a ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_9_eqs__def,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( binary64540844le_eqs @ A )
% 2.22/1.27 = ( ^ [H2: A > int,X2: A] :
% 2.22/1.27 ( collect @ A
% 2.22/1.27 @ ^ [Y: A] :
% 2.22/1.27 ( ( H2 @ Y )
% 2.22/1.27 = ( H2 @ X2 ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_10_h1,axiom,
% 2.22/1.27 ( ( binary1610619414edTree @ a @ h @ t1 )
% 2.22/1.27 => ( ( binary1653327646_setOf @ a @ ( binary1830089824insert @ a @ h @ e @ t1 ) )
% 2.22/1.27 = ( sup_sup @ ( set @ a ) @ ( minus_minus @ ( set @ a ) @ ( binary1653327646_setOf @ a @ t1 ) @ ( binary64540844le_eqs @ a @ h @ e ) ) @ ( insert @ a @ e @ ( bot_bot @ ( set @ a ) ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_11_h2,axiom,
% 2.22/1.27 ( ( binary1610619414edTree @ a @ h @ t2 )
% 2.22/1.27 => ( ( binary1653327646_setOf @ a @ ( binary1830089824insert @ a @ h @ e @ t2 ) )
% 2.22/1.27 = ( sup_sup @ ( set @ a ) @ ( minus_minus @ ( set @ a ) @ ( binary1653327646_setOf @ a @ t2 ) @ ( binary64540844le_eqs @ a @ h @ e ) ) @ ( insert @ a @ e @ ( bot_bot @ ( set @ a ) ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_12__092_060open_062sortedTree_Ah_ATip_A_092_060longrightarrow_062_AsetOf_A_Ibinsert_Ah_Ae_ATip_J_A_061_AsetOf_ATip_A_N_Aeqs_Ah_Ae_A_092_060union_062_A_123e_125_092_060close_062,axiom,
% 2.22/1.27 ( ( binary1610619414edTree @ a @ h @ ( binary1746293266le_Tip @ a ) )
% 2.22/1.27 => ( ( binary1653327646_setOf @ a @ ( binary1830089824insert @ a @ h @ e @ ( binary1746293266le_Tip @ a ) ) )
% 2.22/1.27 = ( sup_sup @ ( set @ a ) @ ( minus_minus @ ( set @ a ) @ ( binary1653327646_setOf @ a @ ( binary1746293266le_Tip @ a ) ) @ ( binary64540844le_eqs @ a @ h @ e ) ) @ ( insert @ a @ e @ ( bot_bot @ ( set @ a ) ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_13_Tree_Oinject,axiom,
% 2.22/1.27 ! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A,Y21: binary1291135688e_Tree @ A,Y22: A,Y23: binary1291135688e_Tree @ A] :
% 2.22/1.27 ( ( ( binary210054475elle_T @ A @ X21 @ X22 @ X23 )
% 2.22/1.27 = ( binary210054475elle_T @ A @ Y21 @ Y22 @ Y23 ) )
% 2.22/1.27 = ( ( X21 = Y21 )
% 2.22/1.27 & ( X22 = Y22 )
% 2.22/1.27 & ( X23 = Y23 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_14_setOf_Osimps_I2_J,axiom,
% 2.22/1.27 ! [A: $tType,T1: binary1291135688e_Tree @ A,X3: A,T2: binary1291135688e_Tree @ A] :
% 2.22/1.27 ( ( binary1653327646_setOf @ A @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
% 2.22/1.27 = ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( binary1653327646_setOf @ A @ T1 ) @ ( binary1653327646_setOf @ A @ T2 ) ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_15_setOf_Osimps_I1_J,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( binary1653327646_setOf @ A @ ( binary1746293266le_Tip @ A ) )
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_16_Tree_Odistinct_I1_J,axiom,
% 2.22/1.27 ! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
% 2.22/1.27 ( ( binary1746293266le_Tip @ A )
% 2.22/1.27 != ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_17_binsert_Osimps_I2_J,axiom,
% 2.22/1.27 ! [A: $tType,H: A > int,E: A,X3: A,T1: binary1291135688e_Tree @ A,T2: binary1291135688e_Tree @ A] :
% 2.22/1.27 ( ( ( ord_less @ int @ ( H @ E ) @ ( H @ X3 ) )
% 2.22/1.27 => ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
% 2.22/1.27 = ( binary210054475elle_T @ A @ ( binary1830089824insert @ A @ H @ E @ T1 ) @ X3 @ T2 ) ) )
% 2.22/1.27 & ( ~ ( ord_less @ int @ ( H @ E ) @ ( H @ X3 ) )
% 2.22/1.27 => ( ( ( ord_less @ int @ ( H @ X3 ) @ ( H @ E ) )
% 2.22/1.27 => ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
% 2.22/1.27 = ( binary210054475elle_T @ A @ T1 @ X3 @ ( binary1830089824insert @ A @ H @ E @ T2 ) ) ) )
% 2.22/1.27 & ( ~ ( ord_less @ int @ ( H @ X3 ) @ ( H @ E ) )
% 2.22/1.27 => ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
% 2.22/1.27 = ( binary210054475elle_T @ A @ T1 @ E @ T2 ) ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_18_binsert_Osimps_I1_J,axiom,
% 2.22/1.27 ! [A: $tType,H: A > int,E: A] :
% 2.22/1.27 ( ( binary1830089824insert @ A @ H @ E @ ( binary1746293266le_Tip @ A ) )
% 2.22/1.27 = ( binary210054475elle_T @ A @ ( binary1746293266le_Tip @ A ) @ E @ ( binary1746293266le_Tip @ A ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_19_sortedTree_Osimps_I2_J,axiom,
% 2.22/1.27 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X3: A,T2: binary1291135688e_Tree @ A] :
% 2.22/1.27 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
% 2.22/1.27 = ( ( binary1610619414edTree @ A @ H @ T1 )
% 2.22/1.27 & ! [X2: A] :
% 2.22/1.27 ( ( member @ A @ X2 @ ( binary1653327646_setOf @ A @ T1 ) )
% 2.22/1.27 => ( ord_less @ int @ ( H @ X2 ) @ ( H @ X3 ) ) )
% 2.22/1.27 & ! [X2: A] :
% 2.22/1.27 ( ( member @ A @ X2 @ ( binary1653327646_setOf @ A @ T2 ) )
% 2.22/1.27 => ( ord_less @ int @ ( H @ X3 ) @ ( H @ X2 ) ) )
% 2.22/1.27 & ( binary1610619414edTree @ A @ H @ T2 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_20_sortedTree_Osimps_I1_J,axiom,
% 2.22/1.27 ! [A: $tType,H: A > int] : ( binary1610619414edTree @ A @ H @ ( binary1746293266le_Tip @ A ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_21_sortLemmaL,axiom,
% 2.22/1.27 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X3: A,T2: binary1291135688e_Tree @ A] :
% 2.22/1.27 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
% 2.22/1.27 => ( binary1610619414edTree @ A @ H @ T1 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_22_sortLemmaR,axiom,
% 2.22/1.27 ! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X3: A,T2: binary1291135688e_Tree @ A] :
% 2.22/1.27 ( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X3 @ T2 ) )
% 2.22/1.27 => ( binary1610619414edTree @ A @ H @ T2 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_23_Tree_Oinduct,axiom,
% 2.22/1.27 ! [A: $tType,P: ( binary1291135688e_Tree @ A ) > $o,Tree: binary1291135688e_Tree @ A] :
% 2.22/1.27 ( ( P @ ( binary1746293266le_Tip @ A ) )
% 2.22/1.27 => ( ! [X1: binary1291135688e_Tree @ A,X24: A,X32: binary1291135688e_Tree @ A] :
% 2.22/1.27 ( ( P @ X1 )
% 2.22/1.27 => ( ( P @ X32 )
% 2.22/1.27 => ( P @ ( binary210054475elle_T @ A @ X1 @ X24 @ X32 ) ) ) )
% 2.22/1.27 => ( P @ Tree ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_24_Tree_Oexhaust,axiom,
% 2.22/1.27 ! [A: $tType,Y2: binary1291135688e_Tree @ A] :
% 2.22/1.27 ( ( Y2
% 2.22/1.27 != ( binary1746293266le_Tip @ A ) )
% 2.22/1.27 => ~ ! [X212: binary1291135688e_Tree @ A,X222: A,X232: binary1291135688e_Tree @ A] :
% 2.22/1.27 ( Y2
% 2.22/1.27 != ( binary210054475elle_T @ A @ X212 @ X222 @ X232 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_25_sorted__distinct__pred__def,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( binary231205461t_pred @ A )
% 2.22/1.27 = ( ^ [H2: A > int,A3: A,B2: A,T3: binary1291135688e_Tree @ A] :
% 2.22/1.27 ( ( ( binary1610619414edTree @ A @ H2 @ T3 )
% 2.22/1.27 & ( member @ A @ A3 @ ( binary1653327646_setOf @ A @ T3 ) )
% 2.22/1.27 & ( member @ A @ B2 @ ( binary1653327646_setOf @ A @ T3 ) )
% 2.22/1.27 & ( ( H2 @ A3 )
% 2.22/1.27 = ( H2 @ B2 ) ) )
% 2.22/1.27 => ( A3 = B2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_26_insert__Diff__single,axiom,
% 2.22/1.27 ! [A: $tType,A2: A,A4: set @ A] :
% 2.22/1.27 ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.22/1.27 = ( insert @ A @ A2 @ A4 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_27_singleton__conv,axiom,
% 2.22/1.27 ! [A: $tType,A2: A] :
% 2.22/1.27 ( ( collect @ A
% 2.22/1.27 @ ^ [X2: A] : ( X2 = A2 ) )
% 2.22/1.27 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_28_singleton__conv2,axiom,
% 2.22/1.27 ! [A: $tType,A2: A] :
% 2.22/1.27 ( ( collect @ A
% 2.22/1.27 @ ( ^ [Y3: A,Z: A] : ( Y3 = Z )
% 2.22/1.27 @ A2 ) )
% 2.22/1.27 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_29_Un__Diff__cancel,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ( sup_sup @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) )
% 2.22/1.27 = ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_30_Un__Diff__cancel2,axiom,
% 2.22/1.27 ! [A: $tType,B3: set @ A,A4: set @ A] :
% 2.22/1.27 ( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) @ A4 )
% 2.22/1.27 = ( sup_sup @ ( set @ A ) @ B3 @ A4 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_31_Diff__insert0,axiom,
% 2.22/1.27 ! [A: $tType,X3: A,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ~ ( member @ A @ X3 @ A4 )
% 2.22/1.27 => ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ B3 ) )
% 2.22/1.27 = ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_32_insert__Diff1,axiom,
% 2.22/1.27 ! [A: $tType,X3: A,B3: set @ A,A4: set @ A] :
% 2.22/1.27 ( ( member @ A @ X3 @ B3 )
% 2.22/1.27 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A4 ) @ B3 )
% 2.22/1.27 = ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_33_Un__insert__left,axiom,
% 2.22/1.27 ! [A: $tType,A2: A,B3: set @ A,C: set @ A] :
% 2.22/1.27 ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C )
% 2.22/1.27 = ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B3 @ C ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_34_Un__insert__right,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,A2: A,B3: set @ A] :
% 2.22/1.27 ( ( sup_sup @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B3 ) )
% 2.22/1.27 = ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_35_Diff__empty,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A] :
% 2.22/1.27 ( ( minus_minus @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 2.22/1.27 = A4 ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_36_empty__Diff,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A] :
% 2.22/1.27 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 )
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_37_empty__Collect__eq,axiom,
% 2.22/1.27 ! [A: $tType,P: A > $o] :
% 2.22/1.27 ( ( ( bot_bot @ ( set @ A ) )
% 2.22/1.27 = ( collect @ A @ P ) )
% 2.22/1.27 = ( ! [X2: A] :
% 2.22/1.27 ~ ( P @ X2 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_38_Collect__empty__eq,axiom,
% 2.22/1.27 ! [A: $tType,P: A > $o] :
% 2.22/1.27 ( ( ( collect @ A @ P )
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) )
% 2.22/1.27 = ( ! [X2: A] :
% 2.22/1.27 ~ ( P @ X2 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_39_all__not__in__conv,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A] :
% 2.22/1.27 ( ( ! [X2: A] :
% 2.22/1.27 ~ ( member @ A @ X2 @ A4 ) )
% 2.22/1.27 = ( A4
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_40_empty__iff,axiom,
% 2.22/1.27 ! [A: $tType,C2: A] :
% 2.22/1.27 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_41_insert__absorb2,axiom,
% 2.22/1.27 ! [A: $tType,X3: A,A4: set @ A] :
% 2.22/1.27 ( ( insert @ A @ X3 @ ( insert @ A @ X3 @ A4 ) )
% 2.22/1.27 = ( insert @ A @ X3 @ A4 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_42_insert__iff,axiom,
% 2.22/1.27 ! [A: $tType,A2: A,B: A,A4: set @ A] :
% 2.22/1.27 ( ( member @ A @ A2 @ ( insert @ A @ B @ A4 ) )
% 2.22/1.27 = ( ( A2 = B )
% 2.22/1.27 | ( member @ A @ A2 @ A4 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_43_insertCI,axiom,
% 2.22/1.27 ! [A: $tType,A2: A,B3: set @ A,B: A] :
% 2.22/1.27 ( ( ~ ( member @ A @ A2 @ B3 )
% 2.22/1.27 => ( A2 = B ) )
% 2.22/1.27 => ( member @ A @ A2 @ ( insert @ A @ B @ B3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_44_Un__iff,axiom,
% 2.22/1.27 ! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 2.22/1.27 = ( ( member @ A @ C2 @ A4 )
% 2.22/1.27 | ( member @ A @ C2 @ B3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_45_mem__Collect__eq,axiom,
% 2.22/1.27 ! [A: $tType,A2: A,P: A > $o] :
% 2.22/1.27 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.22/1.27 = ( P @ A2 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_46_Collect__mem__eq,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A] :
% 2.22/1.27 ( ( collect @ A
% 2.22/1.27 @ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
% 2.22/1.27 = A4 ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_47_Collect__cong,axiom,
% 2.22/1.27 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.22/1.27 ( ! [X4: A] :
% 2.22/1.27 ( ( P @ X4 )
% 2.22/1.27 = ( Q @ X4 ) )
% 2.22/1.27 => ( ( collect @ A @ P )
% 2.22/1.27 = ( collect @ A @ Q ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_48_ext,axiom,
% 2.22/1.27 ! [B4: $tType,A: $tType,F: A > B4,G: A > B4] :
% 2.22/1.27 ( ! [X4: A] :
% 2.22/1.27 ( ( F @ X4 )
% 2.22/1.27 = ( G @ X4 ) )
% 2.22/1.27 => ( F = G ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_49_UnCI,axiom,
% 2.22/1.27 ! [A: $tType,C2: A,B3: set @ A,A4: set @ A] :
% 2.22/1.27 ( ( ~ ( member @ A @ C2 @ B3 )
% 2.22/1.27 => ( member @ A @ C2 @ A4 ) )
% 2.22/1.27 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_50_Diff__idemp,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ B3 )
% 2.22/1.27 = ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_51_Diff__iff,axiom,
% 2.22/1.27 ! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 2.22/1.27 = ( ( member @ A @ C2 @ A4 )
% 2.22/1.27 & ~ ( member @ A @ C2 @ B3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_52_DiffI,axiom,
% 2.22/1.27 ! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ( member @ A @ C2 @ A4 )
% 2.22/1.27 => ( ~ ( member @ A @ C2 @ B3 )
% 2.22/1.27 => ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_53_singletonI,axiom,
% 2.22/1.27 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_54_Un__empty,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ( ( sup_sup @ ( set @ A ) @ A4 @ B3 )
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) )
% 2.22/1.27 = ( ( A4
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) )
% 2.22/1.27 & ( B3
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_55_Diff__cancel,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A] :
% 2.22/1.27 ( ( minus_minus @ ( set @ A ) @ A4 @ A4 )
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_56_ex__in__conv,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A] :
% 2.22/1.27 ( ( ? [X2: A] : ( member @ A @ X2 @ A4 ) )
% 2.22/1.27 = ( A4
% 2.22/1.27 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_57_equals0I,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A] :
% 2.22/1.27 ( ! [Y4: A] :
% 2.22/1.27 ~ ( member @ A @ Y4 @ A4 )
% 2.22/1.27 => ( A4
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_58_equals0D,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,A2: A] :
% 2.22/1.27 ( ( A4
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) )
% 2.22/1.27 => ~ ( member @ A @ A2 @ A4 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_59_emptyE,axiom,
% 2.22/1.27 ! [A: $tType,A2: A] :
% 2.22/1.27 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_60_mk__disjoint__insert,axiom,
% 2.22/1.27 ! [A: $tType,A2: A,A4: set @ A] :
% 2.22/1.27 ( ( member @ A @ A2 @ A4 )
% 2.22/1.27 => ? [B5: set @ A] :
% 2.22/1.27 ( ( A4
% 2.22/1.27 = ( insert @ A @ A2 @ B5 ) )
% 2.22/1.27 & ~ ( member @ A @ A2 @ B5 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_61_insert__commute,axiom,
% 2.22/1.27 ! [A: $tType,X3: A,Y2: A,A4: set @ A] :
% 2.22/1.27 ( ( insert @ A @ X3 @ ( insert @ A @ Y2 @ A4 ) )
% 2.22/1.27 = ( insert @ A @ Y2 @ ( insert @ A @ X3 @ A4 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_62_insert__eq__iff,axiom,
% 2.22/1.27 ! [A: $tType,A2: A,A4: set @ A,B: A,B3: set @ A] :
% 2.22/1.27 ( ~ ( member @ A @ A2 @ A4 )
% 2.22/1.27 => ( ~ ( member @ A @ B @ B3 )
% 2.22/1.27 => ( ( ( insert @ A @ A2 @ A4 )
% 2.22/1.27 = ( insert @ A @ B @ B3 ) )
% 2.22/1.27 = ( ( ( A2 = B )
% 2.22/1.27 => ( A4 = B3 ) )
% 2.22/1.27 & ( ( A2 != B )
% 2.22/1.27 => ? [C3: set @ A] :
% 2.22/1.27 ( ( A4
% 2.22/1.27 = ( insert @ A @ B @ C3 ) )
% 2.22/1.27 & ~ ( member @ A @ B @ C3 )
% 2.22/1.27 & ( B3
% 2.22/1.27 = ( insert @ A @ A2 @ C3 ) )
% 2.22/1.27 & ~ ( member @ A @ A2 @ C3 ) ) ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_63_insert__absorb,axiom,
% 2.22/1.27 ! [A: $tType,A2: A,A4: set @ A] :
% 2.22/1.27 ( ( member @ A @ A2 @ A4 )
% 2.22/1.27 => ( ( insert @ A @ A2 @ A4 )
% 2.22/1.27 = A4 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_64_insert__ident,axiom,
% 2.22/1.27 ! [A: $tType,X3: A,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ~ ( member @ A @ X3 @ A4 )
% 2.22/1.27 => ( ~ ( member @ A @ X3 @ B3 )
% 2.22/1.27 => ( ( ( insert @ A @ X3 @ A4 )
% 2.22/1.27 = ( insert @ A @ X3 @ B3 ) )
% 2.22/1.27 = ( A4 = B3 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_65_Set_Oset__insert,axiom,
% 2.22/1.27 ! [A: $tType,X3: A,A4: set @ A] :
% 2.22/1.27 ( ( member @ A @ X3 @ A4 )
% 2.22/1.27 => ~ ! [B5: set @ A] :
% 2.22/1.27 ( ( A4
% 2.22/1.27 = ( insert @ A @ X3 @ B5 ) )
% 2.22/1.27 => ( member @ A @ X3 @ B5 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_66_insertI2,axiom,
% 2.22/1.27 ! [A: $tType,A2: A,B3: set @ A,B: A] :
% 2.22/1.27 ( ( member @ A @ A2 @ B3 )
% 2.22/1.27 => ( member @ A @ A2 @ ( insert @ A @ B @ B3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_67_insertI1,axiom,
% 2.22/1.27 ! [A: $tType,A2: A,B3: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B3 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_68_insertE,axiom,
% 2.22/1.27 ! [A: $tType,A2: A,B: A,A4: set @ A] :
% 2.22/1.27 ( ( member @ A @ A2 @ ( insert @ A @ B @ A4 ) )
% 2.22/1.27 => ( ( A2 != B )
% 2.22/1.27 => ( member @ A @ A2 @ A4 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_69_Un__left__commute,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,B3: set @ A,C: set @ A] :
% 2.22/1.27 ( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B3 @ C ) )
% 2.22/1.27 = ( sup_sup @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A4 @ C ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_70_Un__left__absorb,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 2.22/1.27 = ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_71_Un__commute,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( sup_sup @ ( set @ A ) )
% 2.22/1.27 = ( ^ [A5: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_72_Un__absorb,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A] :
% 2.22/1.27 ( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
% 2.22/1.27 = A4 ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_73_Un__assoc,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,B3: set @ A,C: set @ A] :
% 2.22/1.27 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C )
% 2.22/1.27 = ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B3 @ C ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_74_ball__Un,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,B3: set @ A,P: A > $o] :
% 2.22/1.27 ( ( ! [X2: A] :
% 2.22/1.27 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 2.22/1.27 => ( P @ X2 ) ) )
% 2.22/1.27 = ( ! [X2: A] :
% 2.22/1.27 ( ( member @ A @ X2 @ A4 )
% 2.22/1.27 => ( P @ X2 ) )
% 2.22/1.27 & ! [X2: A] :
% 2.22/1.27 ( ( member @ A @ X2 @ B3 )
% 2.22/1.27 => ( P @ X2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_75_bex__Un,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,B3: set @ A,P: A > $o] :
% 2.22/1.27 ( ( ? [X2: A] :
% 2.22/1.27 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 2.22/1.27 & ( P @ X2 ) ) )
% 2.22/1.27 = ( ? [X2: A] :
% 2.22/1.27 ( ( member @ A @ X2 @ A4 )
% 2.22/1.27 & ( P @ X2 ) )
% 2.22/1.27 | ? [X2: A] :
% 2.22/1.27 ( ( member @ A @ X2 @ B3 )
% 2.22/1.27 & ( P @ X2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_76_UnI2,axiom,
% 2.22/1.27 ! [A: $tType,C2: A,B3: set @ A,A4: set @ A] :
% 2.22/1.27 ( ( member @ A @ C2 @ B3 )
% 2.22/1.27 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_77_UnI1,axiom,
% 2.22/1.27 ! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ( member @ A @ C2 @ A4 )
% 2.22/1.27 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_78_UnE,axiom,
% 2.22/1.27 ! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 2.22/1.27 => ( ~ ( member @ A @ C2 @ A4 )
% 2.22/1.27 => ( member @ A @ C2 @ B3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_79_DiffD2,axiom,
% 2.22/1.27 ! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 2.22/1.27 => ~ ( member @ A @ C2 @ B3 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_80_DiffD1,axiom,
% 2.22/1.27 ! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 2.22/1.27 => ( member @ A @ C2 @ A4 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_81_DiffE,axiom,
% 2.22/1.27 ! [A: $tType,C2: A,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) )
% 2.22/1.27 => ~ ( ( member @ A @ C2 @ A4 )
% 2.22/1.27 => ( member @ A @ C2 @ B3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_82_empty__def,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( bot_bot @ ( set @ A ) )
% 2.22/1.27 = ( collect @ A
% 2.22/1.27 @ ^ [X2: A] : $false ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_83_insert__Collect,axiom,
% 2.22/1.27 ! [A: $tType,A2: A,P: A > $o] :
% 2.22/1.27 ( ( insert @ A @ A2 @ ( collect @ A @ P ) )
% 2.22/1.27 = ( collect @ A
% 2.22/1.27 @ ^ [U: A] :
% 2.22/1.27 ( ( U != A2 )
% 2.22/1.27 => ( P @ U ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_84_insert__compr,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( insert @ A )
% 2.22/1.27 = ( ^ [A3: A,B6: set @ A] :
% 2.22/1.27 ( collect @ A
% 2.22/1.27 @ ^ [X2: A] :
% 2.22/1.27 ( ( X2 = A3 )
% 2.22/1.27 | ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_85_Collect__disj__eq,axiom,
% 2.22/1.27 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.22/1.27 ( ( collect @ A
% 2.22/1.27 @ ^ [X2: A] :
% 2.22/1.27 ( ( P @ X2 )
% 2.22/1.27 | ( Q @ X2 ) ) )
% 2.22/1.27 = ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_86_Un__def,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( sup_sup @ ( set @ A ) )
% 2.22/1.27 = ( ^ [A5: set @ A,B6: set @ A] :
% 2.22/1.27 ( collect @ A
% 2.22/1.27 @ ^ [X2: A] :
% 2.22/1.27 ( ( member @ A @ X2 @ A5 )
% 2.22/1.27 | ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_87_set__diff__eq,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( minus_minus @ ( set @ A ) )
% 2.22/1.27 = ( ^ [A5: set @ A,B6: set @ A] :
% 2.22/1.27 ( collect @ A
% 2.22/1.27 @ ^ [X2: A] :
% 2.22/1.27 ( ( member @ A @ X2 @ A5 )
% 2.22/1.27 & ~ ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_88_singleton__inject,axiom,
% 2.22/1.27 ! [A: $tType,A2: A,B: A] :
% 2.22/1.27 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 2.22/1.27 = ( insert @ A @ B @ ( bot_bot @ ( set @ A ) ) ) )
% 2.22/1.27 => ( A2 = B ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_89_insert__not__empty,axiom,
% 2.22/1.27 ! [A: $tType,A2: A,A4: set @ A] :
% 2.22/1.27 ( ( insert @ A @ A2 @ A4 )
% 2.22/1.27 != ( bot_bot @ ( set @ A ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_90_doubleton__eq__iff,axiom,
% 2.22/1.27 ! [A: $tType,A2: A,B: A,C2: A,D: A] :
% 2.22/1.27 ( ( ( insert @ A @ A2 @ ( insert @ A @ B @ ( bot_bot @ ( set @ A ) ) ) )
% 2.22/1.27 = ( insert @ A @ C2 @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.22/1.27 = ( ( ( A2 = C2 )
% 2.22/1.27 & ( B = D ) )
% 2.22/1.27 | ( ( A2 = D )
% 2.22/1.27 & ( B = C2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_91_singleton__iff,axiom,
% 2.22/1.27 ! [A: $tType,B: A,A2: A] :
% 2.22/1.27 ( ( member @ A @ B @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.22/1.27 = ( B = A2 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_92_singletonD,axiom,
% 2.22/1.27 ! [A: $tType,B: A,A2: A] :
% 2.22/1.27 ( ( member @ A @ B @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.22/1.27 => ( B = A2 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_93_Un__empty__right,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A] :
% 2.22/1.27 ( ( sup_sup @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 2.22/1.27 = A4 ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_94_Un__empty__left,axiom,
% 2.22/1.27 ! [A: $tType,B3: set @ A] :
% 2.22/1.27 ( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B3 )
% 2.22/1.27 = B3 ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_95_insert__Diff__if,axiom,
% 2.22/1.27 ! [A: $tType,X3: A,B3: set @ A,A4: set @ A] :
% 2.22/1.27 ( ( ( member @ A @ X3 @ B3 )
% 2.22/1.27 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A4 ) @ B3 )
% 2.22/1.27 = ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) )
% 2.22/1.27 & ( ~ ( member @ A @ X3 @ B3 )
% 2.22/1.27 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A4 ) @ B3 )
% 2.22/1.27 = ( insert @ A @ X3 @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_96_Un__Diff,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,B3: set @ A,C: set @ A] :
% 2.22/1.27 ( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C )
% 2.22/1.27 = ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ C ) @ ( minus_minus @ ( set @ A ) @ B3 @ C ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_97_Collect__conv__if2,axiom,
% 2.22/1.27 ! [A: $tType,P: A > $o,A2: A] :
% 2.22/1.27 ( ( ( P @ A2 )
% 2.22/1.27 => ( ( collect @ A
% 2.22/1.27 @ ^ [X2: A] :
% 2.22/1.27 ( ( A2 = X2 )
% 2.22/1.27 & ( P @ X2 ) ) )
% 2.22/1.27 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.22/1.27 & ( ~ ( P @ A2 )
% 2.22/1.27 => ( ( collect @ A
% 2.22/1.27 @ ^ [X2: A] :
% 2.22/1.27 ( ( A2 = X2 )
% 2.22/1.27 & ( P @ X2 ) ) )
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_98_Collect__conv__if,axiom,
% 2.22/1.27 ! [A: $tType,P: A > $o,A2: A] :
% 2.22/1.27 ( ( ( P @ A2 )
% 2.22/1.27 => ( ( collect @ A
% 2.22/1.27 @ ^ [X2: A] :
% 2.22/1.27 ( ( X2 = A2 )
% 2.22/1.27 & ( P @ X2 ) ) )
% 2.22/1.27 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.22/1.27 & ( ~ ( P @ A2 )
% 2.22/1.27 => ( ( collect @ A
% 2.22/1.27 @ ^ [X2: A] :
% 2.22/1.27 ( ( X2 = A2 )
% 2.22/1.27 & ( P @ X2 ) ) )
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_99_insert__def,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( insert @ A )
% 2.22/1.27 = ( ^ [A3: A] :
% 2.22/1.27 ( sup_sup @ ( set @ A )
% 2.22/1.27 @ ( collect @ A
% 2.22/1.27 @ ^ [X2: A] : ( X2 = A3 ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_100_singleton__Un__iff,axiom,
% 2.22/1.27 ! [A: $tType,X3: A,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ( ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
% 2.22/1.27 = ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
% 2.22/1.27 = ( ( ( A4
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) )
% 2.22/1.27 & ( B3
% 2.22/1.27 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.22/1.27 | ( ( A4
% 2.22/1.27 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.22/1.27 & ( B3
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) ) )
% 2.22/1.27 | ( ( A4
% 2.22/1.27 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.22/1.27 & ( B3
% 2.22/1.27 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_101_Un__singleton__iff,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,B3: set @ A,X3: A] :
% 2.22/1.27 ( ( ( sup_sup @ ( set @ A ) @ A4 @ B3 )
% 2.22/1.27 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.22/1.27 = ( ( ( A4
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) )
% 2.22/1.27 & ( B3
% 2.22/1.27 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.22/1.27 | ( ( A4
% 2.22/1.27 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.22/1.27 & ( B3
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) ) )
% 2.22/1.27 | ( ( A4
% 2.22/1.27 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.22/1.27 & ( B3
% 2.22/1.27 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_102_insert__is__Un,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( insert @ A )
% 2.22/1.27 = ( ^ [A3: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_103_Diff__insert__absorb,axiom,
% 2.22/1.27 ! [A: $tType,X3: A,A4: set @ A] :
% 2.22/1.27 ( ~ ( member @ A @ X3 @ A4 )
% 2.22/1.27 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A4 ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.22/1.27 = A4 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_104_Diff__insert2,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,A2: A,B3: set @ A] :
% 2.22/1.27 ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B3 ) )
% 2.22/1.27 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_105_insert__Diff,axiom,
% 2.22/1.27 ! [A: $tType,A2: A,A4: set @ A] :
% 2.22/1.27 ( ( member @ A @ A2 @ A4 )
% 2.22/1.27 => ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.22/1.27 = A4 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_106_Diff__insert,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,A2: A,B3: set @ A] :
% 2.22/1.27 ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B3 ) )
% 2.22/1.27 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B3 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_107_sup__bot_Oright__neutral,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( bounde1808546759up_bot @ A )
% 2.22/1.27 => ! [A2: A] :
% 2.22/1.27 ( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
% 2.22/1.27 = A2 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_108_sup__bot_Oneutr__eq__iff,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( bounde1808546759up_bot @ A )
% 2.22/1.27 => ! [A2: A,B: A] :
% 2.22/1.27 ( ( ( bot_bot @ A )
% 2.22/1.27 = ( sup_sup @ A @ A2 @ B ) )
% 2.22/1.27 = ( ( A2
% 2.22/1.27 = ( bot_bot @ A ) )
% 2.22/1.27 & ( B
% 2.22/1.27 = ( bot_bot @ A ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_109_sup__bot_Oleft__neutral,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( bounde1808546759up_bot @ A )
% 2.22/1.27 => ! [A2: A] :
% 2.22/1.27 ( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
% 2.22/1.27 = A2 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_110_sup__bot_Oeq__neutr__iff,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( bounde1808546759up_bot @ A )
% 2.22/1.27 => ! [A2: A,B: A] :
% 2.22/1.27 ( ( ( sup_sup @ A @ A2 @ B )
% 2.22/1.27 = ( bot_bot @ A ) )
% 2.22/1.27 = ( ( A2
% 2.22/1.27 = ( bot_bot @ A ) )
% 2.22/1.27 & ( B
% 2.22/1.27 = ( bot_bot @ A ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_111_sup__eq__bot__iff,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( bounde1808546759up_bot @ A )
% 2.22/1.27 => ! [X3: A,Y2: A] :
% 2.22/1.27 ( ( ( sup_sup @ A @ X3 @ Y2 )
% 2.22/1.27 = ( bot_bot @ A ) )
% 2.22/1.27 = ( ( X3
% 2.22/1.27 = ( bot_bot @ A ) )
% 2.22/1.27 & ( Y2
% 2.22/1.27 = ( bot_bot @ A ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_112_bot__eq__sup__iff,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( bounde1808546759up_bot @ A )
% 2.22/1.27 => ! [X3: A,Y2: A] :
% 2.22/1.27 ( ( ( bot_bot @ A )
% 2.22/1.27 = ( sup_sup @ A @ X3 @ Y2 ) )
% 2.22/1.27 = ( ( X3
% 2.22/1.27 = ( bot_bot @ A ) )
% 2.22/1.27 & ( Y2
% 2.22/1.27 = ( bot_bot @ A ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_113_sup__bot__right,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( bounde1808546759up_bot @ A )
% 2.22/1.27 => ! [X3: A] :
% 2.22/1.27 ( ( sup_sup @ A @ X3 @ ( bot_bot @ A ) )
% 2.22/1.27 = X3 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_114_sup__bot__left,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( bounde1808546759up_bot @ A )
% 2.22/1.27 => ! [X3: A] :
% 2.22/1.27 ( ( sup_sup @ A @ ( bot_bot @ A ) @ X3 )
% 2.22/1.27 = X3 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_115_memb__spec,axiom,
% 2.22/1.27 ! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,X3: A] :
% 2.22/1.27 ( ( binary1610619414edTree @ A @ H @ T )
% 2.22/1.27 => ( ( binary827270440e_memb @ A @ H @ X3 @ T )
% 2.22/1.27 = ( member @ A @ X3 @ ( binary1653327646_setOf @ A @ T ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_116_minus__apply,axiom,
% 2.22/1.27 ! [B4: $tType,A: $tType] :
% 2.22/1.27 ( ( minus @ B4 )
% 2.22/1.27 => ( ( minus_minus @ ( A > B4 ) )
% 2.22/1.27 = ( ^ [A5: A > B4,B6: A > B4,X2: A] : ( minus_minus @ B4 @ ( A5 @ X2 ) @ ( B6 @ X2 ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_117_sup__apply,axiom,
% 2.22/1.27 ! [B4: $tType,A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ B4 )
% 2.22/1.27 => ( ( sup_sup @ ( A > B4 ) )
% 2.22/1.27 = ( ^ [F2: A > B4,G2: A > B4,X2: A] : ( sup_sup @ B4 @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_118_sup_Oidem,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ! [A2: A] :
% 2.22/1.27 ( ( sup_sup @ A @ A2 @ A2 )
% 2.22/1.27 = A2 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_119_sup__idem,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ! [X3: A] :
% 2.22/1.27 ( ( sup_sup @ A @ X3 @ X3 )
% 2.22/1.27 = X3 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_120_sup_Oleft__idem,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ! [A2: A,B: A] :
% 2.22/1.27 ( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B ) )
% 2.22/1.27 = ( sup_sup @ A @ A2 @ B ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_121_sup__left__idem,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ! [X3: A,Y2: A] :
% 2.22/1.27 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) )
% 2.22/1.27 = ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_122_sup_Oright__idem,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ! [A2: A,B: A] :
% 2.22/1.27 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B ) @ B )
% 2.22/1.27 = ( sup_sup @ A @ A2 @ B ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_123_bot__set__def,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( bot_bot @ ( set @ A ) )
% 2.22/1.27 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_124_sup__set__def,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( sup_sup @ ( set @ A ) )
% 2.22/1.27 = ( ^ [A5: set @ A,B6: set @ A] :
% 2.22/1.27 ( collect @ A
% 2.22/1.27 @ ( sup_sup @ ( A > $o )
% 2.22/1.27 @ ^ [X2: A] : ( member @ A @ X2 @ A5 )
% 2.22/1.27 @ ^ [X2: A] : ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_125_fun__diff__def,axiom,
% 2.22/1.27 ! [B4: $tType,A: $tType] :
% 2.22/1.27 ( ( minus @ B4 )
% 2.22/1.27 => ( ( minus_minus @ ( A > B4 ) )
% 2.22/1.27 = ( ^ [A5: A > B4,B6: A > B4,X2: A] : ( minus_minus @ B4 @ ( A5 @ X2 ) @ ( B6 @ X2 ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_126_inf__sup__aci_I8_J,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( lattice @ A )
% 2.22/1.27 => ! [X3: A,Y2: A] :
% 2.22/1.27 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) )
% 2.22/1.27 = ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_127_inf__sup__aci_I7_J,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( lattice @ A )
% 2.22/1.27 => ! [X3: A,Y2: A,Z2: A] :
% 2.22/1.27 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z2 ) )
% 2.22/1.27 = ( sup_sup @ A @ Y2 @ ( sup_sup @ A @ X3 @ Z2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_128_inf__sup__aci_I6_J,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( lattice @ A )
% 2.22/1.27 => ! [X3: A,Y2: A,Z2: A] :
% 2.22/1.27 ( ( sup_sup @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z2 )
% 2.22/1.27 = ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_129_inf__sup__aci_I5_J,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( lattice @ A )
% 2.22/1.27 => ( ( sup_sup @ A )
% 2.22/1.27 = ( ^ [X2: A,Y: A] : ( sup_sup @ A @ Y @ X2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_130_sup__fun__def,axiom,
% 2.22/1.27 ! [B4: $tType,A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ B4 )
% 2.22/1.27 => ( ( sup_sup @ ( A > B4 ) )
% 2.22/1.27 = ( ^ [F2: A > B4,G2: A > B4,X2: A] : ( sup_sup @ B4 @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_131_boolean__algebra__cancel_Osup1,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ! [A4: A,K: A,A2: A,B: A] :
% 2.22/1.27 ( ( A4
% 2.22/1.27 = ( sup_sup @ A @ K @ A2 ) )
% 2.22/1.27 => ( ( sup_sup @ A @ A4 @ B )
% 2.22/1.27 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_132_boolean__algebra__cancel_Osup2,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ! [B3: A,K: A,B: A,A2: A] :
% 2.22/1.27 ( ( B3
% 2.22/1.27 = ( sup_sup @ A @ K @ B ) )
% 2.22/1.27 => ( ( sup_sup @ A @ A2 @ B3 )
% 2.22/1.27 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_133_sup_Oassoc,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ! [A2: A,B: A,C2: A] :
% 2.22/1.27 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B ) @ C2 )
% 2.22/1.27 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B @ C2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_134_sup__assoc,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ! [X3: A,Y2: A,Z2: A] :
% 2.22/1.27 ( ( sup_sup @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z2 )
% 2.22/1.27 = ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_135_sup_Ocommute,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ( ( sup_sup @ A )
% 2.22/1.27 = ( ^ [A3: A,B2: A] : ( sup_sup @ A @ B2 @ A3 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_136_sup__commute,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ( ( sup_sup @ A )
% 2.22/1.27 = ( ^ [X2: A,Y: A] : ( sup_sup @ A @ Y @ X2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_137_sup_Oleft__commute,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ! [B: A,A2: A,C2: A] :
% 2.22/1.27 ( ( sup_sup @ A @ B @ ( sup_sup @ A @ A2 @ C2 ) )
% 2.22/1.27 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B @ C2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_138_sup__left__commute,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ! [X3: A,Y2: A,Z2: A] :
% 2.22/1.27 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z2 ) )
% 2.22/1.27 = ( sup_sup @ A @ Y2 @ ( sup_sup @ A @ X3 @ Z2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_139_less__supI1,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ! [X3: A,A2: A,B: A] :
% 2.22/1.27 ( ( ord_less @ A @ X3 @ A2 )
% 2.22/1.27 => ( ord_less @ A @ X3 @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_140_less__supI2,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ! [X3: A,B: A,A2: A] :
% 2.22/1.27 ( ( ord_less @ A @ X3 @ B )
% 2.22/1.27 => ( ord_less @ A @ X3 @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_141_sup_Ostrict__boundedE,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ! [B: A,C2: A,A2: A] :
% 2.22/1.27 ( ( ord_less @ A @ ( sup_sup @ A @ B @ C2 ) @ A2 )
% 2.22/1.27 => ~ ( ( ord_less @ A @ B @ A2 )
% 2.22/1.27 => ~ ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_142_sup_Ostrict__order__iff,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ( ( ord_less @ A )
% 2.22/1.27 = ( ^ [B2: A,A3: A] :
% 2.22/1.27 ( ( A3
% 2.22/1.27 = ( sup_sup @ A @ A3 @ B2 ) )
% 2.22/1.27 & ( A3 != B2 ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_143_sup_Ostrict__coboundedI1,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ! [C2: A,A2: A,B: A] :
% 2.22/1.27 ( ( ord_less @ A @ C2 @ A2 )
% 2.22/1.27 => ( ord_less @ A @ C2 @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_144_sup_Ostrict__coboundedI2,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ! [C2: A,B: A,A2: A] :
% 2.22/1.27 ( ( ord_less @ A @ C2 @ B )
% 2.22/1.27 => ( ord_less @ A @ C2 @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_145_the__elem__eq,axiom,
% 2.22/1.27 ! [A: $tType,X3: A] :
% 2.22/1.27 ( ( the_elem @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.22/1.27 = X3 ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_146_bot__apply,axiom,
% 2.22/1.27 ! [C4: $tType,D2: $tType] :
% 2.22/1.27 ( ( bot @ C4 )
% 2.22/1.27 => ( ( bot_bot @ ( D2 > C4 ) )
% 2.22/1.27 = ( ^ [X2: D2] : ( bot_bot @ C4 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_147_is__singletonI,axiom,
% 2.22/1.27 ! [A: $tType,X3: A] : ( is_singleton @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_148_diff__strict__mono,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( ordered_ab_group_add @ A )
% 2.22/1.27 => ! [A2: A,B: A,D: A,C2: A] :
% 2.22/1.27 ( ( ord_less @ A @ A2 @ B )
% 2.22/1.27 => ( ( ord_less @ A @ D @ C2 )
% 2.22/1.27 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_149_sup1CI,axiom,
% 2.22/1.27 ! [A: $tType,B3: A > $o,X3: A,A4: A > $o] :
% 2.22/1.27 ( ( ~ ( B3 @ X3 )
% 2.22/1.27 => ( A4 @ X3 ) )
% 2.22/1.27 => ( sup_sup @ ( A > $o ) @ A4 @ B3 @ X3 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_150_sup1E,axiom,
% 2.22/1.27 ! [A: $tType,A4: A > $o,B3: A > $o,X3: A] :
% 2.22/1.27 ( ( sup_sup @ ( A > $o ) @ A4 @ B3 @ X3 )
% 2.22/1.27 => ( ~ ( A4 @ X3 )
% 2.22/1.27 => ( B3 @ X3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_151_sup1I1,axiom,
% 2.22/1.27 ! [A: $tType,A4: A > $o,X3: A,B3: A > $o] :
% 2.22/1.27 ( ( A4 @ X3 )
% 2.22/1.27 => ( sup_sup @ ( A > $o ) @ A4 @ B3 @ X3 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_152_sup1I2,axiom,
% 2.22/1.27 ! [A: $tType,B3: A > $o,X3: A,A4: A > $o] :
% 2.22/1.27 ( ( B3 @ X3 )
% 2.22/1.27 => ( sup_sup @ ( A > $o ) @ A4 @ B3 @ X3 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_153_not__psubset__empty,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A] :
% 2.22/1.27 ~ ( ord_less @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_154_psubset__imp__ex__mem,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
% 2.22/1.27 => ? [B7: A] : ( member @ A @ B7 @ ( minus_minus @ ( set @ A ) @ B3 @ A4 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_155_minus__set__def,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( minus_minus @ ( set @ A ) )
% 2.22/1.27 = ( ^ [A5: set @ A,B6: set @ A] :
% 2.22/1.27 ( collect @ A
% 2.22/1.27 @ ( minus_minus @ ( A > $o )
% 2.22/1.27 @ ^ [X2: A] : ( member @ A @ X2 @ A5 )
% 2.22/1.27 @ ^ [X2: A] : ( member @ A @ X2 @ B6 ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_156_is__singleton__the__elem,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( is_singleton @ A )
% 2.22/1.27 = ( ^ [A5: set @ A] :
% 2.22/1.27 ( A5
% 2.22/1.27 = ( insert @ A @ ( the_elem @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_157_is__singletonI_H,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A] :
% 2.22/1.27 ( ( A4
% 2.22/1.27 != ( bot_bot @ ( set @ A ) ) )
% 2.22/1.27 => ( ! [X4: A,Y4: A] :
% 2.22/1.27 ( ( member @ A @ X4 @ A4 )
% 2.22/1.27 => ( ( member @ A @ Y4 @ A4 )
% 2.22/1.27 => ( X4 = Y4 ) ) )
% 2.22/1.27 => ( is_singleton @ A @ A4 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_158_dual__order_Ostrict__implies__not__eq,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( order @ A )
% 2.22/1.27 => ! [B: A,A2: A] :
% 2.22/1.27 ( ( ord_less @ A @ B @ A2 )
% 2.22/1.27 => ( A2 != B ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_159_order_Ostrict__implies__not__eq,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( order @ A )
% 2.22/1.27 => ! [A2: A,B: A] :
% 2.22/1.27 ( ( ord_less @ A @ A2 @ B )
% 2.22/1.27 => ( A2 != B ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_160_not__less__iff__gr__or__eq,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( linorder @ A )
% 2.22/1.27 => ! [X3: A,Y2: A] :
% 2.22/1.27 ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 2.22/1.27 = ( ( ord_less @ A @ Y2 @ X3 )
% 2.22/1.27 | ( X3 = Y2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_161_dual__order_Ostrict__trans,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( order @ A )
% 2.22/1.27 => ! [B: A,A2: A,C2: A] :
% 2.22/1.27 ( ( ord_less @ A @ B @ A2 )
% 2.22/1.27 => ( ( ord_less @ A @ C2 @ B )
% 2.22/1.27 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_162_linorder__less__wlog,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( linorder @ A )
% 2.22/1.27 => ! [P: A > A > $o,A2: A,B: A] :
% 2.22/1.27 ( ! [A6: A,B7: A] :
% 2.22/1.27 ( ( ord_less @ A @ A6 @ B7 )
% 2.22/1.27 => ( P @ A6 @ B7 ) )
% 2.22/1.27 => ( ! [A6: A] : ( P @ A6 @ A6 )
% 2.22/1.27 => ( ! [A6: A,B7: A] :
% 2.22/1.27 ( ( P @ B7 @ A6 )
% 2.22/1.27 => ( P @ A6 @ B7 ) )
% 2.22/1.27 => ( P @ A2 @ B ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_163_exists__least__iff,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( wellorder @ A )
% 2.22/1.27 => ( ( ^ [P2: A > $o] :
% 2.22/1.27 ? [X5: A] : ( P2 @ X5 ) )
% 2.22/1.27 = ( ^ [P3: A > $o] :
% 2.22/1.27 ? [N: A] :
% 2.22/1.27 ( ( P3 @ N )
% 2.22/1.27 & ! [M: A] :
% 2.22/1.27 ( ( ord_less @ A @ M @ N )
% 2.22/1.27 => ~ ( P3 @ M ) ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_164_less__imp__not__less,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( preorder @ A )
% 2.22/1.27 => ! [X3: A,Y2: A] :
% 2.22/1.27 ( ( ord_less @ A @ X3 @ Y2 )
% 2.22/1.27 => ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_165_order_Ostrict__trans,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( order @ A )
% 2.22/1.27 => ! [A2: A,B: A,C2: A] :
% 2.22/1.27 ( ( ord_less @ A @ A2 @ B )
% 2.22/1.27 => ( ( ord_less @ A @ B @ C2 )
% 2.22/1.27 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_166_dual__order_Oirrefl,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( order @ A )
% 2.22/1.27 => ! [A2: A] :
% 2.22/1.27 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_167_linorder__cases,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( linorder @ A )
% 2.22/1.27 => ! [X3: A,Y2: A] :
% 2.22/1.27 ( ~ ( ord_less @ A @ X3 @ Y2 )
% 2.22/1.27 => ( ( X3 != Y2 )
% 2.22/1.27 => ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_168_less__imp__triv,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( preorder @ A )
% 2.22/1.27 => ! [X3: A,Y2: A,P: $o] :
% 2.22/1.27 ( ( ord_less @ A @ X3 @ Y2 )
% 2.22/1.27 => ( ( ord_less @ A @ Y2 @ X3 )
% 2.22/1.27 => P ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_169_less__imp__not__eq2,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( order @ A )
% 2.22/1.27 => ! [X3: A,Y2: A] :
% 2.22/1.27 ( ( ord_less @ A @ X3 @ Y2 )
% 2.22/1.27 => ( Y2 != X3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_170_antisym__conv3,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( linorder @ A )
% 2.22/1.27 => ! [Y2: A,X3: A] :
% 2.22/1.27 ( ~ ( ord_less @ A @ Y2 @ X3 )
% 2.22/1.27 => ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 2.22/1.27 = ( X3 = Y2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_171_less__induct,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( wellorder @ A )
% 2.22/1.27 => ! [P: A > $o,A2: A] :
% 2.22/1.27 ( ! [X4: A] :
% 2.22/1.27 ( ! [Y5: A] :
% 2.22/1.27 ( ( ord_less @ A @ Y5 @ X4 )
% 2.22/1.27 => ( P @ Y5 ) )
% 2.22/1.27 => ( P @ X4 ) )
% 2.22/1.27 => ( P @ A2 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_172_less__not__sym,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( preorder @ A )
% 2.22/1.27 => ! [X3: A,Y2: A] :
% 2.22/1.27 ( ( ord_less @ A @ X3 @ Y2 )
% 2.22/1.27 => ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_173_less__imp__not__eq,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( order @ A )
% 2.22/1.27 => ! [X3: A,Y2: A] :
% 2.22/1.27 ( ( ord_less @ A @ X3 @ Y2 )
% 2.22/1.27 => ( X3 != Y2 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_174_dual__order_Oasym,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( order @ A )
% 2.22/1.27 => ! [B: A,A2: A] :
% 2.22/1.27 ( ( ord_less @ A @ B @ A2 )
% 2.22/1.27 => ~ ( ord_less @ A @ A2 @ B ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_175_ord__less__eq__trans,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( ord @ A )
% 2.22/1.27 => ! [A2: A,B: A,C2: A] :
% 2.22/1.27 ( ( ord_less @ A @ A2 @ B )
% 2.22/1.27 => ( ( B = C2 )
% 2.22/1.27 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_176_ord__eq__less__trans,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( ord @ A )
% 2.22/1.27 => ! [A2: A,B: A,C2: A] :
% 2.22/1.27 ( ( A2 = B )
% 2.22/1.27 => ( ( ord_less @ A @ B @ C2 )
% 2.22/1.27 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_177_less__irrefl,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( preorder @ A )
% 2.22/1.27 => ! [X3: A] :
% 2.22/1.27 ~ ( ord_less @ A @ X3 @ X3 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_178_less__linear,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( linorder @ A )
% 2.22/1.27 => ! [X3: A,Y2: A] :
% 2.22/1.27 ( ( ord_less @ A @ X3 @ Y2 )
% 2.22/1.27 | ( X3 = Y2 )
% 2.22/1.27 | ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_179_less__trans,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( preorder @ A )
% 2.22/1.27 => ! [X3: A,Y2: A,Z2: A] :
% 2.22/1.27 ( ( ord_less @ A @ X3 @ Y2 )
% 2.22/1.27 => ( ( ord_less @ A @ Y2 @ Z2 )
% 2.22/1.27 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_180_less__asym_H,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( preorder @ A )
% 2.22/1.27 => ! [A2: A,B: A] :
% 2.22/1.27 ( ( ord_less @ A @ A2 @ B )
% 2.22/1.27 => ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_181_less__asym,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( preorder @ A )
% 2.22/1.27 => ! [X3: A,Y2: A] :
% 2.22/1.27 ( ( ord_less @ A @ X3 @ Y2 )
% 2.22/1.27 => ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_182_less__imp__neq,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( order @ A )
% 2.22/1.27 => ! [X3: A,Y2: A] :
% 2.22/1.27 ( ( ord_less @ A @ X3 @ Y2 )
% 2.22/1.27 => ( X3 != Y2 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_183_dense,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( dense_order @ A )
% 2.22/1.27 => ! [X3: A,Y2: A] :
% 2.22/1.27 ( ( ord_less @ A @ X3 @ Y2 )
% 2.22/1.27 => ? [Z3: A] :
% 2.22/1.27 ( ( ord_less @ A @ X3 @ Z3 )
% 2.22/1.27 & ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_184_order_Oasym,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( order @ A )
% 2.22/1.27 => ! [A2: A,B: A] :
% 2.22/1.27 ( ( ord_less @ A @ A2 @ B )
% 2.22/1.27 => ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_185_neq__iff,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( linorder @ A )
% 2.22/1.27 => ! [X3: A,Y2: A] :
% 2.22/1.27 ( ( X3 != Y2 )
% 2.22/1.27 = ( ( ord_less @ A @ X3 @ Y2 )
% 2.22/1.27 | ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_186_neqE,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( linorder @ A )
% 2.22/1.27 => ! [X3: A,Y2: A] :
% 2.22/1.27 ( ( X3 != Y2 )
% 2.22/1.27 => ( ~ ( ord_less @ A @ X3 @ Y2 )
% 2.22/1.27 => ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_187_gt__ex,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( no_top @ A )
% 2.22/1.27 => ! [X3: A] :
% 2.22/1.27 ? [X_1: A] : ( ord_less @ A @ X3 @ X_1 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_188_lt__ex,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( no_bot @ A )
% 2.22/1.27 => ! [X3: A] :
% 2.22/1.27 ? [Y4: A] : ( ord_less @ A @ Y4 @ X3 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_189_order__less__subst2,axiom,
% 2.22/1.27 ! [A: $tType,C4: $tType] :
% 2.22/1.27 ( ( ( order @ C4 )
% 2.22/1.27 & ( order @ A ) )
% 2.22/1.27 => ! [A2: A,B: A,F: A > C4,C2: C4] :
% 2.22/1.27 ( ( ord_less @ A @ A2 @ B )
% 2.22/1.27 => ( ( ord_less @ C4 @ ( F @ B ) @ C2 )
% 2.22/1.27 => ( ! [X4: A,Y4: A] :
% 2.22/1.27 ( ( ord_less @ A @ X4 @ Y4 )
% 2.22/1.27 => ( ord_less @ C4 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.22/1.27 => ( ord_less @ C4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_190_order__less__subst1,axiom,
% 2.22/1.27 ! [A: $tType,B4: $tType] :
% 2.22/1.27 ( ( ( order @ B4 )
% 2.22/1.27 & ( order @ A ) )
% 2.22/1.27 => ! [A2: A,F: B4 > A,B: B4,C2: B4] :
% 2.22/1.27 ( ( ord_less @ A @ A2 @ ( F @ B ) )
% 2.22/1.27 => ( ( ord_less @ B4 @ B @ C2 )
% 2.22/1.27 => ( ! [X4: B4,Y4: B4] :
% 2.22/1.27 ( ( ord_less @ B4 @ X4 @ Y4 )
% 2.22/1.27 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.22/1.27 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_191_ord__less__eq__subst,axiom,
% 2.22/1.27 ! [A: $tType,B4: $tType] :
% 2.22/1.27 ( ( ( ord @ B4 )
% 2.22/1.27 & ( ord @ A ) )
% 2.22/1.27 => ! [A2: A,B: A,F: A > B4,C2: B4] :
% 2.22/1.27 ( ( ord_less @ A @ A2 @ B )
% 2.22/1.27 => ( ( ( F @ B )
% 2.22/1.27 = C2 )
% 2.22/1.27 => ( ! [X4: A,Y4: A] :
% 2.22/1.27 ( ( ord_less @ A @ X4 @ Y4 )
% 2.22/1.27 => ( ord_less @ B4 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.22/1.27 => ( ord_less @ B4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_192_ord__eq__less__subst,axiom,
% 2.22/1.27 ! [A: $tType,B4: $tType] :
% 2.22/1.27 ( ( ( ord @ B4 )
% 2.22/1.27 & ( ord @ A ) )
% 2.22/1.27 => ! [A2: A,F: B4 > A,B: B4,C2: B4] :
% 2.22/1.27 ( ( A2
% 2.22/1.27 = ( F @ B ) )
% 2.22/1.27 => ( ( ord_less @ B4 @ B @ C2 )
% 2.22/1.27 => ( ! [X4: B4,Y4: B4] :
% 2.22/1.27 ( ( ord_less @ B4 @ X4 @ Y4 )
% 2.22/1.27 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.22/1.27 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_193_bot__fun__def,axiom,
% 2.22/1.27 ! [B4: $tType,A: $tType] :
% 2.22/1.27 ( ( bot @ B4 )
% 2.22/1.27 => ( ( bot_bot @ ( A > B4 ) )
% 2.22/1.27 = ( ^ [X2: A] : ( bot_bot @ B4 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_194_diff__right__commute,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( cancel146912293up_add @ A )
% 2.22/1.27 => ! [A2: A,C2: A,B: A] :
% 2.22/1.27 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B )
% 2.22/1.27 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C2 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_195_diff__eq__diff__eq,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( group_add @ A )
% 2.22/1.27 => ! [A2: A,B: A,C2: A,D: A] :
% 2.22/1.27 ( ( ( minus_minus @ A @ A2 @ B )
% 2.22/1.27 = ( minus_minus @ A @ C2 @ D ) )
% 2.22/1.27 => ( ( A2 = B )
% 2.22/1.27 = ( C2 = D ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_196_is__singleton__def,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( is_singleton @ A )
% 2.22/1.27 = ( ^ [A5: set @ A] :
% 2.22/1.27 ? [X2: A] :
% 2.22/1.27 ( A5
% 2.22/1.27 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_197_is__singletonE,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A] :
% 2.22/1.27 ( ( is_singleton @ A @ A4 )
% 2.22/1.27 => ~ ! [X4: A] :
% 2.22/1.27 ( A4
% 2.22/1.27 != ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_198_bot_Onot__eq__extremum,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( order_bot @ A )
% 2.22/1.27 => ! [A2: A] :
% 2.22/1.27 ( ( A2
% 2.22/1.27 != ( bot_bot @ A ) )
% 2.22/1.27 = ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_199_bot_Oextremum__strict,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( order_bot @ A )
% 2.22/1.27 => ! [A2: A] :
% 2.22/1.27 ~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_200_diff__strict__right__mono,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( ordered_ab_group_add @ A )
% 2.22/1.27 => ! [A2: A,B: A,C2: A] :
% 2.22/1.27 ( ( ord_less @ A @ A2 @ B )
% 2.22/1.27 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B @ C2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_201_diff__strict__left__mono,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( ordered_ab_group_add @ A )
% 2.22/1.27 => ! [B: A,A2: A,C2: A] :
% 2.22/1.27 ( ( ord_less @ A @ B @ A2 )
% 2.22/1.27 => ( ord_less @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_202_diff__eq__diff__less,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( ordered_ab_group_add @ A )
% 2.22/1.27 => ! [A2: A,B: A,C2: A,D: A] :
% 2.22/1.27 ( ( ( minus_minus @ A @ A2 @ B )
% 2.22/1.27 = ( minus_minus @ A @ C2 @ D ) )
% 2.22/1.27 => ( ( ord_less @ A @ A2 @ B )
% 2.22/1.27 = ( ord_less @ A @ C2 @ D ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_203_sup__Un__eq,axiom,
% 2.22/1.27 ! [A: $tType,R: set @ A,S: set @ A] :
% 2.22/1.27 ( ( sup_sup @ ( A > $o )
% 2.22/1.27 @ ^ [X2: A] : ( member @ A @ X2 @ R )
% 2.22/1.27 @ ^ [X2: A] : ( member @ A @ X2 @ S ) )
% 2.22/1.27 = ( ^ [X2: A] : ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ R @ S ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_204_bot__empty__eq,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( bot_bot @ ( A > $o ) )
% 2.22/1.27 = ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_205_Collect__empty__eq__bot,axiom,
% 2.22/1.27 ! [A: $tType,P: A > $o] :
% 2.22/1.27 ( ( ( collect @ A @ P )
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) )
% 2.22/1.27 = ( P
% 2.22/1.27 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_206_psubset__trans,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,B3: set @ A,C: set @ A] :
% 2.22/1.27 ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
% 2.22/1.27 => ( ( ord_less @ ( set @ A ) @ B3 @ C )
% 2.22/1.27 => ( ord_less @ ( set @ A ) @ A4 @ C ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_207_less__set__def,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( ord_less @ ( set @ A ) )
% 2.22/1.27 = ( ^ [A5: set @ A,B6: set @ A] :
% 2.22/1.27 ( ord_less @ ( A > $o )
% 2.22/1.27 @ ^ [X2: A] : ( member @ A @ X2 @ A5 )
% 2.22/1.27 @ ^ [X2: A] : ( member @ A @ X2 @ B6 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_208_psubsetD,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,B3: set @ A,C2: A] :
% 2.22/1.27 ( ( ord_less @ ( set @ A ) @ A4 @ B3 )
% 2.22/1.27 => ( ( member @ A @ C2 @ A4 )
% 2.22/1.27 => ( member @ A @ C2 @ B3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_209_remove__def,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( remove @ A )
% 2.22/1.27 = ( ^ [X2: A,A5: set @ A] : ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_210_Tree_Osimps_I15_J,axiom,
% 2.22/1.27 ! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
% 2.22/1.27 ( ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) )
% 2.22/1.27 = ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( binary2130109271t_Tree @ A @ X21 ) @ ( insert @ A @ X22 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( binary2130109271t_Tree @ A @ X23 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_211_member__remove,axiom,
% 2.22/1.27 ! [A: $tType,X3: A,Y2: A,A4: set @ A] :
% 2.22/1.27 ( ( member @ A @ X3 @ ( remove @ A @ Y2 @ A4 ) )
% 2.22/1.27 = ( ( member @ A @ X3 @ A4 )
% 2.22/1.27 & ( X3 != Y2 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_212_Tree_Oset__intros_I3_J,axiom,
% 2.22/1.27 ! [A: $tType,Ya: A,X23: binary1291135688e_Tree @ A,X21: binary1291135688e_Tree @ A,X22: A] :
% 2.22/1.27 ( ( member @ A @ Ya @ ( binary2130109271t_Tree @ A @ X23 ) )
% 2.22/1.27 => ( member @ A @ Ya @ ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_213_Tree_Oset__intros_I2_J,axiom,
% 2.22/1.27 ! [A: $tType,X22: A,X21: binary1291135688e_Tree @ A,X23: binary1291135688e_Tree @ A] : ( member @ A @ X22 @ ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_214_Tree_Oset__intros_I1_J,axiom,
% 2.22/1.27 ! [A: $tType,Y2: A,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
% 2.22/1.27 ( ( member @ A @ Y2 @ ( binary2130109271t_Tree @ A @ X21 ) )
% 2.22/1.27 => ( member @ A @ Y2 @ ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_215_Tree_Oset__cases,axiom,
% 2.22/1.27 ! [A: $tType,E: A,A2: binary1291135688e_Tree @ A] :
% 2.22/1.27 ( ( member @ A @ E @ ( binary2130109271t_Tree @ A @ A2 ) )
% 2.22/1.27 => ( ! [Z1: binary1291135688e_Tree @ A] :
% 2.22/1.27 ( ? [Z22: A,Z32: binary1291135688e_Tree @ A] :
% 2.22/1.27 ( A2
% 2.22/1.27 = ( binary210054475elle_T @ A @ Z1 @ Z22 @ Z32 ) )
% 2.22/1.27 => ~ ( member @ A @ E @ ( binary2130109271t_Tree @ A @ Z1 ) ) )
% 2.22/1.27 => ( ! [Z1: binary1291135688e_Tree @ A,Z32: binary1291135688e_Tree @ A] :
% 2.22/1.27 ( A2
% 2.22/1.27 != ( binary210054475elle_T @ A @ Z1 @ E @ Z32 ) )
% 2.22/1.27 => ~ ! [Z1: binary1291135688e_Tree @ A,Z22: A,Z32: binary1291135688e_Tree @ A] :
% 2.22/1.27 ( ( A2
% 2.22/1.27 = ( binary210054475elle_T @ A @ Z1 @ Z22 @ Z32 ) )
% 2.22/1.27 => ~ ( member @ A @ E @ ( binary2130109271t_Tree @ A @ Z32 ) ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_216_Tree_Osimps_I14_J,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( binary2130109271t_Tree @ A @ ( binary1746293266le_Tip @ A ) )
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_217_Set_Ois__empty__def,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( is_empty @ A )
% 2.22/1.27 = ( ^ [A5: set @ A] :
% 2.22/1.27 ( A5
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_218_the__elem__def,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( the_elem @ A )
% 2.22/1.27 = ( ^ [X6: set @ A] :
% 2.22/1.27 ( the @ A
% 2.22/1.27 @ ^ [X2: A] :
% 2.22/1.27 ( X6
% 2.22/1.27 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_219_the__sym__eq__trivial,axiom,
% 2.22/1.27 ! [A: $tType,X3: A] :
% 2.22/1.27 ( ( the @ A
% 2.22/1.27 @ ( ^ [Y3: A,Z: A] : ( Y3 = Z )
% 2.22/1.27 @ X3 ) )
% 2.22/1.27 = X3 ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_220_the__eq__trivial,axiom,
% 2.22/1.27 ! [A: $tType,A2: A] :
% 2.22/1.27 ( ( the @ A
% 2.22/1.27 @ ^ [X2: A] : ( X2 = A2 ) )
% 2.22/1.27 = A2 ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_221_the__equality,axiom,
% 2.22/1.27 ! [A: $tType,P: A > $o,A2: A] :
% 2.22/1.27 ( ( P @ A2 )
% 2.22/1.27 => ( ! [X4: A] :
% 2.22/1.27 ( ( P @ X4 )
% 2.22/1.27 => ( X4 = A2 ) )
% 2.22/1.27 => ( ( the @ A @ P )
% 2.22/1.27 = A2 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_222_theI,axiom,
% 2.22/1.27 ! [A: $tType,P: A > $o,A2: A] :
% 2.22/1.27 ( ( P @ A2 )
% 2.22/1.27 => ( ! [X4: A] :
% 2.22/1.27 ( ( P @ X4 )
% 2.22/1.27 => ( X4 = A2 ) )
% 2.22/1.27 => ( P @ ( the @ A @ P ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_223_theI_H,axiom,
% 2.22/1.27 ! [A: $tType,P: A > $o] :
% 2.22/1.27 ( ? [X: A] :
% 2.22/1.27 ( ( P @ X )
% 2.22/1.27 & ! [Y4: A] :
% 2.22/1.27 ( ( P @ Y4 )
% 2.22/1.27 => ( Y4 = X ) ) )
% 2.22/1.27 => ( P @ ( the @ A @ P ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_224_theI2,axiom,
% 2.22/1.27 ! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
% 2.22/1.27 ( ( P @ A2 )
% 2.22/1.27 => ( ! [X4: A] :
% 2.22/1.27 ( ( P @ X4 )
% 2.22/1.27 => ( X4 = A2 ) )
% 2.22/1.27 => ( ! [X4: A] :
% 2.22/1.27 ( ( P @ X4 )
% 2.22/1.27 => ( Q @ X4 ) )
% 2.22/1.27 => ( Q @ ( the @ A @ P ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_225_If__def,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( if @ A )
% 2.22/1.27 = ( ^ [P3: $o,X2: A,Y: A] :
% 2.22/1.27 ( the @ A
% 2.22/1.27 @ ^ [Z4: A] :
% 2.22/1.27 ( ( P3
% 2.22/1.27 => ( Z4 = X2 ) )
% 2.22/1.27 & ( ~ P3
% 2.22/1.27 => ( Z4 = Y ) ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_226_the1I2,axiom,
% 2.22/1.27 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.22/1.27 ( ? [X: A] :
% 2.22/1.27 ( ( P @ X )
% 2.22/1.27 & ! [Y4: A] :
% 2.22/1.27 ( ( P @ Y4 )
% 2.22/1.27 => ( Y4 = X ) ) )
% 2.22/1.27 => ( ! [X4: A] :
% 2.22/1.27 ( ( P @ X4 )
% 2.22/1.27 => ( Q @ X4 ) )
% 2.22/1.27 => ( Q @ ( the @ A @ P ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_227_the1__equality,axiom,
% 2.22/1.27 ! [A: $tType,P: A > $o,A2: A] :
% 2.22/1.27 ( ? [X: A] :
% 2.22/1.27 ( ( P @ X )
% 2.22/1.27 & ! [Y4: A] :
% 2.22/1.27 ( ( P @ Y4 )
% 2.22/1.27 => ( Y4 = X ) ) )
% 2.22/1.27 => ( ( P @ A2 )
% 2.22/1.27 => ( ( the @ A @ P )
% 2.22/1.27 = A2 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_228_psubset__insert__iff,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,X3: A,B3: set @ A] :
% 2.22/1.27 ( ( ord_less @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ B3 ) )
% 2.22/1.27 = ( ( ( member @ A @ X3 @ B3 )
% 2.22/1.27 => ( ord_less @ ( set @ A ) @ A4 @ B3 ) )
% 2.22/1.27 & ( ~ ( member @ A @ X3 @ B3 )
% 2.22/1.27 => ( ( ( member @ A @ X3 @ A4 )
% 2.22/1.27 => ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) )
% 2.22/1.27 & ( ~ ( member @ A @ X3 @ A4 )
% 2.22/1.27 => ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_229_pairwise__alt,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( pairwise @ A )
% 2.22/1.27 = ( ^ [R2: A > A > $o,S2: set @ A] :
% 2.22/1.27 ! [X2: A] :
% 2.22/1.27 ( ( member @ A @ X2 @ S2 )
% 2.22/1.27 => ! [Y: A] :
% 2.22/1.27 ( ( member @ A @ Y @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.22/1.27 => ( R2 @ X2 @ Y ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_230_order__refl,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( preorder @ A )
% 2.22/1.27 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_231_subset__antisym,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 2.22/1.27 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
% 2.22/1.27 => ( A4 = B3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_232_subsetI,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ! [X4: A] :
% 2.22/1.27 ( ( member @ A @ X4 @ A4 )
% 2.22/1.27 => ( member @ A @ X4 @ B3 ) )
% 2.22/1.27 => ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_233_sup_Obounded__iff,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ! [B: A,C2: A,A2: A] :
% 2.22/1.27 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B @ C2 ) @ A2 )
% 2.22/1.27 = ( ( ord_less_eq @ A @ B @ A2 )
% 2.22/1.27 & ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_234_le__sup__iff,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( semilattice_sup @ A )
% 2.22/1.27 => ! [X3: A,Y2: A,Z2: A] :
% 2.22/1.27 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z2 )
% 2.22/1.27 = ( ( ord_less_eq @ A @ X3 @ Z2 )
% 2.22/1.27 & ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_235_empty__subsetI,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_236_subset__empty,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A] :
% 2.22/1.27 ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
% 2.22/1.27 = ( A4
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_237_insert__subset,axiom,
% 2.22/1.27 ! [A: $tType,X3: A,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A4 ) @ B3 )
% 2.22/1.27 = ( ( member @ A @ X3 @ B3 )
% 2.22/1.27 & ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_238_Un__subset__iff,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,B3: set @ A,C: set @ A] :
% 2.22/1.27 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C )
% 2.22/1.27 = ( ( ord_less_eq @ ( set @ A ) @ A4 @ C )
% 2.22/1.27 & ( ord_less_eq @ ( set @ A ) @ B3 @ C ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_239_psubsetI,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 2.22/1.27 => ( ( A4 != B3 )
% 2.22/1.27 => ( ord_less @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_240_singleton__insert__inj__eq_H,axiom,
% 2.22/1.27 ! [A: $tType,A2: A,A4: set @ A,B: A] :
% 2.22/1.27 ( ( ( insert @ A @ A2 @ A4 )
% 2.22/1.27 = ( insert @ A @ B @ ( bot_bot @ ( set @ A ) ) ) )
% 2.22/1.27 = ( ( A2 = B )
% 2.22/1.27 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_241_singleton__insert__inj__eq,axiom,
% 2.22/1.27 ! [A: $tType,B: A,A2: A,A4: set @ A] :
% 2.22/1.27 ( ( ( insert @ A @ B @ ( bot_bot @ ( set @ A ) ) )
% 2.22/1.27 = ( insert @ A @ A2 @ A4 ) )
% 2.22/1.27 = ( ( A2 = B )
% 2.22/1.27 & ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_242_Diff__eq__empty__iff,axiom,
% 2.22/1.27 ! [A: $tType,A4: set @ A,B3: set @ A] :
% 2.22/1.27 ( ( ( minus_minus @ ( set @ A ) @ A4 @ B3 )
% 2.22/1.27 = ( bot_bot @ ( set @ A ) ) )
% 2.22/1.27 = ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_243_leD,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( order @ A )
% 2.22/1.27 => ! [Y2: A,X3: A] :
% 2.22/1.27 ( ( ord_less_eq @ A @ Y2 @ X3 )
% 2.22/1.27 => ~ ( ord_less @ A @ X3 @ Y2 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_244_leI,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( linorder @ A )
% 2.22/1.27 => ! [X3: A,Y2: A] :
% 2.22/1.27 ( ~ ( ord_less @ A @ X3 @ Y2 )
% 2.22/1.27 => ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_245_le__less,axiom,
% 2.22/1.27 ! [A: $tType] :
% 2.22/1.27 ( ( order @ A )
% 2.22/1.27 => ( ( ord_less_eq @ A )
% 2.22/1.27 = ( ^ [X2: A,Y: A] :
% 2.22/1.27 ( ( ord_less @ A @ X2 @ Y )
% 2.22/1.27 | ( X2 = Y ) ) ) ) ) ).
% 2.22/1.27
% 2.22/1.27 thf(fact_246_less__le,axiom,
% 2.22/1.28 ! [A: $tType] :
% 2.22/1.28 ( ( order @ A )
% 2.22/1.28 => ( ( ord_less @ A )
% 2.22/1.28 = ( ^ [X2: A,Y: A] :
% 2.22/1.28 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.22/1.28 & ( X2 != Y ) ) ) ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(fact_247_order__le__less__subst1,axiom,
% 2.22/1.28 ! [A: $tType,B4: $tType] :
% 2.22/1.28 ( ( ( order @ B4 )
% 2.22/1.28 & ( order @ A ) )
% 2.22/1.28 => ! [A2: A,F: B4 > A,B: B4,C2: B4] :
% 2.22/1.28 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 2.22/1.28 => ( ( ord_less @ B4 @ B @ C2 )
% 2.22/1.28 => ( ! [X4: B4,Y4: B4] :
% 2.22/1.28 ( ( ord_less @ B4 @ X4 @ Y4 )
% 2.22/1.28 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.22/1.28 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(fact_248_order__le__less__subst2,axiom,
% 2.22/1.28 ! [A: $tType,C4: $tType] :
% 2.22/1.28 ( ( ( order @ C4 )
% 2.22/1.28 & ( order @ A ) )
% 2.22/1.28 => ! [A2: A,B: A,F: A > C4,C2: C4] :
% 2.22/1.28 ( ( ord_less_eq @ A @ A2 @ B )
% 2.22/1.28 => ( ( ord_less @ C4 @ ( F @ B ) @ C2 )
% 2.22/1.28 => ( ! [X4: A,Y4: A] :
% 2.22/1.28 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 2.22/1.28 => ( ord_less_eq @ C4 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.22/1.28 => ( ord_less @ C4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(fact_249_order__less__le__subst1,axiom,
% 2.22/1.28 ! [A: $tType,B4: $tType] :
% 2.22/1.28 ( ( ( order @ B4 )
% 2.22/1.28 & ( order @ A ) )
% 2.22/1.28 => ! [A2: A,F: B4 > A,B: B4,C2: B4] :
% 2.22/1.28 ( ( ord_less @ A @ A2 @ ( F @ B ) )
% 2.22/1.28 => ( ( ord_less_eq @ B4 @ B @ C2 )
% 2.22/1.28 => ( ! [X4: B4,Y4: B4] :
% 2.22/1.28 ( ( ord_less_eq @ B4 @ X4 @ Y4 )
% 2.22/1.28 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.22/1.28 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(fact_250_order__less__le__subst2,axiom,
% 2.22/1.28 ! [A: $tType,C4: $tType] :
% 2.22/1.28 ( ( ( order @ C4 )
% 2.22/1.28 & ( order @ A ) )
% 2.22/1.28 => ! [A2: A,B: A,F: A > C4,C2: C4] :
% 2.22/1.28 ( ( ord_less @ A @ A2 @ B )
% 2.22/1.28 => ( ( ord_less_eq @ C4 @ ( F @ B ) @ C2 )
% 2.22/1.28 => ( ! [X4: A,Y4: A] :
% 2.22/1.28 ( ( ord_less @ A @ X4 @ Y4 )
% 2.22/1.28 => ( ord_less @ C4 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.22/1.28 => ( ord_less @ C4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(fact_251_not__le,axiom,
% 2.22/1.28 ! [A: $tType] :
% 2.22/1.28 ( ( linorder @ A )
% 2.22/1.28 => ! [X3: A,Y2: A] :
% 2.22/1.28 ( ( ~ ( ord_less_eq @ A @ X3 @ Y2 ) )
% 2.22/1.28 = ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(fact_252_not__less,axiom,
% 2.22/1.28 ! [A: $tType] :
% 2.22/1.28 ( ( linorder @ A )
% 2.22/1.28 => ! [X3: A,Y2: A] :
% 2.22/1.28 ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 2.22/1.28 = ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(fact_253_le__neq__trans,axiom,
% 2.22/1.28 ! [A: $tType] :
% 2.22/1.28 ( ( order @ A )
% 2.22/1.28 => ! [A2: A,B: A] :
% 2.22/1.28 ( ( ord_less_eq @ A @ A2 @ B )
% 2.22/1.28 => ( ( A2 != B )
% 2.22/1.28 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(fact_254_antisym__conv1,axiom,
% 2.22/1.28 ! [A: $tType] :
% 2.22/1.28 ( ( order @ A )
% 2.22/1.28 => ! [X3: A,Y2: A] :
% 2.22/1.28 ( ~ ( ord_less @ A @ X3 @ Y2 )
% 2.22/1.28 => ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.22/1.28 = ( X3 = Y2 ) ) ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(fact_255_antisym__conv2,axiom,
% 2.22/1.28 ! [A: $tType] :
% 2.22/1.28 ( ( order @ A )
% 2.22/1.28 => ! [X3: A,Y2: A] :
% 2.22/1.28 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.22/1.28 => ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 2.22/1.28 = ( X3 = Y2 ) ) ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 2.22/1.28 bounded_lattice @ $o ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 2.22/1.28 ! [A7: $tType] : ( bounded_lattice @ ( set @ A7 ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 2.22/1.28 ! [A7: $tType,A8: $tType] :
% 2.22/1.28 ( ( bounded_lattice @ A8 )
% 2.22/1.28 => ( bounded_lattice @ ( A7 > A8 ) ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
% 2.22/1.28 ! [A7: $tType,A8: $tType] :
% 2.22/1.28 ( ( bounded_lattice @ A8 )
% 2.22/1.28 => ( bounde1808546759up_bot @ ( A7 > A8 ) ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 2.22/1.28 ! [A7: $tType,A8: $tType] :
% 2.22/1.28 ( ( semilattice_sup @ A8 )
% 2.22/1.28 => ( semilattice_sup @ ( A7 > A8 ) ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 2.22/1.28 ! [A7: $tType,A8: $tType] :
% 2.22/1.28 ( ( order_bot @ A8 )
% 2.22/1.28 => ( order_bot @ ( A7 > A8 ) ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.22/1.28 ! [A7: $tType,A8: $tType] :
% 2.22/1.28 ( ( preorder @ A8 )
% 2.22/1.28 => ( preorder @ ( A7 > A8 ) ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_fun___Lattices_Olattice,axiom,
% 2.22/1.28 ! [A7: $tType,A8: $tType] :
% 2.22/1.28 ( ( lattice @ A8 )
% 2.22/1.28 => ( lattice @ ( A7 > A8 ) ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.22/1.28 ! [A7: $tType,A8: $tType] :
% 2.22/1.28 ( ( order @ A8 )
% 2.22/1.28 => ( order @ ( A7 > A8 ) ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_fun___Orderings_Oord,axiom,
% 2.22/1.28 ! [A7: $tType,A8: $tType] :
% 2.22/1.28 ( ( ord @ A8 )
% 2.22/1.28 => ( ord @ ( A7 > A8 ) ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_fun___Orderings_Obot,axiom,
% 2.22/1.28 ! [A7: $tType,A8: $tType] :
% 2.22/1.28 ( ( bot @ A8 )
% 2.22/1.28 => ( bot @ ( A7 > A8 ) ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_fun___Groups_Ominus,axiom,
% 2.22/1.28 ! [A7: $tType,A8: $tType] :
% 2.22/1.28 ( ( minus @ A8 )
% 2.22/1.28 => ( minus @ ( A7 > A8 ) ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
% 2.22/1.28 cancel146912293up_add @ int ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 2.22/1.28 ordered_ab_group_add @ int ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Int_Oint___Lattices_Osemilattice__sup_3,axiom,
% 2.22/1.28 semilattice_sup @ int ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Int_Oint___Orderings_Opreorder_4,axiom,
% 2.22/1.28 preorder @ int ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 2.22/1.28 linorder @ int ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
% 2.22/1.28 no_top @ int ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
% 2.22/1.28 no_bot @ int ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Int_Oint___Lattices_Olattice_5,axiom,
% 2.22/1.28 lattice @ int ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 2.22/1.28 group_add @ int ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Int_Oint___Orderings_Oorder_6,axiom,
% 2.22/1.28 order @ int ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Int_Oint___Orderings_Oord_7,axiom,
% 2.22/1.28 ord @ int ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Int_Oint___Groups_Ominus_8,axiom,
% 2.22/1.28 minus @ int ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_9,axiom,
% 2.22/1.28 ! [A7: $tType] : ( bounde1808546759up_bot @ ( set @ A7 ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_10,axiom,
% 2.22/1.28 ! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Set_Oset___Orderings_Oorder__bot_11,axiom,
% 2.22/1.28 ! [A7: $tType] : ( order_bot @ ( set @ A7 ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Set_Oset___Orderings_Opreorder_12,axiom,
% 2.22/1.28 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Set_Oset___Lattices_Olattice_13,axiom,
% 2.22/1.28 ! [A7: $tType] : ( lattice @ ( set @ A7 ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Set_Oset___Orderings_Oorder_14,axiom,
% 2.22/1.28 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Set_Oset___Orderings_Oord_15,axiom,
% 2.22/1.28 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Set_Oset___Orderings_Obot_16,axiom,
% 2.22/1.28 ! [A7: $tType] : ( bot @ ( set @ A7 ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_Set_Oset___Groups_Ominus_17,axiom,
% 2.22/1.28 ! [A7: $tType] : ( minus @ ( set @ A7 ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_18,axiom,
% 2.22/1.28 bounde1808546759up_bot @ $o ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_19,axiom,
% 2.22/1.28 semilattice_sup @ $o ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_HOL_Obool___Orderings_Oorder__bot_20,axiom,
% 2.22/1.28 order_bot @ $o ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_HOL_Obool___Orderings_Opreorder_21,axiom,
% 2.22/1.28 preorder @ $o ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_HOL_Obool___Orderings_Olinorder_22,axiom,
% 2.22/1.28 linorder @ $o ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_HOL_Obool___Lattices_Olattice_23,axiom,
% 2.22/1.28 lattice @ $o ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_HOL_Obool___Orderings_Oorder_24,axiom,
% 2.22/1.28 order @ $o ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_HOL_Obool___Orderings_Oord_25,axiom,
% 2.22/1.28 ord @ $o ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_HOL_Obool___Orderings_Obot_26,axiom,
% 2.22/1.28 bot @ $o ).
% 2.22/1.28
% 2.22/1.28 thf(tcon_HOL_Obool___Groups_Ominus_27,axiom,
% 2.22/1.28 minus @ $o ).
% 2.22/1.28
% 2.22/1.28 thf(help_If_3_1_T,axiom,
% 2.22/1.28 ! [P: $o] :
% 2.22/1.28 ( ( P = $true )
% 2.22/1.28 | ( P = $false ) ) ).
% 2.22/1.28
% 2.22/1.28 thf(help_If_2_1_T,axiom,
% 2.22/1.28 ! [A: $tType,X3: A,Y2: A] :
% 2.22/1.28 ( ( if @ A @ $false @ X3 @ Y2 )
% 2.22/1.28 = Y2 ) ).
% 2.22/1.28
% 2.22/1.28 thf(help_If_1_1_T,axiom,
% 2.22/1.28 ! [A: $tType,X3: A,Y2: A] :
% 2.22/1.28 ( ( if @ A @ $true @ X3 @ Y2 )
% 2.22/1.28 = X3 ) ).
% 2.22/1.28
% 2.22/1.28 thf(conj_0,conjecture,
% 2.22/1.28 ! [X4: a] :
% 2.22/1.28 ( ( member @ a @ X4 @ ( binary64540844le_eqs @ a @ h @ e ) )
% 2.22/1.28 => ( ( h @ X4 )
% 2.22/1.28 != ( h @ x ) ) ) ).
% 2.22/1.28 ------------------------
% 2.22/1.29 % (4698)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_4426 for (2999ds/2Mi)
% 2.22/1.29 % (4696)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_4426 for (2999ds/27Mi)
% 2.22/1.29 % (4695)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_4426 for (2999ds/4Mi)
% 2.22/1.29 % (4697)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_4426 for (2999ds/2Mi)
% 2.22/1.29 % (4694)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_4426 for (2999ds/183Mi)
% 2.22/1.29 % (4699)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_4426 for (2999ds/275Mi)
% 2.22/1.29 % (4700)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_4426 for (2999ds/18Mi)
% 2.22/1.29 % (4697)Instruction limit reached!
% 2.22/1.29 % (4697)------------------------------
% 2.22/1.29 % (4697)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.22/1.29 % (4697)Termination reason: Unknown
% 2.22/1.29 % (4697)Termination phase: shuffling
% 2.22/1.29
% 2.22/1.29 % (4697)Memory used [KB]: 1279
% 2.22/1.29 % (4697)Time elapsed: 0.002 s
% 2.22/1.29 % (4697)Instructions burned: 2 (million)
% 2.22/1.29 % (4697)------------------------------
% 2.22/1.29 % (4697)------------------------------
% 2.22/1.29 % (4698)Instruction limit reached!
% 2.22/1.29 % (4698)------------------------------
% 2.22/1.29 % (4698)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.22/1.29 % (4698)Termination reason: Unknown
% 2.22/1.29 % (4698)Termination phase: shuffling
% 2.22/1.29
% 2.22/1.29 % (4698)Memory used [KB]: 1279
% 2.22/1.29 % (4698)Time elapsed: 0.002 s
% 2.22/1.29 % (4698)Instructions burned: 2 (million)
% 2.22/1.29 % (4698)------------------------------
% 2.22/1.29 % (4698)------------------------------
% 2.22/1.29 % (4695)Instruction limit reached!
% 2.22/1.29 % (4695)------------------------------
% 2.22/1.29 % (4695)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.22/1.29 % (4695)Termination reason: Unknown
% 2.22/1.29 % (4695)Termination phase: shuffling
% 2.22/1.29
% 2.22/1.29 % (4695)Memory used [KB]: 1407
% 2.22/1.29 % (4695)Time elapsed: 0.003 s
% 2.22/1.29 % (4695)Instructions burned: 5 (million)
% 2.22/1.29 % (4695)------------------------------
% 2.22/1.29 % (4695)------------------------------
% 2.35/1.30 % (4700)Instruction limit reached!
% 2.35/1.30 % (4700)------------------------------
% 2.35/1.30 % (4700)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.30 % (4700)Termination reason: Unknown
% 2.35/1.30 % (4700)Termination phase: Property scanning
% 2.35/1.30
% 2.35/1.30 % (4700)Memory used [KB]: 1663
% 2.35/1.30 % (4700)Time elapsed: 0.008 s
% 2.35/1.30 % (4700)Instructions burned: 20 (million)
% 2.35/1.30 % (4700)------------------------------
% 2.35/1.30 % (4700)------------------------------
% 2.35/1.30 % (4696)Instruction limit reached!
% 2.35/1.30 % (4696)------------------------------
% 2.35/1.30 % (4696)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.30 % (4696)Termination reason: Unknown
% 2.35/1.30 % (4696)Termination phase: Property scanning
% 2.35/1.30
% 2.35/1.30 % (4696)Memory used [KB]: 1791
% 2.35/1.30 % (4696)Time elapsed: 0.010 s
% 2.35/1.30 % (4696)Instructions burned: 27 (million)
% 2.35/1.30 % (4696)------------------------------
% 2.35/1.30 % (4696)------------------------------
% 2.35/1.30 % (4707)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_4426 for (2999ds/3Mi)
% 2.35/1.30 % (4708)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_4426 for (2999ds/37Mi)
% 2.35/1.30 % (4709)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_4426 for (2999ds/15Mi)
% 2.35/1.30 % (4707)Instruction limit reached!
% 2.35/1.30 % (4707)------------------------------
% 2.35/1.30 % (4707)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.30 % (4707)Termination reason: Unknown
% 2.35/1.30 % (4707)Termination phase: shuffling
% 2.35/1.30
% 2.35/1.30 % (4707)Memory used [KB]: 1407
% 2.35/1.30 % (4707)Time elapsed: 0.002 s
% 2.35/1.30 % (4707)Instructions burned: 3 (million)
% 2.35/1.30 % (4707)------------------------------
% 2.35/1.30 % (4707)------------------------------
% 2.35/1.31 % (4709)Instruction limit reached!
% 2.35/1.31 % (4709)------------------------------
% 2.35/1.31 % (4709)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.31 % (4709)Termination reason: Unknown
% 2.35/1.31 % (4709)Termination phase: shuffling
% 2.35/1.31
% 2.35/1.31 % (4709)Memory used [KB]: 1663
% 2.35/1.31 % (4709)Time elapsed: 0.006 s
% 2.35/1.31 % (4709)Instructions burned: 15 (million)
% 2.35/1.31 % (4709)------------------------------
% 2.35/1.31 % (4709)------------------------------
% 2.35/1.31 % (4712)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_4426 for (2999ds/3Mi)
% 2.35/1.31 % (4712)Instruction limit reached!
% 2.35/1.31 % (4712)------------------------------
% 2.35/1.31 % (4712)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.31 % (4712)Termination reason: Unknown
% 2.35/1.31 % (4712)Termination phase: shuffling
% 2.35/1.31
% 2.35/1.31 % (4712)Memory used [KB]: 1279
% 2.35/1.31 % (4712)Time elapsed: 0.002 s
% 2.35/1.31 % (4712)Instructions burned: 3 (million)
% 2.35/1.31 % (4712)------------------------------
% 2.35/1.31 % (4712)------------------------------
% 2.35/1.31 % (4714)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_4426 for (2999ds/1041Mi)
% 2.35/1.31 % (4715)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_4426 for (2999ds/7Mi)
% 2.35/1.31 % (4708)Instruction limit reached!
% 2.35/1.31 % (4708)------------------------------
% 2.35/1.31 % (4708)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.31 % (4708)Termination reason: Unknown
% 2.35/1.31 % (4708)Termination phase: Property scanning
% 2.35/1.31
% 2.35/1.31 % (4708)Memory used [KB]: 2046
% 2.35/1.31 % (4708)Time elapsed: 0.012 s
% 2.35/1.31 % (4708)Instructions burned: 39 (million)
% 2.35/1.31 % (4708)------------------------------
% 2.35/1.31 % (4708)------------------------------
% 2.35/1.32 % (4715)Instruction limit reached!
% 2.35/1.32 % (4715)------------------------------
% 2.35/1.32 % (4715)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.32 % (4715)Termination reason: Unknown
% 2.35/1.32 % (4715)Termination phase: shuffling
% 2.35/1.32
% 2.35/1.32 % (4715)Memory used [KB]: 1535
% 2.35/1.32 % (4715)Time elapsed: 0.004 s
% 2.35/1.32 % (4715)Instructions burned: 8 (million)
% 2.35/1.32 % (4715)------------------------------
% 2.35/1.32 % (4715)------------------------------
% 2.35/1.32 % (4719)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_4426 for (2999ds/16Mi)
% 2.35/1.32 % (4720)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_4426 for (2999ds/3Mi)
% 2.35/1.32 % (4720)Instruction limit reached!
% 2.35/1.32 % (4720)------------------------------
% 2.35/1.32 % (4720)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.32 % (4720)Termination reason: Unknown
% 2.35/1.32 % (4720)Termination phase: shuffling
% 2.35/1.32
% 2.35/1.32 % (4720)Memory used [KB]: 1407
% 2.35/1.32 % (4720)Time elapsed: 0.003 s
% 2.35/1.32 % (4720)Instructions burned: 5 (million)
% 2.35/1.32 % (4720)------------------------------
% 2.35/1.32 % (4720)------------------------------
% 2.35/1.32 % (4723)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_4426 for (2999ds/3Mi)
% 2.35/1.32 % (4719)Instruction limit reached!
% 2.35/1.32 % (4719)------------------------------
% 2.35/1.32 % (4719)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.32 % (4719)Termination reason: Unknown
% 2.35/1.32 % (4719)Termination phase: shuffling
% 2.35/1.32
% 2.35/1.32 % (4719)Memory used [KB]: 1663
% 2.35/1.32 % (4719)Time elapsed: 0.007 s
% 2.35/1.32 % (4719)Instructions burned: 18 (million)
% 2.35/1.32 % (4719)------------------------------
% 2.35/1.32 % (4719)------------------------------
% 2.35/1.32 % (4723)Instruction limit reached!
% 2.35/1.32 % (4723)------------------------------
% 2.35/1.32 % (4723)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.32 % (4723)Termination reason: Unknown
% 2.35/1.32 % (4723)Termination phase: shuffling
% 2.35/1.32
% 2.35/1.32 % (4723)Memory used [KB]: 1407
% 2.35/1.32 % (4723)Time elapsed: 0.002 s
% 2.35/1.32 % (4723)Instructions burned: 3 (million)
% 2.35/1.32 % (4723)------------------------------
% 2.35/1.32 % (4723)------------------------------
% 2.35/1.33 % (4725)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_4426 for (2999ds/7Mi)
% 2.35/1.33 % (4725)Instruction limit reached!
% 2.35/1.33 % (4725)------------------------------
% 2.35/1.33 % (4725)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.33 % (4725)Termination reason: Unknown
% 2.35/1.33 % (4725)Termination phase: shuffling
% 2.35/1.33
% 2.35/1.33 % (4725)Memory used [KB]: 1535
% 2.35/1.33 % (4725)Time elapsed: 0.004 s
% 2.35/1.33 % (4725)Instructions burned: 9 (million)
% 2.35/1.33 % (4725)------------------------------
% 2.35/1.33 % (4725)------------------------------
% 2.35/1.33 % (4728)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_4426 for (2999ds/3Mi)
% 2.35/1.33 % (4728)Instruction limit reached!
% 2.35/1.33 % (4728)------------------------------
% 2.35/1.33 % (4728)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.33 % (4728)Termination reason: Unknown
% 2.35/1.33 % (4728)Termination phase: shuffling
% 2.35/1.33
% 2.35/1.33 % (4728)Memory used [KB]: 1407
% 2.35/1.33 % (4728)Time elapsed: 0.002 s
% 2.35/1.33 % (4728)Instructions burned: 3 (million)
% 2.35/1.33 % (4728)------------------------------
% 2.35/1.33 % (4728)------------------------------
% 2.35/1.33 % (4730)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_4426 for (2999ds/4Mi)
% 2.35/1.33 % (4731)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_4426 for (2999ds/18Mi)
% 2.35/1.34 % (4730)Instruction limit reached!
% 2.35/1.34 % (4730)------------------------------
% 2.35/1.34 % (4730)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.34 % (4730)Termination reason: Unknown
% 2.35/1.34 % (4730)Termination phase: shuffling
% 2.35/1.34
% 2.35/1.34 % (4730)Memory used [KB]: 1407
% 2.35/1.34 % (4730)Time elapsed: 0.003 s
% 2.35/1.34 % (4730)Instructions burned: 6 (million)
% 2.35/1.34 % (4730)------------------------------
% 2.35/1.34 % (4730)------------------------------
% 2.35/1.34 % (4733)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_4426 for (2999ds/710Mi)
% 2.35/1.34 % (4731)Instruction limit reached!
% 2.35/1.34 % (4731)------------------------------
% 2.35/1.34 % (4731)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.34 % (4731)Termination reason: Unknown
% 2.35/1.34 % (4731)Termination phase: shuffling
% 2.35/1.34
% 2.35/1.34 % (4731)Memory used [KB]: 1663
% 2.35/1.34 % (4731)Time elapsed: 0.007 s
% 2.35/1.34 % (4731)Instructions burned: 18 (million)
% 2.35/1.34 % (4731)------------------------------
% 2.35/1.34 % (4731)------------------------------
% 2.35/1.34 % (4736)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_4426 for (2999ds/6Mi)
% 2.35/1.34 % (4736)Instruction limit reached!
% 2.35/1.34 % (4736)------------------------------
% 2.35/1.34 % (4736)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.34 % (4736)Termination reason: Unknown
% 2.35/1.34 % (4736)Termination phase: shuffling
% 2.35/1.34
% 2.35/1.34 % (4736)Memory used [KB]: 1407
% 2.35/1.34 % (4736)Time elapsed: 0.003 s
% 2.35/1.34 % (4736)Instructions burned: 6 (million)
% 2.35/1.34 % (4736)------------------------------
% 2.35/1.34 % (4736)------------------------------
% 2.35/1.34 % (4739)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_4426 for (2999ds/902Mi)
% 2.35/1.35 % (4694)Instruction limit reached!
% 2.35/1.35 % (4694)------------------------------
% 2.35/1.35 % (4694)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.35 % (4694)Termination reason: Unknown
% 2.35/1.35 % (4694)Termination phase: Saturation
% 2.35/1.35
% 2.35/1.35 % (4694)Memory used [KB]: 7036
% 2.35/1.35 % (4694)Time elapsed: 0.056 s
% 2.35/1.35 % (4694)Instructions burned: 185 (million)
% 2.35/1.35 % (4694)------------------------------
% 2.35/1.35 % (4694)------------------------------
% 2.35/1.35 % (4743)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_4426 for (2999ds/21Mi)
% 2.35/1.35 % (4746)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_4426 for (2999ds/5Mi)
% 2.35/1.35 % (4746)Instruction limit reached!
% 2.35/1.35 % (4746)------------------------------
% 2.35/1.35 % (4746)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.35 % (4746)Termination reason: Unknown
% 2.35/1.35 % (4746)Termination phase: shuffling
% 2.35/1.35
% 2.35/1.35 % (4746)Memory used [KB]: 1407
% 2.35/1.35 % (4746)Time elapsed: 0.003 s
% 2.35/1.35 % (4746)Instructions burned: 6 (million)
% 2.35/1.35 % (4746)------------------------------
% 2.35/1.35 % (4746)------------------------------
% 2.35/1.36 % (4743)Instruction limit reached!
% 2.35/1.36 % (4743)------------------------------
% 2.35/1.36 % (4743)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.36 % (4743)Termination reason: Unknown
% 2.35/1.36 % (4743)Termination phase: shuffling
% 2.35/1.36
% 2.35/1.36 % (4743)Memory used [KB]: 1663
% 2.35/1.36 % (4743)Time elapsed: 0.008 s
% 2.35/1.36 % (4743)Instructions burned: 21 (million)
% 2.35/1.36 % (4743)------------------------------
% 2.35/1.36 % (4743)------------------------------
% 2.35/1.36 % (4751)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_4426 for (2999ds/6Mi)
% 2.35/1.36 % (4751)Instruction limit reached!
% 2.35/1.36 % (4751)------------------------------
% 2.35/1.36 % (4751)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.36 % (4751)Termination reason: Unknown
% 2.35/1.36 % (4751)Termination phase: shuffling
% 2.35/1.36
% 2.35/1.36 % (4751)Memory used [KB]: 1407
% 2.35/1.36 % (4751)Time elapsed: 0.003 s
% 2.35/1.36 % (4751)Instructions burned: 6 (million)
% 2.35/1.36 % (4751)------------------------------
% 2.35/1.36 % (4751)------------------------------
% 2.35/1.36 % (4753)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_4426 for (2999ds/377Mi)
% 2.35/1.37 % (4756)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_4426 for (2999ds/779Mi)
% 2.35/1.37 % (4757)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_4426 for (2999ds/19Mi)
% 2.35/1.37 % (4757)Instruction limit reached!
% 2.35/1.37 % (4757)------------------------------
% 2.35/1.37 % (4757)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.37 % (4757)Termination reason: Unknown
% 2.35/1.37 % (4757)Termination phase: shuffling
% 2.35/1.37
% 2.35/1.37 % (4757)Memory used [KB]: 1663
% 2.35/1.37 % (4757)Time elapsed: 0.007 s
% 2.35/1.37 % (4757)Instructions burned: 20 (million)
% 2.35/1.37 % (4757)------------------------------
% 2.35/1.37 % (4757)------------------------------
% 2.35/1.38 % (4699)Instruction limit reached!
% 2.35/1.38 % (4699)------------------------------
% 2.35/1.38 % (4699)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.38 % (4699)Termination reason: Unknown
% 2.35/1.38 % (4699)Termination phase: Saturation
% 2.35/1.38
% 2.35/1.38 % (4699)Memory used [KB]: 7164
% 2.35/1.38 % (4699)Time elapsed: 0.084 s
% 2.35/1.38 % (4699)Instructions burned: 275 (million)
% 2.35/1.38 % (4699)------------------------------
% 2.35/1.38 % (4699)------------------------------
% 2.35/1.38 % (4771)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_4426 for (2999ds/879Mi)
% 2.35/1.39 % (4774)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_4426 for (2998ds/17Mi)
% 2.35/1.39 % (4739)First to succeed.
% 2.35/1.39 % (4774)Instruction limit reached!
% 2.35/1.39 % (4774)------------------------------
% 2.35/1.39 % (4774)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.39 % (4774)Termination reason: Unknown
% 2.35/1.39 % (4774)Termination phase: shuffling
% 2.35/1.39
% 2.35/1.39 % (4774)Memory used [KB]: 1663
% 2.35/1.39 % (4774)Time elapsed: 0.008 s
% 2.35/1.39 % (4774)Instructions burned: 19 (million)
% 2.35/1.39 % (4774)------------------------------
% 2.35/1.39 % (4774)------------------------------
% 2.35/1.39 % (4739)Refutation found. Thanks to Tanya!
% 2.35/1.39 % SZS status Theorem for DTF2THF_4426
% 2.35/1.39 % SZS output start Proof for DTF2THF_4426
% 2.35/1.39 thf(type_def_5, type, binary1291135688e_Tree: ($tType) > $tType).
% 2.35/1.39 thf(type_def_7, type, set: ($tType) > $tType).
% 2.35/1.39 thf(type_def_8, type, int: $tType).
% 2.35/1.39 thf(type_def_9, type, a: $tType).
% 2.35/1.39 thf(func_def_0, type, binary1291135688e_Tree: ($tType) > $tType).
% 2.35/1.39 thf(func_def_1, type, set: ($tType) > $tType).
% 2.35/1.39 thf(func_def_2, type, int: $tType).
% 2.35/1.39 thf(func_def_3, type, a: $tType).
% 2.35/1.39 thf(func_def_4, type, bounded_lattice: !>[X0: $tType]:($o)).
% 2.35/1.39 thf(func_def_5, type, type: !>[X0: $tType]:($o)).
% 2.35/1.39 thf(func_def_6, type, minus: !>[X0: $tType]:($o)).
% 2.35/1.39 thf(func_def_7, type, bot: !>[X0: $tType]:($o)).
% 2.35/1.39 thf(func_def_8, type, ord: !>[X0: $tType]:($o)).
% 2.35/1.39 thf(func_def_9, type, order: !>[X0: $tType]:($o)).
% 2.35/1.39 thf(func_def_10, type, group_add: !>[X0: $tType]:($o)).
% 2.35/1.39 thf(func_def_11, type, lattice: !>[X0: $tType]:($o)).
% 2.35/1.39 thf(func_def_12, type, no_bot: !>[X0: $tType]:($o)).
% 2.35/1.39 thf(func_def_13, type, no_top: !>[X0: $tType]:($o)).
% 2.35/1.39 thf(func_def_14, type, linorder: !>[X0: $tType]:($o)).
% 2.35/1.39 thf(func_def_15, type, preorder: !>[X0: $tType]:($o)).
% 2.35/1.39 thf(func_def_16, type, order_bot: !>[X0: $tType]:($o)).
% 2.35/1.39 thf(func_def_17, type, wellorder: !>[X0: $tType]:($o)).
% 2.35/1.39 thf(func_def_18, type, dense_order: !>[X0: $tType]:($o)).
% 2.35/1.39 thf(func_def_19, type, semilattice_sup: !>[X0: $tType]:($o)).
% 2.35/1.39 thf(func_def_20, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 2.35/1.39 thf(func_def_21, type, cancel146912293up_add: !>[X0: $tType]:($o)).
% 2.35/1.39 thf(func_def_22, type, bounde1808546759up_bot: !>[X0: $tType]:($o)).
% 2.35/1.39 thf(func_def_23, type, binary210054475elle_T: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > binary1291135688e_Tree @ X0 > binary1291135688e_Tree @ X0)).
% 2.35/1.39 thf(func_def_24, type, binary1746293266le_Tip: !>[X0: $tType]:(binary1291135688e_Tree @ X0)).
% 2.35/1.39 thf(func_def_25, type, binary2130109271t_Tree: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > set @ X0)).
% 2.35/1.39 thf(func_def_26, type, binary1830089824insert: !>[X0: $tType]:((X0 > int) > X0 > binary1291135688e_Tree @ X0 > binary1291135688e_Tree @ X0)).
% 2.35/1.39 thf(func_def_27, type, binary64540844le_eqs: !>[X0: $tType]:((X0 > int) > X0 > set @ X0)).
% 2.35/1.39 thf(func_def_28, type, binary827270440e_memb: !>[X0: $tType]:((X0 > int) > X0 > binary1291135688e_Tree @ X0 > $o)).
% 2.35/1.39 thf(func_def_29, type, binary1653327646_setOf: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > set @ X0)).
% 2.35/1.39 thf(func_def_30, type, binary1610619414edTree: !>[X0: $tType]:((X0 > int) > binary1291135688e_Tree @ X0 > $o)).
% 2.35/1.39 thf(func_def_31, type, binary231205461t_pred: !>[X0: $tType]:((X0 > int) > X0 > X0 > binary1291135688e_Tree @ X0 > $o)).
% 2.35/1.39 thf(func_def_32, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.35/1.39 thf(func_def_33, type, the: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.35/1.39 thf(func_def_34, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.35/1.39 thf(func_def_35, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.35/1.39 thf(func_def_36, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.35/1.39 thf(func_def_37, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.35/1.39 thf(func_def_38, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.35/1.39 thf(func_def_39, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.35/1.39 thf(func_def_40, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 2.35/1.39 thf(func_def_41, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 2.35/1.39 thf(func_def_42, type, is_singleton: !>[X0: $tType]:(set @ X0 > $o)).
% 2.35/1.39 thf(func_def_43, type, pairwise: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > $o)).
% 2.35/1.39 thf(func_def_44, type, remove: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 2.35/1.39 thf(func_def_45, type, the_elem: !>[X0: $tType]:(set @ X0 > X0)).
% 2.35/1.39 thf(func_def_46, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.35/1.39 thf(func_def_47, type, e: a).
% 2.35/1.39 thf(func_def_48, type, h: a > int).
% 2.35/1.39 thf(func_def_49, type, t1: binary1291135688e_Tree @ a).
% 2.35/1.39 thf(func_def_50, type, t2: binary1291135688e_Tree @ a).
% 2.35/1.39 thf(func_def_51, type, x: a).
% 2.35/1.39 thf(func_def_67, type, sP0: !>[X0: $tType]:(X0 > set @ X0 > set @ X0 > $o)).
% 2.35/1.39 thf(func_def_68, type, sP1: !>[X0: $tType]:(set @ X0 > X0 > set @ X0 > $o)).
% 2.35/1.39 thf(func_def_69, type, sP2: !>[X0: $tType]:(X0 > set @ X0 > set @ X0 > $o)).
% 2.35/1.39 thf(func_def_70, type, sP3: !>[X0: $tType]:(set @ X0 > set @ X0 > X0 > X0 > $o)).
% 2.35/1.39 thf(func_def_71, type, sP4: !>[X0: $tType]:(set @ X0 > set @ X0 > X0 > X0 > $o)).
% 2.35/1.39 thf(func_def_72, type, sK5: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.35/1.39 thf(func_def_73, type, sK6: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.35/1.39 thf(func_def_74, type, sK7: !>[X0: $tType]:(set @ X0 > X0)).
% 2.35/1.39 thf(func_def_75, type, sK8: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.35/1.39 thf(func_def_76, type, sK9: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.35/1.39 thf(func_def_77, type, sK10: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.35/1.39 thf(func_def_78, type, sK11: !>[X0: $tType]:(X0 > X0)).
% 2.35/1.39 thf(func_def_79, type, sK12: !>[X0: $tType]:(set @ X0 > X0)).
% 2.35/1.39 thf(func_def_80, type, sK13: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.35/1.39 thf(func_def_81, type, sK14: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.35/1.39 thf(func_def_82, type, sK15: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.35/1.39 thf(func_def_83, type, sK16: a).
% 2.35/1.39 thf(func_def_84, type, sK17: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.35/1.39 thf(func_def_85, type, sK18: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.35/1.39 thf(func_def_86, type, sK19: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.35/1.39 thf(func_def_87, type, sK20: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.35/1.39 thf(func_def_88, type, sK21: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > binary1291135688e_Tree @ X0)).
% 2.35/1.39 thf(func_def_89, type, sK22: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > X0)).
% 2.35/1.39 thf(func_def_90, type, sK23: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > binary1291135688e_Tree @ X0)).
% 2.35/1.39 thf(func_def_91, type, sK24: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > binary1291135688e_Tree @ X0)).
% 2.35/1.39 thf(func_def_92, type, sK25: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > binary1291135688e_Tree @ X0)).
% 2.35/1.39 thf(func_def_93, type, sK26: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > binary1291135688e_Tree @ X0)).
% 2.35/1.39 thf(func_def_94, type, sK27: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > X0)).
% 2.35/1.39 thf(func_def_95, type, sK28: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > binary1291135688e_Tree @ X0)).
% 2.35/1.39 thf(func_def_96, type, sK29: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.35/1.39 thf(func_def_97, type, sK30: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.35/1.39 thf(func_def_98, type, sK31: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.35/1.39 thf(func_def_99, type, sK32: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.35/1.39 thf(func_def_100, type, sK33: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.35/1.39 thf(func_def_101, type, sK34: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.35/1.39 thf(func_def_102, type, sK35: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.35/1.39 thf(func_def_103, type, sK36: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.35/1.39 thf(func_def_104, type, sK37: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0 > (X0 > int) > X0)).
% 2.35/1.39 thf(func_def_105, type, sK38: !>[X0: $tType]:(X0 > binary1291135688e_Tree @ X0 > (X0 > int) > X0)).
% 2.35/1.39 thf(func_def_106, type, sK39: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.35/1.39 thf(func_def_107, type, sK40: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.35/1.39 thf(func_def_108, type, sK41: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.35/1.39 thf(func_def_109, type, sK42: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.35/1.39 thf(func_def_110, type, sK43: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.35/1.39 thf(func_def_111, type, sK44: !>[X0: $tType]:((binary1291135688e_Tree @ X0 > $o) > binary1291135688e_Tree @ X0)).
% 2.35/1.39 thf(func_def_112, type, sK45: !>[X0: $tType]:((binary1291135688e_Tree @ X0 > $o) > X0)).
% 2.35/1.39 thf(func_def_113, type, sK46: !>[X0: $tType]:((binary1291135688e_Tree @ X0 > $o) > binary1291135688e_Tree @ X0)).
% 2.35/1.39 thf(func_def_114, type, sK47: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.35/1.39 thf(func_def_115, type, sK48: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.35/1.39 thf(func_def_116, type, sK49: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.35/1.39 thf(func_def_117, type, sK50: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.35/1.39 thf(func_def_118, type, sK51: !>[X0: $tType]:((X0 > $o) > set @ X0 > set @ X0 > X0)).
% 2.35/1.39 thf(func_def_119, type, sK52: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.35/1.39 thf(func_def_120, type, sK53: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.35/1.39 thf(func_def_121, type, sK54: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.35/1.39 thf(func_def_122, type, sK55: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > binary1291135688e_Tree @ X0)).
% 2.35/1.39 thf(func_def_123, type, sK56: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > X0)).
% 2.35/1.39 thf(func_def_124, type, sK57: !>[X0: $tType]:(binary1291135688e_Tree @ X0 > binary1291135688e_Tree @ X0)).
% 2.35/1.39 thf(func_def_125, type, sK58: !>[X0: $tType]:(X0 > X0)).
% 2.35/1.39 thf(func_def_126, type, sK59: !>[X0: $tType]:(set @ X0 > set @ X0 > X0 > X0 > set @ X0)).
% 2.35/1.39 thf(func_def_127, type, sK60: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.35/1.39 thf(func_def_128, type, sK61: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.35/1.39 thf(func_def_129, type, sK62: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.35/1.39 thf(func_def_130, type, sK63: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.35/1.39 thf(func_def_131, type, sK64: !>[X0: $tType]:((X0 > $o) > set @ X0 > set @ X0 > X0)).
% 2.35/1.39 thf(func_def_132, type, sK65: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.35/1.39 thf(func_def_133, type, sK66: !>[X0: $tType]:(set @ X0 > X0)).
% 2.35/1.39 thf(func_def_134, type, sK67: !>[X0: $tType]:(set @ X0 > X0)).
% 2.35/1.39 thf(func_def_135, type, sK68: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.35/1.39 thf(func_def_136, type, sK69: !>[X0: $tType]:(set @ X0 > X0 > set @ X0)).
% 2.35/1.39 thf(func_def_137, type, sK70: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.35/1.39 thf(func_def_138, type, sK71: !>[X0: $tType]:(set @ X0 > X0 > set @ X0)).
% 2.35/1.39 thf(func_def_139, type, sK72: !>[X0: $tType]:(set @ X0 > X0)).
% 2.35/1.39 thf(func_def_140, type, sK73: !>[X0: $tType]:(set @ X0 > X0)).
% 2.35/1.39 thf(func_def_141, type, sK74: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.35/1.39 thf(func_def_142, type, sK75: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.35/1.39 thf(func_def_144, type, ph77: !>[X0: $tType]:(X0)).
% 2.35/1.39 thf(f2013,plain,(
% 2.35/1.39 $false),
% 2.35/1.39 inference(trivial_inequality_removal,[status(thm)],[f2012])).
% 2.35/1.39 thf(f2012,plain,(
% 2.35/1.39 ($true != $true)),
% 2.35/1.39 inference(superposition,[status(thm)],[f1902,f1431])).
% 2.35/1.39 thf(f1431,plain,(
% 2.35/1.39 (order @ int = $true)),
% 2.35/1.39 inference(cnf_transformation,[status(thm)],[f473])).
% 2.35/1.39 thf(f473,plain,(
% 2.35/1.39 (order @ int = $true)),
% 2.35/1.39 inference(fool_elimination,[status(thm)],[f472])).
% 2.35/1.39 thf(f472,plain,(
% 2.35/1.39 order @ int),
% 2.35/1.39 inference(rectify,[status(thm)],[f278])).
% 2.35/1.39 thf(f278,axiom,(
% 2.35/1.39 order @ int),
% 2.35/1.39 file('/export/starexec/sandbox2/tmp/tmp.dsmPofI642/DTF2THF_4426.p',tcon_Int_Oint___Orderings_Oorder_6)).
% 2.35/1.39 thf(f1902,plain,(
% 2.35/1.39 (order @ int != $true)),
% 2.35/1.39 inference(trivial_inequality_removal,[status(thm)],[f1901])).
% 2.35/1.39 thf(f1901,plain,(
% 2.35/1.39 ($true != $true) | (order @ int != $true)),
% 2.35/1.39 inference(duplicate_literal_removal,[status(thm)],[f1892])).
% 2.35/1.39 thf(f1892,plain,(
% 2.35/1.39 ($true != $true) | (order @ int != $true) | ($true != $true)),
% 2.35/1.39 inference(superposition,[status(thm)],[f1520,f1876])).
% 2.35/1.39 thf(f1876,plain,(
% 2.35/1.39 ($true = (ord_less @ int @ (h @ x) @ (h @ x)))),
% 2.35/1.39 inference(forward_demodulation,[status(thm)],[f1875,f1406])).
% 2.35/1.39 thf(f1406,plain,(
% 2.35/1.39 ((h @ x) = (h @ sK16))),
% 2.35/1.39 inference(cnf_transformation,[status(thm)],[f1167])).
% 2.35/1.39 thf(f1167,plain,(
% 2.35/1.39 ((h @ x) = (h @ sK16)) & ($true = (member @ a @ sK16 @ (binary64540844le_eqs @ a @ h @ e)))),
% 2.35/1.39 inference(skolemisation,[status(esa),new_symbols(skolem,[sK16])],[f940,f1166])).
% 2.35/1.39 thf(f1166,plain,(
% 2.35/1.39 ? [X0 : a] : (((h @ x) = (h @ X0)) & ($true = (member @ a @ X0 @ (binary64540844le_eqs @ a @ h @ e)))) => (((h @ x) = (h @ sK16)) & ($true = (member @ a @ sK16 @ (binary64540844le_eqs @ a @ h @ e))))),
% 2.35/1.39 introduced(definition,[],[choice_axiom])).
% 2.35/1.39 thf(f940,plain,(
% 2.35/1.39 ? [X0 : a] : (((h @ x) = (h @ X0)) & ($true = (member @ a @ X0 @ (binary64540844le_eqs @ a @ h @ e))))),
% 2.35/1.39 inference(ennf_transformation,[status(thm)],[f587])).
% 2.35/1.39 thf(f587,plain,(
% 2.35/1.39 ~! [X0 : a] : (($true = (member @ a @ X0 @ (binary64540844le_eqs @ a @ h @ e))) => ((h @ x) != (h @ X0)))),
% 2.35/1.39 inference(fool_elimination,[status(thm)],[f586])).
% 2.35/1.39 thf(f586,plain,(
% 2.35/1.39 ~! [X0 : a] : ((member @ a @ X0 @ (binary64540844le_eqs @ a @ h @ e)) => ((h @ x) != (h @ X0)))),
% 2.35/1.39 inference(rectify,[status(thm)],[f304])).
% 2.35/1.39 thf(f304,negated_conjecture,(
% 2.35/1.39 ~! [X38 : a] : ((member @ a @ X38 @ (binary64540844le_eqs @ a @ h @ e)) => ((h @ x) != (h @ X38)))),
% 2.35/1.39 inference(negated_conjecture,[status(cth)],[f303])).
% 2.35/1.39 thf(f303,conjecture,(
% 2.35/1.39 ! [X38 : a] : ((member @ a @ X38 @ (binary64540844le_eqs @ a @ h @ e)) => ((h @ x) != (h @ X38)))),
% 2.35/1.39 file('/export/starexec/sandbox2/tmp/tmp.dsmPofI642/DTF2THF_4426.p',conj_0)).
% 2.35/1.39 thf(f1875,plain,(
% 2.35/1.39 ($true = (ord_less @ int @ (h @ sK16) @ (h @ x)))),
% 2.35/1.39 inference(trivial_inequality_removal,[status(thm)],[f1874])).
% 2.35/1.39 thf(f1874,plain,(
% 2.35/1.39 ($true = (ord_less @ int @ (h @ sK16) @ (h @ x))) | ($true != $true)),
% 2.35/1.39 inference(superposition,[status(thm)],[f1629,f1405])).
% 2.35/1.39 thf(f1405,plain,(
% 2.35/1.39 ($true = (member @ a @ sK16 @ (binary64540844le_eqs @ a @ h @ e)))),
% 2.35/1.39 inference(cnf_transformation,[status(thm)],[f1167])).
% 2.35/1.39 thf(f1629,plain,(
% 2.35/1.39 ( ! [X0 : a] : (($true != (member @ a @ X0 @ (binary64540844le_eqs @ a @ h @ e))) | ($true = (ord_less @ int @ (h @ X0) @ (h @ x)))) )),
% 2.35/1.39 inference(cnf_transformation,[status(thm)],[f1069])).
% 2.35/1.39 thf(f1069,plain,(
% 2.35/1.39 ! [X0 : a] : (($true = (ord_less @ int @ (h @ X0) @ (h @ x))) | ($true != (member @ a @ X0 @ (binary64540844le_eqs @ a @ h @ e))))),
% 2.35/1.39 inference(ennf_transformation,[status(thm)],[f621])).
% 2.35/1.39 thf(f621,plain,(
% 2.35/1.39 ! [X0 : a] : (($true = (member @ a @ X0 @ (binary64540844le_eqs @ a @ h @ e))) => ($true = (ord_less @ int @ (h @ X0) @ (h @ x))))),
% 2.35/1.39 inference(fool_elimination,[status(thm)],[f620])).
% 2.35/1.39 thf(f620,plain,(
% 2.35/1.39 ! [X0 : a] : ((member @ a @ X0 @ (binary64540844le_eqs @ a @ h @ e)) => (ord_less @ int @ (h @ X0) @ (h @ x)))),
% 2.35/1.39 inference(rectify,[status(thm)],[f1])).
% 2.35/1.39 thf(f1,axiom,(
% 2.35/1.39 ! [X1 : a] : ((member @ a @ X1 @ (binary64540844le_eqs @ a @ h @ e)) => (ord_less @ int @ (h @ X1) @ (h @ x)))),
% 2.35/1.39 file('/export/starexec/sandbox2/tmp/tmp.dsmPofI642/DTF2THF_4426.p',fact_0_eqsLessX)).
% 2.35/1.39 thf(f1520,plain,(
% 2.35/1.39 ( ! [X0 : $tType,X2 : X0,X1 : X0] : (($true != (ord_less @ X0 @ X2 @ X1)) | (order @ X0 != $true) | ((ord_less @ X0 @ X1 @ X2) != $true)) )),
% 2.35/1.39 inference(cnf_transformation,[status(thm)],[f1033])).
% 2.35/1.39 thf(f1033,plain,(
% 2.35/1.39 ! [X0 : $tType] : ((order @ X0 != $true) | ! [X1 : X0,X2 : X0] : (($true != (ord_less @ X0 @ X2 @ X1)) | ((ord_less @ X0 @ X1 @ X2) != $true)))),
% 2.35/1.39 inference(ennf_transformation,[status(thm)],[f906])).
% 2.35/1.39 thf(f906,plain,(
% 2.35/1.39 ! [X0 : $tType] : ((order @ X0 = $true) => ! [X1 : X0,X2 : X0] : (((ord_less @ X0 @ X1 @ X2) = $true) => ($true != (ord_less @ X0 @ X2 @ X1))))),
% 2.35/1.39 inference(flattening,[status(thm)],[f721])).
% 2.35/1.39 thf(f721,plain,(
% 2.35/1.39 ! [X0 : $tType] : ((order @ X0 = $true) => ! [X1 : X0,X2 : X0] : (((ord_less @ X0 @ X1 @ X2) = $true) => ~($true = (ord_less @ X0 @ X2 @ X1))))),
% 2.35/1.39 inference(fool_elimination,[status(thm)],[f720])).
% 2.35/1.39 thf(f720,plain,(
% 2.35/1.39 ! [X0 : $tType] : (order @ X0 => ! [X1 : X0,X2 : X0] : ((ord_less @ X0 @ X1 @ X2) => ~(ord_less @ X0 @ X2 @ X1)))),
% 2.35/1.39 inference(rectify,[status(thm)],[f175])).
% 2.35/1.39 thf(f175,axiom,(
% 2.35/1.39 ! [X0 : $tType] : (order @ X0 => ! [X4 : X0,X3 : X0] : ((ord_less @ X0 @ X4 @ X3) => ~(ord_less @ X0 @ X3 @ X4)))),
% 2.35/1.39 file('/export/starexec/sandbox2/tmp/tmp.dsmPofI642/DTF2THF_4426.p',fact_174_dual__order_Oasym)).
% 2.35/1.39 % SZS output end Proof for DTF2THF_4426
% 2.35/1.39 % (4739)------------------------------
% 2.35/1.39 % (4739)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.35/1.39 % (4739)Termination reason: Refutation
% 2.35/1.39
% 2.35/1.39 % (4739)Memory used [KB]: 7036
% 2.35/1.39 % (4739)Time elapsed: 0.048 s
% 2.35/1.39 % (4739)Instructions burned: 151 (million)
% 2.35/1.39 % (4739)------------------------------
% 2.35/1.39 % (4739)------------------------------
% 2.35/1.39 % (4687)Success in time 0.127 s
%------------------------------------------------------------------------------