%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP124^2 : 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 : n024.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 08:55:45 AM UTC 2026
% Result : Theorem 1.77s 1.12s
% Output : Refutation 1.77s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : ITP124^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.06 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.07/0.24 % Computer : n024.cluster.edu
% 0.07/0.24 % Model : x86_64 x86_64
% 0.07/0.24 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.24 % Memory : 8042.1875MB
% 0.07/0.24 % OS : Linux 3.10.0-693.el7.x86_64
% 0.07/0.24 % CPULimit : 300
% 0.07/0.24 % WCLimit : 300
% 0.07/0.24 % DateTime : Tue Feb 24 03:26:55 EST 2026
% 0.07/0.24 % CPUTime :
% 0.07/0.24 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.07/0.30 ---- Original DTF file ---
% 0.07/0.30 thf(spec,logic,$$dhol).
% 0.07/0.30 %------------------------------------------------------------------------------
% 0.07/0.30 % File : ITP124^2 : TPTP v9.2.1. Released v7.5.0.
% 0.07/0.30 % Domain : Interactive Theorem Proving
% 0.07/0.30 % Problem : Sledgehammer Modular_Distrib_Lattice problem prob_46__3255746_1
% 0.07/0.30 % Version : Especial.
% 0.07/0.30 % English :
% 0.07/0.30
% 0.07/0.30 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.07/0.30 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.07/0.30 % Source : [Des21]
% 0.07/0.30 % Names : Modular_Distrib_Lattice/prob_46__3255746_1 [Des21]
% 0.07/0.30
% 0.07/0.30 % Status : Theorem
% 0.07/0.30 % Rating : 0.00 v7.5.0
% 0.07/0.30 % Syntax : Number of formulae : 303 ( 122 unt; 39 typ; 0 def)
% 0.07/0.30 % Number of atoms : 726 ( 226 equ; 0 cnn)
% 0.07/0.30 % Maximal formula atoms : 8 ( 2 avg)
% 0.07/0.30 % Number of connectives : 2589 ( 52 ~; 7 |; 35 &;2208 @)
% 0.07/0.30 % ( 0 <=>; 287 =>; 0 <=; 0 <~>)
% 0.07/0.30 % Maximal formula depth : 19 ( 6 avg)
% 0.07/0.30 % Number of types : 2 ( 1 usr)
% 0.07/0.30 % Number of type conns : 175 ( 175 >; 0 *; 0 +; 0 <<)
% 0.07/0.30 % Number of symbols : 40 ( 38 usr; 4 con; 0-6 aty)
% 0.07/0.30 % Number of variables : 737 ( 23 ^; 668 !; 12 ?; 737 :)
% 0.07/0.30 % ( 34 !>; 0 ?*; 0 @-; 0 @+)
% 0.07/0.30 % SPC : TH1_THM_EQU_NAR
% 0.07/0.30
% 0.07/0.30 % Comments : This file was generated by Sledgehammer 2021-02-23 16:28:37.161
% 0.07/0.30 %------------------------------------------------------------------------------
% 0.07/0.30 % Could-be-implicit typings (2)
% 0.07/0.30 thf(ty_t_Set_Oset,type,
% 0.07/0.30 set: $tType > $tType ).
% 0.07/0.30
% 0.07/0.30 thf(ty_tf_a,type,
% 0.07/0.30 a: $tType ).
% 0.07/0.30
% 0.07/0.30 % Explicit typings (37)
% 0.07/0.30 thf(sy_cl_HOL_Otype,type,
% 0.07/0.30 type:
% 0.07/0.30 !>[A: $tType] : $o ).
% 0.07/0.30
% 0.07/0.30 thf(sy_cl_Orderings_Oorder,type,
% 0.07/0.30 order:
% 0.07/0.30 !>[A: $tType] : $o ).
% 0.07/0.30
% 0.07/0.30 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.07/0.30 finite_finite:
% 0.07/0.30 !>[A: $tType] : $o ).
% 0.07/0.30
% 0.07/0.30 thf(sy_cl_Orderings_Olinorder,type,
% 0.07/0.30 linorder:
% 0.07/0.30 !>[A: $tType] : $o ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above,type,
% 0.07/0.30 condit2040224947_above:
% 0.07/0.30 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below,type,
% 0.07/0.30 condit1201339847_below:
% 0.07/0.30 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Finite__Set_Ocomp__fun__idem,type,
% 0.07/0.30 finite_comp_fun_idem:
% 0.07/0.30 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Finite__Set_Ofinite,type,
% 0.07/0.30 finite_finite2:
% 0.07/0.30 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Groups_Oabel__semigroup,type,
% 0.07/0.30 abel_semigroup:
% 0.07/0.30 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Groups_Osemigroup,type,
% 0.07/0.30 semigroup:
% 0.07/0.30 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Lattices_Osemilattice,type,
% 0.07/0.30 semilattice:
% 0.07/0.30 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Lattices__Big_Osemilattice__inf_OInf__fin,type,
% 0.07/0.30 lattic1263571978nf_fin:
% 0.07/0.30 !>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Lattices__Big_Osemilattice__set,type,
% 0.07/0.30 lattic35693393ce_set:
% 0.07/0.30 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Lattices__Big_Osemilattice__sup_OSup__fin,type,
% 0.07/0.30 lattic1039401930up_fin:
% 0.07/0.30 !>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Od__aux,type,
% 0.07/0.30 modula1221160330_d_aux:
% 0.07/0.30 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oe__aux,type,
% 0.07/0.30 modula1593167561_e_aux:
% 0.07/0.30 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.07/0.30 bot_bot:
% 0.07/0.30 !>[A: $tType] : A ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Orderings_Oord_OLeast,type,
% 0.07/0.30 least:
% 0.07/0.30 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Orderings_Oord_Omax,type,
% 0.07/0.30 max:
% 0.07/0.30 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Orderings_Oord_Omin,type,
% 0.07/0.30 min:
% 0.07/0.30 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.07/0.30 ord_less_eq:
% 0.07/0.30 !>[A: $tType] : ( A > A > $o ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Orderings_Oorder_OGreatest,type,
% 0.07/0.30 greatest:
% 0.07/0.30 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Orderings_Oorder_Oantimono,type,
% 0.07/0.30 antimono:
% 0.07/0.30 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Orderings_Oorder_Omono,type,
% 0.07/0.30 mono:
% 0.07/0.30 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Relation_Otransp,type,
% 0.07/0.30 transp:
% 0.07/0.30 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Set_OCollect,type,
% 0.07/0.30 collect:
% 0.07/0.30 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Set_Oinsert,type,
% 0.07/0.30 insert:
% 0.07/0.30 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Set__Interval_Oord_OatLeast,type,
% 0.07/0.30 set_atLeast:
% 0.07/0.30 !>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Set__Interval_Oord_OatLeastAtMost,type,
% 0.07/0.30 set_atLeastAtMost:
% 0.07/0.30 !>[A: $tType] : ( ( A > A > $o ) > A > A > ( set @ A ) ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_Set__Interval_Oord_OatMost,type,
% 0.07/0.30 set_atMost:
% 0.07/0.30 !>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_c_member,type,
% 0.07/0.30 member:
% 0.07/0.30 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.07/0.30
% 0.07/0.30 thf(sy_v_a,type,
% 0.07/0.30 a2: a ).
% 0.07/0.30
% 0.07/0.30 thf(sy_v_b,type,
% 0.07/0.30 b: a ).
% 0.07/0.30
% 0.07/0.30 thf(sy_v_c,type,
% 0.07/0.30 c: a ).
% 0.07/0.30
% 0.07/0.30 thf(sy_v_inf,type,
% 0.07/0.30 inf: a > a > a ).
% 0.07/0.30
% 0.07/0.30 thf(sy_v_less__eq,type,
% 0.07/0.30 less_eq: a > a > $o ).
% 0.07/0.30
% 0.07/0.30 thf(sy_v_sup,type,
% 0.07/0.30 sup: a > a > a ).
% 0.07/0.30
% 0.07/0.30 % Relevant facts (256)
% 0.07/0.30 thf(fact_0_local_Oinf_Oassoc,axiom,
% 0.07/0.30 ! [A2: a,B2: a,C: a] :
% 0.07/0.30 ( ( inf @ ( inf @ A2 @ B2 ) @ C )
% 0.07/0.30 = ( inf @ A2 @ ( inf @ B2 @ C ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf.assoc
% 0.07/0.30 thf(fact_1_local_Oinf_Ocommute,axiom,
% 0.07/0.30 ! [A2: a,B2: a] :
% 0.07/0.30 ( ( inf @ A2 @ B2 )
% 0.07/0.30 = ( inf @ B2 @ A2 ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf.commute
% 0.07/0.30 thf(fact_2_local_Oinf_Oleft__commute,axiom,
% 0.07/0.30 ! [B2: a,A2: a,C: a] :
% 0.07/0.30 ( ( inf @ B2 @ ( inf @ A2 @ C ) )
% 0.07/0.30 = ( inf @ A2 @ ( inf @ B2 @ C ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf.left_commute
% 0.07/0.30 thf(fact_3_local_Oinf__assoc,axiom,
% 0.07/0.30 ! [X: a,Y: a,Z: a] :
% 0.07/0.30 ( ( inf @ ( inf @ X @ Y ) @ Z )
% 0.07/0.30 = ( inf @ X @ ( inf @ Y @ Z ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf_assoc
% 0.07/0.30 thf(fact_4_local_Oinf__commute,axiom,
% 0.07/0.30 ! [X: a,Y: a] :
% 0.07/0.30 ( ( inf @ X @ Y )
% 0.07/0.30 = ( inf @ Y @ X ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf_commute
% 0.07/0.30 thf(fact_5_local_Oinf__left__commute,axiom,
% 0.07/0.30 ! [X: a,Y: a,Z: a] :
% 0.07/0.30 ( ( inf @ X @ ( inf @ Y @ Z ) )
% 0.07/0.30 = ( inf @ Y @ ( inf @ X @ Z ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf_left_commute
% 0.07/0.30 thf(fact_6_local_Osup_Oassoc,axiom,
% 0.07/0.30 ! [A2: a,B2: a,C: a] :
% 0.07/0.30 ( ( sup @ ( sup @ A2 @ B2 ) @ C )
% 0.07/0.30 = ( sup @ A2 @ ( sup @ B2 @ C ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.assoc
% 0.07/0.30 thf(fact_7_local_Osup_Ocommute,axiom,
% 0.07/0.30 ! [A2: a,B2: a] :
% 0.07/0.30 ( ( sup @ A2 @ B2 )
% 0.07/0.30 = ( sup @ B2 @ A2 ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.commute
% 0.07/0.30 thf(fact_8_local_Osup_Oleft__commute,axiom,
% 0.07/0.30 ! [B2: a,A2: a,C: a] :
% 0.07/0.30 ( ( sup @ B2 @ ( sup @ A2 @ C ) )
% 0.07/0.30 = ( sup @ A2 @ ( sup @ B2 @ C ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.left_commute
% 0.07/0.30 thf(fact_9_local_Osup__assoc,axiom,
% 0.07/0.30 ! [X: a,Y: a,Z: a] :
% 0.07/0.30 ( ( sup @ ( sup @ X @ Y ) @ Z )
% 0.07/0.30 = ( sup @ X @ ( sup @ Y @ Z ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup_assoc
% 0.07/0.30 thf(fact_10_local_Osup__commute,axiom,
% 0.07/0.30 ! [X: a,Y: a] :
% 0.07/0.30 ( ( sup @ X @ Y )
% 0.07/0.30 = ( sup @ Y @ X ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup_commute
% 0.07/0.30 thf(fact_11_local_Osup__left__commute,axiom,
% 0.07/0.30 ! [X: a,Y: a,Z: a] :
% 0.07/0.30 ( ( sup @ X @ ( sup @ Y @ Z ) )
% 0.07/0.30 = ( sup @ Y @ ( sup @ X @ Z ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup_left_commute
% 0.07/0.30 thf(fact_12_local_Odistrib__imp1,axiom,
% 0.07/0.30 ! [X: a,Y: a,Z: a] :
% 0.07/0.30 ( ! [X2: a,Y2: a,Z2: a] :
% 0.07/0.30 ( ( inf @ X2 @ ( sup @ Y2 @ Z2 ) )
% 0.07/0.30 = ( sup @ ( inf @ X2 @ Y2 ) @ ( inf @ X2 @ Z2 ) ) )
% 0.07/0.30 => ( ( sup @ X @ ( inf @ Y @ Z ) )
% 0.07/0.30 = ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z ) ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.distrib_imp1
% 0.07/0.30 thf(fact_13_local_Odistrib__imp2,axiom,
% 0.07/0.30 ! [X: a,Y: a,Z: a] :
% 0.07/0.30 ( ! [X2: a,Y2: a,Z2: a] :
% 0.07/0.30 ( ( sup @ X2 @ ( inf @ Y2 @ Z2 ) )
% 0.07/0.30 = ( inf @ ( sup @ X2 @ Y2 ) @ ( sup @ X2 @ Z2 ) ) )
% 0.07/0.30 => ( ( inf @ X @ ( sup @ Y @ Z ) )
% 0.07/0.30 = ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z ) ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.distrib_imp2
% 0.07/0.30 thf(fact_14_e__aux__def,axiom,
% 0.07/0.30 ! [A2: a,B2: a,C: a] :
% 0.07/0.30 ( ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 0.07/0.30 = ( inf @ ( inf @ ( sup @ A2 @ B2 ) @ ( sup @ B2 @ C ) ) @ ( sup @ C @ A2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % e_aux_def
% 0.07/0.30 thf(fact_15_local_Oinf_Oidem,axiom,
% 0.07/0.30 ! [A2: a] :
% 0.07/0.30 ( ( inf @ A2 @ A2 )
% 0.07/0.30 = A2 ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf.idem
% 0.07/0.30 thf(fact_16_local_Oinf_Oleft__idem,axiom,
% 0.07/0.30 ! [A2: a,B2: a] :
% 0.07/0.30 ( ( inf @ A2 @ ( inf @ A2 @ B2 ) )
% 0.07/0.30 = ( inf @ A2 @ B2 ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf.left_idem
% 0.07/0.30 thf(fact_17_local_Oinf_Oright__idem,axiom,
% 0.07/0.30 ! [A2: a,B2: a] :
% 0.07/0.30 ( ( inf @ ( inf @ A2 @ B2 ) @ B2 )
% 0.07/0.30 = ( inf @ A2 @ B2 ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf.right_idem
% 0.07/0.30 thf(fact_18_local_Oinf__idem,axiom,
% 0.07/0.30 ! [X: a] :
% 0.07/0.30 ( ( inf @ X @ X )
% 0.07/0.30 = X ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf_idem
% 0.07/0.30 thf(fact_19_local_Oinf__left__idem,axiom,
% 0.07/0.30 ! [X: a,Y: a] :
% 0.07/0.30 ( ( inf @ X @ ( inf @ X @ Y ) )
% 0.07/0.30 = ( inf @ X @ Y ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf_left_idem
% 0.07/0.30 thf(fact_20_local_Oinf__right__idem,axiom,
% 0.07/0.30 ! [X: a,Y: a] :
% 0.07/0.30 ( ( inf @ ( inf @ X @ Y ) @ Y )
% 0.07/0.30 = ( inf @ X @ Y ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf_right_idem
% 0.07/0.30 thf(fact_21_local_Osup_Oidem,axiom,
% 0.07/0.30 ! [A2: a] :
% 0.07/0.30 ( ( sup @ A2 @ A2 )
% 0.07/0.30 = A2 ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.idem
% 0.07/0.30 thf(fact_22_local_Osup_Oleft__idem,axiom,
% 0.07/0.30 ! [A2: a,B2: a] :
% 0.07/0.30 ( ( sup @ A2 @ ( sup @ A2 @ B2 ) )
% 0.07/0.30 = ( sup @ A2 @ B2 ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.left_idem
% 0.07/0.30 thf(fact_23_local_Osup_Oright__idem,axiom,
% 0.07/0.30 ! [A2: a,B2: a] :
% 0.07/0.30 ( ( sup @ ( sup @ A2 @ B2 ) @ B2 )
% 0.07/0.30 = ( sup @ A2 @ B2 ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.right_idem
% 0.07/0.30 thf(fact_24_local_Osup__idem,axiom,
% 0.07/0.30 ! [X: a] :
% 0.07/0.30 ( ( sup @ X @ X )
% 0.07/0.30 = X ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup_idem
% 0.07/0.30 thf(fact_25_local_Osup__left__idem,axiom,
% 0.07/0.30 ! [X: a,Y: a] :
% 0.07/0.30 ( ( sup @ X @ ( sup @ X @ Y ) )
% 0.07/0.30 = ( sup @ X @ Y ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup_left_idem
% 0.07/0.30 thf(fact_26_local_Oinf__sup__absorb,axiom,
% 0.07/0.30 ! [X: a,Y: a] :
% 0.07/0.30 ( ( inf @ X @ ( sup @ X @ Y ) )
% 0.07/0.30 = X ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf_sup_absorb
% 0.07/0.30 thf(fact_27_local_Osup__inf__absorb,axiom,
% 0.07/0.30 ! [X: a,Y: a] :
% 0.07/0.30 ( ( sup @ X @ ( inf @ X @ Y ) )
% 0.07/0.30 = X ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup_inf_absorb
% 0.07/0.30 thf(fact_28_d__aux__def,axiom,
% 0.07/0.30 ! [A2: a,B2: a,C: a] :
% 0.07/0.30 ( ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 0.07/0.30 = ( sup @ ( sup @ ( inf @ A2 @ B2 ) @ ( inf @ B2 @ C ) ) @ ( inf @ C @ A2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % d_aux_def
% 0.07/0.30 thf(fact_29_d__b__c__a,axiom,
% 0.07/0.30 ! [B2: a,C: a,A2: a] :
% 0.07/0.30 ( ( modula1221160330_d_aux @ a @ inf @ sup @ B2 @ C @ A2 )
% 0.07/0.30 = ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% 0.07/0.30
% 0.07/0.30 % d_b_c_a
% 0.07/0.30 thf(fact_30_d__c__a__b,axiom,
% 0.07/0.30 ! [C: a,A2: a,B2: a] :
% 0.07/0.30 ( ( modula1221160330_d_aux @ a @ inf @ sup @ C @ A2 @ B2 )
% 0.07/0.30 = ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% 0.07/0.30
% 0.07/0.30 % d_c_a_b
% 0.07/0.30 thf(fact_31_lattice_Oe__aux_Ocong,axiom,
% 0.07/0.30 ! [A: $tType] :
% 0.07/0.30 ( ( modula1593167561_e_aux @ A )
% 0.07/0.30 = ( modula1593167561_e_aux @ A ) ) ).
% 0.07/0.30
% 0.07/0.30 % lattice.e_aux.cong
% 0.07/0.30 thf(fact_32_local_Ocomp__fun__idem__sup,axiom,
% 0.07/0.30 finite_comp_fun_idem @ a @ a @ sup ).
% 0.07/0.30
% 0.07/0.30 % local.comp_fun_idem_sup
% 0.07/0.30 thf(fact_33_local_Ocomp__fun__idem__inf,axiom,
% 0.07/0.30 finite_comp_fun_idem @ a @ a @ inf ).
% 0.07/0.30
% 0.07/0.30 % local.comp_fun_idem_inf
% 0.07/0.30 thf(fact_34_local_Osup_Osemigroup__axioms,axiom,
% 0.07/0.30 semigroup @ a @ sup ).
% 0.07/0.30
% 0.07/0.30 % local.sup.semigroup_axioms
% 0.07/0.30 thf(fact_35_local_Oinf_Osemigroup__axioms,axiom,
% 0.07/0.30 semigroup @ a @ inf ).
% 0.07/0.30
% 0.07/0.30 % local.inf.semigroup_axioms
% 0.07/0.30 thf(fact_36_local_Osup_Osemilattice__axioms,axiom,
% 0.07/0.30 semilattice @ a @ sup ).
% 0.07/0.30
% 0.07/0.30 % local.sup.semilattice_axioms
% 0.07/0.30 thf(fact_37_local_Oinf_Osemilattice__axioms,axiom,
% 0.07/0.30 semilattice @ a @ inf ).
% 0.07/0.30
% 0.07/0.30 % local.inf.semilattice_axioms
% 0.07/0.30 thf(fact_38_local_Osup_Oabel__semigroup__axioms,axiom,
% 0.07/0.30 abel_semigroup @ a @ sup ).
% 0.07/0.30
% 0.07/0.30 % local.sup.abel_semigroup_axioms
% 0.07/0.30 thf(fact_39_local_Oinf_Oabel__semigroup__axioms,axiom,
% 0.07/0.30 abel_semigroup @ a @ inf ).
% 0.07/0.30
% 0.07/0.30 % local.inf.abel_semigroup_axioms
% 0.07/0.30 thf(fact_40_local_OSup__fin_Osemilattice__set__axioms,axiom,
% 0.07/0.30 lattic35693393ce_set @ a @ sup ).
% 0.07/0.30
% 0.07/0.30 % local.Sup_fin.semilattice_set_axioms
% 0.07/0.30 thf(fact_41_local_OInf__fin_Osemilattice__set__axioms,axiom,
% 0.07/0.30 lattic35693393ce_set @ a @ inf ).
% 0.07/0.30
% 0.07/0.30 % local.Inf_fin.semilattice_set_axioms
% 0.07/0.30 thf(fact_42_local_Odistrib__inf__le,axiom,
% 0.07/0.30 ! [X: a,Y: a,Z: a] : ( less_eq @ ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z ) ) @ ( inf @ X @ ( sup @ Y @ Z ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.distrib_inf_le
% 0.07/0.30 thf(fact_43_local_Odistrib__sup__le,axiom,
% 0.07/0.30 ! [X: a,Y: a,Z: a] : ( less_eq @ ( sup @ X @ ( inf @ Y @ Z ) ) @ ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.distrib_sup_le
% 0.07/0.30 thf(fact_44_local_Oantisym,axiom,
% 0.07/0.30 ! [X: a,Y: a] :
% 0.07/0.30 ( ( less_eq @ X @ Y )
% 0.07/0.30 => ( ( less_eq @ Y @ X )
% 0.07/0.30 => ( X = Y ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.antisym
% 0.07/0.30 thf(fact_45_mem__Collect__eq,axiom,
% 0.07/0.30 ! [A: $tType,A2: A,P: A > $o] :
% 0.07/0.30 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.07/0.30 = ( P @ A2 ) ) ).
% 0.07/0.30
% 0.07/0.30 % mem_Collect_eq
% 0.07/0.30 thf(fact_46_Collect__mem__eq,axiom,
% 0.07/0.30 ! [A: $tType,A3: set @ A] :
% 0.07/0.30 ( ( collect @ A
% 0.07/0.30 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
% 0.07/0.30 = A3 ) ).
% 0.07/0.30
% 0.07/0.30 % Collect_mem_eq
% 0.07/0.30 thf(fact_47_Collect__cong,axiom,
% 0.07/0.30 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.07/0.30 ( ! [X2: A] :
% 0.07/0.30 ( ( P @ X2 )
% 0.07/0.30 = ( Q @ X2 ) )
% 0.07/0.30 => ( ( collect @ A @ P )
% 0.07/0.30 = ( collect @ A @ Q ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % Collect_cong
% 0.07/0.30 thf(fact_48_ext,axiom,
% 0.07/0.30 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.07/0.30 ( ! [X2: A] :
% 0.07/0.30 ( ( F @ X2 )
% 0.07/0.30 = ( G @ X2 ) )
% 0.07/0.30 => ( F = G ) ) ).
% 0.07/0.30
% 0.07/0.30 % ext
% 0.07/0.30 thf(fact_49_local_Oantisym__conv,axiom,
% 0.07/0.30 ! [Y: a,X: a] :
% 0.07/0.30 ( ( less_eq @ Y @ X )
% 0.07/0.30 => ( ( less_eq @ X @ Y )
% 0.07/0.30 = ( X = Y ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.antisym_conv
% 0.07/0.30 thf(fact_50_local_Odual__order_Oantisym,axiom,
% 0.07/0.30 ! [B2: a,A2: a] :
% 0.07/0.30 ( ( less_eq @ B2 @ A2 )
% 0.07/0.30 => ( ( less_eq @ A2 @ B2 )
% 0.07/0.30 => ( A2 = B2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.dual_order.antisym
% 0.07/0.30 thf(fact_51_local_Odual__order_Oeq__iff,axiom,
% 0.07/0.30 ( ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 ) )
% 0.07/0.30 = ( ^ [A4: a,B3: a] :
% 0.07/0.30 ( ( less_eq @ B3 @ A4 )
% 0.07/0.30 & ( less_eq @ A4 @ B3 ) ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.dual_order.eq_iff
% 0.07/0.30 thf(fact_52_local_Odual__order_Otrans,axiom,
% 0.07/0.30 ! [B2: a,A2: a,C: a] :
% 0.07/0.30 ( ( less_eq @ B2 @ A2 )
% 0.07/0.30 => ( ( less_eq @ C @ B2 )
% 0.07/0.30 => ( less_eq @ C @ A2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.dual_order.trans
% 0.07/0.30 thf(fact_53_local_Oeq__iff,axiom,
% 0.07/0.30 ( ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 ) )
% 0.07/0.30 = ( ^ [X3: a,Y4: a] :
% 0.07/0.30 ( ( less_eq @ X3 @ Y4 )
% 0.07/0.30 & ( less_eq @ Y4 @ X3 ) ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.eq_iff
% 0.07/0.30 thf(fact_54_local_Oeq__refl,axiom,
% 0.07/0.30 ! [X: a,Y: a] :
% 0.07/0.30 ( ( X = Y )
% 0.07/0.30 => ( less_eq @ X @ Y ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.eq_refl
% 0.07/0.30 thf(fact_55_local_Oord__eq__le__trans,axiom,
% 0.07/0.30 ! [A2: a,B2: a,C: a] :
% 0.07/0.30 ( ( A2 = B2 )
% 0.07/0.30 => ( ( less_eq @ B2 @ C )
% 0.07/0.30 => ( less_eq @ A2 @ C ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.ord_eq_le_trans
% 0.07/0.30 thf(fact_56_local_Oord__le__eq__trans,axiom,
% 0.07/0.30 ! [A2: a,B2: a,C: a] :
% 0.07/0.30 ( ( less_eq @ A2 @ B2 )
% 0.07/0.30 => ( ( B2 = C )
% 0.07/0.30 => ( less_eq @ A2 @ C ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.ord_le_eq_trans
% 0.07/0.30 thf(fact_57_local_Oorder_Oantisym,axiom,
% 0.07/0.30 ! [A2: a,B2: a] :
% 0.07/0.30 ( ( less_eq @ A2 @ B2 )
% 0.07/0.30 => ( ( less_eq @ B2 @ A2 )
% 0.07/0.30 => ( A2 = B2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.order.antisym
% 0.07/0.30 thf(fact_58_local_Oorder_Oeq__iff,axiom,
% 0.07/0.30 ( ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 ) )
% 0.07/0.30 = ( ^ [A4: a,B3: a] :
% 0.07/0.30 ( ( less_eq @ A4 @ B3 )
% 0.07/0.30 & ( less_eq @ B3 @ A4 ) ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.order.eq_iff
% 0.07/0.30 thf(fact_59_local_Oorder_Otrans,axiom,
% 0.07/0.30 ! [A2: a,B2: a,C: a] :
% 0.07/0.30 ( ( less_eq @ A2 @ B2 )
% 0.07/0.30 => ( ( less_eq @ B2 @ C )
% 0.07/0.30 => ( less_eq @ A2 @ C ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.order.trans
% 0.07/0.30 thf(fact_60_local_Oorder__trans,axiom,
% 0.07/0.30 ! [X: a,Y: a,Z: a] :
% 0.07/0.30 ( ( less_eq @ X @ Y )
% 0.07/0.30 => ( ( less_eq @ Y @ Z )
% 0.07/0.30 => ( less_eq @ X @ Z ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.order_trans
% 0.07/0.30 thf(fact_61_local_Ole__infI2,axiom,
% 0.07/0.30 ! [B2: a,X: a,A2: a] :
% 0.07/0.30 ( ( less_eq @ B2 @ X )
% 0.07/0.30 => ( less_eq @ ( inf @ A2 @ B2 ) @ X ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.le_infI2
% 0.07/0.30 thf(fact_62_local_Ole__infI1,axiom,
% 0.07/0.30 ! [A2: a,X: a,B2: a] :
% 0.07/0.30 ( ( less_eq @ A2 @ X )
% 0.07/0.30 => ( less_eq @ ( inf @ A2 @ B2 ) @ X ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.le_infI1
% 0.07/0.30 thf(fact_63_local_Ole__infI,axiom,
% 0.07/0.30 ! [X: a,A2: a,B2: a] :
% 0.07/0.30 ( ( less_eq @ X @ A2 )
% 0.07/0.30 => ( ( less_eq @ X @ B2 )
% 0.07/0.30 => ( less_eq @ X @ ( inf @ A2 @ B2 ) ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.le_infI
% 0.07/0.30 thf(fact_64_local_Ole__infE,axiom,
% 0.07/0.30 ! [X: a,A2: a,B2: a] :
% 0.07/0.30 ( ( less_eq @ X @ ( inf @ A2 @ B2 ) )
% 0.07/0.30 => ~ ( ( less_eq @ X @ A2 )
% 0.07/0.30 => ~ ( less_eq @ X @ B2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.le_infE
% 0.07/0.30 thf(fact_65_local_Ole__iff__inf,axiom,
% 0.07/0.30 ! [X: a,Y: a] :
% 0.07/0.30 ( ( less_eq @ X @ Y )
% 0.07/0.30 = ( ( inf @ X @ Y )
% 0.07/0.30 = X ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.le_iff_inf
% 0.07/0.30 thf(fact_66_local_Oinf__unique,axiom,
% 0.07/0.30 ! [F: a > a > a,X: a,Y: a] :
% 0.07/0.30 ( ! [X2: a,Y2: a] : ( less_eq @ ( F @ X2 @ Y2 ) @ X2 )
% 0.07/0.30 => ( ! [X2: a,Y2: a] : ( less_eq @ ( F @ X2 @ Y2 ) @ Y2 )
% 0.07/0.30 => ( ! [X2: a,Y2: a,Z2: a] :
% 0.07/0.30 ( ( less_eq @ X2 @ Y2 )
% 0.07/0.30 => ( ( less_eq @ X2 @ Z2 )
% 0.07/0.30 => ( less_eq @ X2 @ ( F @ Y2 @ Z2 ) ) ) )
% 0.07/0.30 => ( ( inf @ X @ Y )
% 0.07/0.30 = ( F @ X @ Y ) ) ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf_unique
% 0.07/0.30 thf(fact_67_local_Oinf__mono,axiom,
% 0.07/0.30 ! [A2: a,C: a,B2: a,D: a] :
% 0.07/0.30 ( ( less_eq @ A2 @ C )
% 0.07/0.30 => ( ( less_eq @ B2 @ D )
% 0.07/0.30 => ( less_eq @ ( inf @ A2 @ B2 ) @ ( inf @ C @ D ) ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf_mono
% 0.07/0.30 thf(fact_68_local_Oinf__le2,axiom,
% 0.07/0.30 ! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ Y ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf_le2
% 0.07/0.30 thf(fact_69_local_Oinf__le1,axiom,
% 0.07/0.30 ! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ X ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf_le1
% 0.07/0.30 thf(fact_70_local_Oinf__greatest,axiom,
% 0.07/0.30 ! [X: a,Y: a,Z: a] :
% 0.07/0.30 ( ( less_eq @ X @ Y )
% 0.07/0.30 => ( ( less_eq @ X @ Z )
% 0.07/0.30 => ( less_eq @ X @ ( inf @ Y @ Z ) ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf_greatest
% 0.07/0.30 thf(fact_71_local_Oinf__absorb2,axiom,
% 0.07/0.30 ! [Y: a,X: a] :
% 0.07/0.30 ( ( less_eq @ Y @ X )
% 0.07/0.30 => ( ( inf @ X @ Y )
% 0.07/0.30 = Y ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf_absorb2
% 0.07/0.30 thf(fact_72_local_Oinf__absorb1,axiom,
% 0.07/0.30 ! [X: a,Y: a] :
% 0.07/0.30 ( ( less_eq @ X @ Y )
% 0.07/0.30 => ( ( inf @ X @ Y )
% 0.07/0.30 = X ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf_absorb1
% 0.07/0.30 thf(fact_73_local_Oinf_Oorder__iff,axiom,
% 0.07/0.30 ! [A2: a,B2: a] :
% 0.07/0.30 ( ( less_eq @ A2 @ B2 )
% 0.07/0.30 = ( A2
% 0.07/0.30 = ( inf @ A2 @ B2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf.order_iff
% 0.07/0.30 thf(fact_74_local_Oinf_OorderI,axiom,
% 0.07/0.30 ! [A2: a,B2: a] :
% 0.07/0.30 ( ( A2
% 0.07/0.30 = ( inf @ A2 @ B2 ) )
% 0.07/0.30 => ( less_eq @ A2 @ B2 ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf.orderI
% 0.07/0.30 thf(fact_75_local_Oinf_OorderE,axiom,
% 0.07/0.30 ! [A2: a,B2: a] :
% 0.07/0.30 ( ( less_eq @ A2 @ B2 )
% 0.07/0.30 => ( A2
% 0.07/0.30 = ( inf @ A2 @ B2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf.orderE
% 0.07/0.30 thf(fact_76_local_Oinf_OcoboundedI2,axiom,
% 0.07/0.30 ! [B2: a,C: a,A2: a] :
% 0.07/0.30 ( ( less_eq @ B2 @ C )
% 0.07/0.30 => ( less_eq @ ( inf @ A2 @ B2 ) @ C ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf.coboundedI2
% 0.07/0.30 thf(fact_77_local_Oinf_OcoboundedI1,axiom,
% 0.07/0.30 ! [A2: a,C: a,B2: a] :
% 0.07/0.30 ( ( less_eq @ A2 @ C )
% 0.07/0.30 => ( less_eq @ ( inf @ A2 @ B2 ) @ C ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf.coboundedI1
% 0.07/0.30 thf(fact_78_local_Oinf_Ocobounded2,axiom,
% 0.07/0.30 ! [A2: a,B2: a] : ( less_eq @ ( inf @ A2 @ B2 ) @ B2 ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf.cobounded2
% 0.07/0.30 thf(fact_79_local_Oinf_Ocobounded1,axiom,
% 0.07/0.30 ! [A2: a,B2: a] : ( less_eq @ ( inf @ A2 @ B2 ) @ A2 ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf.cobounded1
% 0.07/0.30 thf(fact_80_local_Oinf_OboundedI,axiom,
% 0.07/0.30 ! [A2: a,B2: a,C: a] :
% 0.07/0.30 ( ( less_eq @ A2 @ B2 )
% 0.07/0.30 => ( ( less_eq @ A2 @ C )
% 0.07/0.30 => ( less_eq @ A2 @ ( inf @ B2 @ C ) ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf.boundedI
% 0.07/0.30 thf(fact_81_local_Oinf_OboundedE,axiom,
% 0.07/0.30 ! [A2: a,B2: a,C: a] :
% 0.07/0.30 ( ( less_eq @ A2 @ ( inf @ B2 @ C ) )
% 0.07/0.30 => ~ ( ( less_eq @ A2 @ B2 )
% 0.07/0.30 => ~ ( less_eq @ A2 @ C ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf.boundedE
% 0.07/0.30 thf(fact_82_local_Oinf_Oabsorb__iff2,axiom,
% 0.07/0.30 ! [B2: a,A2: a] :
% 0.07/0.30 ( ( less_eq @ B2 @ A2 )
% 0.07/0.30 = ( ( inf @ A2 @ B2 )
% 0.07/0.30 = B2 ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf.absorb_iff2
% 0.07/0.30 thf(fact_83_local_Oinf_Oabsorb__iff1,axiom,
% 0.07/0.30 ! [A2: a,B2: a] :
% 0.07/0.30 ( ( less_eq @ A2 @ B2 )
% 0.07/0.30 = ( ( inf @ A2 @ B2 )
% 0.07/0.30 = A2 ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf.absorb_iff1
% 0.07/0.30 thf(fact_84_local_Oinf_Oabsorb2,axiom,
% 0.07/0.30 ! [B2: a,A2: a] :
% 0.07/0.30 ( ( less_eq @ B2 @ A2 )
% 0.07/0.30 => ( ( inf @ A2 @ B2 )
% 0.07/0.30 = B2 ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf.absorb2
% 0.07/0.30 thf(fact_85_local_Oinf_Oabsorb1,axiom,
% 0.07/0.30 ! [A2: a,B2: a] :
% 0.07/0.30 ( ( less_eq @ A2 @ B2 )
% 0.07/0.30 => ( ( inf @ A2 @ B2 )
% 0.07/0.30 = A2 ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf.absorb1
% 0.07/0.30 thf(fact_86_local_Osup__unique,axiom,
% 0.07/0.30 ! [F: a > a > a,X: a,Y: a] :
% 0.07/0.30 ( ! [X2: a,Y2: a] : ( less_eq @ X2 @ ( F @ X2 @ Y2 ) )
% 0.07/0.30 => ( ! [X2: a,Y2: a] : ( less_eq @ Y2 @ ( F @ X2 @ Y2 ) )
% 0.07/0.30 => ( ! [X2: a,Y2: a,Z2: a] :
% 0.07/0.30 ( ( less_eq @ Y2 @ X2 )
% 0.07/0.30 => ( ( less_eq @ Z2 @ X2 )
% 0.07/0.30 => ( less_eq @ ( F @ Y2 @ Z2 ) @ X2 ) ) )
% 0.07/0.30 => ( ( sup @ X @ Y )
% 0.07/0.30 = ( F @ X @ Y ) ) ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup_unique
% 0.07/0.30 thf(fact_87_local_Osup__mono,axiom,
% 0.07/0.30 ! [A2: a,C: a,B2: a,D: a] :
% 0.07/0.30 ( ( less_eq @ A2 @ C )
% 0.07/0.30 => ( ( less_eq @ B2 @ D )
% 0.07/0.30 => ( less_eq @ ( sup @ A2 @ B2 ) @ ( sup @ C @ D ) ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup_mono
% 0.07/0.30 thf(fact_88_local_Osup__least,axiom,
% 0.07/0.30 ! [Y: a,X: a,Z: a] :
% 0.07/0.30 ( ( less_eq @ Y @ X )
% 0.07/0.30 => ( ( less_eq @ Z @ X )
% 0.07/0.30 => ( less_eq @ ( sup @ Y @ Z ) @ X ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup_least
% 0.07/0.30 thf(fact_89_local_Osup__ge2,axiom,
% 0.07/0.30 ! [Y: a,X: a] : ( less_eq @ Y @ ( sup @ X @ Y ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup_ge2
% 0.07/0.30 thf(fact_90_local_Osup__ge1,axiom,
% 0.07/0.30 ! [X: a,Y: a] : ( less_eq @ X @ ( sup @ X @ Y ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup_ge1
% 0.07/0.30 thf(fact_91_local_Osup__absorb2,axiom,
% 0.07/0.30 ! [X: a,Y: a] :
% 0.07/0.30 ( ( less_eq @ X @ Y )
% 0.07/0.30 => ( ( sup @ X @ Y )
% 0.07/0.30 = Y ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup_absorb2
% 0.07/0.30 thf(fact_92_local_Osup__absorb1,axiom,
% 0.07/0.30 ! [Y: a,X: a] :
% 0.07/0.30 ( ( less_eq @ Y @ X )
% 0.07/0.30 => ( ( sup @ X @ Y )
% 0.07/0.30 = X ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup_absorb1
% 0.07/0.30 thf(fact_93_local_Osup_Oorder__iff,axiom,
% 0.07/0.30 ! [B2: a,A2: a] :
% 0.07/0.30 ( ( less_eq @ B2 @ A2 )
% 0.07/0.30 = ( A2
% 0.07/0.30 = ( sup @ A2 @ B2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.order_iff
% 0.07/0.30 thf(fact_94_local_Osup_OorderI,axiom,
% 0.07/0.30 ! [A2: a,B2: a] :
% 0.07/0.30 ( ( A2
% 0.07/0.30 = ( sup @ A2 @ B2 ) )
% 0.07/0.30 => ( less_eq @ B2 @ A2 ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.orderI
% 0.07/0.30 thf(fact_95_local_Osup_OorderE,axiom,
% 0.07/0.30 ! [B2: a,A2: a] :
% 0.07/0.30 ( ( less_eq @ B2 @ A2 )
% 0.07/0.30 => ( A2
% 0.07/0.30 = ( sup @ A2 @ B2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.orderE
% 0.07/0.30 thf(fact_96_local_Osup_Omono,axiom,
% 0.07/0.30 ! [C: a,A2: a,D: a,B2: a] :
% 0.07/0.30 ( ( less_eq @ C @ A2 )
% 0.07/0.30 => ( ( less_eq @ D @ B2 )
% 0.07/0.30 => ( less_eq @ ( sup @ C @ D ) @ ( sup @ A2 @ B2 ) ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.mono
% 0.07/0.30 thf(fact_97_local_Osup_OcoboundedI2,axiom,
% 0.07/0.30 ! [C: a,B2: a,A2: a] :
% 0.07/0.30 ( ( less_eq @ C @ B2 )
% 0.07/0.30 => ( less_eq @ C @ ( sup @ A2 @ B2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.coboundedI2
% 0.07/0.30 thf(fact_98_local_Osup_OcoboundedI1,axiom,
% 0.07/0.30 ! [C: a,A2: a,B2: a] :
% 0.07/0.30 ( ( less_eq @ C @ A2 )
% 0.07/0.30 => ( less_eq @ C @ ( sup @ A2 @ B2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.coboundedI1
% 0.07/0.30 thf(fact_99_local_Osup_Ocobounded2,axiom,
% 0.07/0.30 ! [B2: a,A2: a] : ( less_eq @ B2 @ ( sup @ A2 @ B2 ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.cobounded2
% 0.07/0.30 thf(fact_100_local_Osup_Ocobounded1,axiom,
% 0.07/0.30 ! [A2: a,B2: a] : ( less_eq @ A2 @ ( sup @ A2 @ B2 ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.cobounded1
% 0.07/0.30 thf(fact_101_local_Osup_OboundedI,axiom,
% 0.07/0.30 ! [B2: a,A2: a,C: a] :
% 0.07/0.30 ( ( less_eq @ B2 @ A2 )
% 0.07/0.30 => ( ( less_eq @ C @ A2 )
% 0.07/0.30 => ( less_eq @ ( sup @ B2 @ C ) @ A2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.boundedI
% 0.07/0.30 thf(fact_102_local_Osup_OboundedE,axiom,
% 0.07/0.30 ! [B2: a,C: a,A2: a] :
% 0.07/0.30 ( ( less_eq @ ( sup @ B2 @ C ) @ A2 )
% 0.07/0.30 => ~ ( ( less_eq @ B2 @ A2 )
% 0.07/0.30 => ~ ( less_eq @ C @ A2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.boundedE
% 0.07/0.30 thf(fact_103_local_Osup_Oabsorb__iff2,axiom,
% 0.07/0.30 ! [A2: a,B2: a] :
% 0.07/0.30 ( ( less_eq @ A2 @ B2 )
% 0.07/0.30 = ( ( sup @ A2 @ B2 )
% 0.07/0.30 = B2 ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.absorb_iff2
% 0.07/0.30 thf(fact_104_local_Osup_Oabsorb__iff1,axiom,
% 0.07/0.30 ! [B2: a,A2: a] :
% 0.07/0.30 ( ( less_eq @ B2 @ A2 )
% 0.07/0.30 = ( ( sup @ A2 @ B2 )
% 0.07/0.30 = A2 ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.absorb_iff1
% 0.07/0.30 thf(fact_105_local_Osup_Oabsorb2,axiom,
% 0.07/0.30 ! [A2: a,B2: a] :
% 0.07/0.30 ( ( less_eq @ A2 @ B2 )
% 0.07/0.30 => ( ( sup @ A2 @ B2 )
% 0.07/0.30 = B2 ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.absorb2
% 0.07/0.30 thf(fact_106_local_Osup_Oabsorb1,axiom,
% 0.07/0.30 ! [B2: a,A2: a] :
% 0.07/0.30 ( ( less_eq @ B2 @ A2 )
% 0.07/0.30 => ( ( sup @ A2 @ B2 )
% 0.07/0.30 = A2 ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.absorb1
% 0.07/0.30 thf(fact_107_local_Ole__supI2,axiom,
% 0.07/0.30 ! [X: a,B2: a,A2: a] :
% 0.07/0.30 ( ( less_eq @ X @ B2 )
% 0.07/0.30 => ( less_eq @ X @ ( sup @ A2 @ B2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.le_supI2
% 0.07/0.30 thf(fact_108_local_Ole__supI1,axiom,
% 0.07/0.30 ! [X: a,A2: a,B2: a] :
% 0.07/0.30 ( ( less_eq @ X @ A2 )
% 0.07/0.30 => ( less_eq @ X @ ( sup @ A2 @ B2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.le_supI1
% 0.07/0.30 thf(fact_109_local_Ole__supI,axiom,
% 0.07/0.30 ! [A2: a,X: a,B2: a] :
% 0.07/0.30 ( ( less_eq @ A2 @ X )
% 0.07/0.30 => ( ( less_eq @ B2 @ X )
% 0.07/0.30 => ( less_eq @ ( sup @ A2 @ B2 ) @ X ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.le_supI
% 0.07/0.30 thf(fact_110_local_Ole__supE,axiom,
% 0.07/0.30 ! [A2: a,B2: a,X: a] :
% 0.07/0.30 ( ( less_eq @ ( sup @ A2 @ B2 ) @ X )
% 0.07/0.30 => ~ ( ( less_eq @ A2 @ X )
% 0.07/0.30 => ~ ( less_eq @ B2 @ X ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.le_supE
% 0.07/0.30 thf(fact_111_local_Ole__iff__sup,axiom,
% 0.07/0.30 ! [X: a,Y: a] :
% 0.07/0.30 ( ( less_eq @ X @ Y )
% 0.07/0.30 = ( ( sup @ X @ Y )
% 0.07/0.30 = Y ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.le_iff_sup
% 0.07/0.30 thf(fact_112_local_Oorder_Orefl,axiom,
% 0.07/0.30 ! [A2: a] : ( less_eq @ A2 @ A2 ) ).
% 0.07/0.30
% 0.07/0.30 % local.order.refl
% 0.07/0.30 thf(fact_113_local_Oorder__refl,axiom,
% 0.07/0.30 ! [X: a] : ( less_eq @ X @ X ) ).
% 0.07/0.30
% 0.07/0.30 % local.order_refl
% 0.07/0.30 thf(fact_114_local_OGreatestI2__order,axiom,
% 0.07/0.30 ! [P: a > $o,X: a,Q: a > $o] :
% 0.07/0.30 ( ( P @ X )
% 0.07/0.30 => ( ! [Y2: a] :
% 0.07/0.30 ( ( P @ Y2 )
% 0.07/0.30 => ( less_eq @ Y2 @ X ) )
% 0.07/0.30 => ( ! [X2: a] :
% 0.07/0.30 ( ( P @ X2 )
% 0.07/0.30 => ( ! [Y5: a] :
% 0.07/0.30 ( ( P @ Y5 )
% 0.07/0.30 => ( less_eq @ Y5 @ X2 ) )
% 0.07/0.30 => ( Q @ X2 ) ) )
% 0.07/0.30 => ( Q @ ( greatest @ a @ less_eq @ P ) ) ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.GreatestI2_order
% 0.07/0.30 thf(fact_115_local_OGreatest__equality,axiom,
% 0.07/0.30 ! [P: a > $o,X: a] :
% 0.07/0.30 ( ( P @ X )
% 0.07/0.30 => ( ! [Y2: a] :
% 0.07/0.30 ( ( P @ Y2 )
% 0.07/0.30 => ( less_eq @ Y2 @ X ) )
% 0.07/0.30 => ( ( greatest @ a @ less_eq @ P )
% 0.07/0.30 = X ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.Greatest_equality
% 0.07/0.30 thf(fact_116_local_Omax__def,axiom,
% 0.07/0.30 ! [A2: a,B2: a] :
% 0.07/0.30 ( ( ( less_eq @ A2 @ B2 )
% 0.07/0.30 => ( ( max @ a @ less_eq @ A2 @ B2 )
% 0.07/0.30 = B2 ) )
% 0.07/0.30 & ( ~ ( less_eq @ A2 @ B2 )
% 0.07/0.30 => ( ( max @ a @ less_eq @ A2 @ B2 )
% 0.07/0.30 = A2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.max_def
% 0.07/0.30 thf(fact_117_local_Omin__def,axiom,
% 0.07/0.30 ! [A2: a,B2: a] :
% 0.07/0.30 ( ( ( less_eq @ A2 @ B2 )
% 0.07/0.30 => ( ( min @ a @ less_eq @ A2 @ B2 )
% 0.07/0.30 = A2 ) )
% 0.07/0.30 & ( ~ ( less_eq @ A2 @ B2 )
% 0.07/0.30 => ( ( min @ a @ less_eq @ A2 @ B2 )
% 0.07/0.30 = B2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.min_def
% 0.07/0.30 thf(fact_118_local_Ole__inf__iff,axiom,
% 0.07/0.30 ! [X: a,Y: a,Z: a] :
% 0.07/0.30 ( ( less_eq @ X @ ( inf @ Y @ Z ) )
% 0.07/0.30 = ( ( less_eq @ X @ Y )
% 0.07/0.30 & ( less_eq @ X @ Z ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.le_inf_iff
% 0.07/0.30 thf(fact_119_local_Oinf_Obounded__iff,axiom,
% 0.07/0.30 ! [A2: a,B2: a,C: a] :
% 0.07/0.30 ( ( less_eq @ A2 @ ( inf @ B2 @ C ) )
% 0.07/0.30 = ( ( less_eq @ A2 @ B2 )
% 0.07/0.30 & ( less_eq @ A2 @ C ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.inf.bounded_iff
% 0.07/0.30 thf(fact_120_local_Osup_Obounded__iff,axiom,
% 0.07/0.30 ! [B2: a,C: a,A2: a] :
% 0.07/0.30 ( ( less_eq @ ( sup @ B2 @ C ) @ A2 )
% 0.07/0.30 = ( ( less_eq @ B2 @ A2 )
% 0.07/0.30 & ( less_eq @ C @ A2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.sup.bounded_iff
% 0.07/0.30 thf(fact_121_local_Ole__sup__iff,axiom,
% 0.07/0.30 ! [X: a,Y: a,Z: a] :
% 0.07/0.30 ( ( less_eq @ ( sup @ X @ Y ) @ Z )
% 0.07/0.30 = ( ( less_eq @ X @ Z )
% 0.07/0.30 & ( less_eq @ Y @ Z ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.le_sup_iff
% 0.07/0.30 thf(fact_122_local_OLeast1I,axiom,
% 0.07/0.30 ! [P: a > $o] :
% 0.07/0.30 ( ? [X4: a] :
% 0.07/0.30 ( ( P @ X4 )
% 0.07/0.30 & ! [Y2: a] :
% 0.07/0.30 ( ( P @ Y2 )
% 0.07/0.30 => ( less_eq @ X4 @ Y2 ) )
% 0.07/0.30 & ! [Y2: a] :
% 0.07/0.30 ( ( ( P @ Y2 )
% 0.07/0.30 & ! [Ya: a] :
% 0.07/0.30 ( ( P @ Ya )
% 0.07/0.30 => ( less_eq @ Y2 @ Ya ) ) )
% 0.07/0.30 => ( Y2 = X4 ) ) )
% 0.07/0.30 => ( P @ ( least @ a @ less_eq @ P ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.Least1I
% 0.07/0.30 thf(fact_123_local_OLeast1__le,axiom,
% 0.07/0.30 ! [P: a > $o,Z: a] :
% 0.07/0.30 ( ? [X4: a] :
% 0.07/0.30 ( ( P @ X4 )
% 0.07/0.30 & ! [Y2: a] :
% 0.07/0.30 ( ( P @ Y2 )
% 0.07/0.30 => ( less_eq @ X4 @ Y2 ) )
% 0.07/0.30 & ! [Y2: a] :
% 0.07/0.30 ( ( ( P @ Y2 )
% 0.07/0.30 & ! [Ya: a] :
% 0.07/0.30 ( ( P @ Ya )
% 0.07/0.30 => ( less_eq @ Y2 @ Ya ) ) )
% 0.07/0.30 => ( Y2 = X4 ) ) )
% 0.07/0.30 => ( ( P @ Z )
% 0.07/0.30 => ( less_eq @ ( least @ a @ less_eq @ P ) @ Z ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.Least1_le
% 0.07/0.30 thf(fact_124_local_OLeastI2__order,axiom,
% 0.07/0.30 ! [P: a > $o,X: a,Q: a > $o] :
% 0.07/0.30 ( ( P @ X )
% 0.07/0.30 => ( ! [Y2: a] :
% 0.07/0.30 ( ( P @ Y2 )
% 0.07/0.30 => ( less_eq @ X @ Y2 ) )
% 0.07/0.30 => ( ! [X2: a] :
% 0.07/0.30 ( ( P @ X2 )
% 0.07/0.30 => ( ! [Y5: a] :
% 0.07/0.30 ( ( P @ Y5 )
% 0.07/0.30 => ( less_eq @ X2 @ Y5 ) )
% 0.07/0.30 => ( Q @ X2 ) ) )
% 0.07/0.30 => ( Q @ ( least @ a @ less_eq @ P ) ) ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.LeastI2_order
% 0.07/0.30 thf(fact_125_local_OLeast__equality,axiom,
% 0.07/0.30 ! [P: a > $o,X: a] :
% 0.07/0.30 ( ( P @ X )
% 0.07/0.30 => ( ! [Y2: a] :
% 0.07/0.30 ( ( P @ Y2 )
% 0.07/0.30 => ( less_eq @ X @ Y2 ) )
% 0.07/0.30 => ( ( least @ a @ less_eq @ P )
% 0.07/0.30 = X ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.Least_equality
% 0.07/0.30 thf(fact_126_lattice_Od__aux_Ocong,axiom,
% 0.07/0.30 ! [A: $tType] :
% 0.07/0.30 ( ( modula1221160330_d_aux @ A )
% 0.07/0.30 = ( modula1221160330_d_aux @ A ) ) ).
% 0.07/0.30
% 0.07/0.30 % lattice.d_aux.cong
% 0.07/0.30 thf(fact_127_abel__semigroup_Oaxioms_I1_J,axiom,
% 0.07/0.30 ! [A: $tType,F: A > A > A] :
% 0.07/0.30 ( ( abel_semigroup @ A @ F )
% 0.07/0.30 => ( semigroup @ A @ F ) ) ).
% 0.07/0.30
% 0.07/0.30 % abel_semigroup.axioms(1)
% 0.07/0.30 thf(fact_128_semilattice__set__def,axiom,
% 0.07/0.30 ! [A: $tType] :
% 0.07/0.30 ( ( lattic35693393ce_set @ A )
% 0.07/0.30 = ( semilattice @ A ) ) ).
% 0.07/0.30
% 0.07/0.30 % semilattice_set_def
% 0.07/0.30 thf(fact_129_semilattice__set_Ointro,axiom,
% 0.07/0.30 ! [A: $tType,F: A > A > A] :
% 0.07/0.30 ( ( semilattice @ A @ F )
% 0.07/0.30 => ( lattic35693393ce_set @ A @ F ) ) ).
% 0.07/0.30
% 0.07/0.30 % semilattice_set.intro
% 0.07/0.30 thf(fact_130_semilattice__set_Oaxioms,axiom,
% 0.07/0.30 ! [A: $tType,F: A > A > A] :
% 0.07/0.30 ( ( lattic35693393ce_set @ A @ F )
% 0.07/0.30 => ( semilattice @ A @ F ) ) ).
% 0.07/0.30
% 0.07/0.30 % semilattice_set.axioms
% 0.07/0.30 thf(fact_131_semilattice_Oaxioms_I1_J,axiom,
% 0.07/0.30 ! [A: $tType,F: A > A > A] :
% 0.07/0.30 ( ( semilattice @ A @ F )
% 0.07/0.30 => ( abel_semigroup @ A @ F ) ) ).
% 0.07/0.30
% 0.07/0.30 % semilattice.axioms(1)
% 0.07/0.30 thf(fact_132_local_Obdd__above__def,axiom,
% 0.07/0.30 ! [A3: set @ a] :
% 0.07/0.30 ( ( condit2040224947_above @ a @ less_eq @ A3 )
% 0.07/0.30 = ( ? [M: a] :
% 0.07/0.30 ! [X3: a] :
% 0.07/0.30 ( ( member @ a @ X3 @ A3 )
% 0.07/0.30 => ( less_eq @ X3 @ M ) ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.bdd_above_def
% 0.07/0.30 thf(fact_133_local_Obdd__below__def,axiom,
% 0.07/0.30 ! [A3: set @ a] :
% 0.07/0.30 ( ( condit1201339847_below @ a @ less_eq @ A3 )
% 0.07/0.30 = ( ? [M2: a] :
% 0.07/0.30 ! [X3: a] :
% 0.07/0.30 ( ( member @ a @ X3 @ A3 )
% 0.07/0.30 => ( less_eq @ M2 @ X3 ) ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.bdd_below_def
% 0.07/0.30 thf(fact_134_local_Otransp__le,axiom,
% 0.07/0.30 transp @ a @ less_eq ).
% 0.07/0.30
% 0.07/0.30 % local.transp_le
% 0.07/0.30 thf(fact_135_local_Obdd__belowI,axiom,
% 0.07/0.30 ! [A3: set @ a,M3: a] :
% 0.07/0.30 ( ! [X2: a] :
% 0.07/0.30 ( ( member @ a @ X2 @ A3 )
% 0.07/0.30 => ( less_eq @ M3 @ X2 ) )
% 0.07/0.30 => ( condit1201339847_below @ a @ less_eq @ A3 ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.bdd_belowI
% 0.07/0.30 thf(fact_136_local_Obdd__aboveI,axiom,
% 0.07/0.30 ! [A3: set @ a,M4: a] :
% 0.07/0.30 ( ! [X2: a] :
% 0.07/0.30 ( ( member @ a @ X2 @ A3 )
% 0.07/0.30 => ( less_eq @ X2 @ M4 ) )
% 0.07/0.30 => ( condit2040224947_above @ a @ less_eq @ A3 ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.bdd_aboveI
% 0.07/0.30 thf(fact_137_abel__semigroup_Oleft__commute,axiom,
% 0.07/0.30 ! [A: $tType,F: A > A > A,B2: A,A2: A,C: A] :
% 0.07/0.30 ( ( abel_semigroup @ A @ F )
% 0.07/0.30 => ( ( F @ B2 @ ( F @ A2 @ C ) )
% 0.07/0.30 = ( F @ A2 @ ( F @ B2 @ C ) ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % abel_semigroup.left_commute
% 0.07/0.30 thf(fact_138_semilattice_Oright__idem,axiom,
% 0.07/0.30 ! [A: $tType,F: A > A > A,A2: A,B2: A] :
% 0.07/0.30 ( ( semilattice @ A @ F )
% 0.07/0.30 => ( ( F @ ( F @ A2 @ B2 ) @ B2 )
% 0.07/0.30 = ( F @ A2 @ B2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % semilattice.right_idem
% 0.07/0.30 thf(fact_139_semilattice_Oleft__idem,axiom,
% 0.07/0.30 ! [A: $tType,F: A > A > A,A2: A,B2: A] :
% 0.07/0.30 ( ( semilattice @ A @ F )
% 0.07/0.30 => ( ( F @ A2 @ ( F @ A2 @ B2 ) )
% 0.07/0.30 = ( F @ A2 @ B2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % semilattice.left_idem
% 0.07/0.30 thf(fact_140_abel__semigroup_Ocommute,axiom,
% 0.07/0.30 ! [A: $tType,F: A > A > A,A2: A,B2: A] :
% 0.07/0.30 ( ( abel_semigroup @ A @ F )
% 0.07/0.30 => ( ( F @ A2 @ B2 )
% 0.07/0.30 = ( F @ B2 @ A2 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % abel_semigroup.commute
% 0.07/0.30 thf(fact_141_semilattice_Oidem,axiom,
% 0.07/0.30 ! [A: $tType,F: A > A > A,A2: A] :
% 0.07/0.30 ( ( semilattice @ A @ F )
% 0.07/0.30 => ( ( F @ A2 @ A2 )
% 0.07/0.30 = A2 ) ) ).
% 0.07/0.30
% 0.07/0.30 % semilattice.idem
% 0.07/0.30 thf(fact_142_semigroup_Ointro,axiom,
% 0.07/0.30 ! [A: $tType,F: A > A > A] :
% 0.07/0.30 ( ! [A5: A,B4: A,C2: A] :
% 0.07/0.30 ( ( F @ ( F @ A5 @ B4 ) @ C2 )
% 0.07/0.30 = ( F @ A5 @ ( F @ B4 @ C2 ) ) )
% 0.07/0.30 => ( semigroup @ A @ F ) ) ).
% 0.07/0.30
% 0.07/0.30 % semigroup.intro
% 0.07/0.30 thf(fact_143_semigroup_Oassoc,axiom,
% 0.07/0.30 ! [A: $tType,F: A > A > A,A2: A,B2: A,C: A] :
% 0.07/0.30 ( ( semigroup @ A @ F )
% 0.07/0.30 => ( ( F @ ( F @ A2 @ B2 ) @ C )
% 0.07/0.30 = ( F @ A2 @ ( F @ B2 @ C ) ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % semigroup.assoc
% 0.07/0.30 thf(fact_144_semigroup__def,axiom,
% 0.07/0.30 ! [A: $tType] :
% 0.07/0.30 ( ( semigroup @ A )
% 0.07/0.30 = ( ^ [F2: A > A > A] :
% 0.07/0.30 ! [A4: A,B3: A,C3: A] :
% 0.07/0.30 ( ( F2 @ ( F2 @ A4 @ B3 ) @ C3 )
% 0.07/0.30 = ( F2 @ A4 @ ( F2 @ B3 @ C3 ) ) ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % semigroup_def
% 0.07/0.30 thf(fact_145_local_Obdd__below__mono,axiom,
% 0.07/0.30 ! [B5: set @ a,A3: set @ a] :
% 0.07/0.30 ( ( condit1201339847_below @ a @ less_eq @ B5 )
% 0.07/0.30 => ( ( ord_less_eq @ ( set @ a ) @ A3 @ B5 )
% 0.07/0.30 => ( condit1201339847_below @ a @ less_eq @ A3 ) ) ) ).
% 0.07/0.30
% 0.07/0.30 % local.bdd_below_mono
% 0.07/0.30 thf(fact_146_local_Obdd__above__mono,axiom,
% 0.07/0.31 ! [B5: set @ a,A3: set @ a] :
% 0.07/0.31 ( ( condit2040224947_above @ a @ less_eq @ B5 )
% 0.07/0.31 => ( ( ord_less_eq @ ( set @ a ) @ A3 @ B5 )
% 0.07/0.31 => ( condit2040224947_above @ a @ less_eq @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.bdd_above_mono
% 0.07/0.31 thf(fact_147_local_Obdd__below__finite,axiom,
% 0.07/0.31 ! [A3: set @ a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( condit1201339847_below @ a @ less_eq @ A3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.bdd_below_finite
% 0.07/0.31 thf(fact_148_local_Obdd__above__finite,axiom,
% 0.07/0.31 ! [A3: set @ a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( condit2040224947_above @ a @ less_eq @ A3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.bdd_above_finite
% 0.07/0.31 thf(fact_149_local_Ofinite__has__maximal2,axiom,
% 0.07/0.31 ! [A3: set @ a,A2: a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( member @ a @ A2 @ A3 )
% 0.07/0.31 => ? [X2: a] :
% 0.07/0.31 ( ( member @ a @ X2 @ A3 )
% 0.07/0.31 & ( less_eq @ A2 @ X2 )
% 0.07/0.31 & ! [Xa: a] :
% 0.07/0.31 ( ( member @ a @ Xa @ A3 )
% 0.07/0.31 => ( ( less_eq @ X2 @ Xa )
% 0.07/0.31 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.finite_has_maximal2
% 0.07/0.31 thf(fact_150_local_Ofinite__has__minimal2,axiom,
% 0.07/0.31 ! [A3: set @ a,A2: a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( member @ a @ A2 @ A3 )
% 0.07/0.31 => ? [X2: a] :
% 0.07/0.31 ( ( member @ a @ X2 @ A3 )
% 0.07/0.31 & ( less_eq @ X2 @ A2 )
% 0.07/0.31 & ! [Xa: a] :
% 0.07/0.31 ( ( member @ a @ Xa @ A3 )
% 0.07/0.31 => ( ( less_eq @ Xa @ X2 )
% 0.07/0.31 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.finite_has_minimal2
% 0.07/0.31 thf(fact_151_local_OantimonoD,axiom,
% 0.07/0.31 ! [B: $tType] :
% 0.07/0.31 ( ( order @ B )
% 0.07/0.31 => ! [F: a > B,X: a,Y: a] :
% 0.07/0.31 ( ( antimono @ a @ B @ less_eq @ F )
% 0.07/0.31 => ( ( less_eq @ X @ Y )
% 0.07/0.31 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.antimonoD
% 0.07/0.31 thf(fact_152_local_OantimonoE,axiom,
% 0.07/0.31 ! [B: $tType] :
% 0.07/0.31 ( ( order @ B )
% 0.07/0.31 => ! [F: a > B,X: a,Y: a] :
% 0.07/0.31 ( ( antimono @ a @ B @ less_eq @ F )
% 0.07/0.31 => ( ( less_eq @ X @ Y )
% 0.07/0.31 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.antimonoE
% 0.07/0.31 thf(fact_153_local_OantimonoI,axiom,
% 0.07/0.31 ! [B: $tType] :
% 0.07/0.31 ( ( order @ B )
% 0.07/0.31 => ! [F: a > B] :
% 0.07/0.31 ( ! [X2: a,Y2: a] :
% 0.07/0.31 ( ( less_eq @ X2 @ Y2 )
% 0.07/0.31 => ( ord_less_eq @ B @ ( F @ Y2 ) @ ( F @ X2 ) ) )
% 0.07/0.31 => ( antimono @ a @ B @ less_eq @ F ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.antimonoI
% 0.07/0.31 thf(fact_154_local_Oantimono__def,axiom,
% 0.07/0.31 ! [B: $tType] :
% 0.07/0.31 ( ( order @ B )
% 0.07/0.31 => ! [F: a > B] :
% 0.07/0.31 ( ( antimono @ a @ B @ less_eq @ F )
% 0.07/0.31 = ( ! [X3: a,Y4: a] :
% 0.07/0.31 ( ( less_eq @ X3 @ Y4 )
% 0.07/0.31 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X3 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.antimono_def
% 0.07/0.31 thf(fact_155_local_OmonoD,axiom,
% 0.07/0.31 ! [B: $tType] :
% 0.07/0.31 ( ( order @ B )
% 0.07/0.31 => ! [F: a > B,X: a,Y: a] :
% 0.07/0.31 ( ( mono @ a @ B @ less_eq @ F )
% 0.07/0.31 => ( ( less_eq @ X @ Y )
% 0.07/0.31 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.monoD
% 0.07/0.31 thf(fact_156_local_OmonoE,axiom,
% 0.07/0.31 ! [B: $tType] :
% 0.07/0.31 ( ( order @ B )
% 0.07/0.31 => ! [F: a > B,X: a,Y: a] :
% 0.07/0.31 ( ( mono @ a @ B @ less_eq @ F )
% 0.07/0.31 => ( ( less_eq @ X @ Y )
% 0.07/0.31 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.monoE
% 0.07/0.31 thf(fact_157_local_OmonoI,axiom,
% 0.07/0.31 ! [B: $tType] :
% 0.07/0.31 ( ( order @ B )
% 0.07/0.31 => ! [F: a > B] :
% 0.07/0.31 ( ! [X2: a,Y2: a] :
% 0.07/0.31 ( ( less_eq @ X2 @ Y2 )
% 0.07/0.31 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.07/0.31 => ( mono @ a @ B @ less_eq @ F ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.monoI
% 0.07/0.31 thf(fact_158_local_Omono__def,axiom,
% 0.07/0.31 ! [B: $tType] :
% 0.07/0.31 ( ( order @ B )
% 0.07/0.31 => ! [F: a > B] :
% 0.07/0.31 ( ( mono @ a @ B @ less_eq @ F )
% 0.07/0.31 = ( ! [X3: a,Y4: a] :
% 0.07/0.31 ( ( less_eq @ X3 @ Y4 )
% 0.07/0.31 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.mono_def
% 0.07/0.31 thf(fact_159_local_OSup__fin_OcoboundedI,axiom,
% 0.07/0.31 ! [A3: set @ a,A2: a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( member @ a @ A2 @ A3 )
% 0.07/0.31 => ( less_eq @ A2 @ ( lattic1039401930up_fin @ a @ sup @ A3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Sup_fin.coboundedI
% 0.07/0.31 thf(fact_160_local_OInf__fin_OcoboundedI,axiom,
% 0.07/0.31 ! [A3: set @ a,A2: a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( member @ a @ A2 @ A3 )
% 0.07/0.31 => ( less_eq @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) @ A2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Inf_fin.coboundedI
% 0.07/0.31 thf(fact_161_local_OSup__fin_Oin__idem,axiom,
% 0.07/0.31 ! [A3: set @ a,X: a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( member @ a @ X @ A3 )
% 0.07/0.31 => ( ( sup @ X @ ( lattic1039401930up_fin @ a @ sup @ A3 ) )
% 0.07/0.31 = ( lattic1039401930up_fin @ a @ sup @ A3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Sup_fin.in_idem
% 0.07/0.31 thf(fact_162_local_OInf__fin_Oin__idem,axiom,
% 0.07/0.31 ! [A3: set @ a,X: a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( member @ a @ X @ A3 )
% 0.07/0.31 => ( ( inf @ X @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) )
% 0.07/0.31 = ( lattic1263571978nf_fin @ a @ inf @ A3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Inf_fin.in_idem
% 0.07/0.31 thf(fact_163_local_Oinf__Sup__absorb,axiom,
% 0.07/0.31 ! [A3: set @ a,A2: a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( member @ a @ A2 @ A3 )
% 0.07/0.31 => ( ( inf @ A2 @ ( lattic1039401930up_fin @ a @ sup @ A3 ) )
% 0.07/0.31 = A2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.inf_Sup_absorb
% 0.07/0.31 thf(fact_164_local_Osup__Inf__absorb,axiom,
% 0.07/0.31 ! [A3: set @ a,A2: a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( member @ a @ A2 @ A3 )
% 0.07/0.31 => ( ( sup @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) @ A2 )
% 0.07/0.31 = A2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.sup_Inf_absorb
% 0.07/0.31 thf(fact_165_semilattice__inf_OInf__fin_Ocong,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( lattic1263571978nf_fin @ A )
% 0.07/0.31 = ( lattic1263571978nf_fin @ A ) ) ).
% 0.07/0.31
% 0.07/0.31 % semilattice_inf.Inf_fin.cong
% 0.07/0.31 thf(fact_166_semilattice__sup_OSup__fin_Ocong,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( lattic1039401930up_fin @ A )
% 0.07/0.31 = ( lattic1039401930up_fin @ A ) ) ).
% 0.07/0.31
% 0.07/0.31 % semilattice_sup.Sup_fin.cong
% 0.07/0.31 thf(fact_167_local_OInf__fin__le__Sup__fin,axiom,
% 0.07/0.31 ! [A3: set @ a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( A3
% 0.07/0.31 != ( bot_bot @ ( set @ a ) ) )
% 0.07/0.31 => ( less_eq @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) @ ( lattic1039401930up_fin @ a @ sup @ A3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Inf_fin_le_Sup_fin
% 0.07/0.31 thf(fact_168_local_OSup__fin_Osubset__imp,axiom,
% 0.07/0.31 ! [A3: set @ a,B5: set @ a] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ a ) @ A3 @ B5 )
% 0.07/0.31 => ( ( A3
% 0.07/0.31 != ( bot_bot @ ( set @ a ) ) )
% 0.07/0.31 => ( ( finite_finite2 @ a @ B5 )
% 0.07/0.31 => ( less_eq @ ( lattic1039401930up_fin @ a @ sup @ A3 ) @ ( lattic1039401930up_fin @ a @ sup @ B5 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Sup_fin.subset_imp
% 0.07/0.31 thf(fact_169_local_OInf__fin_Osubset__imp,axiom,
% 0.07/0.31 ! [A3: set @ a,B5: set @ a] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ a ) @ A3 @ B5 )
% 0.07/0.31 => ( ( A3
% 0.07/0.31 != ( bot_bot @ ( set @ a ) ) )
% 0.07/0.31 => ( ( finite_finite2 @ a @ B5 )
% 0.07/0.31 => ( less_eq @ ( lattic1263571978nf_fin @ a @ inf @ B5 ) @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Inf_fin.subset_imp
% 0.07/0.31 thf(fact_170_local_Ofinite__has__minimal,axiom,
% 0.07/0.31 ! [A3: set @ a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( A3
% 0.07/0.31 != ( bot_bot @ ( set @ a ) ) )
% 0.07/0.31 => ? [X2: a] :
% 0.07/0.31 ( ( member @ a @ X2 @ A3 )
% 0.07/0.31 & ! [Xa: a] :
% 0.07/0.31 ( ( member @ a @ Xa @ A3 )
% 0.07/0.31 => ( ( less_eq @ Xa @ X2 )
% 0.07/0.31 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.finite_has_minimal
% 0.07/0.31 thf(fact_171_local_Ofinite__has__maximal,axiom,
% 0.07/0.31 ! [A3: set @ a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( A3
% 0.07/0.31 != ( bot_bot @ ( set @ a ) ) )
% 0.07/0.31 => ? [X2: a] :
% 0.07/0.31 ( ( member @ a @ X2 @ A3 )
% 0.07/0.31 & ! [Xa: a] :
% 0.07/0.31 ( ( member @ a @ Xa @ A3 )
% 0.07/0.31 => ( ( less_eq @ X2 @ Xa )
% 0.07/0.31 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.finite_has_maximal
% 0.07/0.31 thf(fact_172_local_OInf__fin_Osubset,axiom,
% 0.07/0.31 ! [A3: set @ a,B5: set @ a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( B5
% 0.07/0.31 != ( bot_bot @ ( set @ a ) ) )
% 0.07/0.31 => ( ( ord_less_eq @ ( set @ a ) @ B5 @ A3 )
% 0.07/0.31 => ( ( inf @ ( lattic1263571978nf_fin @ a @ inf @ B5 ) @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) )
% 0.07/0.31 = ( lattic1263571978nf_fin @ a @ inf @ A3 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Inf_fin.subset
% 0.07/0.31 thf(fact_173_local_OSup__fin_Osubset,axiom,
% 0.07/0.31 ! [A3: set @ a,B5: set @ a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( B5
% 0.07/0.31 != ( bot_bot @ ( set @ a ) ) )
% 0.07/0.31 => ( ( ord_less_eq @ ( set @ a ) @ B5 @ A3 )
% 0.07/0.31 => ( ( sup @ ( lattic1039401930up_fin @ a @ sup @ B5 ) @ ( lattic1039401930up_fin @ a @ sup @ A3 ) )
% 0.07/0.31 = ( lattic1039401930up_fin @ a @ sup @ A3 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Sup_fin.subset
% 0.07/0.31 thf(fact_174_local_OInf__fin_Obounded__iff,axiom,
% 0.07/0.31 ! [A3: set @ a,X: a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( A3
% 0.07/0.31 != ( bot_bot @ ( set @ a ) ) )
% 0.07/0.31 => ( ( less_eq @ X @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) )
% 0.07/0.31 = ( ! [X3: a] :
% 0.07/0.31 ( ( member @ a @ X3 @ A3 )
% 0.07/0.31 => ( less_eq @ X @ X3 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Inf_fin.bounded_iff
% 0.07/0.31 thf(fact_175_local_OInf__fin_OboundedI,axiom,
% 0.07/0.31 ! [A3: set @ a,X: a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( A3
% 0.07/0.31 != ( bot_bot @ ( set @ a ) ) )
% 0.07/0.31 => ( ! [A5: a] :
% 0.07/0.31 ( ( member @ a @ A5 @ A3 )
% 0.07/0.31 => ( less_eq @ X @ A5 ) )
% 0.07/0.31 => ( less_eq @ X @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Inf_fin.boundedI
% 0.07/0.31 thf(fact_176_local_OInf__fin_OboundedE,axiom,
% 0.07/0.31 ! [A3: set @ a,X: a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( A3
% 0.07/0.31 != ( bot_bot @ ( set @ a ) ) )
% 0.07/0.31 => ( ( less_eq @ X @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) )
% 0.07/0.31 => ! [A6: a] :
% 0.07/0.31 ( ( member @ a @ A6 @ A3 )
% 0.07/0.31 => ( less_eq @ X @ A6 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Inf_fin.boundedE
% 0.07/0.31 thf(fact_177_local_OSup__fin_Obounded__iff,axiom,
% 0.07/0.31 ! [A3: set @ a,X: a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( A3
% 0.07/0.31 != ( bot_bot @ ( set @ a ) ) )
% 0.07/0.31 => ( ( less_eq @ ( lattic1039401930up_fin @ a @ sup @ A3 ) @ X )
% 0.07/0.31 = ( ! [X3: a] :
% 0.07/0.31 ( ( member @ a @ X3 @ A3 )
% 0.07/0.31 => ( less_eq @ X3 @ X ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Sup_fin.bounded_iff
% 0.07/0.31 thf(fact_178_local_OSup__fin_OboundedI,axiom,
% 0.07/0.31 ! [A3: set @ a,X: a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( A3
% 0.07/0.31 != ( bot_bot @ ( set @ a ) ) )
% 0.07/0.31 => ( ! [A5: a] :
% 0.07/0.31 ( ( member @ a @ A5 @ A3 )
% 0.07/0.31 => ( less_eq @ A5 @ X ) )
% 0.07/0.31 => ( less_eq @ ( lattic1039401930up_fin @ a @ sup @ A3 ) @ X ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Sup_fin.boundedI
% 0.07/0.31 thf(fact_179_local_OSup__fin_OboundedE,axiom,
% 0.07/0.31 ! [A3: set @ a,X: a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( A3
% 0.07/0.31 != ( bot_bot @ ( set @ a ) ) )
% 0.07/0.31 => ( ( less_eq @ ( lattic1039401930up_fin @ a @ sup @ A3 ) @ X )
% 0.07/0.31 => ! [A6: a] :
% 0.07/0.31 ( ( member @ a @ A6 @ A3 )
% 0.07/0.31 => ( less_eq @ A6 @ X ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Sup_fin.boundedE
% 0.07/0.31 thf(fact_180_local_Obdd__above__empty,axiom,
% 0.07/0.31 condit2040224947_above @ a @ less_eq @ ( bot_bot @ ( set @ a ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.bdd_above_empty
% 0.07/0.31 thf(fact_181_local_Obdd__below__empty,axiom,
% 0.07/0.31 condit1201339847_below @ a @ less_eq @ ( bot_bot @ ( set @ a ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.bdd_below_empty
% 0.07/0.31 thf(fact_182_local_Onot__empty__eq__Iic__eq__empty,axiom,
% 0.07/0.31 ! [H: a] :
% 0.07/0.31 ( ( bot_bot @ ( set @ a ) )
% 0.07/0.31 != ( set_atMost @ a @ less_eq @ H ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.not_empty_eq_Iic_eq_empty
% 0.07/0.31 thf(fact_183_local_Onot__empty__eq__Ici__eq__empty,axiom,
% 0.07/0.31 ! [L: a] :
% 0.07/0.31 ( ( bot_bot @ ( set @ a ) )
% 0.07/0.31 != ( set_atLeast @ a @ less_eq @ L ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.not_empty_eq_Ici_eq_empty
% 0.07/0.31 thf(fact_184_subset__empty,axiom,
% 0.07/0.31 ! [A: $tType,A3: set @ A] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.07/0.31 = ( A3
% 0.07/0.31 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_empty
% 0.07/0.31 thf(fact_185_empty__Collect__eq,axiom,
% 0.07/0.31 ! [A: $tType,P: A > $o] :
% 0.07/0.31 ( ( ( bot_bot @ ( set @ A ) )
% 0.07/0.31 = ( collect @ A @ P ) )
% 0.07/0.31 = ( ! [X3: A] :
% 0.07/0.31 ~ ( P @ X3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % empty_Collect_eq
% 0.07/0.31 thf(fact_186_Collect__empty__eq,axiom,
% 0.07/0.31 ! [A: $tType,P: A > $o] :
% 0.07/0.31 ( ( ( collect @ A @ P )
% 0.07/0.31 = ( bot_bot @ ( set @ A ) ) )
% 0.07/0.31 = ( ! [X3: A] :
% 0.07/0.31 ~ ( P @ X3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_empty_eq
% 0.07/0.31 thf(fact_187_all__not__in__conv,axiom,
% 0.07/0.31 ! [A: $tType,A3: set @ A] :
% 0.07/0.31 ( ( ! [X3: A] :
% 0.07/0.31 ~ ( member @ A @ X3 @ A3 ) )
% 0.07/0.31 = ( A3
% 0.07/0.31 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % all_not_in_conv
% 0.07/0.31 thf(fact_188_empty__iff,axiom,
% 0.07/0.31 ! [A: $tType,C: A] :
% 0.07/0.31 ~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % empty_iff
% 0.07/0.31 thf(fact_189_subset__antisym,axiom,
% 0.07/0.31 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 0.07/0.31 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
% 0.07/0.31 => ( A3 = B5 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_antisym
% 0.07/0.31 thf(fact_190_subsetI,axiom,
% 0.07/0.31 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 0.07/0.31 ( ! [X2: A] :
% 0.07/0.31 ( ( member @ A @ X2 @ A3 )
% 0.07/0.31 => ( member @ A @ X2 @ B5 ) )
% 0.07/0.31 => ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% 0.07/0.31
% 0.07/0.31 % subsetI
% 0.07/0.31 thf(fact_191_empty__subsetI,axiom,
% 0.07/0.31 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% 0.07/0.31
% 0.07/0.31 % empty_subsetI
% 0.07/0.31 thf(fact_192_local_OatLeast__iff,axiom,
% 0.07/0.31 ! [I: a,K: a] :
% 0.07/0.31 ( ( member @ a @ I @ ( set_atLeast @ a @ less_eq @ K ) )
% 0.07/0.31 = ( less_eq @ K @ I ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.atLeast_iff
% 0.07/0.31 thf(fact_193_local_OatMost__iff,axiom,
% 0.07/0.31 ! [I: a,K: a] :
% 0.07/0.31 ( ( member @ a @ I @ ( set_atMost @ a @ less_eq @ K ) )
% 0.07/0.31 = ( less_eq @ I @ K ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.atMost_iff
% 0.07/0.31 thf(fact_194_local_Obdd__below__Ici,axiom,
% 0.07/0.31 ! [A2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_atLeast @ a @ less_eq @ A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.bdd_below_Ici
% 0.07/0.31 thf(fact_195_local_Obdd__above__Iic,axiom,
% 0.07/0.31 ! [B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_atMost @ a @ less_eq @ B2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.bdd_above_Iic
% 0.07/0.31 thf(fact_196_ex__in__conv,axiom,
% 0.07/0.31 ! [A: $tType,A3: set @ A] :
% 0.07/0.31 ( ( ? [X3: A] : ( member @ A @ X3 @ A3 ) )
% 0.07/0.31 = ( A3
% 0.07/0.31 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % ex_in_conv
% 0.07/0.31 thf(fact_197_equals0I,axiom,
% 0.07/0.31 ! [A: $tType,A3: set @ A] :
% 0.07/0.31 ( ! [Y2: A] :
% 0.07/0.31 ~ ( member @ A @ Y2 @ A3 )
% 0.07/0.31 => ( A3
% 0.07/0.31 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % equals0I
% 0.07/0.31 thf(fact_198_equals0D,axiom,
% 0.07/0.31 ! [A: $tType,A3: set @ A,A2: A] :
% 0.07/0.31 ( ( A3
% 0.07/0.31 = ( bot_bot @ ( set @ A ) ) )
% 0.07/0.31 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % equals0D
% 0.07/0.31 thf(fact_199_emptyE,axiom,
% 0.07/0.31 ! [A: $tType,A2: A] :
% 0.07/0.31 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % emptyE
% 0.07/0.31 thf(fact_200_Collect__mono__iff,axiom,
% 0.07/0.31 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.07/0.31 = ( ! [X3: A] :
% 0.07/0.31 ( ( P @ X3 )
% 0.07/0.31 => ( Q @ X3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_mono_iff
% 0.07/0.31 thf(fact_201_set__eq__subset,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ^ [Y3: set @ A,Z3: set @ A] : ( Y3 = Z3 ) )
% 0.07/0.31 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 0.07/0.31 & ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % set_eq_subset
% 0.07/0.31 thf(fact_202_subset__trans,axiom,
% 0.07/0.31 ! [A: $tType,A3: set @ A,B5: set @ A,C4: set @ A] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 0.07/0.31 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
% 0.07/0.31 => ( ord_less_eq @ ( set @ A ) @ A3 @ C4 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_trans
% 0.07/0.31 thf(fact_203_Collect__mono,axiom,
% 0.07/0.31 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.07/0.31 ( ! [X2: A] :
% 0.07/0.31 ( ( P @ X2 )
% 0.07/0.31 => ( Q @ X2 ) )
% 0.07/0.31 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_mono
% 0.07/0.31 thf(fact_204_subset__refl,axiom,
% 0.07/0.31 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 0.07/0.31
% 0.07/0.31 % subset_refl
% 0.07/0.31 thf(fact_205_subset__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) )
% 0.07/0.31 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.07/0.31 ! [T: A] :
% 0.07/0.31 ( ( member @ A @ T @ A7 )
% 0.07/0.31 => ( member @ A @ T @ B6 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_iff
% 0.07/0.31 thf(fact_206_equalityD2,axiom,
% 0.07/0.31 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 0.07/0.31 ( ( A3 = B5 )
% 0.07/0.31 => ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % equalityD2
% 0.07/0.31 thf(fact_207_equalityD1,axiom,
% 0.07/0.31 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 0.07/0.31 ( ( A3 = B5 )
% 0.07/0.31 => ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% 0.07/0.31
% 0.07/0.31 % equalityD1
% 0.07/0.31 thf(fact_208_subset__eq,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) )
% 0.07/0.31 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.07/0.31 ! [X3: A] :
% 0.07/0.31 ( ( member @ A @ X3 @ A7 )
% 0.07/0.31 => ( member @ A @ X3 @ B6 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_eq
% 0.07/0.31 thf(fact_209_equalityE,axiom,
% 0.07/0.31 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 0.07/0.31 ( ( A3 = B5 )
% 0.07/0.31 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 0.07/0.31 => ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % equalityE
% 0.07/0.31 thf(fact_210_subsetD,axiom,
% 0.07/0.31 ! [A: $tType,A3: set @ A,B5: set @ A,C: A] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 0.07/0.31 => ( ( member @ A @ C @ A3 )
% 0.07/0.31 => ( member @ A @ C @ B5 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subsetD
% 0.07/0.31 thf(fact_211_in__mono,axiom,
% 0.07/0.31 ! [A: $tType,A3: set @ A,B5: set @ A,X: A] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 0.07/0.31 => ( ( member @ A @ X @ A3 )
% 0.07/0.31 => ( member @ A @ X @ B5 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % in_mono
% 0.07/0.31 thf(fact_212_local_OSup__fin_Oclosed,axiom,
% 0.07/0.31 ! [A3: set @ a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( A3
% 0.07/0.31 != ( bot_bot @ ( set @ a ) ) )
% 0.07/0.31 => ( ! [X2: a,Y2: a] : ( member @ a @ ( sup @ X2 @ Y2 ) @ ( insert @ a @ X2 @ ( insert @ a @ Y2 @ ( bot_bot @ ( set @ a ) ) ) ) )
% 0.07/0.31 => ( member @ a @ ( lattic1039401930up_fin @ a @ sup @ A3 ) @ A3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Sup_fin.closed
% 0.07/0.31 thf(fact_213_local_OSup__fin_Oinsert__not__elem,axiom,
% 0.07/0.31 ! [A3: set @ a,X: a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ~ ( member @ a @ X @ A3 )
% 0.07/0.31 => ( ( A3
% 0.07/0.31 != ( bot_bot @ ( set @ a ) ) )
% 0.07/0.31 => ( ( lattic1039401930up_fin @ a @ sup @ ( insert @ a @ X @ A3 ) )
% 0.07/0.31 = ( sup @ X @ ( lattic1039401930up_fin @ a @ sup @ A3 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Sup_fin.insert_not_elem
% 0.07/0.31 thf(fact_214_local_OInf__fin_Oclosed,axiom,
% 0.07/0.31 ! [A3: set @ a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( A3
% 0.07/0.31 != ( bot_bot @ ( set @ a ) ) )
% 0.07/0.31 => ( ! [X2: a,Y2: a] : ( member @ a @ ( inf @ X2 @ Y2 ) @ ( insert @ a @ X2 @ ( insert @ a @ Y2 @ ( bot_bot @ ( set @ a ) ) ) ) )
% 0.07/0.31 => ( member @ a @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) @ A3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Inf_fin.closed
% 0.07/0.31 thf(fact_215_local_OInf__fin_Oinsert__not__elem,axiom,
% 0.07/0.31 ! [A3: set @ a,X: a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ~ ( member @ a @ X @ A3 )
% 0.07/0.31 => ( ( A3
% 0.07/0.31 != ( bot_bot @ ( set @ a ) ) )
% 0.07/0.31 => ( ( lattic1263571978nf_fin @ a @ inf @ ( insert @ a @ X @ A3 ) )
% 0.07/0.31 = ( inf @ X @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Inf_fin.insert_not_elem
% 0.07/0.31 thf(fact_216_singletonI,axiom,
% 0.07/0.31 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % singletonI
% 0.07/0.31 thf(fact_217_insert__subset,axiom,
% 0.07/0.31 ! [A: $tType,X: A,A3: set @ A,B5: set @ A] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B5 )
% 0.07/0.31 = ( ( member @ A @ X @ B5 )
% 0.07/0.31 & ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % insert_subset
% 0.07/0.31 thf(fact_218_singleton__insert__inj__eq_H,axiom,
% 0.07/0.31 ! [A: $tType,A2: A,A3: set @ A,B2: A] :
% 0.07/0.31 ( ( ( insert @ A @ A2 @ A3 )
% 0.07/0.31 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.07/0.31 = ( ( A2 = B2 )
% 0.07/0.31 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % singleton_insert_inj_eq'
% 0.07/0.31 thf(fact_219_singleton__insert__inj__eq,axiom,
% 0.07/0.31 ! [A: $tType,B2: A,A2: A,A3: set @ A] :
% 0.07/0.31 ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
% 0.07/0.31 = ( insert @ A @ A2 @ A3 ) )
% 0.07/0.31 = ( ( A2 = B2 )
% 0.07/0.31 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % singleton_insert_inj_eq
% 0.07/0.31 thf(fact_220_local_Obdd__above__insert,axiom,
% 0.07/0.31 ! [A2: a,A3: set @ a] :
% 0.07/0.31 ( ( condit2040224947_above @ a @ less_eq @ ( insert @ a @ A2 @ A3 ) )
% 0.07/0.31 = ( condit2040224947_above @ a @ less_eq @ A3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.bdd_above_insert
% 0.07/0.31 thf(fact_221_local_Obdd__below__insert,axiom,
% 0.07/0.31 ! [A2: a,A3: set @ a] :
% 0.07/0.31 ( ( condit1201339847_below @ a @ less_eq @ ( insert @ a @ A2 @ A3 ) )
% 0.07/0.31 = ( condit1201339847_below @ a @ less_eq @ A3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.bdd_below_insert
% 0.07/0.31 thf(fact_222_local_OInf__fin_Osingleton,axiom,
% 0.07/0.31 ! [X: a] :
% 0.07/0.31 ( ( lattic1263571978nf_fin @ a @ inf @ ( insert @ a @ X @ ( bot_bot @ ( set @ a ) ) ) )
% 0.07/0.31 = X ) ).
% 0.07/0.31
% 0.07/0.31 % local.Inf_fin.singleton
% 0.07/0.31 thf(fact_223_local_OSup__fin_Osingleton,axiom,
% 0.07/0.31 ! [X: a] :
% 0.07/0.31 ( ( lattic1039401930up_fin @ a @ sup @ ( insert @ a @ X @ ( bot_bot @ ( set @ a ) ) ) )
% 0.07/0.31 = X ) ).
% 0.07/0.31
% 0.07/0.31 % local.Sup_fin.singleton
% 0.07/0.31 thf(fact_224_local_OInf__fin_Oinsert,axiom,
% 0.07/0.31 ! [A3: set @ a,X: a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( A3
% 0.07/0.31 != ( bot_bot @ ( set @ a ) ) )
% 0.07/0.31 => ( ( lattic1263571978nf_fin @ a @ inf @ ( insert @ a @ X @ A3 ) )
% 0.07/0.31 = ( inf @ X @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Inf_fin.insert
% 0.07/0.31 thf(fact_225_local_OSup__fin_Oinsert,axiom,
% 0.07/0.31 ! [A3: set @ a,X: a] :
% 0.07/0.31 ( ( finite_finite2 @ a @ A3 )
% 0.07/0.31 => ( ( A3
% 0.07/0.31 != ( bot_bot @ ( set @ a ) ) )
% 0.07/0.31 => ( ( lattic1039401930up_fin @ a @ sup @ ( insert @ a @ X @ A3 ) )
% 0.07/0.31 = ( sup @ X @ ( lattic1039401930up_fin @ a @ sup @ A3 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Sup_fin.insert
% 0.07/0.31 thf(fact_226_subset__singletonD,axiom,
% 0.07/0.31 ! [A: $tType,A3: set @ A,X: A] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.07/0.31 => ( ( A3
% 0.07/0.31 = ( bot_bot @ ( set @ A ) ) )
% 0.07/0.31 | ( A3
% 0.07/0.31 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_singletonD
% 0.07/0.31 thf(fact_227_subset__singleton__iff,axiom,
% 0.07/0.31 ! [A: $tType,X5: set @ A,A2: A] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.07/0.31 = ( ( X5
% 0.07/0.31 = ( bot_bot @ ( set @ A ) ) )
% 0.07/0.31 | ( X5
% 0.07/0.31 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_singleton_iff
% 0.07/0.31 thf(fact_228_singletonD,axiom,
% 0.07/0.31 ! [A: $tType,B2: A,A2: A] :
% 0.07/0.31 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.07/0.31 => ( B2 = A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % singletonD
% 0.07/0.31 thf(fact_229_singleton__iff,axiom,
% 0.07/0.31 ! [A: $tType,B2: A,A2: A] :
% 0.07/0.31 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.07/0.31 = ( B2 = A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % singleton_iff
% 0.07/0.31 thf(fact_230_doubleton__eq__iff,axiom,
% 0.07/0.31 ! [A: $tType,A2: A,B2: A,C: A,D: A] :
% 0.07/0.31 ( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.07/0.31 = ( insert @ A @ C @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.07/0.31 = ( ( ( A2 = C )
% 0.07/0.31 & ( B2 = D ) )
% 0.07/0.31 | ( ( A2 = D )
% 0.07/0.31 & ( B2 = C ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % doubleton_eq_iff
% 0.07/0.31 thf(fact_231_insert__not__empty,axiom,
% 0.07/0.31 ! [A: $tType,A2: A,A3: set @ A] :
% 0.07/0.31 ( ( insert @ A @ A2 @ A3 )
% 0.07/0.31 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % insert_not_empty
% 0.07/0.31 thf(fact_232_singleton__inject,axiom,
% 0.07/0.31 ! [A: $tType,A2: A,B2: A] :
% 0.07/0.31 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.07/0.31 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.07/0.31 => ( A2 = B2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % singleton_inject
% 0.07/0.31 thf(fact_233_subset__insertI2,axiom,
% 0.07/0.31 ! [A: $tType,A3: set @ A,B5: set @ A,B2: A] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 0.07/0.31 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_insertI2
% 0.07/0.31 thf(fact_234_subset__insertI,axiom,
% 0.07/0.31 ! [A: $tType,B5: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert @ A @ A2 @ B5 ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_insertI
% 0.07/0.31 thf(fact_235_subset__insert,axiom,
% 0.07/0.31 ! [A: $tType,X: A,A3: set @ A,B5: set @ A] :
% 0.07/0.31 ( ~ ( member @ A @ X @ A3 )
% 0.07/0.31 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B5 ) )
% 0.07/0.31 = ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_insert
% 0.07/0.31 thf(fact_236_insert__mono,axiom,
% 0.07/0.31 ! [A: $tType,C4: set @ A,D2: set @ A,A2: A] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ C4 @ D2 )
% 0.07/0.31 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C4 ) @ ( insert @ A @ A2 @ D2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % insert_mono
% 0.07/0.31 thf(fact_237_finite__ranking__induct,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType] :
% 0.07/0.31 ( ( linorder @ A )
% 0.07/0.31 => ! [S: set @ B,P: ( set @ B ) > $o,F: B > A] :
% 0.07/0.31 ( ( finite_finite2 @ B @ S )
% 0.07/0.31 => ( ( P @ ( bot_bot @ ( set @ B ) ) )
% 0.07/0.31 => ( ! [X2: B,S2: set @ B] :
% 0.07/0.31 ( ( finite_finite2 @ B @ S2 )
% 0.07/0.31 => ( ! [Y5: B] :
% 0.07/0.31 ( ( member @ B @ Y5 @ S2 )
% 0.07/0.31 => ( ord_less_eq @ A @ ( F @ Y5 ) @ ( F @ X2 ) ) )
% 0.07/0.31 => ( ( P @ S2 )
% 0.07/0.31 => ( P @ ( insert @ B @ X2 @ S2 ) ) ) ) )
% 0.07/0.31 => ( P @ S ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % finite_ranking_induct
% 0.07/0.31 thf(fact_238_local_OatLeastAtMost__singleton_H,axiom,
% 0.07/0.31 ! [A2: a,B2: a] :
% 0.07/0.31 ( ( A2 = B2 )
% 0.07/0.31 => ( ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 )
% 0.07/0.31 = ( insert @ a @ A2 @ ( bot_bot @ ( set @ a ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.atLeastAtMost_singleton'
% 0.07/0.31 thf(fact_239_finite__insert,axiom,
% 0.07/0.31 ! [A: $tType,A2: A,A3: set @ A] :
% 0.07/0.31 ( ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) )
% 0.07/0.31 = ( finite_finite2 @ A @ A3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % finite_insert
% 0.07/0.31 thf(fact_240_finite__code,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( finite_finite @ A )
% 0.07/0.31 => ( ( finite_finite2 @ A )
% 0.07/0.31 = ( ^ [A7: set @ A] : $true ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % finite_code
% 0.07/0.31 thf(fact_241_local_OIcc__eq__Icc,axiom,
% 0.07/0.31 ! [L: a,H: a,L2: a,H2: a] :
% 0.07/0.31 ( ( ( set_atLeastAtMost @ a @ less_eq @ L @ H )
% 0.07/0.31 = ( set_atLeastAtMost @ a @ less_eq @ L2 @ H2 ) )
% 0.07/0.31 = ( ( ( L = L2 )
% 0.07/0.31 & ( H = H2 ) )
% 0.07/0.31 | ( ~ ( less_eq @ L @ H )
% 0.07/0.31 & ~ ( less_eq @ L2 @ H2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Icc_eq_Icc
% 0.07/0.31 thf(fact_242_local_OatLeastAtMost__iff,axiom,
% 0.07/0.31 ! [I: a,L: a,U: a] :
% 0.07/0.31 ( ( member @ a @ I @ ( set_atLeastAtMost @ a @ less_eq @ L @ U ) )
% 0.07/0.31 = ( ( less_eq @ L @ I )
% 0.07/0.31 & ( less_eq @ I @ U ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.atLeastAtMost_iff
% 0.07/0.31 thf(fact_243_local_OatLeastatMost__empty__iff,axiom,
% 0.07/0.31 ! [A2: a,B2: a] :
% 0.07/0.31 ( ( ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 )
% 0.07/0.31 = ( bot_bot @ ( set @ a ) ) )
% 0.07/0.31 = ( ~ ( less_eq @ A2 @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.atLeastatMost_empty_iff
% 0.07/0.31 thf(fact_244_local_OatLeastatMost__empty__iff2,axiom,
% 0.07/0.31 ! [A2: a,B2: a] :
% 0.07/0.31 ( ( ( bot_bot @ ( set @ a ) )
% 0.07/0.31 = ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) )
% 0.07/0.31 = ( ~ ( less_eq @ A2 @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.atLeastatMost_empty_iff2
% 0.07/0.31 thf(fact_245_local_Obdd__below__Icc,axiom,
% 0.07/0.31 ! [A2: a,B2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.bdd_below_Icc
% 0.07/0.31 thf(fact_246_local_Obdd__above__Icc,axiom,
% 0.07/0.31 ! [A2: a,B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.bdd_above_Icc
% 0.07/0.31 thf(fact_247_local_OatLeastatMost__subset__iff,axiom,
% 0.07/0.31 ! [A2: a,B2: a,C: a,D: a] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) @ ( set_atLeastAtMost @ a @ less_eq @ C @ D ) )
% 0.07/0.31 = ( ~ ( less_eq @ A2 @ B2 )
% 0.07/0.31 | ( ( less_eq @ C @ A2 )
% 0.07/0.31 & ( less_eq @ B2 @ D ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.atLeastatMost_subset_iff
% 0.07/0.31 thf(fact_248_local_OatLeastAtMost__singleton,axiom,
% 0.07/0.31 ! [A2: a] :
% 0.07/0.31 ( ( set_atLeastAtMost @ a @ less_eq @ A2 @ A2 )
% 0.07/0.31 = ( insert @ a @ A2 @ ( bot_bot @ ( set @ a ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.atLeastAtMost_singleton
% 0.07/0.31 thf(fact_249_local_OatLeastAtMost__singleton__iff,axiom,
% 0.07/0.31 ! [A2: a,B2: a,C: a] :
% 0.07/0.31 ( ( ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 )
% 0.07/0.31 = ( insert @ a @ C @ ( bot_bot @ ( set @ a ) ) ) )
% 0.07/0.31 = ( ( A2 = B2 )
% 0.07/0.31 & ( B2 = C ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.atLeastAtMost_singleton_iff
% 0.07/0.31 thf(fact_250_local_OIcc__subset__Ici__iff,axiom,
% 0.07/0.31 ! [L: a,H: a,L2: a] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ L @ H ) @ ( set_atLeast @ a @ less_eq @ L2 ) )
% 0.07/0.31 = ( ~ ( less_eq @ L @ H )
% 0.07/0.31 | ( less_eq @ L2 @ L ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Icc_subset_Ici_iff
% 0.07/0.31 thf(fact_251_local_OIcc__subset__Iic__iff,axiom,
% 0.07/0.31 ! [L: a,H: a,H2: a] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ L @ H ) @ ( set_atMost @ a @ less_eq @ H2 ) )
% 0.07/0.31 = ( ~ ( less_eq @ L @ H )
% 0.07/0.31 | ( less_eq @ H @ H2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % local.Icc_subset_Iic_iff
% 0.07/0.31 thf(fact_252_finite,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( finite_finite @ A )
% 0.07/0.31 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % finite
% 0.07/0.31 thf(fact_253_finite__set__choice,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
% 0.07/0.31 ( ( finite_finite2 @ A @ A3 )
% 0.07/0.31 => ( ! [X2: A] :
% 0.07/0.31 ( ( member @ A @ X2 @ A3 )
% 0.07/0.31 => ? [X_1: B] : ( P @ X2 @ X_1 ) )
% 0.07/0.31 => ? [F3: A > B] :
% 0.07/0.31 ! [X4: A] :
% 0.07/0.31 ( ( member @ A @ X4 @ A3 )
% 0.07/0.31 => ( P @ X4 @ ( F3 @ X4 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % finite_set_choice
% 0.07/0.31 thf(fact_254_comp__fun__idem_Ofun__left__idem,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType,F: A > B > B,X: A,Z: B] :
% 0.07/0.31 ( ( finite_comp_fun_idem @ A @ B @ F )
% 0.07/0.31 => ( ( F @ X @ ( F @ X @ Z ) )
% 0.07/0.31 = ( F @ X @ Z ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % comp_fun_idem.fun_left_idem
% 0.07/0.31 thf(fact_255_order__class_Ofinite__has__minimal2,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( order @ A )
% 0.07/0.31 => ! [A3: set @ A,A2: A] :
% 0.07/0.31 ( ( finite_finite2 @ A @ A3 )
% 0.07/0.31 => ( ( member @ A @ A2 @ A3 )
% 0.07/0.31 => ? [X2: A] :
% 0.07/0.31 ( ( member @ A @ X2 @ A3 )
% 0.07/0.31 & ( ord_less_eq @ A @ X2 @ A2 )
% 0.07/0.31 & ! [Xa: A] :
% 0.07/0.31 ( ( member @ A @ Xa @ A3 )
% 0.07/0.31 => ( ( ord_less_eq @ A @ Xa @ X2 )
% 0.07/0.31 => ( X2 = Xa ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % order_class.finite_has_minimal2
% 0.07/0.31
% 0.07/0.31 % Type constructors (7)
% 0.07/0.31 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.07/0.31 ! [A8: $tType,A9: $tType] :
% 0.07/0.31 ( ( ( finite_finite @ A8 )
% 0.07/0.31 & ( finite_finite @ A9 ) )
% 0.07/0.31 => ( finite_finite @ ( A8 > A9 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.07/0.31 ! [A8: $tType,A9: $tType] :
% 0.07/0.31 ( ( order @ A9 )
% 0.07/0.31 => ( order @ ( A8 > A9 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(tcon_Set_Oset___Finite__Set_Ofinite_1,axiom,
% 0.07/0.31 ! [A8: $tType] :
% 0.07/0.31 ( ( finite_finite @ A8 )
% 0.07/0.31 => ( finite_finite @ ( set @ A8 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
% 0.07/0.31 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.07/0.31 linorder @ $o ).
% 0.07/0.31
% 0.07/0.31 thf(tcon_HOL_Obool___Finite__Set_Ofinite_3,axiom,
% 0.07/0.31 finite_finite @ $o ).
% 0.07/0.31
% 0.07/0.31 thf(tcon_HOL_Obool___Orderings_Oorder_4,axiom,
% 0.07/0.31 order @ $o ).
% 0.07/0.31
% 0.07/0.31 % Conjectures (1)
% 0.07/0.31 thf(conj_0,conjecture,
% 0.07/0.31 ( ( modula1593167561_e_aux @ a @ inf @ sup @ b @ c @ a2 )
% 0.07/0.31 = ( modula1593167561_e_aux @ a @ inf @ sup @ a2 @ b @ c ) ) ).
% 0.07/0.31
% 0.07/0.31 %------------------------------------------------------------------------------
% 0.07/0.31 ------------------------
% 1.61/1.07 ---- Embedded in THF ---
% 1.61/1.07 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.61/1.07 %%% Generated on Tue Feb 24 03:26:56 EST 2026
% 1.61/1.07 %%% using '$$dhol' embedding, version 1.3.0.
% 1.61/1.07 %%% Logic specification used:
% 1.61/1.07 %%% thf(spec, logic, $$dhol).
% 1.61/1.07
% 1.61/1.07 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.3GJe6d4PX9/DTF2DTF3385.p
% 1.61/1.07 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 1.61/1.07 thf(ty_tf_a, type, a: $tType).
% 1.61/1.07 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 1.61/1.07 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 1.61/1.07 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 1.61/1.07 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 1.61/1.07 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above, type, condit2040224947_above: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > $o))))).
% 1.61/1.07 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below, type, condit1201339847_below: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > $o))))).
% 1.61/1.07 thf(sy_c_Finite__Set_Ocomp__fun__idem, type, finite_comp_fun_idem: (!> [A:$tType,B:$tType]: (((A > (B > B)) > $o)))).
% 1.61/1.07 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))).
% 1.61/1.07 thf(sy_c_Groups_Oabel__semigroup, type, abel_semigroup: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 1.61/1.07 thf(sy_c_Groups_Osemigroup, type, semigroup: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 1.61/1.07 thf(sy_c_Lattices_Osemilattice, type, semilattice: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 1.61/1.07 thf(sy_c_Lattices__Big_Osemilattice__inf_OInf__fin, type, lattic1263571978nf_fin: (!> [A:$tType]: (((A > (A > A)) > ((set @ A) > A))))).
% 1.61/1.07 thf(sy_c_Lattices__Big_Osemilattice__set, type, lattic35693393ce_set: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 1.61/1.07 thf(sy_c_Lattices__Big_Osemilattice__sup_OSup__fin, type, lattic1039401930up_fin: (!> [A:$tType]: (((A > (A > A)) > ((set @ A) > A))))).
% 1.61/1.07 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Od__aux, type, modula1221160330_d_aux: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > A)) > (A > (A > (A > A)))))))).
% 1.61/1.07 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oe__aux, type, modula1593167561_e_aux: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > A)) > (A > (A > (A > A)))))))).
% 1.61/1.07 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 1.61/1.07 thf(sy_c_Orderings_Oord_OLeast, type, least: (!> [A:$tType]: (((A > (A > $o)) > ((A > $o) > A))))).
% 1.61/1.07 thf(sy_c_Orderings_Oord_Omax, type, max: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > A)))))).
% 1.61/1.07 thf(sy_c_Orderings_Oord_Omin, type, min: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > A)))))).
% 1.61/1.07 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 1.61/1.07 thf(sy_c_Orderings_Oorder_OGreatest, type, greatest: (!> [A:$tType]: (((A > (A > $o)) > ((A > $o) > A))))).
% 1.61/1.07 thf(sy_c_Orderings_Oorder_Oantimono, type, antimono: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((A > B) > $o))))).
% 1.61/1.07 thf(sy_c_Orderings_Oorder_Omono, type, mono: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((A > B) > $o))))).
% 1.61/1.07 thf(sy_c_Relation_Otransp, type, transp: (!> [A:$tType]: (((A > (A > $o)) > $o)))).
% 1.61/1.07 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 1.61/1.07 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 1.61/1.07 thf(sy_c_Set__Interval_Oord_OatLeast, type, set_atLeast: (!> [A:$tType]: (((A > (A > $o)) > (A > (set @ A)))))).
% 1.61/1.07 thf(sy_c_Set__Interval_Oord_OatLeastAtMost, type, set_atLeastAtMost: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > (set @ A))))))).
% 1.61/1.07 thf(sy_c_Set__Interval_Oord_OatMost, type, set_atMost: (!> [A:$tType]: (((A > (A > $o)) > (A > (set @ A)))))).
% 1.61/1.07 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 1.61/1.07 thf(sy_v_a, type, a2: a).
% 1.61/1.07 thf(sy_v_b, type, b: a).
% 1.61/1.07 thf(sy_v_c, type, c: a).
% 1.61/1.07 thf(sy_v_inf, type, inf: (a > (a > a))).
% 1.61/1.07 thf(sy_v_less__eq, type, less_eq: (a > (a > $o))).
% 1.61/1.07 thf(sy_v_sup, type, sup: (a > (a > a))).
% 1.61/1.07 thf(fact_0_local_Oinf_Oassoc, axiom, (! [A2:a,B2:a,C:a]: ((((inf @ ((inf @ A2) @ B2)) @ C) = ((inf @ A2) @ ((inf @ B2) @ C)))))).
% 1.61/1.07 thf(fact_1_local_Oinf_Ocommute, axiom, (! [A2:a,B2:a]: ((((inf @ A2) @ B2) = ((inf @ B2) @ A2))))).
% 1.61/1.07 thf(fact_2_local_Oinf_Oleft__commute, axiom, (! [B2:a,A2:a,C:a]: ((((inf @ B2) @ ((inf @ A2) @ C)) = ((inf @ A2) @ ((inf @ B2) @ C)))))).
% 1.61/1.07 thf(fact_3_local_Oinf__assoc, axiom, (! [X:a,Y:a,Z:a]: ((((inf @ ((inf @ X) @ Y)) @ Z) = ((inf @ X) @ ((inf @ Y) @ Z)))))).
% 1.61/1.07 thf(fact_4_local_Oinf__commute, axiom, (! [X:a,Y:a]: ((((inf @ X) @ Y) = ((inf @ Y) @ X))))).
% 1.61/1.07 thf(fact_5_local_Oinf__left__commute, axiom, (! [X:a,Y:a,Z:a]: ((((inf @ X) @ ((inf @ Y) @ Z)) = ((inf @ Y) @ ((inf @ X) @ Z)))))).
% 1.61/1.07 thf(fact_6_local_Osup_Oassoc, axiom, (! [A2:a,B2:a,C:a]: ((((sup @ ((sup @ A2) @ B2)) @ C) = ((sup @ A2) @ ((sup @ B2) @ C)))))).
% 1.61/1.07 thf(fact_7_local_Osup_Ocommute, axiom, (! [A2:a,B2:a]: ((((sup @ A2) @ B2) = ((sup @ B2) @ A2))))).
% 1.61/1.07 thf(fact_8_local_Osup_Oleft__commute, axiom, (! [B2:a,A2:a,C:a]: ((((sup @ B2) @ ((sup @ A2) @ C)) = ((sup @ A2) @ ((sup @ B2) @ C)))))).
% 1.61/1.07 thf(fact_9_local_Osup__assoc, axiom, (! [X:a,Y:a,Z:a]: ((((sup @ ((sup @ X) @ Y)) @ Z) = ((sup @ X) @ ((sup @ Y) @ Z)))))).
% 1.61/1.07 thf(fact_10_local_Osup__commute, axiom, (! [X:a,Y:a]: ((((sup @ X) @ Y) = ((sup @ Y) @ X))))).
% 1.61/1.07 thf(fact_11_local_Osup__left__commute, axiom, (! [X:a,Y:a,Z:a]: ((((sup @ X) @ ((sup @ Y) @ Z)) = ((sup @ Y) @ ((sup @ X) @ Z)))))).
% 1.61/1.07 thf(fact_12_local_Odistrib__imp1, axiom, (! [X:a,Y:a,Z:a]: (((! [X2:a,Y2:a,Z2:a]: ((((inf @ X2) @ ((sup @ Y2) @ Z2)) = ((sup @ ((inf @ X2) @ Y2)) @ ((inf @ X2) @ Z2))))) => (((sup @ X) @ ((inf @ Y) @ Z)) = ((inf @ ((sup @ X) @ Y)) @ ((sup @ X) @ Z))))))).
% 1.61/1.07 thf(fact_13_local_Odistrib__imp2, axiom, (! [X:a,Y:a,Z:a]: (((! [X2:a,Y2:a,Z2:a]: ((((sup @ X2) @ ((inf @ Y2) @ Z2)) = ((inf @ ((sup @ X2) @ Y2)) @ ((sup @ X2) @ Z2))))) => (((inf @ X) @ ((sup @ Y) @ Z)) = ((sup @ ((inf @ X) @ Y)) @ ((inf @ X) @ Z))))))).
% 1.61/1.07 thf(fact_14_e__aux__def, axiom, (! [A2:a,B2:a,C:a]: ((((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C) = ((inf @ ((inf @ ((sup @ A2) @ B2)) @ ((sup @ B2) @ C))) @ ((sup @ C) @ A2)))))).
% 1.61/1.07 thf(fact_15_local_Oinf_Oidem, axiom, (! [A2:a]: ((((inf @ A2) @ A2) = A2)))).
% 1.61/1.07 thf(fact_16_local_Oinf_Oleft__idem, axiom, (! [A2:a,B2:a]: ((((inf @ A2) @ ((inf @ A2) @ B2)) = ((inf @ A2) @ B2))))).
% 1.61/1.07 thf(fact_17_local_Oinf_Oright__idem, axiom, (! [A2:a,B2:a]: ((((inf @ ((inf @ A2) @ B2)) @ B2) = ((inf @ A2) @ B2))))).
% 1.61/1.07 thf(fact_18_local_Oinf__idem, axiom, (! [X:a]: ((((inf @ X) @ X) = X)))).
% 1.61/1.07 thf(fact_19_local_Oinf__left__idem, axiom, (! [X:a,Y:a]: ((((inf @ X) @ ((inf @ X) @ Y)) = ((inf @ X) @ Y))))).
% 1.61/1.07 thf(fact_20_local_Oinf__right__idem, axiom, (! [X:a,Y:a]: ((((inf @ ((inf @ X) @ Y)) @ Y) = ((inf @ X) @ Y))))).
% 1.61/1.07 thf(fact_21_local_Osup_Oidem, axiom, (! [A2:a]: ((((sup @ A2) @ A2) = A2)))).
% 1.61/1.07 thf(fact_22_local_Osup_Oleft__idem, axiom, (! [A2:a,B2:a]: ((((sup @ A2) @ ((sup @ A2) @ B2)) = ((sup @ A2) @ B2))))).
% 1.61/1.07 thf(fact_23_local_Osup_Oright__idem, axiom, (! [A2:a,B2:a]: ((((sup @ ((sup @ A2) @ B2)) @ B2) = ((sup @ A2) @ B2))))).
% 1.61/1.07 thf(fact_24_local_Osup__idem, axiom, (! [X:a]: ((((sup @ X) @ X) = X)))).
% 1.61/1.07 thf(fact_25_local_Osup__left__idem, axiom, (! [X:a,Y:a]: ((((sup @ X) @ ((sup @ X) @ Y)) = ((sup @ X) @ Y))))).
% 1.61/1.07 thf(fact_26_local_Oinf__sup__absorb, axiom, (! [X:a,Y:a]: ((((inf @ X) @ ((sup @ X) @ Y)) = X)))).
% 1.61/1.07 thf(fact_27_local_Osup__inf__absorb, axiom, (! [X:a,Y:a]: ((((sup @ X) @ ((inf @ X) @ Y)) = X)))).
% 1.61/1.07 thf(fact_28_d__aux__def, axiom, (! [A2:a,B2:a,C:a]: ((((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C) = ((sup @ ((sup @ ((inf @ A2) @ B2)) @ ((inf @ B2) @ C))) @ ((inf @ C) @ A2)))))).
% 1.61/1.07 thf(fact_29_d__b__c__a, axiom, (! [B2:a,C:a,A2:a]: ((((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ B2) @ C) @ A2) = ((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C))))).
% 1.61/1.07 thf(fact_30_d__c__a__b, axiom, (! [C:a,A2:a,B2:a]: ((((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ C) @ A2) @ B2) = ((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C))))).
% 1.61/1.07 thf(fact_31_lattice_Oe__aux_Ocong, axiom, (! [A:$tType]: (((modula1593167561_e_aux @ A) = (modula1593167561_e_aux @ A))))).
% 1.61/1.07 thf(fact_32_local_Ocomp__fun__idem__sup, axiom, (((finite_comp_fun_idem @ a) @ a) @ sup)).
% 1.61/1.07 thf(fact_33_local_Ocomp__fun__idem__inf, axiom, (((finite_comp_fun_idem @ a) @ a) @ inf)).
% 1.61/1.07 thf(fact_34_local_Osup_Osemigroup__axioms, axiom, ((semigroup @ a) @ sup)).
% 1.61/1.07 thf(fact_35_local_Oinf_Osemigroup__axioms, axiom, ((semigroup @ a) @ inf)).
% 1.61/1.07 thf(fact_36_local_Osup_Osemilattice__axioms, axiom, ((semilattice @ a) @ sup)).
% 1.61/1.07 thf(fact_37_local_Oinf_Osemilattice__axioms, axiom, ((semilattice @ a) @ inf)).
% 1.61/1.07 thf(fact_38_local_Osup_Oabel__semigroup__axioms, axiom, ((abel_semigroup @ a) @ sup)).
% 1.61/1.07 thf(fact_39_local_Oinf_Oabel__semigroup__axioms, axiom, ((abel_semigroup @ a) @ inf)).
% 1.61/1.07 thf(fact_40_local_OSup__fin_Osemilattice__set__axioms, axiom, ((lattic35693393ce_set @ a) @ sup)).
% 1.61/1.07 thf(fact_41_local_OInf__fin_Osemilattice__set__axioms, axiom, ((lattic35693393ce_set @ a) @ inf)).
% 1.61/1.07 thf(fact_42_local_Odistrib__inf__le, axiom, (! [X:a,Y:a,Z:a]: (((less_eq @ ((sup @ ((inf @ X) @ Y)) @ ((inf @ X) @ Z))) @ ((inf @ X) @ ((sup @ Y) @ Z)))))).
% 1.61/1.07 thf(fact_43_local_Odistrib__sup__le, axiom, (! [X:a,Y:a,Z:a]: (((less_eq @ ((sup @ X) @ ((inf @ Y) @ Z))) @ ((inf @ ((sup @ X) @ Y)) @ ((sup @ X) @ Z)))))).
% 1.61/1.07 thf(fact_44_local_Oantisym, axiom, (! [X:a,Y:a]: ((((less_eq @ X) @ Y) => (((less_eq @ Y) @ X) => (X = Y)))))).
% 1.61/1.07 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 1.61/1.07 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A3)))) = A3)))).
% 1.61/1.07 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) = (Q @ X2)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 1.61/1.07 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X2:A]: (((F @ X2) = (G @ X2)))) => (F = G))))).
% 1.61/1.07 thf(fact_49_local_Oantisym__conv, axiom, (! [Y:a,X:a]: ((((less_eq @ Y) @ X) => (((less_eq @ X) @ Y) = (X = Y)))))).
% 1.61/1.07 thf(fact_50_local_Odual__order_Oantisym, axiom, (! [B2:a,A2:a]: ((((less_eq @ B2) @ A2) => (((less_eq @ A2) @ B2) => (A2 = B2)))))).
% 1.61/1.07 thf(fact_51_local_Odual__order_Oeq__iff, axiom, ((^ [Y3:a,Z3:a]: ((Y3 = Z3))) = (^ [A4:a,B3:a]: ((((less_eq @ B3) @ A4) & ((less_eq @ A4) @ B3)))))).
% 1.61/1.07 thf(fact_52_local_Odual__order_Otrans, axiom, (! [B2:a,A2:a,C:a]: ((((less_eq @ B2) @ A2) => (((less_eq @ C) @ B2) => ((less_eq @ C) @ A2)))))).
% 1.61/1.07 thf(fact_53_local_Oeq__iff, axiom, ((^ [Y3:a,Z3:a]: ((Y3 = Z3))) = (^ [X3:a,Y4:a]: ((((less_eq @ X3) @ Y4) & ((less_eq @ Y4) @ X3)))))).
% 1.61/1.07 thf(fact_54_local_Oeq__refl, axiom, (! [X:a,Y:a]: (((X = Y) => ((less_eq @ X) @ Y))))).
% 1.61/1.07 thf(fact_55_local_Oord__eq__le__trans, axiom, (! [A2:a,B2:a,C:a]: (((A2 = B2) => (((less_eq @ B2) @ C) => ((less_eq @ A2) @ C)))))).
% 1.61/1.07 thf(fact_56_local_Oord__le__eq__trans, axiom, (! [A2:a,B2:a,C:a]: ((((less_eq @ A2) @ B2) => ((B2 = C) => ((less_eq @ A2) @ C)))))).
% 1.61/1.07 thf(fact_57_local_Oorder_Oantisym, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) => (((less_eq @ B2) @ A2) => (A2 = B2)))))).
% 1.61/1.07 thf(fact_58_local_Oorder_Oeq__iff, axiom, ((^ [Y3:a,Z3:a]: ((Y3 = Z3))) = (^ [A4:a,B3:a]: ((((less_eq @ A4) @ B3) & ((less_eq @ B3) @ A4)))))).
% 1.61/1.07 thf(fact_59_local_Oorder_Otrans, axiom, (! [A2:a,B2:a,C:a]: ((((less_eq @ A2) @ B2) => (((less_eq @ B2) @ C) => ((less_eq @ A2) @ C)))))).
% 1.61/1.07 thf(fact_60_local_Oorder__trans, axiom, (! [X:a,Y:a,Z:a]: ((((less_eq @ X) @ Y) => (((less_eq @ Y) @ Z) => ((less_eq @ X) @ Z)))))).
% 1.61/1.07 thf(fact_61_local_Ole__infI2, axiom, (! [B2:a,X:a,A2:a]: ((((less_eq @ B2) @ X) => ((less_eq @ ((inf @ A2) @ B2)) @ X))))).
% 1.61/1.07 thf(fact_62_local_Ole__infI1, axiom, (! [A2:a,X:a,B2:a]: ((((less_eq @ A2) @ X) => ((less_eq @ ((inf @ A2) @ B2)) @ X))))).
% 1.61/1.07 thf(fact_63_local_Ole__infI, axiom, (! [X:a,A2:a,B2:a]: ((((less_eq @ X) @ A2) => (((less_eq @ X) @ B2) => ((less_eq @ X) @ ((inf @ A2) @ B2))))))).
% 1.61/1.07 thf(fact_64_local_Ole__infE, axiom, (! [X:a,A2:a,B2:a]: ((((less_eq @ X) @ ((inf @ A2) @ B2)) => (~ ((((less_eq @ X) @ A2) => (~ (((less_eq @ X) @ B2)))))))))).
% 1.61/1.07 thf(fact_65_local_Ole__iff__inf, axiom, (! [X:a,Y:a]: ((((less_eq @ X) @ Y) = (((inf @ X) @ Y) = X))))).
% 1.61/1.07 thf(fact_66_local_Oinf__unique, axiom, (! [F:(a > (a > a)),X:a,Y:a]: (((! [X2:a,Y2:a]: (((less_eq @ ((F @ X2) @ Y2)) @ X2))) => ((! [X2:a,Y2:a]: (((less_eq @ ((F @ X2) @ Y2)) @ Y2))) => ((! [X2:a,Y2:a,Z2:a]: ((((less_eq @ X2) @ Y2) => (((less_eq @ X2) @ Z2) => ((less_eq @ X2) @ ((F @ Y2) @ Z2)))))) => (((inf @ X) @ Y) = ((F @ X) @ Y)))))))).
% 1.61/1.07 thf(fact_67_local_Oinf__mono, axiom, (! [A2:a,C:a,B2:a,D:a]: ((((less_eq @ A2) @ C) => (((less_eq @ B2) @ D) => ((less_eq @ ((inf @ A2) @ B2)) @ ((inf @ C) @ D))))))).
% 1.61/1.07 thf(fact_68_local_Oinf__le2, axiom, (! [X:a,Y:a]: (((less_eq @ ((inf @ X) @ Y)) @ Y)))).
% 1.61/1.07 thf(fact_69_local_Oinf__le1, axiom, (! [X:a,Y:a]: (((less_eq @ ((inf @ X) @ Y)) @ X)))).
% 1.61/1.07 thf(fact_70_local_Oinf__greatest, axiom, (! [X:a,Y:a,Z:a]: ((((less_eq @ X) @ Y) => (((less_eq @ X) @ Z) => ((less_eq @ X) @ ((inf @ Y) @ Z))))))).
% 1.61/1.07 thf(fact_71_local_Oinf__absorb2, axiom, (! [Y:a,X:a]: ((((less_eq @ Y) @ X) => (((inf @ X) @ Y) = Y))))).
% 1.61/1.07 thf(fact_72_local_Oinf__absorb1, axiom, (! [X:a,Y:a]: ((((less_eq @ X) @ Y) => (((inf @ X) @ Y) = X))))).
% 1.61/1.07 thf(fact_73_local_Oinf_Oorder__iff, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) = (A2 = ((inf @ A2) @ B2)))))).
% 1.61/1.07 thf(fact_74_local_Oinf_OorderI, axiom, (! [A2:a,B2:a]: (((A2 = ((inf @ A2) @ B2)) => ((less_eq @ A2) @ B2))))).
% 1.61/1.07 thf(fact_75_local_Oinf_OorderE, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) => (A2 = ((inf @ A2) @ B2)))))).
% 1.61/1.07 thf(fact_76_local_Oinf_OcoboundedI2, axiom, (! [B2:a,C:a,A2:a]: ((((less_eq @ B2) @ C) => ((less_eq @ ((inf @ A2) @ B2)) @ C))))).
% 1.61/1.07 thf(fact_77_local_Oinf_OcoboundedI1, axiom, (! [A2:a,C:a,B2:a]: ((((less_eq @ A2) @ C) => ((less_eq @ ((inf @ A2) @ B2)) @ C))))).
% 1.61/1.07 thf(fact_78_local_Oinf_Ocobounded2, axiom, (! [A2:a,B2:a]: (((less_eq @ ((inf @ A2) @ B2)) @ B2)))).
% 1.61/1.07 thf(fact_79_local_Oinf_Ocobounded1, axiom, (! [A2:a,B2:a]: (((less_eq @ ((inf @ A2) @ B2)) @ A2)))).
% 1.61/1.07 thf(fact_80_local_Oinf_OboundedI, axiom, (! [A2:a,B2:a,C:a]: ((((less_eq @ A2) @ B2) => (((less_eq @ A2) @ C) => ((less_eq @ A2) @ ((inf @ B2) @ C))))))).
% 1.61/1.07 thf(fact_81_local_Oinf_OboundedE, axiom, (! [A2:a,B2:a,C:a]: ((((less_eq @ A2) @ ((inf @ B2) @ C)) => (~ ((((less_eq @ A2) @ B2) => (~ (((less_eq @ A2) @ C)))))))))).
% 1.61/1.07 thf(fact_82_local_Oinf_Oabsorb__iff2, axiom, (! [B2:a,A2:a]: ((((less_eq @ B2) @ A2) = (((inf @ A2) @ B2) = B2))))).
% 1.61/1.07 thf(fact_83_local_Oinf_Oabsorb__iff1, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) = (((inf @ A2) @ B2) = A2))))).
% 1.61/1.07 thf(fact_84_local_Oinf_Oabsorb2, axiom, (! [B2:a,A2:a]: ((((less_eq @ B2) @ A2) => (((inf @ A2) @ B2) = B2))))).
% 1.61/1.07 thf(fact_85_local_Oinf_Oabsorb1, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) => (((inf @ A2) @ B2) = A2))))).
% 1.61/1.07 thf(fact_86_local_Osup__unique, axiom, (! [F:(a > (a > a)),X:a,Y:a]: (((! [X2:a,Y2:a]: (((less_eq @ X2) @ ((F @ X2) @ Y2)))) => ((! [X2:a,Y2:a]: (((less_eq @ Y2) @ ((F @ X2) @ Y2)))) => ((! [X2:a,Y2:a,Z2:a]: ((((less_eq @ Y2) @ X2) => (((less_eq @ Z2) @ X2) => ((less_eq @ ((F @ Y2) @ Z2)) @ X2))))) => (((sup @ X) @ Y) = ((F @ X) @ Y)))))))).
% 1.61/1.07 thf(fact_87_local_Osup__mono, axiom, (! [A2:a,C:a,B2:a,D:a]: ((((less_eq @ A2) @ C) => (((less_eq @ B2) @ D) => ((less_eq @ ((sup @ A2) @ B2)) @ ((sup @ C) @ D))))))).
% 1.61/1.07 thf(fact_88_local_Osup__least, axiom, (! [Y:a,X:a,Z:a]: ((((less_eq @ Y) @ X) => (((less_eq @ Z) @ X) => ((less_eq @ ((sup @ Y) @ Z)) @ X)))))).
% 1.61/1.07 thf(fact_89_local_Osup__ge2, axiom, (! [Y:a,X:a]: (((less_eq @ Y) @ ((sup @ X) @ Y))))).
% 1.61/1.07 thf(fact_90_local_Osup__ge1, axiom, (! [X:a,Y:a]: (((less_eq @ X) @ ((sup @ X) @ Y))))).
% 1.61/1.07 thf(fact_91_local_Osup__absorb2, axiom, (! [X:a,Y:a]: ((((less_eq @ X) @ Y) => (((sup @ X) @ Y) = Y))))).
% 1.61/1.07 thf(fact_92_local_Osup__absorb1, axiom, (! [Y:a,X:a]: ((((less_eq @ Y) @ X) => (((sup @ X) @ Y) = X))))).
% 1.61/1.07 thf(fact_93_local_Osup_Oorder__iff, axiom, (! [B2:a,A2:a]: ((((less_eq @ B2) @ A2) = (A2 = ((sup @ A2) @ B2)))))).
% 1.61/1.07 thf(fact_94_local_Osup_OorderI, axiom, (! [A2:a,B2:a]: (((A2 = ((sup @ A2) @ B2)) => ((less_eq @ B2) @ A2))))).
% 1.61/1.07 thf(fact_95_local_Osup_OorderE, axiom, (! [B2:a,A2:a]: ((((less_eq @ B2) @ A2) => (A2 = ((sup @ A2) @ B2)))))).
% 1.61/1.07 thf(fact_96_local_Osup_Omono, axiom, (! [C:a,A2:a,D:a,B2:a]: ((((less_eq @ C) @ A2) => (((less_eq @ D) @ B2) => ((less_eq @ ((sup @ C) @ D)) @ ((sup @ A2) @ B2))))))).
% 1.61/1.07 thf(fact_97_local_Osup_OcoboundedI2, axiom, (! [C:a,B2:a,A2:a]: ((((less_eq @ C) @ B2) => ((less_eq @ C) @ ((sup @ A2) @ B2)))))).
% 1.61/1.07 thf(fact_98_local_Osup_OcoboundedI1, axiom, (! [C:a,A2:a,B2:a]: ((((less_eq @ C) @ A2) => ((less_eq @ C) @ ((sup @ A2) @ B2)))))).
% 1.61/1.07 thf(fact_99_local_Osup_Ocobounded2, axiom, (! [B2:a,A2:a]: (((less_eq @ B2) @ ((sup @ A2) @ B2))))).
% 1.61/1.07 thf(fact_100_local_Osup_Ocobounded1, axiom, (! [A2:a,B2:a]: (((less_eq @ A2) @ ((sup @ A2) @ B2))))).
% 1.61/1.07 thf(fact_101_local_Osup_OboundedI, axiom, (! [B2:a,A2:a,C:a]: ((((less_eq @ B2) @ A2) => (((less_eq @ C) @ A2) => ((less_eq @ ((sup @ B2) @ C)) @ A2)))))).
% 1.61/1.07 thf(fact_102_local_Osup_OboundedE, axiom, (! [B2:a,C:a,A2:a]: ((((less_eq @ ((sup @ B2) @ C)) @ A2) => (~ ((((less_eq @ B2) @ A2) => (~ (((less_eq @ C) @ A2)))))))))).
% 1.61/1.07 thf(fact_103_local_Osup_Oabsorb__iff2, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) = (((sup @ A2) @ B2) = B2))))).
% 1.61/1.07 thf(fact_104_local_Osup_Oabsorb__iff1, axiom, (! [B2:a,A2:a]: ((((less_eq @ B2) @ A2) = (((sup @ A2) @ B2) = A2))))).
% 1.61/1.07 thf(fact_105_local_Osup_Oabsorb2, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) => (((sup @ A2) @ B2) = B2))))).
% 1.61/1.07 thf(fact_106_local_Osup_Oabsorb1, axiom, (! [B2:a,A2:a]: ((((less_eq @ B2) @ A2) => (((sup @ A2) @ B2) = A2))))).
% 1.61/1.07 thf(fact_107_local_Ole__supI2, axiom, (! [X:a,B2:a,A2:a]: ((((less_eq @ X) @ B2) => ((less_eq @ X) @ ((sup @ A2) @ B2)))))).
% 1.61/1.07 thf(fact_108_local_Ole__supI1, axiom, (! [X:a,A2:a,B2:a]: ((((less_eq @ X) @ A2) => ((less_eq @ X) @ ((sup @ A2) @ B2)))))).
% 1.61/1.07 thf(fact_109_local_Ole__supI, axiom, (! [A2:a,X:a,B2:a]: ((((less_eq @ A2) @ X) => (((less_eq @ B2) @ X) => ((less_eq @ ((sup @ A2) @ B2)) @ X)))))).
% 1.61/1.07 thf(fact_110_local_Ole__supE, axiom, (! [A2:a,B2:a,X:a]: ((((less_eq @ ((sup @ A2) @ B2)) @ X) => (~ ((((less_eq @ A2) @ X) => (~ (((less_eq @ B2) @ X)))))))))).
% 1.61/1.07 thf(fact_111_local_Ole__iff__sup, axiom, (! [X:a,Y:a]: ((((less_eq @ X) @ Y) = (((sup @ X) @ Y) = Y))))).
% 1.61/1.07 thf(fact_112_local_Oorder_Orefl, axiom, (! [A2:a]: (((less_eq @ A2) @ A2)))).
% 1.61/1.07 thf(fact_113_local_Oorder__refl, axiom, (! [X:a]: (((less_eq @ X) @ X)))).
% 1.61/1.07 thf(fact_114_local_OGreatestI2__order, axiom, (! [P:(a > $o),X:a,Q:(a > $o)]: (((P @ X) => ((! [Y2:a]: (((P @ Y2) => ((less_eq @ Y2) @ X)))) => ((! [X2:a]: (((P @ X2) => ((! [Y5:a]: (((P @ Y5) => ((less_eq @ Y5) @ X2)))) => (Q @ X2))))) => (Q @ (((greatest @ a) @ less_eq) @ P)))))))).
% 1.61/1.07 thf(fact_115_local_OGreatest__equality, axiom, (! [P:(a > $o),X:a]: (((P @ X) => ((! [Y2:a]: (((P @ Y2) => ((less_eq @ Y2) @ X)))) => ((((greatest @ a) @ less_eq) @ P) = X)))))).
% 1.61/1.07 thf(fact_116_local_Omax__def, axiom, (! [A2:a,B2:a]: (((((less_eq @ A2) @ B2) => (((((max @ a) @ less_eq) @ A2) @ B2) = B2)) & ((~ (((less_eq @ A2) @ B2))) => (((((max @ a) @ less_eq) @ A2) @ B2) = A2)))))).
% 1.61/1.07 thf(fact_117_local_Omin__def, axiom, (! [A2:a,B2:a]: (((((less_eq @ A2) @ B2) => (((((min @ a) @ less_eq) @ A2) @ B2) = A2)) & ((~ (((less_eq @ A2) @ B2))) => (((((min @ a) @ less_eq) @ A2) @ B2) = B2)))))).
% 1.61/1.07 thf(fact_118_local_Ole__inf__iff, axiom, (! [X:a,Y:a,Z:a]: ((((less_eq @ X) @ ((inf @ Y) @ Z)) = (((less_eq @ X) @ Y) & ((less_eq @ X) @ Z)))))).
% 1.61/1.07 thf(fact_119_local_Oinf_Obounded__iff, axiom, (! [A2:a,B2:a,C:a]: ((((less_eq @ A2) @ ((inf @ B2) @ C)) = (((less_eq @ A2) @ B2) & ((less_eq @ A2) @ C)))))).
% 1.61/1.07 thf(fact_120_local_Osup_Obounded__iff, axiom, (! [B2:a,C:a,A2:a]: ((((less_eq @ ((sup @ B2) @ C)) @ A2) = (((less_eq @ B2) @ A2) & ((less_eq @ C) @ A2)))))).
% 1.61/1.07 thf(fact_121_local_Ole__sup__iff, axiom, (! [X:a,Y:a,Z:a]: ((((less_eq @ ((sup @ X) @ Y)) @ Z) = (((less_eq @ X) @ Z) & ((less_eq @ Y) @ Z)))))).
% 1.61/1.07 thf(fact_122_local_OLeast1I, axiom, (! [P:(a > $o)]: (((? [X4:a]: (((P @ X4) & ((! [Y2:a]: (((P @ Y2) => ((less_eq @ X4) @ Y2)))) & (! [Y2:a]: ((((P @ Y2) & (! [Ya:a]: (((P @ Ya) => ((less_eq @ Y2) @ Ya))))) => (Y2 = X4)))))))) => (P @ (((least @ a) @ less_eq) @ P)))))).
% 1.61/1.07 thf(fact_123_local_OLeast1__le, axiom, (! [P:(a > $o),Z:a]: (((? [X4:a]: (((P @ X4) & ((! [Y2:a]: (((P @ Y2) => ((less_eq @ X4) @ Y2)))) & (! [Y2:a]: ((((P @ Y2) & (! [Ya:a]: (((P @ Ya) => ((less_eq @ Y2) @ Ya))))) => (Y2 = X4)))))))) => ((P @ Z) => ((less_eq @ (((least @ a) @ less_eq) @ P)) @ Z)))))).
% 1.61/1.07 thf(fact_124_local_OLeastI2__order, axiom, (! [P:(a > $o),X:a,Q:(a > $o)]: (((P @ X) => ((! [Y2:a]: (((P @ Y2) => ((less_eq @ X) @ Y2)))) => ((! [X2:a]: (((P @ X2) => ((! [Y5:a]: (((P @ Y5) => ((less_eq @ X2) @ Y5)))) => (Q @ X2))))) => (Q @ (((least @ a) @ less_eq) @ P)))))))).
% 1.61/1.07 thf(fact_125_local_OLeast__equality, axiom, (! [P:(a > $o),X:a]: (((P @ X) => ((! [Y2:a]: (((P @ Y2) => ((less_eq @ X) @ Y2)))) => ((((least @ a) @ less_eq) @ P) = X)))))).
% 1.61/1.07 thf(fact_126_lattice_Od__aux_Ocong, axiom, (! [A:$tType]: (((modula1221160330_d_aux @ A) = (modula1221160330_d_aux @ A))))).
% 1.61/1.07 thf(fact_127_abel__semigroup_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A))]: ((((abel_semigroup @ A) @ F) => ((semigroup @ A) @ F))))).
% 1.61/1.07 thf(fact_128_semilattice__set__def, axiom, (! [A:$tType]: (((lattic35693393ce_set @ A) = (semilattice @ A))))).
% 1.61/1.07 thf(fact_129_semilattice__set_Ointro, axiom, (! [A:$tType,F:(A > (A > A))]: ((((semilattice @ A) @ F) => ((lattic35693393ce_set @ A) @ F))))).
% 1.61/1.07 thf(fact_130_semilattice__set_Oaxioms, axiom, (! [A:$tType,F:(A > (A > A))]: ((((lattic35693393ce_set @ A) @ F) => ((semilattice @ A) @ F))))).
% 1.61/1.07 thf(fact_131_semilattice_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A))]: ((((semilattice @ A) @ F) => ((abel_semigroup @ A) @ F))))).
% 1.61/1.07 thf(fact_132_local_Obdd__above__def, axiom, (! [A3:(set @ a)]: (((((condit2040224947_above @ a) @ less_eq) @ A3) = (? [M:a]: ((! [X3:a]: (((((member @ a) @ X3) @ A3) => ((less_eq @ X3) @ M)))))))))).
% 1.61/1.07 thf(fact_133_local_Obdd__below__def, axiom, (! [A3:(set @ a)]: (((((condit1201339847_below @ a) @ less_eq) @ A3) = (? [M2:a]: ((! [X3:a]: (((((member @ a) @ X3) @ A3) => ((less_eq @ M2) @ X3)))))))))).
% 1.61/1.07 thf(fact_134_local_Otransp__le, axiom, ((transp @ a) @ less_eq)).
% 1.61/1.07 thf(fact_135_local_Obdd__belowI, axiom, (! [A3:(set @ a),M3:a]: (((! [X2:a]: (((((member @ a) @ X2) @ A3) => ((less_eq @ M3) @ X2)))) => (((condit1201339847_below @ a) @ less_eq) @ A3))))).
% 1.61/1.07 thf(fact_136_local_Obdd__aboveI, axiom, (! [A3:(set @ a),M4:a]: (((! [X2:a]: (((((member @ a) @ X2) @ A3) => ((less_eq @ X2) @ M4)))) => (((condit2040224947_above @ a) @ less_eq) @ A3))))).
% 1.61/1.07 thf(fact_137_abel__semigroup_Oleft__commute, axiom, (! [A:$tType,F:(A > (A > A)),B2:A,A2:A,C:A]: ((((abel_semigroup @ A) @ F) => (((F @ B2) @ ((F @ A2) @ C)) = ((F @ A2) @ ((F @ B2) @ C))))))).
% 1.61/1.07 thf(fact_138_semilattice_Oright__idem, axiom, (! [A:$tType,F:(A > (A > A)),A2:A,B2:A]: ((((semilattice @ A) @ F) => (((F @ ((F @ A2) @ B2)) @ B2) = ((F @ A2) @ B2)))))).
% 1.61/1.07 thf(fact_139_semilattice_Oleft__idem, axiom, (! [A:$tType,F:(A > (A > A)),A2:A,B2:A]: ((((semilattice @ A) @ F) => (((F @ A2) @ ((F @ A2) @ B2)) = ((F @ A2) @ B2)))))).
% 1.61/1.07 thf(fact_140_abel__semigroup_Ocommute, axiom, (! [A:$tType,F:(A > (A > A)),A2:A,B2:A]: ((((abel_semigroup @ A) @ F) => (((F @ A2) @ B2) = ((F @ B2) @ A2)))))).
% 1.61/1.07 thf(fact_141_semilattice_Oidem, axiom, (! [A:$tType,F:(A > (A > A)),A2:A]: ((((semilattice @ A) @ F) => (((F @ A2) @ A2) = A2))))).
% 1.61/1.07 thf(fact_142_semigroup_Ointro, axiom, (! [A:$tType,F:(A > (A > A))]: (((! [A5:A,B4:A,C2:A]: ((((F @ ((F @ A5) @ B4)) @ C2) = ((F @ A5) @ ((F @ B4) @ C2))))) => ((semigroup @ A) @ F))))).
% 1.61/1.07 thf(fact_143_semigroup_Oassoc, axiom, (! [A:$tType,F:(A > (A > A)),A2:A,B2:A,C:A]: ((((semigroup @ A) @ F) => (((F @ ((F @ A2) @ B2)) @ C) = ((F @ A2) @ ((F @ B2) @ C))))))).
% 1.61/1.07 thf(fact_144_semigroup__def, axiom, (! [A:$tType]: (((semigroup @ A) = (^ [F2:(A > (A > A))]: ((! [A4:A,B3:A,C3:A]: ((((F2 @ ((F2 @ A4) @ B3)) @ C3) = ((F2 @ A4) @ ((F2 @ B3) @ C3))))))))))).
% 1.61/1.07 thf(fact_145_local_Obdd__below__mono, axiom, (! [B5:(set @ a),A3:(set @ a)]: (((((condit1201339847_below @ a) @ less_eq) @ B5) => ((((ord_less_eq @ (set @ a)) @ A3) @ B5) => (((condit1201339847_below @ a) @ less_eq) @ A3)))))).
% 1.61/1.07 thf(fact_146_local_Obdd__above__mono, axiom, (! [B5:(set @ a),A3:(set @ a)]: (((((condit2040224947_above @ a) @ less_eq) @ B5) => ((((ord_less_eq @ (set @ a)) @ A3) @ B5) => (((condit2040224947_above @ a) @ less_eq) @ A3)))))).
% 1.61/1.07 thf(fact_147_local_Obdd__below__finite, axiom, (! [A3:(set @ a)]: ((((finite_finite2 @ a) @ A3) => (((condit1201339847_below @ a) @ less_eq) @ A3))))).
% 1.61/1.07 thf(fact_148_local_Obdd__above__finite, axiom, (! [A3:(set @ a)]: ((((finite_finite2 @ a) @ A3) => (((condit2040224947_above @ a) @ less_eq) @ A3))))).
% 1.61/1.07 thf(fact_149_local_Ofinite__has__maximal2, axiom, (! [A3:(set @ a),A2:a]: ((((finite_finite2 @ a) @ A3) => ((((member @ a) @ A2) @ A3) => (? [X2:a]: (((((member @ a) @ X2) @ A3) & (((less_eq @ A2) @ X2) & (! [Xa:a]: (((((member @ a) @ Xa) @ A3) => (((less_eq @ X2) @ Xa) => (X2 = Xa)))))))))))))).
% 1.61/1.07 thf(fact_150_local_Ofinite__has__minimal2, axiom, (! [A3:(set @ a),A2:a]: ((((finite_finite2 @ a) @ A3) => ((((member @ a) @ A2) @ A3) => (? [X2:a]: (((((member @ a) @ X2) @ A3) & (((less_eq @ X2) @ A2) & (! [Xa:a]: (((((member @ a) @ Xa) @ A3) => (((less_eq @ Xa) @ X2) => (X2 = Xa)))))))))))))).
% 1.61/1.07 thf(fact_151_local_OantimonoD, axiom, (! [B:$tType]: (((order @ B) => (! [F:(a > B),X:a,Y:a]: ((((((antimono @ a) @ B) @ less_eq) @ F) => (((less_eq @ X) @ Y) => (((ord_less_eq @ B) @ (F @ Y)) @ (F @ X)))))))))).
% 1.61/1.07 thf(fact_152_local_OantimonoE, axiom, (! [B:$tType]: (((order @ B) => (! [F:(a > B),X:a,Y:a]: ((((((antimono @ a) @ B) @ less_eq) @ F) => (((less_eq @ X) @ Y) => (((ord_less_eq @ B) @ (F @ Y)) @ (F @ X)))))))))).
% 1.61/1.07 thf(fact_153_local_OantimonoI, axiom, (! [B:$tType]: (((order @ B) => (! [F:(a > B)]: (((! [X2:a,Y2:a]: ((((less_eq @ X2) @ Y2) => (((ord_less_eq @ B) @ (F @ Y2)) @ (F @ X2))))) => ((((antimono @ a) @ B) @ less_eq) @ F)))))))).
% 1.61/1.07 thf(fact_154_local_Oantimono__def, axiom, (! [B:$tType]: (((order @ B) => (! [F:(a > B)]: ((((((antimono @ a) @ B) @ less_eq) @ F) = (! [X3:a,Y4:a]: ((((less_eq @ X3) @ Y4) => (((ord_less_eq @ B) @ (F @ Y4)) @ (F @ X3)))))))))))).
% 1.61/1.07 thf(fact_155_local_OmonoD, axiom, (! [B:$tType]: (((order @ B) => (! [F:(a > B),X:a,Y:a]: ((((((mono @ a) @ B) @ less_eq) @ F) => (((less_eq @ X) @ Y) => (((ord_less_eq @ B) @ (F @ X)) @ (F @ Y)))))))))).
% 1.61/1.07 thf(fact_156_local_OmonoE, axiom, (! [B:$tType]: (((order @ B) => (! [F:(a > B),X:a,Y:a]: ((((((mono @ a) @ B) @ less_eq) @ F) => (((less_eq @ X) @ Y) => (((ord_less_eq @ B) @ (F @ X)) @ (F @ Y)))))))))).
% 1.61/1.07 thf(fact_157_local_OmonoI, axiom, (! [B:$tType]: (((order @ B) => (! [F:(a > B)]: (((! [X2:a,Y2:a]: ((((less_eq @ X2) @ Y2) => (((ord_less_eq @ B) @ (F @ X2)) @ (F @ Y2))))) => ((((mono @ a) @ B) @ less_eq) @ F)))))))).
% 1.61/1.07 thf(fact_158_local_Omono__def, axiom, (! [B:$tType]: (((order @ B) => (! [F:(a > B)]: ((((((mono @ a) @ B) @ less_eq) @ F) = (! [X3:a,Y4:a]: ((((less_eq @ X3) @ Y4) => (((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y4)))))))))))).
% 1.61/1.07 thf(fact_159_local_OSup__fin_OcoboundedI, axiom, (! [A3:(set @ a),A2:a]: ((((finite_finite2 @ a) @ A3) => ((((member @ a) @ A2) @ A3) => ((less_eq @ A2) @ (((lattic1039401930up_fin @ a) @ sup) @ A3))))))).
% 1.61/1.07 thf(fact_160_local_OInf__fin_OcoboundedI, axiom, (! [A3:(set @ a),A2:a]: ((((finite_finite2 @ a) @ A3) => ((((member @ a) @ A2) @ A3) => ((less_eq @ (((lattic1263571978nf_fin @ a) @ inf) @ A3)) @ A2)))))).
% 1.61/1.07 thf(fact_161_local_OSup__fin_Oin__idem, axiom, (! [A3:(set @ a),X:a]: ((((finite_finite2 @ a) @ A3) => ((((member @ a) @ X) @ A3) => (((sup @ X) @ (((lattic1039401930up_fin @ a) @ sup) @ A3)) = (((lattic1039401930up_fin @ a) @ sup) @ A3))))))).
% 1.61/1.07 thf(fact_162_local_OInf__fin_Oin__idem, axiom, (! [A3:(set @ a),X:a]: ((((finite_finite2 @ a) @ A3) => ((((member @ a) @ X) @ A3) => (((inf @ X) @ (((lattic1263571978nf_fin @ a) @ inf) @ A3)) = (((lattic1263571978nf_fin @ a) @ inf) @ A3))))))).
% 1.61/1.07 thf(fact_163_local_Oinf__Sup__absorb, axiom, (! [A3:(set @ a),A2:a]: ((((finite_finite2 @ a) @ A3) => ((((member @ a) @ A2) @ A3) => (((inf @ A2) @ (((lattic1039401930up_fin @ a) @ sup) @ A3)) = A2)))))).
% 1.61/1.07 thf(fact_164_local_Osup__Inf__absorb, axiom, (! [A3:(set @ a),A2:a]: ((((finite_finite2 @ a) @ A3) => ((((member @ a) @ A2) @ A3) => (((sup @ (((lattic1263571978nf_fin @ a) @ inf) @ A3)) @ A2) = A2)))))).
% 1.61/1.07 thf(fact_165_semilattice__inf_OInf__fin_Ocong, axiom, (! [A:$tType]: (((lattic1263571978nf_fin @ A) = (lattic1263571978nf_fin @ A))))).
% 1.61/1.07 thf(fact_166_semilattice__sup_OSup__fin_Ocong, axiom, (! [A:$tType]: (((lattic1039401930up_fin @ A) = (lattic1039401930up_fin @ A))))).
% 1.61/1.07 thf(fact_167_local_OInf__fin__le__Sup__fin, axiom, (! [A3:(set @ a)]: ((((finite_finite2 @ a) @ A3) => ((A3 != (bot_bot @ (set @ a))) => ((less_eq @ (((lattic1263571978nf_fin @ a) @ inf) @ A3)) @ (((lattic1039401930up_fin @ a) @ sup) @ A3))))))).
% 1.61/1.07 thf(fact_168_local_OSup__fin_Osubset__imp, axiom, (! [A3:(set @ a),B5:(set @ a)]: (((((ord_less_eq @ (set @ a)) @ A3) @ B5) => ((A3 != (bot_bot @ (set @ a))) => (((finite_finite2 @ a) @ B5) => ((less_eq @ (((lattic1039401930up_fin @ a) @ sup) @ A3)) @ (((lattic1039401930up_fin @ a) @ sup) @ B5)))))))).
% 1.61/1.07 thf(fact_169_local_OInf__fin_Osubset__imp, axiom, (! [A3:(set @ a),B5:(set @ a)]: (((((ord_less_eq @ (set @ a)) @ A3) @ B5) => ((A3 != (bot_bot @ (set @ a))) => (((finite_finite2 @ a) @ B5) => ((less_eq @ (((lattic1263571978nf_fin @ a) @ inf) @ B5)) @ (((lattic1263571978nf_fin @ a) @ inf) @ A3)))))))).
% 1.61/1.07 thf(fact_170_local_Ofinite__has__minimal, axiom, (! [A3:(set @ a)]: ((((finite_finite2 @ a) @ A3) => ((A3 != (bot_bot @ (set @ a))) => (? [X2:a]: (((((member @ a) @ X2) @ A3) & (! [Xa:a]: (((((member @ a) @ Xa) @ A3) => (((less_eq @ Xa) @ X2) => (X2 = Xa))))))))))))).
% 1.61/1.07 thf(fact_171_local_Ofinite__has__maximal, axiom, (! [A3:(set @ a)]: ((((finite_finite2 @ a) @ A3) => ((A3 != (bot_bot @ (set @ a))) => (? [X2:a]: (((((member @ a) @ X2) @ A3) & (! [Xa:a]: (((((member @ a) @ Xa) @ A3) => (((less_eq @ X2) @ Xa) => (X2 = Xa))))))))))))).
% 1.61/1.07 thf(fact_172_local_OInf__fin_Osubset, axiom, (! [A3:(set @ a),B5:(set @ a)]: ((((finite_finite2 @ a) @ A3) => ((B5 != (bot_bot @ (set @ a))) => ((((ord_less_eq @ (set @ a)) @ B5) @ A3) => (((inf @ (((lattic1263571978nf_fin @ a) @ inf) @ B5)) @ (((lattic1263571978nf_fin @ a) @ inf) @ A3)) = (((lattic1263571978nf_fin @ a) @ inf) @ A3)))))))).
% 1.61/1.07 thf(fact_173_local_OSup__fin_Osubset, axiom, (! [A3:(set @ a),B5:(set @ a)]: ((((finite_finite2 @ a) @ A3) => ((B5 != (bot_bot @ (set @ a))) => ((((ord_less_eq @ (set @ a)) @ B5) @ A3) => (((sup @ (((lattic1039401930up_fin @ a) @ sup) @ B5)) @ (((lattic1039401930up_fin @ a) @ sup) @ A3)) = (((lattic1039401930up_fin @ a) @ sup) @ A3)))))))).
% 1.61/1.07 thf(fact_174_local_OInf__fin_Obounded__iff, axiom, (! [A3:(set @ a),X:a]: ((((finite_finite2 @ a) @ A3) => ((A3 != (bot_bot @ (set @ a))) => (((less_eq @ X) @ (((lattic1263571978nf_fin @ a) @ inf) @ A3)) = (! [X3:a]: (((((member @ a) @ X3) @ A3) => ((less_eq @ X) @ X3)))))))))).
% 1.61/1.07 thf(fact_175_local_OInf__fin_OboundedI, axiom, (! [A3:(set @ a),X:a]: ((((finite_finite2 @ a) @ A3) => ((A3 != (bot_bot @ (set @ a))) => ((! [A5:a]: (((((member @ a) @ A5) @ A3) => ((less_eq @ X) @ A5)))) => ((less_eq @ X) @ (((lattic1263571978nf_fin @ a) @ inf) @ A3)))))))).
% 1.61/1.07 thf(fact_176_local_OInf__fin_OboundedE, axiom, (! [A3:(set @ a),X:a]: ((((finite_finite2 @ a) @ A3) => ((A3 != (bot_bot @ (set @ a))) => (((less_eq @ X) @ (((lattic1263571978nf_fin @ a) @ inf) @ A3)) => (! [A6:a]: (((((member @ a) @ A6) @ A3) => ((less_eq @ X) @ A6)))))))))).
% 1.61/1.07 thf(fact_177_local_OSup__fin_Obounded__iff, axiom, (! [A3:(set @ a),X:a]: ((((finite_finite2 @ a) @ A3) => ((A3 != (bot_bot @ (set @ a))) => (((less_eq @ (((lattic1039401930up_fin @ a) @ sup) @ A3)) @ X) = (! [X3:a]: (((((member @ a) @ X3) @ A3) => ((less_eq @ X3) @ X)))))))))).
% 1.61/1.07 thf(fact_178_local_OSup__fin_OboundedI, axiom, (! [A3:(set @ a),X:a]: ((((finite_finite2 @ a) @ A3) => ((A3 != (bot_bot @ (set @ a))) => ((! [A5:a]: (((((member @ a) @ A5) @ A3) => ((less_eq @ A5) @ X)))) => ((less_eq @ (((lattic1039401930up_fin @ a) @ sup) @ A3)) @ X))))))).
% 1.61/1.07 thf(fact_179_local_OSup__fin_OboundedE, axiom, (! [A3:(set @ a),X:a]: ((((finite_finite2 @ a) @ A3) => ((A3 != (bot_bot @ (set @ a))) => (((less_eq @ (((lattic1039401930up_fin @ a) @ sup) @ A3)) @ X) => (! [A6:a]: (((((member @ a) @ A6) @ A3) => ((less_eq @ A6) @ X)))))))))).
% 1.61/1.07 thf(fact_180_local_Obdd__above__empty, axiom, (((condit2040224947_above @ a) @ less_eq) @ (bot_bot @ (set @ a)))).
% 1.61/1.07 thf(fact_181_local_Obdd__below__empty, axiom, (((condit1201339847_below @ a) @ less_eq) @ (bot_bot @ (set @ a)))).
% 1.61/1.07 thf(fact_182_local_Onot__empty__eq__Iic__eq__empty, axiom, (! [H:a]: (((bot_bot @ (set @ a)) != (((set_atMost @ a) @ less_eq) @ H))))).
% 1.61/1.07 thf(fact_183_local_Onot__empty__eq__Ici__eq__empty, axiom, (! [L:a]: (((bot_bot @ (set @ a)) != (((set_atLeast @ a) @ less_eq) @ L))))).
% 1.61/1.07 thf(fact_184_subset__empty, axiom, (! [A:$tType,A3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = (A3 = (bot_bot @ (set @ A))))))).
% 1.61/1.07 thf(fact_185_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X3:A]: ((~ ((P @ X3))))))))).
% 1.61/1.07 thf(fact_186_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X3:A]: ((~ ((P @ X3))))))))).
% 1.61/1.07 thf(fact_187_all__not__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X3:A]: ((~ ((((member @ A) @ X3) @ A3))))) = (A3 = (bot_bot @ (set @ A))))))).
% 1.61/1.07 thf(fact_188_empty__iff, axiom, (! [A:$tType,C:A]: ((~ ((((member @ A) @ C) @ (bot_bot @ (set @ A)))))))).
% 1.61/1.07 thf(fact_189_subset__antisym, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ A3) => (A3 = B5)))))).
% 1.61/1.07 thf(fact_190_subsetI, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A)]: (((! [X2:A]: (((((member @ A) @ X2) @ A3) => (((member @ A) @ X2) @ B5)))) => (((ord_less_eq @ (set @ A)) @ A3) @ B5))))).
% 1.61/1.07 thf(fact_191_empty__subsetI, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A3)))).
% 1.61/1.07 thf(fact_192_local_OatLeast__iff, axiom, (! [I:a,K:a]: (((((member @ a) @ I) @ (((set_atLeast @ a) @ less_eq) @ K)) = ((less_eq @ K) @ I))))).
% 1.61/1.07 thf(fact_193_local_OatMost__iff, axiom, (! [I:a,K:a]: (((((member @ a) @ I) @ (((set_atMost @ a) @ less_eq) @ K)) = ((less_eq @ I) @ K))))).
% 1.61/1.07 thf(fact_194_local_Obdd__below__Ici, axiom, (! [A2:a]: ((((condit1201339847_below @ a) @ less_eq) @ (((set_atLeast @ a) @ less_eq) @ A2))))).
% 1.61/1.07 thf(fact_195_local_Obdd__above__Iic, axiom, (! [B2:a]: ((((condit2040224947_above @ a) @ less_eq) @ (((set_atMost @ a) @ less_eq) @ B2))))).
% 1.61/1.07 thf(fact_196_ex__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((? [X3:A]: ((((member @ A) @ X3) @ A3))) = (A3 != (bot_bot @ (set @ A))))))).
% 1.61/1.07 thf(fact_197_equals0I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [Y2:A]: ((~ ((((member @ A) @ Y2) @ A3))))) => (A3 = (bot_bot @ (set @ A))))))).
% 1.61/1.07 thf(fact_198_equals0D, axiom, (! [A:$tType,A3:(set @ A),A2:A]: (((A3 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A3))))))).
% 1.61/1.07 thf(fact_199_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 1.61/1.07 thf(fact_200_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X3:A]: (((P @ X3) => (Q @ X3)))))))).
% 1.61/1.07 thf(fact_201_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y3:(set @ A),Z3:(set @ A)]: ((Y3 = Z3))) = (^ [A7:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B6) & (((ord_less_eq @ (set @ A)) @ B6) @ A7)))))))).
% 1.61/1.07 thf(fact_202_subset__trans, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ C4) => (((ord_less_eq @ (set @ A)) @ A3) @ C4)))))).
% 1.61/1.07 thf(fact_203_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) => (Q @ X2)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 1.61/1.07 thf(fact_204_subset__refl, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ A3)))).
% 1.61/1.07 thf(fact_205_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B6:(set @ A)]: ((! [T:A]: (((((member @ A) @ T) @ A7) => (((member @ A) @ T) @ B6)))))))))).
% 1.61/1.07 thf(fact_206_equalityD2, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A)]: (((A3 = B5) => (((ord_less_eq @ (set @ A)) @ B5) @ A3))))).
% 1.61/1.07 thf(fact_207_equalityD1, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A)]: (((A3 = B5) => (((ord_less_eq @ (set @ A)) @ A3) @ B5))))).
% 1.61/1.07 thf(fact_208_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B6:(set @ A)]: ((! [X3:A]: (((((member @ A) @ X3) @ A7) => (((member @ A) @ X3) @ B6)))))))))).
% 1.61/1.07 thf(fact_209_equalityE, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A)]: (((A3 = B5) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B5) => (~ ((((ord_less_eq @ (set @ A)) @ B5) @ A3)))))))))).
% 1.61/1.07 thf(fact_210_subsetD, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B5) => ((((member @ A) @ C) @ A3) => (((member @ A) @ C) @ B5)))))).
% 1.61/1.07 thf(fact_211_in__mono, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B5) => ((((member @ A) @ X) @ A3) => (((member @ A) @ X) @ B5)))))).
% 1.61/1.07 thf(fact_212_local_OSup__fin_Oclosed, axiom, (! [A3:(set @ a)]: ((((finite_finite2 @ a) @ A3) => ((A3 != (bot_bot @ (set @ a))) => ((! [X2:a,Y2:a]: ((((member @ a) @ ((sup @ X2) @ Y2)) @ (((insert @ a) @ X2) @ (((insert @ a) @ Y2) @ (bot_bot @ (set @ a))))))) => (((member @ a) @ (((lattic1039401930up_fin @ a) @ sup) @ A3)) @ A3))))))).
% 1.61/1.07 thf(fact_213_local_OSup__fin_Oinsert__not__elem, axiom, (! [A3:(set @ a),X:a]: ((((finite_finite2 @ a) @ A3) => ((~ ((((member @ a) @ X) @ A3))) => ((A3 != (bot_bot @ (set @ a))) => ((((lattic1039401930up_fin @ a) @ sup) @ (((insert @ a) @ X) @ A3)) = ((sup @ X) @ (((lattic1039401930up_fin @ a) @ sup) @ A3))))))))).
% 1.61/1.07 thf(fact_214_local_OInf__fin_Oclosed, axiom, (! [A3:(set @ a)]: ((((finite_finite2 @ a) @ A3) => ((A3 != (bot_bot @ (set @ a))) => ((! [X2:a,Y2:a]: ((((member @ a) @ ((inf @ X2) @ Y2)) @ (((insert @ a) @ X2) @ (((insert @ a) @ Y2) @ (bot_bot @ (set @ a))))))) => (((member @ a) @ (((lattic1263571978nf_fin @ a) @ inf) @ A3)) @ A3))))))).
% 1.61/1.07 thf(fact_215_local_OInf__fin_Oinsert__not__elem, axiom, (! [A3:(set @ a),X:a]: ((((finite_finite2 @ a) @ A3) => ((~ ((((member @ a) @ X) @ A3))) => ((A3 != (bot_bot @ (set @ a))) => ((((lattic1263571978nf_fin @ a) @ inf) @ (((insert @ a) @ X) @ A3)) = ((inf @ X) @ (((lattic1263571978nf_fin @ a) @ inf) @ A3))))))))).
% 1.61/1.07 thf(fact_216_singletonI, axiom, (! [A:$tType,A2:A]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 1.61/1.07 thf(fact_217_insert__subset, axiom, (! [A:$tType,X:A,A3:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X) @ A3)) @ B5) = ((((member @ A) @ X) @ B5) & (((ord_less_eq @ (set @ A)) @ A3) @ B5)))))).
% 1.61/1.07 thf(fact_218_singleton__insert__inj__eq_H, axiom, (! [A:$tType,A2:A,A3:(set @ A),B2:A]: ((((((insert @ A) @ A2) @ A3) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = ((A2 = B2) & (((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))))))).
% 1.61/1.07 thf(fact_219_singleton__insert__inj__eq, axiom, (! [A:$tType,B2:A,A2:A,A3:(set @ A)]: ((((((insert @ A) @ B2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ A2) @ A3)) = ((A2 = B2) & (((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))))))).
% 1.61/1.07 thf(fact_220_local_Obdd__above__insert, axiom, (! [A2:a,A3:(set @ a)]: (((((condit2040224947_above @ a) @ less_eq) @ (((insert @ a) @ A2) @ A3)) = (((condit2040224947_above @ a) @ less_eq) @ A3))))).
% 1.61/1.07 thf(fact_221_local_Obdd__below__insert, axiom, (! [A2:a,A3:(set @ a)]: (((((condit1201339847_below @ a) @ less_eq) @ (((insert @ a) @ A2) @ A3)) = (((condit1201339847_below @ a) @ less_eq) @ A3))))).
% 1.61/1.07 thf(fact_222_local_OInf__fin_Osingleton, axiom, (! [X:a]: (((((lattic1263571978nf_fin @ a) @ inf) @ (((insert @ a) @ X) @ (bot_bot @ (set @ a)))) = X)))).
% 1.61/1.07 thf(fact_223_local_OSup__fin_Osingleton, axiom, (! [X:a]: (((((lattic1039401930up_fin @ a) @ sup) @ (((insert @ a) @ X) @ (bot_bot @ (set @ a)))) = X)))).
% 1.61/1.07 thf(fact_224_local_OInf__fin_Oinsert, axiom, (! [A3:(set @ a),X:a]: ((((finite_finite2 @ a) @ A3) => ((A3 != (bot_bot @ (set @ a))) => ((((lattic1263571978nf_fin @ a) @ inf) @ (((insert @ a) @ X) @ A3)) = ((inf @ X) @ (((lattic1263571978nf_fin @ a) @ inf) @ A3)))))))).
% 1.61/1.07 thf(fact_225_local_OSup__fin_Oinsert, axiom, (! [A3:(set @ a),X:a]: ((((finite_finite2 @ a) @ A3) => ((A3 != (bot_bot @ (set @ a))) => ((((lattic1039401930up_fin @ a) @ sup) @ (((insert @ a) @ X) @ A3)) = ((sup @ X) @ (((lattic1039401930up_fin @ a) @ sup) @ A3)))))))).
% 1.61/1.07 thf(fact_226_subset__singletonD, axiom, (! [A:$tType,A3:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) => ((A3 = (bot_bot @ (set @ A))) | (A3 = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))).
% 1.61/1.07 thf(fact_227_subset__singleton__iff, axiom, (! [A:$tType,X5:(set @ A),A2:A]: (((((ord_less_eq @ (set @ A)) @ X5) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = ((X5 = (bot_bot @ (set @ A))) | (X5 = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))))).
% 1.61/1.07 thf(fact_228_singletonD, axiom, (! [A:$tType,B2:A,A2:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) => (B2 = A2))))).
% 1.61/1.07 thf(fact_229_singleton__iff, axiom, (! [A:$tType,B2:A,A2:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = (B2 = A2))))).
% 1.61/1.07 thf(fact_230_doubleton__eq__iff, axiom, (! [A:$tType,A2:A,B2:A,C:A,D:A]: ((((((insert @ A) @ A2) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C) @ (((insert @ A) @ D) @ (bot_bot @ (set @ A))))) = (((A2 = C) & (B2 = D)) | ((A2 = D) & (B2 = C))))))).
% 1.61/1.07 thf(fact_231_insert__not__empty, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((insert @ A) @ A2) @ A3) != (bot_bot @ (set @ A)))))).
% 1.61/1.07 thf(fact_232_singleton__inject, axiom, (! [A:$tType,A2:A,B2:A]: ((((((insert @ A) @ A2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) => (A2 = B2))))).
% 1.61/1.07 thf(fact_233_subset__insertI2, axiom, (! [A:$tType,A3:(set @ A),B5:(set @ A),B2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B5) => (((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ B2) @ B5)))))).
% 1.61/1.07 thf(fact_234_subset__insertI, axiom, (! [A:$tType,B5:(set @ A),A2:A]: ((((ord_less_eq @ (set @ A)) @ B5) @ (((insert @ A) @ A2) @ B5))))).
% 1.61/1.07 thf(fact_235_subset__insert, axiom, (! [A:$tType,X:A,A3:(set @ A),B5:(set @ A)]: (((~ ((((member @ A) @ X) @ A3))) => ((((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ X) @ B5)) = (((ord_less_eq @ (set @ A)) @ A3) @ B5)))))).
% 1.61/1.07 thf(fact_236_insert__mono, axiom, (! [A:$tType,C4:(set @ A),D2:(set @ A),A2:A]: (((((ord_less_eq @ (set @ A)) @ C4) @ D2) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ A2) @ C4)) @ (((insert @ A) @ A2) @ D2)))))).
% 1.61/1.07 thf(fact_237_finite__ranking__induct, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [S:(set @ B),P:((set @ B) > $o),F:(B > A)]: ((((finite_finite2 @ B) @ S) => ((P @ (bot_bot @ (set @ B))) => ((! [X2:B,S2:(set @ B)]: ((((finite_finite2 @ B) @ S2) => ((! [Y5:B]: (((((member @ B) @ Y5) @ S2) => (((ord_less_eq @ A) @ (F @ Y5)) @ (F @ X2))))) => ((P @ S2) => (P @ (((insert @ B) @ X2) @ S2))))))) => (P @ S)))))))))).
% 1.61/1.07 thf(fact_238_local_OatLeastAtMost__singleton_H, axiom, (! [A2:a,B2:a]: (((A2 = B2) => (((((set_atLeastAtMost @ a) @ less_eq) @ A2) @ B2) = (((insert @ a) @ A2) @ (bot_bot @ (set @ a)))))))).
% 1.61/1.07 thf(fact_239_finite__insert, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: ((((finite_finite2 @ A) @ (((insert @ A) @ A2) @ A3)) = ((finite_finite2 @ A) @ A3))))).
% 1.61/1.07 thf(fact_240_finite__code, axiom, (! [A:$tType]: (((finite_finite @ A) => ((finite_finite2 @ A) = (^ [A7:(set @ A)]: ($true))))))).
% 1.61/1.07 thf(fact_241_local_OIcc__eq__Icc, axiom, (! [L:a,H:a,L2:a,H2:a]: (((((((set_atLeastAtMost @ a) @ less_eq) @ L) @ H) = ((((set_atLeastAtMost @ a) @ less_eq) @ L2) @ H2)) = (((L = L2) & (H = H2)) | ((~ (((less_eq @ L) @ H))) & (~ (((less_eq @ L2) @ H2))))))))).
% 1.61/1.07 thf(fact_242_local_OatLeastAtMost__iff, axiom, (! [I:a,L:a,U:a]: (((((member @ a) @ I) @ ((((set_atLeastAtMost @ a) @ less_eq) @ L) @ U)) = (((less_eq @ L) @ I) & ((less_eq @ I) @ U)))))).
% 1.61/1.07 thf(fact_243_local_OatLeastatMost__empty__iff, axiom, (! [A2:a,B2:a]: (((((((set_atLeastAtMost @ a) @ less_eq) @ A2) @ B2) = (bot_bot @ (set @ a))) = (~ (((less_eq @ A2) @ B2))))))).
% 1.61/1.07 thf(fact_244_local_OatLeastatMost__empty__iff2, axiom, (! [A2:a,B2:a]: ((((bot_bot @ (set @ a)) = ((((set_atLeastAtMost @ a) @ less_eq) @ A2) @ B2)) = (~ (((less_eq @ A2) @ B2))))))).
% 1.61/1.07 thf(fact_245_local_Obdd__below__Icc, axiom, (! [A2:a,B2:a]: ((((condit1201339847_below @ a) @ less_eq) @ ((((set_atLeastAtMost @ a) @ less_eq) @ A2) @ B2))))).
% 1.61/1.07 thf(fact_246_local_Obdd__above__Icc, axiom, (! [A2:a,B2:a]: ((((condit2040224947_above @ a) @ less_eq) @ ((((set_atLeastAtMost @ a) @ less_eq) @ A2) @ B2))))).
% 1.61/1.07 thf(fact_247_local_OatLeastatMost__subset__iff, axiom, (! [A2:a,B2:a,C:a,D:a]: (((((ord_less_eq @ (set @ a)) @ ((((set_atLeastAtMost @ a) @ less_eq) @ A2) @ B2)) @ ((((set_atLeastAtMost @ a) @ less_eq) @ C) @ D)) = ((~ (((less_eq @ A2) @ B2))) | (((less_eq @ C) @ A2) & ((less_eq @ B2) @ D))))))).
% 1.61/1.07 thf(fact_248_local_OatLeastAtMost__singleton, axiom, (! [A2:a]: ((((((set_atLeastAtMost @ a) @ less_eq) @ A2) @ A2) = (((insert @ a) @ A2) @ (bot_bot @ (set @ a))))))).
% 1.61/1.07 thf(fact_249_local_OatLeastAtMost__singleton__iff, axiom, (! [A2:a,B2:a,C:a]: (((((((set_atLeastAtMost @ a) @ less_eq) @ A2) @ B2) = (((insert @ a) @ C) @ (bot_bot @ (set @ a)))) = ((A2 = B2) & (B2 = C)))))).
% 1.61/1.07 thf(fact_250_local_OIcc__subset__Ici__iff, axiom, (! [L:a,H:a,L2:a]: (((((ord_less_eq @ (set @ a)) @ ((((set_atLeastAtMost @ a) @ less_eq) @ L) @ H)) @ (((set_atLeast @ a) @ less_eq) @ L2)) = ((~ (((less_eq @ L) @ H))) | ((less_eq @ L2) @ L)))))).
% 1.61/1.07 thf(fact_251_local_OIcc__subset__Iic__iff, axiom, (! [L:a,H:a,H2:a]: (((((ord_less_eq @ (set @ a)) @ ((((set_atLeastAtMost @ a) @ less_eq) @ L) @ H)) @ (((set_atMost @ a) @ less_eq) @ H2)) = ((~ (((less_eq @ L) @ H))) | ((less_eq @ H) @ H2)))))).
% 1.61/1.07 thf(fact_252_finite, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A3:(set @ A)]: (((finite_finite2 @ A) @ A3))))))).
% 1.61/1.07 thf(fact_253_finite__set__choice, axiom, (! [B:$tType,A:$tType,A3:(set @ A),P:(A > (B > $o))]: ((((finite_finite2 @ A) @ A3) => ((! [X2:A]: (((((member @ A) @ X2) @ A3) => (? [X_1:B]: (((P @ X2) @ X_1)))))) => (? [F3:(A > B)]: ((! [X4:A]: (((((member @ A) @ X4) @ A3) => ((P @ X4) @ (F3 @ X4)))))))))))).
% 1.61/1.07 thf(fact_254_comp__fun__idem_Ofun__left__idem, axiom, (! [A:$tType,B:$tType,F:(A > (B > B)),X:A,Z:B]: (((((finite_comp_fun_idem @ A) @ B) @ F) => (((F @ X) @ ((F @ X) @ Z)) = ((F @ X) @ Z)))))).
% 1.61/1.07 thf(fact_255_order__class_Ofinite__has__minimal2, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ A2) @ A3) => (? [X2:A]: (((((member @ A) @ X2) @ A3) & ((((ord_less_eq @ A) @ X2) @ A2) & (! [Xa:A]: (((((member @ A) @ Xa) @ A3) => ((((ord_less_eq @ A) @ Xa) @ X2) => (X2 = Xa))))))))))))))))).
% 1.61/1.07 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A8:$tType,A9:$tType]: ((((finite_finite @ A8) & (finite_finite @ A9)) => (finite_finite @ (A8 > A9)))))).
% 1.61/1.07 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: (((order @ A9) => (order @ (A8 > A9)))))).
% 1.61/1.07 thf(tcon_Set_Oset___Finite__Set_Ofinite_1, axiom, (! [A8:$tType]: (((finite_finite @ A8) => (finite_finite @ (set @ A8)))))).
% 1.61/1.07 thf(tcon_Set_Oset___Orderings_Oorder_2, axiom, (! [A8:$tType]: ((order @ (set @ A8))))).
% 1.61/1.07 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 1.61/1.07 thf(tcon_HOL_Obool___Finite__Set_Ofinite_3, axiom, (finite_finite @ $o)).
% 1.61/1.07 thf(tcon_HOL_Obool___Orderings_Oorder_4, axiom, (order @ $o)).
% 1.61/1.07 thf(conj_0, conjecture, (((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ b) @ c) @ a2) = ((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ a2) @ b) @ c))).
% 1.61/1.07 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.3GJe6d4PX9/DTF2DTF3385.p
% 1.61/1.07 ------------------------
% 1.61/1.08 ---- Cleaned THF ---
% 1.61/1.09 thf(ty_t_Set_Oset,type,
% 1.61/1.09 set: $tType > $tType ).
% 1.61/1.09
% 1.61/1.09 thf(ty_tf_a,type,
% 1.61/1.09 a: $tType ).
% 1.61/1.09
% 1.61/1.09 thf(sy_cl_HOL_Otype,type,
% 1.61/1.09 type:
% 1.61/1.09 !>[A: $tType] : $o ).
% 1.61/1.09
% 1.61/1.09 thf(sy_cl_Orderings_Oorder,type,
% 1.61/1.09 order:
% 1.61/1.09 !>[A: $tType] : $o ).
% 1.61/1.09
% 1.61/1.09 thf(sy_cl_Finite__Set_Ofinite,type,
% 1.61/1.09 finite_finite:
% 1.61/1.09 !>[A: $tType] : $o ).
% 1.61/1.09
% 1.61/1.09 thf(sy_cl_Orderings_Olinorder,type,
% 1.61/1.09 linorder:
% 1.61/1.09 !>[A: $tType] : $o ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above,type,
% 1.61/1.09 condit2040224947_above:
% 1.61/1.09 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below,type,
% 1.61/1.09 condit1201339847_below:
% 1.61/1.09 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Finite__Set_Ocomp__fun__idem,type,
% 1.61/1.09 finite_comp_fun_idem:
% 1.61/1.09 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Finite__Set_Ofinite,type,
% 1.61/1.09 finite_finite2:
% 1.61/1.09 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Groups_Oabel__semigroup,type,
% 1.61/1.09 abel_semigroup:
% 1.61/1.09 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Groups_Osemigroup,type,
% 1.61/1.09 semigroup:
% 1.61/1.09 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Lattices_Osemilattice,type,
% 1.61/1.09 semilattice:
% 1.61/1.09 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Lattices__Big_Osemilattice__inf_OInf__fin,type,
% 1.61/1.09 lattic1263571978nf_fin:
% 1.61/1.09 !>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Lattices__Big_Osemilattice__set,type,
% 1.61/1.09 lattic35693393ce_set:
% 1.61/1.09 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Lattices__Big_Osemilattice__sup_OSup__fin,type,
% 1.61/1.09 lattic1039401930up_fin:
% 1.61/1.09 !>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Od__aux,type,
% 1.61/1.09 modula1221160330_d_aux:
% 1.61/1.09 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oe__aux,type,
% 1.61/1.09 modula1593167561_e_aux:
% 1.61/1.09 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Orderings_Obot__class_Obot,type,
% 1.61/1.09 bot_bot:
% 1.61/1.09 !>[A: $tType] : A ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Orderings_Oord_OLeast,type,
% 1.61/1.09 least:
% 1.61/1.09 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Orderings_Oord_Omax,type,
% 1.61/1.09 max:
% 1.61/1.09 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Orderings_Oord_Omin,type,
% 1.61/1.09 min:
% 1.61/1.09 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 1.61/1.09 ord_less_eq:
% 1.61/1.09 !>[A: $tType] : ( A > A > $o ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Orderings_Oorder_OGreatest,type,
% 1.61/1.09 greatest:
% 1.61/1.09 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Orderings_Oorder_Oantimono,type,
% 1.61/1.09 antimono:
% 1.61/1.09 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Orderings_Oorder_Omono,type,
% 1.61/1.09 mono:
% 1.61/1.09 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Relation_Otransp,type,
% 1.61/1.09 transp:
% 1.61/1.09 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Set_OCollect,type,
% 1.61/1.09 collect:
% 1.61/1.09 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Set_Oinsert,type,
% 1.61/1.09 insert:
% 1.61/1.09 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Set__Interval_Oord_OatLeast,type,
% 1.61/1.09 set_atLeast:
% 1.61/1.09 !>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Set__Interval_Oord_OatLeastAtMost,type,
% 1.61/1.09 set_atLeastAtMost:
% 1.61/1.09 !>[A: $tType] : ( ( A > A > $o ) > A > A > ( set @ A ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_Set__Interval_Oord_OatMost,type,
% 1.61/1.09 set_atMost:
% 1.61/1.09 !>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_c_member,type,
% 1.61/1.09 member:
% 1.61/1.09 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 1.61/1.09
% 1.61/1.09 thf(sy_v_a,type,
% 1.61/1.09 a2: a ).
% 1.61/1.09
% 1.61/1.09 thf(sy_v_b,type,
% 1.61/1.09 b: a ).
% 1.61/1.09
% 1.61/1.09 thf(sy_v_c,type,
% 1.61/1.09 c: a ).
% 1.61/1.09
% 1.61/1.09 thf(sy_v_inf,type,
% 1.61/1.09 inf: a > a > a ).
% 1.61/1.09
% 1.61/1.09 thf(sy_v_less__eq,type,
% 1.61/1.09 less_eq: a > a > $o ).
% 1.61/1.09
% 1.61/1.09 thf(sy_v_sup,type,
% 1.61/1.09 sup: a > a > a ).
% 1.61/1.09
% 1.61/1.09 thf(fact_0_local_Oinf_Oassoc,axiom,
% 1.61/1.09 ! [A2: a,B2: a,C: a] :
% 1.61/1.09 ( ( inf @ ( inf @ A2 @ B2 ) @ C )
% 1.61/1.09 = ( inf @ A2 @ ( inf @ B2 @ C ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_1_local_Oinf_Ocommute,axiom,
% 1.61/1.09 ! [A2: a,B2: a] :
% 1.61/1.09 ( ( inf @ A2 @ B2 )
% 1.61/1.09 = ( inf @ B2 @ A2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_2_local_Oinf_Oleft__commute,axiom,
% 1.61/1.09 ! [B2: a,A2: a,C: a] :
% 1.61/1.09 ( ( inf @ B2 @ ( inf @ A2 @ C ) )
% 1.61/1.09 = ( inf @ A2 @ ( inf @ B2 @ C ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_3_local_Oinf__assoc,axiom,
% 1.61/1.09 ! [X: a,Y: a,Z: a] :
% 1.61/1.09 ( ( inf @ ( inf @ X @ Y ) @ Z )
% 1.61/1.09 = ( inf @ X @ ( inf @ Y @ Z ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_4_local_Oinf__commute,axiom,
% 1.61/1.09 ! [X: a,Y: a] :
% 1.61/1.09 ( ( inf @ X @ Y )
% 1.61/1.09 = ( inf @ Y @ X ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_5_local_Oinf__left__commute,axiom,
% 1.61/1.09 ! [X: a,Y: a,Z: a] :
% 1.61/1.09 ( ( inf @ X @ ( inf @ Y @ Z ) )
% 1.61/1.09 = ( inf @ Y @ ( inf @ X @ Z ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_6_local_Osup_Oassoc,axiom,
% 1.61/1.09 ! [A2: a,B2: a,C: a] :
% 1.61/1.09 ( ( sup @ ( sup @ A2 @ B2 ) @ C )
% 1.61/1.09 = ( sup @ A2 @ ( sup @ B2 @ C ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_7_local_Osup_Ocommute,axiom,
% 1.61/1.09 ! [A2: a,B2: a] :
% 1.61/1.09 ( ( sup @ A2 @ B2 )
% 1.61/1.09 = ( sup @ B2 @ A2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_8_local_Osup_Oleft__commute,axiom,
% 1.61/1.09 ! [B2: a,A2: a,C: a] :
% 1.61/1.09 ( ( sup @ B2 @ ( sup @ A2 @ C ) )
% 1.61/1.09 = ( sup @ A2 @ ( sup @ B2 @ C ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_9_local_Osup__assoc,axiom,
% 1.61/1.09 ! [X: a,Y: a,Z: a] :
% 1.61/1.09 ( ( sup @ ( sup @ X @ Y ) @ Z )
% 1.61/1.09 = ( sup @ X @ ( sup @ Y @ Z ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_10_local_Osup__commute,axiom,
% 1.61/1.09 ! [X: a,Y: a] :
% 1.61/1.09 ( ( sup @ X @ Y )
% 1.61/1.09 = ( sup @ Y @ X ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_11_local_Osup__left__commute,axiom,
% 1.61/1.09 ! [X: a,Y: a,Z: a] :
% 1.61/1.09 ( ( sup @ X @ ( sup @ Y @ Z ) )
% 1.61/1.09 = ( sup @ Y @ ( sup @ X @ Z ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_12_local_Odistrib__imp1,axiom,
% 1.61/1.09 ! [X: a,Y: a,Z: a] :
% 1.61/1.09 ( ! [X2: a,Y2: a,Z2: a] :
% 1.61/1.09 ( ( inf @ X2 @ ( sup @ Y2 @ Z2 ) )
% 1.61/1.09 = ( sup @ ( inf @ X2 @ Y2 ) @ ( inf @ X2 @ Z2 ) ) )
% 1.61/1.09 => ( ( sup @ X @ ( inf @ Y @ Z ) )
% 1.61/1.09 = ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_13_local_Odistrib__imp2,axiom,
% 1.61/1.09 ! [X: a,Y: a,Z: a] :
% 1.61/1.09 ( ! [X2: a,Y2: a,Z2: a] :
% 1.61/1.09 ( ( sup @ X2 @ ( inf @ Y2 @ Z2 ) )
% 1.61/1.09 = ( inf @ ( sup @ X2 @ Y2 ) @ ( sup @ X2 @ Z2 ) ) )
% 1.61/1.09 => ( ( inf @ X @ ( sup @ Y @ Z ) )
% 1.61/1.09 = ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_14_e__aux__def,axiom,
% 1.61/1.09 ! [A2: a,B2: a,C: a] :
% 1.61/1.09 ( ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 1.61/1.09 = ( inf @ ( inf @ ( sup @ A2 @ B2 ) @ ( sup @ B2 @ C ) ) @ ( sup @ C @ A2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_15_local_Oinf_Oidem,axiom,
% 1.61/1.09 ! [A2: a] :
% 1.61/1.09 ( ( inf @ A2 @ A2 )
% 1.61/1.09 = A2 ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_16_local_Oinf_Oleft__idem,axiom,
% 1.61/1.09 ! [A2: a,B2: a] :
% 1.61/1.09 ( ( inf @ A2 @ ( inf @ A2 @ B2 ) )
% 1.61/1.09 = ( inf @ A2 @ B2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_17_local_Oinf_Oright__idem,axiom,
% 1.61/1.09 ! [A2: a,B2: a] :
% 1.61/1.09 ( ( inf @ ( inf @ A2 @ B2 ) @ B2 )
% 1.61/1.09 = ( inf @ A2 @ B2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_18_local_Oinf__idem,axiom,
% 1.61/1.09 ! [X: a] :
% 1.61/1.09 ( ( inf @ X @ X )
% 1.61/1.09 = X ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_19_local_Oinf__left__idem,axiom,
% 1.61/1.09 ! [X: a,Y: a] :
% 1.61/1.09 ( ( inf @ X @ ( inf @ X @ Y ) )
% 1.61/1.09 = ( inf @ X @ Y ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_20_local_Oinf__right__idem,axiom,
% 1.61/1.09 ! [X: a,Y: a] :
% 1.61/1.09 ( ( inf @ ( inf @ X @ Y ) @ Y )
% 1.61/1.09 = ( inf @ X @ Y ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_21_local_Osup_Oidem,axiom,
% 1.61/1.09 ! [A2: a] :
% 1.61/1.09 ( ( sup @ A2 @ A2 )
% 1.61/1.09 = A2 ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_22_local_Osup_Oleft__idem,axiom,
% 1.61/1.09 ! [A2: a,B2: a] :
% 1.61/1.09 ( ( sup @ A2 @ ( sup @ A2 @ B2 ) )
% 1.61/1.09 = ( sup @ A2 @ B2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_23_local_Osup_Oright__idem,axiom,
% 1.61/1.09 ! [A2: a,B2: a] :
% 1.61/1.09 ( ( sup @ ( sup @ A2 @ B2 ) @ B2 )
% 1.61/1.09 = ( sup @ A2 @ B2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_24_local_Osup__idem,axiom,
% 1.61/1.09 ! [X: a] :
% 1.61/1.09 ( ( sup @ X @ X )
% 1.61/1.09 = X ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_25_local_Osup__left__idem,axiom,
% 1.61/1.09 ! [X: a,Y: a] :
% 1.61/1.09 ( ( sup @ X @ ( sup @ X @ Y ) )
% 1.61/1.09 = ( sup @ X @ Y ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_26_local_Oinf__sup__absorb,axiom,
% 1.61/1.09 ! [X: a,Y: a] :
% 1.61/1.09 ( ( inf @ X @ ( sup @ X @ Y ) )
% 1.61/1.09 = X ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_27_local_Osup__inf__absorb,axiom,
% 1.61/1.09 ! [X: a,Y: a] :
% 1.61/1.09 ( ( sup @ X @ ( inf @ X @ Y ) )
% 1.61/1.09 = X ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_28_d__aux__def,axiom,
% 1.61/1.09 ! [A2: a,B2: a,C: a] :
% 1.61/1.09 ( ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 1.61/1.09 = ( sup @ ( sup @ ( inf @ A2 @ B2 ) @ ( inf @ B2 @ C ) ) @ ( inf @ C @ A2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_29_d__b__c__a,axiom,
% 1.61/1.09 ! [B2: a,C: a,A2: a] :
% 1.61/1.09 ( ( modula1221160330_d_aux @ a @ inf @ sup @ B2 @ C @ A2 )
% 1.61/1.09 = ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_30_d__c__a__b,axiom,
% 1.61/1.09 ! [C: a,A2: a,B2: a] :
% 1.61/1.09 ( ( modula1221160330_d_aux @ a @ inf @ sup @ C @ A2 @ B2 )
% 1.61/1.09 = ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_31_lattice_Oe__aux_Ocong,axiom,
% 1.61/1.09 ! [A: $tType] :
% 1.61/1.09 ( ( modula1593167561_e_aux @ A )
% 1.61/1.09 = ( modula1593167561_e_aux @ A ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_32_local_Ocomp__fun__idem__sup,axiom,
% 1.61/1.09 finite_comp_fun_idem @ a @ a @ sup ).
% 1.61/1.09
% 1.61/1.09 thf(fact_33_local_Ocomp__fun__idem__inf,axiom,
% 1.61/1.09 finite_comp_fun_idem @ a @ a @ inf ).
% 1.61/1.09
% 1.61/1.09 thf(fact_34_local_Osup_Osemigroup__axioms,axiom,
% 1.61/1.09 semigroup @ a @ sup ).
% 1.61/1.09
% 1.61/1.09 thf(fact_35_local_Oinf_Osemigroup__axioms,axiom,
% 1.61/1.09 semigroup @ a @ inf ).
% 1.61/1.09
% 1.61/1.09 thf(fact_36_local_Osup_Osemilattice__axioms,axiom,
% 1.61/1.09 semilattice @ a @ sup ).
% 1.61/1.09
% 1.61/1.09 thf(fact_37_local_Oinf_Osemilattice__axioms,axiom,
% 1.61/1.09 semilattice @ a @ inf ).
% 1.61/1.09
% 1.61/1.09 thf(fact_38_local_Osup_Oabel__semigroup__axioms,axiom,
% 1.61/1.09 abel_semigroup @ a @ sup ).
% 1.61/1.09
% 1.61/1.09 thf(fact_39_local_Oinf_Oabel__semigroup__axioms,axiom,
% 1.61/1.09 abel_semigroup @ a @ inf ).
% 1.61/1.09
% 1.61/1.09 thf(fact_40_local_OSup__fin_Osemilattice__set__axioms,axiom,
% 1.61/1.09 lattic35693393ce_set @ a @ sup ).
% 1.61/1.09
% 1.61/1.09 thf(fact_41_local_OInf__fin_Osemilattice__set__axioms,axiom,
% 1.61/1.09 lattic35693393ce_set @ a @ inf ).
% 1.61/1.09
% 1.61/1.09 thf(fact_42_local_Odistrib__inf__le,axiom,
% 1.61/1.09 ! [X: a,Y: a,Z: a] : ( less_eq @ ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z ) ) @ ( inf @ X @ ( sup @ Y @ Z ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_43_local_Odistrib__sup__le,axiom,
% 1.61/1.09 ! [X: a,Y: a,Z: a] : ( less_eq @ ( sup @ X @ ( inf @ Y @ Z ) ) @ ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_44_local_Oantisym,axiom,
% 1.61/1.09 ! [X: a,Y: a] :
% 1.61/1.09 ( ( less_eq @ X @ Y )
% 1.61/1.09 => ( ( less_eq @ Y @ X )
% 1.61/1.09 => ( X = Y ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_45_mem__Collect__eq,axiom,
% 1.61/1.09 ! [A: $tType,A2: A,P: A > $o] :
% 1.61/1.09 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 1.61/1.09 = ( P @ A2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_46_Collect__mem__eq,axiom,
% 1.61/1.09 ! [A: $tType,A3: set @ A] :
% 1.61/1.09 ( ( collect @ A
% 1.61/1.09 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
% 1.61/1.09 = A3 ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_47_Collect__cong,axiom,
% 1.61/1.09 ! [A: $tType,P: A > $o,Q: A > $o] :
% 1.61/1.09 ( ! [X2: A] :
% 1.61/1.09 ( ( P @ X2 )
% 1.61/1.09 = ( Q @ X2 ) )
% 1.61/1.09 => ( ( collect @ A @ P )
% 1.61/1.09 = ( collect @ A @ Q ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_48_ext,axiom,
% 1.61/1.09 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 1.61/1.09 ( ! [X2: A] :
% 1.61/1.09 ( ( F @ X2 )
% 1.61/1.09 = ( G @ X2 ) )
% 1.61/1.09 => ( F = G ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_49_local_Oantisym__conv,axiom,
% 1.61/1.09 ! [Y: a,X: a] :
% 1.61/1.09 ( ( less_eq @ Y @ X )
% 1.61/1.09 => ( ( less_eq @ X @ Y )
% 1.61/1.09 = ( X = Y ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_50_local_Odual__order_Oantisym,axiom,
% 1.61/1.09 ! [B2: a,A2: a] :
% 1.61/1.09 ( ( less_eq @ B2 @ A2 )
% 1.61/1.09 => ( ( less_eq @ A2 @ B2 )
% 1.61/1.09 => ( A2 = B2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_51_local_Odual__order_Oeq__iff,axiom,
% 1.61/1.09 ( ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 ) )
% 1.61/1.09 = ( ^ [A4: a,B3: a] :
% 1.61/1.09 ( ( less_eq @ B3 @ A4 )
% 1.61/1.09 & ( less_eq @ A4 @ B3 ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_52_local_Odual__order_Otrans,axiom,
% 1.61/1.09 ! [B2: a,A2: a,C: a] :
% 1.61/1.09 ( ( less_eq @ B2 @ A2 )
% 1.61/1.09 => ( ( less_eq @ C @ B2 )
% 1.61/1.09 => ( less_eq @ C @ A2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_53_local_Oeq__iff,axiom,
% 1.61/1.09 ( ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 ) )
% 1.61/1.09 = ( ^ [X3: a,Y4: a] :
% 1.61/1.09 ( ( less_eq @ X3 @ Y4 )
% 1.61/1.09 & ( less_eq @ Y4 @ X3 ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_54_local_Oeq__refl,axiom,
% 1.61/1.09 ! [X: a,Y: a] :
% 1.61/1.09 ( ( X = Y )
% 1.61/1.09 => ( less_eq @ X @ Y ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_55_local_Oord__eq__le__trans,axiom,
% 1.61/1.09 ! [A2: a,B2: a,C: a] :
% 1.61/1.09 ( ( A2 = B2 )
% 1.61/1.09 => ( ( less_eq @ B2 @ C )
% 1.61/1.09 => ( less_eq @ A2 @ C ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_56_local_Oord__le__eq__trans,axiom,
% 1.61/1.09 ! [A2: a,B2: a,C: a] :
% 1.61/1.09 ( ( less_eq @ A2 @ B2 )
% 1.61/1.09 => ( ( B2 = C )
% 1.61/1.09 => ( less_eq @ A2 @ C ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_57_local_Oorder_Oantisym,axiom,
% 1.61/1.09 ! [A2: a,B2: a] :
% 1.61/1.09 ( ( less_eq @ A2 @ B2 )
% 1.61/1.09 => ( ( less_eq @ B2 @ A2 )
% 1.61/1.09 => ( A2 = B2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_58_local_Oorder_Oeq__iff,axiom,
% 1.61/1.09 ( ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 ) )
% 1.61/1.09 = ( ^ [A4: a,B3: a] :
% 1.61/1.09 ( ( less_eq @ A4 @ B3 )
% 1.61/1.09 & ( less_eq @ B3 @ A4 ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_59_local_Oorder_Otrans,axiom,
% 1.61/1.09 ! [A2: a,B2: a,C: a] :
% 1.61/1.09 ( ( less_eq @ A2 @ B2 )
% 1.61/1.09 => ( ( less_eq @ B2 @ C )
% 1.61/1.09 => ( less_eq @ A2 @ C ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_60_local_Oorder__trans,axiom,
% 1.61/1.09 ! [X: a,Y: a,Z: a] :
% 1.61/1.09 ( ( less_eq @ X @ Y )
% 1.61/1.09 => ( ( less_eq @ Y @ Z )
% 1.61/1.09 => ( less_eq @ X @ Z ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_61_local_Ole__infI2,axiom,
% 1.61/1.09 ! [B2: a,X: a,A2: a] :
% 1.61/1.09 ( ( less_eq @ B2 @ X )
% 1.61/1.09 => ( less_eq @ ( inf @ A2 @ B2 ) @ X ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_62_local_Ole__infI1,axiom,
% 1.61/1.09 ! [A2: a,X: a,B2: a] :
% 1.61/1.09 ( ( less_eq @ A2 @ X )
% 1.61/1.09 => ( less_eq @ ( inf @ A2 @ B2 ) @ X ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_63_local_Ole__infI,axiom,
% 1.61/1.09 ! [X: a,A2: a,B2: a] :
% 1.61/1.09 ( ( less_eq @ X @ A2 )
% 1.61/1.09 => ( ( less_eq @ X @ B2 )
% 1.61/1.09 => ( less_eq @ X @ ( inf @ A2 @ B2 ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_64_local_Ole__infE,axiom,
% 1.61/1.09 ! [X: a,A2: a,B2: a] :
% 1.61/1.09 ( ( less_eq @ X @ ( inf @ A2 @ B2 ) )
% 1.61/1.09 => ~ ( ( less_eq @ X @ A2 )
% 1.61/1.09 => ~ ( less_eq @ X @ B2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_65_local_Ole__iff__inf,axiom,
% 1.61/1.09 ! [X: a,Y: a] :
% 1.61/1.09 ( ( less_eq @ X @ Y )
% 1.61/1.09 = ( ( inf @ X @ Y )
% 1.61/1.09 = X ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_66_local_Oinf__unique,axiom,
% 1.61/1.09 ! [F: a > a > a,X: a,Y: a] :
% 1.61/1.09 ( ! [X2: a,Y2: a] : ( less_eq @ ( F @ X2 @ Y2 ) @ X2 )
% 1.61/1.09 => ( ! [X2: a,Y2: a] : ( less_eq @ ( F @ X2 @ Y2 ) @ Y2 )
% 1.61/1.09 => ( ! [X2: a,Y2: a,Z2: a] :
% 1.61/1.09 ( ( less_eq @ X2 @ Y2 )
% 1.61/1.09 => ( ( less_eq @ X2 @ Z2 )
% 1.61/1.09 => ( less_eq @ X2 @ ( F @ Y2 @ Z2 ) ) ) )
% 1.61/1.09 => ( ( inf @ X @ Y )
% 1.61/1.09 = ( F @ X @ Y ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_67_local_Oinf__mono,axiom,
% 1.61/1.09 ! [A2: a,C: a,B2: a,D: a] :
% 1.61/1.09 ( ( less_eq @ A2 @ C )
% 1.61/1.09 => ( ( less_eq @ B2 @ D )
% 1.61/1.09 => ( less_eq @ ( inf @ A2 @ B2 ) @ ( inf @ C @ D ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_68_local_Oinf__le2,axiom,
% 1.61/1.09 ! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ Y ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_69_local_Oinf__le1,axiom,
% 1.61/1.09 ! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ X ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_70_local_Oinf__greatest,axiom,
% 1.61/1.09 ! [X: a,Y: a,Z: a] :
% 1.61/1.09 ( ( less_eq @ X @ Y )
% 1.61/1.09 => ( ( less_eq @ X @ Z )
% 1.61/1.09 => ( less_eq @ X @ ( inf @ Y @ Z ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_71_local_Oinf__absorb2,axiom,
% 1.61/1.09 ! [Y: a,X: a] :
% 1.61/1.09 ( ( less_eq @ Y @ X )
% 1.61/1.09 => ( ( inf @ X @ Y )
% 1.61/1.09 = Y ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_72_local_Oinf__absorb1,axiom,
% 1.61/1.09 ! [X: a,Y: a] :
% 1.61/1.09 ( ( less_eq @ X @ Y )
% 1.61/1.09 => ( ( inf @ X @ Y )
% 1.61/1.09 = X ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_73_local_Oinf_Oorder__iff,axiom,
% 1.61/1.09 ! [A2: a,B2: a] :
% 1.61/1.09 ( ( less_eq @ A2 @ B2 )
% 1.61/1.09 = ( A2
% 1.61/1.09 = ( inf @ A2 @ B2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_74_local_Oinf_OorderI,axiom,
% 1.61/1.09 ! [A2: a,B2: a] :
% 1.61/1.09 ( ( A2
% 1.61/1.09 = ( inf @ A2 @ B2 ) )
% 1.61/1.09 => ( less_eq @ A2 @ B2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_75_local_Oinf_OorderE,axiom,
% 1.61/1.09 ! [A2: a,B2: a] :
% 1.61/1.09 ( ( less_eq @ A2 @ B2 )
% 1.61/1.09 => ( A2
% 1.61/1.09 = ( inf @ A2 @ B2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_76_local_Oinf_OcoboundedI2,axiom,
% 1.61/1.09 ! [B2: a,C: a,A2: a] :
% 1.61/1.09 ( ( less_eq @ B2 @ C )
% 1.61/1.09 => ( less_eq @ ( inf @ A2 @ B2 ) @ C ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_77_local_Oinf_OcoboundedI1,axiom,
% 1.61/1.09 ! [A2: a,C: a,B2: a] :
% 1.61/1.09 ( ( less_eq @ A2 @ C )
% 1.61/1.09 => ( less_eq @ ( inf @ A2 @ B2 ) @ C ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_78_local_Oinf_Ocobounded2,axiom,
% 1.61/1.09 ! [A2: a,B2: a] : ( less_eq @ ( inf @ A2 @ B2 ) @ B2 ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_79_local_Oinf_Ocobounded1,axiom,
% 1.61/1.09 ! [A2: a,B2: a] : ( less_eq @ ( inf @ A2 @ B2 ) @ A2 ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_80_local_Oinf_OboundedI,axiom,
% 1.61/1.09 ! [A2: a,B2: a,C: a] :
% 1.61/1.09 ( ( less_eq @ A2 @ B2 )
% 1.61/1.09 => ( ( less_eq @ A2 @ C )
% 1.61/1.09 => ( less_eq @ A2 @ ( inf @ B2 @ C ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_81_local_Oinf_OboundedE,axiom,
% 1.61/1.09 ! [A2: a,B2: a,C: a] :
% 1.61/1.09 ( ( less_eq @ A2 @ ( inf @ B2 @ C ) )
% 1.61/1.09 => ~ ( ( less_eq @ A2 @ B2 )
% 1.61/1.09 => ~ ( less_eq @ A2 @ C ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_82_local_Oinf_Oabsorb__iff2,axiom,
% 1.61/1.09 ! [B2: a,A2: a] :
% 1.61/1.09 ( ( less_eq @ B2 @ A2 )
% 1.61/1.09 = ( ( inf @ A2 @ B2 )
% 1.61/1.09 = B2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_83_local_Oinf_Oabsorb__iff1,axiom,
% 1.61/1.09 ! [A2: a,B2: a] :
% 1.61/1.09 ( ( less_eq @ A2 @ B2 )
% 1.61/1.09 = ( ( inf @ A2 @ B2 )
% 1.61/1.09 = A2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_84_local_Oinf_Oabsorb2,axiom,
% 1.61/1.09 ! [B2: a,A2: a] :
% 1.61/1.09 ( ( less_eq @ B2 @ A2 )
% 1.61/1.09 => ( ( inf @ A2 @ B2 )
% 1.61/1.09 = B2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_85_local_Oinf_Oabsorb1,axiom,
% 1.61/1.09 ! [A2: a,B2: a] :
% 1.61/1.09 ( ( less_eq @ A2 @ B2 )
% 1.61/1.09 => ( ( inf @ A2 @ B2 )
% 1.61/1.09 = A2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_86_local_Osup__unique,axiom,
% 1.61/1.09 ! [F: a > a > a,X: a,Y: a] :
% 1.61/1.09 ( ! [X2: a,Y2: a] : ( less_eq @ X2 @ ( F @ X2 @ Y2 ) )
% 1.61/1.09 => ( ! [X2: a,Y2: a] : ( less_eq @ Y2 @ ( F @ X2 @ Y2 ) )
% 1.61/1.09 => ( ! [X2: a,Y2: a,Z2: a] :
% 1.61/1.09 ( ( less_eq @ Y2 @ X2 )
% 1.61/1.09 => ( ( less_eq @ Z2 @ X2 )
% 1.61/1.09 => ( less_eq @ ( F @ Y2 @ Z2 ) @ X2 ) ) )
% 1.61/1.09 => ( ( sup @ X @ Y )
% 1.61/1.09 = ( F @ X @ Y ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_87_local_Osup__mono,axiom,
% 1.61/1.09 ! [A2: a,C: a,B2: a,D: a] :
% 1.61/1.09 ( ( less_eq @ A2 @ C )
% 1.61/1.09 => ( ( less_eq @ B2 @ D )
% 1.61/1.09 => ( less_eq @ ( sup @ A2 @ B2 ) @ ( sup @ C @ D ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_88_local_Osup__least,axiom,
% 1.61/1.09 ! [Y: a,X: a,Z: a] :
% 1.61/1.09 ( ( less_eq @ Y @ X )
% 1.61/1.09 => ( ( less_eq @ Z @ X )
% 1.61/1.09 => ( less_eq @ ( sup @ Y @ Z ) @ X ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_89_local_Osup__ge2,axiom,
% 1.61/1.09 ! [Y: a,X: a] : ( less_eq @ Y @ ( sup @ X @ Y ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_90_local_Osup__ge1,axiom,
% 1.61/1.09 ! [X: a,Y: a] : ( less_eq @ X @ ( sup @ X @ Y ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_91_local_Osup__absorb2,axiom,
% 1.61/1.09 ! [X: a,Y: a] :
% 1.61/1.09 ( ( less_eq @ X @ Y )
% 1.61/1.09 => ( ( sup @ X @ Y )
% 1.61/1.09 = Y ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_92_local_Osup__absorb1,axiom,
% 1.61/1.09 ! [Y: a,X: a] :
% 1.61/1.09 ( ( less_eq @ Y @ X )
% 1.61/1.09 => ( ( sup @ X @ Y )
% 1.61/1.09 = X ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_93_local_Osup_Oorder__iff,axiom,
% 1.61/1.09 ! [B2: a,A2: a] :
% 1.61/1.09 ( ( less_eq @ B2 @ A2 )
% 1.61/1.09 = ( A2
% 1.61/1.09 = ( sup @ A2 @ B2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_94_local_Osup_OorderI,axiom,
% 1.61/1.09 ! [A2: a,B2: a] :
% 1.61/1.09 ( ( A2
% 1.61/1.09 = ( sup @ A2 @ B2 ) )
% 1.61/1.09 => ( less_eq @ B2 @ A2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_95_local_Osup_OorderE,axiom,
% 1.61/1.09 ! [B2: a,A2: a] :
% 1.61/1.09 ( ( less_eq @ B2 @ A2 )
% 1.61/1.09 => ( A2
% 1.61/1.09 = ( sup @ A2 @ B2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_96_local_Osup_Omono,axiom,
% 1.61/1.09 ! [C: a,A2: a,D: a,B2: a] :
% 1.61/1.09 ( ( less_eq @ C @ A2 )
% 1.61/1.09 => ( ( less_eq @ D @ B2 )
% 1.61/1.09 => ( less_eq @ ( sup @ C @ D ) @ ( sup @ A2 @ B2 ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_97_local_Osup_OcoboundedI2,axiom,
% 1.61/1.09 ! [C: a,B2: a,A2: a] :
% 1.61/1.09 ( ( less_eq @ C @ B2 )
% 1.61/1.09 => ( less_eq @ C @ ( sup @ A2 @ B2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_98_local_Osup_OcoboundedI1,axiom,
% 1.61/1.09 ! [C: a,A2: a,B2: a] :
% 1.61/1.09 ( ( less_eq @ C @ A2 )
% 1.61/1.09 => ( less_eq @ C @ ( sup @ A2 @ B2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_99_local_Osup_Ocobounded2,axiom,
% 1.61/1.09 ! [B2: a,A2: a] : ( less_eq @ B2 @ ( sup @ A2 @ B2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_100_local_Osup_Ocobounded1,axiom,
% 1.61/1.09 ! [A2: a,B2: a] : ( less_eq @ A2 @ ( sup @ A2 @ B2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_101_local_Osup_OboundedI,axiom,
% 1.61/1.09 ! [B2: a,A2: a,C: a] :
% 1.61/1.09 ( ( less_eq @ B2 @ A2 )
% 1.61/1.09 => ( ( less_eq @ C @ A2 )
% 1.61/1.09 => ( less_eq @ ( sup @ B2 @ C ) @ A2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_102_local_Osup_OboundedE,axiom,
% 1.61/1.09 ! [B2: a,C: a,A2: a] :
% 1.61/1.09 ( ( less_eq @ ( sup @ B2 @ C ) @ A2 )
% 1.61/1.09 => ~ ( ( less_eq @ B2 @ A2 )
% 1.61/1.09 => ~ ( less_eq @ C @ A2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_103_local_Osup_Oabsorb__iff2,axiom,
% 1.61/1.09 ! [A2: a,B2: a] :
% 1.61/1.09 ( ( less_eq @ A2 @ B2 )
% 1.61/1.09 = ( ( sup @ A2 @ B2 )
% 1.61/1.09 = B2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_104_local_Osup_Oabsorb__iff1,axiom,
% 1.61/1.09 ! [B2: a,A2: a] :
% 1.61/1.09 ( ( less_eq @ B2 @ A2 )
% 1.61/1.09 = ( ( sup @ A2 @ B2 )
% 1.61/1.09 = A2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_105_local_Osup_Oabsorb2,axiom,
% 1.61/1.09 ! [A2: a,B2: a] :
% 1.61/1.09 ( ( less_eq @ A2 @ B2 )
% 1.61/1.09 => ( ( sup @ A2 @ B2 )
% 1.61/1.09 = B2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_106_local_Osup_Oabsorb1,axiom,
% 1.61/1.09 ! [B2: a,A2: a] :
% 1.61/1.09 ( ( less_eq @ B2 @ A2 )
% 1.61/1.09 => ( ( sup @ A2 @ B2 )
% 1.61/1.09 = A2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_107_local_Ole__supI2,axiom,
% 1.61/1.09 ! [X: a,B2: a,A2: a] :
% 1.61/1.09 ( ( less_eq @ X @ B2 )
% 1.61/1.09 => ( less_eq @ X @ ( sup @ A2 @ B2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_108_local_Ole__supI1,axiom,
% 1.61/1.09 ! [X: a,A2: a,B2: a] :
% 1.61/1.09 ( ( less_eq @ X @ A2 )
% 1.61/1.09 => ( less_eq @ X @ ( sup @ A2 @ B2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_109_local_Ole__supI,axiom,
% 1.61/1.09 ! [A2: a,X: a,B2: a] :
% 1.61/1.09 ( ( less_eq @ A2 @ X )
% 1.61/1.09 => ( ( less_eq @ B2 @ X )
% 1.61/1.09 => ( less_eq @ ( sup @ A2 @ B2 ) @ X ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_110_local_Ole__supE,axiom,
% 1.61/1.09 ! [A2: a,B2: a,X: a] :
% 1.61/1.09 ( ( less_eq @ ( sup @ A2 @ B2 ) @ X )
% 1.61/1.09 => ~ ( ( less_eq @ A2 @ X )
% 1.61/1.09 => ~ ( less_eq @ B2 @ X ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_111_local_Ole__iff__sup,axiom,
% 1.61/1.09 ! [X: a,Y: a] :
% 1.61/1.09 ( ( less_eq @ X @ Y )
% 1.61/1.09 = ( ( sup @ X @ Y )
% 1.61/1.09 = Y ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_112_local_Oorder_Orefl,axiom,
% 1.61/1.09 ! [A2: a] : ( less_eq @ A2 @ A2 ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_113_local_Oorder__refl,axiom,
% 1.61/1.09 ! [X: a] : ( less_eq @ X @ X ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_114_local_OGreatestI2__order,axiom,
% 1.61/1.09 ! [P: a > $o,X: a,Q: a > $o] :
% 1.61/1.09 ( ( P @ X )
% 1.61/1.09 => ( ! [Y2: a] :
% 1.61/1.09 ( ( P @ Y2 )
% 1.61/1.09 => ( less_eq @ Y2 @ X ) )
% 1.61/1.09 => ( ! [X2: a] :
% 1.61/1.09 ( ( P @ X2 )
% 1.61/1.09 => ( ! [Y5: a] :
% 1.61/1.09 ( ( P @ Y5 )
% 1.61/1.09 => ( less_eq @ Y5 @ X2 ) )
% 1.61/1.09 => ( Q @ X2 ) ) )
% 1.61/1.09 => ( Q @ ( greatest @ a @ less_eq @ P ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_115_local_OGreatest__equality,axiom,
% 1.61/1.09 ! [P: a > $o,X: a] :
% 1.61/1.09 ( ( P @ X )
% 1.61/1.09 => ( ! [Y2: a] :
% 1.61/1.09 ( ( P @ Y2 )
% 1.61/1.09 => ( less_eq @ Y2 @ X ) )
% 1.61/1.09 => ( ( greatest @ a @ less_eq @ P )
% 1.61/1.09 = X ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_116_local_Omax__def,axiom,
% 1.61/1.09 ! [A2: a,B2: a] :
% 1.61/1.09 ( ( ( less_eq @ A2 @ B2 )
% 1.61/1.09 => ( ( max @ a @ less_eq @ A2 @ B2 )
% 1.61/1.09 = B2 ) )
% 1.61/1.09 & ( ~ ( less_eq @ A2 @ B2 )
% 1.61/1.09 => ( ( max @ a @ less_eq @ A2 @ B2 )
% 1.61/1.09 = A2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_117_local_Omin__def,axiom,
% 1.61/1.09 ! [A2: a,B2: a] :
% 1.61/1.09 ( ( ( less_eq @ A2 @ B2 )
% 1.61/1.09 => ( ( min @ a @ less_eq @ A2 @ B2 )
% 1.61/1.09 = A2 ) )
% 1.61/1.09 & ( ~ ( less_eq @ A2 @ B2 )
% 1.61/1.09 => ( ( min @ a @ less_eq @ A2 @ B2 )
% 1.61/1.09 = B2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_118_local_Ole__inf__iff,axiom,
% 1.61/1.09 ! [X: a,Y: a,Z: a] :
% 1.61/1.09 ( ( less_eq @ X @ ( inf @ Y @ Z ) )
% 1.61/1.09 = ( ( less_eq @ X @ Y )
% 1.61/1.09 & ( less_eq @ X @ Z ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_119_local_Oinf_Obounded__iff,axiom,
% 1.61/1.09 ! [A2: a,B2: a,C: a] :
% 1.61/1.09 ( ( less_eq @ A2 @ ( inf @ B2 @ C ) )
% 1.61/1.09 = ( ( less_eq @ A2 @ B2 )
% 1.61/1.09 & ( less_eq @ A2 @ C ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_120_local_Osup_Obounded__iff,axiom,
% 1.61/1.09 ! [B2: a,C: a,A2: a] :
% 1.61/1.09 ( ( less_eq @ ( sup @ B2 @ C ) @ A2 )
% 1.61/1.09 = ( ( less_eq @ B2 @ A2 )
% 1.61/1.09 & ( less_eq @ C @ A2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_121_local_Ole__sup__iff,axiom,
% 1.61/1.09 ! [X: a,Y: a,Z: a] :
% 1.61/1.09 ( ( less_eq @ ( sup @ X @ Y ) @ Z )
% 1.61/1.09 = ( ( less_eq @ X @ Z )
% 1.61/1.09 & ( less_eq @ Y @ Z ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_122_local_OLeast1I,axiom,
% 1.61/1.09 ! [P: a > $o] :
% 1.61/1.09 ( ? [X4: a] :
% 1.61/1.09 ( ( P @ X4 )
% 1.61/1.09 & ! [Y2: a] :
% 1.61/1.09 ( ( P @ Y2 )
% 1.61/1.09 => ( less_eq @ X4 @ Y2 ) )
% 1.61/1.09 & ! [Y2: a] :
% 1.61/1.09 ( ( ( P @ Y2 )
% 1.61/1.09 & ! [Ya: a] :
% 1.61/1.09 ( ( P @ Ya )
% 1.61/1.09 => ( less_eq @ Y2 @ Ya ) ) )
% 1.61/1.09 => ( Y2 = X4 ) ) )
% 1.61/1.09 => ( P @ ( least @ a @ less_eq @ P ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_123_local_OLeast1__le,axiom,
% 1.61/1.09 ! [P: a > $o,Z: a] :
% 1.61/1.09 ( ? [X4: a] :
% 1.61/1.09 ( ( P @ X4 )
% 1.61/1.09 & ! [Y2: a] :
% 1.61/1.09 ( ( P @ Y2 )
% 1.61/1.09 => ( less_eq @ X4 @ Y2 ) )
% 1.61/1.09 & ! [Y2: a] :
% 1.61/1.09 ( ( ( P @ Y2 )
% 1.61/1.09 & ! [Ya: a] :
% 1.61/1.09 ( ( P @ Ya )
% 1.61/1.09 => ( less_eq @ Y2 @ Ya ) ) )
% 1.61/1.09 => ( Y2 = X4 ) ) )
% 1.61/1.09 => ( ( P @ Z )
% 1.61/1.09 => ( less_eq @ ( least @ a @ less_eq @ P ) @ Z ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_124_local_OLeastI2__order,axiom,
% 1.61/1.09 ! [P: a > $o,X: a,Q: a > $o] :
% 1.61/1.09 ( ( P @ X )
% 1.61/1.09 => ( ! [Y2: a] :
% 1.61/1.09 ( ( P @ Y2 )
% 1.61/1.09 => ( less_eq @ X @ Y2 ) )
% 1.61/1.09 => ( ! [X2: a] :
% 1.61/1.09 ( ( P @ X2 )
% 1.61/1.09 => ( ! [Y5: a] :
% 1.61/1.09 ( ( P @ Y5 )
% 1.61/1.09 => ( less_eq @ X2 @ Y5 ) )
% 1.61/1.09 => ( Q @ X2 ) ) )
% 1.61/1.09 => ( Q @ ( least @ a @ less_eq @ P ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_125_local_OLeast__equality,axiom,
% 1.61/1.09 ! [P: a > $o,X: a] :
% 1.61/1.09 ( ( P @ X )
% 1.61/1.09 => ( ! [Y2: a] :
% 1.61/1.09 ( ( P @ Y2 )
% 1.61/1.09 => ( less_eq @ X @ Y2 ) )
% 1.61/1.09 => ( ( least @ a @ less_eq @ P )
% 1.61/1.09 = X ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_126_lattice_Od__aux_Ocong,axiom,
% 1.61/1.09 ! [A: $tType] :
% 1.61/1.09 ( ( modula1221160330_d_aux @ A )
% 1.61/1.09 = ( modula1221160330_d_aux @ A ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_127_abel__semigroup_Oaxioms_I1_J,axiom,
% 1.61/1.09 ! [A: $tType,F: A > A > A] :
% 1.61/1.09 ( ( abel_semigroup @ A @ F )
% 1.61/1.09 => ( semigroup @ A @ F ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_128_semilattice__set__def,axiom,
% 1.61/1.09 ! [A: $tType] :
% 1.61/1.09 ( ( lattic35693393ce_set @ A )
% 1.61/1.09 = ( semilattice @ A ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_129_semilattice__set_Ointro,axiom,
% 1.61/1.09 ! [A: $tType,F: A > A > A] :
% 1.61/1.09 ( ( semilattice @ A @ F )
% 1.61/1.09 => ( lattic35693393ce_set @ A @ F ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_130_semilattice__set_Oaxioms,axiom,
% 1.61/1.09 ! [A: $tType,F: A > A > A] :
% 1.61/1.09 ( ( lattic35693393ce_set @ A @ F )
% 1.61/1.09 => ( semilattice @ A @ F ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_131_semilattice_Oaxioms_I1_J,axiom,
% 1.61/1.09 ! [A: $tType,F: A > A > A] :
% 1.61/1.09 ( ( semilattice @ A @ F )
% 1.61/1.09 => ( abel_semigroup @ A @ F ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_132_local_Obdd__above__def,axiom,
% 1.61/1.09 ! [A3: set @ a] :
% 1.61/1.09 ( ( condit2040224947_above @ a @ less_eq @ A3 )
% 1.61/1.09 = ( ? [M: a] :
% 1.61/1.09 ! [X3: a] :
% 1.61/1.09 ( ( member @ a @ X3 @ A3 )
% 1.61/1.09 => ( less_eq @ X3 @ M ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_133_local_Obdd__below__def,axiom,
% 1.61/1.09 ! [A3: set @ a] :
% 1.61/1.09 ( ( condit1201339847_below @ a @ less_eq @ A3 )
% 1.61/1.09 = ( ? [M2: a] :
% 1.61/1.09 ! [X3: a] :
% 1.61/1.09 ( ( member @ a @ X3 @ A3 )
% 1.61/1.09 => ( less_eq @ M2 @ X3 ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_134_local_Otransp__le,axiom,
% 1.61/1.09 transp @ a @ less_eq ).
% 1.61/1.09
% 1.61/1.09 thf(fact_135_local_Obdd__belowI,axiom,
% 1.61/1.09 ! [A3: set @ a,M3: a] :
% 1.61/1.09 ( ! [X2: a] :
% 1.61/1.09 ( ( member @ a @ X2 @ A3 )
% 1.61/1.09 => ( less_eq @ M3 @ X2 ) )
% 1.61/1.09 => ( condit1201339847_below @ a @ less_eq @ A3 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_136_local_Obdd__aboveI,axiom,
% 1.61/1.09 ! [A3: set @ a,M4: a] :
% 1.61/1.09 ( ! [X2: a] :
% 1.61/1.09 ( ( member @ a @ X2 @ A3 )
% 1.61/1.09 => ( less_eq @ X2 @ M4 ) )
% 1.61/1.09 => ( condit2040224947_above @ a @ less_eq @ A3 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_137_abel__semigroup_Oleft__commute,axiom,
% 1.61/1.09 ! [A: $tType,F: A > A > A,B2: A,A2: A,C: A] :
% 1.61/1.09 ( ( abel_semigroup @ A @ F )
% 1.61/1.09 => ( ( F @ B2 @ ( F @ A2 @ C ) )
% 1.61/1.09 = ( F @ A2 @ ( F @ B2 @ C ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_138_semilattice_Oright__idem,axiom,
% 1.61/1.09 ! [A: $tType,F: A > A > A,A2: A,B2: A] :
% 1.61/1.09 ( ( semilattice @ A @ F )
% 1.61/1.09 => ( ( F @ ( F @ A2 @ B2 ) @ B2 )
% 1.61/1.09 = ( F @ A2 @ B2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_139_semilattice_Oleft__idem,axiom,
% 1.61/1.09 ! [A: $tType,F: A > A > A,A2: A,B2: A] :
% 1.61/1.09 ( ( semilattice @ A @ F )
% 1.61/1.09 => ( ( F @ A2 @ ( F @ A2 @ B2 ) )
% 1.61/1.09 = ( F @ A2 @ B2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_140_abel__semigroup_Ocommute,axiom,
% 1.61/1.09 ! [A: $tType,F: A > A > A,A2: A,B2: A] :
% 1.61/1.09 ( ( abel_semigroup @ A @ F )
% 1.61/1.09 => ( ( F @ A2 @ B2 )
% 1.61/1.09 = ( F @ B2 @ A2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_141_semilattice_Oidem,axiom,
% 1.61/1.09 ! [A: $tType,F: A > A > A,A2: A] :
% 1.61/1.09 ( ( semilattice @ A @ F )
% 1.61/1.09 => ( ( F @ A2 @ A2 )
% 1.61/1.09 = A2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_142_semigroup_Ointro,axiom,
% 1.61/1.09 ! [A: $tType,F: A > A > A] :
% 1.61/1.09 ( ! [A5: A,B4: A,C2: A] :
% 1.61/1.09 ( ( F @ ( F @ A5 @ B4 ) @ C2 )
% 1.61/1.09 = ( F @ A5 @ ( F @ B4 @ C2 ) ) )
% 1.61/1.09 => ( semigroup @ A @ F ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_143_semigroup_Oassoc,axiom,
% 1.61/1.09 ! [A: $tType,F: A > A > A,A2: A,B2: A,C: A] :
% 1.61/1.09 ( ( semigroup @ A @ F )
% 1.61/1.09 => ( ( F @ ( F @ A2 @ B2 ) @ C )
% 1.61/1.09 = ( F @ A2 @ ( F @ B2 @ C ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_144_semigroup__def,axiom,
% 1.61/1.09 ! [A: $tType] :
% 1.61/1.09 ( ( semigroup @ A )
% 1.61/1.09 = ( ^ [F2: A > A > A] :
% 1.61/1.09 ! [A4: A,B3: A,C3: A] :
% 1.61/1.09 ( ( F2 @ ( F2 @ A4 @ B3 ) @ C3 )
% 1.61/1.09 = ( F2 @ A4 @ ( F2 @ B3 @ C3 ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_145_local_Obdd__below__mono,axiom,
% 1.61/1.09 ! [B5: set @ a,A3: set @ a] :
% 1.61/1.09 ( ( condit1201339847_below @ a @ less_eq @ B5 )
% 1.61/1.09 => ( ( ord_less_eq @ ( set @ a ) @ A3 @ B5 )
% 1.61/1.09 => ( condit1201339847_below @ a @ less_eq @ A3 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_146_local_Obdd__above__mono,axiom,
% 1.61/1.09 ! [B5: set @ a,A3: set @ a] :
% 1.61/1.09 ( ( condit2040224947_above @ a @ less_eq @ B5 )
% 1.61/1.09 => ( ( ord_less_eq @ ( set @ a ) @ A3 @ B5 )
% 1.61/1.09 => ( condit2040224947_above @ a @ less_eq @ A3 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_147_local_Obdd__below__finite,axiom,
% 1.61/1.09 ! [A3: set @ a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( condit1201339847_below @ a @ less_eq @ A3 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_148_local_Obdd__above__finite,axiom,
% 1.61/1.09 ! [A3: set @ a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( condit2040224947_above @ a @ less_eq @ A3 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_149_local_Ofinite__has__maximal2,axiom,
% 1.61/1.09 ! [A3: set @ a,A2: a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( member @ a @ A2 @ A3 )
% 1.61/1.09 => ? [X2: a] :
% 1.61/1.09 ( ( member @ a @ X2 @ A3 )
% 1.61/1.09 & ( less_eq @ A2 @ X2 )
% 1.61/1.09 & ! [Xa: a] :
% 1.61/1.09 ( ( member @ a @ Xa @ A3 )
% 1.61/1.09 => ( ( less_eq @ X2 @ Xa )
% 1.61/1.09 => ( X2 = Xa ) ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_150_local_Ofinite__has__minimal2,axiom,
% 1.61/1.09 ! [A3: set @ a,A2: a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( member @ a @ A2 @ A3 )
% 1.61/1.09 => ? [X2: a] :
% 1.61/1.09 ( ( member @ a @ X2 @ A3 )
% 1.61/1.09 & ( less_eq @ X2 @ A2 )
% 1.61/1.09 & ! [Xa: a] :
% 1.61/1.09 ( ( member @ a @ Xa @ A3 )
% 1.61/1.09 => ( ( less_eq @ Xa @ X2 )
% 1.61/1.09 => ( X2 = Xa ) ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_151_local_OantimonoD,axiom,
% 1.61/1.09 ! [B: $tType] :
% 1.61/1.09 ( ( order @ B )
% 1.61/1.09 => ! [F: a > B,X: a,Y: a] :
% 1.61/1.09 ( ( antimono @ a @ B @ less_eq @ F )
% 1.61/1.09 => ( ( less_eq @ X @ Y )
% 1.61/1.09 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_152_local_OantimonoE,axiom,
% 1.61/1.09 ! [B: $tType] :
% 1.61/1.09 ( ( order @ B )
% 1.61/1.09 => ! [F: a > B,X: a,Y: a] :
% 1.61/1.09 ( ( antimono @ a @ B @ less_eq @ F )
% 1.61/1.09 => ( ( less_eq @ X @ Y )
% 1.61/1.09 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_153_local_OantimonoI,axiom,
% 1.61/1.09 ! [B: $tType] :
% 1.61/1.09 ( ( order @ B )
% 1.61/1.09 => ! [F: a > B] :
% 1.61/1.09 ( ! [X2: a,Y2: a] :
% 1.61/1.09 ( ( less_eq @ X2 @ Y2 )
% 1.61/1.09 => ( ord_less_eq @ B @ ( F @ Y2 ) @ ( F @ X2 ) ) )
% 1.61/1.09 => ( antimono @ a @ B @ less_eq @ F ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_154_local_Oantimono__def,axiom,
% 1.61/1.09 ! [B: $tType] :
% 1.61/1.09 ( ( order @ B )
% 1.61/1.09 => ! [F: a > B] :
% 1.61/1.09 ( ( antimono @ a @ B @ less_eq @ F )
% 1.61/1.09 = ( ! [X3: a,Y4: a] :
% 1.61/1.09 ( ( less_eq @ X3 @ Y4 )
% 1.61/1.09 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X3 ) ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_155_local_OmonoD,axiom,
% 1.61/1.09 ! [B: $tType] :
% 1.61/1.09 ( ( order @ B )
% 1.61/1.09 => ! [F: a > B,X: a,Y: a] :
% 1.61/1.09 ( ( mono @ a @ B @ less_eq @ F )
% 1.61/1.09 => ( ( less_eq @ X @ Y )
% 1.61/1.09 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_156_local_OmonoE,axiom,
% 1.61/1.09 ! [B: $tType] :
% 1.61/1.09 ( ( order @ B )
% 1.61/1.09 => ! [F: a > B,X: a,Y: a] :
% 1.61/1.09 ( ( mono @ a @ B @ less_eq @ F )
% 1.61/1.09 => ( ( less_eq @ X @ Y )
% 1.61/1.09 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_157_local_OmonoI,axiom,
% 1.61/1.09 ! [B: $tType] :
% 1.61/1.09 ( ( order @ B )
% 1.61/1.09 => ! [F: a > B] :
% 1.61/1.09 ( ! [X2: a,Y2: a] :
% 1.61/1.09 ( ( less_eq @ X2 @ Y2 )
% 1.61/1.09 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 1.61/1.09 => ( mono @ a @ B @ less_eq @ F ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_158_local_Omono__def,axiom,
% 1.61/1.09 ! [B: $tType] :
% 1.61/1.09 ( ( order @ B )
% 1.61/1.09 => ! [F: a > B] :
% 1.61/1.09 ( ( mono @ a @ B @ less_eq @ F )
% 1.61/1.09 = ( ! [X3: a,Y4: a] :
% 1.61/1.09 ( ( less_eq @ X3 @ Y4 )
% 1.61/1.09 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_159_local_OSup__fin_OcoboundedI,axiom,
% 1.61/1.09 ! [A3: set @ a,A2: a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( member @ a @ A2 @ A3 )
% 1.61/1.09 => ( less_eq @ A2 @ ( lattic1039401930up_fin @ a @ sup @ A3 ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_160_local_OInf__fin_OcoboundedI,axiom,
% 1.61/1.09 ! [A3: set @ a,A2: a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( member @ a @ A2 @ A3 )
% 1.61/1.09 => ( less_eq @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) @ A2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_161_local_OSup__fin_Oin__idem,axiom,
% 1.61/1.09 ! [A3: set @ a,X: a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( member @ a @ X @ A3 )
% 1.61/1.09 => ( ( sup @ X @ ( lattic1039401930up_fin @ a @ sup @ A3 ) )
% 1.61/1.09 = ( lattic1039401930up_fin @ a @ sup @ A3 ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_162_local_OInf__fin_Oin__idem,axiom,
% 1.61/1.09 ! [A3: set @ a,X: a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( member @ a @ X @ A3 )
% 1.61/1.09 => ( ( inf @ X @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) )
% 1.61/1.09 = ( lattic1263571978nf_fin @ a @ inf @ A3 ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_163_local_Oinf__Sup__absorb,axiom,
% 1.61/1.09 ! [A3: set @ a,A2: a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( member @ a @ A2 @ A3 )
% 1.61/1.09 => ( ( inf @ A2 @ ( lattic1039401930up_fin @ a @ sup @ A3 ) )
% 1.61/1.09 = A2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_164_local_Osup__Inf__absorb,axiom,
% 1.61/1.09 ! [A3: set @ a,A2: a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( member @ a @ A2 @ A3 )
% 1.61/1.09 => ( ( sup @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) @ A2 )
% 1.61/1.09 = A2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_165_semilattice__inf_OInf__fin_Ocong,axiom,
% 1.61/1.09 ! [A: $tType] :
% 1.61/1.09 ( ( lattic1263571978nf_fin @ A )
% 1.61/1.09 = ( lattic1263571978nf_fin @ A ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_166_semilattice__sup_OSup__fin_Ocong,axiom,
% 1.61/1.09 ! [A: $tType] :
% 1.61/1.09 ( ( lattic1039401930up_fin @ A )
% 1.61/1.09 = ( lattic1039401930up_fin @ A ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_167_local_OInf__fin__le__Sup__fin,axiom,
% 1.61/1.09 ! [A3: set @ a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( A3
% 1.61/1.09 != ( bot_bot @ ( set @ a ) ) )
% 1.61/1.09 => ( less_eq @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) @ ( lattic1039401930up_fin @ a @ sup @ A3 ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_168_local_OSup__fin_Osubset__imp,axiom,
% 1.61/1.09 ! [A3: set @ a,B5: set @ a] :
% 1.61/1.09 ( ( ord_less_eq @ ( set @ a ) @ A3 @ B5 )
% 1.61/1.09 => ( ( A3
% 1.61/1.09 != ( bot_bot @ ( set @ a ) ) )
% 1.61/1.09 => ( ( finite_finite2 @ a @ B5 )
% 1.61/1.09 => ( less_eq @ ( lattic1039401930up_fin @ a @ sup @ A3 ) @ ( lattic1039401930up_fin @ a @ sup @ B5 ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_169_local_OInf__fin_Osubset__imp,axiom,
% 1.61/1.09 ! [A3: set @ a,B5: set @ a] :
% 1.61/1.09 ( ( ord_less_eq @ ( set @ a ) @ A3 @ B5 )
% 1.61/1.09 => ( ( A3
% 1.61/1.09 != ( bot_bot @ ( set @ a ) ) )
% 1.61/1.09 => ( ( finite_finite2 @ a @ B5 )
% 1.61/1.09 => ( less_eq @ ( lattic1263571978nf_fin @ a @ inf @ B5 ) @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_170_local_Ofinite__has__minimal,axiom,
% 1.61/1.09 ! [A3: set @ a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( A3
% 1.61/1.09 != ( bot_bot @ ( set @ a ) ) )
% 1.61/1.09 => ? [X2: a] :
% 1.61/1.09 ( ( member @ a @ X2 @ A3 )
% 1.61/1.09 & ! [Xa: a] :
% 1.61/1.09 ( ( member @ a @ Xa @ A3 )
% 1.61/1.09 => ( ( less_eq @ Xa @ X2 )
% 1.61/1.09 => ( X2 = Xa ) ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_171_local_Ofinite__has__maximal,axiom,
% 1.61/1.09 ! [A3: set @ a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( A3
% 1.61/1.09 != ( bot_bot @ ( set @ a ) ) )
% 1.61/1.09 => ? [X2: a] :
% 1.61/1.09 ( ( member @ a @ X2 @ A3 )
% 1.61/1.09 & ! [Xa: a] :
% 1.61/1.09 ( ( member @ a @ Xa @ A3 )
% 1.61/1.09 => ( ( less_eq @ X2 @ Xa )
% 1.61/1.09 => ( X2 = Xa ) ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_172_local_OInf__fin_Osubset,axiom,
% 1.61/1.09 ! [A3: set @ a,B5: set @ a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( B5
% 1.61/1.09 != ( bot_bot @ ( set @ a ) ) )
% 1.61/1.09 => ( ( ord_less_eq @ ( set @ a ) @ B5 @ A3 )
% 1.61/1.09 => ( ( inf @ ( lattic1263571978nf_fin @ a @ inf @ B5 ) @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) )
% 1.61/1.09 = ( lattic1263571978nf_fin @ a @ inf @ A3 ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_173_local_OSup__fin_Osubset,axiom,
% 1.61/1.09 ! [A3: set @ a,B5: set @ a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( B5
% 1.61/1.09 != ( bot_bot @ ( set @ a ) ) )
% 1.61/1.09 => ( ( ord_less_eq @ ( set @ a ) @ B5 @ A3 )
% 1.61/1.09 => ( ( sup @ ( lattic1039401930up_fin @ a @ sup @ B5 ) @ ( lattic1039401930up_fin @ a @ sup @ A3 ) )
% 1.61/1.09 = ( lattic1039401930up_fin @ a @ sup @ A3 ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_174_local_OInf__fin_Obounded__iff,axiom,
% 1.61/1.09 ! [A3: set @ a,X: a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( A3
% 1.61/1.09 != ( bot_bot @ ( set @ a ) ) )
% 1.61/1.09 => ( ( less_eq @ X @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) )
% 1.61/1.09 = ( ! [X3: a] :
% 1.61/1.09 ( ( member @ a @ X3 @ A3 )
% 1.61/1.09 => ( less_eq @ X @ X3 ) ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_175_local_OInf__fin_OboundedI,axiom,
% 1.61/1.09 ! [A3: set @ a,X: a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( A3
% 1.61/1.09 != ( bot_bot @ ( set @ a ) ) )
% 1.61/1.09 => ( ! [A5: a] :
% 1.61/1.09 ( ( member @ a @ A5 @ A3 )
% 1.61/1.09 => ( less_eq @ X @ A5 ) )
% 1.61/1.09 => ( less_eq @ X @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_176_local_OInf__fin_OboundedE,axiom,
% 1.61/1.09 ! [A3: set @ a,X: a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( A3
% 1.61/1.09 != ( bot_bot @ ( set @ a ) ) )
% 1.61/1.09 => ( ( less_eq @ X @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) )
% 1.61/1.09 => ! [A6: a] :
% 1.61/1.09 ( ( member @ a @ A6 @ A3 )
% 1.61/1.09 => ( less_eq @ X @ A6 ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_177_local_OSup__fin_Obounded__iff,axiom,
% 1.61/1.09 ! [A3: set @ a,X: a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( A3
% 1.61/1.09 != ( bot_bot @ ( set @ a ) ) )
% 1.61/1.09 => ( ( less_eq @ ( lattic1039401930up_fin @ a @ sup @ A3 ) @ X )
% 1.61/1.09 = ( ! [X3: a] :
% 1.61/1.09 ( ( member @ a @ X3 @ A3 )
% 1.61/1.09 => ( less_eq @ X3 @ X ) ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_178_local_OSup__fin_OboundedI,axiom,
% 1.61/1.09 ! [A3: set @ a,X: a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( A3
% 1.61/1.09 != ( bot_bot @ ( set @ a ) ) )
% 1.61/1.09 => ( ! [A5: a] :
% 1.61/1.09 ( ( member @ a @ A5 @ A3 )
% 1.61/1.09 => ( less_eq @ A5 @ X ) )
% 1.61/1.09 => ( less_eq @ ( lattic1039401930up_fin @ a @ sup @ A3 ) @ X ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_179_local_OSup__fin_OboundedE,axiom,
% 1.61/1.09 ! [A3: set @ a,X: a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( A3
% 1.61/1.09 != ( bot_bot @ ( set @ a ) ) )
% 1.61/1.09 => ( ( less_eq @ ( lattic1039401930up_fin @ a @ sup @ A3 ) @ X )
% 1.61/1.09 => ! [A6: a] :
% 1.61/1.09 ( ( member @ a @ A6 @ A3 )
% 1.61/1.09 => ( less_eq @ A6 @ X ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_180_local_Obdd__above__empty,axiom,
% 1.61/1.09 condit2040224947_above @ a @ less_eq @ ( bot_bot @ ( set @ a ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_181_local_Obdd__below__empty,axiom,
% 1.61/1.09 condit1201339847_below @ a @ less_eq @ ( bot_bot @ ( set @ a ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_182_local_Onot__empty__eq__Iic__eq__empty,axiom,
% 1.61/1.09 ! [H: a] :
% 1.61/1.09 ( ( bot_bot @ ( set @ a ) )
% 1.61/1.09 != ( set_atMost @ a @ less_eq @ H ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_183_local_Onot__empty__eq__Ici__eq__empty,axiom,
% 1.61/1.09 ! [L: a] :
% 1.61/1.09 ( ( bot_bot @ ( set @ a ) )
% 1.61/1.09 != ( set_atLeast @ a @ less_eq @ L ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_184_subset__empty,axiom,
% 1.61/1.09 ! [A: $tType,A3: set @ A] :
% 1.61/1.09 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 1.61/1.09 = ( A3
% 1.61/1.09 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_185_empty__Collect__eq,axiom,
% 1.61/1.09 ! [A: $tType,P: A > $o] :
% 1.61/1.09 ( ( ( bot_bot @ ( set @ A ) )
% 1.61/1.09 = ( collect @ A @ P ) )
% 1.61/1.09 = ( ! [X3: A] :
% 1.61/1.09 ~ ( P @ X3 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_186_Collect__empty__eq,axiom,
% 1.61/1.09 ! [A: $tType,P: A > $o] :
% 1.61/1.09 ( ( ( collect @ A @ P )
% 1.61/1.09 = ( bot_bot @ ( set @ A ) ) )
% 1.61/1.09 = ( ! [X3: A] :
% 1.61/1.09 ~ ( P @ X3 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_187_all__not__in__conv,axiom,
% 1.61/1.09 ! [A: $tType,A3: set @ A] :
% 1.61/1.09 ( ( ! [X3: A] :
% 1.61/1.09 ~ ( member @ A @ X3 @ A3 ) )
% 1.61/1.09 = ( A3
% 1.61/1.09 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_188_empty__iff,axiom,
% 1.61/1.09 ! [A: $tType,C: A] :
% 1.61/1.09 ~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_189_subset__antisym,axiom,
% 1.61/1.09 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 1.61/1.09 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 1.61/1.09 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
% 1.61/1.09 => ( A3 = B5 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_190_subsetI,axiom,
% 1.61/1.09 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 1.61/1.09 ( ! [X2: A] :
% 1.61/1.09 ( ( member @ A @ X2 @ A3 )
% 1.61/1.09 => ( member @ A @ X2 @ B5 ) )
% 1.61/1.09 => ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_191_empty__subsetI,axiom,
% 1.61/1.09 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_192_local_OatLeast__iff,axiom,
% 1.61/1.09 ! [I: a,K: a] :
% 1.61/1.09 ( ( member @ a @ I @ ( set_atLeast @ a @ less_eq @ K ) )
% 1.61/1.09 = ( less_eq @ K @ I ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_193_local_OatMost__iff,axiom,
% 1.61/1.09 ! [I: a,K: a] :
% 1.61/1.09 ( ( member @ a @ I @ ( set_atMost @ a @ less_eq @ K ) )
% 1.61/1.09 = ( less_eq @ I @ K ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_194_local_Obdd__below__Ici,axiom,
% 1.61/1.09 ! [A2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_atLeast @ a @ less_eq @ A2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_195_local_Obdd__above__Iic,axiom,
% 1.61/1.09 ! [B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_atMost @ a @ less_eq @ B2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_196_ex__in__conv,axiom,
% 1.61/1.09 ! [A: $tType,A3: set @ A] :
% 1.61/1.09 ( ( ? [X3: A] : ( member @ A @ X3 @ A3 ) )
% 1.61/1.09 = ( A3
% 1.61/1.09 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_197_equals0I,axiom,
% 1.61/1.09 ! [A: $tType,A3: set @ A] :
% 1.61/1.09 ( ! [Y2: A] :
% 1.61/1.09 ~ ( member @ A @ Y2 @ A3 )
% 1.61/1.09 => ( A3
% 1.61/1.09 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_198_equals0D,axiom,
% 1.61/1.09 ! [A: $tType,A3: set @ A,A2: A] :
% 1.61/1.09 ( ( A3
% 1.61/1.09 = ( bot_bot @ ( set @ A ) ) )
% 1.61/1.09 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_199_emptyE,axiom,
% 1.61/1.09 ! [A: $tType,A2: A] :
% 1.61/1.09 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_200_Collect__mono__iff,axiom,
% 1.61/1.09 ! [A: $tType,P: A > $o,Q: A > $o] :
% 1.61/1.09 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 1.61/1.09 = ( ! [X3: A] :
% 1.61/1.09 ( ( P @ X3 )
% 1.61/1.09 => ( Q @ X3 ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_201_set__eq__subset,axiom,
% 1.61/1.09 ! [A: $tType] :
% 1.61/1.09 ( ( ^ [Y3: set @ A,Z3: set @ A] : ( Y3 = Z3 ) )
% 1.61/1.09 = ( ^ [A7: set @ A,B6: set @ A] :
% 1.61/1.09 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 1.61/1.09 & ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_202_subset__trans,axiom,
% 1.61/1.09 ! [A: $tType,A3: set @ A,B5: set @ A,C4: set @ A] :
% 1.61/1.09 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 1.61/1.09 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
% 1.61/1.09 => ( ord_less_eq @ ( set @ A ) @ A3 @ C4 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_203_Collect__mono,axiom,
% 1.61/1.09 ! [A: $tType,P: A > $o,Q: A > $o] :
% 1.61/1.09 ( ! [X2: A] :
% 1.61/1.09 ( ( P @ X2 )
% 1.61/1.09 => ( Q @ X2 ) )
% 1.61/1.09 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_204_subset__refl,axiom,
% 1.61/1.09 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_205_subset__iff,axiom,
% 1.61/1.09 ! [A: $tType] :
% 1.61/1.09 ( ( ord_less_eq @ ( set @ A ) )
% 1.61/1.09 = ( ^ [A7: set @ A,B6: set @ A] :
% 1.61/1.09 ! [T: A] :
% 1.61/1.09 ( ( member @ A @ T @ A7 )
% 1.61/1.09 => ( member @ A @ T @ B6 ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_206_equalityD2,axiom,
% 1.61/1.09 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 1.61/1.09 ( ( A3 = B5 )
% 1.61/1.09 => ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_207_equalityD1,axiom,
% 1.61/1.09 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 1.61/1.09 ( ( A3 = B5 )
% 1.61/1.09 => ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_208_subset__eq,axiom,
% 1.61/1.09 ! [A: $tType] :
% 1.61/1.09 ( ( ord_less_eq @ ( set @ A ) )
% 1.61/1.09 = ( ^ [A7: set @ A,B6: set @ A] :
% 1.61/1.09 ! [X3: A] :
% 1.61/1.09 ( ( member @ A @ X3 @ A7 )
% 1.61/1.09 => ( member @ A @ X3 @ B6 ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_209_equalityE,axiom,
% 1.61/1.09 ! [A: $tType,A3: set @ A,B5: set @ A] :
% 1.61/1.09 ( ( A3 = B5 )
% 1.61/1.09 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 1.61/1.09 => ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_210_subsetD,axiom,
% 1.61/1.09 ! [A: $tType,A3: set @ A,B5: set @ A,C: A] :
% 1.61/1.09 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 1.61/1.09 => ( ( member @ A @ C @ A3 )
% 1.61/1.09 => ( member @ A @ C @ B5 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_211_in__mono,axiom,
% 1.61/1.09 ! [A: $tType,A3: set @ A,B5: set @ A,X: A] :
% 1.61/1.09 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 1.61/1.09 => ( ( member @ A @ X @ A3 )
% 1.61/1.09 => ( member @ A @ X @ B5 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_212_local_OSup__fin_Oclosed,axiom,
% 1.61/1.09 ! [A3: set @ a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( A3
% 1.61/1.09 != ( bot_bot @ ( set @ a ) ) )
% 1.61/1.09 => ( ! [X2: a,Y2: a] : ( member @ a @ ( sup @ X2 @ Y2 ) @ ( insert @ a @ X2 @ ( insert @ a @ Y2 @ ( bot_bot @ ( set @ a ) ) ) ) )
% 1.61/1.09 => ( member @ a @ ( lattic1039401930up_fin @ a @ sup @ A3 ) @ A3 ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_213_local_OSup__fin_Oinsert__not__elem,axiom,
% 1.61/1.09 ! [A3: set @ a,X: a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ~ ( member @ a @ X @ A3 )
% 1.61/1.09 => ( ( A3
% 1.61/1.09 != ( bot_bot @ ( set @ a ) ) )
% 1.61/1.09 => ( ( lattic1039401930up_fin @ a @ sup @ ( insert @ a @ X @ A3 ) )
% 1.61/1.09 = ( sup @ X @ ( lattic1039401930up_fin @ a @ sup @ A3 ) ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_214_local_OInf__fin_Oclosed,axiom,
% 1.61/1.09 ! [A3: set @ a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( A3
% 1.61/1.09 != ( bot_bot @ ( set @ a ) ) )
% 1.61/1.09 => ( ! [X2: a,Y2: a] : ( member @ a @ ( inf @ X2 @ Y2 ) @ ( insert @ a @ X2 @ ( insert @ a @ Y2 @ ( bot_bot @ ( set @ a ) ) ) ) )
% 1.61/1.09 => ( member @ a @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) @ A3 ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_215_local_OInf__fin_Oinsert__not__elem,axiom,
% 1.61/1.09 ! [A3: set @ a,X: a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ~ ( member @ a @ X @ A3 )
% 1.61/1.09 => ( ( A3
% 1.61/1.09 != ( bot_bot @ ( set @ a ) ) )
% 1.61/1.09 => ( ( lattic1263571978nf_fin @ a @ inf @ ( insert @ a @ X @ A3 ) )
% 1.61/1.09 = ( inf @ X @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_216_singletonI,axiom,
% 1.61/1.09 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_217_insert__subset,axiom,
% 1.61/1.09 ! [A: $tType,X: A,A3: set @ A,B5: set @ A] :
% 1.61/1.09 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B5 )
% 1.61/1.09 = ( ( member @ A @ X @ B5 )
% 1.61/1.09 & ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_218_singleton__insert__inj__eq_H,axiom,
% 1.61/1.09 ! [A: $tType,A2: A,A3: set @ A,B2: A] :
% 1.61/1.09 ( ( ( insert @ A @ A2 @ A3 )
% 1.61/1.09 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 1.61/1.09 = ( ( A2 = B2 )
% 1.61/1.09 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_219_singleton__insert__inj__eq,axiom,
% 1.61/1.09 ! [A: $tType,B2: A,A2: A,A3: set @ A] :
% 1.61/1.09 ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
% 1.61/1.09 = ( insert @ A @ A2 @ A3 ) )
% 1.61/1.09 = ( ( A2 = B2 )
% 1.61/1.09 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_220_local_Obdd__above__insert,axiom,
% 1.61/1.09 ! [A2: a,A3: set @ a] :
% 1.61/1.09 ( ( condit2040224947_above @ a @ less_eq @ ( insert @ a @ A2 @ A3 ) )
% 1.61/1.09 = ( condit2040224947_above @ a @ less_eq @ A3 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_221_local_Obdd__below__insert,axiom,
% 1.61/1.09 ! [A2: a,A3: set @ a] :
% 1.61/1.09 ( ( condit1201339847_below @ a @ less_eq @ ( insert @ a @ A2 @ A3 ) )
% 1.61/1.09 = ( condit1201339847_below @ a @ less_eq @ A3 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_222_local_OInf__fin_Osingleton,axiom,
% 1.61/1.09 ! [X: a] :
% 1.61/1.09 ( ( lattic1263571978nf_fin @ a @ inf @ ( insert @ a @ X @ ( bot_bot @ ( set @ a ) ) ) )
% 1.61/1.09 = X ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_223_local_OSup__fin_Osingleton,axiom,
% 1.61/1.09 ! [X: a] :
% 1.61/1.09 ( ( lattic1039401930up_fin @ a @ sup @ ( insert @ a @ X @ ( bot_bot @ ( set @ a ) ) ) )
% 1.61/1.09 = X ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_224_local_OInf__fin_Oinsert,axiom,
% 1.61/1.09 ! [A3: set @ a,X: a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( A3
% 1.61/1.09 != ( bot_bot @ ( set @ a ) ) )
% 1.61/1.09 => ( ( lattic1263571978nf_fin @ a @ inf @ ( insert @ a @ X @ A3 ) )
% 1.61/1.09 = ( inf @ X @ ( lattic1263571978nf_fin @ a @ inf @ A3 ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_225_local_OSup__fin_Oinsert,axiom,
% 1.61/1.09 ! [A3: set @ a,X: a] :
% 1.61/1.09 ( ( finite_finite2 @ a @ A3 )
% 1.61/1.09 => ( ( A3
% 1.61/1.09 != ( bot_bot @ ( set @ a ) ) )
% 1.61/1.09 => ( ( lattic1039401930up_fin @ a @ sup @ ( insert @ a @ X @ A3 ) )
% 1.61/1.09 = ( sup @ X @ ( lattic1039401930up_fin @ a @ sup @ A3 ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_226_subset__singletonD,axiom,
% 1.61/1.09 ! [A: $tType,A3: set @ A,X: A] :
% 1.61/1.09 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 1.61/1.09 => ( ( A3
% 1.61/1.09 = ( bot_bot @ ( set @ A ) ) )
% 1.61/1.09 | ( A3
% 1.61/1.09 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_227_subset__singleton__iff,axiom,
% 1.61/1.09 ! [A: $tType,X5: set @ A,A2: A] :
% 1.61/1.09 ( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 1.61/1.09 = ( ( X5
% 1.61/1.09 = ( bot_bot @ ( set @ A ) ) )
% 1.61/1.09 | ( X5
% 1.61/1.09 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_228_singletonD,axiom,
% 1.61/1.09 ! [A: $tType,B2: A,A2: A] :
% 1.61/1.09 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 1.61/1.09 => ( B2 = A2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_229_singleton__iff,axiom,
% 1.61/1.09 ! [A: $tType,B2: A,A2: A] :
% 1.61/1.09 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 1.61/1.09 = ( B2 = A2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_230_doubleton__eq__iff,axiom,
% 1.61/1.09 ! [A: $tType,A2: A,B2: A,C: A,D: A] :
% 1.61/1.09 ( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 1.61/1.09 = ( insert @ A @ C @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
% 1.61/1.09 = ( ( ( A2 = C )
% 1.61/1.09 & ( B2 = D ) )
% 1.61/1.09 | ( ( A2 = D )
% 1.61/1.09 & ( B2 = C ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_231_insert__not__empty,axiom,
% 1.61/1.09 ! [A: $tType,A2: A,A3: set @ A] :
% 1.61/1.09 ( ( insert @ A @ A2 @ A3 )
% 1.61/1.09 != ( bot_bot @ ( set @ A ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_232_singleton__inject,axiom,
% 1.61/1.09 ! [A: $tType,A2: A,B2: A] :
% 1.61/1.09 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 1.61/1.09 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 1.61/1.09 => ( A2 = B2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_233_subset__insertI2,axiom,
% 1.61/1.09 ! [A: $tType,A3: set @ A,B5: set @ A,B2: A] :
% 1.61/1.09 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
% 1.61/1.09 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_234_subset__insertI,axiom,
% 1.61/1.09 ! [A: $tType,B5: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert @ A @ A2 @ B5 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_235_subset__insert,axiom,
% 1.61/1.09 ! [A: $tType,X: A,A3: set @ A,B5: set @ A] :
% 1.61/1.09 ( ~ ( member @ A @ X @ A3 )
% 1.61/1.09 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B5 ) )
% 1.61/1.09 = ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_236_insert__mono,axiom,
% 1.61/1.09 ! [A: $tType,C4: set @ A,D2: set @ A,A2: A] :
% 1.61/1.09 ( ( ord_less_eq @ ( set @ A ) @ C4 @ D2 )
% 1.61/1.09 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C4 ) @ ( insert @ A @ A2 @ D2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_237_finite__ranking__induct,axiom,
% 1.61/1.09 ! [A: $tType,B: $tType] :
% 1.61/1.09 ( ( linorder @ A )
% 1.61/1.09 => ! [S: set @ B,P: ( set @ B ) > $o,F: B > A] :
% 1.61/1.09 ( ( finite_finite2 @ B @ S )
% 1.61/1.09 => ( ( P @ ( bot_bot @ ( set @ B ) ) )
% 1.61/1.09 => ( ! [X2: B,S2: set @ B] :
% 1.61/1.09 ( ( finite_finite2 @ B @ S2 )
% 1.61/1.09 => ( ! [Y5: B] :
% 1.61/1.09 ( ( member @ B @ Y5 @ S2 )
% 1.61/1.09 => ( ord_less_eq @ A @ ( F @ Y5 ) @ ( F @ X2 ) ) )
% 1.61/1.09 => ( ( P @ S2 )
% 1.61/1.09 => ( P @ ( insert @ B @ X2 @ S2 ) ) ) ) )
% 1.61/1.09 => ( P @ S ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_238_local_OatLeastAtMost__singleton_H,axiom,
% 1.61/1.09 ! [A2: a,B2: a] :
% 1.61/1.09 ( ( A2 = B2 )
% 1.61/1.09 => ( ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 )
% 1.61/1.09 = ( insert @ a @ A2 @ ( bot_bot @ ( set @ a ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_239_finite__insert,axiom,
% 1.61/1.09 ! [A: $tType,A2: A,A3: set @ A] :
% 1.61/1.09 ( ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) )
% 1.61/1.09 = ( finite_finite2 @ A @ A3 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_240_finite__code,axiom,
% 1.61/1.09 ! [A: $tType] :
% 1.61/1.09 ( ( finite_finite @ A )
% 1.61/1.09 => ( ( finite_finite2 @ A )
% 1.61/1.09 = ( ^ [A7: set @ A] : $true ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_241_local_OIcc__eq__Icc,axiom,
% 1.61/1.09 ! [L: a,H: a,L2: a,H2: a] :
% 1.61/1.09 ( ( ( set_atLeastAtMost @ a @ less_eq @ L @ H )
% 1.61/1.09 = ( set_atLeastAtMost @ a @ less_eq @ L2 @ H2 ) )
% 1.61/1.09 = ( ( ( L = L2 )
% 1.61/1.09 & ( H = H2 ) )
% 1.61/1.09 | ( ~ ( less_eq @ L @ H )
% 1.61/1.09 & ~ ( less_eq @ L2 @ H2 ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_242_local_OatLeastAtMost__iff,axiom,
% 1.61/1.09 ! [I: a,L: a,U: a] :
% 1.61/1.09 ( ( member @ a @ I @ ( set_atLeastAtMost @ a @ less_eq @ L @ U ) )
% 1.61/1.09 = ( ( less_eq @ L @ I )
% 1.61/1.09 & ( less_eq @ I @ U ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_243_local_OatLeastatMost__empty__iff,axiom,
% 1.61/1.09 ! [A2: a,B2: a] :
% 1.61/1.09 ( ( ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 )
% 1.61/1.09 = ( bot_bot @ ( set @ a ) ) )
% 1.61/1.09 = ( ~ ( less_eq @ A2 @ B2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_244_local_OatLeastatMost__empty__iff2,axiom,
% 1.61/1.09 ! [A2: a,B2: a] :
% 1.61/1.09 ( ( ( bot_bot @ ( set @ a ) )
% 1.61/1.09 = ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) )
% 1.61/1.09 = ( ~ ( less_eq @ A2 @ B2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_245_local_Obdd__below__Icc,axiom,
% 1.61/1.09 ! [A2: a,B2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_246_local_Obdd__above__Icc,axiom,
% 1.61/1.09 ! [A2: a,B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_247_local_OatLeastatMost__subset__iff,axiom,
% 1.61/1.09 ! [A2: a,B2: a,C: a,D: a] :
% 1.61/1.09 ( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) @ ( set_atLeastAtMost @ a @ less_eq @ C @ D ) )
% 1.61/1.09 = ( ~ ( less_eq @ A2 @ B2 )
% 1.61/1.09 | ( ( less_eq @ C @ A2 )
% 1.61/1.09 & ( less_eq @ B2 @ D ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_248_local_OatLeastAtMost__singleton,axiom,
% 1.61/1.09 ! [A2: a] :
% 1.61/1.09 ( ( set_atLeastAtMost @ a @ less_eq @ A2 @ A2 )
% 1.61/1.09 = ( insert @ a @ A2 @ ( bot_bot @ ( set @ a ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_249_local_OatLeastAtMost__singleton__iff,axiom,
% 1.61/1.09 ! [A2: a,B2: a,C: a] :
% 1.61/1.09 ( ( ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 )
% 1.61/1.09 = ( insert @ a @ C @ ( bot_bot @ ( set @ a ) ) ) )
% 1.61/1.09 = ( ( A2 = B2 )
% 1.61/1.09 & ( B2 = C ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_250_local_OIcc__subset__Ici__iff,axiom,
% 1.61/1.09 ! [L: a,H: a,L2: a] :
% 1.61/1.09 ( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ L @ H ) @ ( set_atLeast @ a @ less_eq @ L2 ) )
% 1.61/1.09 = ( ~ ( less_eq @ L @ H )
% 1.61/1.09 | ( less_eq @ L2 @ L ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_251_local_OIcc__subset__Iic__iff,axiom,
% 1.61/1.09 ! [L: a,H: a,H2: a] :
% 1.61/1.09 ( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ L @ H ) @ ( set_atMost @ a @ less_eq @ H2 ) )
% 1.61/1.09 = ( ~ ( less_eq @ L @ H )
% 1.61/1.09 | ( less_eq @ H @ H2 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_252_finite,axiom,
% 1.61/1.09 ! [A: $tType] :
% 1.61/1.09 ( ( finite_finite @ A )
% 1.61/1.09 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_253_finite__set__choice,axiom,
% 1.61/1.09 ! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
% 1.61/1.09 ( ( finite_finite2 @ A @ A3 )
% 1.61/1.09 => ( ! [X2: A] :
% 1.61/1.09 ( ( member @ A @ X2 @ A3 )
% 1.61/1.09 => ? [X_1: B] : ( P @ X2 @ X_1 ) )
% 1.61/1.09 => ? [F3: A > B] :
% 1.61/1.09 ! [X4: A] :
% 1.61/1.09 ( ( member @ A @ X4 @ A3 )
% 1.61/1.09 => ( P @ X4 @ ( F3 @ X4 ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_254_comp__fun__idem_Ofun__left__idem,axiom,
% 1.61/1.09 ! [A: $tType,B: $tType,F: A > B > B,X: A,Z: B] :
% 1.61/1.09 ( ( finite_comp_fun_idem @ A @ B @ F )
% 1.61/1.09 => ( ( F @ X @ ( F @ X @ Z ) )
% 1.61/1.09 = ( F @ X @ Z ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(fact_255_order__class_Ofinite__has__minimal2,axiom,
% 1.61/1.09 ! [A: $tType] :
% 1.61/1.09 ( ( order @ A )
% 1.61/1.09 => ! [A3: set @ A,A2: A] :
% 1.61/1.09 ( ( finite_finite2 @ A @ A3 )
% 1.61/1.09 => ( ( member @ A @ A2 @ A3 )
% 1.61/1.09 => ? [X2: A] :
% 1.61/1.09 ( ( member @ A @ X2 @ A3 )
% 1.61/1.09 & ( ord_less_eq @ A @ X2 @ A2 )
% 1.61/1.09 & ! [Xa: A] :
% 1.61/1.09 ( ( member @ A @ Xa @ A3 )
% 1.61/1.09 => ( ( ord_less_eq @ A @ Xa @ X2 )
% 1.61/1.09 => ( X2 = Xa ) ) ) ) ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 1.61/1.09 ! [A8: $tType,A9: $tType] :
% 1.61/1.09 ( ( ( finite_finite @ A8 )
% 1.61/1.09 & ( finite_finite @ A9 ) )
% 1.61/1.09 => ( finite_finite @ ( A8 > A9 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(tcon_fun___Orderings_Oorder,axiom,
% 1.61/1.09 ! [A8: $tType,A9: $tType] :
% 1.61/1.09 ( ( order @ A9 )
% 1.61/1.09 => ( order @ ( A8 > A9 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(tcon_Set_Oset___Finite__Set_Ofinite_1,axiom,
% 1.61/1.09 ! [A8: $tType] :
% 1.61/1.09 ( ( finite_finite @ A8 )
% 1.61/1.09 => ( finite_finite @ ( set @ A8 ) ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
% 1.61/1.09 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 1.61/1.09
% 1.61/1.09 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 1.61/1.09 linorder @ $o ).
% 1.61/1.09
% 1.61/1.09 thf(tcon_HOL_Obool___Finite__Set_Ofinite_3,axiom,
% 1.61/1.09 finite_finite @ $o ).
% 1.61/1.09
% 1.61/1.09 thf(tcon_HOL_Obool___Orderings_Oorder_4,axiom,
% 1.61/1.09 order @ $o ).
% 1.61/1.09
% 1.61/1.09 thf(conj_0,conjecture,
% 1.61/1.09 ( ( modula1593167561_e_aux @ a @ inf @ sup @ b @ c @ a2 )
% 1.61/1.09 = ( modula1593167561_e_aux @ a @ inf @ sup @ a2 @ b @ c ) ) ).
% 1.61/1.09 ------------------------
% 1.61/1.10 % (3486)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_3385 for (2999ds/2Mi)
% 1.61/1.10 % (3483)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_3385 for (2999ds/183Mi)
% 1.61/1.10 % (3488)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_3385 for (2999ds/275Mi)
% 1.61/1.10 % (3487)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_3385 for (2999ds/2Mi)
% 1.61/1.10 % (3484)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_3385 for (2999ds/4Mi)
% 1.61/1.10 % (3485)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_3385 for (2999ds/27Mi)
% 1.61/1.10 % (3489)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_3385 for (2999ds/18Mi)
% 1.61/1.11 % (3486)Instruction limit reached!
% 1.61/1.11 % (3486)------------------------------
% 1.61/1.11 % (3486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.61/1.11 % (3486)Termination reason: Unknown
% 1.61/1.11 % (3486)Termination phase: shuffling
% 1.61/1.11
% 1.61/1.11 % (3486)Memory used [KB]: 1279
% 1.61/1.11 % (3486)Time elapsed: 0.002 s
% 1.61/1.11 % (3486)Instructions burned: 2 (million)
% 1.61/1.11 % (3487)Instruction limit reached!
% 1.61/1.11 % (3487)------------------------------
% 1.61/1.11 % (3487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.61/1.11 % (3487)Termination reason: Unknown
% 1.61/1.11 % (3487)Termination phase: shuffling
% 1.61/1.11
% 1.61/1.11 % (3487)Memory used [KB]: 1279
% 1.61/1.11 % (3487)Time elapsed: 0.002 s
% 1.61/1.11 % (3487)Instructions burned: 3 (million)
% 1.61/1.11 % (3487)------------------------------
% 1.61/1.11 % (3487)------------------------------
% 1.61/1.11 % (3486)------------------------------
% 1.61/1.11 % (3486)------------------------------
% 1.61/1.11 % (3484)Instruction limit reached!
% 1.61/1.11 % (3484)------------------------------
% 1.61/1.11 % (3484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.61/1.11 % (3484)Termination reason: Unknown
% 1.61/1.11 % (3484)Termination phase: shuffling
% 1.61/1.11
% 1.61/1.11 % (3484)Memory used [KB]: 1407
% 1.61/1.11 % (3484)Time elapsed: 0.003 s
% 1.61/1.11 % (3484)Instructions burned: 6 (million)
% 1.61/1.11 % (3484)------------------------------
% 1.61/1.11 % (3484)------------------------------
% 1.61/1.11 % (3489)Instruction limit reached!
% 1.61/1.11 % (3489)------------------------------
% 1.61/1.11 % (3489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.61/1.11 % (3489)Termination reason: Unknown
% 1.61/1.11 % (3489)Termination phase: Property scanning
% 1.61/1.11
% 1.61/1.11 % (3489)Memory used [KB]: 1535
% 1.61/1.11 % (3489)Time elapsed: 0.007 s
% 1.61/1.11 % (3489)Instructions burned: 18 (million)
% 1.61/1.11 % (3489)------------------------------
% 1.61/1.11 % (3489)------------------------------
% 1.77/1.11 % (3485)Instruction limit reached!
% 1.77/1.11 % (3485)------------------------------
% 1.77/1.11 % (3485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.11 % (3485)Termination reason: Unknown
% 1.77/1.11 % (3485)Termination phase: Preprocessing 1
% 1.77/1.11
% 1.77/1.11 % (3485)Memory used [KB]: 1535
% 1.77/1.11 % (3485)Time elapsed: 0.010 s
% 1.77/1.11 % (3485)Instructions burned: 28 (million)
% 1.77/1.11 % (3485)------------------------------
% 1.77/1.11 % (3485)------------------------------
% 1.77/1.11 % (3490)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_3385 for (2999ds/3Mi)
% 1.77/1.11 % (3491)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_3385 for (2999ds/37Mi)
% 1.77/1.12 % (3492)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_3385 for (2999ds/15Mi)
% 1.77/1.12 % (3490)Instruction limit reached!
% 1.77/1.12 % (3490)------------------------------
% 1.77/1.12 % (3490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.12 % (3490)Termination reason: Unknown
% 1.77/1.12 % (3490)Termination phase: shuffling
% 1.77/1.12
% 1.77/1.12 % (3490)Memory used [KB]: 1279
% 1.77/1.12 % (3490)Time elapsed: 0.002 s
% 1.77/1.12 % (3490)Instructions burned: 3 (million)
% 1.77/1.12 % (3490)------------------------------
% 1.77/1.12 % (3490)------------------------------
% 1.77/1.12 % (3493)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_3385 for (2999ds/3Mi)
% 1.77/1.12 % (3492)Instruction limit reached!
% 1.77/1.12 % (3492)------------------------------
% 1.77/1.12 % (3492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.12 % (3492)Termination reason: Unknown
% 1.77/1.12 % (3492)Termination phase: shuffling
% 1.77/1.12
% 1.77/1.12 % (3492)Memory used [KB]: 1535
% 1.77/1.12 % (3492)Time elapsed: 0.006 s
% 1.77/1.12 % (3492)Instructions burned: 15 (million)
% 1.77/1.12 % (3492)------------------------------
% 1.77/1.12 % (3492)------------------------------
% 1.77/1.12 % (3493)Instruction limit reached!
% 1.77/1.12 % (3493)------------------------------
% 1.77/1.12 % (3493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.12 % (3493)Termination reason: Unknown
% 1.77/1.12 % (3493)Termination phase: shuffling
% 1.77/1.12
% 1.77/1.12 % (3493)Memory used [KB]: 1279
% 1.77/1.12 % (3493)Time elapsed: 0.002 s
% 1.77/1.12 % (3493)Instructions burned: 3 (million)
% 1.77/1.12 % (3493)------------------------------
% 1.77/1.12 % (3493)------------------------------
% 1.77/1.12 % (3488)First to succeed.
% 1.77/1.12 % (3488)Refutation found. Thanks to Tanya!
% 1.77/1.12 % SZS status Theorem for DTF2THF_3385
% 1.77/1.12 % SZS output start Proof for DTF2THF_3385
% 1.77/1.12 thf(type_def_6, type, set: ($tType) > $tType).
% 1.77/1.12 thf(type_def_7, type, a: $tType).
% 1.77/1.12 thf(func_def_0, type, set: ($tType) > $tType).
% 1.77/1.12 thf(func_def_1, type, a: $tType).
% 1.77/1.12 thf(func_def_2, type, type: !>[X0: $tType]:($o)).
% 1.77/1.12 thf(func_def_3, type, order: !>[X0: $tType]:($o)).
% 1.77/1.12 thf(func_def_4, type, finite_finite: !>[X0: $tType]:($o)).
% 1.77/1.12 thf(func_def_5, type, linorder: !>[X0: $tType]:($o)).
% 1.77/1.12 thf(func_def_6, type, condit2040224947_above: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > $o)).
% 1.77/1.12 thf(func_def_7, type, condit1201339847_below: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > $o)).
% 1.77/1.12 thf(func_def_8, type, finite_comp_fun_idem: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > $o)).
% 1.77/1.12 thf(func_def_9, type, finite_finite2: !>[X0: $tType]:(set @ X0 > $o)).
% 1.77/1.12 thf(func_def_10, type, abel_semigroup: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 1.77/1.12 thf(func_def_11, type, semigroup: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 1.77/1.12 thf(func_def_12, type, semilattice: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 1.77/1.12 thf(func_def_13, type, lattic1263571978nf_fin: !>[X0: $tType]:((X0 > X0 > X0) > set @ X0 > X0)).
% 1.77/1.12 thf(func_def_14, type, lattic35693393ce_set: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 1.77/1.12 thf(func_def_15, type, lattic1039401930up_fin: !>[X0: $tType]:((X0 > X0 > X0) > set @ X0 > X0)).
% 1.77/1.12 thf(func_def_16, type, modula1221160330_d_aux: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > X0) > X0 > X0 > X0 > X0)).
% 1.77/1.12 thf(func_def_17, type, modula1593167561_e_aux: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > X0) > X0 > X0 > X0 > X0)).
% 1.77/1.12 thf(func_def_18, type, bot_bot: !>[X0: $tType]:(X0)).
% 1.77/1.12 thf(func_def_19, type, least: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > $o) > X0)).
% 1.77/1.12 thf(func_def_20, type, max: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > X0)).
% 1.77/1.12 thf(func_def_21, type, min: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > X0)).
% 1.77/1.12 thf(func_def_22, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 1.77/1.12 thf(func_def_23, type, greatest: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > $o) > X0)).
% 1.77/1.12 thf(func_def_24, type, antimono: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1) > $o)).
% 1.77/1.12 thf(func_def_25, type, mono: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1) > $o)).
% 1.77/1.12 thf(func_def_26, type, transp: !>[X0: $tType]:((X0 > X0 > $o) > $o)).
% 1.77/1.12 thf(func_def_27, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 1.77/1.12 thf(func_def_28, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 1.77/1.12 thf(func_def_29, type, set_atLeast: !>[X0: $tType]:((X0 > X0 > $o) > X0 > set @ X0)).
% 1.77/1.12 thf(func_def_30, type, set_atLeastAtMost: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > set @ X0)).
% 1.77/1.12 thf(func_def_31, type, set_atMost: !>[X0: $tType]:((X0 > X0 > $o) > X0 > set @ X0)).
% 1.77/1.12 thf(func_def_32, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 1.77/1.12 thf(func_def_33, type, a2: a).
% 1.77/1.12 thf(func_def_34, type, b: a).
% 1.77/1.12 thf(func_def_35, type, c: a).
% 1.77/1.12 thf(func_def_36, type, inf: a > a > a).
% 1.77/1.12 thf(func_def_37, type, less_eq: a > a > $o).
% 1.77/1.12 thf(func_def_38, type, sup: a > a > a).
% 1.77/1.12 thf(func_def_51, type, sK0: a).
% 1.77/1.12 thf(func_def_52, type, sK1: a).
% 1.77/1.12 thf(func_def_53, type, sK2: a).
% 1.77/1.12 thf(func_def_54, type, sK3: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 1.77/1.12 thf(func_def_55, type, sK4: (a > a > a) > a).
% 1.77/1.12 thf(func_def_56, type, sK5: (a > a > a) > a).
% 1.77/1.12 thf(func_def_57, type, sK6: (a > a > a) > a).
% 1.77/1.12 thf(func_def_58, type, sK7: (a > a > a) > a).
% 1.77/1.12 thf(func_def_59, type, sK8: (a > a > a) > a).
% 1.77/1.12 thf(func_def_60, type, sK9: (a > a > a) > a).
% 1.77/1.12 thf(func_def_61, type, sK10: (a > a > a) > a).
% 1.77/1.12 thf(func_def_62, type, sK11: a).
% 1.77/1.12 thf(func_def_63, type, sK12: a).
% 1.77/1.12 thf(func_def_64, type, sK13: a).
% 1.77/1.12 thf(func_def_65, type, sK14: (a > a > a) > a).
% 1.77/1.12 thf(func_def_66, type, sK15: (a > a > a) > a).
% 1.77/1.12 thf(func_def_67, type, sK16: (a > a > a) > a).
% 1.77/1.12 thf(func_def_68, type, sK17: (a > a > a) > a).
% 1.77/1.12 thf(func_def_69, type, sK18: (a > a > a) > a).
% 1.77/1.12 thf(func_def_70, type, sK19: (a > a > a) > a).
% 1.77/1.12 thf(func_def_71, type, sK20: (a > a > a) > a).
% 1.77/1.12 thf(func_def_73, type, ph22: !>[X0: $tType]:(X0)).
% 1.77/1.12 thf(f959,plain,(
% 1.77/1.12 $false),
% 1.77/1.12 inference(subsumption_resolution,[status(thm)],[f958,f898])).
% 1.77/1.12 thf(f898,plain,(
% 1.77/1.12 ( ! [X0 : a,X1 : a] : (((inf @ X1 @ X0) = (inf @ X0 @ X1))) )),
% 1.77/1.12 inference(cnf_transformation,[status(thm)],[f695])).
% 1.77/1.12 thf(f695,plain,(
% 1.77/1.12 ! [X0 : a,X1 : a] : ((inf @ X1 @ X0) = (inf @ X0 @ X1))),
% 1.77/1.12 inference(rectify,[status(thm)],[f5])).
% 1.77/1.12 thf(f5,axiom,(
% 1.77/1.12 ! [X5 : a,X6 : a] : ((inf @ X5 @ X6) = (inf @ X6 @ X5))),
% 1.77/1.12 file('/export/starexec/sandbox/tmp/tmp.3GJe6d4PX9/DTF2THF_3385.p',fact_4_local_Oinf__commute)).
% 1.77/1.12 thf(f958,plain,(
% 1.77/1.12 ((inf @ (inf @ (sup @ a2 @ c) @ (sup @ b @ c)) @ (sup @ a2 @ b)) != (inf @ (sup @ a2 @ b) @ (inf @ (sup @ a2 @ c) @ (sup @ b @ c))))),
% 1.77/1.12 inference(forward_demodulation,[status(thm)],[f957,f898])).
% 1.77/1.12 thf(f957,plain,(
% 1.77/1.12 ((inf @ (sup @ a2 @ b) @ (inf @ (sup @ a2 @ c) @ (sup @ b @ c))) != (inf @ (inf @ (sup @ b @ c) @ (sup @ a2 @ c)) @ (sup @ a2 @ b)))),
% 1.77/1.12 inference(forward_demodulation,[status(thm)],[f956,f837])).
% 1.77/1.12 thf(f837,plain,(
% 1.77/1.12 ( ! [X0 : a,X1 : a] : (((sup @ X1 @ X0) = (sup @ X0 @ X1))) )),
% 1.77/1.12 inference(cnf_transformation,[status(thm)],[f715])).
% 1.77/1.12 thf(f715,plain,(
% 1.77/1.12 ! [X0 : a,X1 : a] : ((sup @ X1 @ X0) = (sup @ X0 @ X1))),
% 1.77/1.12 inference(rectify,[status(thm)],[f8])).
% 1.77/1.12 thf(f8,axiom,(
% 1.77/1.12 ! [X2 : a,X3 : a] : ((sup @ X2 @ X3) = (sup @ X3 @ X2))),
% 1.77/1.12 file('/export/starexec/sandbox/tmp/tmp.3GJe6d4PX9/DTF2THF_3385.p',fact_7_local_Osup_Ocommute)).
% 1.77/1.12 thf(f956,plain,(
% 1.77/1.12 ((inf @ (inf @ (sup @ b @ c) @ (sup @ c @ a2)) @ (sup @ a2 @ b)) != (inf @ (sup @ a2 @ b) @ (inf @ (sup @ a2 @ c) @ (sup @ b @ c))))),
% 1.77/1.12 inference(superposition,[status(thm)],[f955,f916])).
% 1.77/1.12 thf(f916,plain,(
% 1.77/1.12 ( ! [X2 : a,X0 : a,X1 : a] : (((inf @ (inf @ (sup @ X0 @ X1) @ (sup @ X1 @ X2)) @ (sup @ X2 @ X0)) = (modula1593167561_e_aux @ a @ inf @ sup @ X0 @ X1 @ X2))) )),
% 1.77/1.12 inference(cnf_transformation,[status(thm)],[f690])).
% 1.77/1.12 thf(f690,plain,(
% 1.77/1.12 ! [X0 : a,X1 : a,X2 : a] : ((inf @ (inf @ (sup @ X0 @ X1) @ (sup @ X1 @ X2)) @ (sup @ X2 @ X0)) = (modula1593167561_e_aux @ a @ inf @ sup @ X0 @ X1 @ X2))),
% 1.77/1.12 inference(rectify,[status(thm)],[f15])).
% 1.77/1.12 thf(f15,axiom,(
% 1.77/1.12 ! [X2 : a,X3 : a,X4 : a] : ((modula1593167561_e_aux @ a @ inf @ sup @ X2 @ X3 @ X4) = (inf @ (inf @ (sup @ X2 @ X3) @ (sup @ X3 @ X4)) @ (sup @ X4 @ X2)))),
% 1.77/1.12 file('/export/starexec/sandbox/tmp/tmp.3GJe6d4PX9/DTF2THF_3385.p',fact_14_e__aux__def)).
% 1.77/1.12 thf(f955,plain,(
% 1.77/1.12 ((modula1593167561_e_aux @ a @ inf @ sup @ b @ c @ a2) != (inf @ (sup @ a2 @ b) @ (inf @ (sup @ a2 @ c) @ (sup @ b @ c))))),
% 1.77/1.12 inference(forward_demodulation,[status(thm)],[f954,f898])).
% 1.77/1.12 thf(f954,plain,(
% 1.77/1.12 ((modula1593167561_e_aux @ a @ inf @ sup @ b @ c @ a2) != (inf @ (sup @ a2 @ b) @ (inf @ (sup @ b @ c) @ (sup @ a2 @ c))))),
% 1.77/1.12 inference(forward_demodulation,[status(thm)],[f953,f837])).
% 1.77/1.12 thf(f953,plain,(
% 1.77/1.12 ((modula1593167561_e_aux @ a @ inf @ sup @ b @ c @ a2) != (inf @ (sup @ a2 @ b) @ (inf @ (sup @ b @ c) @ (sup @ c @ a2))))),
% 1.77/1.12 inference(forward_demodulation,[status(thm)],[f952,f898])).
% 1.77/1.12 thf(f952,plain,(
% 1.77/1.12 ((modula1593167561_e_aux @ a @ inf @ sup @ b @ c @ a2) != (inf @ (sup @ a2 @ b) @ (inf @ (sup @ c @ a2) @ (sup @ b @ c))))),
% 1.77/1.12 inference(forward_demodulation,[status(thm)],[f951,f832])).
% 1.77/1.12 thf(f832,plain,(
% 1.77/1.12 ( ! [X2 : a,X0 : a,X1 : a] : (((inf @ X0 @ (inf @ X1 @ X2)) = (inf @ X1 @ (inf @ X0 @ X2)))) )),
% 1.77/1.12 inference(cnf_transformation,[status(thm)],[f692])).
% 1.77/1.12 thf(f692,plain,(
% 1.77/1.12 ! [X0 : a,X1 : a,X2 : a] : ((inf @ X0 @ (inf @ X1 @ X2)) = (inf @ X1 @ (inf @ X0 @ X2)))),
% 1.77/1.12 inference(rectify,[status(thm)],[f6])).
% 1.77/1.12 thf(f6,axiom,(
% 1.77/1.12 ! [X5 : a,X6 : a,X7 : a] : ((inf @ X5 @ (inf @ X6 @ X7)) = (inf @ X6 @ (inf @ X5 @ X7)))),
% 1.77/1.12 file('/export/starexec/sandbox/tmp/tmp.3GJe6d4PX9/DTF2THF_3385.p',fact_5_local_Oinf__left__commute)).
% 1.77/1.12 thf(f951,plain,(
% 1.77/1.12 ((modula1593167561_e_aux @ a @ inf @ sup @ b @ c @ a2) != (inf @ (sup @ c @ a2) @ (inf @ (sup @ a2 @ b) @ (sup @ b @ c))))),
% 1.77/1.12 inference(forward_demodulation,[status(thm)],[f950,f898])).
% 1.77/1.12 thf(f950,plain,(
% 1.77/1.12 ((modula1593167561_e_aux @ a @ inf @ sup @ b @ c @ a2) != (inf @ (inf @ (sup @ a2 @ b) @ (sup @ b @ c)) @ (sup @ c @ a2)))),
% 1.77/1.12 inference(forward_demodulation,[status(thm)],[f899,f916])).
% 1.77/1.12 thf(f899,plain,(
% 1.77/1.12 ((modula1593167561_e_aux @ a @ inf @ sup @ b @ c @ a2) != (modula1593167561_e_aux @ a @ inf @ sup @ a2 @ b @ c))),
% 1.77/1.12 inference(cnf_transformation,[status(thm)],[f689])).
% 1.77/1.12 thf(f689,plain,(
% 1.77/1.12 ((modula1593167561_e_aux @ a @ inf @ sup @ b @ c @ a2) != (modula1593167561_e_aux @ a @ inf @ sup @ a2 @ b @ c))),
% 1.77/1.12 inference(flattening,[status(thm)],[f265])).
% 1.77/1.12 thf(f265,negated_conjecture,(
% 1.77/1.12 ~((modula1593167561_e_aux @ a @ inf @ sup @ b @ c @ a2) = (modula1593167561_e_aux @ a @ inf @ sup @ a2 @ b @ c))),
% 1.77/1.12 inference(negated_conjecture,[status(cth)],[f264])).
% 1.77/1.12 thf(f264,conjecture,(
% 1.77/1.12 ((modula1593167561_e_aux @ a @ inf @ sup @ b @ c @ a2) = (modula1593167561_e_aux @ a @ inf @ sup @ a2 @ b @ c))),
% 1.77/1.12 file('/export/starexec/sandbox/tmp/tmp.3GJe6d4PX9/DTF2THF_3385.p',conj_0)).
% 1.77/1.12 % SZS output end Proof for DTF2THF_3385
% 1.77/1.12 % (3488)------------------------------
% 1.77/1.12 % (3488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.12 % (3488)Termination reason: Refutation
% 1.77/1.12
% 1.77/1.12 % (3488)Memory used [KB]: 6268
% 1.77/1.12 % (3488)Time elapsed: 0.018 s
% 1.77/1.12 % (3488)Instructions burned: 54 (million)
% 1.77/1.12 % (3488)------------------------------
% 1.77/1.12 % (3488)------------------------------
% 1.77/1.12 % (3482)Success in time 0.024 s
%------------------------------------------------------------------------------