%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP115^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n018.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:40 AM UTC 2026
% Result : Theorem 2.70s 1.60s
% Output : Refutation 2.70s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP115^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n018.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 08:26:31 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.29/0.45 ---- Original DTF file ---
% 0.29/0.45 thf(spec,logic,$$dhol).
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % File : ITP115^2 : TPTP v9.2.1. Released v7.5.0.
% 0.29/0.45 % Domain : Interactive Theorem Proving
% 0.29/0.45 % Problem : Sledgehammer Lower_Semicontinuous problem prob_96__6247558_1
% 0.29/0.45 % Version : Especial.
% 0.29/0.45 % English :
% 0.29/0.45
% 0.29/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.29/0.45 % Source : [Des21]
% 0.29/0.45 % Names : Lower_Semicontinuous/prob_96__6247558_1 [Des21]
% 0.29/0.45
% 0.29/0.45 % Status : Theorem
% 0.29/0.45 % Rating : 0.33 v9.1.0, 0.00 v7.5.0
% 0.29/0.45 % Syntax : Number of formulae : 412 ( 140 unt; 54 typ; 0 def)
% 0.29/0.45 % Number of atoms : 913 ( 256 equ; 0 cnn)
% 0.29/0.45 % Maximal formula atoms : 13 ( 2 avg)
% 0.29/0.45 % Number of connectives : 3781 ( 78 ~; 5 |; 72 &;3254 @)
% 0.29/0.45 % ( 0 <=>; 372 =>; 0 <=; 0 <~>)
% 0.29/0.45 % Maximal formula depth : 17 ( 7 avg)
% 0.29/0.45 % Number of types : 4 ( 3 usr)
% 0.29/0.45 % Number of type conns : 97 ( 97 >; 0 *; 0 +; 0 <<)
% 0.29/0.45 % Number of symbols : 52 ( 51 usr; 3 con; 0-4 aty)
% 0.29/0.45 % Number of variables : 1046 ( 63 ^; 917 !; 20 ?;1046 :)
% 0.29/0.45 % ( 46 !>; 0 ?*; 0 @-; 0 @+)
% 0.29/0.45 % SPC : TH1_THM_EQU_NAR
% 0.29/0.45
% 0.29/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 16:25:13.913
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % Could-be-implicit typings (5)
% 0.29/0.45 thf(ty_t_Filter_Ofilter,type,
% 0.29/0.45 filter: $tType > $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_t_Set_Oset,type,
% 0.29/0.45 set: $tType > $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_t_Nat_Onat,type,
% 0.29/0.45 nat: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_tf_b,type,
% 0.29/0.45 b: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_tf_a,type,
% 0.29/0.45 a: $tType ).
% 0.29/0.45
% 0.29/0.45 % Explicit typings (49)
% 0.29/0.45 thf(sy_cl_Topological__Spaces_Olinorder__topology,type,
% 0.29/0.45 topolo2117631714pology:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
% 0.29/0.45 comple1035589618norder:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.29/0.45 finite_finite:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Lattices_Obounded__lattice,type,
% 0.29/0.45 bounded_lattice:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_HOL_Otype,type,
% 0.29/0.45 type:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Countable_Ocountable,type,
% 0.29/0.45 countable:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Ominus,type,
% 0.29/0.45 minus:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Obot,type,
% 0.29/0.45 bot:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Oord,type,
% 0.29/0.45 ord:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Otop,type,
% 0.29/0.45 top:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Oorder,type,
% 0.29/0.45 order:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Ogroup__add,type,
% 0.29/0.45 group_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Lattices_Olattice,type,
% 0.29/0.45 lattice:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Olinorder,type,
% 0.29/0.45 linorder:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Opreorder,type,
% 0.29/0.45 preorder:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.29/0.45 order_bot:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 0.29/0.45 semilattice_inf:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.29/0.45 ordered_ab_group_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Topological__Spaces_Ot0__space,type,
% 0.29/0.45 topological_t0_space:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Topological__Spaces_Ot1__space,type,
% 0.29/0.45 topological_t1_space:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Topological__Spaces_Ot2__space,type,
% 0.29/0.45 topological_t2_space:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Lattices_Obounded__lattice__bot,type,
% 0.29/0.45 bounded_lattice_bot:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.29/0.45 cancel146912293up_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Topological__Spaces_Operfect__space,type,
% 0.29/0.45 topolo890362671_space:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Topological__Spaces_Ouniform__space,type,
% 0.29/0.45 topolo47006728_space:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Topological__Spaces_Odiscrete__topology,type,
% 0.29/0.45 topolo2133971006pology:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Topological__Spaces_Otopological__space,type,
% 0.29/0.45 topolo503727757_space:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Topological__Spaces_Ofirst__countable__topology,type,
% 0.29/0.45 topolo2135403230pology:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Elementary__Topology_Oclosure,type,
% 0.29/0.45 elementary_closure:
% 0.29/0.45 !>[A: $tType] : ( ( set @ A ) > ( set @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.29/0.45 minus_minus:
% 0.29/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 0.29/0.45 inf_inf:
% 0.29/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc__at,type,
% 0.29/0.45 lower_582600101lsc_at:
% 0.29/0.45 !>[A: $tType,B: $tType] : ( A > ( A > B ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.29/0.45 bot_bot:
% 0.29/0.45 !>[A: $tType] : A ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.29/0.45 ord_less_eq:
% 0.29/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.29/0.45 top_top:
% 0.29/0.45 !>[A: $tType] : A ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_OCollect,type,
% 0.29/0.45 collect:
% 0.29/0.45 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oinsert,type,
% 0.29/0.45 insert:
% 0.29/0.45 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Ois__empty,type,
% 0.29/0.45 is_empty:
% 0.29/0.45 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Ois__singleton,type,
% 0.29/0.45 is_singleton:
% 0.29/0.45 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oremove,type,
% 0.29/0.45 remove:
% 0.29/0.45 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Othe__elem,type,
% 0.29/0.45 the_elem:
% 0.29/0.45 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Topological__Spaces_Oopen__class_Oopen,type,
% 0.29/0.45 topolo1751647064n_open:
% 0.29/0.45 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type,
% 0.29/0.45 topolo507301023within:
% 0.29/0.45 !>[A: $tType] : ( A > ( set @ A ) > ( filter @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type,
% 0.29/0.45 topolo406746546ounded:
% 0.29/0.45 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_member,type,
% 0.29/0.45 member:
% 0.29/0.45 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_A____,type,
% 0.29/0.45 a2: b ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_f,type,
% 0.29/0.45 f: a > b ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_thesis____,type,
% 0.29/0.45 thesis: $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_x0,type,
% 0.29/0.45 x0: a ).
% 0.29/0.45
% 0.29/0.45 % Relevant facts (256)
% 0.29/0.45 thf(fact_0__092_060open_062A_A_092_060noteq_062_Af_Ax0_092_060close_062,axiom,
% 0.29/0.45 ( a2
% 0.29/0.45 != ( f @ x0 ) ) ).
% 0.29/0.45
% 0.29/0.45 % \<open>A \<noteq> f x0\<close>
% 0.29/0.45 thf(fact_1__092_060open_062f_Ax0_A_092_060noteq_062_AA_A_092_060Longrightarrow_062_A_092_060exists_062U_AV_O_Aopen_AU_A_092_060and_062_Aopen_AV_A_092_060and_062_Af_Ax0_A_092_060in_062_AU_A_092_060and_062_AA_A_092_060in_062_AV_A_092_060and_062_AU_A_092_060inter_062_AV_A_061_A_123_125_092_060close_062,axiom,
% 0.29/0.45 ( ( ( f @ x0 )
% 0.29/0.45 != a2 )
% 0.29/0.45 => ? [U: set @ b,V: set @ b] :
% 0.29/0.45 ( ( topolo1751647064n_open @ b @ U )
% 0.29/0.45 & ( topolo1751647064n_open @ b @ V )
% 0.29/0.45 & ( member @ b @ ( f @ x0 ) @ U )
% 0.29/0.45 & ( member @ b @ a2 @ V )
% 0.29/0.45 & ( ( inf_inf @ ( set @ b ) @ U @ V )
% 0.29/0.45 = ( bot_bot @ ( set @ b ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % \<open>f x0 \<noteq> A \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> f x0 \<in> U \<and> A \<in> V \<and> U \<inter> V = {}\<close>
% 0.29/0.45 thf(fact_2_open__Int,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( topolo503727757_space @ A )
% 0.29/0.45 => ! [S: set @ A,T: set @ A] :
% 0.29/0.45 ( ( topolo1751647064n_open @ A @ S )
% 0.29/0.45 => ( ( topolo1751647064n_open @ A @ T )
% 0.29/0.45 => ( topolo1751647064n_open @ A @ ( inf_inf @ ( set @ A ) @ S @ T ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % open_Int
% 0.29/0.45 thf(fact_3_open__empty,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( topolo503727757_space @ A )
% 0.29/0.45 => ( topolo1751647064n_open @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % open_empty
% 0.29/0.45 thf(fact_4_inf__bot__left,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( bounded_lattice_bot @ A )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
% 0.29/0.45 = ( bot_bot @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_bot_left
% 0.29/0.45 thf(fact_5_inf__bot__right,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( bounded_lattice_bot @ A )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
% 0.29/0.45 = ( bot_bot @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_bot_right
% 0.29/0.45 thf(fact_6_hausdorff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( topological_t2_space @ A )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( X != Y )
% 0.29/0.45 => ? [U: set @ A,V: set @ A] :
% 0.29/0.45 ( ( topolo1751647064n_open @ A @ U )
% 0.29/0.45 & ( topolo1751647064n_open @ A @ V )
% 0.29/0.45 & ( member @ A @ X @ U )
% 0.29/0.45 & ( member @ A @ Y @ V )
% 0.29/0.45 & ( ( inf_inf @ ( set @ A ) @ U @ V )
% 0.29/0.45 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hausdorff
% 0.29/0.45 thf(fact_7_separation__t2,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( topological_t2_space @ A )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( X != Y )
% 0.29/0.45 = ( ? [U2: set @ A,V2: set @ A] :
% 0.29/0.45 ( ( topolo1751647064n_open @ A @ U2 )
% 0.29/0.45 & ( topolo1751647064n_open @ A @ V2 )
% 0.29/0.45 & ( member @ A @ X @ U2 )
% 0.29/0.45 & ( member @ A @ Y @ V2 )
% 0.29/0.45 & ( ( inf_inf @ ( set @ A ) @ U2 @ V2 )
% 0.29/0.45 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % separation_t2
% 0.29/0.45 thf(fact_8_IntI,axiom,
% 0.29/0.45 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 0.29/0.45 ( ( member @ A @ C @ A2 )
% 0.29/0.45 => ( ( member @ A @ C @ B2 )
% 0.29/0.45 => ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % IntI
% 0.29/0.45 thf(fact_9_Int__iff,axiom,
% 0.29/0.45 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 0.29/0.45 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
% 0.29/0.45 = ( ( member @ A @ C @ A2 )
% 0.29/0.45 & ( member @ A @ C @ B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Int_iff
% 0.29/0.45 thf(fact_10_inf__apply,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( semilattice_inf @ B )
% 0.29/0.45 => ( ( inf_inf @ ( A > B ) )
% 0.29/0.45 = ( ^ [F: A > B,G: A > B,X2: A] : ( inf_inf @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_apply
% 0.29/0.45 thf(fact_11_inf_Oidem,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_inf @ A )
% 0.29/0.45 => ! [A3: A] :
% 0.29/0.45 ( ( inf_inf @ A @ A3 @ A3 )
% 0.29/0.45 = A3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf.idem
% 0.29/0.45 thf(fact_12_inf__idem,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_inf @ A )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ( ( inf_inf @ A @ X @ X )
% 0.29/0.46 = X ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_idem
% 0.29/0.46 thf(fact_13_inf_Oleft__idem,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [A3: A,B3: A] :
% 0.29/0.46 ( ( inf_inf @ A @ A3 @ ( inf_inf @ A @ A3 @ B3 ) )
% 0.29/0.46 = ( inf_inf @ A @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.left_idem
% 0.29/0.46 thf(fact_14_empty__Collect__eq,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o] :
% 0.29/0.46 ( ( ( bot_bot @ ( set @ A ) )
% 0.29/0.46 = ( collect @ A @ P ) )
% 0.29/0.46 = ( ! [X2: A] :
% 0.29/0.46 ~ ( P @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % empty_Collect_eq
% 0.29/0.46 thf(fact_15_Collect__empty__eq,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o] :
% 0.29/0.46 ( ( ( collect @ A @ P )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 = ( ! [X2: A] :
% 0.29/0.46 ~ ( P @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_empty_eq
% 0.29/0.46 thf(fact_16_all__not__in__conv,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] :
% 0.29/0.46 ( ( ! [X2: A] :
% 0.29/0.46 ~ ( member @ A @ X2 @ A2 ) )
% 0.29/0.46 = ( A2
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % all_not_in_conv
% 0.29/0.46 thf(fact_17_empty__iff,axiom,
% 0.29/0.46 ! [A: $tType,C: A] :
% 0.29/0.46 ~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % empty_iff
% 0.29/0.46 thf(fact_18_inf__right__idem,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Y )
% 0.29/0.46 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_right_idem
% 0.29/0.46 thf(fact_19_inf_Oright__idem,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [A3: A,B3: A] :
% 0.29/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B3 ) @ B3 )
% 0.29/0.46 = ( inf_inf @ A @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.right_idem
% 0.29/0.46 thf(fact_20_inf__left__idem,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 0.29/0.46 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_left_idem
% 0.29/0.46 thf(fact_21__092_060open_062_092_060forall_062S_O_Aopen_AS_A_092_060and_062_Af_Ax0_A_092_060in_062_AS_A_092_060longrightarrow_062_A_I_092_060exists_062T_O_Aopen_AT_A_092_060and_062_Ax0_A_092_060in_062_AT_A_092_060and_062_A_I_092_060forall_062x_H_092_060in_062T_O_Af_Ax_H_A_092_060le_062_Af_Ax0_A_092_060longrightarrow_062_Af_Ax_H_A_092_060in_062_AS_J_J_092_060close_062,axiom,
% 0.29/0.46 ! [S2: set @ b] :
% 0.29/0.46 ( ( ( topolo1751647064n_open @ b @ S2 )
% 0.29/0.46 & ( member @ b @ ( f @ x0 ) @ S2 ) )
% 0.29/0.46 => ? [T2: set @ a] :
% 0.29/0.46 ( ( topolo1751647064n_open @ a @ T2 )
% 0.29/0.46 & ( member @ a @ x0 @ T2 )
% 0.29/0.46 & ! [X3: a] :
% 0.29/0.46 ( ( member @ a @ X3 @ T2 )
% 0.29/0.46 => ( ( ord_less_eq @ b @ ( f @ X3 ) @ ( f @ x0 ) )
% 0.29/0.46 => ( member @ b @ ( f @ X3 ) @ S2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % \<open>\<forall>S. open S \<and> f x0 \<in> S \<longrightarrow> (\<exists>T. open T \<and> x0 \<in> T \<and> (\<forall>x'\<in>T. f x' \<le> f x0 \<longrightarrow> f x' \<in> S))\<close>
% 0.29/0.46 thf(fact_22_bot__set__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( bot_bot @ ( set @ A ) )
% 0.29/0.46 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_set_def
% 0.29/0.46 thf(fact_23_ex__in__conv,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] :
% 0.29/0.46 ( ( ? [X2: A] : ( member @ A @ X2 @ A2 ) )
% 0.29/0.46 = ( A2
% 0.29/0.46 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ex_in_conv
% 0.29/0.46 thf(fact_24_equals0I,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] :
% 0.29/0.46 ( ! [Y2: A] :
% 0.29/0.46 ~ ( member @ A @ Y2 @ A2 )
% 0.29/0.46 => ( A2
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % equals0I
% 0.29/0.46 thf(fact_25_equals0D,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,A3: A] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 => ~ ( member @ A @ A3 @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % equals0D
% 0.29/0.46 thf(fact_26_emptyE,axiom,
% 0.29/0.46 ! [A: $tType,A3: A] :
% 0.29/0.46 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % emptyE
% 0.29/0.46 thf(fact_27_inf__left__commute,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [X: A,Y: A,Z: A] :
% 0.29/0.46 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 0.29/0.46 = ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_left_commute
% 0.29/0.46 thf(fact_28_inf_Oleft__commute,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [B3: A,A3: A,C: A] :
% 0.29/0.46 ( ( inf_inf @ A @ B3 @ ( inf_inf @ A @ A3 @ C ) )
% 0.29/0.46 = ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B3 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.left_commute
% 0.29/0.46 thf(fact_29_inf__commute,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ( ( inf_inf @ A )
% 0.29/0.46 = ( ^ [X2: A,Y3: A] : ( inf_inf @ A @ Y3 @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_commute
% 0.29/0.46 thf(fact_30_inf_Ocommute,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ( ( inf_inf @ A )
% 0.29/0.46 = ( ^ [A4: A,B4: A] : ( inf_inf @ A @ B4 @ A4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.commute
% 0.29/0.46 thf(fact_31_inf__assoc,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [X: A,Y: A,Z: A] :
% 0.29/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
% 0.29/0.46 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_assoc
% 0.29/0.46 thf(fact_32_inf_Oassoc,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [A3: A,B3: A,C: A] :
% 0.29/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C )
% 0.29/0.46 = ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B3 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.assoc
% 0.29/0.46 thf(fact_33_boolean__algebra__cancel_Oinf2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [B2: A,K: A,B3: A,A3: A] :
% 0.29/0.46 ( ( B2
% 0.29/0.46 = ( inf_inf @ A @ K @ B3 ) )
% 0.29/0.46 => ( ( inf_inf @ A @ A3 @ B2 )
% 0.29/0.46 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.inf2
% 0.29/0.46 thf(fact_34_boolean__algebra__cancel_Oinf1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [A2: A,K: A,A3: A,B3: A] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( inf_inf @ A @ K @ A3 ) )
% 0.29/0.46 => ( ( inf_inf @ A @ A2 @ B3 )
% 0.29/0.46 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.inf1
% 0.29/0.46 thf(fact_35_inf__fun__def,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ B )
% 0.29/0.46 => ( ( inf_inf @ ( A > B ) )
% 0.29/0.46 = ( ^ [F: A > B,G: A > B,X2: A] : ( inf_inf @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_fun_def
% 0.29/0.46 thf(fact_36_inf__sup__aci_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( lattice @ A )
% 0.29/0.46 => ( ( inf_inf @ A )
% 0.29/0.46 = ( ^ [X2: A,Y3: A] : ( inf_inf @ A @ Y3 @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(1)
% 0.29/0.46 thf(fact_37_inf__sup__aci_I2_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( lattice @ A )
% 0.29/0.46 => ! [X: A,Y: A,Z: A] :
% 0.29/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
% 0.29/0.46 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(2)
% 0.29/0.46 thf(fact_38_inf__sup__aci_I3_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( lattice @ A )
% 0.29/0.46 => ! [X: A,Y: A,Z: A] :
% 0.29/0.46 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 0.29/0.46 = ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(3)
% 0.29/0.46 thf(fact_39_inf__sup__aci_I4_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( lattice @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 0.29/0.46 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(4)
% 0.29/0.46 thf(fact_40_Int__left__commute,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 0.29/0.46 ( ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) )
% 0.29/0.46 = ( inf_inf @ ( set @ A ) @ B2 @ ( inf_inf @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_left_commute
% 0.29/0.46 thf(fact_41_Int__left__absorb,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
% 0.29/0.46 = ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_left_absorb
% 0.29/0.46 thf(fact_42_Int__commute,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( inf_inf @ ( set @ A ) )
% 0.29/0.46 = ( ^ [A5: set @ A,B5: set @ A] : ( inf_inf @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_commute
% 0.29/0.46 thf(fact_43_Int__absorb,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] :
% 0.29/0.46 ( ( inf_inf @ ( set @ A ) @ A2 @ A2 )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % Int_absorb
% 0.29/0.46 thf(fact_44_Int__assoc,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 0.29/0.46 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ C2 )
% 0.29/0.46 = ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_assoc
% 0.29/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,P: A > $o] :
% 0.29/0.46 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.29/0.46 = ( P @ A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % mem_Collect_eq
% 0.29/0.46 thf(fact_46_Collect__mem__eq,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] :
% 0.29/0.46 ( ( collect @ A
% 0.29/0.46 @ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mem_eq
% 0.29/0.46 thf(fact_47_Collect__cong,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ! [X4: A] :
% 0.29/0.46 ( ( P @ X4 )
% 0.29/0.46 = ( Q @ X4 ) )
% 0.29/0.46 => ( ( collect @ A @ P )
% 0.29/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_cong
% 0.29/0.46 thf(fact_48_ext,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
% 0.29/0.46 ( ! [X4: A] :
% 0.29/0.46 ( ( F2 @ X4 )
% 0.29/0.46 = ( G2 @ X4 ) )
% 0.29/0.46 => ( F2 = G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % ext
% 0.29/0.46 thf(fact_49_IntD2,axiom,
% 0.29/0.46 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
% 0.29/0.46 => ( member @ A @ C @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % IntD2
% 0.29/0.46 thf(fact_50_IntD1,axiom,
% 0.29/0.46 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
% 0.29/0.46 => ( member @ A @ C @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % IntD1
% 0.29/0.46 thf(fact_51_IntE,axiom,
% 0.29/0.46 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
% 0.29/0.46 => ~ ( ( member @ A @ C @ A2 )
% 0.29/0.46 => ~ ( member @ A @ C @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % IntE
% 0.29/0.46 thf(fact_52_open__discrete,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo2133971006pology @ A )
% 0.29/0.46 => ! [A2: set @ A] : ( topolo1751647064n_open @ A @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % open_discrete
% 0.29/0.46 thf(fact_53_separation__t1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topological_t1_space @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( X != Y )
% 0.29/0.46 = ( ? [U2: set @ A] :
% 0.29/0.46 ( ( topolo1751647064n_open @ A @ U2 )
% 0.29/0.46 & ( member @ A @ X @ U2 )
% 0.29/0.46 & ~ ( member @ A @ Y @ U2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % separation_t1
% 0.29/0.46 thf(fact_54_separation__t0,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topological_t0_space @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( X != Y )
% 0.29/0.46 = ( ? [U2: set @ A] :
% 0.29/0.46 ( ( topolo1751647064n_open @ A @ U2 )
% 0.29/0.46 & ( ( member @ A @ X @ U2 )
% 0.29/0.46 != ( member @ A @ Y @ U2 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % separation_t0
% 0.29/0.46 thf(fact_55_t1__space,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topological_t1_space @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( X != Y )
% 0.29/0.46 => ? [U: set @ A] :
% 0.29/0.46 ( ( topolo1751647064n_open @ A @ U )
% 0.29/0.46 & ( member @ A @ X @ U )
% 0.29/0.46 & ~ ( member @ A @ Y @ U ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % t1_space
% 0.29/0.46 thf(fact_56_t0__space,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topological_t0_space @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( X != Y )
% 0.29/0.46 => ? [U: set @ A] :
% 0.29/0.46 ( ( topolo1751647064n_open @ A @ U )
% 0.29/0.46 & ( ( member @ A @ X @ U )
% 0.29/0.46 != ( member @ A @ Y @ U ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % t0_space
% 0.29/0.46 thf(fact_57_disjoint__iff__not__equal,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 = ( ! [X2: A] :
% 0.29/0.46 ( ( member @ A @ X2 @ A2 )
% 0.29/0.46 => ! [Y3: A] :
% 0.29/0.46 ( ( member @ A @ Y3 @ B2 )
% 0.29/0.46 => ( X2 != Y3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % disjoint_iff_not_equal
% 0.29/0.46 thf(fact_58_Int__empty__right,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] :
% 0.29/0.46 ( ( inf_inf @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_empty_right
% 0.29/0.46 thf(fact_59_Int__empty__left,axiom,
% 0.29/0.46 ! [A: $tType,B2: set @ A] :
% 0.29/0.46 ( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_empty_left
% 0.29/0.46 thf(fact_60_disjoint__iff,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 = ( ! [X2: A] :
% 0.29/0.46 ( ( member @ A @ X2 @ A2 )
% 0.29/0.46 => ~ ( member @ A @ X2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % disjoint_iff
% 0.29/0.46 thf(fact_61_Int__emptyI,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ! [X4: A] :
% 0.29/0.46 ( ( member @ A @ X4 @ A2 )
% 0.29/0.46 => ~ ( member @ A @ X4 @ B2 ) )
% 0.29/0.46 => ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_emptyI
% 0.29/0.46 thf(fact_62_bot__apply,axiom,
% 0.29/0.46 ! [C3: $tType,D: $tType] :
% 0.29/0.46 ( ( bot @ C3 )
% 0.29/0.46 => ( ( bot_bot @ ( D > C3 ) )
% 0.29/0.46 = ( ^ [X2: D] : ( bot_bot @ C3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_apply
% 0.29/0.46 thf(fact_63_calculation,axiom,
% 0.29/0.46 ( ~ ! [S2: set @ b] :
% 0.29/0.46 ( ( ( topolo1751647064n_open @ b @ S2 )
% 0.29/0.46 & ( member @ b @ ( f @ x0 ) @ S2 ) )
% 0.29/0.46 => ? [T2: set @ a] :
% 0.29/0.46 ( ( topolo1751647064n_open @ a @ T2 )
% 0.29/0.46 & ( member @ a @ x0 @ T2 )
% 0.29/0.46 & ! [X3: a] :
% 0.29/0.46 ( ( member @ a @ X3 @ T2 )
% 0.29/0.46 => ( ( ord_less_eq @ b @ ( f @ X3 ) @ ( f @ x0 ) )
% 0.29/0.46 => ( member @ b @ ( f @ X3 ) @ S2 ) ) ) ) )
% 0.29/0.46 => ~ ( lower_582600101lsc_at @ a @ b @ x0 @ f ) ) ).
% 0.29/0.46
% 0.29/0.46 % calculation
% 0.29/0.46 thf(fact_64_Set_Ois__empty__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( is_empty @ A )
% 0.29/0.46 = ( ^ [A5: set @ A] :
% 0.29/0.46 ( A5
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Set.is_empty_def
% 0.29/0.46 thf(fact_65_totally__bounded__empty,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo47006728_space @ A )
% 0.29/0.46 => ( topolo406746546ounded @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % totally_bounded_empty
% 0.29/0.46 thf(fact_66_Collect__empty__eq__bot,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o] :
% 0.29/0.46 ( ( ( collect @ A @ P )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 = ( P
% 0.29/0.46 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_empty_eq_bot
% 0.29/0.46 thf(fact_67_bot__empty__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( bot_bot @ ( A > $o ) )
% 0.29/0.46 = ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_empty_eq
% 0.29/0.46 thf(fact_68_bot__fun__def,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( bot @ B )
% 0.29/0.46 => ( ( bot_bot @ ( A > B ) )
% 0.29/0.46 = ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_fun_def
% 0.29/0.46 thf(fact_69_is__singletonI_H,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 => ( ! [X4: A,Y2: A] :
% 0.29/0.46 ( ( member @ A @ X4 @ A2 )
% 0.29/0.46 => ( ( member @ A @ Y2 @ A2 )
% 0.29/0.46 => ( X4 = Y2 ) ) )
% 0.29/0.46 => ( is_singleton @ A @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % is_singletonI'
% 0.29/0.46 thf(fact_70_Diff__disjoint,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( inf_inf @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_disjoint
% 0.29/0.46 thf(fact_71_insert__disjoint_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ A2 ) @ B2 )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 = ( ~ ( member @ A @ A3 @ B2 )
% 0.29/0.46 & ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_disjoint(1)
% 0.29/0.46 thf(fact_72_insert__disjoint_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( ( bot_bot @ ( set @ A ) )
% 0.29/0.46 = ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ A2 ) @ B2 ) )
% 0.29/0.46 = ( ~ ( member @ A @ A3 @ B2 )
% 0.29/0.46 & ( ( bot_bot @ ( set @ A ) )
% 0.29/0.46 = ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_disjoint(2)
% 0.29/0.46 thf(fact_73_order__refl,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A )
% 0.29/0.46 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_refl
% 0.29/0.46 thf(fact_74_minus__apply,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( minus @ B )
% 0.29/0.46 => ( ( minus_minus @ ( A > B ) )
% 0.29/0.46 = ( ^ [A5: A > B,B5: A > B,X2: A] : ( minus_minus @ B @ ( A5 @ X2 ) @ ( B5 @ X2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minus_apply
% 0.29/0.46 thf(fact_75_insertCI,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,B2: set @ A,B3: A] :
% 0.29/0.46 ( ( ~ ( member @ A @ A3 @ B2 )
% 0.29/0.46 => ( A3 = B3 ) )
% 0.29/0.46 => ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertCI
% 0.29/0.46 thf(fact_76_insert__iff,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,B3: A,A2: set @ A] :
% 0.29/0.46 ( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
% 0.29/0.46 = ( ( A3 = B3 )
% 0.29/0.46 | ( member @ A @ A3 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_iff
% 0.29/0.46 thf(fact_77_insert__absorb2,axiom,
% 0.29/0.46 ! [A: $tType,X: A,A2: set @ A] :
% 0.29/0.46 ( ( insert @ A @ X @ ( insert @ A @ X @ A2 ) )
% 0.29/0.46 = ( insert @ A @ X @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_absorb2
% 0.29/0.46 thf(fact_78_DiffI,axiom,
% 0.29/0.46 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( member @ A @ C @ A2 )
% 0.29/0.46 => ( ~ ( member @ A @ C @ B2 )
% 0.29/0.46 => ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % DiffI
% 0.29/0.46 thf(fact_79_Diff__iff,axiom,
% 0.29/0.46 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 0.29/0.46 = ( ( member @ A @ C @ A2 )
% 0.29/0.46 & ~ ( member @ A @ C @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_iff
% 0.29/0.46 thf(fact_80_Diff__idemp,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ B2 )
% 0.29/0.46 = ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_idemp
% 0.29/0.46 thf(fact_81_le__inf__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [X: A,Y: A,Z: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 0.29/0.46 = ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 & ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_inf_iff
% 0.29/0.46 thf(fact_82_inf_Obounded__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [A3: A,B3: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B3 @ C ) )
% 0.29/0.46 = ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 & ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.bounded_iff
% 0.29/0.46 thf(fact_83_singletonI,axiom,
% 0.29/0.46 ! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % singletonI
% 0.29/0.46 thf(fact_84_Int__insert__left__if0,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,C2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ~ ( member @ A @ A3 @ C2 )
% 0.29/0.46 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C2 )
% 0.29/0.46 = ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_left_if0
% 0.29/0.46 thf(fact_85_Int__insert__left__if1,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,C2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( member @ A @ A3 @ C2 )
% 0.29/0.46 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C2 )
% 0.29/0.46 = ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_left_if1
% 0.29/0.46 thf(fact_86_insert__inter__insert,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ A2 ) @ ( insert @ A @ A3 @ B2 ) )
% 0.29/0.46 = ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_inter_insert
% 0.29/0.46 thf(fact_87_Int__insert__right__if0,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ~ ( member @ A @ A3 @ A2 )
% 0.29/0.46 => ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
% 0.29/0.46 = ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_right_if0
% 0.29/0.46 thf(fact_88_Int__insert__right__if1,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( member @ A @ A3 @ A2 )
% 0.29/0.46 => ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
% 0.29/0.46 = ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_right_if1
% 0.29/0.46 thf(fact_89_Diff__empty,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] :
% 0.29/0.46 ( ( minus_minus @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_empty
% 0.29/0.46 thf(fact_90_empty__Diff,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] :
% 0.29/0.46 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % empty_Diff
% 0.29/0.46 thf(fact_91_Diff__cancel,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] :
% 0.29/0.46 ( ( minus_minus @ ( set @ A ) @ A2 @ A2 )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_cancel
% 0.29/0.46 thf(fact_92_Diff__insert0,axiom,
% 0.29/0.46 ! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ~ ( member @ A @ X @ A2 )
% 0.29/0.46 => ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B2 ) )
% 0.29/0.46 = ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_insert0
% 0.29/0.46 thf(fact_93_insert__Diff1,axiom,
% 0.29/0.46 ! [A: $tType,X: A,B2: set @ A,A2: set @ A] :
% 0.29/0.46 ( ( member @ A @ X @ B2 )
% 0.29/0.46 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
% 0.29/0.46 = ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_Diff1
% 0.29/0.46 thf(fact_94_disjoint__insert_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B3: A,B2: set @ A] :
% 0.29/0.46 ( ( ( bot_bot @ ( set @ A ) )
% 0.29/0.46 = ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ B2 ) ) )
% 0.29/0.46 = ( ~ ( member @ A @ B3 @ A2 )
% 0.29/0.46 & ( ( bot_bot @ ( set @ A ) )
% 0.29/0.46 = ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % disjoint_insert(2)
% 0.29/0.46 thf(fact_95_disjoint__insert_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,B2: set @ A,A3: A,A2: set @ A] :
% 0.29/0.46 ( ( ( inf_inf @ ( set @ A ) @ B2 @ ( insert @ A @ A3 @ A2 ) )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 = ( ~ ( member @ A @ A3 @ B2 )
% 0.29/0.46 & ( ( inf_inf @ ( set @ A ) @ B2 @ A2 )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % disjoint_insert(1)
% 0.29/0.46 thf(fact_96_insert__Diff__single,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,A2: set @ A] :
% 0.29/0.46 ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.29/0.46 = ( insert @ A @ A3 @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_Diff_single
% 0.29/0.46 thf(fact_97_is__singletonI,axiom,
% 0.29/0.46 ! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % is_singletonI
% 0.29/0.46 thf(fact_98_DiffE,axiom,
% 0.29/0.46 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 0.29/0.46 => ~ ( ( member @ A @ C @ A2 )
% 0.29/0.46 => ( member @ A @ C @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % DiffE
% 0.29/0.46 thf(fact_99_DiffD1,axiom,
% 0.29/0.46 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 0.29/0.46 => ( member @ A @ C @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % DiffD1
% 0.29/0.46 thf(fact_100_DiffD2,axiom,
% 0.29/0.46 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 0.29/0.46 => ~ ( member @ A @ C @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % DiffD2
% 0.29/0.46 thf(fact_101_insertE,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,B3: A,A2: set @ A] :
% 0.29/0.46 ( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
% 0.29/0.46 => ( ( A3 != B3 )
% 0.29/0.46 => ( member @ A @ A3 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertE
% 0.29/0.46 thf(fact_102_insertI1,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,B2: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertI1
% 0.29/0.46 thf(fact_103_insertI2,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,B2: set @ A,B3: A] :
% 0.29/0.46 ( ( member @ A @ A3 @ B2 )
% 0.29/0.46 => ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertI2
% 0.29/0.46 thf(fact_104_Set_Oset__insert,axiom,
% 0.29/0.46 ! [A: $tType,X: A,A2: set @ A] :
% 0.29/0.46 ( ( member @ A @ X @ A2 )
% 0.29/0.46 => ~ ! [B6: set @ A] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( insert @ A @ X @ B6 ) )
% 0.29/0.46 => ( member @ A @ X @ B6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Set.set_insert
% 0.29/0.46 thf(fact_105_insert__ident,axiom,
% 0.29/0.46 ! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ~ ( member @ A @ X @ A2 )
% 0.29/0.46 => ( ~ ( member @ A @ X @ B2 )
% 0.29/0.46 => ( ( ( insert @ A @ X @ A2 )
% 0.29/0.46 = ( insert @ A @ X @ B2 ) )
% 0.29/0.46 = ( A2 = B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_ident
% 0.29/0.46 thf(fact_106_insert__absorb,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,A2: set @ A] :
% 0.29/0.46 ( ( member @ A @ A3 @ A2 )
% 0.29/0.46 => ( ( insert @ A @ A3 @ A2 )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_absorb
% 0.29/0.46 thf(fact_107_insert__eq__iff,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,A2: set @ A,B3: A,B2: set @ A] :
% 0.29/0.46 ( ~ ( member @ A @ A3 @ A2 )
% 0.29/0.46 => ( ~ ( member @ A @ B3 @ B2 )
% 0.29/0.46 => ( ( ( insert @ A @ A3 @ A2 )
% 0.29/0.46 = ( insert @ A @ B3 @ B2 ) )
% 0.29/0.46 = ( ( ( A3 = B3 )
% 0.29/0.46 => ( A2 = B2 ) )
% 0.29/0.46 & ( ( A3 != B3 )
% 0.29/0.46 => ? [C4: set @ A] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( insert @ A @ B3 @ C4 ) )
% 0.29/0.46 & ~ ( member @ A @ B3 @ C4 )
% 0.29/0.46 & ( B2
% 0.29/0.46 = ( insert @ A @ A3 @ C4 ) )
% 0.29/0.46 & ~ ( member @ A @ A3 @ C4 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_eq_iff
% 0.29/0.46 thf(fact_108_insert__Diff__if,axiom,
% 0.29/0.46 ! [A: $tType,X: A,B2: set @ A,A2: set @ A] :
% 0.29/0.46 ( ( ( member @ A @ X @ B2 )
% 0.29/0.46 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
% 0.29/0.46 = ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) )
% 0.29/0.46 & ( ~ ( member @ A @ X @ B2 )
% 0.29/0.46 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
% 0.29/0.46 = ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_Diff_if
% 0.29/0.46 thf(fact_109_insert__commute,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Y: A,A2: set @ A] :
% 0.29/0.46 ( ( insert @ A @ X @ ( insert @ A @ Y @ A2 ) )
% 0.29/0.46 = ( insert @ A @ Y @ ( insert @ A @ X @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_commute
% 0.29/0.46 thf(fact_110_fun__diff__def,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( minus @ B )
% 0.29/0.46 => ( ( minus_minus @ ( A > B ) )
% 0.29/0.46 = ( ^ [A5: A > B,B5: A > B,X2: A] : ( minus_minus @ B @ ( A5 @ X2 ) @ ( B5 @ X2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % fun_diff_def
% 0.29/0.46 thf(fact_111_mk__disjoint__insert,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,A2: set @ A] :
% 0.29/0.46 ( ( member @ A @ A3 @ A2 )
% 0.29/0.46 => ? [B6: set @ A] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( insert @ A @ A3 @ B6 ) )
% 0.29/0.46 & ~ ( member @ A @ A3 @ B6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mk_disjoint_insert
% 0.29/0.46 thf(fact_112_dual__order_Oantisym,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [B3: A,A3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( A3 = B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.antisym
% 0.29/0.46 thf(fact_113_dual__order_Oeq__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.29/0.46 = ( ^ [A4: A,B4: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.29/0.46 & ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.eq_iff
% 0.29/0.46 thf(fact_114_dual__order_Otrans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [B3: A,A3: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ C @ B3 )
% 0.29/0.46 => ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.trans
% 0.29/0.46 thf(fact_115_linorder__wlog,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [P: A > A > $o,A3: A,B3: A] :
% 0.29/0.46 ( ! [A6: A,B7: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A6 @ B7 )
% 0.29/0.46 => ( P @ A6 @ B7 ) )
% 0.29/0.46 => ( ! [A6: A,B7: A] :
% 0.29/0.46 ( ( P @ B7 @ A6 )
% 0.29/0.46 => ( P @ A6 @ B7 ) )
% 0.29/0.46 => ( P @ A3 @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_wlog
% 0.29/0.46 thf(fact_116_dual__order_Orefl,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.refl
% 0.29/0.46 thf(fact_117_order__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A )
% 0.29/0.46 => ! [X: A,Y: A,Z: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.29/0.46 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_trans
% 0.29/0.46 thf(fact_118_order__class_Oorder_Oantisym,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A3: A,B3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.29/0.46 => ( A3 = B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_class.order.antisym
% 0.29/0.46 thf(fact_119_ord__le__eq__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A )
% 0.29/0.46 => ! [A3: A,B3: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( ( B3 = C )
% 0.29/0.46 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_trans
% 0.29/0.46 thf(fact_120_ord__eq__le__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A )
% 0.29/0.46 => ! [A3: A,B3: A,C: A] :
% 0.29/0.46 ( ( A3 = B3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B3 @ C )
% 0.29/0.46 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_trans
% 0.29/0.46 thf(fact_121_order__class_Oorder_Oeq__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.29/0.46 = ( ^ [A4: A,B4: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.29/0.46 & ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_class.order.eq_iff
% 0.29/0.46 thf(fact_122_antisym__conv,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [Y: A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 = ( X = Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv
% 0.29/0.46 thf(fact_123_le__cases3,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [X: A,Y: A,Z: A] :
% 0.29/0.46 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ X @ Z )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 0.29/0.46 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_cases3
% 0.29/0.46 thf(fact_124_order_Otrans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A3: A,B3: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B3 @ C )
% 0.29/0.46 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.trans
% 0.29/0.46 thf(fact_125_le__cases,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_cases
% 0.29/0.46 thf(fact_126_eq__refl,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( X = Y )
% 0.29/0.46 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_refl
% 0.29/0.46 thf(fact_127_linear,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linear
% 0.29/0.46 thf(fact_128_antisym,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ( X = Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym
% 0.29/0.46 thf(fact_129_eq__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.29/0.46 = ( ^ [X2: A,Y3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.29/0.46 & ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_iff
% 0.29/0.46 thf(fact_130_ord__le__eq__subst,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType] :
% 0.29/0.46 ( ( ( ord @ B )
% 0.29/0.46 & ( ord @ A ) )
% 0.29/0.46 => ! [A3: A,B3: A,F2: A > B,C: B] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( ( ( F2 @ B3 )
% 0.29/0.46 = C )
% 0.29/0.46 => ( ! [X4: A,Y2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.29/0.46 => ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) )
% 0.29/0.46 => ( ord_less_eq @ B @ ( F2 @ A3 ) @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_subst
% 0.29/0.46 thf(fact_131_ord__eq__le__subst,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType] :
% 0.29/0.46 ( ( ( ord @ B )
% 0.29/0.46 & ( ord @ A ) )
% 0.29/0.46 => ! [A3: A,F2: B > A,B3: B,C: B] :
% 0.29/0.46 ( ( A3
% 0.29/0.46 = ( F2 @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq @ B @ B3 @ C )
% 0.29/0.46 => ( ! [X4: B,Y2: B] :
% 0.29/0.46 ( ( ord_less_eq @ B @ X4 @ Y2 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) )
% 0.29/0.46 => ( ord_less_eq @ A @ A3 @ ( F2 @ C ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_subst
% 0.29/0.46 thf(fact_132_order__subst2,axiom,
% 0.29/0.46 ! [A: $tType,C3: $tType] :
% 0.29/0.46 ( ( ( order @ C3 )
% 0.29/0.46 & ( order @ A ) )
% 0.29/0.46 => ! [A3: A,B3: A,F2: A > C3,C: C3] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq @ C3 @ ( F2 @ B3 ) @ C )
% 0.29/0.46 => ( ! [X4: A,Y2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.29/0.46 => ( ord_less_eq @ C3 @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) )
% 0.29/0.46 => ( ord_less_eq @ C3 @ ( F2 @ A3 ) @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst2
% 0.29/0.46 thf(fact_133_order__subst1,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType] :
% 0.29/0.46 ( ( ( order @ B )
% 0.29/0.46 & ( order @ A ) )
% 0.29/0.46 => ! [A3: A,F2: B > A,B3: B,C: B] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ ( F2 @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq @ B @ B3 @ C )
% 0.29/0.46 => ( ! [X4: B,Y2: B] :
% 0.29/0.46 ( ( ord_less_eq @ B @ X4 @ Y2 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) )
% 0.29/0.46 => ( ord_less_eq @ A @ A3 @ ( F2 @ C ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst1
% 0.29/0.46 thf(fact_134_le__fun__def,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B )
% 0.29/0.46 => ( ( ord_less_eq @ ( A > B ) )
% 0.29/0.46 = ( ^ [F: A > B,G: A > B] :
% 0.29/0.46 ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_fun_def
% 0.29/0.46 thf(fact_135_le__funI,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B )
% 0.29/0.46 => ! [F2: A > B,G2: A > B] :
% 0.29/0.46 ( ! [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G2 @ X4 ) )
% 0.29/0.46 => ( ord_less_eq @ ( A > B ) @ F2 @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_funI
% 0.29/0.46 thf(fact_136_le__funE,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B )
% 0.29/0.46 => ! [F2: A > B,G2: A > B,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 0.29/0.46 => ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_funE
% 0.29/0.46 thf(fact_137_le__funD,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B )
% 0.29/0.46 => ! [F2: A > B,G2: A > B,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 0.29/0.46 => ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_funD
% 0.29/0.46 thf(fact_138_Diff__insert__absorb,axiom,
% 0.29/0.46 ! [A: $tType,X: A,A2: set @ A] :
% 0.29/0.46 ( ~ ( member @ A @ X @ A2 )
% 0.29/0.46 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_insert_absorb
% 0.29/0.46 thf(fact_139_Diff__insert2,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,A3: A,B2: set @ A] :
% 0.29/0.46 ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
% 0.29/0.46 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_insert2
% 0.29/0.46 thf(fact_140_insert__Diff,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,A2: set @ A] :
% 0.29/0.46 ( ( member @ A @ A3 @ A2 )
% 0.29/0.46 => ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_Diff
% 0.29/0.46 thf(fact_141_Diff__insert,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,A3: A,B2: set @ A] :
% 0.29/0.46 ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
% 0.29/0.46 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_insert
% 0.29/0.46 thf(fact_142_bot_Oextremum,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order_bot @ A )
% 0.29/0.46 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum
% 0.29/0.46 thf(fact_143_bot_Oextremum__unique,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order_bot @ A )
% 0.29/0.46 => ! [A3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.29/0.46 = ( A3
% 0.29/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum_unique
% 0.29/0.46 thf(fact_144_bot_Oextremum__uniqueI,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order_bot @ A )
% 0.29/0.46 => ! [A3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.29/0.46 => ( A3
% 0.29/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum_uniqueI
% 0.29/0.46 thf(fact_145_is__singleton__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( is_singleton @ A )
% 0.29/0.46 = ( ^ [A5: set @ A] :
% 0.29/0.46 ? [X2: A] :
% 0.29/0.46 ( A5
% 0.29/0.46 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % is_singleton_def
% 0.29/0.46 thf(fact_146_is__singletonE,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] :
% 0.29/0.46 ( ( is_singleton @ A @ A2 )
% 0.29/0.46 => ~ ! [X4: A] :
% 0.29/0.46 ( A2
% 0.29/0.46 != ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % is_singletonE
% 0.29/0.46 thf(fact_147_singletonD,axiom,
% 0.29/0.46 ! [A: $tType,B3: A,A3: A] :
% 0.29/0.46 ( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.29/0.46 => ( B3 = A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % singletonD
% 0.29/0.46 thf(fact_148_singleton__iff,axiom,
% 0.29/0.46 ! [A: $tType,B3: A,A3: A] :
% 0.29/0.46 ( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.29/0.46 = ( B3 = A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % singleton_iff
% 0.29/0.46 thf(fact_149_doubleton__eq__iff,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,B3: A,C: A,D2: A] :
% 0.29/0.46 ( ( ( insert @ A @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.29/0.46 = ( insert @ A @ C @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.29/0.46 = ( ( ( A3 = C )
% 0.29/0.46 & ( B3 = D2 ) )
% 0.29/0.46 | ( ( A3 = D2 )
% 0.29/0.46 & ( B3 = C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % doubleton_eq_iff
% 0.29/0.46 thf(fact_150_insert__not__empty,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,A2: set @ A] :
% 0.29/0.46 ( ( insert @ A @ A3 @ A2 )
% 0.29/0.46 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_not_empty
% 0.29/0.46 thf(fact_151_singleton__inject,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,B3: A] :
% 0.29/0.46 ( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.29/0.46 => ( A3 = B3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % singleton_inject
% 0.29/0.46 thf(fact_152_Int__insert__left,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,C2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( ( member @ A @ A3 @ C2 )
% 0.29/0.46 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C2 )
% 0.29/0.46 = ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) )
% 0.29/0.46 & ( ~ ( member @ A @ A3 @ C2 )
% 0.29/0.46 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C2 )
% 0.29/0.46 = ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_left
% 0.29/0.46 thf(fact_153_Int__insert__right,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( ( member @ A @ A3 @ A2 )
% 0.29/0.46 => ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
% 0.29/0.46 = ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) )
% 0.29/0.46 & ( ~ ( member @ A @ A3 @ A2 )
% 0.29/0.46 => ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
% 0.29/0.46 = ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_right
% 0.29/0.46 thf(fact_154_Int__Diff,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 0.29/0.46 ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ C2 )
% 0.29/0.46 = ( inf_inf @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_Diff
% 0.29/0.46 thf(fact_155_Diff__Int2,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,C2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ C2 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) )
% 0.29/0.46 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ C2 ) @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_Int2
% 0.29/0.46 thf(fact_156_Diff__Diff__Int,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( minus_minus @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 0.29/0.46 = ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_Diff_Int
% 0.29/0.46 thf(fact_157_Diff__Int__distrib,axiom,
% 0.29/0.46 ! [A: $tType,C2: set @ A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( inf_inf @ ( set @ A ) @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 0.29/0.46 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C2 @ A2 ) @ ( inf_inf @ ( set @ A ) @ C2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_Int_distrib
% 0.29/0.46 thf(fact_158_Diff__Int__distrib2,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 0.29/0.46 ( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ C2 )
% 0.29/0.46 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ C2 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_Int_distrib2
% 0.29/0.46 thf(fact_159_inf__sup__ord_I2_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( lattice @ A )
% 0.29/0.46 => ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_ord(2)
% 0.29/0.46 thf(fact_160_inf__sup__ord_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( lattice @ A )
% 0.29/0.46 => ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_ord(1)
% 0.29/0.46 thf(fact_161_inf__le1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_le1
% 0.29/0.46 thf(fact_162_inf__le2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_le2
% 0.29/0.46 thf(fact_163_le__infE,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [X: A,A3: A,B3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B3 ) )
% 0.29/0.46 => ~ ( ( ord_less_eq @ A @ X @ A3 )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ X @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_infE
% 0.29/0.46 thf(fact_164_le__infI,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [X: A,A3: A,B3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ A3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ X @ B3 )
% 0.29/0.46 => ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_infI
% 0.29/0.46 thf(fact_165_inf__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [A3: A,C: A,B3: A,D2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ C )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B3 @ D2 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ ( inf_inf @ A @ C @ D2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_mono
% 0.29/0.46 thf(fact_166_le__infI1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [A3: A,X: A,B3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ X )
% 0.29/0.46 => ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_infI1
% 0.29/0.46 thf(fact_167_le__infI2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [B3: A,X: A,A3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B3 @ X )
% 0.29/0.46 => ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_infI2
% 0.29/0.46 thf(fact_168_inf_OorderE,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [A3: A,B3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( A3
% 0.29/0.46 = ( inf_inf @ A @ A3 @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.orderE
% 0.29/0.46 thf(fact_169_inf_OorderI,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [A3: A,B3: A] :
% 0.29/0.46 ( ( A3
% 0.29/0.46 = ( inf_inf @ A @ A3 @ B3 ) )
% 0.29/0.46 => ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.orderI
% 0.29/0.46 thf(fact_170_inf__unique,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [F2: A > A > A,X: A,Y: A] :
% 0.29/0.46 ( ! [X4: A,Y2: A] : ( ord_less_eq @ A @ ( F2 @ X4 @ Y2 ) @ X4 )
% 0.29/0.46 => ( ! [X4: A,Y2: A] : ( ord_less_eq @ A @ ( F2 @ X4 @ Y2 ) @ Y2 )
% 0.29/0.46 => ( ! [X4: A,Y2: A,Z3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ X4 @ Z3 )
% 0.29/0.46 => ( ord_less_eq @ A @ X4 @ ( F2 @ Y2 @ Z3 ) ) ) )
% 0.29/0.46 => ( ( inf_inf @ A @ X @ Y )
% 0.29/0.46 = ( F2 @ X @ Y ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_unique
% 0.29/0.46 thf(fact_171_le__iff__inf,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ( ( ord_less_eq @ A )
% 0.29/0.46 = ( ^ [X2: A,Y3: A] :
% 0.29/0.46 ( ( inf_inf @ A @ X2 @ Y3 )
% 0.29/0.46 = X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_iff_inf
% 0.29/0.46 thf(fact_172_inf_Oabsorb1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [A3: A,B3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( ( inf_inf @ A @ A3 @ B3 )
% 0.29/0.46 = A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.absorb1
% 0.29/0.46 thf(fact_173_inf_Oabsorb2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [B3: A,A3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.29/0.46 => ( ( inf_inf @ A @ A3 @ B3 )
% 0.29/0.46 = B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.absorb2
% 0.29/0.46 thf(fact_174_inf__absorb1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ( inf_inf @ A @ X @ Y )
% 0.29/0.46 = X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_absorb1
% 0.29/0.46 thf(fact_175_inf__absorb2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [Y: A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ( ( inf_inf @ A @ X @ Y )
% 0.29/0.46 = Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_absorb2
% 0.29/0.46 thf(fact_176_inf_OboundedE,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [A3: A,B3: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B3 @ C ) )
% 0.29/0.46 => ~ ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.boundedE
% 0.29/0.46 thf(fact_177_inf_OboundedI,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [A3: A,B3: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ A3 @ C )
% 0.29/0.46 => ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B3 @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.boundedI
% 0.29/0.46 thf(fact_178_inf__greatest,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [X: A,Y: A,Z: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ( ord_less_eq @ A @ X @ Z )
% 0.29/0.46 => ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_greatest
% 0.29/0.46 thf(fact_179_inf_Oorder__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ( ( ord_less_eq @ A )
% 0.29/0.46 = ( ^ [A4: A,B4: A] :
% 0.29/0.46 ( A4
% 0.29/0.46 = ( inf_inf @ A @ A4 @ B4 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.order_iff
% 0.29/0.46 thf(fact_180_inf_Ocobounded1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.cobounded1
% 0.29/0.46 thf(fact_181_inf_Ocobounded2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ B3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.cobounded2
% 0.29/0.46 thf(fact_182_inf_Oabsorb__iff1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ( ( ord_less_eq @ A )
% 0.29/0.46 = ( ^ [A4: A,B4: A] :
% 0.29/0.46 ( ( inf_inf @ A @ A4 @ B4 )
% 0.29/0.46 = A4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.absorb_iff1
% 0.29/0.46 thf(fact_183_inf_Oabsorb__iff2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ( ( ord_less_eq @ A )
% 0.29/0.46 = ( ^ [B4: A,A4: A] :
% 0.29/0.46 ( ( inf_inf @ A @ A4 @ B4 )
% 0.29/0.46 = B4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.absorb_iff2
% 0.29/0.46 thf(fact_184_inf_OcoboundedI1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [A3: A,C: A,B3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ C )
% 0.29/0.46 => ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.coboundedI1
% 0.29/0.46 thf(fact_185_inf_OcoboundedI2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A )
% 0.29/0.46 => ! [B3: A,C: A,A3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B3 @ C )
% 0.29/0.46 => ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.coboundedI2
% 0.29/0.46 thf(fact_186_not__open__singleton,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo890362671_space @ A )
% 0.29/0.46 => ! [X: A] :
% 0.29/0.46 ~ ( topolo1751647064n_open @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_open_singleton
% 0.29/0.46 thf(fact_187_Diff__triv,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 => ( ( minus_minus @ ( set @ A ) @ A2 @ B2 )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_triv
% 0.29/0.46 thf(fact_188_open__delete,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topological_t1_space @ A )
% 0.29/0.46 => ! [S3: set @ A,X: A] :
% 0.29/0.46 ( ( topolo1751647064n_open @ A @ S3 )
% 0.29/0.46 => ( topolo1751647064n_open @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % open_delete
% 0.29/0.46 thf(fact_189_Int__Diff__disjoint,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_Diff_disjoint
% 0.29/0.46 thf(fact_190_is__singleton__the__elem,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( is_singleton @ A )
% 0.29/0.46 = ( ^ [A5: set @ A] :
% 0.29/0.46 ( A5
% 0.29/0.46 = ( insert @ A @ ( the_elem @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % is_singleton_the_elem
% 0.29/0.46 thf(fact_191_ball__insert,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,B2: set @ A,P: A > $o] :
% 0.29/0.46 ( ( ! [X2: A] :
% 0.29/0.46 ( ( member @ A @ X2 @ ( insert @ A @ A3 @ B2 ) )
% 0.29/0.46 => ( P @ X2 ) ) )
% 0.29/0.46 = ( ( P @ A3 )
% 0.29/0.46 & ! [X2: A] :
% 0.29/0.46 ( ( member @ A @ X2 @ B2 )
% 0.29/0.46 => ( P @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ball_insert
% 0.29/0.46 thf(fact_192_the__elem__eq,axiom,
% 0.29/0.46 ! [A: $tType,X: A] :
% 0.29/0.46 ( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % the_elem_eq
% 0.29/0.46 thf(fact_193_diff__eq__diff__less__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [A3: A,B3: A,C: A,D2: A] :
% 0.29/0.46 ( ( ( minus_minus @ A @ A3 @ B3 )
% 0.29/0.46 = ( minus_minus @ A @ C @ D2 ) )
% 0.29/0.46 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 = ( ord_less_eq @ A @ C @ D2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_eq_diff_less_eq
% 0.29/0.46 thf(fact_194_empty__subsetI,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% 0.29/0.46
% 0.29/0.46 % empty_subsetI
% 0.29/0.46 thf(fact_195_subset__empty,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 = ( A2
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_empty
% 0.29/0.46 thf(fact_196_insert__subset,axiom,
% 0.29/0.46 ! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
% 0.29/0.46 = ( ( member @ A @ X @ B2 )
% 0.29/0.46 & ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_subset
% 0.29/0.46 thf(fact_197_Int__subset__iff,axiom,
% 0.29/0.46 ! [A: $tType,C2: set @ A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ C2 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
% 0.29/0.46 = ( ( ord_less_eq @ ( set @ A ) @ C2 @ A2 )
% 0.29/0.46 & ( ord_less_eq @ ( set @ A ) @ C2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_subset_iff
% 0.29/0.46 thf(fact_198_singleton__insert__inj__eq,axiom,
% 0.29/0.46 ! [A: $tType,B3: A,A3: A,A2: set @ A] :
% 0.29/0.46 ( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 = ( insert @ A @ A3 @ A2 ) )
% 0.29/0.46 = ( ( A3 = B3 )
% 0.29/0.46 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % singleton_insert_inj_eq
% 0.29/0.46 thf(fact_199_singleton__insert__inj__eq_H,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,A2: set @ A,B3: A] :
% 0.29/0.46 ( ( ( insert @ A @ A3 @ A2 )
% 0.29/0.46 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.29/0.46 = ( ( A3 = B3 )
% 0.29/0.46 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % singleton_insert_inj_eq'
% 0.29/0.46 thf(fact_200_Diff__eq__empty__iff,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( ( minus_minus @ ( set @ A ) @ A2 @ B2 )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_eq_empty_iff
% 0.29/0.46 thf(fact_201_subset__insertI2,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,B3: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_insertI2
% 0.29/0.46 thf(fact_202_subset__insertI,axiom,
% 0.29/0.46 ! [A: $tType,B2: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( insert @ A @ A3 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_insertI
% 0.29/0.46 thf(fact_203_subset__insert,axiom,
% 0.29/0.46 ! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ~ ( member @ A @ X @ A2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B2 ) )
% 0.29/0.46 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_insert
% 0.29/0.46 thf(fact_204_insert__mono,axiom,
% 0.29/0.46 ! [A: $tType,C2: set @ A,D3: set @ A,A3: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ C2 @ D3 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C2 ) @ ( insert @ A @ A3 @ D3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_mono
% 0.29/0.46 thf(fact_205_Int__Collect__mono,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.29/0.46 => ( ! [X4: A] :
% 0.29/0.46 ( ( member @ A @ X4 @ A2 )
% 0.29/0.46 => ( ( P @ X4 )
% 0.29/0.46 => ( Q @ X4 ) ) )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B2 @ ( collect @ A @ Q ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_Collect_mono
% 0.29/0.46 thf(fact_206_Int__greatest,axiom,
% 0.29/0.46 ! [A: $tType,C2: set @ A,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ C2 @ A2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ C2 @ B2 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ C2 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_greatest
% 0.29/0.46 thf(fact_207_Int__absorb2,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.29/0.46 => ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_absorb2
% 0.29/0.46 thf(fact_208_Int__absorb1,axiom,
% 0.29/0.46 ! [A: $tType,B2: set @ A,A2: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.29/0.46 => ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
% 0.29/0.46 = B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_absorb1
% 0.29/0.46 thf(fact_209_Int__lower2,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ B2 ) ).
% 0.29/0.46
% 0.29/0.46 % Int_lower2
% 0.29/0.46 thf(fact_210_Int__lower1,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ A2 ) ).
% 0.29/0.46
% 0.29/0.46 % Int_lower1
% 0.29/0.46 thf(fact_211_Int__mono,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,C2: set @ A,B2: set @ A,D3: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ D3 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ ( inf_inf @ ( set @ A ) @ C2 @ D3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_mono
% 0.29/0.46 thf(fact_212_first__countable__basis,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo2135403230pology @ A )
% 0.29/0.46 => ! [X: A] :
% 0.29/0.46 ? [A7: nat > ( set @ A )] :
% 0.29/0.46 ( ! [I: nat] :
% 0.29/0.46 ( ( member @ A @ X @ ( A7 @ I ) )
% 0.29/0.46 & ( topolo1751647064n_open @ A @ ( A7 @ I ) ) )
% 0.29/0.46 & ! [S2: set @ A] :
% 0.29/0.46 ( ( ( topolo1751647064n_open @ A @ S2 )
% 0.29/0.46 & ( member @ A @ X @ S2 ) )
% 0.29/0.46 => ? [I2: nat] : ( ord_less_eq @ ( set @ A ) @ ( A7 @ I2 ) @ S2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % first_countable_basis
% 0.29/0.46 thf(fact_213_open__subopen,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo503727757_space @ A )
% 0.29/0.46 => ( ( topolo1751647064n_open @ A )
% 0.29/0.46 = ( ^ [S4: set @ A] :
% 0.29/0.46 ! [X2: A] :
% 0.29/0.46 ( ( member @ A @ X2 @ S4 )
% 0.29/0.46 => ? [T3: set @ A] :
% 0.29/0.46 ( ( topolo1751647064n_open @ A @ T3 )
% 0.29/0.46 & ( member @ A @ X2 @ T3 )
% 0.29/0.46 & ( ord_less_eq @ ( set @ A ) @ T3 @ S4 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % open_subopen
% 0.29/0.46 thf(fact_214_topological__space__class_OopenI,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo503727757_space @ A )
% 0.29/0.46 => ! [S: set @ A] :
% 0.29/0.46 ( ! [X4: A] :
% 0.29/0.46 ( ( member @ A @ X4 @ S )
% 0.29/0.46 => ? [T4: set @ A] :
% 0.29/0.46 ( ( topolo1751647064n_open @ A @ T4 )
% 0.29/0.46 & ( member @ A @ X4 @ T4 )
% 0.29/0.46 & ( ord_less_eq @ ( set @ A ) @ T4 @ S ) ) )
% 0.29/0.46 => ( topolo1751647064n_open @ A @ S ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % topological_space_class.openI
% 0.29/0.46 thf(fact_215_double__diff,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
% 0.29/0.46 => ( ( minus_minus @ ( set @ A ) @ B2 @ ( minus_minus @ ( set @ A ) @ C2 @ A2 ) )
% 0.29/0.46 = A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % double_diff
% 0.29/0.46 thf(fact_216_Diff__subset,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ A2 ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_subset
% 0.29/0.46 thf(fact_217_Diff__mono,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,C2: set @ A,D3: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ D3 @ B2 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( minus_minus @ ( set @ A ) @ C2 @ D3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_mono
% 0.29/0.46 thf(fact_218_totally__bounded__subset,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo47006728_space @ A )
% 0.29/0.46 => ! [S: set @ A,T: set @ A] :
% 0.29/0.46 ( ( topolo406746546ounded @ A @ S )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ T @ S )
% 0.29/0.46 => ( topolo406746546ounded @ A @ T ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % totally_bounded_subset
% 0.29/0.46 thf(fact_219_subset__singletonD,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.29/0.46 => ( ( A2
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 | ( A2
% 0.29/0.46 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_singletonD
% 0.29/0.46 thf(fact_220_subset__singleton__iff,axiom,
% 0.29/0.46 ! [A: $tType,X5: set @ A,A3: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.29/0.46 = ( ( X5
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 | ( X5
% 0.29/0.46 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_singleton_iff
% 0.29/0.46 thf(fact_221_subset__Diff__insert,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,X: A,C2: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ ( insert @ A @ X @ C2 ) ) )
% 0.29/0.46 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ C2 ) )
% 0.29/0.46 & ~ ( member @ A @ X @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_Diff_insert
% 0.29/0.46 thf(fact_222_diff__eq__diff__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A3: A,B3: A,C: A,D2: A] :
% 0.29/0.46 ( ( ( minus_minus @ A @ A3 @ B3 )
% 0.29/0.46 = ( minus_minus @ A @ C @ D2 ) )
% 0.29/0.46 => ( ( A3 = B3 )
% 0.29/0.46 = ( C = D2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_eq_diff_eq
% 0.29/0.46 thf(fact_223_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( cancel146912293up_add @ A )
% 0.29/0.46 => ! [A3: A,C: A,B3: A] :
% 0.29/0.46 ( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C ) @ B3 )
% 0.29/0.46 = ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % cancel_ab_semigroup_add_class.diff_right_commute
% 0.29/0.46 thf(fact_224_Diff__single__insert,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,X: A,B2: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Diff_single_insert
% 0.29/0.46 thf(fact_225_subset__insert__iff,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,X: A,B2: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B2 ) )
% 0.29/0.46 = ( ( ( member @ A @ X @ A2 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) )
% 0.29/0.46 & ( ~ ( member @ A @ X @ A2 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_insert_iff
% 0.29/0.46 thf(fact_226_diff__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [A3: A,B3: A,D2: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ D2 @ C )
% 0.29/0.46 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ D2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_mono
% 0.29/0.46 thf(fact_227_diff__left__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [B3: A,A3: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A3 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_left_mono
% 0.29/0.46 thf(fact_228_diff__right__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [A3: A,B3: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_right_mono
% 0.29/0.46 thf(fact_229_remove__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( remove @ A )
% 0.29/0.46 = ( ^ [X2: A,A5: set @ A] : ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % remove_def
% 0.29/0.46 thf(fact_230_at__within__nhd,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo503727757_space @ A )
% 0.29/0.46 => ! [X: A,S: set @ A,T: set @ A,U3: set @ A] :
% 0.29/0.46 ( ( member @ A @ X @ S )
% 0.29/0.46 => ( ( topolo1751647064n_open @ A @ S )
% 0.29/0.46 => ( ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ T @ S ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.29/0.46 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ U3 @ S ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.29/0.46 => ( ( topolo507301023within @ A @ X @ T )
% 0.29/0.46 = ( topolo507301023within @ A @ X @ U3 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % at_within_nhd
% 0.29/0.46 thf(fact_231_subsetI,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ! [X4: A] :
% 0.29/0.46 ( ( member @ A @ X4 @ A2 )
% 0.29/0.46 => ( member @ A @ X4 @ B2 ) )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % subsetI
% 0.29/0.46 thf(fact_232_subset__antisym,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.29/0.46 => ( A2 = B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_antisym
% 0.29/0.46 thf(fact_233_member__remove,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Y: A,A2: set @ A] :
% 0.29/0.46 ( ( member @ A @ X @ ( remove @ A @ Y @ A2 ) )
% 0.29/0.46 = ( ( member @ A @ X @ A2 )
% 0.29/0.46 & ( X != Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % member_remove
% 0.29/0.46 thf(fact_234_at__within__empty,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo503727757_space @ A )
% 0.29/0.46 => ! [A3: A] :
% 0.29/0.46 ( ( topolo507301023within @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 = ( bot_bot @ ( filter @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % at_within_empty
% 0.29/0.46 thf(fact_235_in__mono,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.29/0.46 => ( ( member @ A @ X @ A2 )
% 0.29/0.46 => ( member @ A @ X @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % in_mono
% 0.29/0.46 thf(fact_236_subsetD,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.29/0.46 => ( ( member @ A @ C @ A2 )
% 0.29/0.46 => ( member @ A @ C @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subsetD
% 0.29/0.46 thf(fact_237_equalityE,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( A2 = B2 )
% 0.29/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.29/0.46 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityE
% 0.29/0.46 thf(fact_238_subset__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.46 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.29/0.46 ! [X2: A] :
% 0.29/0.46 ( ( member @ A @ X2 @ A5 )
% 0.29/0.46 => ( member @ A @ X2 @ B5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_eq
% 0.29/0.46 thf(fact_239_equalityD1,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( A2 = B2 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityD1
% 0.29/0.46 thf(fact_240_equalityD2,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( A2 = B2 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityD2
% 0.29/0.46 thf(fact_241_subset__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.46 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.29/0.46 ! [T5: A] :
% 0.29/0.46 ( ( member @ A @ T5 @ A5 )
% 0.29/0.46 => ( member @ A @ T5 @ B5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_iff
% 0.29/0.46 thf(fact_242_subset__refl,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 0.29/0.46
% 0.29/0.46 % subset_refl
% 0.29/0.46 thf(fact_243_Collect__mono,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ! [X4: A] :
% 0.29/0.46 ( ( P @ X4 )
% 0.29/0.46 => ( Q @ X4 ) )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mono
% 0.29/0.46 thf(fact_244_subset__trans,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_trans
% 0.29/0.46 thf(fact_245_set__eq__subset,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ^ [Y4: set @ A,Z2: set @ A] : ( Y4 = Z2 ) )
% 0.29/0.46 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.29/0.46 & ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % set_eq_subset
% 0.29/0.46 thf(fact_246_Collect__mono__iff,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.29/0.46 = ( ! [X2: A] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 => ( Q @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mono_iff
% 0.29/0.46 thf(fact_247_at__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo503727757_space @ A )
% 0.29/0.46 => ! [S3: set @ A,T6: set @ A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ S3 @ T6 )
% 0.29/0.46 => ( ord_less_eq @ ( filter @ A ) @ ( topolo507301023within @ A @ X @ S3 ) @ ( topolo507301023within @ A @ X @ T6 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % at_le
% 0.29/0.46 thf(fact_248_at__discrete,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo2133971006pology @ A )
% 0.29/0.46 => ( ( topolo507301023within @ A )
% 0.29/0.46 = ( ^ [X2: A,S4: set @ A] : ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % at_discrete
% 0.29/0.46 thf(fact_249_insert__subsetI,axiom,
% 0.29/0.46 ! [A: $tType,X: A,A2: set @ A,X5: set @ A] :
% 0.29/0.46 ( ( member @ A @ X @ A2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ X5 @ A2 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X5 ) @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_subsetI
% 0.29/0.46 thf(fact_250_subset__emptyI,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] :
% 0.29/0.46 ( ! [X4: A] :
% 0.29/0.46 ~ ( member @ A @ X4 @ A2 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_emptyI
% 0.29/0.46 thf(fact_251_at__eq__bot__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo503727757_space @ A )
% 0.29/0.46 => ! [A3: A] :
% 0.29/0.46 ( ( ( topolo507301023within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
% 0.29/0.46 = ( bot_bot @ ( filter @ A ) ) )
% 0.29/0.46 = ( topolo1751647064n_open @ A @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % at_eq_bot_iff
% 0.29/0.46 thf(fact_252_at__within__eq__bot__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo503727757_space @ A )
% 0.29/0.46 => ! [C: A,A2: set @ A] :
% 0.29/0.46 ( ( ( topolo507301023within @ A @ C @ A2 )
% 0.29/0.46 = ( bot_bot @ ( filter @ A ) ) )
% 0.29/0.46 = ( ~ ( member @ A @ C @ ( elementary_closure @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % at_within_eq_bot_iff
% 0.29/0.46 thf(fact_253_top__apply,axiom,
% 0.29/0.46 ! [C3: $tType,D: $tType] :
% 0.29/0.46 ( ( top @ C3 )
% 0.29/0.46 => ( ( top_top @ ( D > C3 ) )
% 0.29/0.46 = ( ^ [X2: D] : ( top_top @ C3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % top_apply
% 0.29/0.46 thf(fact_254_UNIV__I,axiom,
% 0.29/0.46 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % UNIV_I
% 0.29/0.46 thf(fact_255_closure__closure,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo503727757_space @ A )
% 0.29/0.46 => ! [S: set @ A] :
% 0.29/0.46 ( ( elementary_closure @ A @ ( elementary_closure @ A @ S ) )
% 0.29/0.46 = ( elementary_closure @ A @ S ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % closure_closure
% 0.29/0.46
% 0.29/0.46 % Subclasses (23)
% 0.29/0.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Lattices_Obounded__lattice,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comple1035589618norder @ A )
% 0.29/0.46 => ( bounded_lattice @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___HOL_Otype,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comple1035589618norder @ A )
% 0.29/0.46 => ( type @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Obot,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comple1035589618norder @ A )
% 0.29/0.46 => ( bot @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Oord,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comple1035589618norder @ A )
% 0.29/0.46 => ( ord @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Otop,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comple1035589618norder @ A )
% 0.29/0.46 => ( top @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Oorder,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comple1035589618norder @ A )
% 0.29/0.46 => ( order @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Lattices_Olattice,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comple1035589618norder @ A )
% 0.29/0.46 => ( lattice @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Olinorder,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comple1035589618norder @ A )
% 0.29/0.46 => ( linorder @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Opreorder,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comple1035589618norder @ A )
% 0.29/0.46 => ( preorder @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Oorder__bot,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comple1035589618norder @ A )
% 0.29/0.46 => ( order_bot @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Lattices_Osemilattice__inf,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comple1035589618norder @ A )
% 0.29/0.46 => ( semilattice_inf @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Lattices_Obounded__lattice__bot,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comple1035589618norder @ A )
% 0.29/0.46 => ( bounded_lattice_bot @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Topological__Spaces_Olinorder__topology___HOL_Otype,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo2117631714pology @ A )
% 0.29/0.46 => ( type @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Topological__Spaces_Olinorder__topology___Orderings_Oord,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo2117631714pology @ A )
% 0.29/0.46 => ( ord @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Topological__Spaces_Olinorder__topology___Orderings_Oorder,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo2117631714pology @ A )
% 0.29/0.46 => ( order @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Topological__Spaces_Olinorder__topology___Orderings_Olinorder,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo2117631714pology @ A )
% 0.29/0.46 => ( linorder @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Topological__Spaces_Olinorder__topology___Orderings_Opreorder,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo2117631714pology @ A )
% 0.29/0.46 => ( preorder @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Topological__Spaces_Olinorder__topology___Topological__Spaces_Ot0__space,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo2117631714pology @ A )
% 0.29/0.46 => ( topological_t0_space @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Topological__Spaces_Olinorder__topology___Topological__Spaces_Ot1__space,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo2117631714pology @ A )
% 0.29/0.46 => ( topological_t1_space @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Topological__Spaces_Olinorder__topology___Topological__Spaces_Ot2__space,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo2117631714pology @ A )
% 0.29/0.46 => ( topological_t2_space @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Topological__Spaces_Olinorder__topology___Topological__Spaces_Otopological__space,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo2117631714pology @ A )
% 0.29/0.46 => ( topolo503727757_space @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Topological__Spaces_Ofirst__countable__topology___HOL_Otype,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo2135403230pology @ A )
% 0.29/0.46 => ( type @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subcl_Topological__Spaces_Ofirst__countable__topology___Topological__Spaces_Otopological__space,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo2135403230pology @ A )
% 0.29/0.46 => ( topolo503727757_space @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % Type constructors (74)
% 0.29/0.46 thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice,axiom,
% 0.29/0.46 ! [A8: $tType] : ( bounded_lattice @ ( filter @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Countable_Ocountable,axiom,
% 0.29/0.46 countable @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Lattices_Obounded__lattice_1,axiom,
% 0.29/0.46 bounded_lattice @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Countable_Ocountable_2,axiom,
% 0.29/0.46 ! [A8: $tType] :
% 0.29/0.46 ( ( finite_finite @ A8 )
% 0.29/0.46 => ( countable @ ( set @ A8 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Lattices_Obounded__lattice_3,axiom,
% 0.29/0.46 ! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Countable_Ocountable_4,axiom,
% 0.29/0.46 countable @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Countable_Ocountable_5,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( ( finite_finite @ A8 )
% 0.29/0.46 & ( countable @ A9 ) )
% 0.29/0.46 => ( countable @ ( A8 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Lattices_Obounded__lattice_6,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( bounded_lattice @ A9 )
% 0.29/0.46 => ( bounded_lattice @ ( A8 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( ( finite_finite @ A8 )
% 0.29/0.46 & ( finite_finite @ A9 ) )
% 0.29/0.46 => ( finite_finite @ ( A8 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Finite__Set_Ofinite_7,axiom,
% 0.29/0.46 ! [A8: $tType] :
% 0.29/0.46 ( ( finite_finite @ A8 )
% 0.29/0.46 => ( finite_finite @ ( set @ A8 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Finite__Set_Ofinite_8,axiom,
% 0.29/0.46 finite_finite @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Topological__Spaces_Ofirst__countable__topology,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( ( countable @ A8 )
% 0.29/0.46 & ( topolo2135403230pology @ A9 ) )
% 0.29/0.46 => ( topolo2135403230pology @ ( A8 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( topolo503727757_space @ A9 )
% 0.29/0.46 => ( topolo503727757_space @ ( A8 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( bounded_lattice @ A9 )
% 0.29/0.46 => ( bounded_lattice_bot @ ( A8 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A9 )
% 0.29/0.46 => ( semilattice_inf @ ( A8 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( order_bot @ A9 )
% 0.29/0.46 => ( order_bot @ ( A8 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( preorder @ A9 )
% 0.29/0.46 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( lattice @ A9 )
% 0.29/0.46 => ( lattice @ ( A8 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( order @ A9 )
% 0.29/0.46 => ( order @ ( A8 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Otop,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( top @ A9 )
% 0.29/0.46 => ( top @ ( A8 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Oord,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( ord @ A9 )
% 0.29/0.46 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Obot,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( bot @ A9 )
% 0.29/0.46 => ( bot @ ( A8 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Groups_Ominus,axiom,
% 0.29/0.46 ! [A8: $tType,A9: $tType] :
% 0.29/0.46 ( ( minus @ A9 )
% 0.29/0.46 => ( minus @ ( A8 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Topological__Spaces_Ofirst__countable__topology_9,axiom,
% 0.29/0.46 topolo2135403230pology @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_10,axiom,
% 0.29/0.46 topolo503727757_space @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology,axiom,
% 0.29/0.46 topolo2133971006pology @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.29/0.46 cancel146912293up_add @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space,axiom,
% 0.29/0.46 topological_t2_space @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Topological__Spaces_Ot1__space,axiom,
% 0.29/0.46 topological_t1_space @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Topological__Spaces_Ot0__space,axiom,
% 0.29/0.46 topological_t0_space @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_11,axiom,
% 0.29/0.46 semilattice_inf @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Oorder__bot_12,axiom,
% 0.29/0.46 order_bot @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Opreorder_13,axiom,
% 0.29/0.46 preorder @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.29/0.46 linorder @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Lattices_Olattice_14,axiom,
% 0.29/0.46 lattice @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Oorder_15,axiom,
% 0.29/0.46 order @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Oord_16,axiom,
% 0.29/0.46 ord @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Obot_17,axiom,
% 0.29/0.46 bot @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Ominus_18,axiom,
% 0.29/0.46 minus @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_19,axiom,
% 0.29/0.46 ! [A8: $tType] : ( bounded_lattice_bot @ ( set @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_20,axiom,
% 0.29/0.46 ! [A8: $tType] : ( semilattice_inf @ ( set @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Oorder__bot_21,axiom,
% 0.29/0.46 ! [A8: $tType] : ( order_bot @ ( set @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Opreorder_22,axiom,
% 0.29/0.46 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Lattices_Olattice_23,axiom,
% 0.29/0.46 ! [A8: $tType] : ( lattice @ ( set @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Oorder_24,axiom,
% 0.29/0.46 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Otop_25,axiom,
% 0.29/0.46 ! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Oord_26,axiom,
% 0.29/0.46 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Obot_27,axiom,
% 0.29/0.46 ! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Groups_Ominus_28,axiom,
% 0.29/0.46 ! [A8: $tType] : ( minus @ ( set @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_29,axiom,
% 0.29/0.46 topolo503727757_space @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_30,axiom,
% 0.29/0.46 topolo2133971006pology @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_31,axiom,
% 0.29/0.46 bounded_lattice_bot @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_32,axiom,
% 0.29/0.46 topological_t2_space @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Topological__Spaces_Ot1__space_33,axiom,
% 0.29/0.46 topological_t1_space @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Topological__Spaces_Ot0__space_34,axiom,
% 0.29/0.46 topological_t0_space @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_35,axiom,
% 0.29/0.46 semilattice_inf @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Oorder__bot_36,axiom,
% 0.29/0.46 order_bot @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_37,axiom,
% 0.29/0.46 preorder @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder_38,axiom,
% 0.29/0.46 linorder @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Lattices_Olattice_39,axiom,
% 0.29/0.46 lattice @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_40,axiom,
% 0.29/0.46 order @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Otop_41,axiom,
% 0.29/0.46 top @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Oord_42,axiom,
% 0.29/0.46 ord @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Obot_43,axiom,
% 0.29/0.46 bot @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Groups_Ominus_44,axiom,
% 0.29/0.46 minus @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_45,axiom,
% 0.29/0.46 ! [A8: $tType] : ( bounded_lattice_bot @ ( filter @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_46,axiom,
% 0.29/0.46 ! [A8: $tType] : ( semilattice_inf @ ( filter @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_47,axiom,
% 0.29/0.46 ! [A8: $tType] : ( order_bot @ ( filter @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Filter_Ofilter___Orderings_Opreorder_48,axiom,
% 0.29/0.46 ! [A8: $tType] : ( preorder @ ( filter @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Filter_Ofilter___Lattices_Olattice_49,axiom,
% 0.29/0.46 ! [A8: $tType] : ( lattice @ ( filter @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Filter_Ofilter___Orderings_Oorder_50,axiom,
% 0.29/0.46 ! [A8: $tType] : ( order @ ( filter @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Filter_Ofilter___Orderings_Otop_51,axiom,
% 0.29/0.46 ! [A8: $tType] : ( top @ ( filter @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Filter_Ofilter___Orderings_Oord_52,axiom,
% 0.29/0.46 ! [A8: $tType] : ( ord @ ( filter @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Filter_Ofilter___Orderings_Obot_53,axiom,
% 0.29/0.46 ! [A8: $tType] : ( bot @ ( filter @ A8 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Free types (3)
% 0.29/0.46 thf(tfree_0,hypothesis,
% 0.29/0.46 comple1035589618norder @ b ).
% 0.29/0.46
% 0.29/0.46 thf(tfree_1,hypothesis,
% 0.29/0.46 topolo2117631714pology @ b ).
% 0.29/0.46
% 0.29/0.46 thf(tfree_2,hypothesis,
% 0.29/0.46 topolo2135403230pology @ a ).
% 0.29/0.46
% 0.29/0.46 % Conjectures (2)
% 0.29/0.46 thf(conj_0,hypothesis,
% 0.29/0.46 ! [S2: set @ b,V3: set @ b] :
% 0.29/0.46 ( ( ( topolo1751647064n_open @ b @ S2 )
% 0.29/0.46 & ( topolo1751647064n_open @ b @ V3 )
% 0.29/0.46 & ( member @ b @ ( f @ x0 ) @ S2 )
% 0.29/0.46 & ( member @ b @ a2 @ V3 )
% 0.29/0.46 & ( ( inf_inf @ ( set @ b ) @ S2 @ V3 )
% 0.29/0.46 = ( bot_bot @ ( set @ b ) ) ) )
% 0.29/0.46 => thesis ) ).
% 0.29/0.46
% 0.29/0.46 thf(conj_1,conjecture,
% 0.29/0.46 thesis ).
% 0.29/0.46
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 ------------------------
% 2.45/1.46 ---- Embedded in THF ---
% 2.45/1.46 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.45/1.46 %%% Generated on Tue Feb 24 08:26:32 EST 2026
% 2.45/1.46 %%% using '$$dhol' embedding, version 1.3.0.
% 2.45/1.46 %%% Logic specification used:
% 2.45/1.46 %%% thf(spec, logic, $$dhol).
% 2.45/1.46
% 2.45/1.46 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.wdWkGyYZoX/DTF2DTF5301.p
% 2.45/1.46 thf(ty_t_Filter_Ofilter, type, filter: ($tType > $tType)).
% 2.45/1.46 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.45/1.46 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.45/1.46 thf(ty_tf_b, type, b: $tType).
% 2.45/1.46 thf(ty_tf_a, type, a: $tType).
% 2.45/1.46 thf(sy_cl_Topological__Spaces_Olinorder__topology, type, topolo2117631714pology: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Complete__Lattices_Ocomplete__linorder, type, comple1035589618norder: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Lattices_Obounded__lattice, type, bounded_lattice: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Countable_Ocountable, type, countable: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Groups_Ominus, type, minus: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Lattices_Osemilattice__inf, type, semilattice_inf: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Topological__Spaces_Ot0__space, type, topological_t0_space: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Topological__Spaces_Ot1__space, type, topological_t1_space: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Topological__Spaces_Ot2__space, type, topological_t2_space: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Lattices_Obounded__lattice__bot, type, bounded_lattice_bot: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Topological__Spaces_Operfect__space, type, topolo890362671_space: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Topological__Spaces_Ouniform__space, type, topolo47006728_space: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Topological__Spaces_Odiscrete__topology, type, topolo2133971006pology: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Topological__Spaces_Otopological__space, type, topolo503727757_space: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_cl_Topological__Spaces_Ofirst__countable__topology, type, topolo2135403230pology: (!> [A:$tType]: ($o))).
% 2.45/1.46 thf(sy_c_Elementary__Topology_Oclosure, type, elementary_closure: (!> [A:$tType]: (((set @ A) > (set @ A))))).
% 2.45/1.46 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.45/1.46 thf(sy_c_Lattices_Oinf__class_Oinf, type, inf_inf: (!> [A:$tType]: ((A > (A > A))))).
% 2.45/1.46 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc__at, type, lower_582600101lsc_at: (!> [A:$tType,B:$tType]: ((A > ((A > B) > $o))))).
% 2.45/1.46 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.45/1.46 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.45/1.46 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.45/1.46 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.45/1.46 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.45/1.46 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.45/1.46 thf(sy_c_Set_Ois__singleton, type, is_singleton: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.45/1.46 thf(sy_c_Set_Oremove, type, remove: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.45/1.46 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))).
% 2.45/1.46 thf(sy_c_Topological__Spaces_Oopen__class_Oopen, type, topolo1751647064n_open: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.45/1.46 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within, type, topolo507301023within: (!> [A:$tType]: ((A > ((set @ A) > (filter @ A)))))).
% 2.45/1.46 thf(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded, type, topolo406746546ounded: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.45/1.46 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.45/1.46 thf(sy_v_A____, type, a2: b).
% 2.45/1.46 thf(sy_v_f, type, f: (a > b)).
% 2.45/1.46 thf(sy_v_thesis____, type, thesis: $o).
% 2.45/1.46 thf(sy_v_x0, type, x0: a).
% 2.45/1.46 thf(fact_0__092_060open_062A_A_092_060noteq_062_Af_Ax0_092_060close_062, axiom, (a2 != (f @ x0))).
% 2.45/1.46 thf(fact_1__092_060open_062f_Ax0_A_092_060noteq_062_AA_A_092_060Longrightarrow_062_A_092_060exists_062U_AV_O_Aopen_AU_A_092_060and_062_Aopen_AV_A_092_060and_062_Af_Ax0_A_092_060in_062_AU_A_092_060and_062_AA_A_092_060in_062_AV_A_092_060and_062_AU_A_092_060inter_062_AV_A_061_A_123_125_092_060close_062, axiom, (((f @ x0) != a2) => (? [U:(set @ b),V:(set @ b)]: ((((topolo1751647064n_open @ b) @ U) & (((topolo1751647064n_open @ b) @ V) & ((((member @ b) @ (f @ x0)) @ U) & ((((member @ b) @ a2) @ V) & ((((inf_inf @ (set @ b)) @ U) @ V) = (bot_bot @ (set @ b))))))))))).
% 2.45/1.46 thf(fact_2_open__Int, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [S:(set @ A),T:(set @ A)]: ((((topolo1751647064n_open @ A) @ S) => (((topolo1751647064n_open @ A) @ T) => ((topolo1751647064n_open @ A) @ (((inf_inf @ (set @ A)) @ S) @ T)))))))))).
% 2.45/1.46 thf(fact_3_open__empty, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => ((topolo1751647064n_open @ A) @ (bot_bot @ (set @ A))))))).
% 2.45/1.46 thf(fact_4_inf__bot__left, axiom, (! [A:$tType]: (((bounded_lattice_bot @ A) => (! [X:A]: (((((inf_inf @ A) @ (bot_bot @ A)) @ X) = (bot_bot @ A)))))))).
% 2.45/1.46 thf(fact_5_inf__bot__right, axiom, (! [A:$tType]: (((bounded_lattice_bot @ A) => (! [X:A]: (((((inf_inf @ A) @ X) @ (bot_bot @ A)) = (bot_bot @ A)))))))).
% 2.45/1.46 thf(fact_6_hausdorff, axiom, (! [A:$tType]: (((topological_t2_space @ A) => (! [X:A,Y:A]: (((X != Y) => (? [U:(set @ A),V:(set @ A)]: ((((topolo1751647064n_open @ A) @ U) & (((topolo1751647064n_open @ A) @ V) & ((((member @ A) @ X) @ U) & ((((member @ A) @ Y) @ V) & ((((inf_inf @ (set @ A)) @ U) @ V) = (bot_bot @ (set @ A)))))))))))))))).
% 2.45/1.46 thf(fact_7_separation__t2, axiom, (! [A:$tType]: (((topological_t2_space @ A) => (! [X:A,Y:A]: (((X != Y) = (? [U2:(set @ A),V2:(set @ A)]: ((((topolo1751647064n_open @ A) @ U2) & (((topolo1751647064n_open @ A) @ V2) & ((((member @ A) @ X) @ U2) & ((((member @ A) @ Y) @ V2) & ((((inf_inf @ (set @ A)) @ U2) @ V2) = (bot_bot @ (set @ A)))))))))))))))).
% 2.45/1.46 thf(fact_8_IntI, axiom, (! [A:$tType,C:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ A2) => ((((member @ A) @ C) @ B2) => (((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A2) @ B2))))))).
% 2.45/1.46 thf(fact_9_Int__iff, axiom, (! [A:$tType,C:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A2) @ B2)) = ((((member @ A) @ C) @ A2) & (((member @ A) @ C) @ B2)))))).
% 2.45/1.46 thf(fact_10_inf__apply, axiom, (! [B:$tType,A:$tType]: (((semilattice_inf @ B) => ((inf_inf @ (A > B)) = (^ [F:(A > B),G:(A > B),X2:A]: ((((inf_inf @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 2.45/1.46 thf(fact_11_inf_Oidem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A]: (((((inf_inf @ A) @ A3) @ A3) = A3))))))).
% 2.45/1.46 thf(fact_12_inf__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A]: (((((inf_inf @ A) @ X) @ X) = X))))))).
% 2.45/1.46 thf(fact_13_inf_Oleft__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B3:A]: (((((inf_inf @ A) @ A3) @ (((inf_inf @ A) @ A3) @ B3)) = (((inf_inf @ A) @ A3) @ B3)))))))).
% 2.45/1.46 thf(fact_14_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X2:A]: ((~ ((P @ X2))))))))).
% 2.45/1.46 thf(fact_15_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X2:A]: ((~ ((P @ X2))))))))).
% 2.45/1.46 thf(fact_16_all__not__in__conv, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X2:A]: ((~ ((((member @ A) @ X2) @ A2))))) = (A2 = (bot_bot @ (set @ A))))))).
% 2.45/1.46 thf(fact_17_empty__iff, axiom, (! [A:$tType,C:A]: ((~ ((((member @ A) @ C) @ (bot_bot @ (set @ A)))))))).
% 2.45/1.46 thf(fact_18_inf__right__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y)) @ Y) = (((inf_inf @ A) @ X) @ Y)))))))).
% 2.45/1.46 thf(fact_19_inf_Oright__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B3:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ A3) @ B3)) @ B3) = (((inf_inf @ A) @ A3) @ B3)))))))).
% 2.45/1.46 thf(fact_20_inf__left__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ X) @ Y)) = (((inf_inf @ A) @ X) @ Y)))))))).
% 2.45/1.46 thf(fact_21__092_060open_062_092_060forall_062S_O_Aopen_AS_A_092_060and_062_Af_Ax0_A_092_060in_062_AS_A_092_060longrightarrow_062_A_I_092_060exists_062T_O_Aopen_AT_A_092_060and_062_Ax0_A_092_060in_062_AT_A_092_060and_062_A_I_092_060forall_062x_H_092_060in_062T_O_Af_Ax_H_A_092_060le_062_Af_Ax0_A_092_060longrightarrow_062_Af_Ax_H_A_092_060in_062_AS_J_J_092_060close_062, axiom, (! [S2:(set @ b)]: (((((topolo1751647064n_open @ b) @ S2) & (((member @ b) @ (f @ x0)) @ S2)) => (? [T2:(set @ a)]: ((((topolo1751647064n_open @ a) @ T2) & ((((member @ a) @ x0) @ T2) & (! [X3:a]: (((((member @ a) @ X3) @ T2) => ((((ord_less_eq @ b) @ (f @ X3)) @ (f @ x0)) => (((member @ b) @ (f @ X3)) @ S2))))))))))))).
% 2.45/1.46 thf(fact_22_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 2.45/1.46 thf(fact_23_ex__in__conv, axiom, (! [A:$tType,A2:(set @ A)]: (((? [X2:A]: ((((member @ A) @ X2) @ A2))) = (A2 != (bot_bot @ (set @ A))))))).
% 2.45/1.46 thf(fact_24_equals0I, axiom, (! [A:$tType,A2:(set @ A)]: (((! [Y2:A]: ((~ ((((member @ A) @ Y2) @ A2))))) => (A2 = (bot_bot @ (set @ A))))))).
% 2.45/1.46 thf(fact_25_equals0D, axiom, (! [A:$tType,A2:(set @ A),A3:A]: (((A2 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A3) @ A2))))))).
% 2.45/1.46 thf(fact_26_emptyE, axiom, (! [A:$tType,A3:A]: ((~ ((((member @ A) @ A3) @ (bot_bot @ (set @ A)))))))).
% 2.45/1.46 thf(fact_27_inf__left__commute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y:A,Z:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z)) = (((inf_inf @ A) @ Y) @ (((inf_inf @ A) @ X) @ Z))))))))).
% 2.45/1.46 thf(fact_28_inf_Oleft__commute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B3:A,A3:A,C:A]: (((((inf_inf @ A) @ B3) @ (((inf_inf @ A) @ A3) @ C)) = (((inf_inf @ A) @ A3) @ (((inf_inf @ A) @ B3) @ C))))))))).
% 2.45/1.46 thf(fact_29_inf__commute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((inf_inf @ A) = (^ [X2:A,Y3:A]: ((((inf_inf @ A) @ Y3) @ X2)))))))).
% 2.45/1.46 thf(fact_30_inf_Ocommute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((inf_inf @ A) = (^ [A4:A,B4:A]: ((((inf_inf @ A) @ B4) @ A4)))))))).
% 2.45/1.46 thf(fact_31_inf__assoc, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y:A,Z:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y)) @ Z) = (((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z))))))))).
% 2.45/1.46 thf(fact_32_inf_Oassoc, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B3:A,C:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ A3) @ B3)) @ C) = (((inf_inf @ A) @ A3) @ (((inf_inf @ A) @ B3) @ C))))))))).
% 2.45/1.46 thf(fact_33_boolean__algebra__cancel_Oinf2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,K:A,B3:A,A3:A]: (((B2 = (((inf_inf @ A) @ K) @ B3)) => ((((inf_inf @ A) @ A3) @ B2) = (((inf_inf @ A) @ K) @ (((inf_inf @ A) @ A3) @ B3)))))))))).
% 2.45/1.46 thf(fact_34_boolean__algebra__cancel_Oinf1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,K:A,A3:A,B3:A]: (((A2 = (((inf_inf @ A) @ K) @ A3)) => ((((inf_inf @ A) @ A2) @ B3) = (((inf_inf @ A) @ K) @ (((inf_inf @ A) @ A3) @ B3)))))))))).
% 2.45/1.46 thf(fact_35_inf__fun__def, axiom, (! [B:$tType,A:$tType]: (((semilattice_inf @ B) => ((inf_inf @ (A > B)) = (^ [F:(A > B),G:(A > B),X2:A]: ((((inf_inf @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 2.45/1.46 thf(fact_36_inf__sup__aci_I1_J, axiom, (! [A:$tType]: (((lattice @ A) => ((inf_inf @ A) = (^ [X2:A,Y3:A]: ((((inf_inf @ A) @ Y3) @ X2)))))))).
% 2.45/1.46 thf(fact_37_inf__sup__aci_I2_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A,Z:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y)) @ Z) = (((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z))))))))).
% 2.45/1.46 thf(fact_38_inf__sup__aci_I3_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A,Z:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z)) = (((inf_inf @ A) @ Y) @ (((inf_inf @ A) @ X) @ Z))))))))).
% 2.45/1.46 thf(fact_39_inf__sup__aci_I4_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ X) @ Y)) = (((inf_inf @ A) @ X) @ Y)))))))).
% 2.45/1.46 thf(fact_40_Int__left__commute, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:(set @ A)]: (((((inf_inf @ (set @ A)) @ A2) @ (((inf_inf @ (set @ A)) @ B2) @ C2)) = (((inf_inf @ (set @ A)) @ B2) @ (((inf_inf @ (set @ A)) @ A2) @ C2)))))).
% 2.45/1.46 thf(fact_41_Int__left__absorb, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((inf_inf @ (set @ A)) @ A2) @ (((inf_inf @ (set @ A)) @ A2) @ B2)) = (((inf_inf @ (set @ A)) @ A2) @ B2))))).
% 2.45/1.46 thf(fact_42_Int__commute, axiom, (! [A:$tType]: (((inf_inf @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: ((((inf_inf @ (set @ A)) @ B5) @ A5))))))).
% 2.45/1.46 thf(fact_43_Int__absorb, axiom, (! [A:$tType,A2:(set @ A)]: (((((inf_inf @ (set @ A)) @ A2) @ A2) = A2)))).
% 2.45/1.46 thf(fact_44_Int__assoc, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A2) @ B2)) @ C2) = (((inf_inf @ (set @ A)) @ A2) @ (((inf_inf @ (set @ A)) @ B2) @ C2)))))).
% 2.45/1.46 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 2.45/1.46 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A2)))) = A2)))).
% 2.45/1.46 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) = (Q @ X4)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.45/1.46 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F2:(A > B),G2:(A > B)]: (((! [X4:A]: (((F2 @ X4) = (G2 @ X4)))) => (F2 = G2))))).
% 2.45/1.46 thf(fact_49_IntD2, axiom, (! [A:$tType,C:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A2) @ B2)) => (((member @ A) @ C) @ B2))))).
% 2.45/1.46 thf(fact_50_IntD1, axiom, (! [A:$tType,C:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A2) @ B2)) => (((member @ A) @ C) @ A2))))).
% 2.45/1.46 thf(fact_51_IntE, axiom, (! [A:$tType,C:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A2) @ B2)) => (~ (((((member @ A) @ C) @ A2) => (~ ((((member @ A) @ C) @ B2)))))))))).
% 2.45/1.46 thf(fact_52_open__discrete, axiom, (! [A:$tType]: (((topolo2133971006pology @ A) => (! [A2:(set @ A)]: (((topolo1751647064n_open @ A) @ A2))))))).
% 2.45/1.46 thf(fact_53_separation__t1, axiom, (! [A:$tType]: (((topological_t1_space @ A) => (! [X:A,Y:A]: (((X != Y) = (? [U2:(set @ A)]: ((((topolo1751647064n_open @ A) @ U2) & ((((member @ A) @ X) @ U2) & (~ ((((member @ A) @ Y) @ U2)))))))))))))).
% 2.45/1.46 thf(fact_54_separation__t0, axiom, (! [A:$tType]: (((topological_t0_space @ A) => (! [X:A,Y:A]: (((X != Y) = (? [U2:(set @ A)]: ((((topolo1751647064n_open @ A) @ U2) & ((((member @ A) @ X) @ U2) != (((member @ A) @ Y) @ U2)))))))))))).
% 2.45/1.46 thf(fact_55_t1__space, axiom, (! [A:$tType]: (((topological_t1_space @ A) => (! [X:A,Y:A]: (((X != Y) => (? [U:(set @ A)]: ((((topolo1751647064n_open @ A) @ U) & ((((member @ A) @ X) @ U) & (~ ((((member @ A) @ Y) @ U)))))))))))))).
% 2.45/1.46 thf(fact_56_t0__space, axiom, (! [A:$tType]: (((topological_t0_space @ A) => (! [X:A,Y:A]: (((X != Y) => (? [U:(set @ A)]: ((((topolo1751647064n_open @ A) @ U) & ((((member @ A) @ X) @ U) != (((member @ A) @ Y) @ U)))))))))))).
% 2.45/1.46 thf(fact_57_disjoint__iff__not__equal, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: ((((((inf_inf @ (set @ A)) @ A2) @ B2) = (bot_bot @ (set @ A))) = (! [X2:A]: (((((member @ A) @ X2) @ A2) => (! [Y3:A]: (((((member @ A) @ Y3) @ B2) => (X2 != Y3))))))))))).
% 2.45/1.46 thf(fact_58_Int__empty__right, axiom, (! [A:$tType,A2:(set @ A)]: (((((inf_inf @ (set @ A)) @ A2) @ (bot_bot @ (set @ A))) = (bot_bot @ (set @ A)))))).
% 2.45/1.46 thf(fact_59_Int__empty__left, axiom, (! [A:$tType,B2:(set @ A)]: (((((inf_inf @ (set @ A)) @ (bot_bot @ (set @ A))) @ B2) = (bot_bot @ (set @ A)))))).
% 2.45/1.46 thf(fact_60_disjoint__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: ((((((inf_inf @ (set @ A)) @ A2) @ B2) = (bot_bot @ (set @ A))) = (! [X2:A]: (((((member @ A) @ X2) @ A2) => (~ ((((member @ A) @ X2) @ B2)))))))))).
% 2.45/1.46 thf(fact_61_Int__emptyI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((! [X4:A]: (((((member @ A) @ X4) @ A2) => (~ ((((member @ A) @ X4) @ B2)))))) => ((((inf_inf @ (set @ A)) @ A2) @ B2) = (bot_bot @ (set @ A))))))).
% 2.45/1.46 thf(fact_62_bot__apply, axiom, (! [C3:$tType,D:$tType]: (((bot @ C3) => ((bot_bot @ (D > C3)) = (^ [X2:D]: ((bot_bot @ C3)))))))).
% 2.45/1.46 thf(fact_63_calculation, axiom, ((~ ((! [S2:(set @ b)]: (((((topolo1751647064n_open @ b) @ S2) & (((member @ b) @ (f @ x0)) @ S2)) => (? [T2:(set @ a)]: ((((topolo1751647064n_open @ a) @ T2) & ((((member @ a) @ x0) @ T2) & (! [X3:a]: (((((member @ a) @ X3) @ T2) => ((((ord_less_eq @ b) @ (f @ X3)) @ (f @ x0)) => (((member @ b) @ (f @ X3)) @ S2)))))))))))))) => (~ (((((lower_582600101lsc_at @ a) @ b) @ x0) @ f))))).
% 2.45/1.46 thf(fact_64_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A5:(set @ A)]: ((A5 = (bot_bot @ (set @ A))))))))).
% 2.45/1.46 thf(fact_65_totally__bounded__empty, axiom, (! [A:$tType]: (((topolo47006728_space @ A) => ((topolo406746546ounded @ A) @ (bot_bot @ (set @ A))))))).
% 2.45/1.46 thf(fact_66_Collect__empty__eq__bot, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (P = (bot_bot @ (A > $o))))))).
% 2.45/1.46 thf(fact_67_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X2:A]: ((((member @ A) @ X2) @ (bot_bot @ (set @ A))))))))).
% 2.45/1.46 thf(fact_68_bot__fun__def, axiom, (! [B:$tType,A:$tType]: (((bot @ B) => ((bot_bot @ (A > B)) = (^ [X2:A]: ((bot_bot @ B)))))))).
% 2.45/1.46 thf(fact_69_is__singletonI_H, axiom, (! [A:$tType,A2:(set @ A)]: (((A2 != (bot_bot @ (set @ A))) => ((! [X4:A,Y2:A]: (((((member @ A) @ X4) @ A2) => ((((member @ A) @ Y2) @ A2) => (X4 = Y2))))) => ((is_singleton @ A) @ A2)))))).
% 2.45/1.46 thf(fact_70_Diff__disjoint, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((inf_inf @ (set @ A)) @ A2) @ (((minus_minus @ (set @ A)) @ B2) @ A2)) = (bot_bot @ (set @ A)))))).
% 2.45/1.46 thf(fact_71_insert__disjoint_I1_J, axiom, (! [A:$tType,A3:A,A2:(set @ A),B2:(set @ A)]: ((((((inf_inf @ (set @ A)) @ (((insert @ A) @ A3) @ A2)) @ B2) = (bot_bot @ (set @ A))) = ((~ ((((member @ A) @ A3) @ B2))) & ((((inf_inf @ (set @ A)) @ A2) @ B2) = (bot_bot @ (set @ A)))))))).
% 2.45/1.46 thf(fact_72_insert__disjoint_I2_J, axiom, (! [A:$tType,A3:A,A2:(set @ A),B2:(set @ A)]: ((((bot_bot @ (set @ A)) = (((inf_inf @ (set @ A)) @ (((insert @ A) @ A3) @ A2)) @ B2)) = ((~ ((((member @ A) @ A3) @ B2))) & ((bot_bot @ (set @ A)) = (((inf_inf @ (set @ A)) @ A2) @ B2))))))).
% 2.45/1.46 thf(fact_73_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.45/1.46 thf(fact_74_minus__apply, axiom, (! [B:$tType,A:$tType]: (((minus @ B) => ((minus_minus @ (A > B)) = (^ [A5:(A > B),B5:(A > B),X2:A]: ((((minus_minus @ B) @ (A5 @ X2)) @ (B5 @ X2))))))))).
% 2.45/1.46 thf(fact_75_insertCI, axiom, (! [A:$tType,A3:A,B2:(set @ A),B3:A]: ((((~ ((((member @ A) @ A3) @ B2))) => (A3 = B3)) => (((member @ A) @ A3) @ (((insert @ A) @ B3) @ B2)))))).
% 2.45/1.46 thf(fact_76_insert__iff, axiom, (! [A:$tType,A3:A,B3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ (((insert @ A) @ B3) @ A2)) = ((A3 = B3) | (((member @ A) @ A3) @ A2)))))).
% 2.45/1.46 thf(fact_77_insert__absorb2, axiom, (! [A:$tType,X:A,A2:(set @ A)]: (((((insert @ A) @ X) @ (((insert @ A) @ X) @ A2)) = (((insert @ A) @ X) @ A2))))).
% 2.45/1.46 thf(fact_78_DiffI, axiom, (! [A:$tType,C:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ A2) => ((~ ((((member @ A) @ C) @ B2))) => (((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A2) @ B2))))))).
% 2.45/1.46 thf(fact_79_Diff__iff, axiom, (! [A:$tType,C:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) = ((((member @ A) @ C) @ A2) & (~ ((((member @ A) @ C) @ B2)))))))).
% 2.45/1.46 thf(fact_80_Diff__idemp, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) @ B2) = (((minus_minus @ (set @ A)) @ A2) @ B2))))).
% 2.45/1.46 thf(fact_81_le__inf__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z)) = ((((ord_less_eq @ A) @ X) @ Y) & (((ord_less_eq @ A) @ X) @ Z))))))))).
% 2.45/1.46 thf(fact_82_inf_Obounded__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A3) @ (((inf_inf @ A) @ B3) @ C)) = ((((ord_less_eq @ A) @ A3) @ B3) & (((ord_less_eq @ A) @ A3) @ C))))))))).
% 2.45/1.46 thf(fact_83_singletonI, axiom, (! [A:$tType,A3:A]: ((((member @ A) @ A3) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))).
% 2.45/1.46 thf(fact_84_Int__insert__left__if0, axiom, (! [A:$tType,A3:A,C2:(set @ A),B2:(set @ A)]: (((~ ((((member @ A) @ A3) @ C2))) => ((((inf_inf @ (set @ A)) @ (((insert @ A) @ A3) @ B2)) @ C2) = (((inf_inf @ (set @ A)) @ B2) @ C2)))))).
% 2.45/1.46 thf(fact_85_Int__insert__left__if1, axiom, (! [A:$tType,A3:A,C2:(set @ A),B2:(set @ A)]: (((((member @ A) @ A3) @ C2) => ((((inf_inf @ (set @ A)) @ (((insert @ A) @ A3) @ B2)) @ C2) = (((insert @ A) @ A3) @ (((inf_inf @ (set @ A)) @ B2) @ C2))))))).
% 2.45/1.46 thf(fact_86_insert__inter__insert, axiom, (! [A:$tType,A3:A,A2:(set @ A),B2:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((insert @ A) @ A3) @ A2)) @ (((insert @ A) @ A3) @ B2)) = (((insert @ A) @ A3) @ (((inf_inf @ (set @ A)) @ A2) @ B2)))))).
% 2.45/1.46 thf(fact_87_Int__insert__right__if0, axiom, (! [A:$tType,A3:A,A2:(set @ A),B2:(set @ A)]: (((~ ((((member @ A) @ A3) @ A2))) => ((((inf_inf @ (set @ A)) @ A2) @ (((insert @ A) @ A3) @ B2)) = (((inf_inf @ (set @ A)) @ A2) @ B2)))))).
% 2.45/1.46 thf(fact_88_Int__insert__right__if1, axiom, (! [A:$tType,A3:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ A3) @ A2) => ((((inf_inf @ (set @ A)) @ A2) @ (((insert @ A) @ A3) @ B2)) = (((insert @ A) @ A3) @ (((inf_inf @ (set @ A)) @ A2) @ B2))))))).
% 2.45/1.46 thf(fact_89_Diff__empty, axiom, (! [A:$tType,A2:(set @ A)]: (((((minus_minus @ (set @ A)) @ A2) @ (bot_bot @ (set @ A))) = A2)))).
% 2.45/1.46 thf(fact_90_empty__Diff, axiom, (! [A:$tType,A2:(set @ A)]: (((((minus_minus @ (set @ A)) @ (bot_bot @ (set @ A))) @ A2) = (bot_bot @ (set @ A)))))).
% 2.45/1.46 thf(fact_91_Diff__cancel, axiom, (! [A:$tType,A2:(set @ A)]: (((((minus_minus @ (set @ A)) @ A2) @ A2) = (bot_bot @ (set @ A)))))).
% 2.45/1.46 thf(fact_92_Diff__insert0, axiom, (! [A:$tType,X:A,A2:(set @ A),B2:(set @ A)]: (((~ ((((member @ A) @ X) @ A2))) => ((((minus_minus @ (set @ A)) @ A2) @ (((insert @ A) @ X) @ B2)) = (((minus_minus @ (set @ A)) @ A2) @ B2)))))).
% 2.45/1.46 thf(fact_93_insert__Diff1, axiom, (! [A:$tType,X:A,B2:(set @ A),A2:(set @ A)]: (((((member @ A) @ X) @ B2) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X) @ A2)) @ B2) = (((minus_minus @ (set @ A)) @ A2) @ B2)))))).
% 2.45/1.46 thf(fact_94_disjoint__insert_I2_J, axiom, (! [A:$tType,A2:(set @ A),B3:A,B2:(set @ A)]: ((((bot_bot @ (set @ A)) = (((inf_inf @ (set @ A)) @ A2) @ (((insert @ A) @ B3) @ B2))) = ((~ ((((member @ A) @ B3) @ A2))) & ((bot_bot @ (set @ A)) = (((inf_inf @ (set @ A)) @ A2) @ B2))))))).
% 2.45/1.46 thf(fact_95_disjoint__insert_I1_J, axiom, (! [A:$tType,B2:(set @ A),A3:A,A2:(set @ A)]: ((((((inf_inf @ (set @ A)) @ B2) @ (((insert @ A) @ A3) @ A2)) = (bot_bot @ (set @ A))) = ((~ ((((member @ A) @ A3) @ B2))) & ((((inf_inf @ (set @ A)) @ B2) @ A2) = (bot_bot @ (set @ A)))))))).
% 2.45/1.46 thf(fact_96_insert__Diff__single, axiom, (! [A:$tType,A3:A,A2:(set @ A)]: (((((insert @ A) @ A3) @ (((minus_minus @ (set @ A)) @ A2) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))) = (((insert @ A) @ A3) @ A2))))).
% 2.45/1.46 thf(fact_97_is__singletonI, axiom, (! [A:$tType,X:A]: (((is_singleton @ A) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))).
% 2.45/1.46 thf(fact_98_DiffE, axiom, (! [A:$tType,C:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) => (~ (((((member @ A) @ C) @ A2) => (((member @ A) @ C) @ B2)))))))).
% 2.45/1.46 thf(fact_99_DiffD1, axiom, (! [A:$tType,C:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) => (((member @ A) @ C) @ A2))))).
% 2.45/1.46 thf(fact_100_DiffD2, axiom, (! [A:$tType,C:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) => (~ ((((member @ A) @ C) @ B2))))))).
% 2.45/1.46 thf(fact_101_insertE, axiom, (! [A:$tType,A3:A,B3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ (((insert @ A) @ B3) @ A2)) => ((A3 != B3) => (((member @ A) @ A3) @ A2)))))).
% 2.45/1.46 thf(fact_102_insertI1, axiom, (! [A:$tType,A3:A,B2:(set @ A)]: ((((member @ A) @ A3) @ (((insert @ A) @ A3) @ B2))))).
% 2.45/1.46 thf(fact_103_insertI2, axiom, (! [A:$tType,A3:A,B2:(set @ A),B3:A]: (((((member @ A) @ A3) @ B2) => (((member @ A) @ A3) @ (((insert @ A) @ B3) @ B2)))))).
% 2.45/1.46 thf(fact_104_Set_Oset__insert, axiom, (! [A:$tType,X:A,A2:(set @ A)]: (((((member @ A) @ X) @ A2) => (~ ((! [B6:(set @ A)]: (((A2 = (((insert @ A) @ X) @ B6)) => (((member @ A) @ X) @ B6)))))))))).
% 2.45/1.46 thf(fact_105_insert__ident, axiom, (! [A:$tType,X:A,A2:(set @ A),B2:(set @ A)]: (((~ ((((member @ A) @ X) @ A2))) => ((~ ((((member @ A) @ X) @ B2))) => (((((insert @ A) @ X) @ A2) = (((insert @ A) @ X) @ B2)) = (A2 = B2))))))).
% 2.45/1.46 thf(fact_106_insert__absorb, axiom, (! [A:$tType,A3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ A2) => ((((insert @ A) @ A3) @ A2) = A2))))).
% 2.45/1.46 thf(fact_107_insert__eq__iff, axiom, (! [A:$tType,A3:A,A2:(set @ A),B3:A,B2:(set @ A)]: (((~ ((((member @ A) @ A3) @ A2))) => ((~ ((((member @ A) @ B3) @ B2))) => (((((insert @ A) @ A3) @ A2) = (((insert @ A) @ B3) @ B2)) = (((A3 = B3) => (A2 = B2)) & ((A3 != B3) => (? [C4:(set @ A)]: (((A2 = (((insert @ A) @ B3) @ C4)) & ((~ ((((member @ A) @ B3) @ C4))) & ((B2 = (((insert @ A) @ A3) @ C4)) & (~ ((((member @ A) @ A3) @ C4)))))))))))))))).
% 2.45/1.46 thf(fact_108_insert__Diff__if, axiom, (! [A:$tType,X:A,B2:(set @ A),A2:(set @ A)]: ((((((member @ A) @ X) @ B2) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X) @ A2)) @ B2) = (((minus_minus @ (set @ A)) @ A2) @ B2))) & ((~ ((((member @ A) @ X) @ B2))) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X) @ A2)) @ B2) = (((insert @ A) @ X) @ (((minus_minus @ (set @ A)) @ A2) @ B2)))))))).
% 2.45/1.46 thf(fact_109_insert__commute, axiom, (! [A:$tType,X:A,Y:A,A2:(set @ A)]: (((((insert @ A) @ X) @ (((insert @ A) @ Y) @ A2)) = (((insert @ A) @ Y) @ (((insert @ A) @ X) @ A2)))))).
% 2.45/1.46 thf(fact_110_fun__diff__def, axiom, (! [B:$tType,A:$tType]: (((minus @ B) => ((minus_minus @ (A > B)) = (^ [A5:(A > B),B5:(A > B),X2:A]: ((((minus_minus @ B) @ (A5 @ X2)) @ (B5 @ X2))))))))).
% 2.45/1.46 thf(fact_111_mk__disjoint__insert, axiom, (! [A:$tType,A3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ A2) => (? [B6:(set @ A)]: (((A2 = (((insert @ A) @ A3) @ B6)) & (~ ((((member @ A) @ A3) @ B6)))))))))).
% 2.45/1.46 thf(fact_112_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ A3) @ B3) => (A3 = B3))))))))).
% 2.45/1.46 thf(fact_113_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ B4) @ A4) & (((ord_less_eq @ A) @ A4) @ B4))))))))).
% 2.45/1.46 thf(fact_114_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A,C:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ C) @ B3) => (((ord_less_eq @ A) @ C) @ A3))))))))).
% 2.45/1.46 thf(fact_115_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A3:A,B3:A]: (((! [A6:A,B7:A]: (((((ord_less_eq @ A) @ A6) @ B7) => ((P @ A6) @ B7)))) => ((! [A6:A,B7:A]: ((((P @ B7) @ A6) => ((P @ A6) @ B7)))) => ((P @ A3) @ B3))))))))).
% 2.45/1.46 thf(fact_116_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 2.45/1.46 thf(fact_117_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less_eq @ A) @ X) @ Z))))))))).
% 2.45/1.46 thf(fact_118_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ A3) => (A3 = B3))))))))).
% 2.45/1.46 thf(fact_119_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((B3 = C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 2.45/1.46 thf(fact_120_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C:A]: (((A3 = B3) => ((((ord_less_eq @ A) @ B3) @ C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 2.45/1.46 thf(fact_121_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) & (((ord_less_eq @ A) @ B4) @ A4))))))))).
% 2.45/1.46 thf(fact_122_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.45/1.46 thf(fact_123_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A,Z:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z)))) => (((((ord_less_eq @ A) @ X) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y)))) => (((((ord_less_eq @ A) @ Z) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X)))) => (~ (((((ord_less_eq @ A) @ Z) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 2.45/1.46 thf(fact_124_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 2.45/1.46 thf(fact_125_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.45/1.46 thf(fact_126_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.45/1.46 thf(fact_127_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.45/1.46 thf(fact_128_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 2.45/1.46 thf(fact_129_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) & (((ord_less_eq @ A) @ Y3) @ X2))))))))).
% 2.45/1.46 thf(fact_130_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,B3:A,F2:(A > B),C:B]: (((((ord_less_eq @ A) @ A3) @ B3) => (((F2 @ B3) = C) => ((! [X4:A,Y2:A]: (((((ord_less_eq @ A) @ X4) @ Y2) => (((ord_less_eq @ B) @ (F2 @ X4)) @ (F2 @ Y2))))) => (((ord_less_eq @ B) @ (F2 @ A3)) @ C)))))))))).
% 2.45/1.46 thf(fact_131_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,F2:(B > A),B3:B,C:B]: (((A3 = (F2 @ B3)) => ((((ord_less_eq @ B) @ B3) @ C) => ((! [X4:B,Y2:B]: (((((ord_less_eq @ B) @ X4) @ Y2) => (((ord_less_eq @ A) @ (F2 @ X4)) @ (F2 @ Y2))))) => (((ord_less_eq @ A) @ A3) @ (F2 @ C))))))))))).
% 2.45/1.46 thf(fact_132_order__subst2, axiom, (! [A:$tType,C3:$tType]: ((((order @ C3) & (order @ A)) => (! [A3:A,B3:A,F2:(A > C3),C:C3]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ C3) @ (F2 @ B3)) @ C) => ((! [X4:A,Y2:A]: (((((ord_less_eq @ A) @ X4) @ Y2) => (((ord_less_eq @ C3) @ (F2 @ X4)) @ (F2 @ Y2))))) => (((ord_less_eq @ C3) @ (F2 @ A3)) @ C)))))))))).
% 2.45/1.46 thf(fact_133_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F2:(B > A),B3:B,C:B]: (((((ord_less_eq @ A) @ A3) @ (F2 @ B3)) => ((((ord_less_eq @ B) @ B3) @ C) => ((! [X4:B,Y2:B]: (((((ord_less_eq @ B) @ X4) @ Y2) => (((ord_less_eq @ A) @ (F2 @ X4)) @ (F2 @ Y2))))) => (((ord_less_eq @ A) @ A3) @ (F2 @ C))))))))))).
% 2.45/1.46 thf(fact_134_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F:(A > B),G:(A > B)]: ((! [X2:A]: ((((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2))))))))))).
% 2.45/1.46 thf(fact_135_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G2:(A > B)]: (((! [X4:A]: ((((ord_less_eq @ B) @ (F2 @ X4)) @ (G2 @ X4)))) => (((ord_less_eq @ (A > B)) @ F2) @ G2)))))))).
% 2.45/1.46 thf(fact_136_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G2:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F2) @ G2) => (((ord_less_eq @ B) @ (F2 @ X)) @ (G2 @ X))))))))).
% 2.45/1.46 thf(fact_137_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G2:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F2) @ G2) => (((ord_less_eq @ B) @ (F2 @ X)) @ (G2 @ X))))))))).
% 2.45/1.46 thf(fact_138_Diff__insert__absorb, axiom, (! [A:$tType,X:A,A2:(set @ A)]: (((~ ((((member @ A) @ X) @ A2))) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X) @ A2)) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = A2))))).
% 2.45/1.46 thf(fact_139_Diff__insert2, axiom, (! [A:$tType,A2:(set @ A),A3:A,B2:(set @ A)]: (((((minus_minus @ (set @ A)) @ A2) @ (((insert @ A) @ A3) @ B2)) = (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))) @ B2))))).
% 2.45/1.46 thf(fact_140_insert__Diff, axiom, (! [A:$tType,A3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ A2) => ((((insert @ A) @ A3) @ (((minus_minus @ (set @ A)) @ A2) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))) = A2))))).
% 2.45/1.46 thf(fact_141_Diff__insert, axiom, (! [A:$tType,A2:(set @ A),A3:A,B2:(set @ A)]: (((((minus_minus @ (set @ A)) @ A2) @ (((insert @ A) @ A3) @ B2)) = (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))))))).
% 2.45/1.46 thf(fact_142_bot_Oextremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A3))))))).
% 2.45/1.46 thf(fact_143_bot_Oextremum__unique, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) = (A3 = (bot_bot @ A))))))))).
% 2.45/1.46 thf(fact_144_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) => (A3 = (bot_bot @ A))))))))).
% 2.45/1.46 thf(fact_145_is__singleton__def, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A5:(set @ A)]: ((? [X2:A]: ((A5 = (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))))))))))).
% 2.45/1.46 thf(fact_146_is__singletonE, axiom, (! [A:$tType,A2:(set @ A)]: ((((is_singleton @ A) @ A2) => (~ ((! [X4:A]: ((A2 != (((insert @ A) @ X4) @ (bot_bot @ (set @ A)))))))))))).
% 2.45/1.46 thf(fact_147_singletonD, axiom, (! [A:$tType,B3:A,A3:A]: (((((member @ A) @ B3) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) => (B3 = A3))))).
% 2.45/1.46 thf(fact_148_singleton__iff, axiom, (! [A:$tType,B3:A,A3:A]: (((((member @ A) @ B3) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) = (B3 = A3))))).
% 2.45/1.46 thf(fact_149_doubleton__eq__iff, axiom, (! [A:$tType,A3:A,B3:A,C:A,D2:A]: ((((((insert @ A) @ A3) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C) @ (((insert @ A) @ D2) @ (bot_bot @ (set @ A))))) = (((A3 = C) & (B3 = D2)) | ((A3 = D2) & (B3 = C))))))).
% 2.45/1.46 thf(fact_150_insert__not__empty, axiom, (! [A:$tType,A3:A,A2:(set @ A)]: (((((insert @ A) @ A3) @ A2) != (bot_bot @ (set @ A)))))).
% 2.45/1.46 thf(fact_151_singleton__inject, axiom, (! [A:$tType,A3:A,B3:A]: ((((((insert @ A) @ A3) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))) => (A3 = B3))))).
% 2.45/1.46 thf(fact_152_Int__insert__left, axiom, (! [A:$tType,A3:A,C2:(set @ A),B2:(set @ A)]: ((((((member @ A) @ A3) @ C2) => ((((inf_inf @ (set @ A)) @ (((insert @ A) @ A3) @ B2)) @ C2) = (((insert @ A) @ A3) @ (((inf_inf @ (set @ A)) @ B2) @ C2)))) & ((~ ((((member @ A) @ A3) @ C2))) => ((((inf_inf @ (set @ A)) @ (((insert @ A) @ A3) @ B2)) @ C2) = (((inf_inf @ (set @ A)) @ B2) @ C2))))))).
% 2.45/1.46 thf(fact_153_Int__insert__right, axiom, (! [A:$tType,A3:A,A2:(set @ A),B2:(set @ A)]: ((((((member @ A) @ A3) @ A2) => ((((inf_inf @ (set @ A)) @ A2) @ (((insert @ A) @ A3) @ B2)) = (((insert @ A) @ A3) @ (((inf_inf @ (set @ A)) @ A2) @ B2)))) & ((~ ((((member @ A) @ A3) @ A2))) => ((((inf_inf @ (set @ A)) @ A2) @ (((insert @ A) @ A3) @ B2)) = (((inf_inf @ (set @ A)) @ A2) @ B2))))))).
% 2.45/1.46 thf(fact_154_Int__Diff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A2) @ B2)) @ C2) = (((inf_inf @ (set @ A)) @ A2) @ (((minus_minus @ (set @ A)) @ B2) @ C2)))))).
% 2.45/1.46 thf(fact_155_Diff__Int2, axiom, (! [A:$tType,A2:(set @ A),C2:(set @ A),B2:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A2) @ C2)) @ (((inf_inf @ (set @ A)) @ B2) @ C2)) = (((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A2) @ C2)) @ B2))))).
% 2.45/1.46 thf(fact_156_Diff__Diff__Int, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((minus_minus @ (set @ A)) @ A2) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) = (((inf_inf @ (set @ A)) @ A2) @ B2))))).
% 2.45/1.46 thf(fact_157_Diff__Int__distrib, axiom, (! [A:$tType,C2:(set @ A),A2:(set @ A),B2:(set @ A)]: (((((inf_inf @ (set @ A)) @ C2) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) = (((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ C2) @ A2)) @ (((inf_inf @ (set @ A)) @ C2) @ B2)))))).
% 2.45/1.46 thf(fact_158_Diff__Int__distrib2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) @ C2) = (((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A2) @ C2)) @ (((inf_inf @ (set @ A)) @ B2) @ C2)))))).
% 2.45/1.46 thf(fact_159_inf__sup__ord_I2_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X) @ Y)) @ Y))))))).
% 2.45/1.46 thf(fact_160_inf__sup__ord_I1_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X) @ Y)) @ X))))))).
% 2.45/1.46 thf(fact_161_inf__le1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X) @ Y)) @ X))))))).
% 2.45/1.46 thf(fact_162_inf__le2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X) @ Y)) @ Y))))))).
% 2.45/1.46 thf(fact_163_le__infE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,A3:A,B3:A]: (((((ord_less_eq @ A) @ X) @ (((inf_inf @ A) @ A3) @ B3)) => (~ (((((ord_less_eq @ A) @ X) @ A3) => (~ ((((ord_less_eq @ A) @ X) @ B3))))))))))))).
% 2.45/1.46 thf(fact_164_le__infI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,A3:A,B3:A]: (((((ord_less_eq @ A) @ X) @ A3) => ((((ord_less_eq @ A) @ X) @ B3) => (((ord_less_eq @ A) @ X) @ (((inf_inf @ A) @ A3) @ B3)))))))))).
% 2.45/1.46 thf(fact_165_inf__mono, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,C:A,B3:A,D2:A]: (((((ord_less_eq @ A) @ A3) @ C) => ((((ord_less_eq @ A) @ B3) @ D2) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A3) @ B3)) @ (((inf_inf @ A) @ C) @ D2)))))))))).
% 2.45/1.46 thf(fact_166_le__infI1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,X:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ X) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A3) @ B3)) @ X)))))))).
% 2.45/1.46 thf(fact_167_le__infI2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B3:A,X:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ X) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A3) @ B3)) @ X)))))))).
% 2.45/1.46 thf(fact_168_inf_OorderE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => (A3 = (((inf_inf @ A) @ A3) @ B3))))))))).
% 2.45/1.46 thf(fact_169_inf_OorderI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B3:A]: (((A3 = (((inf_inf @ A) @ A3) @ B3)) => (((ord_less_eq @ A) @ A3) @ B3)))))))).
% 2.45/1.46 thf(fact_170_inf__unique, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [F2:(A > (A > A)),X:A,Y:A]: (((! [X4:A,Y2:A]: ((((ord_less_eq @ A) @ ((F2 @ X4) @ Y2)) @ X4))) => ((! [X4:A,Y2:A]: ((((ord_less_eq @ A) @ ((F2 @ X4) @ Y2)) @ Y2))) => ((! [X4:A,Y2:A,Z3:A]: (((((ord_less_eq @ A) @ X4) @ Y2) => ((((ord_less_eq @ A) @ X4) @ Z3) => (((ord_less_eq @ A) @ X4) @ ((F2 @ Y2) @ Z3)))))) => ((((inf_inf @ A) @ X) @ Y) = ((F2 @ X) @ Y))))))))))).
% 2.45/1.46 thf(fact_171_le__iff__inf, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less_eq @ A) = (^ [X2:A,Y3:A]: (((((inf_inf @ A) @ X2) @ Y3) = X2)))))))).
% 2.45/1.46 thf(fact_172_inf_Oabsorb1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((inf_inf @ A) @ A3) @ B3) = A3)))))))).
% 2.45/1.46 thf(fact_173_inf_Oabsorb2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((inf_inf @ A) @ A3) @ B3) = B3)))))))).
% 2.45/1.46 thf(fact_174_inf__absorb1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((inf_inf @ A) @ X) @ Y) = X)))))))).
% 2.45/1.46 thf(fact_175_inf__absorb2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((inf_inf @ A) @ X) @ Y) = Y)))))))).
% 2.45/1.46 thf(fact_176_inf_OboundedE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A3) @ (((inf_inf @ A) @ B3) @ C)) => (~ (((((ord_less_eq @ A) @ A3) @ B3) => (~ ((((ord_less_eq @ A) @ A3) @ C))))))))))))).
% 2.45/1.46 thf(fact_177_inf_OboundedI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ A3) @ C) => (((ord_less_eq @ A) @ A3) @ (((inf_inf @ A) @ B3) @ C)))))))))).
% 2.45/1.46 thf(fact_178_inf__greatest, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ X) @ Z) => (((ord_less_eq @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z)))))))))).
% 2.45/1.46 thf(fact_179_inf_Oorder__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less_eq @ A) = (^ [A4:A,B4:A]: ((A4 = (((inf_inf @ A) @ A4) @ B4))))))))).
% 2.45/1.46 thf(fact_180_inf_Ocobounded1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B3:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ A3) @ B3)) @ A3))))))).
% 2.45/1.46 thf(fact_181_inf_Ocobounded2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B3:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ A3) @ B3)) @ B3))))))).
% 2.45/1.46 thf(fact_182_inf_Oabsorb__iff1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less_eq @ A) = (^ [A4:A,B4:A]: (((((inf_inf @ A) @ A4) @ B4) = A4)))))))).
% 2.45/1.46 thf(fact_183_inf_Oabsorb__iff2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less_eq @ A) = (^ [B4:A,A4:A]: (((((inf_inf @ A) @ A4) @ B4) = B4)))))))).
% 2.45/1.46 thf(fact_184_inf_OcoboundedI1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,C:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ C) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A3) @ B3)) @ C)))))))).
% 2.45/1.46 thf(fact_185_inf_OcoboundedI2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B3:A,C:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ C) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A3) @ B3)) @ C)))))))).
% 2.45/1.46 thf(fact_186_not__open__singleton, axiom, (! [A:$tType]: (((topolo890362671_space @ A) => (! [X:A]: ((~ (((topolo1751647064n_open @ A) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))))))).
% 2.45/1.46 thf(fact_187_Diff__triv, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: ((((((inf_inf @ (set @ A)) @ A2) @ B2) = (bot_bot @ (set @ A))) => ((((minus_minus @ (set @ A)) @ A2) @ B2) = A2))))).
% 2.45/1.46 thf(fact_188_open__delete, axiom, (! [A:$tType]: (((topological_t1_space @ A) => (! [S3:(set @ A),X:A]: ((((topolo1751647064n_open @ A) @ S3) => ((topolo1751647064n_open @ A) @ (((minus_minus @ (set @ A)) @ S3) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))))))).
% 2.45/1.46 thf(fact_189_Int__Diff__disjoint, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A2) @ B2)) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) = (bot_bot @ (set @ A)))))).
% 2.45/1.46 thf(fact_190_is__singleton__the__elem, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A5:(set @ A)]: ((A5 = (((insert @ A) @ ((the_elem @ A) @ A5)) @ (bot_bot @ (set @ A)))))))))).
% 2.45/1.46 thf(fact_191_ball__insert, axiom, (! [A:$tType,A3:A,B2:(set @ A),P:(A > $o)]: (((! [X2:A]: (((((member @ A) @ X2) @ (((insert @ A) @ A3) @ B2)) => (P @ X2)))) = ((P @ A3) & (! [X2:A]: (((((member @ A) @ X2) @ B2) => (P @ X2))))))))).
% 2.45/1.46 thf(fact_192_the__elem__eq, axiom, (! [A:$tType,X:A]: ((((the_elem @ A) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = X)))).
% 2.45/1.46 thf(fact_193_diff__eq__diff__less__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B3:A,C:A,D2:A]: ((((((minus_minus @ A) @ A3) @ B3) = (((minus_minus @ A) @ C) @ D2)) => ((((ord_less_eq @ A) @ A3) @ B3) = (((ord_less_eq @ A) @ C) @ D2))))))))).
% 2.45/1.46 thf(fact_194_empty__subsetI, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A2)))).
% 2.45/1.46 thf(fact_195_subset__empty, axiom, (! [A:$tType,A2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ (bot_bot @ (set @ A))) = (A2 = (bot_bot @ (set @ A))))))).
% 2.45/1.46 thf(fact_196_insert__subset, axiom, (! [A:$tType,X:A,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X) @ A2)) @ B2) = ((((member @ A) @ X) @ B2) & (((ord_less_eq @ (set @ A)) @ A2) @ B2)))))).
% 2.45/1.46 thf(fact_197_Int__subset__iff, axiom, (! [A:$tType,C2:(set @ A),A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ C2) @ (((inf_inf @ (set @ A)) @ A2) @ B2)) = ((((ord_less_eq @ (set @ A)) @ C2) @ A2) & (((ord_less_eq @ (set @ A)) @ C2) @ B2)))))).
% 2.45/1.46 thf(fact_198_singleton__insert__inj__eq, axiom, (! [A:$tType,B3:A,A3:A,A2:(set @ A)]: ((((((insert @ A) @ B3) @ (bot_bot @ (set @ A))) = (((insert @ A) @ A3) @ A2)) = ((A3 = B3) & (((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A))))))))).
% 2.45/1.46 thf(fact_199_singleton__insert__inj__eq_H, axiom, (! [A:$tType,A3:A,A2:(set @ A),B3:A]: ((((((insert @ A) @ A3) @ A2) = (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))) = ((A3 = B3) & (((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A))))))))).
% 2.45/1.46 thf(fact_200_Diff__eq__empty__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: ((((((minus_minus @ (set @ A)) @ A2) @ B2) = (bot_bot @ (set @ A))) = (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.45/1.46 thf(fact_201_subset__insertI2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),B3:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ B3) @ B2)))))).
% 2.45/1.46 thf(fact_202_subset__insertI, axiom, (! [A:$tType,B2:(set @ A),A3:A]: ((((ord_less_eq @ (set @ A)) @ B2) @ (((insert @ A) @ A3) @ B2))))).
% 2.45/1.46 thf(fact_203_subset__insert, axiom, (! [A:$tType,X:A,A2:(set @ A),B2:(set @ A)]: (((~ ((((member @ A) @ X) @ A2))) => ((((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ X) @ B2)) = (((ord_less_eq @ (set @ A)) @ A2) @ B2)))))).
% 2.45/1.46 thf(fact_204_insert__mono, axiom, (! [A:$tType,C2:(set @ A),D3:(set @ A),A3:A]: (((((ord_less_eq @ (set @ A)) @ C2) @ D3) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ A3) @ C2)) @ (((insert @ A) @ A3) @ D3)))))).
% 2.45/1.46 thf(fact_205_Int__Collect__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((! [X4:A]: (((((member @ A) @ X4) @ A2) => ((P @ X4) => (Q @ X4))))) => (((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A2) @ ((collect @ A) @ P))) @ (((inf_inf @ (set @ A)) @ B2) @ ((collect @ A) @ Q)))))))).
% 2.45/1.46 thf(fact_206_Int__greatest, axiom, (! [A:$tType,C2:(set @ A),A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ C2) @ A2) => ((((ord_less_eq @ (set @ A)) @ C2) @ B2) => (((ord_less_eq @ (set @ A)) @ C2) @ (((inf_inf @ (set @ A)) @ A2) @ B2))))))).
% 2.45/1.46 thf(fact_207_Int__absorb2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((inf_inf @ (set @ A)) @ A2) @ B2) = A2))))).
% 2.45/1.46 thf(fact_208_Int__absorb1, axiom, (! [A:$tType,B2:(set @ A),A2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B2) @ A2) => ((((inf_inf @ (set @ A)) @ A2) @ B2) = B2))))).
% 2.45/1.46 thf(fact_209_Int__lower2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A2) @ B2)) @ B2)))).
% 2.45/1.46 thf(fact_210_Int__lower1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A2) @ B2)) @ A2)))).
% 2.45/1.46 thf(fact_211_Int__mono, axiom, (! [A:$tType,A2:(set @ A),C2:(set @ A),B2:(set @ A),D3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ C2) => ((((ord_less_eq @ (set @ A)) @ B2) @ D3) => (((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A2) @ B2)) @ (((inf_inf @ (set @ A)) @ C2) @ D3))))))).
% 2.45/1.46 thf(fact_212_first__countable__basis, axiom, (! [A:$tType]: (((topolo2135403230pology @ A) => (! [X:A]: ((? [A7:(nat > (set @ A))]: (((! [I:nat]: (((((member @ A) @ X) @ (A7 @ I)) & ((topolo1751647064n_open @ A) @ (A7 @ I))))) & (! [S2:(set @ A)]: (((((topolo1751647064n_open @ A) @ S2) & (((member @ A) @ X) @ S2)) => (? [I2:nat]: ((((ord_less_eq @ (set @ A)) @ (A7 @ I2)) @ S2))))))))))))))).
% 2.45/1.46 thf(fact_213_open__subopen, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => ((topolo1751647064n_open @ A) = (^ [S4:(set @ A)]: ((! [X2:A]: (((((member @ A) @ X2) @ S4) => (? [T3:(set @ A)]: ((((topolo1751647064n_open @ A) @ T3) & ((((member @ A) @ X2) @ T3) & (((ord_less_eq @ (set @ A)) @ T3) @ S4))))))))))))))).
% 2.45/1.46 thf(fact_214_topological__space__class_OopenI, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [S:(set @ A)]: (((! [X4:A]: (((((member @ A) @ X4) @ S) => (? [T4:(set @ A)]: ((((topolo1751647064n_open @ A) @ T4) & ((((member @ A) @ X4) @ T4) & (((ord_less_eq @ (set @ A)) @ T4) @ S)))))))) => ((topolo1751647064n_open @ A) @ S)))))))).
% 2.45/1.46 thf(fact_215_double__diff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C2) => ((((minus_minus @ (set @ A)) @ B2) @ (((minus_minus @ (set @ A)) @ C2) @ A2)) = A2)))))).
% 2.45/1.46 thf(fact_216_Diff__subset, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) @ A2)))).
% 2.45/1.46 thf(fact_217_Diff__mono, axiom, (! [A:$tType,A2:(set @ A),C2:(set @ A),D3:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ C2) => ((((ord_less_eq @ (set @ A)) @ D3) @ B2) => (((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) @ (((minus_minus @ (set @ A)) @ C2) @ D3))))))).
% 2.45/1.46 thf(fact_218_totally__bounded__subset, axiom, (! [A:$tType]: (((topolo47006728_space @ A) => (! [S:(set @ A),T:(set @ A)]: ((((topolo406746546ounded @ A) @ S) => ((((ord_less_eq @ (set @ A)) @ T) @ S) => ((topolo406746546ounded @ A) @ T))))))))).
% 2.45/1.46 thf(fact_219_subset__singletonD, axiom, (! [A:$tType,A2:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) => ((A2 = (bot_bot @ (set @ A))) | (A2 = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))).
% 2.45/1.46 thf(fact_220_subset__singleton__iff, axiom, (! [A:$tType,X5:(set @ A),A3:A]: (((((ord_less_eq @ (set @ A)) @ X5) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) = ((X5 = (bot_bot @ (set @ A))) | (X5 = (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))))).
% 2.45/1.46 thf(fact_221_subset__Diff__insert, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X:A,C2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ (((minus_minus @ (set @ A)) @ B2) @ (((insert @ A) @ X) @ C2))) = ((((ord_less_eq @ (set @ A)) @ A2) @ (((minus_minus @ (set @ A)) @ B2) @ C2)) & (~ ((((member @ A) @ X) @ A2)))))))).
% 2.45/1.46 thf(fact_222_diff__eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B3:A,C:A,D2:A]: ((((((minus_minus @ A) @ A3) @ B3) = (((minus_minus @ A) @ C) @ D2)) => ((A3 = B3) = (C = D2))))))))).
% 2.45/1.46 thf(fact_223_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A3:A,C:A,B3:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A3) @ C)) @ B3) = (((minus_minus @ A) @ (((minus_minus @ A) @ A3) @ B3)) @ C)))))))).
% 2.45/1.46 thf(fact_224_Diff__single__insert, axiom, (! [A:$tType,A2:(set @ A),X:A,B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) @ B2) => (((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ X) @ B2)))))).
% 2.45/1.46 thf(fact_225_subset__insert__iff, axiom, (! [A:$tType,A2:(set @ A),X:A,B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ X) @ B2)) = (((((member @ A) @ X) @ A2) => (((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) @ B2)) & ((~ ((((member @ A) @ X) @ A2))) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))))).
% 2.45/1.46 thf(fact_226_diff__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B3:A,D2:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ D2) @ C) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A3) @ C)) @ (((minus_minus @ A) @ B3) @ D2)))))))))).
% 2.45/1.46 thf(fact_227_diff__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B3:A,A3:A,C:A]: (((((ord_less_eq @ A) @ B3) @ A3) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C) @ A3)) @ (((minus_minus @ A) @ C) @ B3))))))))).
% 2.45/1.46 thf(fact_228_diff__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B3) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A3) @ C)) @ (((minus_minus @ A) @ B3) @ C))))))))).
% 2.45/1.46 thf(fact_229_remove__def, axiom, (! [A:$tType]: (((remove @ A) = (^ [X2:A,A5:(set @ A)]: ((((minus_minus @ (set @ A)) @ A5) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))))))))).
% 2.45/1.46 thf(fact_230_at__within__nhd, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [X:A,S:(set @ A),T:(set @ A),U3:(set @ A)]: (((((member @ A) @ X) @ S) => (((topolo1751647064n_open @ A) @ S) => (((((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ T) @ S)) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = (((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ U3) @ S)) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) => ((((topolo507301023within @ A) @ X) @ T) = (((topolo507301023within @ A) @ X) @ U3))))))))))).
% 2.45/1.46 thf(fact_231_subsetI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((! [X4:A]: (((((member @ A) @ X4) @ A2) => (((member @ A) @ X4) @ B2)))) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.45/1.46 thf(fact_232_subset__antisym, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A2) => (A2 = B2)))))).
% 2.45/1.46 thf(fact_233_member__remove, axiom, (! [A:$tType,X:A,Y:A,A2:(set @ A)]: (((((member @ A) @ X) @ (((remove @ A) @ Y) @ A2)) = ((((member @ A) @ X) @ A2) & (X != Y)))))).
% 2.45/1.46 thf(fact_234_at__within__empty, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [A3:A]: (((((topolo507301023within @ A) @ A3) @ (bot_bot @ (set @ A))) = (bot_bot @ (filter @ A))))))))).
% 2.45/1.46 thf(fact_235_in__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ X) @ A2) => (((member @ A) @ X) @ B2)))))).
% 2.45/1.46 thf(fact_236_subsetD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ C) @ A2) => (((member @ A) @ C) @ B2)))))).
% 2.45/1.46 thf(fact_237_equalityE, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (~ ((((ord_less_eq @ (set @ A)) @ B2) @ A2)))))))))).
% 2.45/1.46 thf(fact_238_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: ((! [X2:A]: (((((member @ A) @ X2) @ A5) => (((member @ A) @ X2) @ B5)))))))))).
% 2.45/1.46 thf(fact_239_equalityD1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.45/1.46 thf(fact_240_equalityD2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A2))))).
% 2.45/1.46 thf(fact_241_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: ((! [T5:A]: (((((member @ A) @ T5) @ A5) => (((member @ A) @ T5) @ B5)))))))))).
% 2.45/1.46 thf(fact_242_subset__refl, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ A2)))).
% 2.45/1.46 thf(fact_243_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) => (Q @ X4)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.45/1.46 thf(fact_244_subset__trans, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C2) => (((ord_less_eq @ (set @ A)) @ A2) @ C2)))))).
% 2.45/1.46 thf(fact_245_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y4:(set @ A),Z2:(set @ A)]: ((Y4 = Z2))) = (^ [A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) & (((ord_less_eq @ (set @ A)) @ B5) @ A5)))))))).
% 2.45/1.46 thf(fact_246_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X2:A]: (((P @ X2) => (Q @ X2)))))))).
% 2.45/1.46 thf(fact_247_at__le, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [S3:(set @ A),T6:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ S3) @ T6) => (((ord_less_eq @ (filter @ A)) @ (((topolo507301023within @ A) @ X) @ S3)) @ (((topolo507301023within @ A) @ X) @ T6))))))))).
% 2.45/1.46 thf(fact_248_at__discrete, axiom, (! [A:$tType]: (((topolo2133971006pology @ A) => ((topolo507301023within @ A) = (^ [X2:A,S4:(set @ A)]: ((bot_bot @ (filter @ A))))))))).
% 2.45/1.46 thf(fact_249_insert__subsetI, axiom, (! [A:$tType,X:A,A2:(set @ A),X5:(set @ A)]: (((((member @ A) @ X) @ A2) => ((((ord_less_eq @ (set @ A)) @ X5) @ A2) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X) @ X5)) @ A2)))))).
% 2.45/1.46 thf(fact_250_subset__emptyI, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X4:A]: ((~ ((((member @ A) @ X4) @ A2))))) => (((ord_less_eq @ (set @ A)) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.45/1.46 thf(fact_251_at__eq__bot__iff, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [A3:A]: ((((((topolo507301023within @ A) @ A3) @ (top_top @ (set @ A))) = (bot_bot @ (filter @ A))) = ((topolo1751647064n_open @ A) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))))))).
% 2.45/1.46 thf(fact_252_at__within__eq__bot__iff, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [C:A,A2:(set @ A)]: ((((((topolo507301023within @ A) @ C) @ A2) = (bot_bot @ (filter @ A))) = (~ ((((member @ A) @ C) @ ((elementary_closure @ A) @ (((minus_minus @ (set @ A)) @ A2) @ (((insert @ A) @ C) @ (bot_bot @ (set @ A))))))))))))))).
% 2.45/1.46 thf(fact_253_top__apply, axiom, (! [C3:$tType,D:$tType]: (((top @ C3) => ((top_top @ (D > C3)) = (^ [X2:D]: ((top_top @ C3)))))))).
% 2.45/1.46 thf(fact_254_UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 2.45/1.46 thf(fact_255_closure__closure, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [S:(set @ A)]: ((((elementary_closure @ A) @ ((elementary_closure @ A) @ S)) = ((elementary_closure @ A) @ S)))))))).
% 2.45/1.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Lattices_Obounded__lattice, axiom, (! [A:$tType]: (((comple1035589618norder @ A) => (bounded_lattice @ A))))).
% 2.45/1.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___HOL_Otype, axiom, (! [A:$tType]: (((comple1035589618norder @ A) => (type @ A))))).
% 2.45/1.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Obot, axiom, (! [A:$tType]: (((comple1035589618norder @ A) => (bot @ A))))).
% 2.45/1.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Oord, axiom, (! [A:$tType]: (((comple1035589618norder @ A) => (ord @ A))))).
% 2.45/1.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Otop, axiom, (! [A:$tType]: (((comple1035589618norder @ A) => (top @ A))))).
% 2.45/1.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Oorder, axiom, (! [A:$tType]: (((comple1035589618norder @ A) => (order @ A))))).
% 2.45/1.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Lattices_Olattice, axiom, (! [A:$tType]: (((comple1035589618norder @ A) => (lattice @ A))))).
% 2.45/1.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Olinorder, axiom, (! [A:$tType]: (((comple1035589618norder @ A) => (linorder @ A))))).
% 2.45/1.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Opreorder, axiom, (! [A:$tType]: (((comple1035589618norder @ A) => (preorder @ A))))).
% 2.45/1.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Oorder__bot, axiom, (! [A:$tType]: (((comple1035589618norder @ A) => (order_bot @ A))))).
% 2.45/1.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Lattices_Osemilattice__inf, axiom, (! [A:$tType]: (((comple1035589618norder @ A) => (semilattice_inf @ A))))).
% 2.45/1.46 thf(subcl_Complete__Lattices_Ocomplete__linorder___Lattices_Obounded__lattice__bot, axiom, (! [A:$tType]: (((comple1035589618norder @ A) => (bounded_lattice_bot @ A))))).
% 2.45/1.46 thf(subcl_Topological__Spaces_Olinorder__topology___HOL_Otype, axiom, (! [A:$tType]: (((topolo2117631714pology @ A) => (type @ A))))).
% 2.45/1.46 thf(subcl_Topological__Spaces_Olinorder__topology___Orderings_Oord, axiom, (! [A:$tType]: (((topolo2117631714pology @ A) => (ord @ A))))).
% 2.45/1.46 thf(subcl_Topological__Spaces_Olinorder__topology___Orderings_Oorder, axiom, (! [A:$tType]: (((topolo2117631714pology @ A) => (order @ A))))).
% 2.45/1.46 thf(subcl_Topological__Spaces_Olinorder__topology___Orderings_Olinorder, axiom, (! [A:$tType]: (((topolo2117631714pology @ A) => (linorder @ A))))).
% 2.45/1.46 thf(subcl_Topological__Spaces_Olinorder__topology___Orderings_Opreorder, axiom, (! [A:$tType]: (((topolo2117631714pology @ A) => (preorder @ A))))).
% 2.45/1.46 thf(subcl_Topological__Spaces_Olinorder__topology___Topological__Spaces_Ot0__space, axiom, (! [A:$tType]: (((topolo2117631714pology @ A) => (topological_t0_space @ A))))).
% 2.45/1.46 thf(subcl_Topological__Spaces_Olinorder__topology___Topological__Spaces_Ot1__space, axiom, (! [A:$tType]: (((topolo2117631714pology @ A) => (topological_t1_space @ A))))).
% 2.45/1.46 thf(subcl_Topological__Spaces_Olinorder__topology___Topological__Spaces_Ot2__space, axiom, (! [A:$tType]: (((topolo2117631714pology @ A) => (topological_t2_space @ A))))).
% 2.45/1.46 thf(subcl_Topological__Spaces_Olinorder__topology___Topological__Spaces_Otopological__space, axiom, (! [A:$tType]: (((topolo2117631714pology @ A) => (topolo503727757_space @ A))))).
% 2.45/1.46 thf(subcl_Topological__Spaces_Ofirst__countable__topology___HOL_Otype, axiom, (! [A:$tType]: (((topolo2135403230pology @ A) => (type @ A))))).
% 2.45/1.46 thf(subcl_Topological__Spaces_Ofirst__countable__topology___Topological__Spaces_Otopological__space, axiom, (! [A:$tType]: (((topolo2135403230pology @ A) => (topolo503727757_space @ A))))).
% 2.45/1.46 thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice, axiom, (! [A8:$tType]: ((bounded_lattice @ (filter @ A8))))).
% 2.45/1.46 thf(tcon_HOL_Obool___Countable_Ocountable, axiom, (countable @ $o)).
% 2.45/1.46 thf(tcon_HOL_Obool___Lattices_Obounded__lattice_1, axiom, (bounded_lattice @ $o)).
% 2.45/1.46 thf(tcon_Set_Oset___Countable_Ocountable_2, axiom, (! [A8:$tType]: (((finite_finite @ A8) => (countable @ (set @ A8)))))).
% 2.45/1.46 thf(tcon_Set_Oset___Lattices_Obounded__lattice_3, axiom, (! [A8:$tType]: ((bounded_lattice @ (set @ A8))))).
% 2.45/1.46 thf(tcon_Nat_Onat___Countable_Ocountable_4, axiom, (countable @ nat)).
% 2.45/1.46 thf(tcon_fun___Countable_Ocountable_5, axiom, (! [A8:$tType,A9:$tType]: ((((finite_finite @ A8) & (countable @ A9)) => (countable @ (A8 > A9)))))).
% 2.45/1.46 thf(tcon_fun___Lattices_Obounded__lattice_6, axiom, (! [A8:$tType,A9:$tType]: (((bounded_lattice @ A9) => (bounded_lattice @ (A8 > A9)))))).
% 2.45/1.46 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A8:$tType,A9:$tType]: ((((finite_finite @ A8) & (finite_finite @ A9)) => (finite_finite @ (A8 > A9)))))).
% 2.45/1.46 thf(tcon_Set_Oset___Finite__Set_Ofinite_7, axiom, (! [A8:$tType]: (((finite_finite @ A8) => (finite_finite @ (set @ A8)))))).
% 2.45/1.46 thf(tcon_HOL_Obool___Finite__Set_Ofinite_8, axiom, (finite_finite @ $o)).
% 2.45/1.46 thf(tcon_fun___Topological__Spaces_Ofirst__countable__topology, axiom, (! [A8:$tType,A9:$tType]: ((((countable @ A8) & (topolo2135403230pology @ A9)) => (topolo2135403230pology @ (A8 > A9)))))).
% 2.45/1.46 thf(tcon_fun___Topological__Spaces_Otopological__space, axiom, (! [A8:$tType,A9:$tType]: (((topolo503727757_space @ A9) => (topolo503727757_space @ (A8 > A9)))))).
% 2.45/1.46 thf(tcon_fun___Lattices_Obounded__lattice__bot, axiom, (! [A8:$tType,A9:$tType]: (((bounded_lattice @ A9) => (bounded_lattice_bot @ (A8 > A9)))))).
% 2.45/1.46 thf(tcon_fun___Lattices_Osemilattice__inf, axiom, (! [A8:$tType,A9:$tType]: (((semilattice_inf @ A9) => (semilattice_inf @ (A8 > A9)))))).
% 2.45/1.46 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A8:$tType,A9:$tType]: (((order_bot @ A9) => (order_bot @ (A8 > A9)))))).
% 2.45/1.46 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A8 > A9)))))).
% 2.45/1.46 thf(tcon_fun___Lattices_Olattice, axiom, (! [A8:$tType,A9:$tType]: (((lattice @ A9) => (lattice @ (A8 > A9)))))).
% 2.45/1.46 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: (((order @ A9) => (order @ (A8 > A9)))))).
% 2.45/1.46 thf(tcon_fun___Orderings_Otop, axiom, (! [A8:$tType,A9:$tType]: (((top @ A9) => (top @ (A8 > A9)))))).
% 2.45/1.46 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: (((ord @ A9) => (ord @ (A8 > A9)))))).
% 2.45/1.46 thf(tcon_fun___Orderings_Obot, axiom, (! [A8:$tType,A9:$tType]: (((bot @ A9) => (bot @ (A8 > A9)))))).
% 2.45/1.46 thf(tcon_fun___Groups_Ominus, axiom, (! [A8:$tType,A9:$tType]: (((minus @ A9) => (minus @ (A8 > A9)))))).
% 2.45/1.46 thf(tcon_Nat_Onat___Topological__Spaces_Ofirst__countable__topology_9, axiom, (topolo2135403230pology @ nat)).
% 2.45/1.46 thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_10, axiom, (topolo503727757_space @ nat)).
% 2.45/1.46 thf(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology, axiom, (topolo2133971006pology @ nat)).
% 2.45/1.46 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add, axiom, (cancel146912293up_add @ nat)).
% 2.45/1.46 thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space, axiom, (topological_t2_space @ nat)).
% 2.45/1.46 thf(tcon_Nat_Onat___Topological__Spaces_Ot1__space, axiom, (topological_t1_space @ nat)).
% 2.45/1.46 thf(tcon_Nat_Onat___Topological__Spaces_Ot0__space, axiom, (topological_t0_space @ nat)).
% 2.45/1.46 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_11, axiom, (semilattice_inf @ nat)).
% 2.45/1.46 thf(tcon_Nat_Onat___Orderings_Oorder__bot_12, axiom, (order_bot @ nat)).
% 2.45/1.46 thf(tcon_Nat_Onat___Orderings_Opreorder_13, axiom, (preorder @ nat)).
% 2.45/1.46 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 2.45/1.46 thf(tcon_Nat_Onat___Lattices_Olattice_14, axiom, (lattice @ nat)).
% 2.45/1.46 thf(tcon_Nat_Onat___Orderings_Oorder_15, axiom, (order @ nat)).
% 2.45/1.46 thf(tcon_Nat_Onat___Orderings_Oord_16, axiom, (ord @ nat)).
% 2.45/1.46 thf(tcon_Nat_Onat___Orderings_Obot_17, axiom, (bot @ nat)).
% 2.45/1.46 thf(tcon_Nat_Onat___Groups_Ominus_18, axiom, (minus @ nat)).
% 2.45/1.46 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_19, axiom, (! [A8:$tType]: ((bounded_lattice_bot @ (set @ A8))))).
% 2.45/1.46 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_20, axiom, (! [A8:$tType]: ((semilattice_inf @ (set @ A8))))).
% 2.45/1.46 thf(tcon_Set_Oset___Orderings_Oorder__bot_21, axiom, (! [A8:$tType]: ((order_bot @ (set @ A8))))).
% 2.45/1.46 thf(tcon_Set_Oset___Orderings_Opreorder_22, axiom, (! [A8:$tType]: ((preorder @ (set @ A8))))).
% 2.45/1.46 thf(tcon_Set_Oset___Lattices_Olattice_23, axiom, (! [A8:$tType]: ((lattice @ (set @ A8))))).
% 2.45/1.46 thf(tcon_Set_Oset___Orderings_Oorder_24, axiom, (! [A8:$tType]: ((order @ (set @ A8))))).
% 2.45/1.46 thf(tcon_Set_Oset___Orderings_Otop_25, axiom, (! [A8:$tType]: ((top @ (set @ A8))))).
% 2.45/1.46 thf(tcon_Set_Oset___Orderings_Oord_26, axiom, (! [A8:$tType]: ((ord @ (set @ A8))))).
% 2.45/1.46 thf(tcon_Set_Oset___Orderings_Obot_27, axiom, (! [A8:$tType]: ((bot @ (set @ A8))))).
% 2.45/1.46 thf(tcon_Set_Oset___Groups_Ominus_28, axiom, (! [A8:$tType]: ((minus @ (set @ A8))))).
% 2.45/1.46 thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_29, axiom, (topolo503727757_space @ $o)).
% 2.45/1.46 thf(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_30, axiom, (topolo2133971006pology @ $o)).
% 2.45/1.46 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_31, axiom, (bounded_lattice_bot @ $o)).
% 2.45/1.46 thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_32, axiom, (topological_t2_space @ $o)).
% 2.45/1.46 thf(tcon_HOL_Obool___Topological__Spaces_Ot1__space_33, axiom, (topological_t1_space @ $o)).
% 2.45/1.46 thf(tcon_HOL_Obool___Topological__Spaces_Ot0__space_34, axiom, (topological_t0_space @ $o)).
% 2.45/1.46 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_35, axiom, (semilattice_inf @ $o)).
% 2.45/1.46 thf(tcon_HOL_Obool___Orderings_Oorder__bot_36, axiom, (order_bot @ $o)).
% 2.45/1.46 thf(tcon_HOL_Obool___Orderings_Opreorder_37, axiom, (preorder @ $o)).
% 2.45/1.46 thf(tcon_HOL_Obool___Orderings_Olinorder_38, axiom, (linorder @ $o)).
% 2.45/1.46 thf(tcon_HOL_Obool___Lattices_Olattice_39, axiom, (lattice @ $o)).
% 2.45/1.46 thf(tcon_HOL_Obool___Orderings_Oorder_40, axiom, (order @ $o)).
% 2.45/1.46 thf(tcon_HOL_Obool___Orderings_Otop_41, axiom, (top @ $o)).
% 2.45/1.46 thf(tcon_HOL_Obool___Orderings_Oord_42, axiom, (ord @ $o)).
% 2.45/1.46 thf(tcon_HOL_Obool___Orderings_Obot_43, axiom, (bot @ $o)).
% 2.45/1.46 thf(tcon_HOL_Obool___Groups_Ominus_44, axiom, (minus @ $o)).
% 2.45/1.46 thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_45, axiom, (! [A8:$tType]: ((bounded_lattice_bot @ (filter @ A8))))).
% 2.45/1.46 thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_46, axiom, (! [A8:$tType]: ((semilattice_inf @ (filter @ A8))))).
% 2.45/1.46 thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_47, axiom, (! [A8:$tType]: ((order_bot @ (filter @ A8))))).
% 2.45/1.46 thf(tcon_Filter_Ofilter___Orderings_Opreorder_48, axiom, (! [A8:$tType]: ((preorder @ (filter @ A8))))).
% 2.45/1.46 thf(tcon_Filter_Ofilter___Lattices_Olattice_49, axiom, (! [A8:$tType]: ((lattice @ (filter @ A8))))).
% 2.45/1.46 thf(tcon_Filter_Ofilter___Orderings_Oorder_50, axiom, (! [A8:$tType]: ((order @ (filter @ A8))))).
% 2.45/1.46 thf(tcon_Filter_Ofilter___Orderings_Otop_51, axiom, (! [A8:$tType]: ((top @ (filter @ A8))))).
% 2.45/1.46 thf(tcon_Filter_Ofilter___Orderings_Oord_52, axiom, (! [A8:$tType]: ((ord @ (filter @ A8))))).
% 2.45/1.46 thf(tcon_Filter_Ofilter___Orderings_Obot_53, axiom, (! [A8:$tType]: ((bot @ (filter @ A8))))).
% 2.45/1.46 thf(tfree_0, hypothesis, (comple1035589618norder @ b)).
% 2.45/1.46 thf(tfree_1, hypothesis, (topolo2117631714pology @ b)).
% 2.45/1.46 thf(tfree_2, hypothesis, (topolo2135403230pology @ a)).
% 2.45/1.46 thf(conj_0, hypothesis, (! [S2:(set @ b),V3:(set @ b)]: (((((topolo1751647064n_open @ b) @ S2) & (((topolo1751647064n_open @ b) @ V3) & ((((member @ b) @ (f @ x0)) @ S2) & ((((member @ b) @ a2) @ V3) & ((((inf_inf @ (set @ b)) @ S2) @ V3) = (bot_bot @ (set @ b))))))) => thesis)))).
% 2.45/1.46 thf(conj_1, conjecture, thesis).
% 2.45/1.46 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.wdWkGyYZoX/DTF2DTF5301.p
% 2.45/1.46 ------------------------
% 2.64/1.51 ---- Cleaned THF ---
% 2.64/1.51 thf(ty_t_Filter_Ofilter,type,
% 2.64/1.51 filter: $tType > $tType ).
% 2.64/1.51
% 2.64/1.51 thf(ty_t_Set_Oset,type,
% 2.64/1.51 set: $tType > $tType ).
% 2.64/1.51
% 2.64/1.51 thf(ty_t_Nat_Onat,type,
% 2.64/1.51 nat: $tType ).
% 2.64/1.51
% 2.64/1.51 thf(ty_tf_b,type,
% 2.64/1.51 b: $tType ).
% 2.64/1.51
% 2.64/1.51 thf(ty_tf_a,type,
% 2.64/1.51 a: $tType ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Topological__Spaces_Olinorder__topology,type,
% 2.64/1.51 topolo2117631714pology:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
% 2.64/1.51 comple1035589618norder:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Finite__Set_Ofinite,type,
% 2.64/1.51 finite_finite:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Lattices_Obounded__lattice,type,
% 2.64/1.51 bounded_lattice:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_HOL_Otype,type,
% 2.64/1.51 type:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Countable_Ocountable,type,
% 2.64/1.51 countable:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Groups_Ominus,type,
% 2.64/1.51 minus:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Orderings_Obot,type,
% 2.64/1.51 bot:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Orderings_Oord,type,
% 2.64/1.51 ord:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Orderings_Otop,type,
% 2.64/1.51 top:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Orderings_Oorder,type,
% 2.64/1.51 order:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Groups_Ogroup__add,type,
% 2.64/1.51 group_add:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Lattices_Olattice,type,
% 2.64/1.51 lattice:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Orderings_Olinorder,type,
% 2.64/1.51 linorder:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Orderings_Opreorder,type,
% 2.64/1.51 preorder:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Orderings_Oorder__bot,type,
% 2.64/1.51 order_bot:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 2.64/1.51 semilattice_inf:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.64/1.51 ordered_ab_group_add:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Topological__Spaces_Ot0__space,type,
% 2.64/1.51 topological_t0_space:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Topological__Spaces_Ot1__space,type,
% 2.64/1.51 topological_t1_space:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Topological__Spaces_Ot2__space,type,
% 2.64/1.51 topological_t2_space:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Lattices_Obounded__lattice__bot,type,
% 2.64/1.51 bounded_lattice_bot:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.64/1.51 cancel146912293up_add:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Topological__Spaces_Operfect__space,type,
% 2.64/1.51 topolo890362671_space:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Topological__Spaces_Ouniform__space,type,
% 2.64/1.51 topolo47006728_space:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Topological__Spaces_Odiscrete__topology,type,
% 2.64/1.51 topolo2133971006pology:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Topological__Spaces_Otopological__space,type,
% 2.64/1.51 topolo503727757_space:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_cl_Topological__Spaces_Ofirst__countable__topology,type,
% 2.64/1.51 topolo2135403230pology:
% 2.64/1.51 !>[A: $tType] : $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_c_Elementary__Topology_Oclosure,type,
% 2.64/1.51 elementary_closure:
% 2.64/1.51 !>[A: $tType] : ( ( set @ A ) > ( set @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.64/1.51 minus_minus:
% 2.64/1.51 !>[A: $tType] : ( A > A > A ) ).
% 2.64/1.51
% 2.64/1.51 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 2.64/1.51 inf_inf:
% 2.64/1.51 !>[A: $tType] : ( A > A > A ) ).
% 2.64/1.51
% 2.64/1.51 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc__at,type,
% 2.64/1.51 lower_582600101lsc_at:
% 2.64/1.51 !>[A: $tType,B: $tType] : ( A > ( A > B ) > $o ) ).
% 2.64/1.51
% 2.64/1.51 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.64/1.51 bot_bot:
% 2.64/1.51 !>[A: $tType] : A ).
% 2.64/1.51
% 2.64/1.51 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.64/1.51 ord_less_eq:
% 2.64/1.51 !>[A: $tType] : ( A > A > $o ) ).
% 2.64/1.51
% 2.64/1.51 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.64/1.51 top_top:
% 2.64/1.51 !>[A: $tType] : A ).
% 2.64/1.51
% 2.64/1.51 thf(sy_c_Set_OCollect,type,
% 2.64/1.51 collect:
% 2.64/1.51 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(sy_c_Set_Oinsert,type,
% 2.64/1.51 insert:
% 2.64/1.51 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(sy_c_Set_Ois__empty,type,
% 2.64/1.51 is_empty:
% 2.64/1.51 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.64/1.51
% 2.64/1.51 thf(sy_c_Set_Ois__singleton,type,
% 2.64/1.51 is_singleton:
% 2.64/1.51 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.64/1.51
% 2.64/1.51 thf(sy_c_Set_Oremove,type,
% 2.64/1.51 remove:
% 2.64/1.51 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(sy_c_Set_Othe__elem,type,
% 2.64/1.51 the_elem:
% 2.64/1.51 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.64/1.51
% 2.64/1.51 thf(sy_c_Topological__Spaces_Oopen__class_Oopen,type,
% 2.64/1.51 topolo1751647064n_open:
% 2.64/1.51 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.64/1.51
% 2.64/1.51 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type,
% 2.64/1.51 topolo507301023within:
% 2.64/1.51 !>[A: $tType] : ( A > ( set @ A ) > ( filter @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type,
% 2.64/1.51 topolo406746546ounded:
% 2.64/1.51 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.64/1.51
% 2.64/1.51 thf(sy_c_member,type,
% 2.64/1.51 member:
% 2.64/1.51 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.64/1.51
% 2.64/1.51 thf(sy_v_A____,type,
% 2.64/1.51 a2: b ).
% 2.64/1.51
% 2.64/1.51 thf(sy_v_f,type,
% 2.64/1.51 f: a > b ).
% 2.64/1.51
% 2.64/1.51 thf(sy_v_thesis____,type,
% 2.64/1.51 thesis: $o ).
% 2.64/1.51
% 2.64/1.51 thf(sy_v_x0,type,
% 2.64/1.51 x0: a ).
% 2.64/1.51
% 2.64/1.51 thf(fact_0__092_060open_062A_A_092_060noteq_062_Af_Ax0_092_060close_062,axiom,
% 2.64/1.51 ( a2
% 2.64/1.51 != ( f @ x0 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_1__092_060open_062f_Ax0_A_092_060noteq_062_AA_A_092_060Longrightarrow_062_A_092_060exists_062U_AV_O_Aopen_AU_A_092_060and_062_Aopen_AV_A_092_060and_062_Af_Ax0_A_092_060in_062_AU_A_092_060and_062_AA_A_092_060in_062_AV_A_092_060and_062_AU_A_092_060inter_062_AV_A_061_A_123_125_092_060close_062,axiom,
% 2.64/1.51 ( ( ( f @ x0 )
% 2.64/1.51 != a2 )
% 2.64/1.51 => ? [U: set @ b,V: set @ b] :
% 2.64/1.51 ( ( topolo1751647064n_open @ b @ U )
% 2.64/1.51 & ( topolo1751647064n_open @ b @ V )
% 2.64/1.51 & ( member @ b @ ( f @ x0 ) @ U )
% 2.64/1.51 & ( member @ b @ a2 @ V )
% 2.64/1.51 & ( ( inf_inf @ ( set @ b ) @ U @ V )
% 2.64/1.51 = ( bot_bot @ ( set @ b ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_2_open__Int,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo503727757_space @ A )
% 2.64/1.51 => ! [S: set @ A,T: set @ A] :
% 2.64/1.51 ( ( topolo1751647064n_open @ A @ S )
% 2.64/1.51 => ( ( topolo1751647064n_open @ A @ T )
% 2.64/1.51 => ( topolo1751647064n_open @ A @ ( inf_inf @ ( set @ A ) @ S @ T ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_3_open__empty,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo503727757_space @ A )
% 2.64/1.51 => ( topolo1751647064n_open @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_4_inf__bot__left,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( bounded_lattice_bot @ A )
% 2.64/1.51 => ! [X: A] :
% 2.64/1.51 ( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
% 2.64/1.51 = ( bot_bot @ A ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_5_inf__bot__right,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( bounded_lattice_bot @ A )
% 2.64/1.51 => ! [X: A] :
% 2.64/1.51 ( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
% 2.64/1.51 = ( bot_bot @ A ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_6_hausdorff,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topological_t2_space @ A )
% 2.64/1.51 => ! [X: A,Y: A] :
% 2.64/1.51 ( ( X != Y )
% 2.64/1.51 => ? [U: set @ A,V: set @ A] :
% 2.64/1.51 ( ( topolo1751647064n_open @ A @ U )
% 2.64/1.51 & ( topolo1751647064n_open @ A @ V )
% 2.64/1.51 & ( member @ A @ X @ U )
% 2.64/1.51 & ( member @ A @ Y @ V )
% 2.64/1.51 & ( ( inf_inf @ ( set @ A ) @ U @ V )
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_7_separation__t2,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topological_t2_space @ A )
% 2.64/1.51 => ! [X: A,Y: A] :
% 2.64/1.51 ( ( X != Y )
% 2.64/1.51 = ( ? [U2: set @ A,V2: set @ A] :
% 2.64/1.51 ( ( topolo1751647064n_open @ A @ U2 )
% 2.64/1.51 & ( topolo1751647064n_open @ A @ V2 )
% 2.64/1.51 & ( member @ A @ X @ U2 )
% 2.64/1.51 & ( member @ A @ Y @ V2 )
% 2.64/1.51 & ( ( inf_inf @ ( set @ A ) @ U2 @ V2 )
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_8_IntI,axiom,
% 2.64/1.51 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( member @ A @ C @ A2 )
% 2.64/1.51 => ( ( member @ A @ C @ B2 )
% 2.64/1.51 => ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_9_Int__iff,axiom,
% 2.64/1.51 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
% 2.64/1.51 = ( ( member @ A @ C @ A2 )
% 2.64/1.51 & ( member @ A @ C @ B2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_10_inf__apply,axiom,
% 2.64/1.51 ! [B: $tType,A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ B )
% 2.64/1.51 => ( ( inf_inf @ ( A > B ) )
% 2.64/1.51 = ( ^ [F: A > B,G: A > B,X2: A] : ( inf_inf @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_11_inf_Oidem,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [A3: A] :
% 2.64/1.51 ( ( inf_inf @ A @ A3 @ A3 )
% 2.64/1.51 = A3 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_12_inf__idem,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [X: A] :
% 2.64/1.51 ( ( inf_inf @ A @ X @ X )
% 2.64/1.51 = X ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_13_inf_Oleft__idem,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [A3: A,B3: A] :
% 2.64/1.51 ( ( inf_inf @ A @ A3 @ ( inf_inf @ A @ A3 @ B3 ) )
% 2.64/1.51 = ( inf_inf @ A @ A3 @ B3 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_14_empty__Collect__eq,axiom,
% 2.64/1.51 ! [A: $tType,P: A > $o] :
% 2.64/1.51 ( ( ( bot_bot @ ( set @ A ) )
% 2.64/1.51 = ( collect @ A @ P ) )
% 2.64/1.51 = ( ! [X2: A] :
% 2.64/1.51 ~ ( P @ X2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_15_Collect__empty__eq,axiom,
% 2.64/1.51 ! [A: $tType,P: A > $o] :
% 2.64/1.51 ( ( ( collect @ A @ P )
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.64/1.51 = ( ! [X2: A] :
% 2.64/1.51 ~ ( P @ X2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_16_all__not__in__conv,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A] :
% 2.64/1.51 ( ( ! [X2: A] :
% 2.64/1.51 ~ ( member @ A @ X2 @ A2 ) )
% 2.64/1.51 = ( A2
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_17_empty__iff,axiom,
% 2.64/1.51 ! [A: $tType,C: A] :
% 2.64/1.51 ~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_18_inf__right__idem,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [X: A,Y: A] :
% 2.64/1.51 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Y )
% 2.64/1.51 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_19_inf_Oright__idem,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [A3: A,B3: A] :
% 2.64/1.51 ( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B3 ) @ B3 )
% 2.64/1.51 = ( inf_inf @ A @ A3 @ B3 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_20_inf__left__idem,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [X: A,Y: A] :
% 2.64/1.51 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 2.64/1.51 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_21__092_060open_062_092_060forall_062S_O_Aopen_AS_A_092_060and_062_Af_Ax0_A_092_060in_062_AS_A_092_060longrightarrow_062_A_I_092_060exists_062T_O_Aopen_AT_A_092_060and_062_Ax0_A_092_060in_062_AT_A_092_060and_062_A_I_092_060forall_062x_H_092_060in_062T_O_Af_Ax_H_A_092_060le_062_Af_Ax0_A_092_060longrightarrow_062_Af_Ax_H_A_092_060in_062_AS_J_J_092_060close_062,axiom,
% 2.64/1.51 ! [S2: set @ b] :
% 2.64/1.51 ( ( ( topolo1751647064n_open @ b @ S2 )
% 2.64/1.51 & ( member @ b @ ( f @ x0 ) @ S2 ) )
% 2.64/1.51 => ? [T2: set @ a] :
% 2.64/1.51 ( ( topolo1751647064n_open @ a @ T2 )
% 2.64/1.51 & ( member @ a @ x0 @ T2 )
% 2.64/1.51 & ! [X3: a] :
% 2.64/1.51 ( ( member @ a @ X3 @ T2 )
% 2.64/1.51 => ( ( ord_less_eq @ b @ ( f @ X3 ) @ ( f @ x0 ) )
% 2.64/1.51 => ( member @ b @ ( f @ X3 ) @ S2 ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_22_bot__set__def,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( bot_bot @ ( set @ A ) )
% 2.64/1.51 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_23_ex__in__conv,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A] :
% 2.64/1.51 ( ( ? [X2: A] : ( member @ A @ X2 @ A2 ) )
% 2.64/1.51 = ( A2
% 2.64/1.51 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_24_equals0I,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A] :
% 2.64/1.51 ( ! [Y2: A] :
% 2.64/1.51 ~ ( member @ A @ Y2 @ A2 )
% 2.64/1.51 => ( A2
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_25_equals0D,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,A3: A] :
% 2.64/1.51 ( ( A2
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.64/1.51 => ~ ( member @ A @ A3 @ A2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_26_emptyE,axiom,
% 2.64/1.51 ! [A: $tType,A3: A] :
% 2.64/1.51 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_27_inf__left__commute,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [X: A,Y: A,Z: A] :
% 2.64/1.51 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 2.64/1.51 = ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_28_inf_Oleft__commute,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [B3: A,A3: A,C: A] :
% 2.64/1.51 ( ( inf_inf @ A @ B3 @ ( inf_inf @ A @ A3 @ C ) )
% 2.64/1.51 = ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B3 @ C ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_29_inf__commute,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ( ( inf_inf @ A )
% 2.64/1.51 = ( ^ [X2: A,Y3: A] : ( inf_inf @ A @ Y3 @ X2 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_30_inf_Ocommute,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ( ( inf_inf @ A )
% 2.64/1.51 = ( ^ [A4: A,B4: A] : ( inf_inf @ A @ B4 @ A4 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_31_inf__assoc,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [X: A,Y: A,Z: A] :
% 2.64/1.51 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
% 2.64/1.51 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_32_inf_Oassoc,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [A3: A,B3: A,C: A] :
% 2.64/1.51 ( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C )
% 2.64/1.51 = ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B3 @ C ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_33_boolean__algebra__cancel_Oinf2,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [B2: A,K: A,B3: A,A3: A] :
% 2.64/1.51 ( ( B2
% 2.64/1.51 = ( inf_inf @ A @ K @ B3 ) )
% 2.64/1.51 => ( ( inf_inf @ A @ A3 @ B2 )
% 2.64/1.51 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B3 ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_34_boolean__algebra__cancel_Oinf1,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [A2: A,K: A,A3: A,B3: A] :
% 2.64/1.51 ( ( A2
% 2.64/1.51 = ( inf_inf @ A @ K @ A3 ) )
% 2.64/1.51 => ( ( inf_inf @ A @ A2 @ B3 )
% 2.64/1.51 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B3 ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_35_inf__fun__def,axiom,
% 2.64/1.51 ! [B: $tType,A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ B )
% 2.64/1.51 => ( ( inf_inf @ ( A > B ) )
% 2.64/1.51 = ( ^ [F: A > B,G: A > B,X2: A] : ( inf_inf @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_36_inf__sup__aci_I1_J,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( lattice @ A )
% 2.64/1.51 => ( ( inf_inf @ A )
% 2.64/1.51 = ( ^ [X2: A,Y3: A] : ( inf_inf @ A @ Y3 @ X2 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_37_inf__sup__aci_I2_J,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( lattice @ A )
% 2.64/1.51 => ! [X: A,Y: A,Z: A] :
% 2.64/1.51 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
% 2.64/1.51 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_38_inf__sup__aci_I3_J,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( lattice @ A )
% 2.64/1.51 => ! [X: A,Y: A,Z: A] :
% 2.64/1.51 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 2.64/1.51 = ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_39_inf__sup__aci_I4_J,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( lattice @ A )
% 2.64/1.51 => ! [X: A,Y: A] :
% 2.64/1.51 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 2.64/1.51 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_40_Int__left__commute,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 2.64/1.51 ( ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) )
% 2.64/1.51 = ( inf_inf @ ( set @ A ) @ B2 @ ( inf_inf @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_41_Int__left__absorb,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
% 2.64/1.51 = ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_42_Int__commute,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( inf_inf @ ( set @ A ) )
% 2.64/1.51 = ( ^ [A5: set @ A,B5: set @ A] : ( inf_inf @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_43_Int__absorb,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A] :
% 2.64/1.51 ( ( inf_inf @ ( set @ A ) @ A2 @ A2 )
% 2.64/1.51 = A2 ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_44_Int__assoc,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 2.64/1.51 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ C2 )
% 2.64/1.51 = ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_45_mem__Collect__eq,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,P: A > $o] :
% 2.64/1.51 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 2.64/1.51 = ( P @ A3 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_46_Collect__mem__eq,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A] :
% 2.64/1.51 ( ( collect @ A
% 2.64/1.51 @ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
% 2.64/1.51 = A2 ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_47_Collect__cong,axiom,
% 2.64/1.51 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.64/1.51 ( ! [X4: A] :
% 2.64/1.51 ( ( P @ X4 )
% 2.64/1.51 = ( Q @ X4 ) )
% 2.64/1.51 => ( ( collect @ A @ P )
% 2.64/1.51 = ( collect @ A @ Q ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_48_ext,axiom,
% 2.64/1.51 ! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
% 2.64/1.51 ( ! [X4: A] :
% 2.64/1.51 ( ( F2 @ X4 )
% 2.64/1.51 = ( G2 @ X4 ) )
% 2.64/1.51 => ( F2 = G2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_49_IntD2,axiom,
% 2.64/1.51 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
% 2.64/1.51 => ( member @ A @ C @ B2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_50_IntD1,axiom,
% 2.64/1.51 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
% 2.64/1.51 => ( member @ A @ C @ A2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_51_IntE,axiom,
% 2.64/1.51 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
% 2.64/1.51 => ~ ( ( member @ A @ C @ A2 )
% 2.64/1.51 => ~ ( member @ A @ C @ B2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_52_open__discrete,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo2133971006pology @ A )
% 2.64/1.51 => ! [A2: set @ A] : ( topolo1751647064n_open @ A @ A2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_53_separation__t1,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topological_t1_space @ A )
% 2.64/1.51 => ! [X: A,Y: A] :
% 2.64/1.51 ( ( X != Y )
% 2.64/1.51 = ( ? [U2: set @ A] :
% 2.64/1.51 ( ( topolo1751647064n_open @ A @ U2 )
% 2.64/1.51 & ( member @ A @ X @ U2 )
% 2.64/1.51 & ~ ( member @ A @ Y @ U2 ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_54_separation__t0,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topological_t0_space @ A )
% 2.64/1.51 => ! [X: A,Y: A] :
% 2.64/1.51 ( ( X != Y )
% 2.64/1.51 = ( ? [U2: set @ A] :
% 2.64/1.51 ( ( topolo1751647064n_open @ A @ U2 )
% 2.64/1.51 & ( ( member @ A @ X @ U2 )
% 2.64/1.51 != ( member @ A @ Y @ U2 ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_55_t1__space,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topological_t1_space @ A )
% 2.64/1.51 => ! [X: A,Y: A] :
% 2.64/1.51 ( ( X != Y )
% 2.64/1.51 => ? [U: set @ A] :
% 2.64/1.51 ( ( topolo1751647064n_open @ A @ U )
% 2.64/1.51 & ( member @ A @ X @ U )
% 2.64/1.51 & ~ ( member @ A @ Y @ U ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_56_t0__space,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topological_t0_space @ A )
% 2.64/1.51 => ! [X: A,Y: A] :
% 2.64/1.51 ( ( X != Y )
% 2.64/1.51 => ? [U: set @ A] :
% 2.64/1.51 ( ( topolo1751647064n_open @ A @ U )
% 2.64/1.51 & ( ( member @ A @ X @ U )
% 2.64/1.51 != ( member @ A @ Y @ U ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_57_disjoint__iff__not__equal,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.64/1.51 = ( ! [X2: A] :
% 2.64/1.51 ( ( member @ A @ X2 @ A2 )
% 2.64/1.51 => ! [Y3: A] :
% 2.64/1.51 ( ( member @ A @ Y3 @ B2 )
% 2.64/1.51 => ( X2 != Y3 ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_58_Int__empty__right,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A] :
% 2.64/1.51 ( ( inf_inf @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_59_Int__empty__left,axiom,
% 2.64/1.51 ! [A: $tType,B2: set @ A] :
% 2.64/1.51 ( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_60_disjoint__iff,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.64/1.51 = ( ! [X2: A] :
% 2.64/1.51 ( ( member @ A @ X2 @ A2 )
% 2.64/1.51 => ~ ( member @ A @ X2 @ B2 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_61_Int__emptyI,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ! [X4: A] :
% 2.64/1.51 ( ( member @ A @ X4 @ A2 )
% 2.64/1.51 => ~ ( member @ A @ X4 @ B2 ) )
% 2.64/1.51 => ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_62_bot__apply,axiom,
% 2.64/1.51 ! [C3: $tType,D: $tType] :
% 2.64/1.51 ( ( bot @ C3 )
% 2.64/1.51 => ( ( bot_bot @ ( D > C3 ) )
% 2.64/1.51 = ( ^ [X2: D] : ( bot_bot @ C3 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_63_calculation,axiom,
% 2.64/1.51 ( ~ ! [S2: set @ b] :
% 2.64/1.51 ( ( ( topolo1751647064n_open @ b @ S2 )
% 2.64/1.51 & ( member @ b @ ( f @ x0 ) @ S2 ) )
% 2.64/1.51 => ? [T2: set @ a] :
% 2.64/1.51 ( ( topolo1751647064n_open @ a @ T2 )
% 2.64/1.51 & ( member @ a @ x0 @ T2 )
% 2.64/1.51 & ! [X3: a] :
% 2.64/1.51 ( ( member @ a @ X3 @ T2 )
% 2.64/1.51 => ( ( ord_less_eq @ b @ ( f @ X3 ) @ ( f @ x0 ) )
% 2.64/1.51 => ( member @ b @ ( f @ X3 ) @ S2 ) ) ) ) )
% 2.64/1.51 => ~ ( lower_582600101lsc_at @ a @ b @ x0 @ f ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_64_Set_Ois__empty__def,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( is_empty @ A )
% 2.64/1.51 = ( ^ [A5: set @ A] :
% 2.64/1.51 ( A5
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_65_totally__bounded__empty,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo47006728_space @ A )
% 2.64/1.51 => ( topolo406746546ounded @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_66_Collect__empty__eq__bot,axiom,
% 2.64/1.51 ! [A: $tType,P: A > $o] :
% 2.64/1.51 ( ( ( collect @ A @ P )
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.64/1.51 = ( P
% 2.64/1.51 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_67_bot__empty__eq,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( bot_bot @ ( A > $o ) )
% 2.64/1.51 = ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_68_bot__fun__def,axiom,
% 2.64/1.51 ! [B: $tType,A: $tType] :
% 2.64/1.51 ( ( bot @ B )
% 2.64/1.51 => ( ( bot_bot @ ( A > B ) )
% 2.64/1.51 = ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_69_is__singletonI_H,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A] :
% 2.64/1.51 ( ( A2
% 2.64/1.51 != ( bot_bot @ ( set @ A ) ) )
% 2.64/1.51 => ( ! [X4: A,Y2: A] :
% 2.64/1.51 ( ( member @ A @ X4 @ A2 )
% 2.64/1.51 => ( ( member @ A @ Y2 @ A2 )
% 2.64/1.51 => ( X4 = Y2 ) ) )
% 2.64/1.51 => ( is_singleton @ A @ A2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_70_Diff__disjoint,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( inf_inf @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) )
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_71_insert__disjoint_I1_J,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ A2 ) @ B2 )
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.64/1.51 = ( ~ ( member @ A @ A3 @ B2 )
% 2.64/1.51 & ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_72_insert__disjoint_I2_J,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( ( bot_bot @ ( set @ A ) )
% 2.64/1.51 = ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ A2 ) @ B2 ) )
% 2.64/1.51 = ( ~ ( member @ A @ A3 @ B2 )
% 2.64/1.51 & ( ( bot_bot @ ( set @ A ) )
% 2.64/1.51 = ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_73_order__refl,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( preorder @ A )
% 2.64/1.51 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_74_minus__apply,axiom,
% 2.64/1.51 ! [B: $tType,A: $tType] :
% 2.64/1.51 ( ( minus @ B )
% 2.64/1.51 => ( ( minus_minus @ ( A > B ) )
% 2.64/1.51 = ( ^ [A5: A > B,B5: A > B,X2: A] : ( minus_minus @ B @ ( A5 @ X2 ) @ ( B5 @ X2 ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_75_insertCI,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,B2: set @ A,B3: A] :
% 2.64/1.51 ( ( ~ ( member @ A @ A3 @ B2 )
% 2.64/1.51 => ( A3 = B3 ) )
% 2.64/1.51 => ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_76_insert__iff,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,B3: A,A2: set @ A] :
% 2.64/1.51 ( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
% 2.64/1.51 = ( ( A3 = B3 )
% 2.64/1.51 | ( member @ A @ A3 @ A2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_77_insert__absorb2,axiom,
% 2.64/1.51 ! [A: $tType,X: A,A2: set @ A] :
% 2.64/1.51 ( ( insert @ A @ X @ ( insert @ A @ X @ A2 ) )
% 2.64/1.51 = ( insert @ A @ X @ A2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_78_DiffI,axiom,
% 2.64/1.51 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( member @ A @ C @ A2 )
% 2.64/1.51 => ( ~ ( member @ A @ C @ B2 )
% 2.64/1.51 => ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_79_Diff__iff,axiom,
% 2.64/1.51 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 2.64/1.51 = ( ( member @ A @ C @ A2 )
% 2.64/1.51 & ~ ( member @ A @ C @ B2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_80_Diff__idemp,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ B2 )
% 2.64/1.51 = ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_81_le__inf__iff,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [X: A,Y: A,Z: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 2.64/1.51 = ( ( ord_less_eq @ A @ X @ Y )
% 2.64/1.51 & ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_82_inf_Obounded__iff,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [A3: A,B3: A,C: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B3 @ C ) )
% 2.64/1.51 = ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.64/1.51 & ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_83_singletonI,axiom,
% 2.64/1.51 ! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_84_Int__insert__left__if0,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,C2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ~ ( member @ A @ A3 @ C2 )
% 2.64/1.51 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C2 )
% 2.64/1.51 = ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_85_Int__insert__left__if1,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,C2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( member @ A @ A3 @ C2 )
% 2.64/1.51 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C2 )
% 2.64/1.51 = ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_86_insert__inter__insert,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ A2 ) @ ( insert @ A @ A3 @ B2 ) )
% 2.64/1.51 = ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_87_Int__insert__right__if0,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ~ ( member @ A @ A3 @ A2 )
% 2.64/1.51 => ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
% 2.64/1.51 = ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_88_Int__insert__right__if1,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( member @ A @ A3 @ A2 )
% 2.64/1.51 => ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
% 2.64/1.51 = ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_89_Diff__empty,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A] :
% 2.64/1.51 ( ( minus_minus @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 2.64/1.51 = A2 ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_90_empty__Diff,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A] :
% 2.64/1.51 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 )
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_91_Diff__cancel,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A] :
% 2.64/1.51 ( ( minus_minus @ ( set @ A ) @ A2 @ A2 )
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_92_Diff__insert0,axiom,
% 2.64/1.51 ! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ~ ( member @ A @ X @ A2 )
% 2.64/1.51 => ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B2 ) )
% 2.64/1.51 = ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_93_insert__Diff1,axiom,
% 2.64/1.51 ! [A: $tType,X: A,B2: set @ A,A2: set @ A] :
% 2.64/1.51 ( ( member @ A @ X @ B2 )
% 2.64/1.51 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
% 2.64/1.51 = ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_94_disjoint__insert_I2_J,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B3: A,B2: set @ A] :
% 2.64/1.51 ( ( ( bot_bot @ ( set @ A ) )
% 2.64/1.51 = ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ B2 ) ) )
% 2.64/1.51 = ( ~ ( member @ A @ B3 @ A2 )
% 2.64/1.51 & ( ( bot_bot @ ( set @ A ) )
% 2.64/1.51 = ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_95_disjoint__insert_I1_J,axiom,
% 2.64/1.51 ! [A: $tType,B2: set @ A,A3: A,A2: set @ A] :
% 2.64/1.51 ( ( ( inf_inf @ ( set @ A ) @ B2 @ ( insert @ A @ A3 @ A2 ) )
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.64/1.51 = ( ~ ( member @ A @ A3 @ B2 )
% 2.64/1.51 & ( ( inf_inf @ ( set @ A ) @ B2 @ A2 )
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_96_insert__Diff__single,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,A2: set @ A] :
% 2.64/1.51 ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.64/1.51 = ( insert @ A @ A3 @ A2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_97_is__singletonI,axiom,
% 2.64/1.51 ! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_98_DiffE,axiom,
% 2.64/1.51 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 2.64/1.51 => ~ ( ( member @ A @ C @ A2 )
% 2.64/1.51 => ( member @ A @ C @ B2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_99_DiffD1,axiom,
% 2.64/1.51 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 2.64/1.51 => ( member @ A @ C @ A2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_100_DiffD2,axiom,
% 2.64/1.51 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 2.64/1.51 => ~ ( member @ A @ C @ B2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_101_insertE,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,B3: A,A2: set @ A] :
% 2.64/1.51 ( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
% 2.64/1.51 => ( ( A3 != B3 )
% 2.64/1.51 => ( member @ A @ A3 @ A2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_102_insertI1,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,B2: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_103_insertI2,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,B2: set @ A,B3: A] :
% 2.64/1.51 ( ( member @ A @ A3 @ B2 )
% 2.64/1.51 => ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_104_Set_Oset__insert,axiom,
% 2.64/1.51 ! [A: $tType,X: A,A2: set @ A] :
% 2.64/1.51 ( ( member @ A @ X @ A2 )
% 2.64/1.51 => ~ ! [B6: set @ A] :
% 2.64/1.51 ( ( A2
% 2.64/1.51 = ( insert @ A @ X @ B6 ) )
% 2.64/1.51 => ( member @ A @ X @ B6 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_105_insert__ident,axiom,
% 2.64/1.51 ! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ~ ( member @ A @ X @ A2 )
% 2.64/1.51 => ( ~ ( member @ A @ X @ B2 )
% 2.64/1.51 => ( ( ( insert @ A @ X @ A2 )
% 2.64/1.51 = ( insert @ A @ X @ B2 ) )
% 2.64/1.51 = ( A2 = B2 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_106_insert__absorb,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,A2: set @ A] :
% 2.64/1.51 ( ( member @ A @ A3 @ A2 )
% 2.64/1.51 => ( ( insert @ A @ A3 @ A2 )
% 2.64/1.51 = A2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_107_insert__eq__iff,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,A2: set @ A,B3: A,B2: set @ A] :
% 2.64/1.51 ( ~ ( member @ A @ A3 @ A2 )
% 2.64/1.51 => ( ~ ( member @ A @ B3 @ B2 )
% 2.64/1.51 => ( ( ( insert @ A @ A3 @ A2 )
% 2.64/1.51 = ( insert @ A @ B3 @ B2 ) )
% 2.64/1.51 = ( ( ( A3 = B3 )
% 2.64/1.51 => ( A2 = B2 ) )
% 2.64/1.51 & ( ( A3 != B3 )
% 2.64/1.51 => ? [C4: set @ A] :
% 2.64/1.51 ( ( A2
% 2.64/1.51 = ( insert @ A @ B3 @ C4 ) )
% 2.64/1.51 & ~ ( member @ A @ B3 @ C4 )
% 2.64/1.51 & ( B2
% 2.64/1.51 = ( insert @ A @ A3 @ C4 ) )
% 2.64/1.51 & ~ ( member @ A @ A3 @ C4 ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_108_insert__Diff__if,axiom,
% 2.64/1.51 ! [A: $tType,X: A,B2: set @ A,A2: set @ A] :
% 2.64/1.51 ( ( ( member @ A @ X @ B2 )
% 2.64/1.51 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
% 2.64/1.51 = ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) )
% 2.64/1.51 & ( ~ ( member @ A @ X @ B2 )
% 2.64/1.51 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
% 2.64/1.51 = ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_109_insert__commute,axiom,
% 2.64/1.51 ! [A: $tType,X: A,Y: A,A2: set @ A] :
% 2.64/1.51 ( ( insert @ A @ X @ ( insert @ A @ Y @ A2 ) )
% 2.64/1.51 = ( insert @ A @ Y @ ( insert @ A @ X @ A2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_110_fun__diff__def,axiom,
% 2.64/1.51 ! [B: $tType,A: $tType] :
% 2.64/1.51 ( ( minus @ B )
% 2.64/1.51 => ( ( minus_minus @ ( A > B ) )
% 2.64/1.51 = ( ^ [A5: A > B,B5: A > B,X2: A] : ( minus_minus @ B @ ( A5 @ X2 ) @ ( B5 @ X2 ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_111_mk__disjoint__insert,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,A2: set @ A] :
% 2.64/1.51 ( ( member @ A @ A3 @ A2 )
% 2.64/1.51 => ? [B6: set @ A] :
% 2.64/1.51 ( ( A2
% 2.64/1.51 = ( insert @ A @ A3 @ B6 ) )
% 2.64/1.51 & ~ ( member @ A @ A3 @ B6 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_112_dual__order_Oantisym,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( order @ A )
% 2.64/1.51 => ! [B3: A,A3: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.64/1.51 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.64/1.51 => ( A3 = B3 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_113_dual__order_Oeq__iff,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( order @ A )
% 2.64/1.51 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 2.64/1.51 = ( ^ [A4: A,B4: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.64/1.51 & ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_114_dual__order_Otrans,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( order @ A )
% 2.64/1.51 => ! [B3: A,A3: A,C: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.64/1.51 => ( ( ord_less_eq @ A @ C @ B3 )
% 2.64/1.51 => ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_115_linorder__wlog,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( linorder @ A )
% 2.64/1.51 => ! [P: A > A > $o,A3: A,B3: A] :
% 2.64/1.51 ( ! [A6: A,B7: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ A6 @ B7 )
% 2.64/1.51 => ( P @ A6 @ B7 ) )
% 2.64/1.51 => ( ! [A6: A,B7: A] :
% 2.64/1.51 ( ( P @ B7 @ A6 )
% 2.64/1.51 => ( P @ A6 @ B7 ) )
% 2.64/1.51 => ( P @ A3 @ B3 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_116_dual__order_Orefl,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( order @ A )
% 2.64/1.51 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_117_order__trans,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( preorder @ A )
% 2.64/1.51 => ! [X: A,Y: A,Z: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ X @ Y )
% 2.64/1.51 => ( ( ord_less_eq @ A @ Y @ Z )
% 2.64/1.51 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_118_order__class_Oorder_Oantisym,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( order @ A )
% 2.64/1.51 => ! [A3: A,B3: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.64/1.51 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.64/1.51 => ( A3 = B3 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_119_ord__le__eq__trans,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( ord @ A )
% 2.64/1.51 => ! [A3: A,B3: A,C: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.64/1.51 => ( ( B3 = C )
% 2.64/1.51 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_120_ord__eq__le__trans,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( ord @ A )
% 2.64/1.51 => ! [A3: A,B3: A,C: A] :
% 2.64/1.51 ( ( A3 = B3 )
% 2.64/1.51 => ( ( ord_less_eq @ A @ B3 @ C )
% 2.64/1.51 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_121_order__class_Oorder_Oeq__iff,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( order @ A )
% 2.64/1.51 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 2.64/1.51 = ( ^ [A4: A,B4: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.64/1.51 & ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_122_antisym__conv,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( order @ A )
% 2.64/1.51 => ! [Y: A,X: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ Y @ X )
% 2.64/1.51 => ( ( ord_less_eq @ A @ X @ Y )
% 2.64/1.51 = ( X = Y ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_123_le__cases3,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( linorder @ A )
% 2.64/1.51 => ! [X: A,Y: A,Z: A] :
% 2.64/1.51 ( ( ( ord_less_eq @ A @ X @ Y )
% 2.64/1.51 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 2.64/1.51 => ( ( ( ord_less_eq @ A @ Y @ X )
% 2.64/1.51 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 2.64/1.51 => ( ( ( ord_less_eq @ A @ X @ Z )
% 2.64/1.51 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 2.64/1.51 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 2.64/1.51 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 2.64/1.51 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 2.64/1.51 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 2.64/1.51 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 2.64/1.51 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_124_order_Otrans,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( order @ A )
% 2.64/1.51 => ! [A3: A,B3: A,C: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.64/1.51 => ( ( ord_less_eq @ A @ B3 @ C )
% 2.64/1.51 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_125_le__cases,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( linorder @ A )
% 2.64/1.51 => ! [X: A,Y: A] :
% 2.64/1.51 ( ~ ( ord_less_eq @ A @ X @ Y )
% 2.64/1.51 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_126_eq__refl,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( preorder @ A )
% 2.64/1.51 => ! [X: A,Y: A] :
% 2.64/1.51 ( ( X = Y )
% 2.64/1.51 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_127_linear,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( linorder @ A )
% 2.64/1.51 => ! [X: A,Y: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ X @ Y )
% 2.64/1.51 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_128_antisym,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( order @ A )
% 2.64/1.51 => ! [X: A,Y: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ X @ Y )
% 2.64/1.51 => ( ( ord_less_eq @ A @ Y @ X )
% 2.64/1.51 => ( X = Y ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_129_eq__iff,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( order @ A )
% 2.64/1.51 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 2.64/1.51 = ( ^ [X2: A,Y3: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.64/1.51 & ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_130_ord__le__eq__subst,axiom,
% 2.64/1.51 ! [A: $tType,B: $tType] :
% 2.64/1.51 ( ( ( ord @ B )
% 2.64/1.51 & ( ord @ A ) )
% 2.64/1.51 => ! [A3: A,B3: A,F2: A > B,C: B] :
% 2.64/1.51 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.64/1.51 => ( ( ( F2 @ B3 )
% 2.64/1.51 = C )
% 2.64/1.51 => ( ! [X4: A,Y2: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 2.64/1.51 => ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) )
% 2.64/1.51 => ( ord_less_eq @ B @ ( F2 @ A3 ) @ C ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_131_ord__eq__le__subst,axiom,
% 2.64/1.51 ! [A: $tType,B: $tType] :
% 2.64/1.51 ( ( ( ord @ B )
% 2.64/1.51 & ( ord @ A ) )
% 2.64/1.51 => ! [A3: A,F2: B > A,B3: B,C: B] :
% 2.64/1.51 ( ( A3
% 2.64/1.51 = ( F2 @ B3 ) )
% 2.64/1.51 => ( ( ord_less_eq @ B @ B3 @ C )
% 2.64/1.51 => ( ! [X4: B,Y2: B] :
% 2.64/1.51 ( ( ord_less_eq @ B @ X4 @ Y2 )
% 2.64/1.51 => ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) )
% 2.64/1.51 => ( ord_less_eq @ A @ A3 @ ( F2 @ C ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_132_order__subst2,axiom,
% 2.64/1.51 ! [A: $tType,C3: $tType] :
% 2.64/1.51 ( ( ( order @ C3 )
% 2.64/1.51 & ( order @ A ) )
% 2.64/1.51 => ! [A3: A,B3: A,F2: A > C3,C: C3] :
% 2.64/1.51 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.64/1.51 => ( ( ord_less_eq @ C3 @ ( F2 @ B3 ) @ C )
% 2.64/1.51 => ( ! [X4: A,Y2: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 2.64/1.51 => ( ord_less_eq @ C3 @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) )
% 2.64/1.51 => ( ord_less_eq @ C3 @ ( F2 @ A3 ) @ C ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_133_order__subst1,axiom,
% 2.64/1.51 ! [A: $tType,B: $tType] :
% 2.64/1.51 ( ( ( order @ B )
% 2.64/1.51 & ( order @ A ) )
% 2.64/1.51 => ! [A3: A,F2: B > A,B3: B,C: B] :
% 2.64/1.51 ( ( ord_less_eq @ A @ A3 @ ( F2 @ B3 ) )
% 2.64/1.51 => ( ( ord_less_eq @ B @ B3 @ C )
% 2.64/1.51 => ( ! [X4: B,Y2: B] :
% 2.64/1.51 ( ( ord_less_eq @ B @ X4 @ Y2 )
% 2.64/1.51 => ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) )
% 2.64/1.51 => ( ord_less_eq @ A @ A3 @ ( F2 @ C ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_134_le__fun__def,axiom,
% 2.64/1.51 ! [B: $tType,A: $tType] :
% 2.64/1.51 ( ( ord @ B )
% 2.64/1.51 => ( ( ord_less_eq @ ( A > B ) )
% 2.64/1.51 = ( ^ [F: A > B,G: A > B] :
% 2.64/1.51 ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_135_le__funI,axiom,
% 2.64/1.51 ! [B: $tType,A: $tType] :
% 2.64/1.51 ( ( ord @ B )
% 2.64/1.51 => ! [F2: A > B,G2: A > B] :
% 2.64/1.51 ( ! [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G2 @ X4 ) )
% 2.64/1.51 => ( ord_less_eq @ ( A > B ) @ F2 @ G2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_136_le__funE,axiom,
% 2.64/1.51 ! [B: $tType,A: $tType] :
% 2.64/1.51 ( ( ord @ B )
% 2.64/1.51 => ! [F2: A > B,G2: A > B,X: A] :
% 2.64/1.51 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 2.64/1.51 => ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_137_le__funD,axiom,
% 2.64/1.51 ! [B: $tType,A: $tType] :
% 2.64/1.51 ( ( ord @ B )
% 2.64/1.51 => ! [F2: A > B,G2: A > B,X: A] :
% 2.64/1.51 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 2.64/1.51 => ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_138_Diff__insert__absorb,axiom,
% 2.64/1.51 ! [A: $tType,X: A,A2: set @ A] :
% 2.64/1.51 ( ~ ( member @ A @ X @ A2 )
% 2.64/1.51 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 2.64/1.51 = A2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_139_Diff__insert2,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,A3: A,B2: set @ A] :
% 2.64/1.51 ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
% 2.64/1.51 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_140_insert__Diff,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,A2: set @ A] :
% 2.64/1.51 ( ( member @ A @ A3 @ A2 )
% 2.64/1.51 => ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.64/1.51 = A2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_141_Diff__insert,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,A3: A,B2: set @ A] :
% 2.64/1.51 ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
% 2.64/1.51 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_142_bot_Oextremum,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( order_bot @ A )
% 2.64/1.51 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_143_bot_Oextremum__unique,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( order_bot @ A )
% 2.64/1.51 => ! [A3: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 2.64/1.51 = ( A3
% 2.64/1.51 = ( bot_bot @ A ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_144_bot_Oextremum__uniqueI,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( order_bot @ A )
% 2.64/1.51 => ! [A3: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 2.64/1.51 => ( A3
% 2.64/1.51 = ( bot_bot @ A ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_145_is__singleton__def,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( is_singleton @ A )
% 2.64/1.51 = ( ^ [A5: set @ A] :
% 2.64/1.51 ? [X2: A] :
% 2.64/1.51 ( A5
% 2.64/1.51 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_146_is__singletonE,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A] :
% 2.64/1.51 ( ( is_singleton @ A @ A2 )
% 2.64/1.51 => ~ ! [X4: A] :
% 2.64/1.51 ( A2
% 2.64/1.51 != ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_147_singletonD,axiom,
% 2.64/1.51 ! [A: $tType,B3: A,A3: A] :
% 2.64/1.51 ( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.64/1.51 => ( B3 = A3 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_148_singleton__iff,axiom,
% 2.64/1.51 ! [A: $tType,B3: A,A3: A] :
% 2.64/1.51 ( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.64/1.51 = ( B3 = A3 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_149_doubleton__eq__iff,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,B3: A,C: A,D2: A] :
% 2.64/1.51 ( ( ( insert @ A @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.64/1.51 = ( insert @ A @ C @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.64/1.51 = ( ( ( A3 = C )
% 2.64/1.51 & ( B3 = D2 ) )
% 2.64/1.51 | ( ( A3 = D2 )
% 2.64/1.51 & ( B3 = C ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_150_insert__not__empty,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,A2: set @ A] :
% 2.64/1.51 ( ( insert @ A @ A3 @ A2 )
% 2.64/1.51 != ( bot_bot @ ( set @ A ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_151_singleton__inject,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,B3: A] :
% 2.64/1.51 ( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 2.64/1.51 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.64/1.51 => ( A3 = B3 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_152_Int__insert__left,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,C2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( ( member @ A @ A3 @ C2 )
% 2.64/1.51 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C2 )
% 2.64/1.51 = ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) )
% 2.64/1.51 & ( ~ ( member @ A @ A3 @ C2 )
% 2.64/1.51 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C2 )
% 2.64/1.51 = ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_153_Int__insert__right,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( ( member @ A @ A3 @ A2 )
% 2.64/1.51 => ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
% 2.64/1.51 = ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) )
% 2.64/1.51 & ( ~ ( member @ A @ A3 @ A2 )
% 2.64/1.51 => ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
% 2.64/1.51 = ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_154_Int__Diff,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 2.64/1.51 ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ C2 )
% 2.64/1.51 = ( inf_inf @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_155_Diff__Int2,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,C2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ C2 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) )
% 2.64/1.51 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ C2 ) @ B2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_156_Diff__Diff__Int,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( minus_minus @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 2.64/1.51 = ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_157_Diff__Int__distrib,axiom,
% 2.64/1.51 ! [A: $tType,C2: set @ A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( inf_inf @ ( set @ A ) @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 2.64/1.51 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C2 @ A2 ) @ ( inf_inf @ ( set @ A ) @ C2 @ B2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_158_Diff__Int__distrib2,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 2.64/1.51 ( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ C2 )
% 2.64/1.51 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ C2 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_159_inf__sup__ord_I2_J,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( lattice @ A )
% 2.64/1.51 => ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_160_inf__sup__ord_I1_J,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( lattice @ A )
% 2.64/1.51 => ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_161_inf__le1,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_162_inf__le2,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_163_le__infE,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [X: A,A3: A,B3: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B3 ) )
% 2.64/1.51 => ~ ( ( ord_less_eq @ A @ X @ A3 )
% 2.64/1.51 => ~ ( ord_less_eq @ A @ X @ B3 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_164_le__infI,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [X: A,A3: A,B3: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ X @ A3 )
% 2.64/1.51 => ( ( ord_less_eq @ A @ X @ B3 )
% 2.64/1.51 => ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B3 ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_165_inf__mono,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [A3: A,C: A,B3: A,D2: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ A3 @ C )
% 2.64/1.51 => ( ( ord_less_eq @ A @ B3 @ D2 )
% 2.64/1.51 => ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ ( inf_inf @ A @ C @ D2 ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_166_le__infI1,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [A3: A,X: A,B3: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ A3 @ X )
% 2.64/1.51 => ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ X ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_167_le__infI2,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [B3: A,X: A,A3: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ B3 @ X )
% 2.64/1.51 => ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ X ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_168_inf_OorderE,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [A3: A,B3: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.64/1.51 => ( A3
% 2.64/1.51 = ( inf_inf @ A @ A3 @ B3 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_169_inf_OorderI,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [A3: A,B3: A] :
% 2.64/1.51 ( ( A3
% 2.64/1.51 = ( inf_inf @ A @ A3 @ B3 ) )
% 2.64/1.51 => ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_170_inf__unique,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [F2: A > A > A,X: A,Y: A] :
% 2.64/1.51 ( ! [X4: A,Y2: A] : ( ord_less_eq @ A @ ( F2 @ X4 @ Y2 ) @ X4 )
% 2.64/1.51 => ( ! [X4: A,Y2: A] : ( ord_less_eq @ A @ ( F2 @ X4 @ Y2 ) @ Y2 )
% 2.64/1.51 => ( ! [X4: A,Y2: A,Z3: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 2.64/1.51 => ( ( ord_less_eq @ A @ X4 @ Z3 )
% 2.64/1.51 => ( ord_less_eq @ A @ X4 @ ( F2 @ Y2 @ Z3 ) ) ) )
% 2.64/1.51 => ( ( inf_inf @ A @ X @ Y )
% 2.64/1.51 = ( F2 @ X @ Y ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_171_le__iff__inf,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ( ( ord_less_eq @ A )
% 2.64/1.51 = ( ^ [X2: A,Y3: A] :
% 2.64/1.51 ( ( inf_inf @ A @ X2 @ Y3 )
% 2.64/1.51 = X2 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_172_inf_Oabsorb1,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [A3: A,B3: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.64/1.51 => ( ( inf_inf @ A @ A3 @ B3 )
% 2.64/1.51 = A3 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_173_inf_Oabsorb2,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [B3: A,A3: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.64/1.51 => ( ( inf_inf @ A @ A3 @ B3 )
% 2.64/1.51 = B3 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_174_inf__absorb1,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [X: A,Y: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ X @ Y )
% 2.64/1.51 => ( ( inf_inf @ A @ X @ Y )
% 2.64/1.51 = X ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_175_inf__absorb2,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [Y: A,X: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ Y @ X )
% 2.64/1.51 => ( ( inf_inf @ A @ X @ Y )
% 2.64/1.51 = Y ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_176_inf_OboundedE,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [A3: A,B3: A,C: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B3 @ C ) )
% 2.64/1.51 => ~ ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.64/1.51 => ~ ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_177_inf_OboundedI,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [A3: A,B3: A,C: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.64/1.51 => ( ( ord_less_eq @ A @ A3 @ C )
% 2.64/1.51 => ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B3 @ C ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_178_inf__greatest,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [X: A,Y: A,Z: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ X @ Y )
% 2.64/1.51 => ( ( ord_less_eq @ A @ X @ Z )
% 2.64/1.51 => ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_179_inf_Oorder__iff,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ( ( ord_less_eq @ A )
% 2.64/1.51 = ( ^ [A4: A,B4: A] :
% 2.64/1.51 ( A4
% 2.64/1.51 = ( inf_inf @ A @ A4 @ B4 ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_180_inf_Ocobounded1,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ A3 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_181_inf_Ocobounded2,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ B3 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_182_inf_Oabsorb__iff1,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ( ( ord_less_eq @ A )
% 2.64/1.51 = ( ^ [A4: A,B4: A] :
% 2.64/1.51 ( ( inf_inf @ A @ A4 @ B4 )
% 2.64/1.51 = A4 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_183_inf_Oabsorb__iff2,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ( ( ord_less_eq @ A )
% 2.64/1.51 = ( ^ [B4: A,A4: A] :
% 2.64/1.51 ( ( inf_inf @ A @ A4 @ B4 )
% 2.64/1.51 = B4 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_184_inf_OcoboundedI1,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [A3: A,C: A,B3: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ A3 @ C )
% 2.64/1.51 => ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_185_inf_OcoboundedI2,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A )
% 2.64/1.51 => ! [B3: A,C: A,A3: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ B3 @ C )
% 2.64/1.51 => ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_186_not__open__singleton,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo890362671_space @ A )
% 2.64/1.51 => ! [X: A] :
% 2.64/1.51 ~ ( topolo1751647064n_open @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_187_Diff__triv,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.64/1.51 => ( ( minus_minus @ ( set @ A ) @ A2 @ B2 )
% 2.64/1.51 = A2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_188_open__delete,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topological_t1_space @ A )
% 2.64/1.51 => ! [S3: set @ A,X: A] :
% 2.64/1.51 ( ( topolo1751647064n_open @ A @ S3 )
% 2.64/1.51 => ( topolo1751647064n_open @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_189_Int__Diff__disjoint,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_190_is__singleton__the__elem,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( is_singleton @ A )
% 2.64/1.51 = ( ^ [A5: set @ A] :
% 2.64/1.51 ( A5
% 2.64/1.51 = ( insert @ A @ ( the_elem @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_191_ball__insert,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,B2: set @ A,P: A > $o] :
% 2.64/1.51 ( ( ! [X2: A] :
% 2.64/1.51 ( ( member @ A @ X2 @ ( insert @ A @ A3 @ B2 ) )
% 2.64/1.51 => ( P @ X2 ) ) )
% 2.64/1.51 = ( ( P @ A3 )
% 2.64/1.51 & ! [X2: A] :
% 2.64/1.51 ( ( member @ A @ X2 @ B2 )
% 2.64/1.51 => ( P @ X2 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_192_the__elem__eq,axiom,
% 2.64/1.51 ! [A: $tType,X: A] :
% 2.64/1.51 ( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 2.64/1.51 = X ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_193_diff__eq__diff__less__eq,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( ordered_ab_group_add @ A )
% 2.64/1.51 => ! [A3: A,B3: A,C: A,D2: A] :
% 2.64/1.51 ( ( ( minus_minus @ A @ A3 @ B3 )
% 2.64/1.51 = ( minus_minus @ A @ C @ D2 ) )
% 2.64/1.51 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.64/1.51 = ( ord_less_eq @ A @ C @ D2 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_194_empty__subsetI,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_195_subset__empty,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 2.64/1.51 = ( A2
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_196_insert__subset,axiom,
% 2.64/1.51 ! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
% 2.64/1.51 = ( ( member @ A @ X @ B2 )
% 2.64/1.51 & ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_197_Int__subset__iff,axiom,
% 2.64/1.51 ! [A: $tType,C2: set @ A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ C2 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
% 2.64/1.51 = ( ( ord_less_eq @ ( set @ A ) @ C2 @ A2 )
% 2.64/1.51 & ( ord_less_eq @ ( set @ A ) @ C2 @ B2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_198_singleton__insert__inj__eq,axiom,
% 2.64/1.51 ! [A: $tType,B3: A,A3: A,A2: set @ A] :
% 2.64/1.51 ( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
% 2.64/1.51 = ( insert @ A @ A3 @ A2 ) )
% 2.64/1.51 = ( ( A3 = B3 )
% 2.64/1.51 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_199_singleton__insert__inj__eq_H,axiom,
% 2.64/1.51 ! [A: $tType,A3: A,A2: set @ A,B3: A] :
% 2.64/1.51 ( ( ( insert @ A @ A3 @ A2 )
% 2.64/1.51 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.64/1.51 = ( ( A3 = B3 )
% 2.64/1.51 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_200_Diff__eq__empty__iff,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( ( minus_minus @ ( set @ A ) @ A2 @ B2 )
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.64/1.51 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_201_subset__insertI2,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,B3: A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.64/1.51 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_202_subset__insertI,axiom,
% 2.64/1.51 ! [A: $tType,B2: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( insert @ A @ A3 @ B2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_203_subset__insert,axiom,
% 2.64/1.51 ! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ~ ( member @ A @ X @ A2 )
% 2.64/1.51 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B2 ) )
% 2.64/1.51 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_204_insert__mono,axiom,
% 2.64/1.51 ! [A: $tType,C2: set @ A,D3: set @ A,A3: A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ C2 @ D3 )
% 2.64/1.51 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C2 ) @ ( insert @ A @ A3 @ D3 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_205_Int__Collect__mono,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o,Q: A > $o] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.64/1.51 => ( ! [X4: A] :
% 2.64/1.51 ( ( member @ A @ X4 @ A2 )
% 2.64/1.51 => ( ( P @ X4 )
% 2.64/1.51 => ( Q @ X4 ) ) )
% 2.64/1.51 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B2 @ ( collect @ A @ Q ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_206_Int__greatest,axiom,
% 2.64/1.51 ! [A: $tType,C2: set @ A,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ C2 @ A2 )
% 2.64/1.51 => ( ( ord_less_eq @ ( set @ A ) @ C2 @ B2 )
% 2.64/1.51 => ( ord_less_eq @ ( set @ A ) @ C2 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_207_Int__absorb2,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.64/1.51 => ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
% 2.64/1.51 = A2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_208_Int__absorb1,axiom,
% 2.64/1.51 ! [A: $tType,B2: set @ A,A2: set @ A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.64/1.51 => ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
% 2.64/1.51 = B2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_209_Int__lower2,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ B2 ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_210_Int__lower1,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ A2 ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_211_Int__mono,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,C2: set @ A,B2: set @ A,D3: set @ A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
% 2.64/1.51 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ D3 )
% 2.64/1.51 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ ( inf_inf @ ( set @ A ) @ C2 @ D3 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_212_first__countable__basis,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo2135403230pology @ A )
% 2.64/1.51 => ! [X: A] :
% 2.64/1.51 ? [A7: nat > ( set @ A )] :
% 2.64/1.51 ( ! [I: nat] :
% 2.64/1.51 ( ( member @ A @ X @ ( A7 @ I ) )
% 2.64/1.51 & ( topolo1751647064n_open @ A @ ( A7 @ I ) ) )
% 2.64/1.51 & ! [S2: set @ A] :
% 2.64/1.51 ( ( ( topolo1751647064n_open @ A @ S2 )
% 2.64/1.51 & ( member @ A @ X @ S2 ) )
% 2.64/1.51 => ? [I2: nat] : ( ord_less_eq @ ( set @ A ) @ ( A7 @ I2 ) @ S2 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_213_open__subopen,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo503727757_space @ A )
% 2.64/1.51 => ( ( topolo1751647064n_open @ A )
% 2.64/1.51 = ( ^ [S4: set @ A] :
% 2.64/1.51 ! [X2: A] :
% 2.64/1.51 ( ( member @ A @ X2 @ S4 )
% 2.64/1.51 => ? [T3: set @ A] :
% 2.64/1.51 ( ( topolo1751647064n_open @ A @ T3 )
% 2.64/1.51 & ( member @ A @ X2 @ T3 )
% 2.64/1.51 & ( ord_less_eq @ ( set @ A ) @ T3 @ S4 ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_214_topological__space__class_OopenI,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo503727757_space @ A )
% 2.64/1.51 => ! [S: set @ A] :
% 2.64/1.51 ( ! [X4: A] :
% 2.64/1.51 ( ( member @ A @ X4 @ S )
% 2.64/1.51 => ? [T4: set @ A] :
% 2.64/1.51 ( ( topolo1751647064n_open @ A @ T4 )
% 2.64/1.51 & ( member @ A @ X4 @ T4 )
% 2.64/1.51 & ( ord_less_eq @ ( set @ A ) @ T4 @ S ) ) )
% 2.64/1.51 => ( topolo1751647064n_open @ A @ S ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_215_double__diff,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.64/1.51 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
% 2.64/1.51 => ( ( minus_minus @ ( set @ A ) @ B2 @ ( minus_minus @ ( set @ A ) @ C2 @ A2 ) )
% 2.64/1.51 = A2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_216_Diff__subset,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ A2 ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_217_Diff__mono,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,C2: set @ A,D3: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
% 2.64/1.51 => ( ( ord_less_eq @ ( set @ A ) @ D3 @ B2 )
% 2.64/1.51 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( minus_minus @ ( set @ A ) @ C2 @ D3 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_218_totally__bounded__subset,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo47006728_space @ A )
% 2.64/1.51 => ! [S: set @ A,T: set @ A] :
% 2.64/1.51 ( ( topolo406746546ounded @ A @ S )
% 2.64/1.51 => ( ( ord_less_eq @ ( set @ A ) @ T @ S )
% 2.64/1.51 => ( topolo406746546ounded @ A @ T ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_219_subset__singletonD,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,X: A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 2.64/1.51 => ( ( A2
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.64/1.51 | ( A2
% 2.64/1.51 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_220_subset__singleton__iff,axiom,
% 2.64/1.51 ! [A: $tType,X5: set @ A,A3: A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.64/1.51 = ( ( X5
% 2.64/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.64/1.51 | ( X5
% 2.64/1.51 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_221_subset__Diff__insert,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,X: A,C2: set @ A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ ( insert @ A @ X @ C2 ) ) )
% 2.64/1.51 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ C2 ) )
% 2.64/1.51 & ~ ( member @ A @ X @ A2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_222_diff__eq__diff__eq,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( group_add @ A )
% 2.64/1.51 => ! [A3: A,B3: A,C: A,D2: A] :
% 2.64/1.51 ( ( ( minus_minus @ A @ A3 @ B3 )
% 2.64/1.51 = ( minus_minus @ A @ C @ D2 ) )
% 2.64/1.51 => ( ( A3 = B3 )
% 2.64/1.51 = ( C = D2 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_223_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( cancel146912293up_add @ A )
% 2.64/1.51 => ! [A3: A,C: A,B3: A] :
% 2.64/1.51 ( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C ) @ B3 )
% 2.64/1.51 = ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_224_Diff__single__insert,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,X: A,B2: set @ A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 )
% 2.64/1.51 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_225_subset__insert__iff,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,X: A,B2: set @ A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B2 ) )
% 2.64/1.51 = ( ( ( member @ A @ X @ A2 )
% 2.64/1.51 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) )
% 2.64/1.51 & ( ~ ( member @ A @ X @ A2 )
% 2.64/1.51 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_226_diff__mono,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( ordered_ab_group_add @ A )
% 2.64/1.51 => ! [A3: A,B3: A,D2: A,C: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.64/1.51 => ( ( ord_less_eq @ A @ D2 @ C )
% 2.64/1.51 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ D2 ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_227_diff__left__mono,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( ordered_ab_group_add @ A )
% 2.64/1.51 => ! [B3: A,A3: A,C: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.64/1.51 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A3 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_228_diff__right__mono,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( ordered_ab_group_add @ A )
% 2.64/1.51 => ! [A3: A,B3: A,C: A] :
% 2.64/1.51 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.64/1.51 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_229_remove__def,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( remove @ A )
% 2.64/1.51 = ( ^ [X2: A,A5: set @ A] : ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_230_at__within__nhd,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo503727757_space @ A )
% 2.64/1.51 => ! [X: A,S: set @ A,T: set @ A,U3: set @ A] :
% 2.64/1.51 ( ( member @ A @ X @ S )
% 2.64/1.51 => ( ( topolo1751647064n_open @ A @ S )
% 2.64/1.51 => ( ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ T @ S ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 2.64/1.51 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ U3 @ S ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.64/1.51 => ( ( topolo507301023within @ A @ X @ T )
% 2.64/1.51 = ( topolo507301023within @ A @ X @ U3 ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_231_subsetI,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ! [X4: A] :
% 2.64/1.51 ( ( member @ A @ X4 @ A2 )
% 2.64/1.51 => ( member @ A @ X4 @ B2 ) )
% 2.64/1.51 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_232_subset__antisym,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.64/1.51 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.64/1.51 => ( A2 = B2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_233_member__remove,axiom,
% 2.64/1.51 ! [A: $tType,X: A,Y: A,A2: set @ A] :
% 2.64/1.51 ( ( member @ A @ X @ ( remove @ A @ Y @ A2 ) )
% 2.64/1.51 = ( ( member @ A @ X @ A2 )
% 2.64/1.51 & ( X != Y ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_234_at__within__empty,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo503727757_space @ A )
% 2.64/1.51 => ! [A3: A] :
% 2.64/1.51 ( ( topolo507301023within @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 2.64/1.51 = ( bot_bot @ ( filter @ A ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_235_in__mono,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.64/1.51 => ( ( member @ A @ X @ A2 )
% 2.64/1.51 => ( member @ A @ X @ B2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_236_subsetD,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.64/1.51 => ( ( member @ A @ C @ A2 )
% 2.64/1.51 => ( member @ A @ C @ B2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_237_equalityE,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( A2 = B2 )
% 2.64/1.51 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.64/1.51 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_238_subset__eq,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) )
% 2.64/1.51 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.64/1.51 ! [X2: A] :
% 2.64/1.51 ( ( member @ A @ X2 @ A5 )
% 2.64/1.51 => ( member @ A @ X2 @ B5 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_239_equalityD1,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( A2 = B2 )
% 2.64/1.51 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_240_equalityD2,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.64/1.51 ( ( A2 = B2 )
% 2.64/1.51 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_241_subset__iff,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) )
% 2.64/1.51 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.64/1.51 ! [T5: A] :
% 2.64/1.51 ( ( member @ A @ T5 @ A5 )
% 2.64/1.51 => ( member @ A @ T5 @ B5 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_242_subset__refl,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_243_Collect__mono,axiom,
% 2.64/1.51 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.64/1.51 ( ! [X4: A] :
% 2.64/1.51 ( ( P @ X4 )
% 2.64/1.51 => ( Q @ X4 ) )
% 2.64/1.51 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_244_subset__trans,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.64/1.51 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
% 2.64/1.51 => ( ord_less_eq @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_245_set__eq__subset,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( ^ [Y4: set @ A,Z2: set @ A] : ( Y4 = Z2 ) )
% 2.64/1.51 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.64/1.51 & ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_246_Collect__mono__iff,axiom,
% 2.64/1.51 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.64/1.51 = ( ! [X2: A] :
% 2.64/1.51 ( ( P @ X2 )
% 2.64/1.51 => ( Q @ X2 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_247_at__le,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo503727757_space @ A )
% 2.64/1.51 => ! [S3: set @ A,T6: set @ A,X: A] :
% 2.64/1.51 ( ( ord_less_eq @ ( set @ A ) @ S3 @ T6 )
% 2.64/1.51 => ( ord_less_eq @ ( filter @ A ) @ ( topolo507301023within @ A @ X @ S3 ) @ ( topolo507301023within @ A @ X @ T6 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_248_at__discrete,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo2133971006pology @ A )
% 2.64/1.51 => ( ( topolo507301023within @ A )
% 2.64/1.51 = ( ^ [X2: A,S4: set @ A] : ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_249_insert__subsetI,axiom,
% 2.64/1.51 ! [A: $tType,X: A,A2: set @ A,X5: set @ A] :
% 2.64/1.51 ( ( member @ A @ X @ A2 )
% 2.64/1.51 => ( ( ord_less_eq @ ( set @ A ) @ X5 @ A2 )
% 2.64/1.51 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X5 ) @ A2 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_250_subset__emptyI,axiom,
% 2.64/1.51 ! [A: $tType,A2: set @ A] :
% 2.64/1.51 ( ! [X4: A] :
% 2.64/1.51 ~ ( member @ A @ X4 @ A2 )
% 2.64/1.51 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_251_at__eq__bot__iff,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo503727757_space @ A )
% 2.64/1.51 => ! [A3: A] :
% 2.64/1.51 ( ( ( topolo507301023within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
% 2.64/1.51 = ( bot_bot @ ( filter @ A ) ) )
% 2.64/1.51 = ( topolo1751647064n_open @ A @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_252_at__within__eq__bot__iff,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo503727757_space @ A )
% 2.64/1.51 => ! [C: A,A2: set @ A] :
% 2.64/1.51 ( ( ( topolo507301023within @ A @ C @ A2 )
% 2.64/1.51 = ( bot_bot @ ( filter @ A ) ) )
% 2.64/1.51 = ( ~ ( member @ A @ C @ ( elementary_closure @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_253_top__apply,axiom,
% 2.64/1.51 ! [C3: $tType,D: $tType] :
% 2.64/1.51 ( ( top @ C3 )
% 2.64/1.51 => ( ( top_top @ ( D > C3 ) )
% 2.64/1.51 = ( ^ [X2: D] : ( top_top @ C3 ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_254_UNIV__I,axiom,
% 2.64/1.51 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(fact_255_closure__closure,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo503727757_space @ A )
% 2.64/1.51 => ! [S: set @ A] :
% 2.64/1.51 ( ( elementary_closure @ A @ ( elementary_closure @ A @ S ) )
% 2.64/1.51 = ( elementary_closure @ A @ S ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Complete__Lattices_Ocomplete__linorder___Lattices_Obounded__lattice,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( comple1035589618norder @ A )
% 2.64/1.51 => ( bounded_lattice @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Complete__Lattices_Ocomplete__linorder___HOL_Otype,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( comple1035589618norder @ A )
% 2.64/1.51 => ( type @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Obot,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( comple1035589618norder @ A )
% 2.64/1.51 => ( bot @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Oord,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( comple1035589618norder @ A )
% 2.64/1.51 => ( ord @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Otop,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( comple1035589618norder @ A )
% 2.64/1.51 => ( top @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Oorder,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( comple1035589618norder @ A )
% 2.64/1.51 => ( order @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Complete__Lattices_Ocomplete__linorder___Lattices_Olattice,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( comple1035589618norder @ A )
% 2.64/1.51 => ( lattice @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Olinorder,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( comple1035589618norder @ A )
% 2.64/1.51 => ( linorder @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Opreorder,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( comple1035589618norder @ A )
% 2.64/1.51 => ( preorder @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Oorder__bot,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( comple1035589618norder @ A )
% 2.64/1.51 => ( order_bot @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Complete__Lattices_Ocomplete__linorder___Lattices_Osemilattice__inf,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( comple1035589618norder @ A )
% 2.64/1.51 => ( semilattice_inf @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Complete__Lattices_Ocomplete__linorder___Lattices_Obounded__lattice__bot,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( comple1035589618norder @ A )
% 2.64/1.51 => ( bounded_lattice_bot @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Topological__Spaces_Olinorder__topology___HOL_Otype,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo2117631714pology @ A )
% 2.64/1.51 => ( type @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Topological__Spaces_Olinorder__topology___Orderings_Oord,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo2117631714pology @ A )
% 2.64/1.51 => ( ord @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Topological__Spaces_Olinorder__topology___Orderings_Oorder,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo2117631714pology @ A )
% 2.64/1.51 => ( order @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Topological__Spaces_Olinorder__topology___Orderings_Olinorder,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo2117631714pology @ A )
% 2.64/1.51 => ( linorder @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Topological__Spaces_Olinorder__topology___Orderings_Opreorder,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo2117631714pology @ A )
% 2.64/1.51 => ( preorder @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Topological__Spaces_Olinorder__topology___Topological__Spaces_Ot0__space,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo2117631714pology @ A )
% 2.64/1.51 => ( topological_t0_space @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Topological__Spaces_Olinorder__topology___Topological__Spaces_Ot1__space,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo2117631714pology @ A )
% 2.64/1.51 => ( topological_t1_space @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Topological__Spaces_Olinorder__topology___Topological__Spaces_Ot2__space,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo2117631714pology @ A )
% 2.64/1.51 => ( topological_t2_space @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Topological__Spaces_Olinorder__topology___Topological__Spaces_Otopological__space,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo2117631714pology @ A )
% 2.64/1.51 => ( topolo503727757_space @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Topological__Spaces_Ofirst__countable__topology___HOL_Otype,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo2135403230pology @ A )
% 2.64/1.51 => ( type @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subcl_Topological__Spaces_Ofirst__countable__topology___Topological__Spaces_Otopological__space,axiom,
% 2.64/1.51 ! [A: $tType] :
% 2.64/1.51 ( ( topolo2135403230pology @ A )
% 2.64/1.51 => ( topolo503727757_space @ A ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice,axiom,
% 2.64/1.51 ! [A8: $tType] : ( bounded_lattice @ ( filter @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_HOL_Obool___Countable_Ocountable,axiom,
% 2.64/1.51 countable @ $o ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_HOL_Obool___Lattices_Obounded__lattice_1,axiom,
% 2.64/1.51 bounded_lattice @ $o ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Set_Oset___Countable_Ocountable_2,axiom,
% 2.64/1.51 ! [A8: $tType] :
% 2.64/1.51 ( ( finite_finite @ A8 )
% 2.64/1.51 => ( countable @ ( set @ A8 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Set_Oset___Lattices_Obounded__lattice_3,axiom,
% 2.64/1.51 ! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Nat_Onat___Countable_Ocountable_4,axiom,
% 2.64/1.51 countable @ nat ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_fun___Countable_Ocountable_5,axiom,
% 2.64/1.51 ! [A8: $tType,A9: $tType] :
% 2.64/1.51 ( ( ( finite_finite @ A8 )
% 2.64/1.51 & ( countable @ A9 ) )
% 2.64/1.51 => ( countable @ ( A8 > A9 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_fun___Lattices_Obounded__lattice_6,axiom,
% 2.64/1.51 ! [A8: $tType,A9: $tType] :
% 2.64/1.51 ( ( bounded_lattice @ A9 )
% 2.64/1.51 => ( bounded_lattice @ ( A8 > A9 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 2.64/1.51 ! [A8: $tType,A9: $tType] :
% 2.64/1.51 ( ( ( finite_finite @ A8 )
% 2.64/1.51 & ( finite_finite @ A9 ) )
% 2.64/1.51 => ( finite_finite @ ( A8 > A9 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Set_Oset___Finite__Set_Ofinite_7,axiom,
% 2.64/1.51 ! [A8: $tType] :
% 2.64/1.51 ( ( finite_finite @ A8 )
% 2.64/1.51 => ( finite_finite @ ( set @ A8 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_HOL_Obool___Finite__Set_Ofinite_8,axiom,
% 2.64/1.51 finite_finite @ $o ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_fun___Topological__Spaces_Ofirst__countable__topology,axiom,
% 2.64/1.51 ! [A8: $tType,A9: $tType] :
% 2.64/1.51 ( ( ( countable @ A8 )
% 2.64/1.51 & ( topolo2135403230pology @ A9 ) )
% 2.64/1.51 => ( topolo2135403230pology @ ( A8 > A9 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
% 2.64/1.51 ! [A8: $tType,A9: $tType] :
% 2.64/1.51 ( ( topolo503727757_space @ A9 )
% 2.64/1.51 => ( topolo503727757_space @ ( A8 > A9 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
% 2.64/1.51 ! [A8: $tType,A9: $tType] :
% 2.64/1.51 ( ( bounded_lattice @ A9 )
% 2.64/1.51 => ( bounded_lattice_bot @ ( A8 > A9 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 2.64/1.51 ! [A8: $tType,A9: $tType] :
% 2.64/1.51 ( ( semilattice_inf @ A9 )
% 2.64/1.51 => ( semilattice_inf @ ( A8 > A9 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 2.64/1.51 ! [A8: $tType,A9: $tType] :
% 2.64/1.51 ( ( order_bot @ A9 )
% 2.64/1.51 => ( order_bot @ ( A8 > A9 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.64/1.51 ! [A8: $tType,A9: $tType] :
% 2.64/1.51 ( ( preorder @ A9 )
% 2.64/1.51 => ( preorder @ ( A8 > A9 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_fun___Lattices_Olattice,axiom,
% 2.64/1.51 ! [A8: $tType,A9: $tType] :
% 2.64/1.51 ( ( lattice @ A9 )
% 2.64/1.51 => ( lattice @ ( A8 > A9 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.64/1.51 ! [A8: $tType,A9: $tType] :
% 2.64/1.51 ( ( order @ A9 )
% 2.64/1.51 => ( order @ ( A8 > A9 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_fun___Orderings_Otop,axiom,
% 2.64/1.51 ! [A8: $tType,A9: $tType] :
% 2.64/1.51 ( ( top @ A9 )
% 2.64/1.51 => ( top @ ( A8 > A9 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_fun___Orderings_Oord,axiom,
% 2.64/1.51 ! [A8: $tType,A9: $tType] :
% 2.64/1.51 ( ( ord @ A9 )
% 2.64/1.51 => ( ord @ ( A8 > A9 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_fun___Orderings_Obot,axiom,
% 2.64/1.51 ! [A8: $tType,A9: $tType] :
% 2.64/1.51 ( ( bot @ A9 )
% 2.64/1.51 => ( bot @ ( A8 > A9 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_fun___Groups_Ominus,axiom,
% 2.64/1.51 ! [A8: $tType,A9: $tType] :
% 2.64/1.51 ( ( minus @ A9 )
% 2.64/1.51 => ( minus @ ( A8 > A9 ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Nat_Onat___Topological__Spaces_Ofirst__countable__topology_9,axiom,
% 2.64/1.51 topolo2135403230pology @ nat ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_10,axiom,
% 2.64/1.51 topolo503727757_space @ nat ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology,axiom,
% 2.64/1.51 topolo2133971006pology @ nat ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 2.64/1.51 cancel146912293up_add @ nat ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space,axiom,
% 2.64/1.51 topological_t2_space @ nat ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Nat_Onat___Topological__Spaces_Ot1__space,axiom,
% 2.64/1.51 topological_t1_space @ nat ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Nat_Onat___Topological__Spaces_Ot0__space,axiom,
% 2.64/1.51 topological_t0_space @ nat ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_11,axiom,
% 2.64/1.51 semilattice_inf @ nat ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Nat_Onat___Orderings_Oorder__bot_12,axiom,
% 2.64/1.51 order_bot @ nat ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Nat_Onat___Orderings_Opreorder_13,axiom,
% 2.64/1.51 preorder @ nat ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.64/1.51 linorder @ nat ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Nat_Onat___Lattices_Olattice_14,axiom,
% 2.64/1.51 lattice @ nat ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Nat_Onat___Orderings_Oorder_15,axiom,
% 2.64/1.51 order @ nat ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Nat_Onat___Orderings_Oord_16,axiom,
% 2.64/1.51 ord @ nat ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Nat_Onat___Orderings_Obot_17,axiom,
% 2.64/1.51 bot @ nat ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Nat_Onat___Groups_Ominus_18,axiom,
% 2.64/1.51 minus @ nat ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_19,axiom,
% 2.64/1.51 ! [A8: $tType] : ( bounded_lattice_bot @ ( set @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_20,axiom,
% 2.64/1.51 ! [A8: $tType] : ( semilattice_inf @ ( set @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Set_Oset___Orderings_Oorder__bot_21,axiom,
% 2.64/1.51 ! [A8: $tType] : ( order_bot @ ( set @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Set_Oset___Orderings_Opreorder_22,axiom,
% 2.64/1.51 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Set_Oset___Lattices_Olattice_23,axiom,
% 2.64/1.51 ! [A8: $tType] : ( lattice @ ( set @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Set_Oset___Orderings_Oorder_24,axiom,
% 2.64/1.51 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Set_Oset___Orderings_Otop_25,axiom,
% 2.64/1.51 ! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Set_Oset___Orderings_Oord_26,axiom,
% 2.64/1.51 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Set_Oset___Orderings_Obot_27,axiom,
% 2.64/1.51 ! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Set_Oset___Groups_Ominus_28,axiom,
% 2.64/1.51 ! [A8: $tType] : ( minus @ ( set @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_29,axiom,
% 2.64/1.51 topolo503727757_space @ $o ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_30,axiom,
% 2.64/1.51 topolo2133971006pology @ $o ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_31,axiom,
% 2.64/1.51 bounded_lattice_bot @ $o ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_32,axiom,
% 2.64/1.51 topological_t2_space @ $o ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_HOL_Obool___Topological__Spaces_Ot1__space_33,axiom,
% 2.64/1.51 topological_t1_space @ $o ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_HOL_Obool___Topological__Spaces_Ot0__space_34,axiom,
% 2.64/1.51 topological_t0_space @ $o ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_35,axiom,
% 2.64/1.51 semilattice_inf @ $o ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_HOL_Obool___Orderings_Oorder__bot_36,axiom,
% 2.64/1.51 order_bot @ $o ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_HOL_Obool___Orderings_Opreorder_37,axiom,
% 2.64/1.51 preorder @ $o ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_HOL_Obool___Orderings_Olinorder_38,axiom,
% 2.64/1.51 linorder @ $o ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_HOL_Obool___Lattices_Olattice_39,axiom,
% 2.64/1.51 lattice @ $o ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_HOL_Obool___Orderings_Oorder_40,axiom,
% 2.64/1.51 order @ $o ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_HOL_Obool___Orderings_Otop_41,axiom,
% 2.64/1.51 top @ $o ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_HOL_Obool___Orderings_Oord_42,axiom,
% 2.64/1.51 ord @ $o ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_HOL_Obool___Orderings_Obot_43,axiom,
% 2.64/1.51 bot @ $o ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_HOL_Obool___Groups_Ominus_44,axiom,
% 2.64/1.51 minus @ $o ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_45,axiom,
% 2.64/1.51 ! [A8: $tType] : ( bounded_lattice_bot @ ( filter @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_46,axiom,
% 2.64/1.51 ! [A8: $tType] : ( semilattice_inf @ ( filter @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_47,axiom,
% 2.64/1.51 ! [A8: $tType] : ( order_bot @ ( filter @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Filter_Ofilter___Orderings_Opreorder_48,axiom,
% 2.64/1.51 ! [A8: $tType] : ( preorder @ ( filter @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Filter_Ofilter___Lattices_Olattice_49,axiom,
% 2.64/1.51 ! [A8: $tType] : ( lattice @ ( filter @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Filter_Ofilter___Orderings_Oorder_50,axiom,
% 2.64/1.51 ! [A8: $tType] : ( order @ ( filter @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Filter_Ofilter___Orderings_Otop_51,axiom,
% 2.64/1.51 ! [A8: $tType] : ( top @ ( filter @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Filter_Ofilter___Orderings_Oord_52,axiom,
% 2.64/1.51 ! [A8: $tType] : ( ord @ ( filter @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tcon_Filter_Ofilter___Orderings_Obot_53,axiom,
% 2.64/1.51 ! [A8: $tType] : ( bot @ ( filter @ A8 ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(tfree_0,hypothesis,
% 2.64/1.51 comple1035589618norder @ b ).
% 2.64/1.51
% 2.64/1.51 thf(tfree_1,hypothesis,
% 2.64/1.51 topolo2117631714pology @ b ).
% 2.64/1.51
% 2.64/1.51 thf(tfree_2,hypothesis,
% 2.64/1.51 topolo2135403230pology @ a ).
% 2.64/1.51
% 2.64/1.51 thf(conj_0,hypothesis,
% 2.64/1.51 ! [S2: set @ b,V3: set @ b] :
% 2.64/1.51 ( ( ( topolo1751647064n_open @ b @ S2 )
% 2.64/1.51 & ( topolo1751647064n_open @ b @ V3 )
% 2.64/1.51 & ( member @ b @ ( f @ x0 ) @ S2 )
% 2.64/1.51 & ( member @ b @ a2 @ V3 )
% 2.64/1.51 & ( ( inf_inf @ ( set @ b ) @ S2 @ V3 )
% 2.64/1.51 = ( bot_bot @ ( set @ b ) ) ) )
% 2.64/1.51 => thesis ) ).
% 2.64/1.51
% 2.64/1.51 thf(conj_1,conjecture,
% 2.64/1.51 thesis ).
% 2.64/1.51 ------------------------
% 2.64/1.53 % (5403)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_5301 for (2999ds/2Mi)
% 2.64/1.53 % (5406)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_5301 for (2999ds/18Mi)
% 2.64/1.53 % (5402)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_5301 for (2999ds/27Mi)
% 2.64/1.53 % (5400)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_5301 for (2999ds/183Mi)
% 2.64/1.53 % (5403)Instruction limit reached!
% 2.64/1.53 % (5403)------------------------------
% 2.64/1.53 % (5403)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.53 % (5403)Termination reason: Unknown
% 2.64/1.53 % (5403)Termination phase: shuffling
% 2.64/1.53
% 2.64/1.53 % (5403)Memory used [KB]: 1407
% 2.64/1.53 % (5403)Time elapsed: 0.002 s
% 2.64/1.53 % (5403)Instructions burned: 2 (million)
% 2.64/1.53 % (5403)------------------------------
% 2.64/1.53 % (5403)------------------------------
% 2.64/1.53 % (5406)Instruction limit reached!
% 2.64/1.53 % (5406)------------------------------
% 2.64/1.53 % (5406)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.53 % (5406)Termination reason: Unknown
% 2.64/1.53 % (5406)Termination phase: Property scanning
% 2.64/1.53
% 2.64/1.53 % (5406)Memory used [KB]: 1791
% 2.64/1.53 % (5406)Time elapsed: 0.008 s
% 2.64/1.53 % (5406)Instructions burned: 20 (million)
% 2.64/1.53 % (5406)------------------------------
% 2.64/1.53 % (5406)------------------------------
% 2.64/1.54 % (5401)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_5301 for (2999ds/4Mi)
% 2.64/1.54 % (5404)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_5301 for (2999ds/2Mi)
% 2.64/1.54 % (5405)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_5301 for (2999ds/275Mi)
% 2.64/1.54 % (5404)Instruction limit reached!
% 2.64/1.54 % (5404)------------------------------
% 2.64/1.54 % (5404)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.54 % (5404)Termination reason: Unknown
% 2.64/1.54 % (5404)Termination phase: shuffling
% 2.64/1.54
% 2.64/1.54 % (5404)Memory used [KB]: 1407
% 2.64/1.54 % (5404)Time elapsed: 0.003 s
% 2.64/1.54 % (5404)Instructions burned: 2 (million)
% 2.64/1.54 % (5404)------------------------------
% 2.64/1.54 % (5404)------------------------------
% 2.64/1.54 % (5401)Instruction limit reached!
% 2.64/1.54 % (5401)------------------------------
% 2.64/1.54 % (5401)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.54 % (5402)Instruction limit reached!
% 2.64/1.54 % (5402)------------------------------
% 2.64/1.54 % (5402)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.54 % (5402)Termination reason: Unknown
% 2.64/1.54 % (5402)Termination phase: Property scanning
% 2.64/1.54
% 2.64/1.54 % (5402)Memory used [KB]: 1791
% 2.64/1.54 % (5402)Time elapsed: 0.010 s
% 2.64/1.54 % (5402)Instructions burned: 27 (million)
% 2.64/1.54 % (5402)------------------------------
% 2.64/1.54 % (5402)------------------------------
% 2.64/1.54 % (5401)Termination reason: Unknown
% 2.64/1.54 % (5401)Termination phase: shuffling
% 2.64/1.54
% 2.64/1.54 % (5401)Memory used [KB]: 1407
% 2.64/1.54 % (5401)Time elapsed: 0.004 s
% 2.64/1.54 % (5401)Instructions burned: 4 (million)
% 2.64/1.54 % (5401)------------------------------
% 2.64/1.54 % (5401)------------------------------
% 2.64/1.54 % (5407)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_5301 for (2999ds/3Mi)
% 2.64/1.54 % (5407)Instruction limit reached!
% 2.64/1.54 % (5407)------------------------------
% 2.64/1.54 % (5407)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.54 % (5407)Termination reason: Unknown
% 2.64/1.54 % (5407)Termination phase: shuffling
% 2.64/1.54
% 2.64/1.54 % (5407)Memory used [KB]: 1407
% 2.64/1.54 % (5407)Time elapsed: 0.002 s
% 2.64/1.54 % (5407)Instructions burned: 3 (million)
% 2.64/1.54 % (5407)------------------------------
% 2.64/1.54 % (5407)------------------------------
% 2.64/1.54 % (5408)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_5301 for (2999ds/37Mi)
% 2.64/1.55 % (5410)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_5301 for (2999ds/3Mi)
% 2.64/1.55 % (5410)Instruction limit reached!
% 2.64/1.55 % (5410)------------------------------
% 2.64/1.55 % (5410)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.55 % (5410)Termination reason: Unknown
% 2.64/1.55 % (5410)Termination phase: shuffling
% 2.64/1.55
% 2.64/1.55 % (5410)Memory used [KB]: 1407
% 2.64/1.55 % (5410)Time elapsed: 0.003 s
% 2.64/1.55 % (5410)Instructions burned: 3 (million)
% 2.64/1.55 % (5410)------------------------------
% 2.64/1.55 % (5410)------------------------------
% 2.64/1.55 % (5409)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_5301 for (2999ds/15Mi)
% 2.64/1.56 % (5408)Instruction limit reached!
% 2.64/1.56 % (5408)------------------------------
% 2.64/1.56 % (5408)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.56 % (5408)Termination reason: Unknown
% 2.64/1.56 % (5408)Termination phase: Property scanning
% 2.64/1.56
% 2.64/1.56 % (5408)Memory used [KB]: 2174
% 2.64/1.56 % (5408)Time elapsed: 0.012 s
% 2.64/1.56 % (5408)Instructions burned: 38 (million)
% 2.64/1.56 % (5408)------------------------------
% 2.64/1.56 % (5408)------------------------------
% 2.64/1.56 % (5411)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_5301 for (2999ds/1041Mi)
% 2.64/1.56 % (5412)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_5301 for (2999ds/7Mi)
% 2.64/1.56 % (5413)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_5301 for (2999ds/16Mi)
% 2.64/1.56 % (5412)Instruction limit reached!
% 2.64/1.56 % (5412)------------------------------
% 2.64/1.56 % (5412)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.56 % (5412)Termination reason: Unknown
% 2.64/1.56 % (5412)Termination phase: shuffling
% 2.64/1.56
% 2.64/1.56 % (5412)Memory used [KB]: 1535
% 2.64/1.56 % (5412)Time elapsed: 0.006 s
% 2.64/1.56 % (5412)Instructions burned: 7 (million)
% 2.64/1.56 % (5412)------------------------------
% 2.64/1.56 % (5412)------------------------------
% 2.64/1.56 % (5409)Instruction limit reached!
% 2.64/1.56 % (5409)------------------------------
% 2.64/1.56 % (5409)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.56 % (5409)Termination reason: Unknown
% 2.64/1.56 % (5409)Termination phase: shuffling
% 2.64/1.56
% 2.64/1.56 % (5409)Memory used [KB]: 1791
% 2.64/1.56 % (5409)Time elapsed: 0.012 s
% 2.64/1.56 % (5409)Instructions burned: 16 (million)
% 2.64/1.56 % (5409)------------------------------
% 2.64/1.56 % (5409)------------------------------
% 2.64/1.56 % (5414)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_5301 for (2999ds/3Mi)
% 2.64/1.57 % (5413)Instruction limit reached!
% 2.64/1.57 % (5413)------------------------------
% 2.64/1.57 % (5413)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.57 % (5413)Termination reason: Unknown
% 2.64/1.57 % (5413)Termination phase: shuffling
% 2.64/1.57
% 2.64/1.57 % (5413)Memory used [KB]: 1791
% 2.64/1.57 % (5413)Time elapsed: 0.008 s
% 2.64/1.57 % (5413)Instructions burned: 17 (million)
% 2.64/1.57 % (5413)------------------------------
% 2.64/1.57 % (5413)------------------------------
% 2.64/1.57 % (5414)Instruction limit reached!
% 2.64/1.57 % (5414)------------------------------
% 2.64/1.57 % (5414)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.57 % (5414)Termination reason: Unknown
% 2.64/1.57 % (5414)Termination phase: shuffling
% 2.64/1.57
% 2.64/1.57 % (5414)Memory used [KB]: 1407
% 2.64/1.57 % (5414)Time elapsed: 0.002 s
% 2.64/1.57 % (5414)Instructions burned: 3 (million)
% 2.64/1.57 % (5414)------------------------------
% 2.64/1.57 % (5414)------------------------------
% 2.64/1.57 % (5415)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_5301 for (2999ds/3Mi)
% 2.64/1.57 % (5415)Instruction limit reached!
% 2.64/1.57 % (5415)------------------------------
% 2.64/1.57 % (5415)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.57 % (5415)Termination reason: Unknown
% 2.64/1.57 % (5415)Termination phase: shuffling
% 2.64/1.57
% 2.64/1.57 % (5415)Memory used [KB]: 1407
% 2.64/1.57 % (5415)Time elapsed: 0.002 s
% 2.64/1.57 % (5415)Instructions burned: 3 (million)
% 2.64/1.57 % (5415)------------------------------
% 2.64/1.57 % (5415)------------------------------
% 2.64/1.57 % (5418)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_5301 for (2999ds/4Mi)
% 2.64/1.58 % (5417)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_5301 for (2999ds/3Mi)
% 2.64/1.58 % (5417)Instruction limit reached!
% 2.64/1.58 % (5417)------------------------------
% 2.64/1.58 % (5417)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.58 % (5417)Termination reason: Unknown
% 2.64/1.58 % (5417)Termination phase: shuffling
% 2.64/1.58
% 2.64/1.58 % (5417)Memory used [KB]: 1407
% 2.64/1.58 % (5417)Time elapsed: 0.003 s
% 2.64/1.58 % (5417)Instructions burned: 4 (million)
% 2.64/1.58 % (5417)------------------------------
% 2.64/1.58 % (5417)------------------------------
% 2.64/1.58 % (5418)Instruction limit reached!
% 2.64/1.58 % (5418)------------------------------
% 2.64/1.58 % (5418)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.58 % (5418)Termination reason: Unknown
% 2.64/1.58 % (5418)Termination phase: shuffling
% 2.64/1.58
% 2.64/1.58 % (5418)Memory used [KB]: 1535
% 2.64/1.58 % (5418)Time elapsed: 0.003 s
% 2.64/1.58 % (5418)Instructions burned: 6 (million)
% 2.64/1.58 % (5418)------------------------------
% 2.64/1.58 % (5418)------------------------------
% 2.64/1.58 % (5416)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_5301 for (2999ds/7Mi)
% 2.64/1.58 % (5419)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_5301 for (2999ds/18Mi)
% 2.64/1.59 % (5416)Instruction limit reached!
% 2.64/1.59 % (5416)------------------------------
% 2.64/1.59 % (5416)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.59 % (5416)Termination reason: Unknown
% 2.64/1.59 % (5416)Termination phase: shuffling
% 2.64/1.59
% 2.64/1.59 % (5416)Memory used [KB]: 1535
% 2.64/1.59 % (5416)Time elapsed: 0.006 s
% 2.64/1.59 % (5416)Instructions burned: 7 (million)
% 2.64/1.59 % (5416)------------------------------
% 2.64/1.59 % (5416)------------------------------
% 2.64/1.59 % (5421)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_5301 for (2999ds/6Mi)
% 2.64/1.59 % (5420)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_5301 for (2999ds/710Mi)
% 2.64/1.59 % (5419)Instruction limit reached!
% 2.64/1.59 % (5419)------------------------------
% 2.64/1.59 % (5419)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.59 % (5419)Termination reason: Unknown
% 2.64/1.59 % (5419)Termination phase: shuffling
% 2.64/1.59
% 2.64/1.59 % (5419)Memory used [KB]: 1791
% 2.64/1.59 % (5419)Time elapsed: 0.007 s
% 2.64/1.59 % (5419)Instructions burned: 19 (million)
% 2.64/1.59 % (5419)------------------------------
% 2.64/1.59 % (5419)------------------------------
% 2.64/1.59 % (5421)Instruction limit reached!
% 2.64/1.59 % (5421)------------------------------
% 2.64/1.59 % (5421)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.59 % (5421)Termination reason: Unknown
% 2.64/1.59 % (5421)Termination phase: shuffling
% 2.64/1.59
% 2.64/1.59 % (5421)Memory used [KB]: 1535
% 2.64/1.59 % (5421)Time elapsed: 0.003 s
% 2.64/1.59 % (5421)Instructions burned: 6 (million)
% 2.64/1.59 % (5421)------------------------------
% 2.64/1.59 % (5421)------------------------------
% 2.64/1.59 % (5400)Instruction limit reached!
% 2.64/1.59 % (5400)------------------------------
% 2.64/1.59 % (5400)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.59 % (5400)Termination reason: Unknown
% 2.64/1.59 % (5400)Termination phase: Saturation
% 2.64/1.59
% 2.64/1.59 % (5400)Memory used [KB]: 7036
% 2.64/1.59 % (5400)Time elapsed: 0.062 s
% 2.64/1.59 % (5400)Instructions burned: 185 (million)
% 2.64/1.59 % (5400)------------------------------
% 2.64/1.59 % (5400)------------------------------
% 2.70/1.60 % (5422)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_5301 for (2999ds/902Mi)
% 2.70/1.60 % (5423)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_5301 for (2999ds/21Mi)
% 2.70/1.60 % (5424)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_5301 for (2999ds/5Mi)
% 2.70/1.60 % (5424)Instruction limit reached!
% 2.70/1.60 % (5424)------------------------------
% 2.70/1.60 % (5424)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.60 % (5424)Termination reason: Unknown
% 2.70/1.60 % (5424)Termination phase: shuffling
% 2.70/1.60
% 2.70/1.60 % (5424)Memory used [KB]: 1535
% 2.70/1.60 % (5424)Time elapsed: 0.003 s
% 2.70/1.60 % (5424)Instructions burned: 7 (million)
% 2.70/1.60 % (5424)------------------------------
% 2.70/1.60 % (5424)------------------------------
% 2.70/1.60 % (5425)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_5301 for (2999ds/6Mi)
% 2.70/1.60 % (5405)First to succeed.
% 2.70/1.60 % (5425)Instruction limit reached!
% 2.70/1.60 % (5425)------------------------------
% 2.70/1.60 % (5425)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.60 % (5425)Termination reason: Unknown
% 2.70/1.60 % (5425)Termination phase: shuffling
% 2.70/1.60
% 2.70/1.60 % (5425)Memory used [KB]: 1535
% 2.70/1.60 % (5425)Time elapsed: 0.004 s
% 2.70/1.60 % (5425)Instructions burned: 7 (million)
% 2.70/1.60 % (5425)------------------------------
% 2.70/1.60 % (5425)------------------------------
% 2.70/1.60 % (5405)Refutation found. Thanks to Tanya!
% 2.70/1.60 % SZS status Theorem for DTF2THF_5301
% 2.70/1.60 % SZS output start Proof for DTF2THF_5301
% 2.70/1.60 thf(type_def_5, type, set: ($tType) > $tType).
% 2.70/1.60 thf(type_def_7, type, filter: ($tType) > $tType).
% 2.70/1.60 thf(type_def_8, type, b: $tType).
% 2.70/1.60 thf(type_def_9, type, a: $tType).
% 2.70/1.60 thf(type_def_10, type, nat: $tType).
% 2.70/1.60 thf(func_def_0, type, filter: ($tType) > $tType).
% 2.70/1.60 thf(func_def_1, type, set: ($tType) > $tType).
% 2.70/1.60 thf(func_def_2, type, nat: $tType).
% 2.70/1.60 thf(func_def_3, type, b: $tType).
% 2.70/1.60 thf(func_def_4, type, a: $tType).
% 2.70/1.60 thf(func_def_5, type, topolo2117631714pology: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_6, type, comple1035589618norder: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_7, type, finite_finite: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_8, type, bounded_lattice: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_9, type, type: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_10, type, countable: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_11, type, minus: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_12, type, bot: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_13, type, ord: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_14, type, top: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_15, type, order: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_16, type, group_add: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_17, type, lattice: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_18, type, linorder: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_19, type, preorder: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_20, type, order_bot: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_21, type, semilattice_inf: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_22, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_23, type, topological_t0_space: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_24, type, topological_t1_space: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_25, type, topological_t2_space: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_26, type, bounded_lattice_bot: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_27, type, cancel146912293up_add: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_28, type, topolo890362671_space: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_29, type, topolo47006728_space: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_30, type, topolo2133971006pology: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_31, type, topolo503727757_space: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_32, type, topolo2135403230pology: !>[X0: $tType]:($o)).
% 2.70/1.60 thf(func_def_33, type, elementary_closure: !>[X0: $tType]:(set @ X0 > set @ X0)).
% 2.70/1.60 thf(func_def_34, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.70/1.60 thf(func_def_35, type, inf_inf: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.70/1.60 thf(func_def_36, type, lower_582600101lsc_at: !>[X0: $tType, X1: $tType]:(X0 > (X0 > X1) > $o)).
% 2.70/1.60 thf(func_def_37, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.70/1.60 thf(func_def_38, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.70/1.60 thf(func_def_39, type, top_top: !>[X0: $tType]:(X0)).
% 2.70/1.60 thf(func_def_40, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.70/1.60 thf(func_def_41, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 2.70/1.60 thf(func_def_42, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 2.70/1.60 thf(func_def_43, type, is_singleton: !>[X0: $tType]:(set @ X0 > $o)).
% 2.70/1.60 thf(func_def_44, type, remove: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 2.70/1.60 thf(func_def_45, type, the_elem: !>[X0: $tType]:(set @ X0 > X0)).
% 2.70/1.60 thf(func_def_46, type, topolo1751647064n_open: !>[X0: $tType]:(set @ X0 > $o)).
% 2.70/1.60 thf(func_def_47, type, topolo507301023within: !>[X0: $tType]:(X0 > set @ X0 > filter @ X0)).
% 2.70/1.60 thf(func_def_48, type, topolo406746546ounded: !>[X0: $tType]:(set @ X0 > $o)).
% 2.70/1.60 thf(func_def_49, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.70/1.60 thf(func_def_50, type, a2: b).
% 2.70/1.60 thf(func_def_51, type, f: a > b).
% 2.70/1.60 thf(func_def_53, type, x0: a).
% 2.70/1.60 thf(func_def_66, type, sK0: !>[X0: $tType]:(X0 > nat > set @ X0)).
% 2.70/1.60 thf(func_def_67, type, sK1: !>[X0: $tType]:(set @ X0 > X0 > nat)).
% 2.70/1.60 thf(func_def_68, type, sK2: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.70/1.60 thf(func_def_69, type, sK3: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.70/1.60 thf(func_def_70, type, sK4: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.70/1.60 thf(func_def_71, type, sK5: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.70/1.60 thf(func_def_72, type, sK6: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.70/1.60 thf(func_def_73, type, sK7: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.70/1.60 thf(func_def_74, type, sK8: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 2.70/1.60 thf(func_def_75, type, sK9: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.70/1.60 thf(func_def_76, type, sK10: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.70/1.60 thf(func_def_77, type, sK11: !>[X0: $tType]:(set @ X0 > X0)).
% 2.70/1.60 thf(func_def_78, type, sK12: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.70/1.60 thf(func_def_79, type, sK13: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.70/1.60 thf(func_def_80, type, sK14: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.70/1.60 thf(func_def_81, type, sK15: !>[X0: $tType]:(set @ X0 > X0)).
% 2.70/1.60 thf(func_def_82, type, sK16: !>[X0: $tType]:(set @ X0 > X0)).
% 2.70/1.60 thf(func_def_83, type, sK17: !>[X0: $tType]:(set @ X0 > X0)).
% 2.70/1.60 thf(func_def_84, type, sK18: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.70/1.60 thf(func_def_85, type, sK19: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.70/1.60 thf(func_def_86, type, sK20: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.70/1.60 thf(func_def_87, type, sK21: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.70/1.60 thf(func_def_88, type, sK22: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.70/1.60 thf(func_def_89, type, sK23: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.70/1.60 thf(func_def_90, type, sK24: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.70/1.60 thf(func_def_91, type, sK25: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.70/1.60 thf(func_def_92, type, sK26: set @ b > set @ a).
% 2.70/1.60 thf(func_def_93, type, sK27: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.70/1.60 thf(func_def_94, type, sK28: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.70/1.60 thf(func_def_95, type, sK29: set @ b).
% 2.70/1.60 thf(func_def_96, type, sK30: set @ b).
% 2.70/1.60 thf(func_def_97, type, sK31: !>[X0: $tType]:(set @ X0 > X0)).
% 2.70/1.60 thf(func_def_99, type, ph33: !>[X0: $tType]:(X0)).
% 2.70/1.60 thf(f1516,plain,(
% 2.70/1.60 $false),
% 2.70/1.60 inference(avatar_sat_refutation,[status(thm)],[f1420,f1421,f1476,f1485,f1489,f1499,f1515])).
% 2.70/1.60 thf(f1515,plain,(
% 2.70/1.60 spl32_6),
% 2.70/1.60 inference(avatar_contradiction_clause,[status(thm)],[f1514])).
% 2.70/1.60 thf(f1514,plain,(
% 2.70/1.60 $false | spl32_6),
% 2.70/1.60 inference(subsumption_resolution,[status(thm)],[f1513,f1318])).
% 2.70/1.60 thf(f1318,plain,(
% 2.70/1.60 (a2 != (f @ x0))),
% 2.70/1.60 inference(cnf_transformation,[status(thm)],[f1])).
% 2.70/1.60 thf(f1,axiom,(
% 2.70/1.60 (a2 != (f @ x0))),
% 2.70/1.60 file('/export/starexec/sandbox/tmp/tmp.wdWkGyYZoX/DTF2THF_5301.p',fact_0__092_060open_062A_A_092_060noteq_062_Af_Ax0_092_060close_062)).
% 2.70/1.60 thf(f1513,plain,(
% 2.70/1.60 (a2 = (f @ x0)) | spl32_6),
% 2.70/1.60 inference(trivial_inequality_removal,[status(thm)],[f1512])).
% 2.70/1.60 thf(f1512,plain,(
% 2.70/1.60 (a2 = (f @ x0)) | ($true != $true) | spl32_6),
% 2.70/1.60 inference(superposition,[status(thm)],[f1475,f1387])).
% 2.70/1.60 thf(f1387,plain,(
% 2.70/1.60 ($true = (member @ b @ (f @ x0) @ sK29)) | (a2 = (f @ x0))),
% 2.70/1.60 inference(cnf_transformation,[status(thm)],[f1216])).
% 2.70/1.60 thf(f1216,plain,(
% 2.70/1.60 (a2 = (f @ x0)) | (($true = (member @ b @ a2 @ sK30)) & ((topolo1751647064n_open @ b @ sK29) = $true) & ($true = (topolo1751647064n_open @ b @ sK30)) & (bot_bot @ set @ b = (inf_inf @ set @ b @ sK29 @ sK30)) & ($true = (member @ b @ (f @ x0) @ sK29)))),
% 2.70/1.60 inference(skolemisation,[status(esa),new_symbols(skolem,[sK29,sK30])],[f1082,f1215])).
% 2.70/1.60 thf(f1215,plain,(
% 2.70/1.60 ? [X0 : set @ b,X1 : set @ b] : (($true = (member @ b @ a2 @ X1)) & ($true = (topolo1751647064n_open @ b @ X0)) & ($true = (topolo1751647064n_open @ b @ X1)) & (bot_bot @ set @ b = (inf_inf @ set @ b @ X0 @ X1)) & ($true = (member @ b @ (f @ x0) @ X0))) => (($true = (member @ b @ a2 @ sK30)) & ((topolo1751647064n_open @ b @ sK29) = $true) & ($true = (topolo1751647064n_open @ b @ sK30)) & (bot_bot @ set @ b = (inf_inf @ set @ b @ sK29 @ sK30)) & ($true = (member @ b @ (f @ x0) @ sK29)))),
% 2.70/1.60 introduced(definition,[],[choice_axiom])).
% 2.70/1.60 thf(f1082,plain,(
% 2.70/1.60 (a2 = (f @ x0)) | ? [X0 : set @ b,X1 : set @ b] : (($true = (member @ b @ a2 @ X1)) & ($true = (topolo1751647064n_open @ b @ X0)) & ($true = (topolo1751647064n_open @ b @ X1)) & (bot_bot @ set @ b = (inf_inf @ set @ b @ X0 @ X1)) & ($true = (member @ b @ (f @ x0) @ X0)))),
% 2.70/1.60 inference(ennf_transformation,[status(thm)],[f918])).
% 2.70/1.60 thf(f918,plain,(
% 2.70/1.60 (a2 != (f @ x0)) => ? [X0 : set @ b,X1 : set @ b] : (($true = (member @ b @ a2 @ X1)) & ($true = (topolo1751647064n_open @ b @ X0)) & ($true = (topolo1751647064n_open @ b @ X1)) & (bot_bot @ set @ b = (inf_inf @ set @ b @ X0 @ X1)) & ($true = (member @ b @ (f @ x0) @ X0)))),
% 2.70/1.60 inference(fool_elimination,[status(thm)],[f917])).
% 2.70/1.60 thf(f917,plain,(
% 2.70/1.60 (a2 != (f @ x0)) => ? [X0 : set @ b,X1 : set @ b] : ((topolo1751647064n_open @ b @ X1) & (member @ b @ (f @ x0) @ X0) & (member @ b @ a2 @ X1) & (topolo1751647064n_open @ b @ X0) & (bot_bot @ set @ b = (inf_inf @ set @ b @ X0 @ X1)))),
% 2.70/1.60 inference(rectify,[status(thm)],[f2])).
% 2.70/1.60 thf(f2,axiom,(
% 2.70/1.60 (a2 != (f @ x0)) => ? [X2 : set @ b,X3 : set @ b] : ((topolo1751647064n_open @ b @ X3) & (member @ b @ (f @ x0) @ X2) & (member @ b @ a2 @ X3) & (topolo1751647064n_open @ b @ X2) & ((inf_inf @ set @ b @ X2 @ X3) = bot_bot @ set @ b))),
% 2.70/1.60 file('/export/starexec/sandbox/tmp/tmp.wdWkGyYZoX/DTF2THF_5301.p',fact_1__092_060open_062f_Ax0_A_092_060noteq_062_AA_A_092_060Longrightarrow_062_A_092_060exists_062U_AV_O_Aopen_AU_A_092_060and_062_Aopen_AV_A_092_060and_062_Af_Ax0_A_092_060in_062_AU_A_092_060and_062_AA_A_092_060in_062_AV_A_092_060and_062_AU_A_092_060inter_062_AV_A_061_A_123_125_092_060close_062)).
% 2.70/1.60 thf(f1475,plain,(
% 2.70/1.60 ($true != (member @ b @ (f @ x0) @ sK29)) | spl32_6),
% 2.70/1.60 inference(avatar_component_clause,[status(thm)],[f1473])).
% 2.70/1.60 thf(f1473,definition,(
% 2.70/1.60 spl32_6 <=> ($true = (member @ b @ (f @ x0) @ sK29))),
% 2.70/1.60 introduced(definition,[new_symbols(naming,[spl32_6])],[avatar_definition])).
% 2.70/1.60 thf(f1499,plain,(
% 2.70/1.60 spl32_5),
% 2.70/1.60 inference(avatar_contradiction_clause,[status(thm)],[f1498])).
% 2.70/1.60 thf(f1498,plain,(
% 2.70/1.60 $false | spl32_5),
% 2.70/1.60 inference(subsumption_resolution,[status(thm)],[f1497,f1318])).
% 2.70/1.60 thf(f1497,plain,(
% 2.70/1.60 (a2 = (f @ x0)) | spl32_5),
% 2.70/1.60 inference(trivial_inequality_removal,[status(thm)],[f1496])).
% 2.70/1.60 thf(f1496,plain,(
% 2.70/1.60 (a2 = (f @ x0)) | ($true != $true) | spl32_5),
% 2.70/1.60 inference(superposition,[status(thm)],[f1471,f1389])).
% 2.70/1.60 thf(f1389,plain,(
% 2.70/1.60 ($true = (topolo1751647064n_open @ b @ sK30)) | (a2 = (f @ x0))),
% 2.70/1.60 inference(cnf_transformation,[status(thm)],[f1216])).
% 2.70/1.60 thf(f1471,plain,(
% 2.70/1.60 ($true != (topolo1751647064n_open @ b @ sK30)) | spl32_5),
% 2.70/1.60 inference(avatar_component_clause,[status(thm)],[f1469])).
% 2.70/1.60 thf(f1469,definition,(
% 2.70/1.60 spl32_5 <=> ($true = (topolo1751647064n_open @ b @ sK30))),
% 2.70/1.60 introduced(definition,[new_symbols(naming,[spl32_5])],[avatar_definition])).
% 2.70/1.60 thf(f1489,plain,(
% 2.70/1.60 spl32_4),
% 2.70/1.60 inference(avatar_contradiction_clause,[status(thm)],[f1488])).
% 2.70/1.60 thf(f1488,plain,(
% 2.70/1.60 $false | spl32_4),
% 2.70/1.60 inference(subsumption_resolution,[status(thm)],[f1487,f1318])).
% 2.70/1.60 thf(f1487,plain,(
% 2.70/1.60 (a2 = (f @ x0)) | spl32_4),
% 2.70/1.60 inference(trivial_inequality_removal,[status(thm)],[f1486])).
% 2.70/1.60 thf(f1486,plain,(
% 2.70/1.60 (a2 = (f @ x0)) | ($true != $true) | spl32_4),
% 2.70/1.60 inference(superposition,[status(thm)],[f1467,f1390])).
% 2.70/1.60 thf(f1390,plain,(
% 2.70/1.60 ((topolo1751647064n_open @ b @ sK29) = $true) | (a2 = (f @ x0))),
% 2.70/1.60 inference(cnf_transformation,[status(thm)],[f1216])).
% 2.70/1.60 thf(f1467,plain,(
% 2.70/1.60 ((topolo1751647064n_open @ b @ sK29) != $true) | spl32_4),
% 2.70/1.60 inference(avatar_component_clause,[status(thm)],[f1465])).
% 2.70/1.60 thf(f1465,definition,(
% 2.70/1.60 spl32_4 <=> ((topolo1751647064n_open @ b @ sK29) = $true)),
% 2.70/1.60 introduced(definition,[new_symbols(naming,[spl32_4])],[avatar_definition])).
% 2.70/1.60 thf(f1485,plain,(
% 2.70/1.60 spl32_3),
% 2.70/1.60 inference(avatar_contradiction_clause,[status(thm)],[f1484])).
% 2.70/1.60 thf(f1484,plain,(
% 2.70/1.60 $false | spl32_3),
% 2.70/1.60 inference(subsumption_resolution,[status(thm)],[f1483,f1318])).
% 2.70/1.60 thf(f1483,plain,(
% 2.70/1.60 (a2 = (f @ x0)) | spl32_3),
% 2.70/1.60 inference(trivial_inequality_removal,[status(thm)],[f1482])).
% 2.70/1.60 thf(f1482,plain,(
% 2.70/1.60 (a2 = (f @ x0)) | ($true != $true) | spl32_3),
% 2.70/1.60 inference(superposition,[status(thm)],[f1463,f1391])).
% 2.70/1.60 thf(f1391,plain,(
% 2.70/1.60 ($true = (member @ b @ a2 @ sK30)) | (a2 = (f @ x0))),
% 2.70/1.60 inference(cnf_transformation,[status(thm)],[f1216])).
% 2.70/1.60 thf(f1463,plain,(
% 2.70/1.60 ($true != (member @ b @ a2 @ sK30)) | spl32_3),
% 2.70/1.60 inference(avatar_component_clause,[status(thm)],[f1461])).
% 2.70/1.60 thf(f1461,definition,(
% 2.70/1.60 spl32_3 <=> ($true = (member @ b @ a2 @ sK30))),
% 2.70/1.60 introduced(definition,[new_symbols(naming,[spl32_3])],[avatar_definition])).
% 2.70/1.60 thf(f1476,plain,(
% 2.70/1.60 ~spl32_3 | ~spl32_4 | ~spl32_5 | ~spl32_6 | ~spl32_2),
% 2.70/1.60 inference(avatar_split_clause,[status(thm)],[f1459,f1418,f1473,f1469,f1465,f1461])).
% 2.70/1.60 thf(f1418,definition,(
% 2.70/1.60 spl32_2 <=> ! [X0 : set @ b,X1 : set @ b] : (($true != (member @ b @ a2 @ X1)) | ($true != (member @ b @ (f @ x0) @ X0)) | ($true != (topolo1751647064n_open @ b @ X1)) | ($true != (topolo1751647064n_open @ b @ X0)) | (bot_bot @ set @ b != (inf_inf @ set @ b @ X0 @ X1)))),
% 2.70/1.60 introduced(definition,[new_symbols(naming,[spl32_2])],[avatar_definition])).
% 2.70/1.60 thf(f1459,plain,(
% 2.70/1.60 ($true != (topolo1751647064n_open @ b @ sK30)) | ((topolo1751647064n_open @ b @ sK29) != $true) | ($true != (member @ b @ (f @ x0) @ sK29)) | ($true != (member @ b @ a2 @ sK30)) | ~spl32_2),
% 2.70/1.60 inference(subsumption_resolution,[status(thm)],[f1457,f1318])).
% 2.70/1.60 thf(f1457,plain,(
% 2.70/1.60 ((topolo1751647064n_open @ b @ sK29) != $true) | ($true != (member @ b @ (f @ x0) @ sK29)) | ($true != (topolo1751647064n_open @ b @ sK30)) | (a2 = (f @ x0)) | ($true != (member @ b @ a2 @ sK30)) | ~spl32_2),
% 2.70/1.60 inference(trivial_inequality_removal,[status(thm)],[f1422])).
% 2.70/1.60 thf(f1422,plain,(
% 2.70/1.60 (a2 = (f @ x0)) | ((topolo1751647064n_open @ b @ sK29) != $true) | ($true != (topolo1751647064n_open @ b @ sK30)) | (bot_bot @ set @ b != bot_bot @ set @ b) | ($true != (member @ b @ a2 @ sK30)) | ($true != (member @ b @ (f @ x0) @ sK29)) | ~spl32_2),
% 2.70/1.60 inference(superposition,[status(thm)],[f1419,f1388])).
% 2.70/1.60 thf(f1388,plain,(
% 2.70/1.60 (bot_bot @ set @ b = (inf_inf @ set @ b @ sK29 @ sK30)) | (a2 = (f @ x0))),
% 2.70/1.60 inference(cnf_transformation,[status(thm)],[f1216])).
% 2.70/1.60 thf(f1419,plain,(
% 2.70/1.60 ( ! [X0 : set @ b,X1 : set @ b] : ((bot_bot @ set @ b != (inf_inf @ set @ b @ X0 @ X1)) | ($true != (member @ b @ (f @ x0) @ X0)) | ($true != (topolo1751647064n_open @ b @ X0)) | ($true != (member @ b @ a2 @ X1)) | ($true != (topolo1751647064n_open @ b @ X1))) ) | ~spl32_2),
% 2.70/1.60 inference(avatar_component_clause,[status(thm)],[f1418])).
% 2.70/1.60 thf(f1421,plain,(
% 2.70/1.60 ~spl32_1),
% 2.70/1.60 inference(avatar_split_clause,[status(thm)],[f1262,f1414])).
% 2.70/1.60 thf(f1414,definition,(
% 2.70/1.60 spl32_1 <=> (thesis = $true)),
% 2.70/1.60 introduced(definition,[new_symbols(naming,[spl32_1])],[avatar_definition])).
% 2.70/1.60 thf(f1262,plain,(
% 2.70/1.60 (thesis != $true)),
% 2.70/1.60 inference(cnf_transformation,[status(thm)],[f1013])).
% 2.70/1.60 thf(f1013,plain,(
% 2.70/1.60 (thesis != $true)),
% 2.70/1.60 inference(flattening,[status(thm)],[f502])).
% 2.70/1.60 thf(f502,plain,(
% 2.70/1.60 ~(thesis = $true)),
% 2.70/1.60 inference(fool_elimination,[status(thm)],[f501])).
% 2.70/1.60 thf(f501,plain,(
% 2.70/1.60 ~thesis),
% 2.70/1.60 inference(rectify,[status(thm)],[f359])).
% 2.70/1.60 thf(f359,negated_conjecture,(
% 2.70/1.60 ~thesis),
% 2.70/1.60 inference(negated_conjecture,[status(cth)],[f358])).
% 2.70/1.60 thf(f358,conjecture,(
% 2.70/1.60 thesis),
% 2.70/1.60 file('/export/starexec/sandbox/tmp/tmp.wdWkGyYZoX/DTF2THF_5301.p',conj_1)).
% 2.70/1.60 thf(f1420,plain,(
% 2.70/1.60 spl32_1 | spl32_2),
% 2.70/1.60 inference(avatar_split_clause,[status(thm)],[f1328,f1418,f1414])).
% 2.70/1.60 thf(f1328,plain,(
% 2.70/1.60 ( ! [X0 : set @ b,X1 : set @ b] : (($true != (member @ b @ a2 @ X1)) | (thesis = $true) | (bot_bot @ set @ b != (inf_inf @ set @ b @ X0 @ X1)) | ($true != (topolo1751647064n_open @ b @ X0)) | ($true != (topolo1751647064n_open @ b @ X1)) | ($true != (member @ b @ (f @ x0) @ X0))) )),
% 2.70/1.60 inference(cnf_transformation,[status(thm)],[f1094])).
% 2.70/1.60 thf(f1094,plain,(
% 2.70/1.60 ! [X0 : set @ b,X1 : set @ b] : ((bot_bot @ set @ b != (inf_inf @ set @ b @ X0 @ X1)) | ($true != (member @ b @ a2 @ X1)) | (thesis = $true) | ($true != (topolo1751647064n_open @ b @ X0)) | ($true != (topolo1751647064n_open @ b @ X1)) | ($true != (member @ b @ (f @ x0) @ X0)))),
% 2.70/1.60 inference(flattening,[status(thm)],[f1093])).
% 2.70/1.60 thf(f1093,plain,(
% 2.70/1.60 ! [X0 : set @ b,X1 : set @ b] : ((thesis = $true) | (($true != (topolo1751647064n_open @ b @ X0)) | (bot_bot @ set @ b != (inf_inf @ set @ b @ X0 @ X1)) | ($true != (member @ b @ a2 @ X1)) | ($true != (member @ b @ (f @ x0) @ X0)) | ($true != (topolo1751647064n_open @ b @ X1))))),
% 2.70/1.60 inference(ennf_transformation,[status(thm)],[f624])).
% 2.70/1.60 thf(f624,plain,(
% 2.70/1.60 ! [X0 : set @ b,X1 : set @ b] : ((($true = (topolo1751647064n_open @ b @ X0)) & (bot_bot @ set @ b = (inf_inf @ set @ b @ X0 @ X1)) & ($true = (member @ b @ a2 @ X1)) & ($true = (member @ b @ (f @ x0) @ X0)) & ($true = (topolo1751647064n_open @ b @ X1))) => (thesis = $true))),
% 2.70/1.60 inference(fool_elimination,[status(thm)],[f623])).
% 2.70/1.60 thf(f623,plain,(
% 2.70/1.60 ! [X0 : set @ b,X1 : set @ b] : (((member @ b @ a2 @ X1) & (topolo1751647064n_open @ b @ X1) & (topolo1751647064n_open @ b @ X0) & (bot_bot @ set @ b = (inf_inf @ set @ b @ X0 @ X1)) & (member @ b @ (f @ x0) @ X0)) => thesis)),
% 2.70/1.60 inference(rectify,[status(thm)],[f357])).
% 2.70/1.60 thf(f357,axiom,(
% 2.70/1.60 ! [X19 : set @ b,X59 : set @ b] : (((member @ b @ a2 @ X59) & (topolo1751647064n_open @ b @ X59) & (topolo1751647064n_open @ b @ X19) & (bot_bot @ set @ b = (inf_inf @ set @ b @ X19 @ X59)) & (member @ b @ (f @ x0) @ X19)) => thesis)),
% 2.70/1.60 file('/export/starexec/sandbox/tmp/tmp.wdWkGyYZoX/DTF2THF_5301.p',conj_0)).
% 2.70/1.60 % SZS output end Proof for DTF2THF_5301
% 2.70/1.60 % (5405)------------------------------
% 2.70/1.60 % (5405)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.60 % (5405)Termination reason: Refutation
% 2.70/1.60
% 2.70/1.60 % (5405)Memory used [KB]: 6780
% 2.70/1.60 % (5405)Time elapsed: 0.068 s
% 2.70/1.60 % (5405)Instructions burned: 108 (million)
% 2.70/1.60 % (5405)------------------------------
% 2.70/1.60 % (5405)------------------------------
% 2.70/1.60 % (5399)Success in time 0.085 s
%------------------------------------------------------------------------------