%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP124^1 : 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 : n027.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 08:55:44 AM UTC 2026
% Result : Theorem 3.24s 1.82s
% Output : Refutation 3.24s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 5
% Syntax : Number of formulae : 25 ( 25 unt; 0 typ; 0 def)
% Number of atoms : 25 ( 24 equ; 0 cnn)
% Maximal formula atoms : 1 ( 1 avg)
% Number of connectives : 280 ( 12 ~; 0 |; 0 &; 268 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 4 ( 3 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 62 ( 60 usr; 12 con; 0-5 aty)
% Number of variables : 27 ( 0 ^; 27 !; 0 ?; 27 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
a: $tType ).
thf(type_def_7,type,
set_a: $tType ).
thf(type_def_8,type,
set_set_a: $tType ).
thf(func_def_0,type,
set_set_a: $tType ).
thf(func_def_1,type,
set_a: $tType ).
thf(func_def_2,type,
a: $tType ).
thf(func_def_3,type,
condit1627435690bove_a: ( a > a > $o ) > set_a > $o ).
thf(func_def_4,type,
condit1001553558elow_a: ( a > a > $o ) > set_a > $o ).
thf(func_def_5,type,
finite40241356em_a_a: ( a > a > a ) > $o ).
thf(func_def_6,type,
finite_finite_set_a: set_set_a > $o ).
thf(func_def_7,type,
finite_finite_a: set_a > $o ).
thf(func_def_8,type,
abel_semigroup_a: ( a > a > a ) > $o ).
thf(func_def_9,type,
semigroup_a: ( a > a > a ) > $o ).
thf(func_def_10,type,
semilattice_a: ( a > a > a ) > $o ).
thf(func_def_11,type,
lattic247676691_fin_a: ( a > a > a ) > set_a > a ).
thf(func_def_12,type,
lattic1885654924_set_a: ( a > a > a ) > $o ).
thf(func_def_13,type,
lattic1781219155_fin_a: ( a > a > a ) > set_a > a ).
thf(func_def_14,type,
modula1936294176_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
thf(func_def_15,type,
modula1144073633_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
thf(func_def_16,type,
bot_bot_set_set_a: set_set_a ).
thf(func_def_17,type,
bot_bot_set_a: set_a ).
thf(func_def_18,type,
least_a: ( a > a > $o ) > ( a > $o ) > a ).
thf(func_def_19,type,
max_a: ( a > a > $o ) > a > a > a ).
thf(func_def_20,type,
min_a: ( a > a > $o ) > a > a > a ).
thf(func_def_21,type,
ord_le318720350_set_a: set_set_a > set_set_a > $o ).
thf(func_def_22,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_23,type,
greatest_a: ( a > a > $o ) > ( a > $o ) > a ).
thf(func_def_24,type,
antimono_a_set_a: ( a > a > $o ) > ( a > set_a ) > $o ).
thf(func_def_25,type,
mono_a_set_a: ( a > a > $o ) > ( a > set_a ) > $o ).
thf(func_def_26,type,
transp_a: ( a > a > $o ) > $o ).
thf(func_def_27,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(func_def_28,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_29,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(func_def_30,type,
insert_a: a > set_a > set_a ).
thf(func_def_31,type,
set_atLeastAtMost_a: ( a > a > $o ) > a > a > set_a ).
thf(func_def_32,type,
set_atLeast_a: ( a > a > $o ) > a > set_a ).
thf(func_def_33,type,
set_atMost_a: ( a > a > $o ) > a > set_a ).
thf(func_def_34,type,
member_set_a: set_a > set_set_a > $o ).
thf(func_def_35,type,
member_a: a > set_a > $o ).
thf(func_def_36,type,
a2: a ).
thf(func_def_37,type,
b: a ).
thf(func_def_38,type,
c: a ).
thf(func_def_39,type,
inf: a > a > a ).
thf(func_def_40,type,
less_eq: a > a > $o ).
thf(func_def_41,type,
sup: a > a > a ).
thf(func_def_54,type,
sK0: a ).
thf(func_def_55,type,
sK1: a ).
thf(func_def_56,type,
sK2: a ).
thf(func_def_57,type,
sK3: ( a > a > a ) > a ).
thf(func_def_58,type,
sK4: ( a > a > a ) > a ).
thf(func_def_59,type,
sK5: ( a > a > a ) > a ).
thf(func_def_60,type,
sK6: ( a > a > a ) > a ).
thf(func_def_61,type,
sK7: ( a > a > a ) > a ).
thf(func_def_62,type,
sK8: ( a > a > a ) > a ).
thf(func_def_63,type,
sK9: ( a > a > a ) > a ).
thf(func_def_64,type,
sK10: ( a > a > a ) > a ).
thf(func_def_65,type,
sK11: ( a > a > a ) > a ).
thf(func_def_66,type,
sK12: ( a > a > a ) > a ).
thf(func_def_67,type,
sK13: ( a > a > a ) > a ).
thf(func_def_68,type,
sK14: ( a > a > a ) > a ).
thf(func_def_69,type,
sK15: ( a > a > a ) > a ).
thf(func_def_70,type,
sK16: ( a > a > a ) > a ).
thf(func_def_71,type,
sK17: a ).
thf(func_def_72,type,
sK18: a ).
thf(func_def_73,type,
sK19: a ).
thf(func_def_75,type,
ph21:
!>[X0: $tType] : X0 ).
thf(f964,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f963,f853]) ).
thf(f853,plain,
! [X0: a,X1: a] :
( ( inf @ X0 @ X1 )
= ( inf @ X1 @ X0 ) ),
inference(cnf_transformation,[status(thm)],[f709]) ).
thf(f709,plain,
! [X0: a,X1: a] :
( ( inf @ X0 @ X1 )
= ( inf @ X1 @ X0 ) ),
inference(rectify,[status(thm)],[f5]) ).
thf(f5,axiom,
! [X4: a,X3: a] :
( ( inf @ X3 @ X4 )
= ( inf @ X4 @ X3 ) ),
file('/export/starexec/sandbox/tmp/tmp.yhjYV0Pxgo/DTF2THF_24647.p',fact_4_local_Oinf__commute) ).
thf(f963,plain,
( ( inf @ ( inf @ ( sup @ a2 @ c ) @ ( sup @ b @ c ) ) @ ( sup @ a2 @ b ) )
!= ( inf @ ( sup @ a2 @ b ) @ ( inf @ ( sup @ a2 @ c ) @ ( sup @ b @ c ) ) ) ),
inference(forward_demodulation,[status(thm)],[f962,f853]) ).
thf(f962,plain,
( ( inf @ ( inf @ ( sup @ b @ c ) @ ( sup @ a2 @ c ) ) @ ( sup @ a2 @ b ) )
!= ( inf @ ( sup @ a2 @ b ) @ ( inf @ ( sup @ a2 @ c ) @ ( sup @ b @ c ) ) ) ),
inference(forward_demodulation,[status(thm)],[f961,f867]) ).
thf(f867,plain,
! [X0: a,X1: a] :
( ( sup @ X0 @ X1 )
= ( sup @ X1 @ X0 ) ),
inference(cnf_transformation,[status(thm)],[f722]) ).
thf(f722,plain,
! [X0: a,X1: a] :
( ( sup @ X0 @ X1 )
= ( sup @ X1 @ X0 ) ),
inference(rectify,[status(thm)],[f11]) ).
thf(f11,axiom,
! [X3: a,X4: a] :
( ( sup @ X3 @ X4 )
= ( sup @ X4 @ X3 ) ),
file('/export/starexec/sandbox/tmp/tmp.yhjYV0Pxgo/DTF2THF_24647.p',fact_10_local_Osup__commute) ).
thf(f961,plain,
( ( inf @ ( inf @ ( sup @ b @ c ) @ ( sup @ c @ a2 ) ) @ ( sup @ a2 @ b ) )
!= ( inf @ ( sup @ a2 @ b ) @ ( inf @ ( sup @ a2 @ c ) @ ( sup @ b @ c ) ) ) ),
inference(superposition,[status(thm)],[f960,f892]) ).
thf(f892,plain,
! [X2: a,X0: a,X1: a] :
( ( modula1144073633_aux_a @ inf @ sup @ X0 @ X1 @ X2 )
= ( inf @ ( inf @ ( sup @ X0 @ X1 ) @ ( sup @ X1 @ X2 ) ) @ ( sup @ X2 @ X0 ) ) ),
inference(cnf_transformation,[status(thm)],[f15]) ).
thf(f15,axiom,
! [X0: a,X1: a,X2: a] :
( ( modula1144073633_aux_a @ inf @ sup @ X0 @ X1 @ X2 )
= ( inf @ ( inf @ ( sup @ X0 @ X1 ) @ ( sup @ X1 @ X2 ) ) @ ( sup @ X2 @ X0 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.yhjYV0Pxgo/DTF2THF_24647.p',fact_14_e__aux__def) ).
thf(f960,plain,
( ( modula1144073633_aux_a @ inf @ sup @ b @ c @ a2 )
!= ( inf @ ( sup @ a2 @ b ) @ ( inf @ ( sup @ a2 @ c ) @ ( sup @ b @ c ) ) ) ),
inference(forward_demodulation,[status(thm)],[f959,f853]) ).
thf(f959,plain,
( ( modula1144073633_aux_a @ inf @ sup @ b @ c @ a2 )
!= ( inf @ ( sup @ a2 @ b ) @ ( inf @ ( sup @ b @ c ) @ ( sup @ a2 @ c ) ) ) ),
inference(forward_demodulation,[status(thm)],[f958,f867]) ).
thf(f958,plain,
( ( modula1144073633_aux_a @ inf @ sup @ b @ c @ a2 )
!= ( inf @ ( sup @ a2 @ b ) @ ( inf @ ( sup @ b @ c ) @ ( sup @ c @ a2 ) ) ) ),
inference(forward_demodulation,[status(thm)],[f957,f853]) ).
thf(f957,plain,
( ( modula1144073633_aux_a @ inf @ sup @ b @ c @ a2 )
!= ( inf @ ( sup @ a2 @ b ) @ ( inf @ ( sup @ c @ a2 ) @ ( sup @ b @ c ) ) ) ),
inference(forward_demodulation,[status(thm)],[f956,f949]) ).
thf(f949,plain,
! [X2: a,X0: a,X1: a] :
( ( inf @ X0 @ ( inf @ X1 @ X2 ) )
= ( inf @ X1 @ ( inf @ X0 @ X2 ) ) ),
inference(cnf_transformation,[status(thm)],[f703]) ).
thf(f703,plain,
! [X0: a,X1: a,X2: a] :
( ( inf @ X0 @ ( inf @ X1 @ X2 ) )
= ( inf @ X1 @ ( inf @ X0 @ X2 ) ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
! [X1: a,X0: a,X2: a] :
( ( inf @ X0 @ ( inf @ X1 @ X2 ) )
= ( inf @ X1 @ ( inf @ X0 @ X2 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.yhjYV0Pxgo/DTF2THF_24647.p',fact_2_local_Oinf_Oleft__commute) ).
thf(f956,plain,
( ( modula1144073633_aux_a @ inf @ sup @ b @ c @ a2 )
!= ( inf @ ( sup @ c @ a2 ) @ ( inf @ ( sup @ a2 @ b ) @ ( sup @ b @ c ) ) ) ),
inference(forward_demodulation,[status(thm)],[f955,f853]) ).
thf(f955,plain,
( ( modula1144073633_aux_a @ inf @ sup @ b @ c @ a2 )
!= ( inf @ ( inf @ ( sup @ a2 @ b ) @ ( sup @ b @ c ) ) @ ( sup @ c @ a2 ) ) ),
inference(forward_demodulation,[status(thm)],[f830,f892]) ).
thf(f830,plain,
( ( modula1144073633_aux_a @ inf @ sup @ b @ c @ a2 )
!= ( modula1144073633_aux_a @ inf @ sup @ a2 @ b @ c ) ),
inference(cnf_transformation,[status(thm)],[f702]) ).
thf(f702,plain,
( ( modula1144073633_aux_a @ inf @ sup @ b @ c @ a2 )
!= ( modula1144073633_aux_a @ inf @ sup @ a2 @ b @ c ) ),
inference(flattening,[status(thm)],[f272]) ).
thf(f272,negated_conjecture,
( ( modula1144073633_aux_a @ inf @ sup @ b @ c @ a2 )
!= ( modula1144073633_aux_a @ inf @ sup @ a2 @ b @ c ) ),
inference(negated_conjecture,[status(cth)],[f271]) ).
thf(f271,conjecture,
( ( modula1144073633_aux_a @ inf @ sup @ b @ c @ a2 )
= ( modula1144073633_aux_a @ inf @ sup @ a2 @ b @ c ) ),
file('/export/starexec/sandbox/tmp/tmp.yhjYV0Pxgo/DTF2THF_24647.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP124^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n027.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 08:27:22 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.44 ---- Original DTF file ---
% 0.18/0.45 thf(spec,logic,$$dhol).
% 0.18/0.45 %------------------------------------------------------------------------------
% 0.18/0.45 % File : ITP124^1 : TPTP v9.2.1. Released v7.5.0.
% 0.18/0.45 % Domain : Interactive Theorem Proving
% 0.18/0.45 % Problem : Sledgehammer Modular_Distrib_Lattice problem prob_46__3255746_1
% 0.18/0.45 % Version : Especial.
% 0.18/0.45 % English :
% 0.18/0.45
% 0.18/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.18/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.18/0.45 % Source : [Des21]
% 0.18/0.45 % Names : Modular_Distrib_Lattice/prob_46__3255746_1 [Des21]
% 0.18/0.45
% 0.18/0.45 % Status : Theorem
% 0.18/0.45 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.18 v7.5.0
% 0.18/0.45 % Syntax : Number of formulae : 313 ( 132 unt; 42 typ; 0 def)
% 0.18/0.45 % Number of atoms : 740 ( 239 equ; 0 cnn)
% 0.18/0.45 % Maximal formula atoms : 7 ( 2 avg)
% 0.18/0.45 % Number of connectives : 1966 ( 59 ~; 7 |; 35 &;1593 @)
% 0.18/0.45 % ( 0 <=>; 272 =>; 0 <=; 0 <~>)
% 0.18/0.45 % Maximal formula depth : 16 ( 6 avg)
% 0.18/0.45 % Number of types : 4 ( 3 usr)
% 0.18/0.45 % Number of type conns : 175 ( 175 >; 0 *; 0 +; 0 <<)
% 0.18/0.45 % Number of symbols : 40 ( 39 usr; 5 con; 0-5 aty)
% 0.18/0.45 % Number of variables : 643 ( 27 ^; 605 !; 11 ?; 643 :)
% 0.18/0.45 % SPC : TH0_THM_EQU_NAR
% 0.18/0.45
% 0.18/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 15:43:25.476
% 0.18/0.45 %------------------------------------------------------------------------------
% 0.18/0.45 % Could-be-implicit typings (3)
% 0.18/0.45 thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.18/0.45 set_set_a: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.18/0.45 set_a: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_tf__a,type,
% 0.18/0.45 a: $tType ).
% 0.18/0.45
% 0.18/0.45 % Explicit typings (39)
% 0.18/0.45 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above_001tf__a,type,
% 0.18/0.45 condit1627435690bove_a: ( a > a > $o ) > set_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below_001tf__a,type,
% 0.18/0.45 condit1001553558elow_a: ( a > a > $o ) > set_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Finite__Set_Ocomp__fun__idem_001tf__a_001tf__a,type,
% 0.18/0.45 finite40241356em_a_a: ( a > a > a ) > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
% 0.18/0.45 finite_finite_set_a: set_set_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
% 0.18/0.45 finite_finite_a: set_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Groups_Oabel__semigroup_001tf__a,type,
% 0.18/0.45 abel_semigroup_a: ( a > a > a ) > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Groups_Osemigroup_001tf__a,type,
% 0.18/0.45 semigroup_a: ( a > a > a ) > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Lattices_Osemilattice_001tf__a,type,
% 0.18/0.45 semilattice_a: ( a > a > a ) > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Lattices__Big_Osemilattice__inf_OInf__fin_001tf__a,type,
% 0.18/0.45 lattic247676691_fin_a: ( a > a > a ) > set_a > a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Lattices__Big_Osemilattice__set_001tf__a,type,
% 0.18/0.45 lattic1885654924_set_a: ( a > a > a ) > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Lattices__Big_Osemilattice__sup_OSup__fin_001tf__a,type,
% 0.18/0.45 lattic1781219155_fin_a: ( a > a > a ) > set_a > a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Od__aux_001tf__a,type,
% 0.18/0.45 modula1936294176_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Oe__aux_001tf__a,type,
% 0.18/0.45 modula1144073633_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.18/0.45 bot_bot_set_set_a: set_set_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
% 0.18/0.45 bot_bot_set_a: set_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oord_OLeast_001tf__a,type,
% 0.18/0.45 least_a: ( a > a > $o ) > ( a > $o ) > a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oord_Omax_001tf__a,type,
% 0.18/0.45 max_a: ( a > a > $o ) > a > a > a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oord_Omin_001tf__a,type,
% 0.18/0.45 min_a: ( a > a > $o ) > a > a > a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.18/0.45 ord_le318720350_set_a: set_set_a > set_set_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 0.18/0.45 ord_less_eq_set_a: set_a > set_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oorder_OGreatest_001tf__a,type,
% 0.18/0.45 greatest_a: ( a > a > $o ) > ( a > $o ) > a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oorder_Oantimono_001tf__a_001t__Set__Oset_Itf__a_J,type,
% 0.18/0.45 antimono_a_set_a: ( a > a > $o ) > ( a > set_a ) > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oorder_Omono_001tf__a_001t__Set__Oset_Itf__a_J,type,
% 0.18/0.45 mono_a_set_a: ( a > a > $o ) > ( a > set_a ) > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Relation_Otransp_001tf__a,type,
% 0.18/0.45 transp_a: ( a > a > $o ) > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
% 0.18/0.45 collect_set_a: ( set_a > $o ) > set_set_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.18/0.45 collect_a: ( a > $o ) > set_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
% 0.18/0.45 insert_set_a: set_a > set_set_a > set_set_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_Oinsert_001tf__a,type,
% 0.18/0.45 insert_a: a > set_a > set_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set__Interval_Oord_OatLeastAtMost_001tf__a,type,
% 0.18/0.45 set_atLeastAtMost_a: ( a > a > $o ) > a > a > set_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set__Interval_Oord_OatLeast_001tf__a,type,
% 0.18/0.45 set_atLeast_a: ( a > a > $o ) > a > set_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set__Interval_Oord_OatMost_001tf__a,type,
% 0.18/0.45 set_atMost_a: ( a > a > $o ) > a > set_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
% 0.18/0.45 member_set_a: set_a > set_set_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001tf__a,type,
% 0.18/0.45 member_a: a > set_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_a,type,
% 0.18/0.45 a2: a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_b,type,
% 0.18/0.45 b: a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_c,type,
% 0.18/0.45 c: a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_inf,type,
% 0.18/0.45 inf: a > a > a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_less__eq,type,
% 0.18/0.45 less_eq: a > a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_sup,type,
% 0.18/0.45 sup: a > a > a ).
% 0.18/0.45
% 0.18/0.45 % Relevant facts (270)
% 0.18/0.45 thf(fact_0_local_Oinf_Oassoc,axiom,
% 0.18/0.45 ! [A: a,B: a,C: a] :
% 0.18/0.45 ( ( inf @ ( inf @ A @ B ) @ C )
% 0.18/0.45 = ( inf @ A @ ( inf @ B @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf.assoc
% 0.18/0.45 thf(fact_1_local_Oinf_Ocommute,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( inf @ A @ B )
% 0.18/0.45 = ( inf @ B @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf.commute
% 0.18/0.45 thf(fact_2_local_Oinf_Oleft__commute,axiom,
% 0.18/0.45 ! [B: a,A: a,C: a] :
% 0.18/0.45 ( ( inf @ B @ ( inf @ A @ C ) )
% 0.18/0.45 = ( inf @ A @ ( inf @ B @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf.left_commute
% 0.18/0.45 thf(fact_3_local_Oinf__assoc,axiom,
% 0.18/0.45 ! [X: a,Y: a,Z: a] :
% 0.18/0.45 ( ( inf @ ( inf @ X @ Y ) @ Z )
% 0.18/0.45 = ( inf @ X @ ( inf @ Y @ Z ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf_assoc
% 0.18/0.45 thf(fact_4_local_Oinf__commute,axiom,
% 0.18/0.45 ! [X: a,Y: a] :
% 0.18/0.45 ( ( inf @ X @ Y )
% 0.18/0.45 = ( inf @ Y @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf_commute
% 0.18/0.45 thf(fact_5_local_Oinf__left__commute,axiom,
% 0.18/0.45 ! [X: a,Y: a,Z: a] :
% 0.18/0.45 ( ( inf @ X @ ( inf @ Y @ Z ) )
% 0.18/0.45 = ( inf @ Y @ ( inf @ X @ Z ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf_left_commute
% 0.18/0.45 thf(fact_6_local_Osup_Oassoc,axiom,
% 0.18/0.45 ! [A: a,B: a,C: a] :
% 0.18/0.45 ( ( sup @ ( sup @ A @ B ) @ C )
% 0.18/0.45 = ( sup @ A @ ( sup @ B @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.assoc
% 0.18/0.45 thf(fact_7_local_Osup_Ocommute,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( sup @ A @ B )
% 0.18/0.45 = ( sup @ B @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.commute
% 0.18/0.45 thf(fact_8_local_Osup_Oleft__commute,axiom,
% 0.18/0.45 ! [B: a,A: a,C: a] :
% 0.18/0.45 ( ( sup @ B @ ( sup @ A @ C ) )
% 0.18/0.45 = ( sup @ A @ ( sup @ B @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.left_commute
% 0.18/0.45 thf(fact_9_local_Osup__assoc,axiom,
% 0.18/0.45 ! [X: a,Y: a,Z: a] :
% 0.18/0.45 ( ( sup @ ( sup @ X @ Y ) @ Z )
% 0.18/0.45 = ( sup @ X @ ( sup @ Y @ Z ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup_assoc
% 0.18/0.45 thf(fact_10_local_Osup__commute,axiom,
% 0.18/0.45 ! [X: a,Y: a] :
% 0.18/0.45 ( ( sup @ X @ Y )
% 0.18/0.45 = ( sup @ Y @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup_commute
% 0.18/0.45 thf(fact_11_local_Osup__left__commute,axiom,
% 0.18/0.45 ! [X: a,Y: a,Z: a] :
% 0.18/0.45 ( ( sup @ X @ ( sup @ Y @ Z ) )
% 0.18/0.45 = ( sup @ Y @ ( sup @ X @ Z ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup_left_commute
% 0.18/0.45 thf(fact_12_local_Odistrib__imp1,axiom,
% 0.18/0.45 ! [X: a,Y: a,Z: a] :
% 0.18/0.45 ( ! [X2: a,Y2: a,Z2: a] :
% 0.18/0.45 ( ( inf @ X2 @ ( sup @ Y2 @ Z2 ) )
% 0.18/0.45 = ( sup @ ( inf @ X2 @ Y2 ) @ ( inf @ X2 @ Z2 ) ) )
% 0.18/0.45 => ( ( sup @ X @ ( inf @ Y @ Z ) )
% 0.18/0.45 = ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.distrib_imp1
% 0.18/0.45 thf(fact_13_local_Odistrib__imp2,axiom,
% 0.18/0.45 ! [X: a,Y: a,Z: a] :
% 0.18/0.45 ( ! [X2: a,Y2: a,Z2: a] :
% 0.18/0.45 ( ( sup @ X2 @ ( inf @ Y2 @ Z2 ) )
% 0.18/0.45 = ( inf @ ( sup @ X2 @ Y2 ) @ ( sup @ X2 @ Z2 ) ) )
% 0.18/0.45 => ( ( inf @ X @ ( sup @ Y @ Z ) )
% 0.18/0.45 = ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.distrib_imp2
% 0.18/0.45 thf(fact_14_e__aux__def,axiom,
% 0.18/0.45 ! [A: a,B: a,C: a] :
% 0.18/0.45 ( ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C )
% 0.18/0.45 = ( inf @ ( inf @ ( sup @ A @ B ) @ ( sup @ B @ C ) ) @ ( sup @ C @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % e_aux_def
% 0.18/0.45 thf(fact_15_local_Oinf_Oidem,axiom,
% 0.18/0.45 ! [A: a] :
% 0.18/0.45 ( ( inf @ A @ A )
% 0.18/0.45 = A ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf.idem
% 0.18/0.45 thf(fact_16_local_Oinf_Oleft__idem,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( inf @ A @ ( inf @ A @ B ) )
% 0.18/0.45 = ( inf @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf.left_idem
% 0.18/0.45 thf(fact_17_local_Oinf_Oright__idem,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( inf @ ( inf @ A @ B ) @ B )
% 0.18/0.45 = ( inf @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf.right_idem
% 0.18/0.45 thf(fact_18_local_Oinf__idem,axiom,
% 0.18/0.45 ! [X: a] :
% 0.18/0.45 ( ( inf @ X @ X )
% 0.18/0.45 = X ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf_idem
% 0.18/0.45 thf(fact_19_local_Oinf__left__idem,axiom,
% 0.18/0.45 ! [X: a,Y: a] :
% 0.18/0.45 ( ( inf @ X @ ( inf @ X @ Y ) )
% 0.18/0.45 = ( inf @ X @ Y ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf_left_idem
% 0.18/0.45 thf(fact_20_local_Oinf__right__idem,axiom,
% 0.18/0.45 ! [X: a,Y: a] :
% 0.18/0.45 ( ( inf @ ( inf @ X @ Y ) @ Y )
% 0.18/0.45 = ( inf @ X @ Y ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf_right_idem
% 0.18/0.45 thf(fact_21_local_Osup_Oidem,axiom,
% 0.18/0.45 ! [A: a] :
% 0.18/0.45 ( ( sup @ A @ A )
% 0.18/0.45 = A ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.idem
% 0.18/0.45 thf(fact_22_local_Osup_Oleft__idem,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( sup @ A @ ( sup @ A @ B ) )
% 0.18/0.45 = ( sup @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.left_idem
% 0.18/0.45 thf(fact_23_local_Osup_Oright__idem,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( sup @ ( sup @ A @ B ) @ B )
% 0.18/0.45 = ( sup @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.right_idem
% 0.18/0.45 thf(fact_24_local_Osup__idem,axiom,
% 0.18/0.45 ! [X: a] :
% 0.18/0.45 ( ( sup @ X @ X )
% 0.18/0.45 = X ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup_idem
% 0.18/0.45 thf(fact_25_local_Osup__left__idem,axiom,
% 0.18/0.45 ! [X: a,Y: a] :
% 0.18/0.45 ( ( sup @ X @ ( sup @ X @ Y ) )
% 0.18/0.45 = ( sup @ X @ Y ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup_left_idem
% 0.18/0.45 thf(fact_26_local_Oinf__sup__absorb,axiom,
% 0.18/0.45 ! [X: a,Y: a] :
% 0.18/0.45 ( ( inf @ X @ ( sup @ X @ Y ) )
% 0.18/0.45 = X ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf_sup_absorb
% 0.18/0.45 thf(fact_27_local_Osup__inf__absorb,axiom,
% 0.18/0.45 ! [X: a,Y: a] :
% 0.18/0.45 ( ( sup @ X @ ( inf @ X @ Y ) )
% 0.18/0.45 = X ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup_inf_absorb
% 0.18/0.45 thf(fact_28_d__aux__def,axiom,
% 0.18/0.45 ! [A: a,B: a,C: a] :
% 0.18/0.45 ( ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C )
% 0.18/0.45 = ( sup @ ( sup @ ( inf @ A @ B ) @ ( inf @ B @ C ) ) @ ( inf @ C @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % d_aux_def
% 0.18/0.45 thf(fact_29_d__b__c__a,axiom,
% 0.18/0.45 ! [B: a,C: a,A: a] :
% 0.18/0.45 ( ( modula1936294176_aux_a @ inf @ sup @ B @ C @ A )
% 0.18/0.45 = ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) ) ).
% 0.18/0.45
% 0.18/0.45 % d_b_c_a
% 0.18/0.45 thf(fact_30_d__c__a__b,axiom,
% 0.18/0.45 ! [C: a,A: a,B: a] :
% 0.18/0.45 ( ( modula1936294176_aux_a @ inf @ sup @ C @ A @ B )
% 0.18/0.45 = ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) ) ).
% 0.18/0.45
% 0.18/0.45 % d_c_a_b
% 0.18/0.45 thf(fact_31_lattice_Oe__aux_Ocong,axiom,
% 0.18/0.45 modula1144073633_aux_a = modula1144073633_aux_a ).
% 0.18/0.45
% 0.18/0.45 % lattice.e_aux.cong
% 0.18/0.45 thf(fact_32_local_Ocomp__fun__idem__sup,axiom,
% 0.18/0.45 finite40241356em_a_a @ sup ).
% 0.18/0.45
% 0.18/0.45 % local.comp_fun_idem_sup
% 0.18/0.45 thf(fact_33_local_Ocomp__fun__idem__inf,axiom,
% 0.18/0.45 finite40241356em_a_a @ inf ).
% 0.18/0.45
% 0.18/0.45 % local.comp_fun_idem_inf
% 0.18/0.45 thf(fact_34_local_Osup_Osemigroup__axioms,axiom,
% 0.18/0.45 semigroup_a @ sup ).
% 0.18/0.45
% 0.18/0.45 % local.sup.semigroup_axioms
% 0.18/0.45 thf(fact_35_local_Oinf_Osemigroup__axioms,axiom,
% 0.18/0.45 semigroup_a @ inf ).
% 0.18/0.45
% 0.18/0.45 % local.inf.semigroup_axioms
% 0.18/0.45 thf(fact_36_local_Osup_Osemilattice__axioms,axiom,
% 0.18/0.45 semilattice_a @ sup ).
% 0.18/0.45
% 0.18/0.45 % local.sup.semilattice_axioms
% 0.18/0.45 thf(fact_37_local_Oinf_Osemilattice__axioms,axiom,
% 0.18/0.45 semilattice_a @ inf ).
% 0.18/0.45
% 0.18/0.45 % local.inf.semilattice_axioms
% 0.18/0.45 thf(fact_38_local_Osup_Oabel__semigroup__axioms,axiom,
% 0.18/0.45 abel_semigroup_a @ sup ).
% 0.18/0.45
% 0.18/0.45 % local.sup.abel_semigroup_axioms
% 0.18/0.45 thf(fact_39_local_Oinf_Oabel__semigroup__axioms,axiom,
% 0.18/0.45 abel_semigroup_a @ inf ).
% 0.18/0.45
% 0.18/0.45 % local.inf.abel_semigroup_axioms
% 0.18/0.45 thf(fact_40_local_OSup__fin_Osemilattice__set__axioms,axiom,
% 0.18/0.45 lattic1885654924_set_a @ sup ).
% 0.18/0.45
% 0.18/0.45 % local.Sup_fin.semilattice_set_axioms
% 0.18/0.45 thf(fact_41_local_OInf__fin_Osemilattice__set__axioms,axiom,
% 0.18/0.45 lattic1885654924_set_a @ inf ).
% 0.18/0.45
% 0.18/0.45 % local.Inf_fin.semilattice_set_axioms
% 0.18/0.45 thf(fact_42_local_Odistrib__inf__le,axiom,
% 0.18/0.45 ! [X: a,Y: a,Z: a] : ( less_eq @ ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z ) ) @ ( inf @ X @ ( sup @ Y @ Z ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.distrib_inf_le
% 0.18/0.45 thf(fact_43_local_Odistrib__sup__le,axiom,
% 0.18/0.45 ! [X: a,Y: a,Z: a] : ( less_eq @ ( sup @ X @ ( inf @ Y @ Z ) ) @ ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.distrib_sup_le
% 0.18/0.45 thf(fact_44_local_Oantisym,axiom,
% 0.18/0.45 ! [X: a,Y: a] :
% 0.18/0.45 ( ( less_eq @ X @ Y )
% 0.18/0.45 => ( ( less_eq @ Y @ X )
% 0.18/0.45 => ( X = Y ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.antisym
% 0.18/0.45 thf(fact_45_mem__Collect__eq,axiom,
% 0.18/0.45 ! [A: set_a,P: set_a > $o] :
% 0.18/0.45 ( ( member_set_a @ A @ ( collect_set_a @ P ) )
% 0.18/0.45 = ( P @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_Collect_eq
% 0.18/0.45 thf(fact_46_mem__Collect__eq,axiom,
% 0.18/0.45 ! [A: a,P: a > $o] :
% 0.18/0.45 ( ( member_a @ A @ ( collect_a @ P ) )
% 0.18/0.45 = ( P @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_Collect_eq
% 0.18/0.45 thf(fact_47_Collect__mem__eq,axiom,
% 0.18/0.45 ! [A2: set_set_a] :
% 0.18/0.45 ( ( collect_set_a
% 0.18/0.45 @ ^ [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
% 0.18/0.45 = A2 ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_mem_eq
% 0.18/0.45 thf(fact_48_Collect__mem__eq,axiom,
% 0.18/0.45 ! [A2: set_a] :
% 0.18/0.45 ( ( collect_a
% 0.18/0.45 @ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
% 0.18/0.45 = A2 ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_mem_eq
% 0.18/0.45 thf(fact_49_Collect__cong,axiom,
% 0.18/0.45 ! [P: a > $o,Q: a > $o] :
% 0.18/0.45 ( ! [X2: a] :
% 0.18/0.45 ( ( P @ X2 )
% 0.18/0.45 = ( Q @ X2 ) )
% 0.18/0.45 => ( ( collect_a @ P )
% 0.18/0.45 = ( collect_a @ Q ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_cong
% 0.18/0.45 thf(fact_50_local_Oantisym__conv,axiom,
% 0.18/0.45 ! [Y: a,X: a] :
% 0.18/0.45 ( ( less_eq @ Y @ X )
% 0.18/0.45 => ( ( less_eq @ X @ Y )
% 0.18/0.45 = ( X = Y ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.antisym_conv
% 0.18/0.45 thf(fact_51_local_Odual__order_Oantisym,axiom,
% 0.18/0.45 ! [B: a,A: a] :
% 0.18/0.45 ( ( less_eq @ B @ A )
% 0.18/0.45 => ( ( less_eq @ A @ B )
% 0.18/0.45 => ( A = B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.dual_order.antisym
% 0.18/0.45 thf(fact_52_local_Odual__order_Oeq__iff,axiom,
% 0.18/0.45 ( ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 ) )
% 0.18/0.45 = ( ^ [A3: a,B2: a] :
% 0.18/0.45 ( ( less_eq @ B2 @ A3 )
% 0.18/0.45 & ( less_eq @ A3 @ B2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.dual_order.eq_iff
% 0.18/0.45 thf(fact_53_local_Odual__order_Otrans,axiom,
% 0.18/0.45 ! [B: a,A: a,C: a] :
% 0.18/0.45 ( ( less_eq @ B @ A )
% 0.18/0.45 => ( ( less_eq @ C @ B )
% 0.18/0.45 => ( less_eq @ C @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.dual_order.trans
% 0.18/0.45 thf(fact_54_local_Oeq__iff,axiom,
% 0.18/0.45 ( ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 ) )
% 0.18/0.45 = ( ^ [X3: a,Y4: a] :
% 0.18/0.45 ( ( less_eq @ X3 @ Y4 )
% 0.18/0.45 & ( less_eq @ Y4 @ X3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.eq_iff
% 0.18/0.45 thf(fact_55_local_Oeq__refl,axiom,
% 0.18/0.45 ! [X: a,Y: a] :
% 0.18/0.45 ( ( X = Y )
% 0.18/0.45 => ( less_eq @ X @ Y ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.eq_refl
% 0.18/0.45 thf(fact_56_local_Oord__eq__le__trans,axiom,
% 0.18/0.45 ! [A: a,B: a,C: a] :
% 0.18/0.45 ( ( A = B )
% 0.18/0.45 => ( ( less_eq @ B @ C )
% 0.18/0.45 => ( less_eq @ A @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.ord_eq_le_trans
% 0.18/0.45 thf(fact_57_local_Oord__le__eq__trans,axiom,
% 0.18/0.45 ! [A: a,B: a,C: a] :
% 0.18/0.45 ( ( less_eq @ A @ B )
% 0.18/0.45 => ( ( B = C )
% 0.18/0.45 => ( less_eq @ A @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.ord_le_eq_trans
% 0.18/0.45 thf(fact_58_local_Oorder_Oantisym,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( less_eq @ A @ B )
% 0.18/0.45 => ( ( less_eq @ B @ A )
% 0.18/0.45 => ( A = B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.order.antisym
% 0.18/0.45 thf(fact_59_local_Oorder_Oeq__iff,axiom,
% 0.18/0.45 ( ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 ) )
% 0.18/0.45 = ( ^ [A3: a,B2: a] :
% 0.18/0.45 ( ( less_eq @ A3 @ B2 )
% 0.18/0.45 & ( less_eq @ B2 @ A3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.order.eq_iff
% 0.18/0.45 thf(fact_60_local_Oorder_Otrans,axiom,
% 0.18/0.45 ! [A: a,B: a,C: a] :
% 0.18/0.45 ( ( less_eq @ A @ B )
% 0.18/0.45 => ( ( less_eq @ B @ C )
% 0.18/0.45 => ( less_eq @ A @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.order.trans
% 0.18/0.45 thf(fact_61_local_Oorder__trans,axiom,
% 0.18/0.45 ! [X: a,Y: a,Z: a] :
% 0.18/0.45 ( ( less_eq @ X @ Y )
% 0.18/0.45 => ( ( less_eq @ Y @ Z )
% 0.18/0.45 => ( less_eq @ X @ Z ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.order_trans
% 0.18/0.45 thf(fact_62_local_Ole__infI2,axiom,
% 0.18/0.45 ! [B: a,X: a,A: a] :
% 0.18/0.45 ( ( less_eq @ B @ X )
% 0.18/0.45 => ( less_eq @ ( inf @ A @ B ) @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.le_infI2
% 0.18/0.45 thf(fact_63_local_Ole__infI1,axiom,
% 0.18/0.45 ! [A: a,X: a,B: a] :
% 0.18/0.45 ( ( less_eq @ A @ X )
% 0.18/0.45 => ( less_eq @ ( inf @ A @ B ) @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.le_infI1
% 0.18/0.45 thf(fact_64_local_Ole__infI,axiom,
% 0.18/0.45 ! [X: a,A: a,B: a] :
% 0.18/0.45 ( ( less_eq @ X @ A )
% 0.18/0.45 => ( ( less_eq @ X @ B )
% 0.18/0.45 => ( less_eq @ X @ ( inf @ A @ B ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.le_infI
% 0.18/0.45 thf(fact_65_local_Ole__infE,axiom,
% 0.18/0.45 ! [X: a,A: a,B: a] :
% 0.18/0.45 ( ( less_eq @ X @ ( inf @ A @ B ) )
% 0.18/0.45 => ~ ( ( less_eq @ X @ A )
% 0.18/0.45 => ~ ( less_eq @ X @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.le_infE
% 0.18/0.45 thf(fact_66_local_Ole__iff__inf,axiom,
% 0.18/0.45 ! [X: a,Y: a] :
% 0.18/0.45 ( ( less_eq @ X @ Y )
% 0.18/0.45 = ( ( inf @ X @ Y )
% 0.18/0.45 = X ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.le_iff_inf
% 0.18/0.45 thf(fact_67_local_Oinf__unique,axiom,
% 0.18/0.45 ! [F: a > a > a,X: a,Y: a] :
% 0.18/0.45 ( ! [X2: a,Y2: a] : ( less_eq @ ( F @ X2 @ Y2 ) @ X2 )
% 0.18/0.45 => ( ! [X2: a,Y2: a] : ( less_eq @ ( F @ X2 @ Y2 ) @ Y2 )
% 0.18/0.45 => ( ! [X2: a,Y2: a,Z2: a] :
% 0.18/0.45 ( ( less_eq @ X2 @ Y2 )
% 0.18/0.45 => ( ( less_eq @ X2 @ Z2 )
% 0.18/0.45 => ( less_eq @ X2 @ ( F @ Y2 @ Z2 ) ) ) )
% 0.18/0.45 => ( ( inf @ X @ Y )
% 0.18/0.45 = ( F @ X @ Y ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf_unique
% 0.18/0.45 thf(fact_68_local_Oinf__mono,axiom,
% 0.18/0.45 ! [A: a,C: a,B: a,D: a] :
% 0.18/0.45 ( ( less_eq @ A @ C )
% 0.18/0.45 => ( ( less_eq @ B @ D )
% 0.18/0.45 => ( less_eq @ ( inf @ A @ B ) @ ( inf @ C @ D ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf_mono
% 0.18/0.45 thf(fact_69_local_Oinf__le2,axiom,
% 0.18/0.45 ! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ Y ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf_le2
% 0.18/0.45 thf(fact_70_local_Oinf__le1,axiom,
% 0.18/0.45 ! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ X ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf_le1
% 0.18/0.45 thf(fact_71_local_Oinf__greatest,axiom,
% 0.18/0.45 ! [X: a,Y: a,Z: a] :
% 0.18/0.45 ( ( less_eq @ X @ Y )
% 0.18/0.45 => ( ( less_eq @ X @ Z )
% 0.18/0.45 => ( less_eq @ X @ ( inf @ Y @ Z ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf_greatest
% 0.18/0.45 thf(fact_72_local_Oinf__absorb2,axiom,
% 0.18/0.45 ! [Y: a,X: a] :
% 0.18/0.45 ( ( less_eq @ Y @ X )
% 0.18/0.45 => ( ( inf @ X @ Y )
% 0.18/0.45 = Y ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf_absorb2
% 0.18/0.45 thf(fact_73_local_Oinf__absorb1,axiom,
% 0.18/0.45 ! [X: a,Y: a] :
% 0.18/0.45 ( ( less_eq @ X @ Y )
% 0.18/0.45 => ( ( inf @ X @ Y )
% 0.18/0.45 = X ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf_absorb1
% 0.18/0.45 thf(fact_74_local_Oinf_Oorder__iff,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( less_eq @ A @ B )
% 0.18/0.45 = ( A
% 0.18/0.45 = ( inf @ A @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf.order_iff
% 0.18/0.45 thf(fact_75_local_Oinf_OorderI,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( A
% 0.18/0.45 = ( inf @ A @ B ) )
% 0.18/0.45 => ( less_eq @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf.orderI
% 0.18/0.45 thf(fact_76_local_Oinf_OorderE,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( less_eq @ A @ B )
% 0.18/0.45 => ( A
% 0.18/0.45 = ( inf @ A @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf.orderE
% 0.18/0.45 thf(fact_77_local_Oinf_OcoboundedI2,axiom,
% 0.18/0.45 ! [B: a,C: a,A: a] :
% 0.18/0.45 ( ( less_eq @ B @ C )
% 0.18/0.45 => ( less_eq @ ( inf @ A @ B ) @ C ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf.coboundedI2
% 0.18/0.45 thf(fact_78_local_Oinf_OcoboundedI1,axiom,
% 0.18/0.45 ! [A: a,C: a,B: a] :
% 0.18/0.45 ( ( less_eq @ A @ C )
% 0.18/0.45 => ( less_eq @ ( inf @ A @ B ) @ C ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf.coboundedI1
% 0.18/0.45 thf(fact_79_local_Oinf_Ocobounded2,axiom,
% 0.18/0.45 ! [A: a,B: a] : ( less_eq @ ( inf @ A @ B ) @ B ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf.cobounded2
% 0.18/0.45 thf(fact_80_local_Oinf_Ocobounded1,axiom,
% 0.18/0.45 ! [A: a,B: a] : ( less_eq @ ( inf @ A @ B ) @ A ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf.cobounded1
% 0.18/0.45 thf(fact_81_local_Oinf_OboundedI,axiom,
% 0.18/0.45 ! [A: a,B: a,C: a] :
% 0.18/0.45 ( ( less_eq @ A @ B )
% 0.18/0.45 => ( ( less_eq @ A @ C )
% 0.18/0.45 => ( less_eq @ A @ ( inf @ B @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf.boundedI
% 0.18/0.45 thf(fact_82_local_Oinf_OboundedE,axiom,
% 0.18/0.45 ! [A: a,B: a,C: a] :
% 0.18/0.45 ( ( less_eq @ A @ ( inf @ B @ C ) )
% 0.18/0.45 => ~ ( ( less_eq @ A @ B )
% 0.18/0.45 => ~ ( less_eq @ A @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf.boundedE
% 0.18/0.45 thf(fact_83_local_Oinf_Oabsorb__iff2,axiom,
% 0.18/0.45 ! [B: a,A: a] :
% 0.18/0.45 ( ( less_eq @ B @ A )
% 0.18/0.45 = ( ( inf @ A @ B )
% 0.18/0.45 = B ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf.absorb_iff2
% 0.18/0.45 thf(fact_84_local_Oinf_Oabsorb__iff1,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( less_eq @ A @ B )
% 0.18/0.45 = ( ( inf @ A @ B )
% 0.18/0.45 = A ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf.absorb_iff1
% 0.18/0.45 thf(fact_85_local_Oinf_Oabsorb2,axiom,
% 0.18/0.45 ! [B: a,A: a] :
% 0.18/0.45 ( ( less_eq @ B @ A )
% 0.18/0.45 => ( ( inf @ A @ B )
% 0.18/0.45 = B ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf.absorb2
% 0.18/0.45 thf(fact_86_local_Oinf_Oabsorb1,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( less_eq @ A @ B )
% 0.18/0.45 => ( ( inf @ A @ B )
% 0.18/0.45 = A ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf.absorb1
% 0.18/0.45 thf(fact_87_local_Osup__unique,axiom,
% 0.18/0.45 ! [F: a > a > a,X: a,Y: a] :
% 0.18/0.45 ( ! [X2: a,Y2: a] : ( less_eq @ X2 @ ( F @ X2 @ Y2 ) )
% 0.18/0.45 => ( ! [X2: a,Y2: a] : ( less_eq @ Y2 @ ( F @ X2 @ Y2 ) )
% 0.18/0.45 => ( ! [X2: a,Y2: a,Z2: a] :
% 0.18/0.45 ( ( less_eq @ Y2 @ X2 )
% 0.18/0.45 => ( ( less_eq @ Z2 @ X2 )
% 0.18/0.45 => ( less_eq @ ( F @ Y2 @ Z2 ) @ X2 ) ) )
% 0.18/0.45 => ( ( sup @ X @ Y )
% 0.18/0.45 = ( F @ X @ Y ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup_unique
% 0.18/0.45 thf(fact_88_local_Osup__mono,axiom,
% 0.18/0.45 ! [A: a,C: a,B: a,D: a] :
% 0.18/0.45 ( ( less_eq @ A @ C )
% 0.18/0.45 => ( ( less_eq @ B @ D )
% 0.18/0.45 => ( less_eq @ ( sup @ A @ B ) @ ( sup @ C @ D ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup_mono
% 0.18/0.45 thf(fact_89_local_Osup__least,axiom,
% 0.18/0.45 ! [Y: a,X: a,Z: a] :
% 0.18/0.45 ( ( less_eq @ Y @ X )
% 0.18/0.45 => ( ( less_eq @ Z @ X )
% 0.18/0.45 => ( less_eq @ ( sup @ Y @ Z ) @ X ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup_least
% 0.18/0.45 thf(fact_90_local_Osup__ge2,axiom,
% 0.18/0.45 ! [Y: a,X: a] : ( less_eq @ Y @ ( sup @ X @ Y ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup_ge2
% 0.18/0.45 thf(fact_91_local_Osup__ge1,axiom,
% 0.18/0.45 ! [X: a,Y: a] : ( less_eq @ X @ ( sup @ X @ Y ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup_ge1
% 0.18/0.45 thf(fact_92_local_Osup__absorb2,axiom,
% 0.18/0.45 ! [X: a,Y: a] :
% 0.18/0.45 ( ( less_eq @ X @ Y )
% 0.18/0.45 => ( ( sup @ X @ Y )
% 0.18/0.45 = Y ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup_absorb2
% 0.18/0.45 thf(fact_93_local_Osup__absorb1,axiom,
% 0.18/0.45 ! [Y: a,X: a] :
% 0.18/0.45 ( ( less_eq @ Y @ X )
% 0.18/0.45 => ( ( sup @ X @ Y )
% 0.18/0.45 = X ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup_absorb1
% 0.18/0.45 thf(fact_94_local_Osup_Oorder__iff,axiom,
% 0.18/0.45 ! [B: a,A: a] :
% 0.18/0.45 ( ( less_eq @ B @ A )
% 0.18/0.45 = ( A
% 0.18/0.45 = ( sup @ A @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.order_iff
% 0.18/0.45 thf(fact_95_local_Osup_OorderI,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( A
% 0.18/0.45 = ( sup @ A @ B ) )
% 0.18/0.45 => ( less_eq @ B @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.orderI
% 0.18/0.45 thf(fact_96_local_Osup_OorderE,axiom,
% 0.18/0.45 ! [B: a,A: a] :
% 0.18/0.45 ( ( less_eq @ B @ A )
% 0.18/0.45 => ( A
% 0.18/0.45 = ( sup @ A @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.orderE
% 0.18/0.45 thf(fact_97_local_Osup_Omono,axiom,
% 0.18/0.45 ! [C: a,A: a,D: a,B: a] :
% 0.18/0.45 ( ( less_eq @ C @ A )
% 0.18/0.45 => ( ( less_eq @ D @ B )
% 0.18/0.45 => ( less_eq @ ( sup @ C @ D ) @ ( sup @ A @ B ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.mono
% 0.18/0.45 thf(fact_98_local_Osup_OcoboundedI2,axiom,
% 0.18/0.45 ! [C: a,B: a,A: a] :
% 0.18/0.45 ( ( less_eq @ C @ B )
% 0.18/0.45 => ( less_eq @ C @ ( sup @ A @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.coboundedI2
% 0.18/0.45 thf(fact_99_local_Osup_OcoboundedI1,axiom,
% 0.18/0.45 ! [C: a,A: a,B: a] :
% 0.18/0.45 ( ( less_eq @ C @ A )
% 0.18/0.45 => ( less_eq @ C @ ( sup @ A @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.coboundedI1
% 0.18/0.45 thf(fact_100_local_Osup_Ocobounded2,axiom,
% 0.18/0.45 ! [B: a,A: a] : ( less_eq @ B @ ( sup @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.cobounded2
% 0.18/0.45 thf(fact_101_local_Osup_Ocobounded1,axiom,
% 0.18/0.45 ! [A: a,B: a] : ( less_eq @ A @ ( sup @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.cobounded1
% 0.18/0.45 thf(fact_102_local_Osup_OboundedI,axiom,
% 0.18/0.45 ! [B: a,A: a,C: a] :
% 0.18/0.45 ( ( less_eq @ B @ A )
% 0.18/0.45 => ( ( less_eq @ C @ A )
% 0.18/0.45 => ( less_eq @ ( sup @ B @ C ) @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.boundedI
% 0.18/0.45 thf(fact_103_local_Osup_OboundedE,axiom,
% 0.18/0.45 ! [B: a,C: a,A: a] :
% 0.18/0.45 ( ( less_eq @ ( sup @ B @ C ) @ A )
% 0.18/0.45 => ~ ( ( less_eq @ B @ A )
% 0.18/0.45 => ~ ( less_eq @ C @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.boundedE
% 0.18/0.45 thf(fact_104_local_Osup_Oabsorb__iff2,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( less_eq @ A @ B )
% 0.18/0.45 = ( ( sup @ A @ B )
% 0.18/0.45 = B ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.absorb_iff2
% 0.18/0.45 thf(fact_105_local_Osup_Oabsorb__iff1,axiom,
% 0.18/0.45 ! [B: a,A: a] :
% 0.18/0.45 ( ( less_eq @ B @ A )
% 0.18/0.45 = ( ( sup @ A @ B )
% 0.18/0.45 = A ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.absorb_iff1
% 0.18/0.45 thf(fact_106_local_Osup_Oabsorb2,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( less_eq @ A @ B )
% 0.18/0.45 => ( ( sup @ A @ B )
% 0.18/0.45 = B ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.absorb2
% 0.18/0.45 thf(fact_107_local_Osup_Oabsorb1,axiom,
% 0.18/0.45 ! [B: a,A: a] :
% 0.18/0.45 ( ( less_eq @ B @ A )
% 0.18/0.45 => ( ( sup @ A @ B )
% 0.18/0.45 = A ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.absorb1
% 0.18/0.45 thf(fact_108_local_Ole__supI2,axiom,
% 0.18/0.45 ! [X: a,B: a,A: a] :
% 0.18/0.45 ( ( less_eq @ X @ B )
% 0.18/0.45 => ( less_eq @ X @ ( sup @ A @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.le_supI2
% 0.18/0.45 thf(fact_109_local_Ole__supI1,axiom,
% 0.18/0.45 ! [X: a,A: a,B: a] :
% 0.18/0.45 ( ( less_eq @ X @ A )
% 0.18/0.45 => ( less_eq @ X @ ( sup @ A @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.le_supI1
% 0.18/0.45 thf(fact_110_local_Ole__supI,axiom,
% 0.18/0.45 ! [A: a,X: a,B: a] :
% 0.18/0.45 ( ( less_eq @ A @ X )
% 0.18/0.45 => ( ( less_eq @ B @ X )
% 0.18/0.45 => ( less_eq @ ( sup @ A @ B ) @ X ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.le_supI
% 0.18/0.45 thf(fact_111_local_Ole__supE,axiom,
% 0.18/0.45 ! [A: a,B: a,X: a] :
% 0.18/0.45 ( ( less_eq @ ( sup @ A @ B ) @ X )
% 0.18/0.45 => ~ ( ( less_eq @ A @ X )
% 0.18/0.45 => ~ ( less_eq @ B @ X ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.le_supE
% 0.18/0.45 thf(fact_112_local_Ole__iff__sup,axiom,
% 0.18/0.45 ! [X: a,Y: a] :
% 0.18/0.45 ( ( less_eq @ X @ Y )
% 0.18/0.45 = ( ( sup @ X @ Y )
% 0.18/0.45 = Y ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.le_iff_sup
% 0.18/0.45 thf(fact_113_local_Oorder_Orefl,axiom,
% 0.18/0.45 ! [A: a] : ( less_eq @ A @ A ) ).
% 0.18/0.45
% 0.18/0.45 % local.order.refl
% 0.18/0.45 thf(fact_114_local_Oorder__refl,axiom,
% 0.18/0.45 ! [X: a] : ( less_eq @ X @ X ) ).
% 0.18/0.45
% 0.18/0.45 % local.order_refl
% 0.18/0.45 thf(fact_115_local_OGreatestI2__order,axiom,
% 0.18/0.45 ! [P: a > $o,X: a,Q: a > $o] :
% 0.18/0.45 ( ( P @ X )
% 0.18/0.45 => ( ! [Y2: a] :
% 0.18/0.45 ( ( P @ Y2 )
% 0.18/0.45 => ( less_eq @ Y2 @ X ) )
% 0.18/0.45 => ( ! [X2: a] :
% 0.18/0.45 ( ( P @ X2 )
% 0.18/0.45 => ( ! [Y5: a] :
% 0.18/0.45 ( ( P @ Y5 )
% 0.18/0.45 => ( less_eq @ Y5 @ X2 ) )
% 0.18/0.45 => ( Q @ X2 ) ) )
% 0.18/0.45 => ( Q @ ( greatest_a @ less_eq @ P ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.GreatestI2_order
% 0.18/0.45 thf(fact_116_local_OGreatest__equality,axiom,
% 0.18/0.45 ! [P: a > $o,X: a] :
% 0.18/0.45 ( ( P @ X )
% 0.18/0.45 => ( ! [Y2: a] :
% 0.18/0.45 ( ( P @ Y2 )
% 0.18/0.45 => ( less_eq @ Y2 @ X ) )
% 0.18/0.45 => ( ( greatest_a @ less_eq @ P )
% 0.18/0.45 = X ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Greatest_equality
% 0.18/0.45 thf(fact_117_local_Omax__def,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( ( less_eq @ A @ B )
% 0.18/0.45 => ( ( max_a @ less_eq @ A @ B )
% 0.18/0.45 = B ) )
% 0.18/0.45 & ( ~ ( less_eq @ A @ B )
% 0.18/0.45 => ( ( max_a @ less_eq @ A @ B )
% 0.18/0.45 = A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.max_def
% 0.18/0.45 thf(fact_118_local_Omin__def,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( ( less_eq @ A @ B )
% 0.18/0.45 => ( ( min_a @ less_eq @ A @ B )
% 0.18/0.45 = A ) )
% 0.18/0.45 & ( ~ ( less_eq @ A @ B )
% 0.18/0.45 => ( ( min_a @ less_eq @ A @ B )
% 0.18/0.45 = B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.min_def
% 0.18/0.45 thf(fact_119_local_Ole__inf__iff,axiom,
% 0.18/0.45 ! [X: a,Y: a,Z: a] :
% 0.18/0.45 ( ( less_eq @ X @ ( inf @ Y @ Z ) )
% 0.18/0.45 = ( ( less_eq @ X @ Y )
% 0.18/0.45 & ( less_eq @ X @ Z ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.le_inf_iff
% 0.18/0.45 thf(fact_120_local_Oinf_Obounded__iff,axiom,
% 0.18/0.45 ! [A: a,B: a,C: a] :
% 0.18/0.45 ( ( less_eq @ A @ ( inf @ B @ C ) )
% 0.18/0.45 = ( ( less_eq @ A @ B )
% 0.18/0.45 & ( less_eq @ A @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf.bounded_iff
% 0.18/0.45 thf(fact_121_local_Osup_Obounded__iff,axiom,
% 0.18/0.45 ! [B: a,C: a,A: a] :
% 0.18/0.45 ( ( less_eq @ ( sup @ B @ C ) @ A )
% 0.18/0.45 = ( ( less_eq @ B @ A )
% 0.18/0.45 & ( less_eq @ C @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup.bounded_iff
% 0.18/0.45 thf(fact_122_local_Ole__sup__iff,axiom,
% 0.18/0.45 ! [X: a,Y: a,Z: a] :
% 0.18/0.45 ( ( less_eq @ ( sup @ X @ Y ) @ Z )
% 0.18/0.45 = ( ( less_eq @ X @ Z )
% 0.18/0.45 & ( less_eq @ Y @ Z ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.le_sup_iff
% 0.18/0.45 thf(fact_123_local_OLeast1I,axiom,
% 0.18/0.45 ! [P: a > $o] :
% 0.18/0.45 ( ? [X4: a] :
% 0.18/0.45 ( ( P @ X4 )
% 0.18/0.45 & ! [Y2: a] :
% 0.18/0.45 ( ( P @ Y2 )
% 0.18/0.45 => ( less_eq @ X4 @ Y2 ) )
% 0.18/0.45 & ! [Y2: a] :
% 0.18/0.45 ( ( ( P @ Y2 )
% 0.18/0.45 & ! [Ya: a] :
% 0.18/0.45 ( ( P @ Ya )
% 0.18/0.45 => ( less_eq @ Y2 @ Ya ) ) )
% 0.18/0.45 => ( Y2 = X4 ) ) )
% 0.18/0.45 => ( P @ ( least_a @ less_eq @ P ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Least1I
% 0.18/0.45 thf(fact_124_local_OLeast1__le,axiom,
% 0.18/0.45 ! [P: a > $o,Z: a] :
% 0.18/0.45 ( ? [X4: a] :
% 0.18/0.45 ( ( P @ X4 )
% 0.18/0.45 & ! [Y2: a] :
% 0.18/0.45 ( ( P @ Y2 )
% 0.18/0.45 => ( less_eq @ X4 @ Y2 ) )
% 0.18/0.45 & ! [Y2: a] :
% 0.18/0.45 ( ( ( P @ Y2 )
% 0.18/0.45 & ! [Ya: a] :
% 0.18/0.45 ( ( P @ Ya )
% 0.18/0.45 => ( less_eq @ Y2 @ Ya ) ) )
% 0.18/0.45 => ( Y2 = X4 ) ) )
% 0.18/0.45 => ( ( P @ Z )
% 0.18/0.45 => ( less_eq @ ( least_a @ less_eq @ P ) @ Z ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Least1_le
% 0.18/0.45 thf(fact_125_local_OLeastI2__order,axiom,
% 0.18/0.45 ! [P: a > $o,X: a,Q: a > $o] :
% 0.18/0.45 ( ( P @ X )
% 0.18/0.45 => ( ! [Y2: a] :
% 0.18/0.45 ( ( P @ Y2 )
% 0.18/0.45 => ( less_eq @ X @ Y2 ) )
% 0.18/0.45 => ( ! [X2: a] :
% 0.18/0.45 ( ( P @ X2 )
% 0.18/0.45 => ( ! [Y5: a] :
% 0.18/0.45 ( ( P @ Y5 )
% 0.18/0.45 => ( less_eq @ X2 @ Y5 ) )
% 0.18/0.45 => ( Q @ X2 ) ) )
% 0.18/0.45 => ( Q @ ( least_a @ less_eq @ P ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.LeastI2_order
% 0.18/0.45 thf(fact_126_local_OLeast__equality,axiom,
% 0.18/0.45 ! [P: a > $o,X: a] :
% 0.18/0.45 ( ( P @ X )
% 0.18/0.45 => ( ! [Y2: a] :
% 0.18/0.45 ( ( P @ Y2 )
% 0.18/0.45 => ( less_eq @ X @ Y2 ) )
% 0.18/0.45 => ( ( least_a @ less_eq @ P )
% 0.18/0.45 = X ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Least_equality
% 0.18/0.45 thf(fact_127_lattice_Od__aux_Ocong,axiom,
% 0.18/0.45 modula1936294176_aux_a = modula1936294176_aux_a ).
% 0.18/0.45
% 0.18/0.45 % lattice.d_aux.cong
% 0.18/0.45 thf(fact_128_abel__semigroup_Oaxioms_I1_J,axiom,
% 0.18/0.45 ! [F: a > a > a] :
% 0.18/0.45 ( ( abel_semigroup_a @ F )
% 0.18/0.45 => ( semigroup_a @ F ) ) ).
% 0.18/0.45
% 0.18/0.45 % abel_semigroup.axioms(1)
% 0.18/0.45 thf(fact_129_semilattice__set__def,axiom,
% 0.18/0.45 lattic1885654924_set_a = semilattice_a ).
% 0.18/0.45
% 0.18/0.45 % semilattice_set_def
% 0.18/0.45 thf(fact_130_semilattice__set_Ointro,axiom,
% 0.18/0.45 ! [F: a > a > a] :
% 0.18/0.45 ( ( semilattice_a @ F )
% 0.18/0.45 => ( lattic1885654924_set_a @ F ) ) ).
% 0.18/0.45
% 0.18/0.45 % semilattice_set.intro
% 0.18/0.45 thf(fact_131_semilattice__set_Oaxioms,axiom,
% 0.18/0.45 ! [F: a > a > a] :
% 0.18/0.45 ( ( lattic1885654924_set_a @ F )
% 0.18/0.45 => ( semilattice_a @ F ) ) ).
% 0.18/0.45
% 0.18/0.45 % semilattice_set.axioms
% 0.18/0.45 thf(fact_132_semilattice_Oaxioms_I1_J,axiom,
% 0.18/0.45 ! [F: a > a > a] :
% 0.18/0.45 ( ( semilattice_a @ F )
% 0.18/0.45 => ( abel_semigroup_a @ F ) ) ).
% 0.18/0.45
% 0.18/0.45 % semilattice.axioms(1)
% 0.18/0.45 thf(fact_133_local_Obdd__above__def,axiom,
% 0.18/0.45 ! [A2: set_a] :
% 0.18/0.45 ( ( condit1627435690bove_a @ less_eq @ A2 )
% 0.18/0.45 = ( ? [M: a] :
% 0.18/0.45 ! [X3: a] :
% 0.18/0.45 ( ( member_a @ X3 @ A2 )
% 0.18/0.45 => ( less_eq @ X3 @ M ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.bdd_above_def
% 0.18/0.45 thf(fact_134_local_Obdd__below__def,axiom,
% 0.18/0.45 ! [A2: set_a] :
% 0.18/0.45 ( ( condit1001553558elow_a @ less_eq @ A2 )
% 0.18/0.45 = ( ? [M2: a] :
% 0.18/0.45 ! [X3: a] :
% 0.18/0.45 ( ( member_a @ X3 @ A2 )
% 0.18/0.45 => ( less_eq @ M2 @ X3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.bdd_below_def
% 0.18/0.45 thf(fact_135_local_Otransp__le,axiom,
% 0.18/0.45 transp_a @ less_eq ).
% 0.18/0.45
% 0.18/0.45 % local.transp_le
% 0.18/0.45 thf(fact_136_local_Obdd__belowI,axiom,
% 0.18/0.45 ! [A2: set_a,M3: a] :
% 0.18/0.45 ( ! [X2: a] :
% 0.18/0.45 ( ( member_a @ X2 @ A2 )
% 0.18/0.45 => ( less_eq @ M3 @ X2 ) )
% 0.18/0.45 => ( condit1001553558elow_a @ less_eq @ A2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.bdd_belowI
% 0.18/0.45 thf(fact_137_local_Obdd__aboveI,axiom,
% 0.18/0.45 ! [A2: set_a,M4: a] :
% 0.18/0.45 ( ! [X2: a] :
% 0.18/0.45 ( ( member_a @ X2 @ A2 )
% 0.18/0.45 => ( less_eq @ X2 @ M4 ) )
% 0.18/0.45 => ( condit1627435690bove_a @ less_eq @ A2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.bdd_aboveI
% 0.18/0.45 thf(fact_138_abel__semigroup_Oleft__commute,axiom,
% 0.18/0.45 ! [F: a > a > a,B: a,A: a,C: a] :
% 0.18/0.45 ( ( abel_semigroup_a @ F )
% 0.18/0.45 => ( ( F @ B @ ( F @ A @ C ) )
% 0.18/0.45 = ( F @ A @ ( F @ B @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % abel_semigroup.left_commute
% 0.18/0.45 thf(fact_139_semilattice_Oright__idem,axiom,
% 0.18/0.45 ! [F: a > a > a,A: a,B: a] :
% 0.18/0.45 ( ( semilattice_a @ F )
% 0.18/0.45 => ( ( F @ ( F @ A @ B ) @ B )
% 0.18/0.45 = ( F @ A @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % semilattice.right_idem
% 0.18/0.45 thf(fact_140_semilattice_Oleft__idem,axiom,
% 0.18/0.45 ! [F: a > a > a,A: a,B: a] :
% 0.18/0.45 ( ( semilattice_a @ F )
% 0.18/0.45 => ( ( F @ A @ ( F @ A @ B ) )
% 0.18/0.45 = ( F @ A @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % semilattice.left_idem
% 0.18/0.45 thf(fact_141_abel__semigroup_Ocommute,axiom,
% 0.18/0.45 ! [F: a > a > a,A: a,B: a] :
% 0.18/0.45 ( ( abel_semigroup_a @ F )
% 0.18/0.45 => ( ( F @ A @ B )
% 0.18/0.45 = ( F @ B @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % abel_semigroup.commute
% 0.18/0.45 thf(fact_142_semilattice_Oidem,axiom,
% 0.18/0.45 ! [F: a > a > a,A: a] :
% 0.18/0.45 ( ( semilattice_a @ F )
% 0.18/0.45 => ( ( F @ A @ A )
% 0.18/0.45 = A ) ) ).
% 0.18/0.45
% 0.18/0.45 % semilattice.idem
% 0.18/0.45 thf(fact_143_semigroup_Ointro,axiom,
% 0.18/0.45 ! [F: a > a > a] :
% 0.18/0.45 ( ! [A4: a,B3: a,C2: a] :
% 0.18/0.45 ( ( F @ ( F @ A4 @ B3 ) @ C2 )
% 0.18/0.45 = ( F @ A4 @ ( F @ B3 @ C2 ) ) )
% 0.18/0.45 => ( semigroup_a @ F ) ) ).
% 0.18/0.45
% 0.18/0.45 % semigroup.intro
% 0.18/0.45 thf(fact_144_semigroup_Oassoc,axiom,
% 0.18/0.45 ! [F: a > a > a,A: a,B: a,C: a] :
% 0.18/0.45 ( ( semigroup_a @ F )
% 0.18/0.45 => ( ( F @ ( F @ A @ B ) @ C )
% 0.18/0.45 = ( F @ A @ ( F @ B @ C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % semigroup.assoc
% 0.18/0.45 thf(fact_145_semigroup__def,axiom,
% 0.18/0.45 ( semigroup_a
% 0.18/0.45 = ( ^ [F2: a > a > a] :
% 0.18/0.45 ! [A3: a,B2: a,C3: a] :
% 0.18/0.45 ( ( F2 @ ( F2 @ A3 @ B2 ) @ C3 )
% 0.18/0.45 = ( F2 @ A3 @ ( F2 @ B2 @ C3 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % semigroup_def
% 0.18/0.45 thf(fact_146_local_Obdd__below__mono,axiom,
% 0.18/0.45 ! [B4: set_a,A2: set_a] :
% 0.18/0.45 ( ( condit1001553558elow_a @ less_eq @ B4 )
% 0.18/0.45 => ( ( ord_less_eq_set_a @ A2 @ B4 )
% 0.18/0.45 => ( condit1001553558elow_a @ less_eq @ A2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.bdd_below_mono
% 0.18/0.45 thf(fact_147_local_Obdd__above__mono,axiom,
% 0.18/0.45 ! [B4: set_a,A2: set_a] :
% 0.18/0.45 ( ( condit1627435690bove_a @ less_eq @ B4 )
% 0.18/0.45 => ( ( ord_less_eq_set_a @ A2 @ B4 )
% 0.18/0.45 => ( condit1627435690bove_a @ less_eq @ A2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.bdd_above_mono
% 0.18/0.45 thf(fact_148_local_Obdd__below__finite,axiom,
% 0.18/0.45 ! [A2: set_a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( condit1001553558elow_a @ less_eq @ A2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.bdd_below_finite
% 0.18/0.45 thf(fact_149_local_Obdd__above__finite,axiom,
% 0.18/0.45 ! [A2: set_a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( condit1627435690bove_a @ less_eq @ A2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.bdd_above_finite
% 0.18/0.45 thf(fact_150_local_Ofinite__has__maximal2,axiom,
% 0.18/0.45 ! [A2: set_a,A: a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( member_a @ A @ A2 )
% 0.18/0.45 => ? [X2: a] :
% 0.18/0.45 ( ( member_a @ X2 @ A2 )
% 0.18/0.45 & ( less_eq @ A @ X2 )
% 0.18/0.45 & ! [Xa: a] :
% 0.18/0.45 ( ( member_a @ Xa @ A2 )
% 0.18/0.45 => ( ( less_eq @ X2 @ Xa )
% 0.18/0.45 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.finite_has_maximal2
% 0.18/0.45 thf(fact_151_local_Ofinite__has__minimal2,axiom,
% 0.18/0.45 ! [A2: set_a,A: a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( member_a @ A @ A2 )
% 0.18/0.45 => ? [X2: a] :
% 0.18/0.45 ( ( member_a @ X2 @ A2 )
% 0.18/0.45 & ( less_eq @ X2 @ A )
% 0.18/0.45 & ! [Xa: a] :
% 0.18/0.45 ( ( member_a @ Xa @ A2 )
% 0.18/0.45 => ( ( less_eq @ Xa @ X2 )
% 0.18/0.45 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.finite_has_minimal2
% 0.18/0.45 thf(fact_152_local_OantimonoD,axiom,
% 0.18/0.45 ! [F: a > set_a,X: a,Y: a] :
% 0.18/0.45 ( ( antimono_a_set_a @ less_eq @ F )
% 0.18/0.45 => ( ( less_eq @ X @ Y )
% 0.18/0.45 => ( ord_less_eq_set_a @ ( F @ Y ) @ ( F @ X ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.antimonoD
% 0.18/0.45 thf(fact_153_local_OantimonoE,axiom,
% 0.18/0.45 ! [F: a > set_a,X: a,Y: a] :
% 0.18/0.45 ( ( antimono_a_set_a @ less_eq @ F )
% 0.18/0.45 => ( ( less_eq @ X @ Y )
% 0.18/0.45 => ( ord_less_eq_set_a @ ( F @ Y ) @ ( F @ X ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.antimonoE
% 0.18/0.45 thf(fact_154_local_OantimonoI,axiom,
% 0.18/0.45 ! [F: a > set_a] :
% 0.18/0.45 ( ! [X2: a,Y2: a] :
% 0.18/0.45 ( ( less_eq @ X2 @ Y2 )
% 0.18/0.45 => ( ord_less_eq_set_a @ ( F @ Y2 ) @ ( F @ X2 ) ) )
% 0.18/0.45 => ( antimono_a_set_a @ less_eq @ F ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.antimonoI
% 0.18/0.45 thf(fact_155_local_Oantimono__def,axiom,
% 0.18/0.45 ! [F: a > set_a] :
% 0.18/0.45 ( ( antimono_a_set_a @ less_eq @ F )
% 0.18/0.45 = ( ! [X3: a,Y4: a] :
% 0.18/0.45 ( ( less_eq @ X3 @ Y4 )
% 0.18/0.45 => ( ord_less_eq_set_a @ ( F @ Y4 ) @ ( F @ X3 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.antimono_def
% 0.18/0.45 thf(fact_156_local_OmonoD,axiom,
% 0.18/0.45 ! [F: a > set_a,X: a,Y: a] :
% 0.18/0.45 ( ( mono_a_set_a @ less_eq @ F )
% 0.18/0.45 => ( ( less_eq @ X @ Y )
% 0.18/0.45 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.monoD
% 0.18/0.45 thf(fact_157_local_OmonoE,axiom,
% 0.18/0.45 ! [F: a > set_a,X: a,Y: a] :
% 0.18/0.45 ( ( mono_a_set_a @ less_eq @ F )
% 0.18/0.45 => ( ( less_eq @ X @ Y )
% 0.18/0.45 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.monoE
% 0.18/0.45 thf(fact_158_local_OmonoI,axiom,
% 0.18/0.45 ! [F: a > set_a] :
% 0.18/0.45 ( ! [X2: a,Y2: a] :
% 0.18/0.45 ( ( less_eq @ X2 @ Y2 )
% 0.18/0.45 => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.45 => ( mono_a_set_a @ less_eq @ F ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.monoI
% 0.18/0.45 thf(fact_159_local_Omono__def,axiom,
% 0.18/0.45 ! [F: a > set_a] :
% 0.18/0.45 ( ( mono_a_set_a @ less_eq @ F )
% 0.18/0.45 = ( ! [X3: a,Y4: a] :
% 0.18/0.45 ( ( less_eq @ X3 @ Y4 )
% 0.18/0.45 => ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.mono_def
% 0.18/0.45 thf(fact_160_local_OSup__fin_OcoboundedI,axiom,
% 0.18/0.45 ! [A2: set_a,A: a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( member_a @ A @ A2 )
% 0.18/0.45 => ( less_eq @ A @ ( lattic1781219155_fin_a @ sup @ A2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Sup_fin.coboundedI
% 0.18/0.45 thf(fact_161_local_OInf__fin_OcoboundedI,axiom,
% 0.18/0.45 ! [A2: set_a,A: a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( member_a @ A @ A2 )
% 0.18/0.45 => ( less_eq @ ( lattic247676691_fin_a @ inf @ A2 ) @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Inf_fin.coboundedI
% 0.18/0.45 thf(fact_162_local_OSup__fin_Oin__idem,axiom,
% 0.18/0.45 ! [A2: set_a,X: a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( member_a @ X @ A2 )
% 0.18/0.45 => ( ( sup @ X @ ( lattic1781219155_fin_a @ sup @ A2 ) )
% 0.18/0.45 = ( lattic1781219155_fin_a @ sup @ A2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Sup_fin.in_idem
% 0.18/0.45 thf(fact_163_local_OInf__fin_Oin__idem,axiom,
% 0.18/0.45 ! [A2: set_a,X: a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( member_a @ X @ A2 )
% 0.18/0.45 => ( ( inf @ X @ ( lattic247676691_fin_a @ inf @ A2 ) )
% 0.18/0.45 = ( lattic247676691_fin_a @ inf @ A2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Inf_fin.in_idem
% 0.18/0.45 thf(fact_164_local_Oinf__Sup__absorb,axiom,
% 0.18/0.45 ! [A2: set_a,A: a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( member_a @ A @ A2 )
% 0.18/0.45 => ( ( inf @ A @ ( lattic1781219155_fin_a @ sup @ A2 ) )
% 0.18/0.45 = A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.inf_Sup_absorb
% 0.18/0.45 thf(fact_165_local_Osup__Inf__absorb,axiom,
% 0.18/0.45 ! [A2: set_a,A: a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( member_a @ A @ A2 )
% 0.18/0.45 => ( ( sup @ ( lattic247676691_fin_a @ inf @ A2 ) @ A )
% 0.18/0.45 = A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.sup_Inf_absorb
% 0.18/0.45 thf(fact_166_semilattice__inf_OInf__fin_Ocong,axiom,
% 0.18/0.45 lattic247676691_fin_a = lattic247676691_fin_a ).
% 0.18/0.45
% 0.18/0.45 % semilattice_inf.Inf_fin.cong
% 0.18/0.45 thf(fact_167_semilattice__sup_OSup__fin_Ocong,axiom,
% 0.18/0.45 lattic1781219155_fin_a = lattic1781219155_fin_a ).
% 0.18/0.45
% 0.18/0.45 % semilattice_sup.Sup_fin.cong
% 0.18/0.45 thf(fact_168_local_OInf__fin__le__Sup__fin,axiom,
% 0.18/0.45 ! [A2: set_a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( A2 != bot_bot_set_a )
% 0.18/0.45 => ( less_eq @ ( lattic247676691_fin_a @ inf @ A2 ) @ ( lattic1781219155_fin_a @ sup @ A2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Inf_fin_le_Sup_fin
% 0.18/0.45 thf(fact_169_local_OSup__fin_Osubset__imp,axiom,
% 0.18/0.45 ! [A2: set_a,B4: set_a] :
% 0.18/0.45 ( ( ord_less_eq_set_a @ A2 @ B4 )
% 0.18/0.45 => ( ( A2 != bot_bot_set_a )
% 0.18/0.45 => ( ( finite_finite_a @ B4 )
% 0.18/0.45 => ( less_eq @ ( lattic1781219155_fin_a @ sup @ A2 ) @ ( lattic1781219155_fin_a @ sup @ B4 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Sup_fin.subset_imp
% 0.18/0.45 thf(fact_170_local_OInf__fin_Osubset__imp,axiom,
% 0.18/0.45 ! [A2: set_a,B4: set_a] :
% 0.18/0.45 ( ( ord_less_eq_set_a @ A2 @ B4 )
% 0.18/0.45 => ( ( A2 != bot_bot_set_a )
% 0.18/0.45 => ( ( finite_finite_a @ B4 )
% 0.18/0.45 => ( less_eq @ ( lattic247676691_fin_a @ inf @ B4 ) @ ( lattic247676691_fin_a @ inf @ A2 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Inf_fin.subset_imp
% 0.18/0.45 thf(fact_171_local_Ofinite__has__minimal,axiom,
% 0.18/0.45 ! [A2: set_a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( A2 != bot_bot_set_a )
% 0.18/0.45 => ? [X2: a] :
% 0.18/0.45 ( ( member_a @ X2 @ A2 )
% 0.18/0.45 & ! [Xa: a] :
% 0.18/0.45 ( ( member_a @ Xa @ A2 )
% 0.18/0.45 => ( ( less_eq @ Xa @ X2 )
% 0.18/0.45 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.finite_has_minimal
% 0.18/0.45 thf(fact_172_local_Ofinite__has__maximal,axiom,
% 0.18/0.45 ! [A2: set_a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( A2 != bot_bot_set_a )
% 0.18/0.45 => ? [X2: a] :
% 0.18/0.45 ( ( member_a @ X2 @ A2 )
% 0.18/0.45 & ! [Xa: a] :
% 0.18/0.45 ( ( member_a @ Xa @ A2 )
% 0.18/0.45 => ( ( less_eq @ X2 @ Xa )
% 0.18/0.45 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.finite_has_maximal
% 0.18/0.45 thf(fact_173_local_OInf__fin_Osubset,axiom,
% 0.18/0.45 ! [A2: set_a,B4: set_a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( B4 != bot_bot_set_a )
% 0.18/0.45 => ( ( ord_less_eq_set_a @ B4 @ A2 )
% 0.18/0.45 => ( ( inf @ ( lattic247676691_fin_a @ inf @ B4 ) @ ( lattic247676691_fin_a @ inf @ A2 ) )
% 0.18/0.45 = ( lattic247676691_fin_a @ inf @ A2 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Inf_fin.subset
% 0.18/0.45 thf(fact_174_local_OSup__fin_Osubset,axiom,
% 0.18/0.45 ! [A2: set_a,B4: set_a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( B4 != bot_bot_set_a )
% 0.18/0.45 => ( ( ord_less_eq_set_a @ B4 @ A2 )
% 0.18/0.45 => ( ( sup @ ( lattic1781219155_fin_a @ sup @ B4 ) @ ( lattic1781219155_fin_a @ sup @ A2 ) )
% 0.18/0.45 = ( lattic1781219155_fin_a @ sup @ A2 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Sup_fin.subset
% 0.18/0.45 thf(fact_175_local_OInf__fin_Obounded__iff,axiom,
% 0.18/0.45 ! [A2: set_a,X: a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( A2 != bot_bot_set_a )
% 0.18/0.45 => ( ( less_eq @ X @ ( lattic247676691_fin_a @ inf @ A2 ) )
% 0.18/0.45 = ( ! [X3: a] :
% 0.18/0.45 ( ( member_a @ X3 @ A2 )
% 0.18/0.45 => ( less_eq @ X @ X3 ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Inf_fin.bounded_iff
% 0.18/0.45 thf(fact_176_local_OInf__fin_OboundedI,axiom,
% 0.18/0.45 ! [A2: set_a,X: a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( A2 != bot_bot_set_a )
% 0.18/0.45 => ( ! [A4: a] :
% 0.18/0.45 ( ( member_a @ A4 @ A2 )
% 0.18/0.45 => ( less_eq @ X @ A4 ) )
% 0.18/0.45 => ( less_eq @ X @ ( lattic247676691_fin_a @ inf @ A2 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Inf_fin.boundedI
% 0.18/0.45 thf(fact_177_local_OInf__fin_OboundedE,axiom,
% 0.18/0.45 ! [A2: set_a,X: a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( A2 != bot_bot_set_a )
% 0.18/0.45 => ( ( less_eq @ X @ ( lattic247676691_fin_a @ inf @ A2 ) )
% 0.18/0.45 => ! [A5: a] :
% 0.18/0.45 ( ( member_a @ A5 @ A2 )
% 0.18/0.45 => ( less_eq @ X @ A5 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Inf_fin.boundedE
% 0.18/0.45 thf(fact_178_local_OSup__fin_Obounded__iff,axiom,
% 0.18/0.45 ! [A2: set_a,X: a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( A2 != bot_bot_set_a )
% 0.18/0.45 => ( ( less_eq @ ( lattic1781219155_fin_a @ sup @ A2 ) @ X )
% 0.18/0.45 = ( ! [X3: a] :
% 0.18/0.45 ( ( member_a @ X3 @ A2 )
% 0.18/0.45 => ( less_eq @ X3 @ X ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Sup_fin.bounded_iff
% 0.18/0.45 thf(fact_179_local_OSup__fin_OboundedI,axiom,
% 0.18/0.45 ! [A2: set_a,X: a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( A2 != bot_bot_set_a )
% 0.18/0.45 => ( ! [A4: a] :
% 0.18/0.45 ( ( member_a @ A4 @ A2 )
% 0.18/0.45 => ( less_eq @ A4 @ X ) )
% 0.18/0.45 => ( less_eq @ ( lattic1781219155_fin_a @ sup @ A2 ) @ X ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Sup_fin.boundedI
% 0.18/0.45 thf(fact_180_local_OSup__fin_OboundedE,axiom,
% 0.18/0.45 ! [A2: set_a,X: a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( A2 != bot_bot_set_a )
% 0.18/0.45 => ( ( less_eq @ ( lattic1781219155_fin_a @ sup @ A2 ) @ X )
% 0.18/0.45 => ! [A5: a] :
% 0.18/0.45 ( ( member_a @ A5 @ A2 )
% 0.18/0.45 => ( less_eq @ A5 @ X ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Sup_fin.boundedE
% 0.18/0.45 thf(fact_181_local_Obdd__above__empty,axiom,
% 0.18/0.45 condit1627435690bove_a @ less_eq @ bot_bot_set_a ).
% 0.18/0.45
% 0.18/0.45 % local.bdd_above_empty
% 0.18/0.45 thf(fact_182_local_Obdd__below__empty,axiom,
% 0.18/0.45 condit1001553558elow_a @ less_eq @ bot_bot_set_a ).
% 0.18/0.45
% 0.18/0.45 % local.bdd_below_empty
% 0.18/0.45 thf(fact_183_local_Onot__empty__eq__Iic__eq__empty,axiom,
% 0.18/0.45 ! [H: a] :
% 0.18/0.45 ( bot_bot_set_a
% 0.18/0.45 != ( set_atMost_a @ less_eq @ H ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.not_empty_eq_Iic_eq_empty
% 0.18/0.45 thf(fact_184_local_Onot__empty__eq__Ici__eq__empty,axiom,
% 0.18/0.45 ! [L: a] :
% 0.18/0.45 ( bot_bot_set_a
% 0.18/0.45 != ( set_atLeast_a @ less_eq @ L ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.not_empty_eq_Ici_eq_empty
% 0.18/0.45 thf(fact_185_subset__empty,axiom,
% 0.18/0.45 ! [A2: set_a] :
% 0.18/0.45 ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
% 0.18/0.45 = ( A2 = bot_bot_set_a ) ) ).
% 0.18/0.45
% 0.18/0.45 % subset_empty
% 0.18/0.45 thf(fact_186_empty__Collect__eq,axiom,
% 0.18/0.45 ! [P: a > $o] :
% 0.18/0.45 ( ( bot_bot_set_a
% 0.18/0.45 = ( collect_a @ P ) )
% 0.18/0.45 = ( ! [X3: a] :
% 0.18/0.45 ~ ( P @ X3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % empty_Collect_eq
% 0.18/0.45 thf(fact_187_Collect__empty__eq,axiom,
% 0.18/0.45 ! [P: a > $o] :
% 0.18/0.45 ( ( ( collect_a @ P )
% 0.18/0.45 = bot_bot_set_a )
% 0.18/0.45 = ( ! [X3: a] :
% 0.18/0.45 ~ ( P @ X3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_empty_eq
% 0.18/0.45 thf(fact_188_all__not__in__conv,axiom,
% 0.18/0.45 ! [A2: set_set_a] :
% 0.18/0.45 ( ( ! [X3: set_a] :
% 0.18/0.45 ~ ( member_set_a @ X3 @ A2 ) )
% 0.18/0.45 = ( A2 = bot_bot_set_set_a ) ) ).
% 0.18/0.45
% 0.18/0.45 % all_not_in_conv
% 0.18/0.45 thf(fact_189_all__not__in__conv,axiom,
% 0.18/0.45 ! [A2: set_a] :
% 0.18/0.45 ( ( ! [X3: a] :
% 0.18/0.45 ~ ( member_a @ X3 @ A2 ) )
% 0.18/0.45 = ( A2 = bot_bot_set_a ) ) ).
% 0.18/0.45
% 0.18/0.45 % all_not_in_conv
% 0.18/0.45 thf(fact_190_empty__iff,axiom,
% 0.18/0.45 ! [C: set_a] :
% 0.18/0.45 ~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% 0.18/0.45
% 0.18/0.45 % empty_iff
% 0.18/0.45 thf(fact_191_empty__iff,axiom,
% 0.18/0.45 ! [C: a] :
% 0.18/0.45 ~ ( member_a @ C @ bot_bot_set_a ) ).
% 0.18/0.45
% 0.18/0.45 % empty_iff
% 0.18/0.45 thf(fact_192_subset__antisym,axiom,
% 0.18/0.45 ! [A2: set_a,B4: set_a] :
% 0.18/0.45 ( ( ord_less_eq_set_a @ A2 @ B4 )
% 0.18/0.45 => ( ( ord_less_eq_set_a @ B4 @ A2 )
% 0.18/0.45 => ( A2 = B4 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subset_antisym
% 0.18/0.45 thf(fact_193_subsetI,axiom,
% 0.18/0.45 ! [A2: set_set_a,B4: set_set_a] :
% 0.18/0.45 ( ! [X2: set_a] :
% 0.18/0.45 ( ( member_set_a @ X2 @ A2 )
% 0.18/0.45 => ( member_set_a @ X2 @ B4 ) )
% 0.18/0.45 => ( ord_le318720350_set_a @ A2 @ B4 ) ) ).
% 0.18/0.45
% 0.18/0.45 % subsetI
% 0.18/0.45 thf(fact_194_subsetI,axiom,
% 0.18/0.45 ! [A2: set_a,B4: set_a] :
% 0.18/0.45 ( ! [X2: a] :
% 0.18/0.45 ( ( member_a @ X2 @ A2 )
% 0.18/0.45 => ( member_a @ X2 @ B4 ) )
% 0.18/0.45 => ( ord_less_eq_set_a @ A2 @ B4 ) ) ).
% 0.18/0.45
% 0.18/0.45 % subsetI
% 0.18/0.45 thf(fact_195_empty__subsetI,axiom,
% 0.18/0.45 ! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% 0.18/0.45
% 0.18/0.45 % empty_subsetI
% 0.18/0.45 thf(fact_196_local_OatLeast__iff,axiom,
% 0.18/0.45 ! [I: a,K: a] :
% 0.18/0.45 ( ( member_a @ I @ ( set_atLeast_a @ less_eq @ K ) )
% 0.18/0.45 = ( less_eq @ K @ I ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.atLeast_iff
% 0.18/0.45 thf(fact_197_local_OatMost__iff,axiom,
% 0.18/0.45 ! [I: a,K: a] :
% 0.18/0.45 ( ( member_a @ I @ ( set_atMost_a @ less_eq @ K ) )
% 0.18/0.45 = ( less_eq @ I @ K ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.atMost_iff
% 0.18/0.45 thf(fact_198_local_Obdd__below__Ici,axiom,
% 0.18/0.45 ! [A: a] : ( condit1001553558elow_a @ less_eq @ ( set_atLeast_a @ less_eq @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.bdd_below_Ici
% 0.18/0.45 thf(fact_199_local_Obdd__above__Iic,axiom,
% 0.18/0.45 ! [B: a] : ( condit1627435690bove_a @ less_eq @ ( set_atMost_a @ less_eq @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.bdd_above_Iic
% 0.18/0.45 thf(fact_200_ex__in__conv,axiom,
% 0.18/0.45 ! [A2: set_set_a] :
% 0.18/0.45 ( ( ? [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
% 0.18/0.45 = ( A2 != bot_bot_set_set_a ) ) ).
% 0.18/0.45
% 0.18/0.45 % ex_in_conv
% 0.18/0.45 thf(fact_201_ex__in__conv,axiom,
% 0.18/0.45 ! [A2: set_a] :
% 0.18/0.45 ( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
% 0.18/0.45 = ( A2 != bot_bot_set_a ) ) ).
% 0.18/0.45
% 0.18/0.45 % ex_in_conv
% 0.18/0.45 thf(fact_202_equals0I,axiom,
% 0.18/0.45 ! [A2: set_set_a] :
% 0.18/0.45 ( ! [Y2: set_a] :
% 0.18/0.45 ~ ( member_set_a @ Y2 @ A2 )
% 0.18/0.45 => ( A2 = bot_bot_set_set_a ) ) ).
% 0.18/0.45
% 0.18/0.45 % equals0I
% 0.18/0.45 thf(fact_203_equals0I,axiom,
% 0.18/0.45 ! [A2: set_a] :
% 0.18/0.45 ( ! [Y2: a] :
% 0.18/0.45 ~ ( member_a @ Y2 @ A2 )
% 0.18/0.45 => ( A2 = bot_bot_set_a ) ) ).
% 0.18/0.45
% 0.18/0.45 % equals0I
% 0.18/0.45 thf(fact_204_equals0D,axiom,
% 0.18/0.45 ! [A2: set_set_a,A: set_a] :
% 0.18/0.45 ( ( A2 = bot_bot_set_set_a )
% 0.18/0.45 => ~ ( member_set_a @ A @ A2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % equals0D
% 0.18/0.45 thf(fact_205_equals0D,axiom,
% 0.18/0.45 ! [A2: set_a,A: a] :
% 0.18/0.45 ( ( A2 = bot_bot_set_a )
% 0.18/0.45 => ~ ( member_a @ A @ A2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % equals0D
% 0.18/0.45 thf(fact_206_emptyE,axiom,
% 0.18/0.45 ! [A: set_a] :
% 0.18/0.45 ~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% 0.18/0.45
% 0.18/0.45 % emptyE
% 0.18/0.45 thf(fact_207_emptyE,axiom,
% 0.18/0.45 ! [A: a] :
% 0.18/0.45 ~ ( member_a @ A @ bot_bot_set_a ) ).
% 0.18/0.45
% 0.18/0.45 % emptyE
% 0.18/0.45 thf(fact_208_Collect__mono__iff,axiom,
% 0.18/0.45 ! [P: a > $o,Q: a > $o] :
% 0.18/0.45 ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
% 0.18/0.45 = ( ! [X3: a] :
% 0.18/0.45 ( ( P @ X3 )
% 0.18/0.45 => ( Q @ X3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_mono_iff
% 0.18/0.45 thf(fact_209_set__eq__subset,axiom,
% 0.18/0.45 ( ( ^ [Y3: set_a,Z3: set_a] : ( Y3 = Z3 ) )
% 0.18/0.45 = ( ^ [A6: set_a,B5: set_a] :
% 0.18/0.45 ( ( ord_less_eq_set_a @ A6 @ B5 )
% 0.18/0.45 & ( ord_less_eq_set_a @ B5 @ A6 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % set_eq_subset
% 0.18/0.45 thf(fact_210_subset__trans,axiom,
% 0.18/0.45 ! [A2: set_a,B4: set_a,C4: set_a] :
% 0.18/0.45 ( ( ord_less_eq_set_a @ A2 @ B4 )
% 0.18/0.45 => ( ( ord_less_eq_set_a @ B4 @ C4 )
% 0.18/0.45 => ( ord_less_eq_set_a @ A2 @ C4 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subset_trans
% 0.18/0.45 thf(fact_211_Collect__mono,axiom,
% 0.18/0.45 ! [P: a > $o,Q: a > $o] :
% 0.18/0.45 ( ! [X2: a] :
% 0.18/0.45 ( ( P @ X2 )
% 0.18/0.45 => ( Q @ X2 ) )
% 0.18/0.45 => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_mono
% 0.18/0.45 thf(fact_212_subset__refl,axiom,
% 0.18/0.45 ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% 0.18/0.45
% 0.18/0.45 % subset_refl
% 0.18/0.45 thf(fact_213_subset__iff,axiom,
% 0.18/0.45 ( ord_le318720350_set_a
% 0.18/0.45 = ( ^ [A6: set_set_a,B5: set_set_a] :
% 0.18/0.45 ! [T: set_a] :
% 0.18/0.45 ( ( member_set_a @ T @ A6 )
% 0.18/0.45 => ( member_set_a @ T @ B5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subset_iff
% 0.18/0.45 thf(fact_214_subset__iff,axiom,
% 0.18/0.45 ( ord_less_eq_set_a
% 0.18/0.45 = ( ^ [A6: set_a,B5: set_a] :
% 0.18/0.45 ! [T: a] :
% 0.18/0.45 ( ( member_a @ T @ A6 )
% 0.18/0.45 => ( member_a @ T @ B5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subset_iff
% 0.18/0.45 thf(fact_215_equalityD2,axiom,
% 0.18/0.45 ! [A2: set_a,B4: set_a] :
% 0.18/0.45 ( ( A2 = B4 )
% 0.18/0.45 => ( ord_less_eq_set_a @ B4 @ A2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % equalityD2
% 0.18/0.45 thf(fact_216_equalityD1,axiom,
% 0.18/0.45 ! [A2: set_a,B4: set_a] :
% 0.18/0.45 ( ( A2 = B4 )
% 0.18/0.45 => ( ord_less_eq_set_a @ A2 @ B4 ) ) ).
% 0.18/0.45
% 0.18/0.45 % equalityD1
% 0.18/0.45 thf(fact_217_subset__eq,axiom,
% 0.18/0.45 ( ord_le318720350_set_a
% 0.18/0.45 = ( ^ [A6: set_set_a,B5: set_set_a] :
% 0.18/0.45 ! [X3: set_a] :
% 0.18/0.45 ( ( member_set_a @ X3 @ A6 )
% 0.18/0.45 => ( member_set_a @ X3 @ B5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subset_eq
% 0.18/0.45 thf(fact_218_subset__eq,axiom,
% 0.18/0.45 ( ord_less_eq_set_a
% 0.18/0.45 = ( ^ [A6: set_a,B5: set_a] :
% 0.18/0.45 ! [X3: a] :
% 0.18/0.45 ( ( member_a @ X3 @ A6 )
% 0.18/0.45 => ( member_a @ X3 @ B5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subset_eq
% 0.18/0.45 thf(fact_219_equalityE,axiom,
% 0.18/0.45 ! [A2: set_a,B4: set_a] :
% 0.18/0.45 ( ( A2 = B4 )
% 0.18/0.45 => ~ ( ( ord_less_eq_set_a @ A2 @ B4 )
% 0.18/0.45 => ~ ( ord_less_eq_set_a @ B4 @ A2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % equalityE
% 0.18/0.45 thf(fact_220_subsetD,axiom,
% 0.18/0.45 ! [A2: set_set_a,B4: set_set_a,C: set_a] :
% 0.18/0.45 ( ( ord_le318720350_set_a @ A2 @ B4 )
% 0.18/0.45 => ( ( member_set_a @ C @ A2 )
% 0.18/0.45 => ( member_set_a @ C @ B4 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subsetD
% 0.18/0.45 thf(fact_221_subsetD,axiom,
% 0.18/0.45 ! [A2: set_a,B4: set_a,C: a] :
% 0.18/0.45 ( ( ord_less_eq_set_a @ A2 @ B4 )
% 0.18/0.45 => ( ( member_a @ C @ A2 )
% 0.18/0.45 => ( member_a @ C @ B4 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subsetD
% 0.18/0.45 thf(fact_222_in__mono,axiom,
% 0.18/0.45 ! [A2: set_set_a,B4: set_set_a,X: set_a] :
% 0.18/0.45 ( ( ord_le318720350_set_a @ A2 @ B4 )
% 0.18/0.45 => ( ( member_set_a @ X @ A2 )
% 0.18/0.45 => ( member_set_a @ X @ B4 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % in_mono
% 0.18/0.45 thf(fact_223_in__mono,axiom,
% 0.18/0.45 ! [A2: set_a,B4: set_a,X: a] :
% 0.18/0.45 ( ( ord_less_eq_set_a @ A2 @ B4 )
% 0.18/0.45 => ( ( member_a @ X @ A2 )
% 0.18/0.45 => ( member_a @ X @ B4 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % in_mono
% 0.18/0.45 thf(fact_224_local_OSup__fin_Oclosed,axiom,
% 0.18/0.45 ! [A2: set_a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( A2 != bot_bot_set_a )
% 0.18/0.45 => ( ! [X2: a,Y2: a] : ( member_a @ ( sup @ X2 @ Y2 ) @ ( insert_a @ X2 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) )
% 0.18/0.45 => ( member_a @ ( lattic1781219155_fin_a @ sup @ A2 ) @ A2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Sup_fin.closed
% 0.18/0.45 thf(fact_225_local_OSup__fin_Oinsert__not__elem,axiom,
% 0.18/0.45 ! [A2: set_a,X: a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ~ ( member_a @ X @ A2 )
% 0.18/0.45 => ( ( A2 != bot_bot_set_a )
% 0.18/0.45 => ( ( lattic1781219155_fin_a @ sup @ ( insert_a @ X @ A2 ) )
% 0.18/0.45 = ( sup @ X @ ( lattic1781219155_fin_a @ sup @ A2 ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Sup_fin.insert_not_elem
% 0.18/0.45 thf(fact_226_local_OInf__fin_Oclosed,axiom,
% 0.18/0.45 ! [A2: set_a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( A2 != bot_bot_set_a )
% 0.18/0.45 => ( ! [X2: a,Y2: a] : ( member_a @ ( inf @ X2 @ Y2 ) @ ( insert_a @ X2 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) )
% 0.18/0.45 => ( member_a @ ( lattic247676691_fin_a @ inf @ A2 ) @ A2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Inf_fin.closed
% 0.18/0.45 thf(fact_227_local_OInf__fin_Oinsert__not__elem,axiom,
% 0.18/0.45 ! [A2: set_a,X: a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ~ ( member_a @ X @ A2 )
% 0.18/0.45 => ( ( A2 != bot_bot_set_a )
% 0.18/0.45 => ( ( lattic247676691_fin_a @ inf @ ( insert_a @ X @ A2 ) )
% 0.18/0.45 = ( inf @ X @ ( lattic247676691_fin_a @ inf @ A2 ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Inf_fin.insert_not_elem
% 0.18/0.45 thf(fact_228_singletonI,axiom,
% 0.18/0.45 ! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% 0.18/0.45
% 0.18/0.45 % singletonI
% 0.18/0.45 thf(fact_229_singletonI,axiom,
% 0.18/0.45 ! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% 0.18/0.45
% 0.18/0.45 % singletonI
% 0.18/0.45 thf(fact_230_insert__subset,axiom,
% 0.18/0.45 ! [X: set_a,A2: set_set_a,B4: set_set_a] :
% 0.18/0.45 ( ( ord_le318720350_set_a @ ( insert_set_a @ X @ A2 ) @ B4 )
% 0.18/0.45 = ( ( member_set_a @ X @ B4 )
% 0.18/0.45 & ( ord_le318720350_set_a @ A2 @ B4 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % insert_subset
% 0.18/0.45 thf(fact_231_insert__subset,axiom,
% 0.18/0.45 ! [X: a,A2: set_a,B4: set_a] :
% 0.18/0.45 ( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B4 )
% 0.18/0.45 = ( ( member_a @ X @ B4 )
% 0.18/0.45 & ( ord_less_eq_set_a @ A2 @ B4 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % insert_subset
% 0.18/0.45 thf(fact_232_singleton__insert__inj__eq_H,axiom,
% 0.18/0.45 ! [A: a,A2: set_a,B: a] :
% 0.18/0.45 ( ( ( insert_a @ A @ A2 )
% 0.18/0.45 = ( insert_a @ B @ bot_bot_set_a ) )
% 0.18/0.45 = ( ( A = B )
% 0.18/0.45 & ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % singleton_insert_inj_eq'
% 0.18/0.45 thf(fact_233_singleton__insert__inj__eq,axiom,
% 0.18/0.45 ! [B: a,A: a,A2: set_a] :
% 0.18/0.45 ( ( ( insert_a @ B @ bot_bot_set_a )
% 0.18/0.45 = ( insert_a @ A @ A2 ) )
% 0.18/0.45 = ( ( A = B )
% 0.18/0.45 & ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % singleton_insert_inj_eq
% 0.18/0.45 thf(fact_234_local_Obdd__above__insert,axiom,
% 0.18/0.45 ! [A: a,A2: set_a] :
% 0.18/0.45 ( ( condit1627435690bove_a @ less_eq @ ( insert_a @ A @ A2 ) )
% 0.18/0.45 = ( condit1627435690bove_a @ less_eq @ A2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.bdd_above_insert
% 0.18/0.45 thf(fact_235_local_Obdd__below__insert,axiom,
% 0.18/0.45 ! [A: a,A2: set_a] :
% 0.18/0.45 ( ( condit1001553558elow_a @ less_eq @ ( insert_a @ A @ A2 ) )
% 0.18/0.45 = ( condit1001553558elow_a @ less_eq @ A2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.bdd_below_insert
% 0.18/0.45 thf(fact_236_local_OInf__fin_Osingleton,axiom,
% 0.18/0.45 ! [X: a] :
% 0.18/0.45 ( ( lattic247676691_fin_a @ inf @ ( insert_a @ X @ bot_bot_set_a ) )
% 0.18/0.45 = X ) ).
% 0.18/0.45
% 0.18/0.45 % local.Inf_fin.singleton
% 0.18/0.45 thf(fact_237_local_OSup__fin_Osingleton,axiom,
% 0.18/0.45 ! [X: a] :
% 0.18/0.45 ( ( lattic1781219155_fin_a @ sup @ ( insert_a @ X @ bot_bot_set_a ) )
% 0.18/0.45 = X ) ).
% 0.18/0.45
% 0.18/0.45 % local.Sup_fin.singleton
% 0.18/0.45 thf(fact_238_local_OInf__fin_Oinsert,axiom,
% 0.18/0.45 ! [A2: set_a,X: a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( A2 != bot_bot_set_a )
% 0.18/0.45 => ( ( lattic247676691_fin_a @ inf @ ( insert_a @ X @ A2 ) )
% 0.18/0.45 = ( inf @ X @ ( lattic247676691_fin_a @ inf @ A2 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Inf_fin.insert
% 0.18/0.45 thf(fact_239_local_OSup__fin_Oinsert,axiom,
% 0.18/0.45 ! [A2: set_a,X: a] :
% 0.18/0.45 ( ( finite_finite_a @ A2 )
% 0.18/0.45 => ( ( A2 != bot_bot_set_a )
% 0.18/0.45 => ( ( lattic1781219155_fin_a @ sup @ ( insert_a @ X @ A2 ) )
% 0.18/0.45 = ( sup @ X @ ( lattic1781219155_fin_a @ sup @ A2 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Sup_fin.insert
% 0.18/0.45 thf(fact_240_subset__singletonD,axiom,
% 0.18/0.45 ! [A2: set_a,X: a] :
% 0.18/0.45 ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
% 0.18/0.45 => ( ( A2 = bot_bot_set_a )
% 0.18/0.45 | ( A2
% 0.18/0.45 = ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subset_singletonD
% 0.18/0.45 thf(fact_241_subset__singleton__iff,axiom,
% 0.18/0.45 ! [X5: set_a,A: a] :
% 0.18/0.45 ( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
% 0.18/0.45 = ( ( X5 = bot_bot_set_a )
% 0.18/0.45 | ( X5
% 0.18/0.45 = ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subset_singleton_iff
% 0.18/0.45 thf(fact_242_singletonD,axiom,
% 0.18/0.45 ! [B: set_a,A: set_a] :
% 0.18/0.45 ( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
% 0.18/0.45 => ( B = A ) ) ).
% 0.18/0.45
% 0.18/0.45 % singletonD
% 0.18/0.45 thf(fact_243_singletonD,axiom,
% 0.18/0.45 ! [B: a,A: a] :
% 0.18/0.45 ( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
% 0.18/0.45 => ( B = A ) ) ).
% 0.18/0.45
% 0.18/0.45 % singletonD
% 0.18/0.45 thf(fact_244_singleton__iff,axiom,
% 0.18/0.45 ! [B: set_a,A: set_a] :
% 0.18/0.45 ( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
% 0.18/0.45 = ( B = A ) ) ).
% 0.18/0.45
% 0.18/0.45 % singleton_iff
% 0.18/0.45 thf(fact_245_singleton__iff,axiom,
% 0.18/0.45 ! [B: a,A: a] :
% 0.18/0.45 ( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
% 0.18/0.45 = ( B = A ) ) ).
% 0.18/0.45
% 0.18/0.45 % singleton_iff
% 0.18/0.45 thf(fact_246_doubleton__eq__iff,axiom,
% 0.18/0.45 ! [A: a,B: a,C: a,D: a] :
% 0.18/0.45 ( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
% 0.18/0.45 = ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
% 0.18/0.45 = ( ( ( A = C )
% 0.18/0.45 & ( B = D ) )
% 0.18/0.45 | ( ( A = D )
% 0.18/0.45 & ( B = C ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % doubleton_eq_iff
% 0.18/0.45 thf(fact_247_insert__not__empty,axiom,
% 0.18/0.45 ! [A: a,A2: set_a] :
% 0.18/0.45 ( ( insert_a @ A @ A2 )
% 0.18/0.45 != bot_bot_set_a ) ).
% 0.18/0.45
% 0.18/0.45 % insert_not_empty
% 0.18/0.45 thf(fact_248_singleton__inject,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( ( insert_a @ A @ bot_bot_set_a )
% 0.18/0.45 = ( insert_a @ B @ bot_bot_set_a ) )
% 0.18/0.45 => ( A = B ) ) ).
% 0.18/0.45
% 0.18/0.45 % singleton_inject
% 0.18/0.45 thf(fact_249_subset__insertI2,axiom,
% 0.18/0.45 ! [A2: set_a,B4: set_a,B: a] :
% 0.18/0.45 ( ( ord_less_eq_set_a @ A2 @ B4 )
% 0.18/0.45 => ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B4 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subset_insertI2
% 0.18/0.45 thf(fact_250_subset__insertI,axiom,
% 0.18/0.45 ! [B4: set_a,A: a] : ( ord_less_eq_set_a @ B4 @ ( insert_a @ A @ B4 ) ) ).
% 0.18/0.45
% 0.18/0.45 % subset_insertI
% 0.18/0.45 thf(fact_251_subset__insert,axiom,
% 0.18/0.45 ! [X: set_a,A2: set_set_a,B4: set_set_a] :
% 0.18/0.45 ( ~ ( member_set_a @ X @ A2 )
% 0.18/0.45 => ( ( ord_le318720350_set_a @ A2 @ ( insert_set_a @ X @ B4 ) )
% 0.18/0.45 = ( ord_le318720350_set_a @ A2 @ B4 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subset_insert
% 0.18/0.45 thf(fact_252_subset__insert,axiom,
% 0.18/0.45 ! [X: a,A2: set_a,B4: set_a] :
% 0.18/0.45 ( ~ ( member_a @ X @ A2 )
% 0.18/0.45 => ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B4 ) )
% 0.18/0.45 = ( ord_less_eq_set_a @ A2 @ B4 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subset_insert
% 0.18/0.45 thf(fact_253_insert__mono,axiom,
% 0.18/0.45 ! [C4: set_a,D2: set_a,A: a] :
% 0.18/0.45 ( ( ord_less_eq_set_a @ C4 @ D2 )
% 0.18/0.45 => ( ord_less_eq_set_a @ ( insert_a @ A @ C4 ) @ ( insert_a @ A @ D2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % insert_mono
% 0.18/0.45 thf(fact_254_local_OatLeastAtMost__singleton_H,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( A = B )
% 0.18/0.45 => ( ( set_atLeastAtMost_a @ less_eq @ A @ B )
% 0.18/0.45 = ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.atLeastAtMost_singleton'
% 0.18/0.45 thf(fact_255_finite__insert,axiom,
% 0.18/0.45 ! [A: set_a,A2: set_set_a] :
% 0.18/0.45 ( ( finite_finite_set_a @ ( insert_set_a @ A @ A2 ) )
% 0.18/0.45 = ( finite_finite_set_a @ A2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_insert
% 0.18/0.45 thf(fact_256_finite__insert,axiom,
% 0.18/0.45 ! [A: a,A2: set_a] :
% 0.18/0.45 ( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
% 0.18/0.45 = ( finite_finite_a @ A2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_insert
% 0.18/0.45 thf(fact_257_local_OIcc__eq__Icc,axiom,
% 0.18/0.45 ! [L: a,H: a,L2: a,H2: a] :
% 0.18/0.45 ( ( ( set_atLeastAtMost_a @ less_eq @ L @ H )
% 0.18/0.45 = ( set_atLeastAtMost_a @ less_eq @ L2 @ H2 ) )
% 0.18/0.45 = ( ( ( L = L2 )
% 0.18/0.45 & ( H = H2 ) )
% 0.18/0.45 | ( ~ ( less_eq @ L @ H )
% 0.18/0.45 & ~ ( less_eq @ L2 @ H2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Icc_eq_Icc
% 0.18/0.45 thf(fact_258_local_OatLeastAtMost__iff,axiom,
% 0.18/0.45 ! [I: a,L: a,U: a] :
% 0.18/0.45 ( ( member_a @ I @ ( set_atLeastAtMost_a @ less_eq @ L @ U ) )
% 0.18/0.45 = ( ( less_eq @ L @ I )
% 0.18/0.45 & ( less_eq @ I @ U ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.atLeastAtMost_iff
% 0.18/0.45 thf(fact_259_local_OatLeastatMost__empty__iff,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( ( set_atLeastAtMost_a @ less_eq @ A @ B )
% 0.18/0.45 = bot_bot_set_a )
% 0.18/0.45 = ( ~ ( less_eq @ A @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.atLeastatMost_empty_iff
% 0.18/0.45 thf(fact_260_local_OatLeastatMost__empty__iff2,axiom,
% 0.18/0.45 ! [A: a,B: a] :
% 0.18/0.45 ( ( bot_bot_set_a
% 0.18/0.45 = ( set_atLeastAtMost_a @ less_eq @ A @ B ) )
% 0.18/0.45 = ( ~ ( less_eq @ A @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.atLeastatMost_empty_iff2
% 0.18/0.45 thf(fact_261_local_Obdd__below__Icc,axiom,
% 0.18/0.45 ! [A: a,B: a] : ( condit1001553558elow_a @ less_eq @ ( set_atLeastAtMost_a @ less_eq @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.bdd_below_Icc
% 0.18/0.45 thf(fact_262_local_Obdd__above__Icc,axiom,
% 0.18/0.45 ! [A: a,B: a] : ( condit1627435690bove_a @ less_eq @ ( set_atLeastAtMost_a @ less_eq @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.bdd_above_Icc
% 0.18/0.45 thf(fact_263_local_OatLeastatMost__subset__iff,axiom,
% 0.18/0.45 ! [A: a,B: a,C: a,D: a] :
% 0.18/0.45 ( ( ord_less_eq_set_a @ ( set_atLeastAtMost_a @ less_eq @ A @ B ) @ ( set_atLeastAtMost_a @ less_eq @ C @ D ) )
% 0.18/0.45 = ( ~ ( less_eq @ A @ B )
% 0.18/0.45 | ( ( less_eq @ C @ A )
% 0.18/0.45 & ( less_eq @ B @ D ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.atLeastatMost_subset_iff
% 0.18/0.45 thf(fact_264_local_OatLeastAtMost__singleton,axiom,
% 0.18/0.45 ! [A: a] :
% 0.18/0.45 ( ( set_atLeastAtMost_a @ less_eq @ A @ A )
% 0.18/0.45 = ( insert_a @ A @ bot_bot_set_a ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.atLeastAtMost_singleton
% 0.18/0.45 thf(fact_265_local_OatLeastAtMost__singleton__iff,axiom,
% 0.18/0.45 ! [A: a,B: a,C: a] :
% 0.18/0.45 ( ( ( set_atLeastAtMost_a @ less_eq @ A @ B )
% 0.18/0.45 = ( insert_a @ C @ bot_bot_set_a ) )
% 0.18/0.45 = ( ( A = B )
% 0.18/0.45 & ( B = C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.atLeastAtMost_singleton_iff
% 0.18/0.45 thf(fact_266_local_OIcc__subset__Ici__iff,axiom,
% 0.18/0.45 ! [L: a,H: a,L2: a] :
% 0.18/0.45 ( ( ord_less_eq_set_a @ ( set_atLeastAtMost_a @ less_eq @ L @ H ) @ ( set_atLeast_a @ less_eq @ L2 ) )
% 0.18/0.45 = ( ~ ( less_eq @ L @ H )
% 0.18/0.45 | ( less_eq @ L2 @ L ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Icc_subset_Ici_iff
% 0.18/0.45 thf(fact_267_local_OIcc__subset__Iic__iff,axiom,
% 0.18/0.45 ! [L: a,H: a,H2: a] :
% 0.18/0.45 ( ( ord_less_eq_set_a @ ( set_atLeastAtMost_a @ less_eq @ L @ H ) @ ( set_atMost_a @ less_eq @ H2 ) )
% 0.18/0.45 = ( ~ ( less_eq @ L @ H )
% 0.18/0.45 | ( less_eq @ H @ H2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % local.Icc_subset_Iic_iff
% 0.18/0.45 thf(fact_268_comp__fun__idem_Ofun__left__idem,axiom,
% 0.18/0.45 ! [F: a > a > a,X: a,Z: a] :
% 0.18/0.45 ( ( finite40241356em_a_a @ F )
% 0.18/0.45 => ( ( F @ X @ ( F @ X @ Z ) )
% 0.18/0.45 = ( F @ X @ Z ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % comp_fun_idem.fun_left_idem
% 0.18/0.45 thf(fact_269_order__class_Ofinite__has__minimal2,axiom,
% 0.18/0.45 ! [A2: set_set_a,A: set_a] :
% 0.18/0.45 ( ( finite_finite_set_a @ A2 )
% 0.18/0.45 => ( ( member_set_a @ A @ A2 )
% 0.18/0.45 => ? [X2: set_a] :
% 0.18/0.45 ( ( member_set_a @ X2 @ A2 )
% 0.18/0.45 & ( ord_less_eq_set_a @ X2 @ A )
% 0.18/0.45 & ! [Xa: set_a] :
% 0.18/0.45 ( ( member_set_a @ Xa @ A2 )
% 0.18/0.45 => ( ( ord_less_eq_set_a @ Xa @ X2 )
% 0.18/0.45 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_class.finite_has_minimal2
% 0.18/0.45
% 0.18/0.45 % Conjectures (1)
% 0.18/0.45 thf(conj_0,conjecture,
% 0.18/0.45 ( ( modula1144073633_aux_a @ inf @ sup @ b @ c @ a2 )
% 0.18/0.45 = ( modula1144073633_aux_a @ inf @ sup @ a2 @ b @ c ) ) ).
% 0.18/0.45
% 0.18/0.45 %------------------------------------------------------------------------------
% 0.18/0.45 ------------------------
% 3.18/1.71 ---- Embedded in THF ---
% 3.18/1.71 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.18/1.71 %%% Generated on Tue Feb 24 08:27:23 EST 2026
% 3.18/1.71 %%% using '$$dhol' embedding, version 1.3.0.
% 3.18/1.71 %%% Logic specification used:
% 3.18/1.71 %%% thf(spec, logic, $$dhol).
% 3.18/1.71
% 3.18/1.71 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.yhjYV0Pxgo/DTF2DTF24647.p
% See solution above
% 3.18/1.71 ------------------------
% 3.18/1.75 ---- Cleaned THF ---
% 3.18/1.75 thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 3.18/1.75 set_set_a: $tType ).
% 3.18/1.75
% 3.18/1.75 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 3.18/1.75 set_a: $tType ).
% 3.18/1.75
% 3.18/1.75 thf(ty_n_tf__a,type,
% 3.18/1.75 a: $tType ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above_001tf__a,type,
% 3.18/1.75 condit1627435690bove_a: ( a > a > $o ) > set_a > $o ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below_001tf__a,type,
% 3.18/1.75 condit1001553558elow_a: ( a > a > $o ) > set_a > $o ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Finite__Set_Ocomp__fun__idem_001tf__a_001tf__a,type,
% 3.18/1.75 finite40241356em_a_a: ( a > a > a ) > $o ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
% 3.18/1.75 finite_finite_set_a: set_set_a > $o ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
% 3.18/1.75 finite_finite_a: set_a > $o ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Groups_Oabel__semigroup_001tf__a,type,
% 3.18/1.75 abel_semigroup_a: ( a > a > a ) > $o ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Groups_Osemigroup_001tf__a,type,
% 3.18/1.75 semigroup_a: ( a > a > a ) > $o ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Lattices_Osemilattice_001tf__a,type,
% 3.18/1.75 semilattice_a: ( a > a > a ) > $o ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Lattices__Big_Osemilattice__inf_OInf__fin_001tf__a,type,
% 3.18/1.75 lattic247676691_fin_a: ( a > a > a ) > set_a > a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Lattices__Big_Osemilattice__set_001tf__a,type,
% 3.18/1.75 lattic1885654924_set_a: ( a > a > a ) > $o ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Lattices__Big_Osemilattice__sup_OSup__fin_001tf__a,type,
% 3.18/1.75 lattic1781219155_fin_a: ( a > a > a ) > set_a > a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Od__aux_001tf__a,type,
% 3.18/1.75 modula1936294176_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Oe__aux_001tf__a,type,
% 3.18/1.75 modula1144073633_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 3.18/1.75 bot_bot_set_set_a: set_set_a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
% 3.18/1.75 bot_bot_set_a: set_a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Orderings_Oord_OLeast_001tf__a,type,
% 3.18/1.75 least_a: ( a > a > $o ) > ( a > $o ) > a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Orderings_Oord_Omax_001tf__a,type,
% 3.18/1.75 max_a: ( a > a > $o ) > a > a > a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Orderings_Oord_Omin_001tf__a,type,
% 3.18/1.75 min_a: ( a > a > $o ) > a > a > a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 3.18/1.75 ord_le318720350_set_a: set_set_a > set_set_a > $o ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 3.18/1.75 ord_less_eq_set_a: set_a > set_a > $o ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Orderings_Oorder_OGreatest_001tf__a,type,
% 3.18/1.75 greatest_a: ( a > a > $o ) > ( a > $o ) > a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Orderings_Oorder_Oantimono_001tf__a_001t__Set__Oset_Itf__a_J,type,
% 3.18/1.75 antimono_a_set_a: ( a > a > $o ) > ( a > set_a ) > $o ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Orderings_Oorder_Omono_001tf__a_001t__Set__Oset_Itf__a_J,type,
% 3.18/1.75 mono_a_set_a: ( a > a > $o ) > ( a > set_a ) > $o ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Relation_Otransp_001tf__a,type,
% 3.18/1.75 transp_a: ( a > a > $o ) > $o ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
% 3.18/1.75 collect_set_a: ( set_a > $o ) > set_set_a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Set_OCollect_001tf__a,type,
% 3.18/1.75 collect_a: ( a > $o ) > set_a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
% 3.18/1.75 insert_set_a: set_a > set_set_a > set_set_a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Set_Oinsert_001tf__a,type,
% 3.18/1.75 insert_a: a > set_a > set_a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Set__Interval_Oord_OatLeastAtMost_001tf__a,type,
% 3.18/1.75 set_atLeastAtMost_a: ( a > a > $o ) > a > a > set_a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Set__Interval_Oord_OatLeast_001tf__a,type,
% 3.18/1.75 set_atLeast_a: ( a > a > $o ) > a > set_a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_Set__Interval_Oord_OatMost_001tf__a,type,
% 3.18/1.75 set_atMost_a: ( a > a > $o ) > a > set_a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
% 3.18/1.75 member_set_a: set_a > set_set_a > $o ).
% 3.18/1.75
% 3.18/1.75 thf(sy_c_member_001tf__a,type,
% 3.18/1.75 member_a: a > set_a > $o ).
% 3.18/1.75
% 3.18/1.75 thf(sy_v_a,type,
% 3.18/1.75 a2: a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_v_b,type,
% 3.18/1.75 b: a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_v_c,type,
% 3.18/1.75 c: a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_v_inf,type,
% 3.18/1.75 inf: a > a > a ).
% 3.18/1.75
% 3.18/1.75 thf(sy_v_less__eq,type,
% 3.18/1.75 less_eq: a > a > $o ).
% 3.18/1.75
% 3.18/1.75 thf(sy_v_sup,type,
% 3.18/1.75 sup: a > a > a ).
% 3.18/1.75
% 3.18/1.75 thf(fact_0_local_Oinf_Oassoc,axiom,
% 3.18/1.75 ! [A: a,B: a,C: a] :
% 3.18/1.75 ( ( inf @ ( inf @ A @ B ) @ C )
% 3.18/1.75 = ( inf @ A @ ( inf @ B @ C ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_1_local_Oinf_Ocommute,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( inf @ A @ B )
% 3.18/1.75 = ( inf @ B @ A ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_2_local_Oinf_Oleft__commute,axiom,
% 3.18/1.75 ! [B: a,A: a,C: a] :
% 3.18/1.75 ( ( inf @ B @ ( inf @ A @ C ) )
% 3.18/1.75 = ( inf @ A @ ( inf @ B @ C ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_3_local_Oinf__assoc,axiom,
% 3.18/1.75 ! [X: a,Y: a,Z: a] :
% 3.18/1.75 ( ( inf @ ( inf @ X @ Y ) @ Z )
% 3.18/1.75 = ( inf @ X @ ( inf @ Y @ Z ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_4_local_Oinf__commute,axiom,
% 3.18/1.75 ! [X: a,Y: a] :
% 3.18/1.75 ( ( inf @ X @ Y )
% 3.18/1.75 = ( inf @ Y @ X ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_5_local_Oinf__left__commute,axiom,
% 3.18/1.75 ! [X: a,Y: a,Z: a] :
% 3.18/1.75 ( ( inf @ X @ ( inf @ Y @ Z ) )
% 3.18/1.75 = ( inf @ Y @ ( inf @ X @ Z ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_6_local_Osup_Oassoc,axiom,
% 3.18/1.75 ! [A: a,B: a,C: a] :
% 3.18/1.75 ( ( sup @ ( sup @ A @ B ) @ C )
% 3.18/1.75 = ( sup @ A @ ( sup @ B @ C ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_7_local_Osup_Ocommute,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( sup @ A @ B )
% 3.18/1.75 = ( sup @ B @ A ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_8_local_Osup_Oleft__commute,axiom,
% 3.18/1.75 ! [B: a,A: a,C: a] :
% 3.18/1.75 ( ( sup @ B @ ( sup @ A @ C ) )
% 3.18/1.75 = ( sup @ A @ ( sup @ B @ C ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_9_local_Osup__assoc,axiom,
% 3.18/1.75 ! [X: a,Y: a,Z: a] :
% 3.18/1.75 ( ( sup @ ( sup @ X @ Y ) @ Z )
% 3.18/1.75 = ( sup @ X @ ( sup @ Y @ Z ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_10_local_Osup__commute,axiom,
% 3.18/1.75 ! [X: a,Y: a] :
% 3.18/1.75 ( ( sup @ X @ Y )
% 3.18/1.75 = ( sup @ Y @ X ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_11_local_Osup__left__commute,axiom,
% 3.18/1.75 ! [X: a,Y: a,Z: a] :
% 3.18/1.75 ( ( sup @ X @ ( sup @ Y @ Z ) )
% 3.18/1.75 = ( sup @ Y @ ( sup @ X @ Z ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_12_local_Odistrib__imp1,axiom,
% 3.18/1.75 ! [X: a,Y: a,Z: a] :
% 3.18/1.75 ( ! [X2: a,Y2: a,Z2: a] :
% 3.18/1.75 ( ( inf @ X2 @ ( sup @ Y2 @ Z2 ) )
% 3.18/1.75 = ( sup @ ( inf @ X2 @ Y2 ) @ ( inf @ X2 @ Z2 ) ) )
% 3.18/1.75 => ( ( sup @ X @ ( inf @ Y @ Z ) )
% 3.18/1.75 = ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_13_local_Odistrib__imp2,axiom,
% 3.18/1.75 ! [X: a,Y: a,Z: a] :
% 3.18/1.75 ( ! [X2: a,Y2: a,Z2: a] :
% 3.18/1.75 ( ( sup @ X2 @ ( inf @ Y2 @ Z2 ) )
% 3.18/1.75 = ( inf @ ( sup @ X2 @ Y2 ) @ ( sup @ X2 @ Z2 ) ) )
% 3.18/1.75 => ( ( inf @ X @ ( sup @ Y @ Z ) )
% 3.18/1.75 = ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_14_e__aux__def,axiom,
% 3.18/1.75 ! [A: a,B: a,C: a] :
% 3.18/1.75 ( ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C )
% 3.18/1.75 = ( inf @ ( inf @ ( sup @ A @ B ) @ ( sup @ B @ C ) ) @ ( sup @ C @ A ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_15_local_Oinf_Oidem,axiom,
% 3.18/1.75 ! [A: a] :
% 3.18/1.75 ( ( inf @ A @ A )
% 3.18/1.75 = A ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_16_local_Oinf_Oleft__idem,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( inf @ A @ ( inf @ A @ B ) )
% 3.18/1.75 = ( inf @ A @ B ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_17_local_Oinf_Oright__idem,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( inf @ ( inf @ A @ B ) @ B )
% 3.18/1.75 = ( inf @ A @ B ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_18_local_Oinf__idem,axiom,
% 3.18/1.75 ! [X: a] :
% 3.18/1.75 ( ( inf @ X @ X )
% 3.18/1.75 = X ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_19_local_Oinf__left__idem,axiom,
% 3.18/1.75 ! [X: a,Y: a] :
% 3.18/1.75 ( ( inf @ X @ ( inf @ X @ Y ) )
% 3.18/1.75 = ( inf @ X @ Y ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_20_local_Oinf__right__idem,axiom,
% 3.18/1.75 ! [X: a,Y: a] :
% 3.18/1.75 ( ( inf @ ( inf @ X @ Y ) @ Y )
% 3.18/1.75 = ( inf @ X @ Y ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_21_local_Osup_Oidem,axiom,
% 3.18/1.75 ! [A: a] :
% 3.18/1.75 ( ( sup @ A @ A )
% 3.18/1.75 = A ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_22_local_Osup_Oleft__idem,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( sup @ A @ ( sup @ A @ B ) )
% 3.18/1.75 = ( sup @ A @ B ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_23_local_Osup_Oright__idem,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( sup @ ( sup @ A @ B ) @ B )
% 3.18/1.75 = ( sup @ A @ B ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_24_local_Osup__idem,axiom,
% 3.18/1.75 ! [X: a] :
% 3.18/1.75 ( ( sup @ X @ X )
% 3.18/1.75 = X ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_25_local_Osup__left__idem,axiom,
% 3.18/1.75 ! [X: a,Y: a] :
% 3.18/1.75 ( ( sup @ X @ ( sup @ X @ Y ) )
% 3.18/1.75 = ( sup @ X @ Y ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_26_local_Oinf__sup__absorb,axiom,
% 3.18/1.75 ! [X: a,Y: a] :
% 3.18/1.75 ( ( inf @ X @ ( sup @ X @ Y ) )
% 3.18/1.75 = X ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_27_local_Osup__inf__absorb,axiom,
% 3.18/1.75 ! [X: a,Y: a] :
% 3.18/1.75 ( ( sup @ X @ ( inf @ X @ Y ) )
% 3.18/1.75 = X ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_28_d__aux__def,axiom,
% 3.18/1.75 ! [A: a,B: a,C: a] :
% 3.18/1.75 ( ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C )
% 3.18/1.75 = ( sup @ ( sup @ ( inf @ A @ B ) @ ( inf @ B @ C ) ) @ ( inf @ C @ A ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_29_d__b__c__a,axiom,
% 3.18/1.75 ! [B: a,C: a,A: a] :
% 3.18/1.75 ( ( modula1936294176_aux_a @ inf @ sup @ B @ C @ A )
% 3.18/1.75 = ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_30_d__c__a__b,axiom,
% 3.18/1.75 ! [C: a,A: a,B: a] :
% 3.18/1.75 ( ( modula1936294176_aux_a @ inf @ sup @ C @ A @ B )
% 3.18/1.75 = ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_31_lattice_Oe__aux_Ocong,axiom,
% 3.18/1.75 modula1144073633_aux_a = modula1144073633_aux_a ).
% 3.18/1.75
% 3.18/1.75 thf(fact_32_local_Ocomp__fun__idem__sup,axiom,
% 3.18/1.75 finite40241356em_a_a @ sup ).
% 3.18/1.75
% 3.18/1.75 thf(fact_33_local_Ocomp__fun__idem__inf,axiom,
% 3.18/1.75 finite40241356em_a_a @ inf ).
% 3.18/1.75
% 3.18/1.75 thf(fact_34_local_Osup_Osemigroup__axioms,axiom,
% 3.18/1.75 semigroup_a @ sup ).
% 3.18/1.75
% 3.18/1.75 thf(fact_35_local_Oinf_Osemigroup__axioms,axiom,
% 3.18/1.75 semigroup_a @ inf ).
% 3.18/1.75
% 3.18/1.75 thf(fact_36_local_Osup_Osemilattice__axioms,axiom,
% 3.18/1.75 semilattice_a @ sup ).
% 3.18/1.75
% 3.18/1.75 thf(fact_37_local_Oinf_Osemilattice__axioms,axiom,
% 3.18/1.75 semilattice_a @ inf ).
% 3.18/1.75
% 3.18/1.75 thf(fact_38_local_Osup_Oabel__semigroup__axioms,axiom,
% 3.18/1.75 abel_semigroup_a @ sup ).
% 3.18/1.75
% 3.18/1.75 thf(fact_39_local_Oinf_Oabel__semigroup__axioms,axiom,
% 3.18/1.75 abel_semigroup_a @ inf ).
% 3.18/1.75
% 3.18/1.75 thf(fact_40_local_OSup__fin_Osemilattice__set__axioms,axiom,
% 3.18/1.75 lattic1885654924_set_a @ sup ).
% 3.18/1.75
% 3.18/1.75 thf(fact_41_local_OInf__fin_Osemilattice__set__axioms,axiom,
% 3.18/1.75 lattic1885654924_set_a @ inf ).
% 3.18/1.75
% 3.18/1.75 thf(fact_42_local_Odistrib__inf__le,axiom,
% 3.18/1.75 ! [X: a,Y: a,Z: a] : ( less_eq @ ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z ) ) @ ( inf @ X @ ( sup @ Y @ Z ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_43_local_Odistrib__sup__le,axiom,
% 3.18/1.75 ! [X: a,Y: a,Z: a] : ( less_eq @ ( sup @ X @ ( inf @ Y @ Z ) ) @ ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_44_local_Oantisym,axiom,
% 3.18/1.75 ! [X: a,Y: a] :
% 3.18/1.75 ( ( less_eq @ X @ Y )
% 3.18/1.75 => ( ( less_eq @ Y @ X )
% 3.18/1.75 => ( X = Y ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_45_mem__Collect__eq,axiom,
% 3.18/1.75 ! [A: set_a,P: set_a > $o] :
% 3.18/1.75 ( ( member_set_a @ A @ ( collect_set_a @ P ) )
% 3.18/1.75 = ( P @ A ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_46_mem__Collect__eq,axiom,
% 3.18/1.75 ! [A: a,P: a > $o] :
% 3.18/1.75 ( ( member_a @ A @ ( collect_a @ P ) )
% 3.18/1.75 = ( P @ A ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_47_Collect__mem__eq,axiom,
% 3.18/1.75 ! [A2: set_set_a] :
% 3.18/1.75 ( ( collect_set_a
% 3.18/1.75 @ ^ [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
% 3.18/1.75 = A2 ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_48_Collect__mem__eq,axiom,
% 3.18/1.75 ! [A2: set_a] :
% 3.18/1.75 ( ( collect_a
% 3.18/1.75 @ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
% 3.18/1.75 = A2 ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_49_Collect__cong,axiom,
% 3.18/1.75 ! [P: a > $o,Q: a > $o] :
% 3.18/1.75 ( ! [X2: a] :
% 3.18/1.75 ( ( P @ X2 )
% 3.18/1.75 = ( Q @ X2 ) )
% 3.18/1.75 => ( ( collect_a @ P )
% 3.18/1.75 = ( collect_a @ Q ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_50_local_Oantisym__conv,axiom,
% 3.18/1.75 ! [Y: a,X: a] :
% 3.18/1.75 ( ( less_eq @ Y @ X )
% 3.18/1.75 => ( ( less_eq @ X @ Y )
% 3.18/1.75 = ( X = Y ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_51_local_Odual__order_Oantisym,axiom,
% 3.18/1.75 ! [B: a,A: a] :
% 3.18/1.75 ( ( less_eq @ B @ A )
% 3.18/1.75 => ( ( less_eq @ A @ B )
% 3.18/1.75 => ( A = B ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_52_local_Odual__order_Oeq__iff,axiom,
% 3.18/1.75 ( ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 ) )
% 3.18/1.75 = ( ^ [A3: a,B2: a] :
% 3.18/1.75 ( ( less_eq @ B2 @ A3 )
% 3.18/1.75 & ( less_eq @ A3 @ B2 ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_53_local_Odual__order_Otrans,axiom,
% 3.18/1.75 ! [B: a,A: a,C: a] :
% 3.18/1.75 ( ( less_eq @ B @ A )
% 3.18/1.75 => ( ( less_eq @ C @ B )
% 3.18/1.75 => ( less_eq @ C @ A ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_54_local_Oeq__iff,axiom,
% 3.18/1.75 ( ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 ) )
% 3.18/1.75 = ( ^ [X3: a,Y4: a] :
% 3.18/1.75 ( ( less_eq @ X3 @ Y4 )
% 3.18/1.75 & ( less_eq @ Y4 @ X3 ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_55_local_Oeq__refl,axiom,
% 3.18/1.75 ! [X: a,Y: a] :
% 3.18/1.75 ( ( X = Y )
% 3.18/1.75 => ( less_eq @ X @ Y ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_56_local_Oord__eq__le__trans,axiom,
% 3.18/1.75 ! [A: a,B: a,C: a] :
% 3.18/1.75 ( ( A = B )
% 3.18/1.75 => ( ( less_eq @ B @ C )
% 3.18/1.75 => ( less_eq @ A @ C ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_57_local_Oord__le__eq__trans,axiom,
% 3.18/1.75 ! [A: a,B: a,C: a] :
% 3.18/1.75 ( ( less_eq @ A @ B )
% 3.18/1.75 => ( ( B = C )
% 3.18/1.75 => ( less_eq @ A @ C ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_58_local_Oorder_Oantisym,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( less_eq @ A @ B )
% 3.18/1.75 => ( ( less_eq @ B @ A )
% 3.18/1.75 => ( A = B ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_59_local_Oorder_Oeq__iff,axiom,
% 3.18/1.75 ( ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 ) )
% 3.18/1.75 = ( ^ [A3: a,B2: a] :
% 3.18/1.75 ( ( less_eq @ A3 @ B2 )
% 3.18/1.75 & ( less_eq @ B2 @ A3 ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_60_local_Oorder_Otrans,axiom,
% 3.18/1.75 ! [A: a,B: a,C: a] :
% 3.18/1.75 ( ( less_eq @ A @ B )
% 3.18/1.75 => ( ( less_eq @ B @ C )
% 3.18/1.75 => ( less_eq @ A @ C ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_61_local_Oorder__trans,axiom,
% 3.18/1.75 ! [X: a,Y: a,Z: a] :
% 3.18/1.75 ( ( less_eq @ X @ Y )
% 3.18/1.75 => ( ( less_eq @ Y @ Z )
% 3.18/1.75 => ( less_eq @ X @ Z ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_62_local_Ole__infI2,axiom,
% 3.18/1.75 ! [B: a,X: a,A: a] :
% 3.18/1.75 ( ( less_eq @ B @ X )
% 3.18/1.75 => ( less_eq @ ( inf @ A @ B ) @ X ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_63_local_Ole__infI1,axiom,
% 3.18/1.75 ! [A: a,X: a,B: a] :
% 3.18/1.75 ( ( less_eq @ A @ X )
% 3.18/1.75 => ( less_eq @ ( inf @ A @ B ) @ X ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_64_local_Ole__infI,axiom,
% 3.18/1.75 ! [X: a,A: a,B: a] :
% 3.18/1.75 ( ( less_eq @ X @ A )
% 3.18/1.75 => ( ( less_eq @ X @ B )
% 3.18/1.75 => ( less_eq @ X @ ( inf @ A @ B ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_65_local_Ole__infE,axiom,
% 3.18/1.75 ! [X: a,A: a,B: a] :
% 3.18/1.75 ( ( less_eq @ X @ ( inf @ A @ B ) )
% 3.18/1.75 => ~ ( ( less_eq @ X @ A )
% 3.18/1.75 => ~ ( less_eq @ X @ B ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_66_local_Ole__iff__inf,axiom,
% 3.18/1.75 ! [X: a,Y: a] :
% 3.18/1.75 ( ( less_eq @ X @ Y )
% 3.18/1.75 = ( ( inf @ X @ Y )
% 3.18/1.75 = X ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_67_local_Oinf__unique,axiom,
% 3.18/1.75 ! [F: a > a > a,X: a,Y: a] :
% 3.18/1.75 ( ! [X2: a,Y2: a] : ( less_eq @ ( F @ X2 @ Y2 ) @ X2 )
% 3.18/1.75 => ( ! [X2: a,Y2: a] : ( less_eq @ ( F @ X2 @ Y2 ) @ Y2 )
% 3.18/1.75 => ( ! [X2: a,Y2: a,Z2: a] :
% 3.18/1.75 ( ( less_eq @ X2 @ Y2 )
% 3.18/1.75 => ( ( less_eq @ X2 @ Z2 )
% 3.18/1.75 => ( less_eq @ X2 @ ( F @ Y2 @ Z2 ) ) ) )
% 3.18/1.75 => ( ( inf @ X @ Y )
% 3.18/1.75 = ( F @ X @ Y ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_68_local_Oinf__mono,axiom,
% 3.18/1.75 ! [A: a,C: a,B: a,D: a] :
% 3.18/1.75 ( ( less_eq @ A @ C )
% 3.18/1.75 => ( ( less_eq @ B @ D )
% 3.18/1.75 => ( less_eq @ ( inf @ A @ B ) @ ( inf @ C @ D ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_69_local_Oinf__le2,axiom,
% 3.18/1.75 ! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ Y ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_70_local_Oinf__le1,axiom,
% 3.18/1.75 ! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ X ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_71_local_Oinf__greatest,axiom,
% 3.18/1.75 ! [X: a,Y: a,Z: a] :
% 3.18/1.75 ( ( less_eq @ X @ Y )
% 3.18/1.75 => ( ( less_eq @ X @ Z )
% 3.18/1.75 => ( less_eq @ X @ ( inf @ Y @ Z ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_72_local_Oinf__absorb2,axiom,
% 3.18/1.75 ! [Y: a,X: a] :
% 3.18/1.75 ( ( less_eq @ Y @ X )
% 3.18/1.75 => ( ( inf @ X @ Y )
% 3.18/1.75 = Y ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_73_local_Oinf__absorb1,axiom,
% 3.18/1.75 ! [X: a,Y: a] :
% 3.18/1.75 ( ( less_eq @ X @ Y )
% 3.18/1.75 => ( ( inf @ X @ Y )
% 3.18/1.75 = X ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_74_local_Oinf_Oorder__iff,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( less_eq @ A @ B )
% 3.18/1.75 = ( A
% 3.18/1.75 = ( inf @ A @ B ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_75_local_Oinf_OorderI,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( A
% 3.18/1.75 = ( inf @ A @ B ) )
% 3.18/1.75 => ( less_eq @ A @ B ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_76_local_Oinf_OorderE,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( less_eq @ A @ B )
% 3.18/1.75 => ( A
% 3.18/1.75 = ( inf @ A @ B ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_77_local_Oinf_OcoboundedI2,axiom,
% 3.18/1.75 ! [B: a,C: a,A: a] :
% 3.18/1.75 ( ( less_eq @ B @ C )
% 3.18/1.75 => ( less_eq @ ( inf @ A @ B ) @ C ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_78_local_Oinf_OcoboundedI1,axiom,
% 3.18/1.75 ! [A: a,C: a,B: a] :
% 3.18/1.75 ( ( less_eq @ A @ C )
% 3.18/1.75 => ( less_eq @ ( inf @ A @ B ) @ C ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_79_local_Oinf_Ocobounded2,axiom,
% 3.18/1.75 ! [A: a,B: a] : ( less_eq @ ( inf @ A @ B ) @ B ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_80_local_Oinf_Ocobounded1,axiom,
% 3.18/1.75 ! [A: a,B: a] : ( less_eq @ ( inf @ A @ B ) @ A ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_81_local_Oinf_OboundedI,axiom,
% 3.18/1.75 ! [A: a,B: a,C: a] :
% 3.18/1.75 ( ( less_eq @ A @ B )
% 3.18/1.75 => ( ( less_eq @ A @ C )
% 3.18/1.75 => ( less_eq @ A @ ( inf @ B @ C ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_82_local_Oinf_OboundedE,axiom,
% 3.18/1.75 ! [A: a,B: a,C: a] :
% 3.18/1.75 ( ( less_eq @ A @ ( inf @ B @ C ) )
% 3.18/1.75 => ~ ( ( less_eq @ A @ B )
% 3.18/1.75 => ~ ( less_eq @ A @ C ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_83_local_Oinf_Oabsorb__iff2,axiom,
% 3.18/1.75 ! [B: a,A: a] :
% 3.18/1.75 ( ( less_eq @ B @ A )
% 3.18/1.75 = ( ( inf @ A @ B )
% 3.18/1.75 = B ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_84_local_Oinf_Oabsorb__iff1,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( less_eq @ A @ B )
% 3.18/1.75 = ( ( inf @ A @ B )
% 3.18/1.75 = A ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_85_local_Oinf_Oabsorb2,axiom,
% 3.18/1.75 ! [B: a,A: a] :
% 3.18/1.75 ( ( less_eq @ B @ A )
% 3.18/1.75 => ( ( inf @ A @ B )
% 3.18/1.75 = B ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_86_local_Oinf_Oabsorb1,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( less_eq @ A @ B )
% 3.18/1.75 => ( ( inf @ A @ B )
% 3.18/1.75 = A ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_87_local_Osup__unique,axiom,
% 3.18/1.75 ! [F: a > a > a,X: a,Y: a] :
% 3.18/1.75 ( ! [X2: a,Y2: a] : ( less_eq @ X2 @ ( F @ X2 @ Y2 ) )
% 3.18/1.75 => ( ! [X2: a,Y2: a] : ( less_eq @ Y2 @ ( F @ X2 @ Y2 ) )
% 3.18/1.75 => ( ! [X2: a,Y2: a,Z2: a] :
% 3.18/1.75 ( ( less_eq @ Y2 @ X2 )
% 3.18/1.75 => ( ( less_eq @ Z2 @ X2 )
% 3.18/1.75 => ( less_eq @ ( F @ Y2 @ Z2 ) @ X2 ) ) )
% 3.18/1.75 => ( ( sup @ X @ Y )
% 3.18/1.75 = ( F @ X @ Y ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_88_local_Osup__mono,axiom,
% 3.18/1.75 ! [A: a,C: a,B: a,D: a] :
% 3.18/1.75 ( ( less_eq @ A @ C )
% 3.18/1.75 => ( ( less_eq @ B @ D )
% 3.18/1.75 => ( less_eq @ ( sup @ A @ B ) @ ( sup @ C @ D ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_89_local_Osup__least,axiom,
% 3.18/1.75 ! [Y: a,X: a,Z: a] :
% 3.18/1.75 ( ( less_eq @ Y @ X )
% 3.18/1.75 => ( ( less_eq @ Z @ X )
% 3.18/1.75 => ( less_eq @ ( sup @ Y @ Z ) @ X ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_90_local_Osup__ge2,axiom,
% 3.18/1.75 ! [Y: a,X: a] : ( less_eq @ Y @ ( sup @ X @ Y ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_91_local_Osup__ge1,axiom,
% 3.18/1.75 ! [X: a,Y: a] : ( less_eq @ X @ ( sup @ X @ Y ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_92_local_Osup__absorb2,axiom,
% 3.18/1.75 ! [X: a,Y: a] :
% 3.18/1.75 ( ( less_eq @ X @ Y )
% 3.18/1.75 => ( ( sup @ X @ Y )
% 3.18/1.75 = Y ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_93_local_Osup__absorb1,axiom,
% 3.18/1.75 ! [Y: a,X: a] :
% 3.18/1.75 ( ( less_eq @ Y @ X )
% 3.18/1.75 => ( ( sup @ X @ Y )
% 3.18/1.75 = X ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_94_local_Osup_Oorder__iff,axiom,
% 3.18/1.75 ! [B: a,A: a] :
% 3.18/1.75 ( ( less_eq @ B @ A )
% 3.18/1.75 = ( A
% 3.18/1.75 = ( sup @ A @ B ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_95_local_Osup_OorderI,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( A
% 3.18/1.75 = ( sup @ A @ B ) )
% 3.18/1.75 => ( less_eq @ B @ A ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_96_local_Osup_OorderE,axiom,
% 3.18/1.75 ! [B: a,A: a] :
% 3.18/1.75 ( ( less_eq @ B @ A )
% 3.18/1.75 => ( A
% 3.18/1.75 = ( sup @ A @ B ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_97_local_Osup_Omono,axiom,
% 3.18/1.75 ! [C: a,A: a,D: a,B: a] :
% 3.18/1.75 ( ( less_eq @ C @ A )
% 3.18/1.75 => ( ( less_eq @ D @ B )
% 3.18/1.75 => ( less_eq @ ( sup @ C @ D ) @ ( sup @ A @ B ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_98_local_Osup_OcoboundedI2,axiom,
% 3.18/1.75 ! [C: a,B: a,A: a] :
% 3.18/1.75 ( ( less_eq @ C @ B )
% 3.18/1.75 => ( less_eq @ C @ ( sup @ A @ B ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_99_local_Osup_OcoboundedI1,axiom,
% 3.18/1.75 ! [C: a,A: a,B: a] :
% 3.18/1.75 ( ( less_eq @ C @ A )
% 3.18/1.75 => ( less_eq @ C @ ( sup @ A @ B ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_100_local_Osup_Ocobounded2,axiom,
% 3.18/1.75 ! [B: a,A: a] : ( less_eq @ B @ ( sup @ A @ B ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_101_local_Osup_Ocobounded1,axiom,
% 3.18/1.75 ! [A: a,B: a] : ( less_eq @ A @ ( sup @ A @ B ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_102_local_Osup_OboundedI,axiom,
% 3.18/1.75 ! [B: a,A: a,C: a] :
% 3.18/1.75 ( ( less_eq @ B @ A )
% 3.18/1.75 => ( ( less_eq @ C @ A )
% 3.18/1.75 => ( less_eq @ ( sup @ B @ C ) @ A ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_103_local_Osup_OboundedE,axiom,
% 3.18/1.75 ! [B: a,C: a,A: a] :
% 3.18/1.75 ( ( less_eq @ ( sup @ B @ C ) @ A )
% 3.18/1.75 => ~ ( ( less_eq @ B @ A )
% 3.18/1.75 => ~ ( less_eq @ C @ A ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_104_local_Osup_Oabsorb__iff2,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( less_eq @ A @ B )
% 3.18/1.75 = ( ( sup @ A @ B )
% 3.18/1.75 = B ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_105_local_Osup_Oabsorb__iff1,axiom,
% 3.18/1.75 ! [B: a,A: a] :
% 3.18/1.75 ( ( less_eq @ B @ A )
% 3.18/1.75 = ( ( sup @ A @ B )
% 3.18/1.75 = A ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_106_local_Osup_Oabsorb2,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( less_eq @ A @ B )
% 3.18/1.75 => ( ( sup @ A @ B )
% 3.18/1.75 = B ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_107_local_Osup_Oabsorb1,axiom,
% 3.18/1.75 ! [B: a,A: a] :
% 3.18/1.75 ( ( less_eq @ B @ A )
% 3.18/1.75 => ( ( sup @ A @ B )
% 3.18/1.75 = A ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_108_local_Ole__supI2,axiom,
% 3.18/1.75 ! [X: a,B: a,A: a] :
% 3.18/1.75 ( ( less_eq @ X @ B )
% 3.18/1.75 => ( less_eq @ X @ ( sup @ A @ B ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_109_local_Ole__supI1,axiom,
% 3.18/1.75 ! [X: a,A: a,B: a] :
% 3.18/1.75 ( ( less_eq @ X @ A )
% 3.18/1.75 => ( less_eq @ X @ ( sup @ A @ B ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_110_local_Ole__supI,axiom,
% 3.18/1.75 ! [A: a,X: a,B: a] :
% 3.18/1.75 ( ( less_eq @ A @ X )
% 3.18/1.75 => ( ( less_eq @ B @ X )
% 3.18/1.75 => ( less_eq @ ( sup @ A @ B ) @ X ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_111_local_Ole__supE,axiom,
% 3.18/1.75 ! [A: a,B: a,X: a] :
% 3.18/1.75 ( ( less_eq @ ( sup @ A @ B ) @ X )
% 3.18/1.75 => ~ ( ( less_eq @ A @ X )
% 3.18/1.75 => ~ ( less_eq @ B @ X ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_112_local_Ole__iff__sup,axiom,
% 3.18/1.75 ! [X: a,Y: a] :
% 3.18/1.75 ( ( less_eq @ X @ Y )
% 3.18/1.75 = ( ( sup @ X @ Y )
% 3.18/1.75 = Y ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_113_local_Oorder_Orefl,axiom,
% 3.18/1.75 ! [A: a] : ( less_eq @ A @ A ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_114_local_Oorder__refl,axiom,
% 3.18/1.75 ! [X: a] : ( less_eq @ X @ X ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_115_local_OGreatestI2__order,axiom,
% 3.18/1.75 ! [P: a > $o,X: a,Q: a > $o] :
% 3.18/1.75 ( ( P @ X )
% 3.18/1.75 => ( ! [Y2: a] :
% 3.18/1.75 ( ( P @ Y2 )
% 3.18/1.75 => ( less_eq @ Y2 @ X ) )
% 3.18/1.75 => ( ! [X2: a] :
% 3.18/1.75 ( ( P @ X2 )
% 3.18/1.75 => ( ! [Y5: a] :
% 3.18/1.75 ( ( P @ Y5 )
% 3.18/1.75 => ( less_eq @ Y5 @ X2 ) )
% 3.18/1.75 => ( Q @ X2 ) ) )
% 3.18/1.75 => ( Q @ ( greatest_a @ less_eq @ P ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_116_local_OGreatest__equality,axiom,
% 3.18/1.75 ! [P: a > $o,X: a] :
% 3.18/1.75 ( ( P @ X )
% 3.18/1.75 => ( ! [Y2: a] :
% 3.18/1.75 ( ( P @ Y2 )
% 3.18/1.75 => ( less_eq @ Y2 @ X ) )
% 3.18/1.75 => ( ( greatest_a @ less_eq @ P )
% 3.18/1.75 = X ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_117_local_Omax__def,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( ( less_eq @ A @ B )
% 3.18/1.75 => ( ( max_a @ less_eq @ A @ B )
% 3.18/1.75 = B ) )
% 3.18/1.75 & ( ~ ( less_eq @ A @ B )
% 3.18/1.75 => ( ( max_a @ less_eq @ A @ B )
% 3.18/1.75 = A ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_118_local_Omin__def,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( ( less_eq @ A @ B )
% 3.18/1.75 => ( ( min_a @ less_eq @ A @ B )
% 3.18/1.75 = A ) )
% 3.18/1.75 & ( ~ ( less_eq @ A @ B )
% 3.18/1.75 => ( ( min_a @ less_eq @ A @ B )
% 3.18/1.75 = B ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_119_local_Ole__inf__iff,axiom,
% 3.18/1.75 ! [X: a,Y: a,Z: a] :
% 3.18/1.75 ( ( less_eq @ X @ ( inf @ Y @ Z ) )
% 3.18/1.75 = ( ( less_eq @ X @ Y )
% 3.18/1.75 & ( less_eq @ X @ Z ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_120_local_Oinf_Obounded__iff,axiom,
% 3.18/1.75 ! [A: a,B: a,C: a] :
% 3.18/1.75 ( ( less_eq @ A @ ( inf @ B @ C ) )
% 3.18/1.75 = ( ( less_eq @ A @ B )
% 3.18/1.75 & ( less_eq @ A @ C ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_121_local_Osup_Obounded__iff,axiom,
% 3.18/1.75 ! [B: a,C: a,A: a] :
% 3.18/1.75 ( ( less_eq @ ( sup @ B @ C ) @ A )
% 3.18/1.75 = ( ( less_eq @ B @ A )
% 3.18/1.75 & ( less_eq @ C @ A ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_122_local_Ole__sup__iff,axiom,
% 3.18/1.75 ! [X: a,Y: a,Z: a] :
% 3.18/1.75 ( ( less_eq @ ( sup @ X @ Y ) @ Z )
% 3.18/1.75 = ( ( less_eq @ X @ Z )
% 3.18/1.75 & ( less_eq @ Y @ Z ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_123_local_OLeast1I,axiom,
% 3.18/1.75 ! [P: a > $o] :
% 3.18/1.75 ( ? [X4: a] :
% 3.18/1.75 ( ( P @ X4 )
% 3.18/1.75 & ! [Y2: a] :
% 3.18/1.75 ( ( P @ Y2 )
% 3.18/1.75 => ( less_eq @ X4 @ Y2 ) )
% 3.18/1.75 & ! [Y2: a] :
% 3.18/1.75 ( ( ( P @ Y2 )
% 3.18/1.75 & ! [Ya: a] :
% 3.18/1.75 ( ( P @ Ya )
% 3.18/1.75 => ( less_eq @ Y2 @ Ya ) ) )
% 3.18/1.75 => ( Y2 = X4 ) ) )
% 3.18/1.75 => ( P @ ( least_a @ less_eq @ P ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_124_local_OLeast1__le,axiom,
% 3.18/1.75 ! [P: a > $o,Z: a] :
% 3.18/1.75 ( ? [X4: a] :
% 3.18/1.75 ( ( P @ X4 )
% 3.18/1.75 & ! [Y2: a] :
% 3.18/1.75 ( ( P @ Y2 )
% 3.18/1.75 => ( less_eq @ X4 @ Y2 ) )
% 3.18/1.75 & ! [Y2: a] :
% 3.18/1.75 ( ( ( P @ Y2 )
% 3.18/1.75 & ! [Ya: a] :
% 3.18/1.75 ( ( P @ Ya )
% 3.18/1.75 => ( less_eq @ Y2 @ Ya ) ) )
% 3.18/1.75 => ( Y2 = X4 ) ) )
% 3.18/1.75 => ( ( P @ Z )
% 3.18/1.75 => ( less_eq @ ( least_a @ less_eq @ P ) @ Z ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_125_local_OLeastI2__order,axiom,
% 3.18/1.75 ! [P: a > $o,X: a,Q: a > $o] :
% 3.18/1.75 ( ( P @ X )
% 3.18/1.75 => ( ! [Y2: a] :
% 3.18/1.75 ( ( P @ Y2 )
% 3.18/1.75 => ( less_eq @ X @ Y2 ) )
% 3.18/1.75 => ( ! [X2: a] :
% 3.18/1.75 ( ( P @ X2 )
% 3.18/1.75 => ( ! [Y5: a] :
% 3.18/1.75 ( ( P @ Y5 )
% 3.18/1.75 => ( less_eq @ X2 @ Y5 ) )
% 3.18/1.75 => ( Q @ X2 ) ) )
% 3.18/1.75 => ( Q @ ( least_a @ less_eq @ P ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_126_local_OLeast__equality,axiom,
% 3.18/1.75 ! [P: a > $o,X: a] :
% 3.18/1.75 ( ( P @ X )
% 3.18/1.75 => ( ! [Y2: a] :
% 3.18/1.75 ( ( P @ Y2 )
% 3.18/1.75 => ( less_eq @ X @ Y2 ) )
% 3.18/1.75 => ( ( least_a @ less_eq @ P )
% 3.18/1.75 = X ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_127_lattice_Od__aux_Ocong,axiom,
% 3.18/1.75 modula1936294176_aux_a = modula1936294176_aux_a ).
% 3.18/1.75
% 3.18/1.75 thf(fact_128_abel__semigroup_Oaxioms_I1_J,axiom,
% 3.18/1.75 ! [F: a > a > a] :
% 3.18/1.75 ( ( abel_semigroup_a @ F )
% 3.18/1.75 => ( semigroup_a @ F ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_129_semilattice__set__def,axiom,
% 3.18/1.75 lattic1885654924_set_a = semilattice_a ).
% 3.18/1.75
% 3.18/1.75 thf(fact_130_semilattice__set_Ointro,axiom,
% 3.18/1.75 ! [F: a > a > a] :
% 3.18/1.75 ( ( semilattice_a @ F )
% 3.18/1.75 => ( lattic1885654924_set_a @ F ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_131_semilattice__set_Oaxioms,axiom,
% 3.18/1.75 ! [F: a > a > a] :
% 3.18/1.75 ( ( lattic1885654924_set_a @ F )
% 3.18/1.75 => ( semilattice_a @ F ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_132_semilattice_Oaxioms_I1_J,axiom,
% 3.18/1.75 ! [F: a > a > a] :
% 3.18/1.75 ( ( semilattice_a @ F )
% 3.18/1.75 => ( abel_semigroup_a @ F ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_133_local_Obdd__above__def,axiom,
% 3.18/1.75 ! [A2: set_a] :
% 3.18/1.75 ( ( condit1627435690bove_a @ less_eq @ A2 )
% 3.18/1.75 = ( ? [M: a] :
% 3.18/1.75 ! [X3: a] :
% 3.18/1.75 ( ( member_a @ X3 @ A2 )
% 3.18/1.75 => ( less_eq @ X3 @ M ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_134_local_Obdd__below__def,axiom,
% 3.18/1.75 ! [A2: set_a] :
% 3.18/1.75 ( ( condit1001553558elow_a @ less_eq @ A2 )
% 3.18/1.75 = ( ? [M2: a] :
% 3.18/1.75 ! [X3: a] :
% 3.18/1.75 ( ( member_a @ X3 @ A2 )
% 3.18/1.75 => ( less_eq @ M2 @ X3 ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_135_local_Otransp__le,axiom,
% 3.18/1.75 transp_a @ less_eq ).
% 3.18/1.75
% 3.18/1.75 thf(fact_136_local_Obdd__belowI,axiom,
% 3.18/1.75 ! [A2: set_a,M3: a] :
% 3.18/1.75 ( ! [X2: a] :
% 3.18/1.75 ( ( member_a @ X2 @ A2 )
% 3.18/1.75 => ( less_eq @ M3 @ X2 ) )
% 3.18/1.75 => ( condit1001553558elow_a @ less_eq @ A2 ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_137_local_Obdd__aboveI,axiom,
% 3.18/1.75 ! [A2: set_a,M4: a] :
% 3.18/1.75 ( ! [X2: a] :
% 3.18/1.75 ( ( member_a @ X2 @ A2 )
% 3.18/1.75 => ( less_eq @ X2 @ M4 ) )
% 3.18/1.75 => ( condit1627435690bove_a @ less_eq @ A2 ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_138_abel__semigroup_Oleft__commute,axiom,
% 3.18/1.75 ! [F: a > a > a,B: a,A: a,C: a] :
% 3.18/1.75 ( ( abel_semigroup_a @ F )
% 3.18/1.75 => ( ( F @ B @ ( F @ A @ C ) )
% 3.18/1.75 = ( F @ A @ ( F @ B @ C ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_139_semilattice_Oright__idem,axiom,
% 3.18/1.75 ! [F: a > a > a,A: a,B: a] :
% 3.18/1.75 ( ( semilattice_a @ F )
% 3.18/1.75 => ( ( F @ ( F @ A @ B ) @ B )
% 3.18/1.75 = ( F @ A @ B ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_140_semilattice_Oleft__idem,axiom,
% 3.18/1.75 ! [F: a > a > a,A: a,B: a] :
% 3.18/1.75 ( ( semilattice_a @ F )
% 3.18/1.75 => ( ( F @ A @ ( F @ A @ B ) )
% 3.18/1.75 = ( F @ A @ B ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_141_abel__semigroup_Ocommute,axiom,
% 3.18/1.75 ! [F: a > a > a,A: a,B: a] :
% 3.18/1.75 ( ( abel_semigroup_a @ F )
% 3.18/1.75 => ( ( F @ A @ B )
% 3.18/1.75 = ( F @ B @ A ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_142_semilattice_Oidem,axiom,
% 3.18/1.75 ! [F: a > a > a,A: a] :
% 3.18/1.75 ( ( semilattice_a @ F )
% 3.18/1.75 => ( ( F @ A @ A )
% 3.18/1.75 = A ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_143_semigroup_Ointro,axiom,
% 3.18/1.75 ! [F: a > a > a] :
% 3.18/1.75 ( ! [A4: a,B3: a,C2: a] :
% 3.18/1.75 ( ( F @ ( F @ A4 @ B3 ) @ C2 )
% 3.18/1.75 = ( F @ A4 @ ( F @ B3 @ C2 ) ) )
% 3.18/1.75 => ( semigroup_a @ F ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_144_semigroup_Oassoc,axiom,
% 3.18/1.75 ! [F: a > a > a,A: a,B: a,C: a] :
% 3.18/1.75 ( ( semigroup_a @ F )
% 3.18/1.75 => ( ( F @ ( F @ A @ B ) @ C )
% 3.18/1.75 = ( F @ A @ ( F @ B @ C ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_145_semigroup__def,axiom,
% 3.18/1.75 ( semigroup_a
% 3.18/1.75 = ( ^ [F2: a > a > a] :
% 3.18/1.75 ! [A3: a,B2: a,C3: a] :
% 3.18/1.75 ( ( F2 @ ( F2 @ A3 @ B2 ) @ C3 )
% 3.18/1.75 = ( F2 @ A3 @ ( F2 @ B2 @ C3 ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_146_local_Obdd__below__mono,axiom,
% 3.18/1.75 ! [B4: set_a,A2: set_a] :
% 3.18/1.75 ( ( condit1001553558elow_a @ less_eq @ B4 )
% 3.18/1.75 => ( ( ord_less_eq_set_a @ A2 @ B4 )
% 3.18/1.75 => ( condit1001553558elow_a @ less_eq @ A2 ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_147_local_Obdd__above__mono,axiom,
% 3.18/1.75 ! [B4: set_a,A2: set_a] :
% 3.18/1.75 ( ( condit1627435690bove_a @ less_eq @ B4 )
% 3.18/1.75 => ( ( ord_less_eq_set_a @ A2 @ B4 )
% 3.18/1.75 => ( condit1627435690bove_a @ less_eq @ A2 ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_148_local_Obdd__below__finite,axiom,
% 3.18/1.75 ! [A2: set_a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( condit1001553558elow_a @ less_eq @ A2 ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_149_local_Obdd__above__finite,axiom,
% 3.18/1.75 ! [A2: set_a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( condit1627435690bove_a @ less_eq @ A2 ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_150_local_Ofinite__has__maximal2,axiom,
% 3.18/1.75 ! [A2: set_a,A: a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( member_a @ A @ A2 )
% 3.18/1.75 => ? [X2: a] :
% 3.18/1.75 ( ( member_a @ X2 @ A2 )
% 3.18/1.75 & ( less_eq @ A @ X2 )
% 3.18/1.75 & ! [Xa: a] :
% 3.18/1.75 ( ( member_a @ Xa @ A2 )
% 3.18/1.75 => ( ( less_eq @ X2 @ Xa )
% 3.18/1.75 => ( X2 = Xa ) ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_151_local_Ofinite__has__minimal2,axiom,
% 3.18/1.75 ! [A2: set_a,A: a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( member_a @ A @ A2 )
% 3.18/1.75 => ? [X2: a] :
% 3.18/1.75 ( ( member_a @ X2 @ A2 )
% 3.18/1.75 & ( less_eq @ X2 @ A )
% 3.18/1.75 & ! [Xa: a] :
% 3.18/1.75 ( ( member_a @ Xa @ A2 )
% 3.18/1.75 => ( ( less_eq @ Xa @ X2 )
% 3.18/1.75 => ( X2 = Xa ) ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_152_local_OantimonoD,axiom,
% 3.18/1.75 ! [F: a > set_a,X: a,Y: a] :
% 3.18/1.75 ( ( antimono_a_set_a @ less_eq @ F )
% 3.18/1.75 => ( ( less_eq @ X @ Y )
% 3.18/1.75 => ( ord_less_eq_set_a @ ( F @ Y ) @ ( F @ X ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_153_local_OantimonoE,axiom,
% 3.18/1.75 ! [F: a > set_a,X: a,Y: a] :
% 3.18/1.75 ( ( antimono_a_set_a @ less_eq @ F )
% 3.18/1.75 => ( ( less_eq @ X @ Y )
% 3.18/1.75 => ( ord_less_eq_set_a @ ( F @ Y ) @ ( F @ X ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_154_local_OantimonoI,axiom,
% 3.18/1.75 ! [F: a > set_a] :
% 3.18/1.75 ( ! [X2: a,Y2: a] :
% 3.18/1.75 ( ( less_eq @ X2 @ Y2 )
% 3.18/1.75 => ( ord_less_eq_set_a @ ( F @ Y2 ) @ ( F @ X2 ) ) )
% 3.18/1.75 => ( antimono_a_set_a @ less_eq @ F ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_155_local_Oantimono__def,axiom,
% 3.18/1.75 ! [F: a > set_a] :
% 3.18/1.75 ( ( antimono_a_set_a @ less_eq @ F )
% 3.18/1.75 = ( ! [X3: a,Y4: a] :
% 3.18/1.75 ( ( less_eq @ X3 @ Y4 )
% 3.18/1.75 => ( ord_less_eq_set_a @ ( F @ Y4 ) @ ( F @ X3 ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_156_local_OmonoD,axiom,
% 3.18/1.75 ! [F: a > set_a,X: a,Y: a] :
% 3.18/1.75 ( ( mono_a_set_a @ less_eq @ F )
% 3.18/1.75 => ( ( less_eq @ X @ Y )
% 3.18/1.75 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_157_local_OmonoE,axiom,
% 3.18/1.75 ! [F: a > set_a,X: a,Y: a] :
% 3.18/1.75 ( ( mono_a_set_a @ less_eq @ F )
% 3.18/1.75 => ( ( less_eq @ X @ Y )
% 3.18/1.75 => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_158_local_OmonoI,axiom,
% 3.18/1.75 ! [F: a > set_a] :
% 3.18/1.75 ( ! [X2: a,Y2: a] :
% 3.18/1.75 ( ( less_eq @ X2 @ Y2 )
% 3.18/1.75 => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 3.18/1.75 => ( mono_a_set_a @ less_eq @ F ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_159_local_Omono__def,axiom,
% 3.18/1.75 ! [F: a > set_a] :
% 3.18/1.75 ( ( mono_a_set_a @ less_eq @ F )
% 3.18/1.75 = ( ! [X3: a,Y4: a] :
% 3.18/1.75 ( ( less_eq @ X3 @ Y4 )
% 3.18/1.75 => ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_160_local_OSup__fin_OcoboundedI,axiom,
% 3.18/1.75 ! [A2: set_a,A: a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( member_a @ A @ A2 )
% 3.18/1.75 => ( less_eq @ A @ ( lattic1781219155_fin_a @ sup @ A2 ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_161_local_OInf__fin_OcoboundedI,axiom,
% 3.18/1.75 ! [A2: set_a,A: a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( member_a @ A @ A2 )
% 3.18/1.75 => ( less_eq @ ( lattic247676691_fin_a @ inf @ A2 ) @ A ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_162_local_OSup__fin_Oin__idem,axiom,
% 3.18/1.75 ! [A2: set_a,X: a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( member_a @ X @ A2 )
% 3.18/1.75 => ( ( sup @ X @ ( lattic1781219155_fin_a @ sup @ A2 ) )
% 3.18/1.75 = ( lattic1781219155_fin_a @ sup @ A2 ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_163_local_OInf__fin_Oin__idem,axiom,
% 3.18/1.75 ! [A2: set_a,X: a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( member_a @ X @ A2 )
% 3.18/1.75 => ( ( inf @ X @ ( lattic247676691_fin_a @ inf @ A2 ) )
% 3.18/1.75 = ( lattic247676691_fin_a @ inf @ A2 ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_164_local_Oinf__Sup__absorb,axiom,
% 3.18/1.75 ! [A2: set_a,A: a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( member_a @ A @ A2 )
% 3.18/1.75 => ( ( inf @ A @ ( lattic1781219155_fin_a @ sup @ A2 ) )
% 3.18/1.75 = A ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_165_local_Osup__Inf__absorb,axiom,
% 3.18/1.75 ! [A2: set_a,A: a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( member_a @ A @ A2 )
% 3.18/1.75 => ( ( sup @ ( lattic247676691_fin_a @ inf @ A2 ) @ A )
% 3.18/1.75 = A ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_166_semilattice__inf_OInf__fin_Ocong,axiom,
% 3.18/1.75 lattic247676691_fin_a = lattic247676691_fin_a ).
% 3.18/1.75
% 3.18/1.75 thf(fact_167_semilattice__sup_OSup__fin_Ocong,axiom,
% 3.18/1.75 lattic1781219155_fin_a = lattic1781219155_fin_a ).
% 3.18/1.75
% 3.18/1.75 thf(fact_168_local_OInf__fin__le__Sup__fin,axiom,
% 3.18/1.75 ! [A2: set_a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( A2 != bot_bot_set_a )
% 3.18/1.75 => ( less_eq @ ( lattic247676691_fin_a @ inf @ A2 ) @ ( lattic1781219155_fin_a @ sup @ A2 ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_169_local_OSup__fin_Osubset__imp,axiom,
% 3.18/1.75 ! [A2: set_a,B4: set_a] :
% 3.18/1.75 ( ( ord_less_eq_set_a @ A2 @ B4 )
% 3.18/1.75 => ( ( A2 != bot_bot_set_a )
% 3.18/1.75 => ( ( finite_finite_a @ B4 )
% 3.18/1.75 => ( less_eq @ ( lattic1781219155_fin_a @ sup @ A2 ) @ ( lattic1781219155_fin_a @ sup @ B4 ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_170_local_OInf__fin_Osubset__imp,axiom,
% 3.18/1.75 ! [A2: set_a,B4: set_a] :
% 3.18/1.75 ( ( ord_less_eq_set_a @ A2 @ B4 )
% 3.18/1.75 => ( ( A2 != bot_bot_set_a )
% 3.18/1.75 => ( ( finite_finite_a @ B4 )
% 3.18/1.75 => ( less_eq @ ( lattic247676691_fin_a @ inf @ B4 ) @ ( lattic247676691_fin_a @ inf @ A2 ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_171_local_Ofinite__has__minimal,axiom,
% 3.18/1.75 ! [A2: set_a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( A2 != bot_bot_set_a )
% 3.18/1.75 => ? [X2: a] :
% 3.18/1.75 ( ( member_a @ X2 @ A2 )
% 3.18/1.75 & ! [Xa: a] :
% 3.18/1.75 ( ( member_a @ Xa @ A2 )
% 3.18/1.75 => ( ( less_eq @ Xa @ X2 )
% 3.18/1.75 => ( X2 = Xa ) ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_172_local_Ofinite__has__maximal,axiom,
% 3.18/1.75 ! [A2: set_a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( A2 != bot_bot_set_a )
% 3.18/1.75 => ? [X2: a] :
% 3.18/1.75 ( ( member_a @ X2 @ A2 )
% 3.18/1.75 & ! [Xa: a] :
% 3.18/1.75 ( ( member_a @ Xa @ A2 )
% 3.18/1.75 => ( ( less_eq @ X2 @ Xa )
% 3.18/1.75 => ( X2 = Xa ) ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_173_local_OInf__fin_Osubset,axiom,
% 3.18/1.75 ! [A2: set_a,B4: set_a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( B4 != bot_bot_set_a )
% 3.18/1.75 => ( ( ord_less_eq_set_a @ B4 @ A2 )
% 3.18/1.75 => ( ( inf @ ( lattic247676691_fin_a @ inf @ B4 ) @ ( lattic247676691_fin_a @ inf @ A2 ) )
% 3.18/1.75 = ( lattic247676691_fin_a @ inf @ A2 ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_174_local_OSup__fin_Osubset,axiom,
% 3.18/1.75 ! [A2: set_a,B4: set_a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( B4 != bot_bot_set_a )
% 3.18/1.75 => ( ( ord_less_eq_set_a @ B4 @ A2 )
% 3.18/1.75 => ( ( sup @ ( lattic1781219155_fin_a @ sup @ B4 ) @ ( lattic1781219155_fin_a @ sup @ A2 ) )
% 3.18/1.75 = ( lattic1781219155_fin_a @ sup @ A2 ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_175_local_OInf__fin_Obounded__iff,axiom,
% 3.18/1.75 ! [A2: set_a,X: a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( A2 != bot_bot_set_a )
% 3.18/1.75 => ( ( less_eq @ X @ ( lattic247676691_fin_a @ inf @ A2 ) )
% 3.18/1.75 = ( ! [X3: a] :
% 3.18/1.75 ( ( member_a @ X3 @ A2 )
% 3.18/1.75 => ( less_eq @ X @ X3 ) ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_176_local_OInf__fin_OboundedI,axiom,
% 3.18/1.75 ! [A2: set_a,X: a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( A2 != bot_bot_set_a )
% 3.18/1.75 => ( ! [A4: a] :
% 3.18/1.75 ( ( member_a @ A4 @ A2 )
% 3.18/1.75 => ( less_eq @ X @ A4 ) )
% 3.18/1.75 => ( less_eq @ X @ ( lattic247676691_fin_a @ inf @ A2 ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_177_local_OInf__fin_OboundedE,axiom,
% 3.18/1.75 ! [A2: set_a,X: a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( A2 != bot_bot_set_a )
% 3.18/1.75 => ( ( less_eq @ X @ ( lattic247676691_fin_a @ inf @ A2 ) )
% 3.18/1.75 => ! [A5: a] :
% 3.18/1.75 ( ( member_a @ A5 @ A2 )
% 3.18/1.75 => ( less_eq @ X @ A5 ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_178_local_OSup__fin_Obounded__iff,axiom,
% 3.18/1.75 ! [A2: set_a,X: a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( A2 != bot_bot_set_a )
% 3.18/1.75 => ( ( less_eq @ ( lattic1781219155_fin_a @ sup @ A2 ) @ X )
% 3.18/1.75 = ( ! [X3: a] :
% 3.18/1.75 ( ( member_a @ X3 @ A2 )
% 3.18/1.75 => ( less_eq @ X3 @ X ) ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_179_local_OSup__fin_OboundedI,axiom,
% 3.18/1.75 ! [A2: set_a,X: a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( A2 != bot_bot_set_a )
% 3.18/1.75 => ( ! [A4: a] :
% 3.18/1.75 ( ( member_a @ A4 @ A2 )
% 3.18/1.75 => ( less_eq @ A4 @ X ) )
% 3.18/1.75 => ( less_eq @ ( lattic1781219155_fin_a @ sup @ A2 ) @ X ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_180_local_OSup__fin_OboundedE,axiom,
% 3.18/1.75 ! [A2: set_a,X: a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( A2 != bot_bot_set_a )
% 3.18/1.75 => ( ( less_eq @ ( lattic1781219155_fin_a @ sup @ A2 ) @ X )
% 3.18/1.75 => ! [A5: a] :
% 3.18/1.75 ( ( member_a @ A5 @ A2 )
% 3.18/1.75 => ( less_eq @ A5 @ X ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_181_local_Obdd__above__empty,axiom,
% 3.18/1.75 condit1627435690bove_a @ less_eq @ bot_bot_set_a ).
% 3.18/1.75
% 3.18/1.75 thf(fact_182_local_Obdd__below__empty,axiom,
% 3.18/1.75 condit1001553558elow_a @ less_eq @ bot_bot_set_a ).
% 3.18/1.75
% 3.18/1.75 thf(fact_183_local_Onot__empty__eq__Iic__eq__empty,axiom,
% 3.18/1.75 ! [H: a] :
% 3.18/1.75 ( bot_bot_set_a
% 3.18/1.75 != ( set_atMost_a @ less_eq @ H ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_184_local_Onot__empty__eq__Ici__eq__empty,axiom,
% 3.18/1.75 ! [L: a] :
% 3.18/1.75 ( bot_bot_set_a
% 3.18/1.75 != ( set_atLeast_a @ less_eq @ L ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_185_subset__empty,axiom,
% 3.18/1.75 ! [A2: set_a] :
% 3.18/1.75 ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
% 3.18/1.75 = ( A2 = bot_bot_set_a ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_186_empty__Collect__eq,axiom,
% 3.18/1.75 ! [P: a > $o] :
% 3.18/1.75 ( ( bot_bot_set_a
% 3.18/1.75 = ( collect_a @ P ) )
% 3.18/1.75 = ( ! [X3: a] :
% 3.18/1.75 ~ ( P @ X3 ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_187_Collect__empty__eq,axiom,
% 3.18/1.75 ! [P: a > $o] :
% 3.18/1.75 ( ( ( collect_a @ P )
% 3.18/1.75 = bot_bot_set_a )
% 3.18/1.75 = ( ! [X3: a] :
% 3.18/1.75 ~ ( P @ X3 ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_188_all__not__in__conv,axiom,
% 3.18/1.75 ! [A2: set_set_a] :
% 3.18/1.75 ( ( ! [X3: set_a] :
% 3.18/1.75 ~ ( member_set_a @ X3 @ A2 ) )
% 3.18/1.75 = ( A2 = bot_bot_set_set_a ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_189_all__not__in__conv,axiom,
% 3.18/1.75 ! [A2: set_a] :
% 3.18/1.75 ( ( ! [X3: a] :
% 3.18/1.75 ~ ( member_a @ X3 @ A2 ) )
% 3.18/1.75 = ( A2 = bot_bot_set_a ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_190_empty__iff,axiom,
% 3.18/1.75 ! [C: set_a] :
% 3.18/1.75 ~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_191_empty__iff,axiom,
% 3.18/1.75 ! [C: a] :
% 3.18/1.75 ~ ( member_a @ C @ bot_bot_set_a ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_192_subset__antisym,axiom,
% 3.18/1.75 ! [A2: set_a,B4: set_a] :
% 3.18/1.75 ( ( ord_less_eq_set_a @ A2 @ B4 )
% 3.18/1.75 => ( ( ord_less_eq_set_a @ B4 @ A2 )
% 3.18/1.75 => ( A2 = B4 ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_193_subsetI,axiom,
% 3.18/1.75 ! [A2: set_set_a,B4: set_set_a] :
% 3.18/1.75 ( ! [X2: set_a] :
% 3.18/1.75 ( ( member_set_a @ X2 @ A2 )
% 3.18/1.75 => ( member_set_a @ X2 @ B4 ) )
% 3.18/1.75 => ( ord_le318720350_set_a @ A2 @ B4 ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_194_subsetI,axiom,
% 3.18/1.75 ! [A2: set_a,B4: set_a] :
% 3.18/1.75 ( ! [X2: a] :
% 3.18/1.75 ( ( member_a @ X2 @ A2 )
% 3.18/1.75 => ( member_a @ X2 @ B4 ) )
% 3.18/1.75 => ( ord_less_eq_set_a @ A2 @ B4 ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_195_empty__subsetI,axiom,
% 3.18/1.75 ! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_196_local_OatLeast__iff,axiom,
% 3.18/1.75 ! [I: a,K: a] :
% 3.18/1.75 ( ( member_a @ I @ ( set_atLeast_a @ less_eq @ K ) )
% 3.18/1.75 = ( less_eq @ K @ I ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_197_local_OatMost__iff,axiom,
% 3.18/1.75 ! [I: a,K: a] :
% 3.18/1.75 ( ( member_a @ I @ ( set_atMost_a @ less_eq @ K ) )
% 3.18/1.75 = ( less_eq @ I @ K ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_198_local_Obdd__below__Ici,axiom,
% 3.18/1.75 ! [A: a] : ( condit1001553558elow_a @ less_eq @ ( set_atLeast_a @ less_eq @ A ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_199_local_Obdd__above__Iic,axiom,
% 3.18/1.75 ! [B: a] : ( condit1627435690bove_a @ less_eq @ ( set_atMost_a @ less_eq @ B ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_200_ex__in__conv,axiom,
% 3.18/1.75 ! [A2: set_set_a] :
% 3.18/1.75 ( ( ? [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
% 3.18/1.75 = ( A2 != bot_bot_set_set_a ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_201_ex__in__conv,axiom,
% 3.18/1.75 ! [A2: set_a] :
% 3.18/1.75 ( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
% 3.18/1.75 = ( A2 != bot_bot_set_a ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_202_equals0I,axiom,
% 3.18/1.75 ! [A2: set_set_a] :
% 3.18/1.75 ( ! [Y2: set_a] :
% 3.18/1.75 ~ ( member_set_a @ Y2 @ A2 )
% 3.18/1.75 => ( A2 = bot_bot_set_set_a ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_203_equals0I,axiom,
% 3.18/1.75 ! [A2: set_a] :
% 3.18/1.75 ( ! [Y2: a] :
% 3.18/1.75 ~ ( member_a @ Y2 @ A2 )
% 3.18/1.75 => ( A2 = bot_bot_set_a ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_204_equals0D,axiom,
% 3.18/1.75 ! [A2: set_set_a,A: set_a] :
% 3.18/1.75 ( ( A2 = bot_bot_set_set_a )
% 3.18/1.75 => ~ ( member_set_a @ A @ A2 ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_205_equals0D,axiom,
% 3.18/1.75 ! [A2: set_a,A: a] :
% 3.18/1.75 ( ( A2 = bot_bot_set_a )
% 3.18/1.75 => ~ ( member_a @ A @ A2 ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_206_emptyE,axiom,
% 3.18/1.75 ! [A: set_a] :
% 3.18/1.75 ~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_207_emptyE,axiom,
% 3.18/1.75 ! [A: a] :
% 3.18/1.75 ~ ( member_a @ A @ bot_bot_set_a ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_208_Collect__mono__iff,axiom,
% 3.18/1.75 ! [P: a > $o,Q: a > $o] :
% 3.18/1.75 ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
% 3.18/1.75 = ( ! [X3: a] :
% 3.18/1.75 ( ( P @ X3 )
% 3.18/1.75 => ( Q @ X3 ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_209_set__eq__subset,axiom,
% 3.18/1.75 ( ( ^ [Y3: set_a,Z3: set_a] : ( Y3 = Z3 ) )
% 3.18/1.75 = ( ^ [A6: set_a,B5: set_a] :
% 3.18/1.75 ( ( ord_less_eq_set_a @ A6 @ B5 )
% 3.18/1.75 & ( ord_less_eq_set_a @ B5 @ A6 ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_210_subset__trans,axiom,
% 3.18/1.75 ! [A2: set_a,B4: set_a,C4: set_a] :
% 3.18/1.75 ( ( ord_less_eq_set_a @ A2 @ B4 )
% 3.18/1.75 => ( ( ord_less_eq_set_a @ B4 @ C4 )
% 3.18/1.75 => ( ord_less_eq_set_a @ A2 @ C4 ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_211_Collect__mono,axiom,
% 3.18/1.75 ! [P: a > $o,Q: a > $o] :
% 3.18/1.75 ( ! [X2: a] :
% 3.18/1.75 ( ( P @ X2 )
% 3.18/1.75 => ( Q @ X2 ) )
% 3.18/1.75 => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_212_subset__refl,axiom,
% 3.18/1.75 ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_213_subset__iff,axiom,
% 3.18/1.75 ( ord_le318720350_set_a
% 3.18/1.75 = ( ^ [A6: set_set_a,B5: set_set_a] :
% 3.18/1.75 ! [T: set_a] :
% 3.18/1.75 ( ( member_set_a @ T @ A6 )
% 3.18/1.75 => ( member_set_a @ T @ B5 ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_214_subset__iff,axiom,
% 3.18/1.75 ( ord_less_eq_set_a
% 3.18/1.75 = ( ^ [A6: set_a,B5: set_a] :
% 3.18/1.75 ! [T: a] :
% 3.18/1.75 ( ( member_a @ T @ A6 )
% 3.18/1.75 => ( member_a @ T @ B5 ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_215_equalityD2,axiom,
% 3.18/1.75 ! [A2: set_a,B4: set_a] :
% 3.18/1.75 ( ( A2 = B4 )
% 3.18/1.75 => ( ord_less_eq_set_a @ B4 @ A2 ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_216_equalityD1,axiom,
% 3.18/1.75 ! [A2: set_a,B4: set_a] :
% 3.18/1.75 ( ( A2 = B4 )
% 3.18/1.75 => ( ord_less_eq_set_a @ A2 @ B4 ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_217_subset__eq,axiom,
% 3.18/1.75 ( ord_le318720350_set_a
% 3.18/1.75 = ( ^ [A6: set_set_a,B5: set_set_a] :
% 3.18/1.75 ! [X3: set_a] :
% 3.18/1.75 ( ( member_set_a @ X3 @ A6 )
% 3.18/1.75 => ( member_set_a @ X3 @ B5 ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_218_subset__eq,axiom,
% 3.18/1.75 ( ord_less_eq_set_a
% 3.18/1.75 = ( ^ [A6: set_a,B5: set_a] :
% 3.18/1.75 ! [X3: a] :
% 3.18/1.75 ( ( member_a @ X3 @ A6 )
% 3.18/1.75 => ( member_a @ X3 @ B5 ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_219_equalityE,axiom,
% 3.18/1.75 ! [A2: set_a,B4: set_a] :
% 3.18/1.75 ( ( A2 = B4 )
% 3.18/1.75 => ~ ( ( ord_less_eq_set_a @ A2 @ B4 )
% 3.18/1.75 => ~ ( ord_less_eq_set_a @ B4 @ A2 ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_220_subsetD,axiom,
% 3.18/1.75 ! [A2: set_set_a,B4: set_set_a,C: set_a] :
% 3.18/1.75 ( ( ord_le318720350_set_a @ A2 @ B4 )
% 3.18/1.75 => ( ( member_set_a @ C @ A2 )
% 3.18/1.75 => ( member_set_a @ C @ B4 ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_221_subsetD,axiom,
% 3.18/1.75 ! [A2: set_a,B4: set_a,C: a] :
% 3.18/1.75 ( ( ord_less_eq_set_a @ A2 @ B4 )
% 3.18/1.75 => ( ( member_a @ C @ A2 )
% 3.18/1.75 => ( member_a @ C @ B4 ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_222_in__mono,axiom,
% 3.18/1.75 ! [A2: set_set_a,B4: set_set_a,X: set_a] :
% 3.18/1.75 ( ( ord_le318720350_set_a @ A2 @ B4 )
% 3.18/1.75 => ( ( member_set_a @ X @ A2 )
% 3.18/1.75 => ( member_set_a @ X @ B4 ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_223_in__mono,axiom,
% 3.18/1.75 ! [A2: set_a,B4: set_a,X: a] :
% 3.18/1.75 ( ( ord_less_eq_set_a @ A2 @ B4 )
% 3.18/1.75 => ( ( member_a @ X @ A2 )
% 3.18/1.75 => ( member_a @ X @ B4 ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_224_local_OSup__fin_Oclosed,axiom,
% 3.18/1.75 ! [A2: set_a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( A2 != bot_bot_set_a )
% 3.18/1.75 => ( ! [X2: a,Y2: a] : ( member_a @ ( sup @ X2 @ Y2 ) @ ( insert_a @ X2 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) )
% 3.18/1.75 => ( member_a @ ( lattic1781219155_fin_a @ sup @ A2 ) @ A2 ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_225_local_OSup__fin_Oinsert__not__elem,axiom,
% 3.18/1.75 ! [A2: set_a,X: a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ~ ( member_a @ X @ A2 )
% 3.18/1.75 => ( ( A2 != bot_bot_set_a )
% 3.18/1.75 => ( ( lattic1781219155_fin_a @ sup @ ( insert_a @ X @ A2 ) )
% 3.18/1.75 = ( sup @ X @ ( lattic1781219155_fin_a @ sup @ A2 ) ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_226_local_OInf__fin_Oclosed,axiom,
% 3.18/1.75 ! [A2: set_a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( A2 != bot_bot_set_a )
% 3.18/1.75 => ( ! [X2: a,Y2: a] : ( member_a @ ( inf @ X2 @ Y2 ) @ ( insert_a @ X2 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) )
% 3.18/1.75 => ( member_a @ ( lattic247676691_fin_a @ inf @ A2 ) @ A2 ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_227_local_OInf__fin_Oinsert__not__elem,axiom,
% 3.18/1.75 ! [A2: set_a,X: a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ~ ( member_a @ X @ A2 )
% 3.18/1.75 => ( ( A2 != bot_bot_set_a )
% 3.18/1.75 => ( ( lattic247676691_fin_a @ inf @ ( insert_a @ X @ A2 ) )
% 3.18/1.75 = ( inf @ X @ ( lattic247676691_fin_a @ inf @ A2 ) ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_228_singletonI,axiom,
% 3.18/1.75 ! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_229_singletonI,axiom,
% 3.18/1.75 ! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_230_insert__subset,axiom,
% 3.18/1.75 ! [X: set_a,A2: set_set_a,B4: set_set_a] :
% 3.18/1.75 ( ( ord_le318720350_set_a @ ( insert_set_a @ X @ A2 ) @ B4 )
% 3.18/1.75 = ( ( member_set_a @ X @ B4 )
% 3.18/1.75 & ( ord_le318720350_set_a @ A2 @ B4 ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_231_insert__subset,axiom,
% 3.18/1.75 ! [X: a,A2: set_a,B4: set_a] :
% 3.18/1.75 ( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B4 )
% 3.18/1.75 = ( ( member_a @ X @ B4 )
% 3.18/1.75 & ( ord_less_eq_set_a @ A2 @ B4 ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_232_singleton__insert__inj__eq_H,axiom,
% 3.18/1.75 ! [A: a,A2: set_a,B: a] :
% 3.18/1.75 ( ( ( insert_a @ A @ A2 )
% 3.18/1.75 = ( insert_a @ B @ bot_bot_set_a ) )
% 3.18/1.75 = ( ( A = B )
% 3.18/1.75 & ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_233_singleton__insert__inj__eq,axiom,
% 3.18/1.75 ! [B: a,A: a,A2: set_a] :
% 3.18/1.75 ( ( ( insert_a @ B @ bot_bot_set_a )
% 3.18/1.75 = ( insert_a @ A @ A2 ) )
% 3.18/1.75 = ( ( A = B )
% 3.18/1.75 & ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_234_local_Obdd__above__insert,axiom,
% 3.18/1.75 ! [A: a,A2: set_a] :
% 3.18/1.75 ( ( condit1627435690bove_a @ less_eq @ ( insert_a @ A @ A2 ) )
% 3.18/1.75 = ( condit1627435690bove_a @ less_eq @ A2 ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_235_local_Obdd__below__insert,axiom,
% 3.18/1.75 ! [A: a,A2: set_a] :
% 3.18/1.75 ( ( condit1001553558elow_a @ less_eq @ ( insert_a @ A @ A2 ) )
% 3.18/1.75 = ( condit1001553558elow_a @ less_eq @ A2 ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_236_local_OInf__fin_Osingleton,axiom,
% 3.18/1.75 ! [X: a] :
% 3.18/1.75 ( ( lattic247676691_fin_a @ inf @ ( insert_a @ X @ bot_bot_set_a ) )
% 3.18/1.75 = X ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_237_local_OSup__fin_Osingleton,axiom,
% 3.18/1.75 ! [X: a] :
% 3.18/1.75 ( ( lattic1781219155_fin_a @ sup @ ( insert_a @ X @ bot_bot_set_a ) )
% 3.18/1.75 = X ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_238_local_OInf__fin_Oinsert,axiom,
% 3.18/1.75 ! [A2: set_a,X: a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( A2 != bot_bot_set_a )
% 3.18/1.75 => ( ( lattic247676691_fin_a @ inf @ ( insert_a @ X @ A2 ) )
% 3.18/1.75 = ( inf @ X @ ( lattic247676691_fin_a @ inf @ A2 ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_239_local_OSup__fin_Oinsert,axiom,
% 3.18/1.75 ! [A2: set_a,X: a] :
% 3.18/1.75 ( ( finite_finite_a @ A2 )
% 3.18/1.75 => ( ( A2 != bot_bot_set_a )
% 3.18/1.75 => ( ( lattic1781219155_fin_a @ sup @ ( insert_a @ X @ A2 ) )
% 3.18/1.75 = ( sup @ X @ ( lattic1781219155_fin_a @ sup @ A2 ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_240_subset__singletonD,axiom,
% 3.18/1.75 ! [A2: set_a,X: a] :
% 3.18/1.75 ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
% 3.18/1.75 => ( ( A2 = bot_bot_set_a )
% 3.18/1.75 | ( A2
% 3.18/1.75 = ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_241_subset__singleton__iff,axiom,
% 3.18/1.75 ! [X5: set_a,A: a] :
% 3.18/1.75 ( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
% 3.18/1.75 = ( ( X5 = bot_bot_set_a )
% 3.18/1.75 | ( X5
% 3.18/1.75 = ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_242_singletonD,axiom,
% 3.18/1.75 ! [B: set_a,A: set_a] :
% 3.18/1.75 ( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
% 3.18/1.75 => ( B = A ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_243_singletonD,axiom,
% 3.18/1.75 ! [B: a,A: a] :
% 3.18/1.75 ( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
% 3.18/1.75 => ( B = A ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_244_singleton__iff,axiom,
% 3.18/1.75 ! [B: set_a,A: set_a] :
% 3.18/1.75 ( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
% 3.18/1.75 = ( B = A ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_245_singleton__iff,axiom,
% 3.18/1.75 ! [B: a,A: a] :
% 3.18/1.75 ( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
% 3.18/1.75 = ( B = A ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_246_doubleton__eq__iff,axiom,
% 3.18/1.75 ! [A: a,B: a,C: a,D: a] :
% 3.18/1.75 ( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
% 3.18/1.75 = ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
% 3.18/1.75 = ( ( ( A = C )
% 3.18/1.75 & ( B = D ) )
% 3.18/1.75 | ( ( A = D )
% 3.18/1.75 & ( B = C ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_247_insert__not__empty,axiom,
% 3.18/1.75 ! [A: a,A2: set_a] :
% 3.18/1.75 ( ( insert_a @ A @ A2 )
% 3.18/1.75 != bot_bot_set_a ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_248_singleton__inject,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( ( insert_a @ A @ bot_bot_set_a )
% 3.18/1.75 = ( insert_a @ B @ bot_bot_set_a ) )
% 3.18/1.75 => ( A = B ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_249_subset__insertI2,axiom,
% 3.18/1.75 ! [A2: set_a,B4: set_a,B: a] :
% 3.18/1.75 ( ( ord_less_eq_set_a @ A2 @ B4 )
% 3.18/1.75 => ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B4 ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_250_subset__insertI,axiom,
% 3.18/1.75 ! [B4: set_a,A: a] : ( ord_less_eq_set_a @ B4 @ ( insert_a @ A @ B4 ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_251_subset__insert,axiom,
% 3.18/1.75 ! [X: set_a,A2: set_set_a,B4: set_set_a] :
% 3.18/1.75 ( ~ ( member_set_a @ X @ A2 )
% 3.18/1.75 => ( ( ord_le318720350_set_a @ A2 @ ( insert_set_a @ X @ B4 ) )
% 3.18/1.75 = ( ord_le318720350_set_a @ A2 @ B4 ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_252_subset__insert,axiom,
% 3.18/1.75 ! [X: a,A2: set_a,B4: set_a] :
% 3.18/1.75 ( ~ ( member_a @ X @ A2 )
% 3.18/1.75 => ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B4 ) )
% 3.18/1.75 = ( ord_less_eq_set_a @ A2 @ B4 ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_253_insert__mono,axiom,
% 3.18/1.75 ! [C4: set_a,D2: set_a,A: a] :
% 3.18/1.75 ( ( ord_less_eq_set_a @ C4 @ D2 )
% 3.18/1.75 => ( ord_less_eq_set_a @ ( insert_a @ A @ C4 ) @ ( insert_a @ A @ D2 ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_254_local_OatLeastAtMost__singleton_H,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( A = B )
% 3.18/1.75 => ( ( set_atLeastAtMost_a @ less_eq @ A @ B )
% 3.18/1.75 = ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_255_finite__insert,axiom,
% 3.18/1.75 ! [A: set_a,A2: set_set_a] :
% 3.18/1.75 ( ( finite_finite_set_a @ ( insert_set_a @ A @ A2 ) )
% 3.18/1.75 = ( finite_finite_set_a @ A2 ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_256_finite__insert,axiom,
% 3.18/1.75 ! [A: a,A2: set_a] :
% 3.18/1.75 ( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
% 3.18/1.75 = ( finite_finite_a @ A2 ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_257_local_OIcc__eq__Icc,axiom,
% 3.18/1.75 ! [L: a,H: a,L2: a,H2: a] :
% 3.18/1.75 ( ( ( set_atLeastAtMost_a @ less_eq @ L @ H )
% 3.18/1.75 = ( set_atLeastAtMost_a @ less_eq @ L2 @ H2 ) )
% 3.18/1.75 = ( ( ( L = L2 )
% 3.18/1.75 & ( H = H2 ) )
% 3.18/1.75 | ( ~ ( less_eq @ L @ H )
% 3.18/1.75 & ~ ( less_eq @ L2 @ H2 ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_258_local_OatLeastAtMost__iff,axiom,
% 3.18/1.75 ! [I: a,L: a,U: a] :
% 3.18/1.75 ( ( member_a @ I @ ( set_atLeastAtMost_a @ less_eq @ L @ U ) )
% 3.18/1.75 = ( ( less_eq @ L @ I )
% 3.18/1.75 & ( less_eq @ I @ U ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_259_local_OatLeastatMost__empty__iff,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( ( set_atLeastAtMost_a @ less_eq @ A @ B )
% 3.18/1.75 = bot_bot_set_a )
% 3.18/1.75 = ( ~ ( less_eq @ A @ B ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_260_local_OatLeastatMost__empty__iff2,axiom,
% 3.18/1.75 ! [A: a,B: a] :
% 3.18/1.75 ( ( bot_bot_set_a
% 3.18/1.75 = ( set_atLeastAtMost_a @ less_eq @ A @ B ) )
% 3.18/1.75 = ( ~ ( less_eq @ A @ B ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_261_local_Obdd__below__Icc,axiom,
% 3.18/1.75 ! [A: a,B: a] : ( condit1001553558elow_a @ less_eq @ ( set_atLeastAtMost_a @ less_eq @ A @ B ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_262_local_Obdd__above__Icc,axiom,
% 3.18/1.75 ! [A: a,B: a] : ( condit1627435690bove_a @ less_eq @ ( set_atLeastAtMost_a @ less_eq @ A @ B ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_263_local_OatLeastatMost__subset__iff,axiom,
% 3.18/1.75 ! [A: a,B: a,C: a,D: a] :
% 3.18/1.75 ( ( ord_less_eq_set_a @ ( set_atLeastAtMost_a @ less_eq @ A @ B ) @ ( set_atLeastAtMost_a @ less_eq @ C @ D ) )
% 3.18/1.75 = ( ~ ( less_eq @ A @ B )
% 3.18/1.75 | ( ( less_eq @ C @ A )
% 3.18/1.75 & ( less_eq @ B @ D ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_264_local_OatLeastAtMost__singleton,axiom,
% 3.18/1.75 ! [A: a] :
% 3.18/1.75 ( ( set_atLeastAtMost_a @ less_eq @ A @ A )
% 3.18/1.75 = ( insert_a @ A @ bot_bot_set_a ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_265_local_OatLeastAtMost__singleton__iff,axiom,
% 3.18/1.75 ! [A: a,B: a,C: a] :
% 3.18/1.75 ( ( ( set_atLeastAtMost_a @ less_eq @ A @ B )
% 3.18/1.75 = ( insert_a @ C @ bot_bot_set_a ) )
% 3.18/1.75 = ( ( A = B )
% 3.18/1.75 & ( B = C ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_266_local_OIcc__subset__Ici__iff,axiom,
% 3.18/1.75 ! [L: a,H: a,L2: a] :
% 3.18/1.75 ( ( ord_less_eq_set_a @ ( set_atLeastAtMost_a @ less_eq @ L @ H ) @ ( set_atLeast_a @ less_eq @ L2 ) )
% 3.18/1.75 = ( ~ ( less_eq @ L @ H )
% 3.18/1.75 | ( less_eq @ L2 @ L ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_267_local_OIcc__subset__Iic__iff,axiom,
% 3.18/1.75 ! [L: a,H: a,H2: a] :
% 3.18/1.75 ( ( ord_less_eq_set_a @ ( set_atLeastAtMost_a @ less_eq @ L @ H ) @ ( set_atMost_a @ less_eq @ H2 ) )
% 3.18/1.75 = ( ~ ( less_eq @ L @ H )
% 3.18/1.75 | ( less_eq @ H @ H2 ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_268_comp__fun__idem_Ofun__left__idem,axiom,
% 3.18/1.75 ! [F: a > a > a,X: a,Z: a] :
% 3.18/1.75 ( ( finite40241356em_a_a @ F )
% 3.18/1.75 => ( ( F @ X @ ( F @ X @ Z ) )
% 3.18/1.75 = ( F @ X @ Z ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(fact_269_order__class_Ofinite__has__minimal2,axiom,
% 3.18/1.75 ! [A2: set_set_a,A: set_a] :
% 3.18/1.75 ( ( finite_finite_set_a @ A2 )
% 3.18/1.75 => ( ( member_set_a @ A @ A2 )
% 3.18/1.75 => ? [X2: set_a] :
% 3.18/1.75 ( ( member_set_a @ X2 @ A2 )
% 3.18/1.75 & ( ord_less_eq_set_a @ X2 @ A )
% 3.18/1.75 & ! [Xa: set_a] :
% 3.18/1.75 ( ( member_set_a @ Xa @ A2 )
% 3.18/1.75 => ( ( ord_less_eq_set_a @ Xa @ X2 )
% 3.18/1.75 => ( X2 = Xa ) ) ) ) ) ) ).
% 3.18/1.75
% 3.18/1.75 thf(conj_0,conjecture,
% 3.18/1.75 ( ( modula1144073633_aux_a @ inf @ sup @ b @ c @ a2 )
% 3.18/1.75 = ( modula1144073633_aux_a @ inf @ sup @ a2 @ b @ c ) ) ).
% 3.18/1.75 ------------------------
% 3.18/1.79 % (24768)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_24647 for (2999ds/183Mi)
% 3.18/1.79 % (24770)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_24647 for (2999ds/27Mi)
% 3.18/1.79 % (24769)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_24647 for (2999ds/4Mi)
% 3.18/1.79 % (24771)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_24647 for (2999ds/2Mi)
% 3.18/1.79 % (24773)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_24647 for (2999ds/275Mi)
% 3.18/1.79 % (24774)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_24647 for (2999ds/18Mi)
% 3.18/1.79 % (24772)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_24647 for (2999ds/2Mi)
% 3.18/1.79 % (24771)Instruction limit reached!
% 3.18/1.79 % (24771)------------------------------
% 3.18/1.79 % (24771)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.79 % (24771)Termination reason: Unknown
% 3.18/1.79 % (24772)Instruction limit reached!
% 3.18/1.79 % (24772)------------------------------
% 3.18/1.79 % (24772)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.79 % (24771)Termination phase: shuffling
% 3.18/1.79
% 3.18/1.79 % (24771)Memory used [KB]: 1279
% 3.18/1.79 % (24771)Time elapsed: 0.003 s
% 3.18/1.79 % (24771)Instructions burned: 2 (million)
% 3.18/1.79 % (24771)------------------------------
% 3.18/1.79 % (24771)------------------------------
% 3.18/1.79 % (24772)Termination reason: Unknown
% 3.18/1.79 % (24772)Termination phase: shuffling
% 3.18/1.79
% 3.18/1.79 % (24772)Memory used [KB]: 1279
% 3.18/1.79 % (24772)Time elapsed: 0.003 s
% 3.18/1.79 % (24772)Instructions burned: 2 (million)
% 3.18/1.79 % (24772)------------------------------
% 3.18/1.79 % (24772)------------------------------
% 3.18/1.79 % (24769)Instruction limit reached!
% 3.18/1.79 % (24769)------------------------------
% 3.18/1.79 % (24769)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.79 % (24769)Termination reason: Unknown
% 3.18/1.79 % (24769)Termination phase: shuffling
% 3.18/1.79
% 3.18/1.79 % (24769)Memory used [KB]: 1279
% 3.18/1.79 % (24769)Time elapsed: 0.005 s
% 3.18/1.79 % (24769)Instructions burned: 5 (million)
% 3.18/1.79 % (24769)------------------------------
% 3.18/1.79 % (24769)------------------------------
% 3.24/1.80 % (24774)Instruction limit reached!
% 3.24/1.80 % (24774)------------------------------
% 3.24/1.80 % (24774)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.80 % (24774)Termination reason: Unknown
% 3.24/1.80 % (24774)Termination phase: Property scanning
% 3.24/1.80
% 3.24/1.80 % (24774)Memory used [KB]: 1535
% 3.24/1.80 % (24774)Time elapsed: 0.012 s
% 3.24/1.80 % (24774)Instructions burned: 19 (million)
% 3.24/1.80 % (24774)------------------------------
% 3.24/1.80 % (24774)------------------------------
% 3.24/1.80 % (24770)Instruction limit reached!
% 3.24/1.80 % (24770)------------------------------
% 3.24/1.80 % (24770)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.80 % (24770)Termination reason: Unknown
% 3.24/1.80 % (24770)Termination phase: Property scanning
% 3.24/1.80
% 3.24/1.80 % (24770)Memory used [KB]: 1535
% 3.24/1.80 % (24770)Time elapsed: 0.016 s
% 3.24/1.80 % (24770)Instructions burned: 27 (million)
% 3.24/1.80 % (24770)------------------------------
% 3.24/1.80 % (24770)------------------------------
% 3.24/1.81 % (24776)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_24647 for (2999ds/37Mi)
% 3.24/1.81 % (24775)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_24647 for (2999ds/3Mi)
% 3.24/1.81 % (24777)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_24647 for (2999ds/15Mi)
% 3.24/1.81 % (24775)Instruction limit reached!
% 3.24/1.81 % (24775)------------------------------
% 3.24/1.81 % (24775)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.81 % (24775)Termination reason: Unknown
% 3.24/1.81 % (24775)Termination phase: shuffling
% 3.24/1.81
% 3.24/1.81 % (24775)Memory used [KB]: 1279
% 3.24/1.81 % (24775)Time elapsed: 0.004 s
% 3.24/1.81 % (24775)Instructions burned: 4 (million)
% 3.24/1.81 % (24775)------------------------------
% 3.24/1.81 % (24775)------------------------------
% 3.24/1.82 % (24778)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_24647 for (2999ds/3Mi)
% 3.24/1.82 % (24777)Instruction limit reached!
% 3.24/1.82 % (24777)------------------------------
% 3.24/1.82 % (24777)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.82 % (24777)Termination reason: Unknown
% 3.24/1.82 % (24777)Termination phase: shuffling
% 3.24/1.82
% 3.24/1.82 % (24777)Memory used [KB]: 1535
% 3.24/1.82 % (24777)Time elapsed: 0.010 s
% 3.24/1.82 % (24777)Instructions burned: 15 (million)
% 3.24/1.82 % (24777)------------------------------
% 3.24/1.82 % (24777)------------------------------
% 3.24/1.82 % (24773)First to succeed.
% 3.24/1.82 % (24778)Instruction limit reached!
% 3.24/1.82 % (24778)------------------------------
% 3.24/1.82 % (24778)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.82 % (24778)Termination reason: Unknown
% 3.24/1.82 % (24778)Termination phase: shuffling
% 3.24/1.82
% 3.24/1.82 % (24778)Memory used [KB]: 1279
% 3.24/1.82 % (24778)Time elapsed: 0.004 s
% 3.24/1.82 % (24778)Instructions burned: 4 (million)
% 3.24/1.82 % (24778)------------------------------
% 3.24/1.82 % (24778)------------------------------
% 3.24/1.82 % (24773)Refutation found. Thanks to Tanya!
% 3.24/1.82 % SZS status Theorem for DTF2THF_24647
% 3.24/1.82 % SZS output start Proof for DTF2THF_24647
% 3.24/1.82 thf(type_def_5, type, a: $tType).
% 3.24/1.82 thf(type_def_7, type, set_a: $tType).
% 3.24/1.82 thf(type_def_8, type, set_set_a: $tType).
% 3.24/1.82 thf(func_def_0, type, set_set_a: $tType).
% 3.24/1.82 thf(func_def_1, type, set_a: $tType).
% 3.24/1.82 thf(func_def_2, type, a: $tType).
% 3.24/1.82 thf(func_def_3, type, condit1627435690bove_a: (a > a > $o) > set_a > $o).
% 3.24/1.82 thf(func_def_4, type, condit1001553558elow_a: (a > a > $o) > set_a > $o).
% 3.24/1.82 thf(func_def_5, type, finite40241356em_a_a: (a > a > a) > $o).
% 3.24/1.82 thf(func_def_6, type, finite_finite_set_a: set_set_a > $o).
% 3.24/1.82 thf(func_def_7, type, finite_finite_a: set_a > $o).
% 3.24/1.82 thf(func_def_8, type, abel_semigroup_a: (a > a > a) > $o).
% 3.24/1.82 thf(func_def_9, type, semigroup_a: (a > a > a) > $o).
% 3.24/1.82 thf(func_def_10, type, semilattice_a: (a > a > a) > $o).
% 3.24/1.82 thf(func_def_11, type, lattic247676691_fin_a: (a > a > a) > set_a > a).
% 3.24/1.82 thf(func_def_12, type, lattic1885654924_set_a: (a > a > a) > $o).
% 3.24/1.82 thf(func_def_13, type, lattic1781219155_fin_a: (a > a > a) > set_a > a).
% 3.24/1.82 thf(func_def_14, type, modula1936294176_aux_a: (a > a > a) > (a > a > a) > a > a > a > a).
% 3.24/1.82 thf(func_def_15, type, modula1144073633_aux_a: (a > a > a) > (a > a > a) > a > a > a > a).
% 3.24/1.82 thf(func_def_16, type, bot_bot_set_set_a: set_set_a).
% 3.24/1.82 thf(func_def_17, type, bot_bot_set_a: set_a).
% 3.24/1.82 thf(func_def_18, type, least_a: (a > a > $o) > (a > $o) > a).
% 3.24/1.82 thf(func_def_19, type, max_a: (a > a > $o) > a > a > a).
% 3.24/1.82 thf(func_def_20, type, min_a: (a > a > $o) > a > a > a).
% 3.24/1.82 thf(func_def_21, type, ord_le318720350_set_a: set_set_a > set_set_a > $o).
% 3.24/1.82 thf(func_def_22, type, ord_less_eq_set_a: set_a > set_a > $o).
% 3.24/1.82 thf(func_def_23, type, greatest_a: (a > a > $o) > (a > $o) > a).
% 3.24/1.82 thf(func_def_24, type, antimono_a_set_a: (a > a > $o) > (a > set_a) > $o).
% 3.24/1.82 thf(func_def_25, type, mono_a_set_a: (a > a > $o) > (a > set_a) > $o).
% 3.24/1.82 thf(func_def_26, type, transp_a: (a > a > $o) > $o).
% 3.24/1.82 thf(func_def_27, type, collect_set_a: (set_a > $o) > set_set_a).
% 3.24/1.82 thf(func_def_28, type, collect_a: (a > $o) > set_a).
% 3.24/1.82 thf(func_def_29, type, insert_set_a: set_a > set_set_a > set_set_a).
% 3.24/1.82 thf(func_def_30, type, insert_a: a > set_a > set_a).
% 3.24/1.82 thf(func_def_31, type, set_atLeastAtMost_a: (a > a > $o) > a > a > set_a).
% 3.24/1.82 thf(func_def_32, type, set_atLeast_a: (a > a > $o) > a > set_a).
% 3.24/1.82 thf(func_def_33, type, set_atMost_a: (a > a > $o) > a > set_a).
% 3.24/1.82 thf(func_def_34, type, member_set_a: set_a > set_set_a > $o).
% 3.24/1.82 thf(func_def_35, type, member_a: a > set_a > $o).
% 3.24/1.82 thf(func_def_36, type, a2: a).
% 3.24/1.82 thf(func_def_37, type, b: a).
% 3.24/1.82 thf(func_def_38, type, c: a).
% 3.24/1.82 thf(func_def_39, type, inf: a > a > a).
% 3.24/1.82 thf(func_def_40, type, less_eq: a > a > $o).
% 3.24/1.82 thf(func_def_41, type, sup: a > a > a).
% 3.24/1.82 thf(func_def_54, type, sK0: a).
% 3.24/1.82 thf(func_def_55, type, sK1: a).
% 3.24/1.82 thf(func_def_56, type, sK2: a).
% 3.24/1.82 thf(func_def_57, type, sK3: (a > a > a) > a).
% 3.24/1.82 thf(func_def_58, type, sK4: (a > a > a) > a).
% 3.24/1.82 thf(func_def_59, type, sK5: (a > a > a) > a).
% 3.24/1.82 thf(func_def_60, type, sK6: (a > a > a) > a).
% 3.24/1.82 thf(func_def_61, type, sK7: (a > a > a) > a).
% 3.24/1.82 thf(func_def_62, type, sK8: (a > a > a) > a).
% 3.24/1.82 thf(func_def_63, type, sK9: (a > a > a) > a).
% 3.24/1.82 thf(func_def_64, type, sK10: (a > a > a) > a).
% 3.24/1.82 thf(func_def_65, type, sK11: (a > a > a) > a).
% 3.24/1.82 thf(func_def_66, type, sK12: (a > a > a) > a).
% 3.24/1.82 thf(func_def_67, type, sK13: (a > a > a) > a).
% 3.24/1.82 thf(func_def_68, type, sK14: (a > a > a) > a).
% 3.24/1.82 thf(func_def_69, type, sK15: (a > a > a) > a).
% 3.24/1.82 thf(func_def_70, type, sK16: (a > a > a) > a).
% 3.24/1.82 thf(func_def_71, type, sK17: a).
% 3.24/1.82 thf(func_def_72, type, sK18: a).
% 3.24/1.82 thf(func_def_73, type, sK19: a).
% 3.24/1.82 thf(func_def_75, type, ph21: !>[X0: $tType]:(X0)).
% 3.24/1.82 thf(f964,plain,(
% 3.24/1.82 $false),
% 3.24/1.82 inference(subsumption_resolution,[status(thm)],[f963,f853])).
% 3.24/1.82 thf(f853,plain,(
% 3.24/1.82 ( ! [X0 : a,X1 : a] : (((inf @ X0 @ X1) = (inf @ X1 @ X0))) )),
% 3.24/1.82 inference(cnf_transformation,[status(thm)],[f709])).
% 3.24/1.82 thf(f709,plain,(
% 3.24/1.82 ! [X0 : a,X1 : a] : ((inf @ X0 @ X1) = (inf @ X1 @ X0))),
% 3.24/1.82 inference(rectify,[status(thm)],[f5])).
% 3.24/1.82 thf(f5,axiom,(
% 3.24/1.82 ! [X4 : a,X3 : a] : ((inf @ X3 @ X4) = (inf @ X4 @ X3))),
% 3.24/1.82 file('/export/starexec/sandbox/tmp/tmp.yhjYV0Pxgo/DTF2THF_24647.p',fact_4_local_Oinf__commute)).
% 3.24/1.82 thf(f963,plain,(
% 3.24/1.82 ((inf @ (inf @ (sup @ a2 @ c) @ (sup @ b @ c)) @ (sup @ a2 @ b)) != (inf @ (sup @ a2 @ b) @ (inf @ (sup @ a2 @ c) @ (sup @ b @ c))))),
% 3.24/1.82 inference(forward_demodulation,[status(thm)],[f962,f853])).
% 3.24/1.82 thf(f962,plain,(
% 3.24/1.82 ((inf @ (inf @ (sup @ b @ c) @ (sup @ a2 @ c)) @ (sup @ a2 @ b)) != (inf @ (sup @ a2 @ b) @ (inf @ (sup @ a2 @ c) @ (sup @ b @ c))))),
% 3.24/1.82 inference(forward_demodulation,[status(thm)],[f961,f867])).
% 3.24/1.82 thf(f867,plain,(
% 3.24/1.82 ( ! [X0 : a,X1 : a] : (((sup @ X0 @ X1) = (sup @ X1 @ X0))) )),
% 3.24/1.82 inference(cnf_transformation,[status(thm)],[f722])).
% 3.24/1.82 thf(f722,plain,(
% 3.24/1.82 ! [X0 : a,X1 : a] : ((sup @ X0 @ X1) = (sup @ X1 @ X0))),
% 3.24/1.82 inference(rectify,[status(thm)],[f11])).
% 3.24/1.82 thf(f11,axiom,(
% 3.24/1.82 ! [X3 : a,X4 : a] : ((sup @ X3 @ X4) = (sup @ X4 @ X3))),
% 3.24/1.82 file('/export/starexec/sandbox/tmp/tmp.yhjYV0Pxgo/DTF2THF_24647.p',fact_10_local_Osup__commute)).
% 3.24/1.82 thf(f961,plain,(
% 3.24/1.82 ((inf @ (inf @ (sup @ b @ c) @ (sup @ c @ a2)) @ (sup @ a2 @ b)) != (inf @ (sup @ a2 @ b) @ (inf @ (sup @ a2 @ c) @ (sup @ b @ c))))),
% 3.24/1.82 inference(superposition,[status(thm)],[f960,f892])).
% 3.24/1.82 thf(f892,plain,(
% 3.24/1.82 ( ! [X2 : a,X0 : a,X1 : a] : (((modula1144073633_aux_a @ inf @ sup @ X0 @ X1 @ X2) = (inf @ (inf @ (sup @ X0 @ X1) @ (sup @ X1 @ X2)) @ (sup @ X2 @ X0)))) )),
% 3.24/1.82 inference(cnf_transformation,[status(thm)],[f15])).
% 3.24/1.82 thf(f15,axiom,(
% 3.24/1.82 ! [X0 : a,X1 : a,X2 : a] : ((modula1144073633_aux_a @ inf @ sup @ X0 @ X1 @ X2) = (inf @ (inf @ (sup @ X0 @ X1) @ (sup @ X1 @ X2)) @ (sup @ X2 @ X0)))),
% 3.24/1.82 file('/export/starexec/sandbox/tmp/tmp.yhjYV0Pxgo/DTF2THF_24647.p',fact_14_e__aux__def)).
% 3.24/1.82 thf(f960,plain,(
% 3.24/1.82 ((modula1144073633_aux_a @ inf @ sup @ b @ c @ a2) != (inf @ (sup @ a2 @ b) @ (inf @ (sup @ a2 @ c) @ (sup @ b @ c))))),
% 3.24/1.82 inference(forward_demodulation,[status(thm)],[f959,f853])).
% 3.24/1.82 thf(f959,plain,(
% 3.24/1.82 ((modula1144073633_aux_a @ inf @ sup @ b @ c @ a2) != (inf @ (sup @ a2 @ b) @ (inf @ (sup @ b @ c) @ (sup @ a2 @ c))))),
% 3.24/1.82 inference(forward_demodulation,[status(thm)],[f958,f867])).
% 3.24/1.82 thf(f958,plain,(
% 3.24/1.82 ((modula1144073633_aux_a @ inf @ sup @ b @ c @ a2) != (inf @ (sup @ a2 @ b) @ (inf @ (sup @ b @ c) @ (sup @ c @ a2))))),
% 3.24/1.82 inference(forward_demodulation,[status(thm)],[f957,f853])).
% 3.24/1.82 thf(f957,plain,(
% 3.24/1.82 ((modula1144073633_aux_a @ inf @ sup @ b @ c @ a2) != (inf @ (sup @ a2 @ b) @ (inf @ (sup @ c @ a2) @ (sup @ b @ c))))),
% 3.24/1.82 inference(forward_demodulation,[status(thm)],[f956,f949])).
% 3.24/1.82 thf(f949,plain,(
% 3.24/1.82 ( ! [X2 : a,X0 : a,X1 : a] : (((inf @ X0 @ (inf @ X1 @ X2)) = (inf @ X1 @ (inf @ X0 @ X2)))) )),
% 3.24/1.82 inference(cnf_transformation,[status(thm)],[f703])).
% 3.24/1.82 thf(f703,plain,(
% 3.24/1.82 ! [X0 : a,X1 : a,X2 : a] : ((inf @ X0 @ (inf @ X1 @ X2)) = (inf @ X1 @ (inf @ X0 @ X2)))),
% 3.24/1.82 inference(rectify,[status(thm)],[f3])).
% 3.24/1.82 thf(f3,axiom,(
% 3.24/1.82 ! [X1 : a,X0 : a,X2 : a] : ((inf @ X0 @ (inf @ X1 @ X2)) = (inf @ X1 @ (inf @ X0 @ X2)))),
% 3.24/1.82 file('/export/starexec/sandbox/tmp/tmp.yhjYV0Pxgo/DTF2THF_24647.p',fact_2_local_Oinf_Oleft__commute)).
% 3.24/1.82 thf(f956,plain,(
% 3.24/1.82 ((modula1144073633_aux_a @ inf @ sup @ b @ c @ a2) != (inf @ (sup @ c @ a2) @ (inf @ (sup @ a2 @ b) @ (sup @ b @ c))))),
% 3.24/1.82 inference(forward_demodulation,[status(thm)],[f955,f853])).
% 3.24/1.82 thf(f955,plain,(
% 3.24/1.82 ((modula1144073633_aux_a @ inf @ sup @ b @ c @ a2) != (inf @ (inf @ (sup @ a2 @ b) @ (sup @ b @ c)) @ (sup @ c @ a2)))),
% 3.24/1.82 inference(forward_demodulation,[status(thm)],[f830,f892])).
% 3.24/1.82 thf(f830,plain,(
% 3.24/1.82 ((modula1144073633_aux_a @ inf @ sup @ b @ c @ a2) != (modula1144073633_aux_a @ inf @ sup @ a2 @ b @ c))),
% 3.24/1.82 inference(cnf_transformation,[status(thm)],[f702])).
% 3.24/1.82 thf(f702,plain,(
% 3.24/1.82 ((modula1144073633_aux_a @ inf @ sup @ b @ c @ a2) != (modula1144073633_aux_a @ inf @ sup @ a2 @ b @ c))),
% 3.24/1.82 inference(flattening,[status(thm)],[f272])).
% 3.24/1.82 thf(f272,negated_conjecture,(
% 3.24/1.82 ~((modula1144073633_aux_a @ inf @ sup @ b @ c @ a2) = (modula1144073633_aux_a @ inf @ sup @ a2 @ b @ c))),
% 3.24/1.82 inference(negated_conjecture,[status(cth)],[f271])).
% 3.24/1.82 thf(f271,conjecture,(
% 3.24/1.82 ((modula1144073633_aux_a @ inf @ sup @ b @ c @ a2) = (modula1144073633_aux_a @ inf @ sup @ a2 @ b @ c))),
% 3.24/1.82 file('/export/starexec/sandbox/tmp/tmp.yhjYV0Pxgo/DTF2THF_24647.p',conj_0)).
% 3.24/1.82 % SZS output end Proof for DTF2THF_24647
% 3.24/1.82 % (24773)------------------------------
% 3.24/1.82 % (24773)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.82 % (24773)Termination reason: Refutation
% 3.24/1.82
% 3.24/1.82 % (24773)Memory used [KB]: 6140
% 3.24/1.82 % (24773)Time elapsed: 0.033 s
% 3.24/1.82 % (24773)Instructions burned: 55 (million)
% 3.24/1.82 % (24773)------------------------------
% 3.24/1.82 % (24773)------------------------------
% 3.24/1.82 % (24759)Success in time 0.061 s
%------------------------------------------------------------------------------