TPTP Problem File: SLH0463^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Query_Optimization/0008_Directed_Tree_Additions/prob_00626_027259__15030246_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1425 ( 653 unt; 148 typ; 0 def)
% Number of atoms : 3374 (1276 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10682 ( 405 ~; 72 |; 354 &;8539 @)
% ( 0 <=>;1312 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 17 ( 16 usr)
% Number of type conns : 482 ( 482 >; 0 *; 0 +; 0 <<)
% Number of symbols : 134 ( 132 usr; 13 con; 0-5 aty)
% Number of variables : 3177 ( 370 ^;2681 !; 126 ?;3177 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:05:57.342
%------------------------------------------------------------------------------
% Could-be-implicit typings (16)
thf(ty_n_t__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J,type,
pre_pr7278220950009878019t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
set_set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Extended____Real__Oereal_J,type,
set_Extended_ereal: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
set_set_b: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (132)
thf(sy_c_Bidirected__Digraph_Obidirected__digraph_001tf__a_001tf__b,type,
bidire6463457107099887885ph_a_b: pre_pr7278220950009878019t_unit > ( b > b ) > $o ).
thf(sy_c_Digraph_Odigraph_001tf__a_001tf__b,type,
digraph_a_b: pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Digraph_Ofin__digraph_001tf__a_001tf__b,type,
fin_digraph_a_b: pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Digraph_Ograph_001tf__a_001tf__b,type,
graph_a_b: pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Digraph_Oin__arcs_001tf__a_001tf__b,type,
in_arcs_a_b: pre_pr7278220950009878019t_unit > a > set_b ).
thf(sy_c_Digraph_Oloopfree__digraph_001tf__a_001tf__b,type,
loopfree_digraph_a_b: pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Digraph_Onomulti__digraph_001tf__a_001tf__b,type,
nomulti_digraph_a_b: pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Digraph_Oout__arcs_001tf__a_001tf__b,type,
out_arcs_a_b: pre_pr7278220950009878019t_unit > a > set_b ).
thf(sy_c_Digraph_Opre__digraph_Oadd__arc_001tf__a_001tf__b,type,
pre_add_arc_a_b: pre_pr7278220950009878019t_unit > b > pre_pr7278220950009878019t_unit ).
thf(sy_c_Digraph_Opre__digraph_Oadd__vert_001tf__a_001tf__b,type,
pre_add_vert_a_b: pre_pr7278220950009878019t_unit > a > pre_pr7278220950009878019t_unit ).
thf(sy_c_Digraph_Opre__digraph_Oarcs_001tf__a_001tf__b_001t__Product____Type__Ounit,type,
pre_ar1395965042833527383t_unit: pre_pr7278220950009878019t_unit > set_b ).
thf(sy_c_Digraph_Opre__digraph_Odel__arc_001tf__a_001tf__b,type,
pre_del_arc_a_b: pre_pr7278220950009878019t_unit > b > pre_pr7278220950009878019t_unit ).
thf(sy_c_Digraph_Opre__digraph_Odel__vert_001tf__a_001tf__b,type,
pre_del_vert_a_b: pre_pr7278220950009878019t_unit > a > pre_pr7278220950009878019t_unit ).
thf(sy_c_Digraph_Opre__digraph_Ohead_001tf__a_001tf__b_001t__Product____Type__Ounit,type,
pre_he5236287464308401016t_unit: pre_pr7278220950009878019t_unit > b > a ).
thf(sy_c_Digraph_Opre__digraph_Otail_001tf__a_001tf__b_001t__Product____Type__Ounit,type,
pre_ta4931606617599662728t_unit: pre_pr7278220950009878019t_unit > b > a ).
thf(sy_c_Digraph_Opre__digraph_Overts_001tf__a_001tf__b_001t__Product____Type__Ounit,type,
pre_ve642382030648772252t_unit: pre_pr7278220950009878019t_unit > set_a ).
thf(sy_c_Digraph_Opseudo__graph_001tf__a_001tf__b,type,
pseudo_graph_a_b: pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Digraph_Oreachable_001tf__a_001tf__b,type,
reachable_a_b: pre_pr7278220950009878019t_unit > a > a > $o ).
thf(sy_c_Digraph_Osymmetric_001tf__a_001tf__b,type,
symmetric_a_b: pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Digraph__Component_Oconnected_001tf__a_001tf__b,type,
digrap8783888973171253482ed_a_b: pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Digraph__Component_Opre__digraph_Oscc__of_001tf__a_001tf__b,type,
digrap2937667069914300949of_a_b: pre_pr7278220950009878019t_unit > a > set_a ).
thf(sy_c_Digraph__Component_Opre__digraph_Osccs__verts_001tf__a_001tf__b,type,
digrap2871191568752656621ts_a_b: pre_pr7278220950009878019t_unit > set_set_a ).
thf(sy_c_Digraph__Component_Ospanning_001tf__a_001tf__b,type,
digraph_spanning_a_b: pre_pr7278220950009878019t_unit > pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Digraph__Component_Ospanning__tree_001tf__a_001tf__b,type,
digrap5718416180170401981ee_a_b: pre_pr7278220950009878019t_unit > pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Digraph__Component_Ostrongly__connected_001tf__a_001tf__b,type,
digrap8691851296217657702ed_a_b: pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_Itf__a_J,type,
finite_card_list_a: set_list_a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_Itf__a_J,type,
finite_card_set_a: set_set_a > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__b,type,
finite_card_b: set_b > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Extended____Real__Oereal,type,
finite7198162374296863863_ereal: set_Extended_ereal > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_Itf__a_J,type,
finite_finite_list_a: set_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
finite5282473924520328461list_a: set_set_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
finite7209287970140883943_set_a: set_set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__b_J,type,
finite_finite_set_b: set_set_b > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__b,type,
finite_finite_b: set_b > $o ).
thf(sy_c_Graph__Additions_Owf__digraph_Obranching__points_001tf__a_001tf__b,type,
graph_4596510882073158607ts_a_b: pre_pr7278220950009878019t_unit > set_a ).
thf(sy_c_Graph__Additions_Owf__digraph_Ois__chain_001tf__a_001tf__b,type,
graph_3890552050688490787in_a_b: pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Graph__Additions_Owf__digraph_Ois__chain_H_001tf__a_001tf__b,type,
graph_8150681439568091980in_a_b: pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Graph__Additions_Owf__digraph_Olast__branching__points_001tf__a_001tf__b,type,
graph_1747835947655717337ts_a_b: pre_pr7278220950009878019t_unit > set_a ).
thf(sy_c_Graph__Additions_Owf__digraph_Olast__merging__points_001tf__a_001tf__b,type,
graph_2659413520663303054ts_a_b: pre_pr7278220950009878019t_unit > set_a ).
thf(sy_c_Graph__Additions_Owf__digraph_Omerging__points_001tf__a_001tf__b,type,
graph_2957805489637798020ts_a_b: pre_pr7278220950009878019t_unit > set_a ).
thf(sy_c_Graph__Definitions_Owf__digraph_Ofin__sp__costs_001tf__a_001tf__b,type,
graph_7485366578106294827ts_a_b: pre_pr7278220950009878019t_unit > ( b > real ) > set_Extended_ereal ).
thf(sy_c_Graph__Definitions_Owf__digraph_Ok__neighborhood_001tf__a_001tf__b,type,
graph_3921080825633621230od_a_b: pre_pr7278220950009878019t_unit > ( b > real ) > a > real > set_a ).
thf(sy_c_Graph__Definitions_Owf__digraph_On__nearest__verts_001tf__a_001tf__b,type,
graph_3148032005746981223ts_a_b: pre_pr7278220950009878019t_unit > ( b > real ) > a > nat > set_a > $o ).
thf(sy_c_Graph__Definitions_Owf__digraph_Osp__costs_001tf__a_001tf__b,type,
graph_1574344591923819902ts_a_b: pre_pr7278220950009878019t_unit > ( b > real ) > set_Extended_ereal ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__b_J,type,
minus_minus_set_b: set_b > set_b > set_b ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
inf_inf_list_a_o: ( list_a > $o ) > ( list_a > $o ) > list_a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
inf_inf_set_a_o: ( set_a > $o ) > ( set_a > $o ) > set_a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__a_M_Eo_J,type,
inf_inf_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__b_M_Eo_J,type,
inf_inf_b_o: ( b > $o ) > ( b > $o ) > b > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
inf_inf_set_list_a: set_list_a > set_list_a > set_list_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__b_J,type,
inf_inf_set_b: set_b > set_b > set_b ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
sup_sup_list_a_o: ( list_a > $o ) > ( list_a > $o ) > list_a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
sup_sup_set_a_o: ( set_a > $o ) > ( set_a > $o ) > set_a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__b_M_Eo_J,type,
sup_sup_b_o: ( b > $o ) > ( b > $o ) > b > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
sup_sup_set_list_a: set_list_a > set_list_a > set_list_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__b_J,type,
sup_sup_set_b: set_b > set_b > set_b ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
bot_bot_list_a_o: list_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__b_M_Eo_J,type,
bot_bot_b_o: b > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
bot_bot_set_list_a: set_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
bot_bot_set_set_b: set_set_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
bot_bot_set_b: set_b ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
ord_less_eq_set_a_o: ( set_a > $o ) > ( set_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__b_M_Eo_J,type,
ord_less_eq_b_o: ( b > $o ) > ( b > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(sy_c_Set_OBex_001t__Set__Oset_Itf__a_J,type,
bex_set_a: set_set_a > ( set_a > $o ) > $o ).
thf(sy_c_Set_OBex_001tf__a,type,
bex_a: set_a > ( a > $o ) > $o ).
thf(sy_c_Set_OBex_001tf__b,type,
bex_b: set_b > ( b > $o ) > $o ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
collect_set_list_a: ( set_list_a > $o ) > set_set_list_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
collect_set_set_a: ( set_set_a > $o ) > set_set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__b_J,type,
collect_set_b: ( set_b > $o ) > set_set_b ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__b_001tf__a,type,
image_b_a: ( b > a ) > set_b > set_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Oinsert_001tf__b,type,
insert_b: b > set_b > set_b ).
thf(sy_c_Shortest__Path__Tree_Opre__digraph_Oleaf_001tf__a_001tf__b,type,
shorte1213025427933718126af_a_b: pre_pr7278220950009878019t_unit > a > $o ).
thf(sy_c_Shortest__Path__Tree_Osubgraph_001tf__a_001tf__b,type,
shorte3657265928840388360ph_a_b: pre_pr7278220950009878019t_unit > pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Vertex__Walk_Ovpath_001tf__a_001tf__b,type,
vertex_vpath_a_b: list_a > pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__b_J,type,
member_set_b: set_b > set_set_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_G,type,
g: pre_pr7278220950009878019t_unit ).
thf(sy_v_T,type,
t: pre_pr7278220950009878019t_unit ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_source,type,
source: a ).
thf(sy_v_w,type,
w: b > real ).
% Relevant facts (1276)
thf(fact_0_two__in__arcs__contr,axiom,
! [E1: b,E2: b] :
( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( E1 != E2 )
=> ( ( pre_he5236287464308401016t_unit @ t @ E1 )
!= ( pre_he5236287464308401016t_unit @ t @ E2 ) ) ) ) ) ).
% two_in_arcs_contr
thf(fact_1__092_060open_062_123e2_A_092_060in_062_Aarcs_AG_O_A_092_060exists_062e1_092_060in_062arcs_AT_O_Ahead_AG_Ae2_A_061_Atail_AG_Ae1_A_092_060and_062_Ahead_AG_Ae1_A_061_Atail_AG_Ae2_125_A_092_060inter_062_Aarcs_AT_A_061_A_123_125_092_060close_062,axiom,
( ( inf_inf_set_b
@ ( collect_b
@ ^ [E22: b] :
( ( member_b @ E22 @ ( pre_ar1395965042833527383t_unit @ g ) )
& ? [X: b] :
( ( member_b @ X @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( pre_he5236287464308401016t_unit @ g @ E22 )
= ( pre_ta4931606617599662728t_unit @ g @ X ) )
& ( ( pre_he5236287464308401016t_unit @ g @ X )
= ( pre_ta4931606617599662728t_unit @ g @ E22 ) ) ) ) )
@ ( pre_ar1395965042833527383t_unit @ t ) )
= bot_bot_set_b ) ).
% \<open>{e2 \<in> arcs G. \<exists>e1\<in>arcs T. head G e2 = tail G e1 \<and> head G e1 = tail G e2} \<inter> arcs T = {}\<close>
thf(fact_2_graph__axioms,axiom,
graph_a_b @ g ).
% graph_axioms
thf(fact_3_headT__eq__headG,axiom,
( ( pre_he5236287464308401016t_unit @ t )
= ( pre_he5236287464308401016t_unit @ g ) ) ).
% headT_eq_headG
thf(fact_4_tailT__eq__tailG,axiom,
( ( pre_ta4931606617599662728t_unit @ t )
= ( pre_ta4931606617599662728t_unit @ g ) ) ).
% tailT_eq_tailG
thf(fact_5_no__loops,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( pre_ta4931606617599662728t_unit @ g @ E )
!= ( pre_he5236287464308401016t_unit @ g @ E ) ) ) ).
% no_loops
thf(fact_6_no__multi__alt,axiom,
! [E1: b,E2: b] :
( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( E1 != E2 )
=> ( ( ( pre_he5236287464308401016t_unit @ g @ E1 )
!= ( pre_he5236287464308401016t_unit @ g @ E2 ) )
| ( ( pre_ta4931606617599662728t_unit @ g @ E1 )
!= ( pre_ta4931606617599662728t_unit @ g @ E2 ) ) ) ) ) ) ).
% no_multi_alt
thf(fact_7_loopfree_Ono__loops,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( pre_ta4931606617599662728t_unit @ t @ E )
!= ( pre_he5236287464308401016t_unit @ t @ E ) ) ) ).
% loopfree.no_loops
thf(fact_8_nomulti_Ono__multi__alt,axiom,
! [E1: b,E2: b] :
( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( E1 != E2 )
=> ( ( ( pre_he5236287464308401016t_unit @ t @ E1 )
!= ( pre_he5236287464308401016t_unit @ t @ E2 ) )
| ( ( pre_ta4931606617599662728t_unit @ t @ E1 )
!= ( pre_ta4931606617599662728t_unit @ t @ E2 ) ) ) ) ) ) ).
% nomulti.no_multi_alt
thf(fact_9__092_060open_062_092_060forall_062e1_092_060in_062arcs_AT_O_A_092_060exists_062e2_092_060in_062arcs_AG_O_Ahead_AG_Ae2_A_061_Atail_AG_Ae1_A_092_060and_062_Ahead_AG_Ae1_A_061_Atail_AG_Ae2_092_060close_062,axiom,
! [X2: b] :
( ( member_b @ X2 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ? [Xa: b] :
( ( member_b @ Xa @ ( pre_ar1395965042833527383t_unit @ g ) )
& ( ( pre_he5236287464308401016t_unit @ g @ Xa )
= ( pre_ta4931606617599662728t_unit @ g @ X2 ) )
& ( ( pre_he5236287464308401016t_unit @ g @ X2 )
= ( pre_ta4931606617599662728t_unit @ g @ Xa ) ) ) ) ).
% \<open>\<forall>e1\<in>arcs T. \<exists>e2\<in>arcs G. head G e2 = tail G e1 \<and> head G e1 = tail G e2\<close>
thf(fact_10_no__multi__T__G,axiom,
! [E1: b,E2: b] :
( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( E1 != E2 )
=> ( ( ( pre_he5236287464308401016t_unit @ g @ E1 )
!= ( pre_he5236287464308401016t_unit @ g @ E2 ) )
| ( ( pre_ta4931606617599662728t_unit @ g @ E1 )
!= ( pre_ta4931606617599662728t_unit @ g @ E2 ) ) ) ) ) ) ).
% no_multi_T_G
thf(fact_11_reverse__arc__notin__T,axiom,
! [E1: b,E2: b] :
( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( ( pre_he5236287464308401016t_unit @ g @ E2 )
= ( pre_ta4931606617599662728t_unit @ g @ E1 ) )
=> ( ( ( pre_he5236287464308401016t_unit @ g @ E1 )
= ( pre_ta4931606617599662728t_unit @ g @ E2 ) )
=> ~ ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) ) ) ) ) ).
% reverse_arc_notin_T
thf(fact_12_pseudo__graph__axioms,axiom,
pseudo_graph_a_b @ g ).
% pseudo_graph_axioms
thf(fact_13_loopfree_Oloopfree__digraph__axioms,axiom,
loopfree_digraph_a_b @ t ).
% loopfree.loopfree_digraph_axioms
thf(fact_14_loopfree__digraph__axioms,axiom,
loopfree_digraph_a_b @ g ).
% loopfree_digraph_axioms
thf(fact_15_nomulti_Onomulti__digraph__axioms,axiom,
nomulti_digraph_a_b @ t ).
% nomulti.nomulti_digraph_axioms
thf(fact_16_nomulti__digraph__axioms,axiom,
nomulti_digraph_a_b @ g ).
% nomulti_digraph_axioms
thf(fact_17_merging__points__def,axiom,
( ( graph_2957805489637798020ts_a_b @ t )
= ( collect_a
@ ^ [X: a] :
? [Y: b] :
( ( member_b @ Y @ ( pre_ar1395965042833527383t_unit @ t ) )
& ? [Z: b] :
( ( member_b @ Z @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( Y != Z )
& ( ( pre_he5236287464308401016t_unit @ t @ Y )
= X )
& ( ( pre_he5236287464308401016t_unit @ t @ Z )
= X ) ) ) ) ) ).
% merging_points_def
thf(fact_18_G_Omerging__points__def,axiom,
( ( graph_2957805489637798020ts_a_b @ g )
= ( collect_a
@ ^ [X: a] :
? [Y: b] :
( ( member_b @ Y @ ( pre_ar1395965042833527383t_unit @ g ) )
& ? [Z: b] :
( ( member_b @ Z @ ( pre_ar1395965042833527383t_unit @ g ) )
& ( Y != Z )
& ( ( pre_he5236287464308401016t_unit @ g @ Y )
= X )
& ( ( pre_he5236287464308401016t_unit @ g @ Z )
= X ) ) ) ) ) ).
% G.merging_points_def
thf(fact_19_reverse__arc__in__G,axiom,
! [E1: b] :
( ( graph_a_b @ g )
=> ( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ? [X3: b] :
( ( member_b @ X3 @ ( pre_ar1395965042833527383t_unit @ g ) )
& ( ( pre_he5236287464308401016t_unit @ g @ X3 )
= ( pre_ta4931606617599662728t_unit @ g @ E1 ) )
& ( ( pre_he5236287464308401016t_unit @ g @ E1 )
= ( pre_ta4931606617599662728t_unit @ g @ X3 ) ) ) ) ) ).
% reverse_arc_in_G
thf(fact_20_reverse__arc__in__G__only,axiom,
! [E1: b] :
( ( graph_a_b @ g )
=> ( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ? [X3: b] :
( ( member_b @ X3 @ ( pre_ar1395965042833527383t_unit @ g ) )
& ( ( pre_he5236287464308401016t_unit @ g @ X3 )
= ( pre_ta4931606617599662728t_unit @ g @ E1 ) )
& ( ( pre_he5236287464308401016t_unit @ g @ E1 )
= ( pre_ta4931606617599662728t_unit @ g @ X3 ) )
& ~ ( member_b @ X3 @ ( pre_ar1395965042833527383t_unit @ t ) ) ) ) ) ).
% reverse_arc_in_G_only
thf(fact_21_digraph__axioms,axiom,
digraph_a_b @ g ).
% digraph_axioms
thf(fact_22_subgraph__axioms,axiom,
shorte3657265928840388360ph_a_b @ t @ g ).
% subgraph_axioms
thf(fact_23_head__add__vert,axiom,
! [U: a] :
( ( pre_he5236287464308401016t_unit @ ( pre_add_vert_a_b @ t @ U ) )
= ( pre_he5236287464308401016t_unit @ t ) ) ).
% head_add_vert
thf(fact_24_tail__add__vert,axiom,
! [U: a] :
( ( pre_ta4931606617599662728t_unit @ ( pre_add_vert_a_b @ t @ U ) )
= ( pre_ta4931606617599662728t_unit @ t ) ) ).
% tail_add_vert
thf(fact_25_G_Oarcs__add__vert,axiom,
! [U: a] :
( ( pre_ar1395965042833527383t_unit @ ( pre_add_vert_a_b @ g @ U ) )
= ( pre_ar1395965042833527383t_unit @ g ) ) ).
% G.arcs_add_vert
thf(fact_26_arcs__add__vert,axiom,
! [U: a] :
( ( pre_ar1395965042833527383t_unit @ ( pre_add_vert_a_b @ t @ U ) )
= ( pre_ar1395965042833527383t_unit @ t ) ) ).
% arcs_add_vert
thf(fact_27_G_Otail__add__vert,axiom,
! [U: a] :
( ( pre_ta4931606617599662728t_unit @ ( pre_add_vert_a_b @ g @ U ) )
= ( pre_ta4931606617599662728t_unit @ g ) ) ).
% G.tail_add_vert
thf(fact_28_G_Ohead__add__vert,axiom,
! [U: a] :
( ( pre_he5236287464308401016t_unit @ ( pre_add_vert_a_b @ g @ U ) )
= ( pre_he5236287464308401016t_unit @ g ) ) ).
% G.head_add_vert
thf(fact_29_G_Omerge__in__supergraph,axiom,
! [C: pre_pr7278220950009878019t_unit,X4: a] :
( ( shorte3657265928840388360ph_a_b @ C @ g )
=> ( ( member_a @ X4 @ ( graph_2957805489637798020ts_a_b @ C ) )
=> ( member_a @ X4 @ ( graph_2957805489637798020ts_a_b @ g ) ) ) ) ).
% G.merge_in_supergraph
thf(fact_30_merge__in__supergraph,axiom,
! [C: pre_pr7278220950009878019t_unit,X4: a] :
( ( shorte3657265928840388360ph_a_b @ C @ t )
=> ( ( member_a @ X4 @ ( graph_2957805489637798020ts_a_b @ C ) )
=> ( member_a @ X4 @ ( graph_2957805489637798020ts_a_b @ t ) ) ) ) ).
% merge_in_supergraph
thf(fact_31_G_Olast__merge__is__merge,axiom,
! [Y2: a] :
( ( member_a @ Y2 @ ( graph_2659413520663303054ts_a_b @ g ) )
=> ( member_a @ Y2 @ ( graph_2957805489637798020ts_a_b @ g ) ) ) ).
% G.last_merge_is_merge
thf(fact_32_last__merge__is__merge,axiom,
! [Y2: a] :
( ( member_a @ Y2 @ ( graph_2659413520663303054ts_a_b @ t ) )
=> ( member_a @ Y2 @ ( graph_2957805489637798020ts_a_b @ t ) ) ) ).
% last_merge_is_merge
thf(fact_33_T__arcs__compl__card__eq,axiom,
! [Es: set_b] :
( ( graph_a_b @ g )
=> ( ( ord_less_eq_set_b @ Es @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( finite_card_b
@ ( collect_b
@ ^ [E22: b] :
( ( member_b @ E22 @ ( pre_ar1395965042833527383t_unit @ g ) )
& ? [X: b] :
( ( member_b @ X @ Es )
& ( ( pre_he5236287464308401016t_unit @ g @ E22 )
= ( pre_ta4931606617599662728t_unit @ g @ X ) )
& ( ( pre_he5236287464308401016t_unit @ g @ X )
= ( pre_ta4931606617599662728t_unit @ g @ E22 ) ) ) ) ) )
= ( finite_card_b @ Es ) ) ) ) ).
% T_arcs_compl_card_eq
thf(fact_34_Un__Int__eq_I1_J,axiom,
! [S: set_b,T: set_b] :
( ( inf_inf_set_b @ ( sup_sup_set_b @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_35_Un__Int__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_36_Un__Int__eq_I2_J,axiom,
! [S: set_b,T: set_b] :
( ( inf_inf_set_b @ ( sup_sup_set_b @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_37_Un__Int__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_38_Un__Int__eq_I3_J,axiom,
! [S: set_b,T: set_b] :
( ( inf_inf_set_b @ S @ ( sup_sup_set_b @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_39_Un__Int__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_40_Un__Int__eq_I4_J,axiom,
! [T: set_b,S: set_b] :
( ( inf_inf_set_b @ T @ ( sup_sup_set_b @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_41_Un__Int__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( inf_inf_set_a @ T @ ( sup_sup_set_a @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_42_Int__Un__eq_I1_J,axiom,
! [S: set_b,T: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_43_Int__Un__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_44_Int__Un__eq_I2_J,axiom,
! [S: set_b,T: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_45_Int__Un__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_46_Int__Un__eq_I3_J,axiom,
! [S: set_b,T: set_b] :
( ( sup_sup_set_b @ S @ ( inf_inf_set_b @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_47_Int__Un__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_48_Int__Un__eq_I4_J,axiom,
! [T: set_b,S: set_b] :
( ( sup_sup_set_b @ T @ ( inf_inf_set_b @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_49_Int__Un__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( sup_sup_set_a @ T @ ( inf_inf_set_a @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_50_bex__empty,axiom,
! [P: b > $o] :
~ ? [X2: b] :
( ( member_b @ X2 @ bot_bot_set_b )
& ( P @ X2 ) ) ).
% bex_empty
thf(fact_51_bex__empty,axiom,
! [P: a > $o] :
~ ? [X2: a] :
( ( member_a @ X2 @ bot_bot_set_a )
& ( P @ X2 ) ) ).
% bex_empty
thf(fact_52_sup__inf__absorb,axiom,
! [X4: set_b,Y2: set_b] :
( ( sup_sup_set_b @ X4 @ ( inf_inf_set_b @ X4 @ Y2 ) )
= X4 ) ).
% sup_inf_absorb
thf(fact_53_sup__inf__absorb,axiom,
! [X4: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y2 ) )
= X4 ) ).
% sup_inf_absorb
thf(fact_54_empty__Collect__eq,axiom,
! [P: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P ) )
= ( ! [X: list_a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_55_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_56_empty__Collect__eq,axiom,
! [P: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P ) )
= ( ! [X: b] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_57_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_58_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_59_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_60_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
! [A: list_a,P: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_63_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X: set_a] : ( member_set_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_64_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X: b] : ( member_b @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_65_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X: list_a] : ( member_list_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_68_Collect__cong,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X3: b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_b @ P )
= ( collect_b @ Q ) ) ) ).
% Collect_cong
thf(fact_69_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_70_Collect__cong,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X3: list_a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_list_a @ P )
= ( collect_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_71_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_72_Collect__empty__eq,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( ! [X: list_a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_73_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_74_Collect__empty__eq,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( ! [X: b] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_75_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_76_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X: set_a] :
~ ( member_set_a @ X @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_77_all__not__in__conv,axiom,
! [A2: set_b] :
( ( ! [X: b] :
~ ( member_b @ X @ A2 ) )
= ( A2 = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_78_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_79_empty__iff,axiom,
! [C2: set_a] :
~ ( member_set_a @ C2 @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_80_empty__iff,axiom,
! [C2: b] :
~ ( member_b @ C2 @ bot_bot_set_b ) ).
% empty_iff
thf(fact_81_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_82_subset__antisym,axiom,
! [A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( ord_less_eq_set_b @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_83_subset__antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_84_subsetI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_set_a @ X3 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_85_subsetI,axiom,
! [A2: set_b,B: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( member_b @ X3 @ B ) )
=> ( ord_less_eq_set_b @ A2 @ B ) ) ).
% subsetI
thf(fact_86_subsetI,axiom,
! [A2: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_87_inf__right__idem,axiom,
! [X4: set_b,Y2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ X4 @ Y2 ) @ Y2 )
= ( inf_inf_set_b @ X4 @ Y2 ) ) ).
% inf_right_idem
thf(fact_88_inf__right__idem,axiom,
! [X4: set_a,Y2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y2 ) @ Y2 )
= ( inf_inf_set_a @ X4 @ Y2 ) ) ).
% inf_right_idem
thf(fact_89_inf_Oright__idem,axiom,
! [A: set_b,B2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A @ B2 ) @ B2 )
= ( inf_inf_set_b @ A @ B2 ) ) ).
% inf.right_idem
thf(fact_90_inf_Oright__idem,axiom,
! [A: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B2 ) @ B2 )
= ( inf_inf_set_a @ A @ B2 ) ) ).
% inf.right_idem
thf(fact_91_inf__left__idem,axiom,
! [X4: set_b,Y2: set_b] :
( ( inf_inf_set_b @ X4 @ ( inf_inf_set_b @ X4 @ Y2 ) )
= ( inf_inf_set_b @ X4 @ Y2 ) ) ).
% inf_left_idem
thf(fact_92_inf__left__idem,axiom,
! [X4: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y2 ) )
= ( inf_inf_set_a @ X4 @ Y2 ) ) ).
% inf_left_idem
thf(fact_93_inf_Oleft__idem,axiom,
! [A: set_b,B2: set_b] :
( ( inf_inf_set_b @ A @ ( inf_inf_set_b @ A @ B2 ) )
= ( inf_inf_set_b @ A @ B2 ) ) ).
% inf.left_idem
thf(fact_94_inf_Oleft__idem,axiom,
! [A: set_a,B2: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B2 ) )
= ( inf_inf_set_a @ A @ B2 ) ) ).
% inf.left_idem
thf(fact_95_inf__idem,axiom,
! [X4: set_b] :
( ( inf_inf_set_b @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_96_inf__idem,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_97_inf_Oidem,axiom,
! [A: set_b] :
( ( inf_inf_set_b @ A @ A )
= A ) ).
% inf.idem
thf(fact_98_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_99_sup_Oidem,axiom,
! [A: set_b] :
( ( sup_sup_set_b @ A @ A )
= A ) ).
% sup.idem
thf(fact_100_sup_Oidem,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_101_sup__idem,axiom,
! [X4: set_b] :
( ( sup_sup_set_b @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_102_sup__idem,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_103_sup_Oleft__idem,axiom,
! [A: set_b,B2: set_b] :
( ( sup_sup_set_b @ A @ ( sup_sup_set_b @ A @ B2 ) )
= ( sup_sup_set_b @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_104_sup_Oleft__idem,axiom,
! [A: set_a,B2: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B2 ) )
= ( sup_sup_set_a @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_105_sup__left__idem,axiom,
! [X4: set_b,Y2: set_b] :
( ( sup_sup_set_b @ X4 @ ( sup_sup_set_b @ X4 @ Y2 ) )
= ( sup_sup_set_b @ X4 @ Y2 ) ) ).
% sup_left_idem
thf(fact_106_sup__left__idem,axiom,
! [X4: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X4 @ ( sup_sup_set_a @ X4 @ Y2 ) )
= ( sup_sup_set_a @ X4 @ Y2 ) ) ).
% sup_left_idem
thf(fact_107_sup_Oright__idem,axiom,
! [A: set_b,B2: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ A @ B2 ) @ B2 )
= ( sup_sup_set_b @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_108_sup_Oright__idem,axiom,
! [A: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B2 ) @ B2 )
= ( sup_sup_set_a @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_109_Int__iff,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
= ( ( member_set_a @ C2 @ A2 )
& ( member_set_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_110_Int__iff,axiom,
! [C2: b,A2: set_b,B: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A2 @ B ) )
= ( ( member_b @ C2 @ A2 )
& ( member_b @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_111_Int__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
& ( member_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_112_IntI,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A2 )
=> ( ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_113_IntI,axiom,
! [C2: b,A2: set_b,B: set_b] :
( ( member_b @ C2 @ A2 )
=> ( ( member_b @ C2 @ B )
=> ( member_b @ C2 @ ( inf_inf_set_b @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_114_IntI,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_115_UnCI,axiom,
! [C2: set_a,B: set_set_a,A2: set_set_a] :
( ( ~ ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ A2 ) )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_116_UnCI,axiom,
! [C2: b,B: set_b,A2: set_b] :
( ( ~ ( member_b @ C2 @ B )
=> ( member_b @ C2 @ A2 ) )
=> ( member_b @ C2 @ ( sup_sup_set_b @ A2 @ B ) ) ) ).
% UnCI
thf(fact_117_UnCI,axiom,
! [C2: a,B: set_a,A2: set_a] :
( ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ A2 ) )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_118_Un__iff,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( ( member_set_a @ C2 @ A2 )
| ( member_set_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_119_Un__iff,axiom,
! [C2: b,A2: set_b,B: set_b] :
( ( member_b @ C2 @ ( sup_sup_set_b @ A2 @ B ) )
= ( ( member_b @ C2 @ A2 )
| ( member_b @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_120_Un__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
| ( member_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_121_inf_Obounded__iff,axiom,
! [A: set_b,B2: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ A @ ( inf_inf_set_b @ B2 @ C2 ) )
= ( ( ord_less_eq_set_b @ A @ B2 )
& ( ord_less_eq_set_b @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_122_inf_Obounded__iff,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_nat @ A @ B2 )
& ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_123_inf_Obounded__iff,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( ( ord_less_eq_set_a @ A @ B2 )
& ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_124_le__inf__iff,axiom,
! [X4: set_b,Y2: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ X4 @ ( inf_inf_set_b @ Y2 @ Z2 ) )
= ( ( ord_less_eq_set_b @ X4 @ Y2 )
& ( ord_less_eq_set_b @ X4 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_125_le__inf__iff,axiom,
! [X4: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ Y2 @ Z2 ) )
= ( ( ord_less_eq_nat @ X4 @ Y2 )
& ( ord_less_eq_nat @ X4 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_126_le__inf__iff,axiom,
! [X4: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y2 @ Z2 ) )
= ( ( ord_less_eq_set_a @ X4 @ Y2 )
& ( ord_less_eq_set_a @ X4 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_127_le__sup__iff,axiom,
! [X4: set_b,Y2: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ X4 @ Y2 ) @ Z2 )
= ( ( ord_less_eq_set_b @ X4 @ Z2 )
& ( ord_less_eq_set_b @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_128_le__sup__iff,axiom,
! [X4: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X4 @ Y2 ) @ Z2 )
= ( ( ord_less_eq_nat @ X4 @ Z2 )
& ( ord_less_eq_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_129_le__sup__iff,axiom,
! [X4: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X4 @ Y2 ) @ Z2 )
= ( ( ord_less_eq_set_a @ X4 @ Z2 )
& ( ord_less_eq_set_a @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_130_sup_Obounded__iff,axiom,
! [B2: set_b,C2: set_b,A: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ B2 @ C2 ) @ A )
= ( ( ord_less_eq_set_b @ B2 @ A )
& ( ord_less_eq_set_b @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_131_sup_Obounded__iff,axiom,
! [B2: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_132_sup_Obounded__iff,axiom,
! [B2: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A )
= ( ( ord_less_eq_set_a @ B2 @ A )
& ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_133_empty__subsetI,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A2 ) ).
% empty_subsetI
thf(fact_134_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_135_subset__empty,axiom,
! [A2: set_b] :
( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
= ( A2 = bot_bot_set_b ) ) ).
% subset_empty
thf(fact_136_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_137_inf__bot__right,axiom,
! [X4: set_b] :
( ( inf_inf_set_b @ X4 @ bot_bot_set_b )
= bot_bot_set_b ) ).
% inf_bot_right
thf(fact_138_inf__bot__right,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_139_inf__bot__left,axiom,
! [X4: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ X4 )
= bot_bot_set_b ) ).
% inf_bot_left
thf(fact_140_inf__bot__left,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X4 )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_141_sup__bot__left,axiom,
! [X4: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_142_sup__bot__left,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_143_sup__bot__right,axiom,
! [X4: set_b] :
( ( sup_sup_set_b @ X4 @ bot_bot_set_b )
= X4 ) ).
% sup_bot_right
thf(fact_144_sup__bot__right,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ X4 @ bot_bot_set_a )
= X4 ) ).
% sup_bot_right
thf(fact_145_bot__eq__sup__iff,axiom,
! [X4: set_b,Y2: set_b] :
( ( bot_bot_set_b
= ( sup_sup_set_b @ X4 @ Y2 ) )
= ( ( X4 = bot_bot_set_b )
& ( Y2 = bot_bot_set_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_146_bot__eq__sup__iff,axiom,
! [X4: set_a,Y2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X4 @ Y2 ) )
= ( ( X4 = bot_bot_set_a )
& ( Y2 = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_147_sup__eq__bot__iff,axiom,
! [X4: set_b,Y2: set_b] :
( ( ( sup_sup_set_b @ X4 @ Y2 )
= bot_bot_set_b )
= ( ( X4 = bot_bot_set_b )
& ( Y2 = bot_bot_set_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_148_sup__eq__bot__iff,axiom,
! [X4: set_a,Y2: set_a] :
( ( ( sup_sup_set_a @ X4 @ Y2 )
= bot_bot_set_a )
= ( ( X4 = bot_bot_set_a )
& ( Y2 = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_149_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_b,B2: set_b] :
( ( ( sup_sup_set_b @ A @ B2 )
= bot_bot_set_b )
= ( ( A = bot_bot_set_b )
& ( B2 = bot_bot_set_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_150_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A @ B2 )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_151_sup__bot_Oleft__neutral,axiom,
! [A: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_152_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_153_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_b,B2: set_b] :
( ( bot_bot_set_b
= ( sup_sup_set_b @ A @ B2 ) )
= ( ( A = bot_bot_set_b )
& ( B2 = bot_bot_set_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_154_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B2 ) )
= ( ( A = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_155_sup__bot_Oright__neutral,axiom,
! [A: set_b] :
( ( sup_sup_set_b @ A @ bot_bot_set_b )
= A ) ).
% sup_bot.right_neutral
thf(fact_156_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_157_Un__empty,axiom,
! [A2: set_b,B: set_b] :
( ( ( sup_sup_set_b @ A2 @ B )
= bot_bot_set_b )
= ( ( A2 = bot_bot_set_b )
& ( B = bot_bot_set_b ) ) ) ).
% Un_empty
thf(fact_158_Un__empty,axiom,
! [A2: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_159_inf__sup__absorb,axiom,
! [X4: set_b,Y2: set_b] :
( ( inf_inf_set_b @ X4 @ ( sup_sup_set_b @ X4 @ Y2 ) )
= X4 ) ).
% inf_sup_absorb
thf(fact_160_inf__sup__absorb,axiom,
! [X4: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X4 @ ( sup_sup_set_a @ X4 @ Y2 ) )
= X4 ) ).
% inf_sup_absorb
thf(fact_161_Int__subset__iff,axiom,
! [C: set_b,A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ C @ ( inf_inf_set_b @ A2 @ B ) )
= ( ( ord_less_eq_set_b @ C @ A2 )
& ( ord_less_eq_set_b @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_162_Int__subset__iff,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_eq_set_a @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_163_Un__subset__iff,axiom,
! [A2: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ A2 @ B ) @ C )
= ( ( ord_less_eq_set_b @ A2 @ C )
& ( ord_less_eq_set_b @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_164_Un__subset__iff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( ( ord_less_eq_set_a @ A2 @ C )
& ( ord_less_eq_set_a @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_165_G_Ois__chain_H__def,axiom,
( ( graph_8150681439568091980in_a_b @ g )
= ( ( graph_2957805489637798020ts_a_b @ g )
= bot_bot_set_a ) ) ).
% G.is_chain'_def
thf(fact_166_is__chain_H__def,axiom,
( ( graph_8150681439568091980in_a_b @ t )
= ( ( graph_2957805489637798020ts_a_b @ t )
= bot_bot_set_a ) ) ).
% is_chain'_def
thf(fact_167_Collect__mono__iff,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) )
= ( ! [X: list_a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_168_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_169_Collect__mono__iff,axiom,
! [P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) )
= ( ! [X: b] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_170_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_171_Collect__subset,axiom,
! [A2: set_set_a,P: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X: set_a] :
( ( member_set_a @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_172_Collect__subset,axiom,
! [A2: set_list_a,P: list_a > $o] :
( ord_le8861187494160871172list_a
@ ( collect_list_a
@ ^ [X: list_a] :
( ( member_list_a @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_173_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_174_Collect__subset,axiom,
! [A2: set_b,P: b > $o] :
( ord_less_eq_set_b
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_175_Collect__subset,axiom,
! [A2: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_176_set__eq__subset,axiom,
( ( ^ [Y3: set_b,Z3: set_b] : ( Y3 = Z3 ) )
= ( ^ [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
& ( ord_less_eq_set_b @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_177_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z3: set_a] : ( Y3 = Z3 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_178_subset__trans,axiom,
! [A2: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ord_less_eq_set_b @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_179_subset__trans,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_180_Collect__mono,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X3: list_a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_mono
thf(fact_181_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_182_Collect__mono,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X3: b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_mono
thf(fact_183_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_184_subset__refl,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ A2 @ A2 ) ).
% subset_refl
thf(fact_185_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_186_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [T2: set_a] :
( ( member_set_a @ T2 @ A3 )
=> ( member_set_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_187_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B3: set_b] :
! [T2: b] :
( ( member_b @ T2 @ A3 )
=> ( member_b @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_188_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A3 )
=> ( member_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_189_equalityD2,axiom,
! [A2: set_b,B: set_b] :
( ( A2 = B )
=> ( ord_less_eq_set_b @ B @ A2 ) ) ).
% equalityD2
thf(fact_190_equalityD2,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_191_equalityD1,axiom,
! [A2: set_b,B: set_b] :
( ( A2 = B )
=> ( ord_less_eq_set_b @ A2 @ B ) ) ).
% equalityD1
thf(fact_192_equalityD1,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_193_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [X: set_a] :
( ( member_set_a @ X @ A3 )
=> ( member_set_a @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_194_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B3: set_b] :
! [X: b] :
( ( member_b @ X @ A3 )
=> ( member_b @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_195_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X: a] :
( ( member_a @ X @ A3 )
=> ( member_a @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_196_equalityE,axiom,
! [A2: set_b,B: set_b] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_b @ A2 @ B )
=> ~ ( ord_less_eq_set_b @ B @ A2 ) ) ) ).
% equalityE
thf(fact_197_equalityE,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_198_subsetD,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_199_subsetD,axiom,
! [A2: set_b,B: set_b,C2: b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( member_b @ C2 @ A2 )
=> ( member_b @ C2 @ B ) ) ) ).
% subsetD
thf(fact_200_subsetD,axiom,
! [A2: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_201_in__mono,axiom,
! [A2: set_set_a,B: set_set_a,X4: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ X4 @ A2 )
=> ( member_set_a @ X4 @ B ) ) ) ).
% in_mono
thf(fact_202_in__mono,axiom,
! [A2: set_b,B: set_b,X4: b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( member_b @ X4 @ A2 )
=> ( member_b @ X4 @ B ) ) ) ).
% in_mono
thf(fact_203_in__mono,axiom,
! [A2: set_a,B: set_a,X4: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ X4 @ A2 )
=> ( member_a @ X4 @ B ) ) ) ).
% in_mono
thf(fact_204_inf_OcoboundedI2,axiom,
! [B2: set_b,C2: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B2 @ C2 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_205_inf_OcoboundedI2,axiom,
! [B2: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_206_inf_OcoboundedI2,axiom,
! [B2: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_207_inf_OcoboundedI1,axiom,
! [A: set_b,C2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A @ C2 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_208_inf_OcoboundedI1,axiom,
! [A: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_209_inf_OcoboundedI1,axiom,
! [A: set_a,C2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_210_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_b
= ( ^ [B4: set_b,A4: set_b] :
( ( inf_inf_set_b @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_211_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_212_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_213_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B4: set_b] :
( ( inf_inf_set_b @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_214_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_215_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( inf_inf_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_216_inf_Ocobounded2,axiom,
! [A: set_b,B2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_217_inf_Ocobounded2,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_218_inf_Ocobounded2,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_219_inf_Ocobounded1,axiom,
! [A: set_b,B2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_220_inf_Ocobounded1,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_221_inf_Ocobounded1,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_222_inf_Oorder__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B4: set_b] :
( A4
= ( inf_inf_set_b @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_223_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( A4
= ( inf_inf_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_224_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_225_inf__greatest,axiom,
! [X4: set_b,Y2: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y2 )
=> ( ( ord_less_eq_set_b @ X4 @ Z2 )
=> ( ord_less_eq_set_b @ X4 @ ( inf_inf_set_b @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_226_inf__greatest,axiom,
! [X4: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ( ord_less_eq_nat @ X4 @ Z2 )
=> ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_227_inf__greatest,axiom,
! [X4: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ( ord_less_eq_set_a @ X4 @ Z2 )
=> ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_228_inf_OboundedI,axiom,
! [A: set_b,B2: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ A @ B2 )
=> ( ( ord_less_eq_set_b @ A @ C2 )
=> ( ord_less_eq_set_b @ A @ ( inf_inf_set_b @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_229_inf_OboundedI,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_230_inf_OboundedI,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_231_inf_OboundedE,axiom,
! [A: set_b,B2: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ A @ ( inf_inf_set_b @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_set_b @ A @ B2 )
=> ~ ( ord_less_eq_set_b @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_232_inf_OboundedE,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A @ B2 )
=> ~ ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_233_inf_OboundedE,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B2 )
=> ~ ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_234_inf__absorb2,axiom,
! [Y2: set_b,X4: set_b] :
( ( ord_less_eq_set_b @ Y2 @ X4 )
=> ( ( inf_inf_set_b @ X4 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_235_inf__absorb2,axiom,
! [Y2: nat,X4: nat] :
( ( ord_less_eq_nat @ Y2 @ X4 )
=> ( ( inf_inf_nat @ X4 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_236_inf__absorb2,axiom,
! [Y2: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X4 )
=> ( ( inf_inf_set_a @ X4 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_237_inf__absorb1,axiom,
! [X4: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y2 )
=> ( ( inf_inf_set_b @ X4 @ Y2 )
= X4 ) ) ).
% inf_absorb1
thf(fact_238_inf__absorb1,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ( inf_inf_nat @ X4 @ Y2 )
= X4 ) ) ).
% inf_absorb1
thf(fact_239_inf__absorb1,axiom,
! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ( inf_inf_set_a @ X4 @ Y2 )
= X4 ) ) ).
% inf_absorb1
thf(fact_240_inf_Oabsorb2,axiom,
! [B2: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B2 @ A )
=> ( ( inf_inf_set_b @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_241_inf_Oabsorb2,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( inf_inf_nat @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_242_inf_Oabsorb2,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( inf_inf_set_a @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_243_inf_Oabsorb1,axiom,
! [A: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A @ B2 )
=> ( ( inf_inf_set_b @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_244_inf_Oabsorb1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( inf_inf_nat @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_245_inf_Oabsorb1,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( inf_inf_set_a @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_246_le__iff__inf,axiom,
( ord_less_eq_set_b
= ( ^ [X: set_b,Y: set_b] :
( ( inf_inf_set_b @ X @ Y )
= X ) ) ) ).
% le_iff_inf
thf(fact_247_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ Y )
= X ) ) ) ).
% le_iff_inf
thf(fact_248_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ Y )
= X ) ) ) ).
% le_iff_inf
thf(fact_249_inf__unique,axiom,
! [F: set_b > set_b > set_b,X4: set_b,Y2: set_b] :
( ! [X3: set_b,Y4: set_b] : ( ord_less_eq_set_b @ ( F @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: set_b,Y4: set_b] : ( ord_less_eq_set_b @ ( F @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: set_b,Y4: set_b,Z4: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y4 )
=> ( ( ord_less_eq_set_b @ X3 @ Z4 )
=> ( ord_less_eq_set_b @ X3 @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_inf_set_b @ X4 @ Y2 )
= ( F @ X4 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_250_inf__unique,axiom,
! [F: nat > nat > nat,X4: nat,Y2: nat] :
( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: nat,Y4: nat,Z4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_nat @ X3 @ Z4 )
=> ( ord_less_eq_nat @ X3 @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_inf_nat @ X4 @ Y2 )
= ( F @ X4 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_251_inf__unique,axiom,
! [F: set_a > set_a > set_a,X4: set_a,Y2: set_a] :
( ! [X3: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: set_a,Y4: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ( ord_less_eq_set_a @ X3 @ Z4 )
=> ( ord_less_eq_set_a @ X3 @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_inf_set_a @ X4 @ Y2 )
= ( F @ X4 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_252_inf_OorderI,axiom,
! [A: set_b,B2: set_b] :
( ( A
= ( inf_inf_set_b @ A @ B2 ) )
=> ( ord_less_eq_set_b @ A @ B2 ) ) ).
% inf.orderI
thf(fact_253_inf_OorderI,axiom,
! [A: nat,B2: nat] :
( ( A
= ( inf_inf_nat @ A @ B2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% inf.orderI
thf(fact_254_inf_OorderI,axiom,
! [A: set_a,B2: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B2 ) )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% inf.orderI
thf(fact_255_inf_OorderE,axiom,
! [A: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A @ B2 )
=> ( A
= ( inf_inf_set_b @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_256_inf_OorderE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( A
= ( inf_inf_nat @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_257_inf_OorderE,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( A
= ( inf_inf_set_a @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_258_le__infI2,axiom,
! [B2: set_b,X4: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B2 @ X4 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B2 ) @ X4 ) ) ).
% le_infI2
thf(fact_259_le__infI2,axiom,
! [B2: nat,X4: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ X4 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ X4 ) ) ).
% le_infI2
thf(fact_260_le__infI2,axiom,
! [B2: set_a,X4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ X4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ X4 ) ) ).
% le_infI2
thf(fact_261_le__infI1,axiom,
! [A: set_b,X4: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A @ X4 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B2 ) @ X4 ) ) ).
% le_infI1
thf(fact_262_le__infI1,axiom,
! [A: nat,X4: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ X4 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ X4 ) ) ).
% le_infI1
thf(fact_263_le__infI1,axiom,
! [A: set_a,X4: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ X4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ X4 ) ) ).
% le_infI1
thf(fact_264_inf__mono,axiom,
! [A: set_b,C2: set_b,B2: set_b,D: set_b] :
( ( ord_less_eq_set_b @ A @ C2 )
=> ( ( ord_less_eq_set_b @ B2 @ D )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B2 ) @ ( inf_inf_set_b @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_265_inf__mono,axiom,
! [A: nat,C2: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ ( inf_inf_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_266_inf__mono,axiom,
! [A: set_a,C2: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_267_le__infI,axiom,
! [X4: set_b,A: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ X4 @ A )
=> ( ( ord_less_eq_set_b @ X4 @ B2 )
=> ( ord_less_eq_set_b @ X4 @ ( inf_inf_set_b @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_268_le__infI,axiom,
! [X4: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X4 @ A )
=> ( ( ord_less_eq_nat @ X4 @ B2 )
=> ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_269_le__infI,axiom,
! [X4: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X4 @ A )
=> ( ( ord_less_eq_set_a @ X4 @ B2 )
=> ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_270_le__infE,axiom,
! [X4: set_b,A: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ X4 @ ( inf_inf_set_b @ A @ B2 ) )
=> ~ ( ( ord_less_eq_set_b @ X4 @ A )
=> ~ ( ord_less_eq_set_b @ X4 @ B2 ) ) ) ).
% le_infE
thf(fact_271_le__infE,axiom,
! [X4: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ A @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X4 @ A )
=> ~ ( ord_less_eq_nat @ X4 @ B2 ) ) ) ).
% le_infE
thf(fact_272_le__infE,axiom,
! [X4: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ A @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X4 @ A )
=> ~ ( ord_less_eq_set_a @ X4 @ B2 ) ) ) ).
% le_infE
thf(fact_273_inf__le2,axiom,
! [X4: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X4 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_274_inf__le2,axiom,
! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_275_inf__le2,axiom,
! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_276_inf__le1,axiom,
! [X4: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X4 @ Y2 ) @ X4 ) ).
% inf_le1
thf(fact_277_inf__le1,axiom,
! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y2 ) @ X4 ) ).
% inf_le1
thf(fact_278_inf__le1,axiom,
! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y2 ) @ X4 ) ).
% inf_le1
thf(fact_279_inf__sup__ord_I1_J,axiom,
! [X4: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X4 @ Y2 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_280_inf__sup__ord_I1_J,axiom,
! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y2 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_281_inf__sup__ord_I1_J,axiom,
! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y2 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_282_inf__sup__ord_I2_J,axiom,
! [X4: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X4 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_283_inf__sup__ord_I2_J,axiom,
! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_284_inf__sup__ord_I2_J,axiom,
! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_285_inf__sup__ord_I4_J,axiom,
! [Y2: set_b,X4: set_b] : ( ord_less_eq_set_b @ Y2 @ ( sup_sup_set_b @ X4 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_286_inf__sup__ord_I4_J,axiom,
! [Y2: nat,X4: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X4 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_287_inf__sup__ord_I4_J,axiom,
! [Y2: set_a,X4: set_a] : ( ord_less_eq_set_a @ Y2 @ ( sup_sup_set_a @ X4 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_288_inf__sup__ord_I3_J,axiom,
! [X4: set_b,Y2: set_b] : ( ord_less_eq_set_b @ X4 @ ( sup_sup_set_b @ X4 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_289_inf__sup__ord_I3_J,axiom,
! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ X4 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_290_inf__sup__ord_I3_J,axiom,
! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X4 @ ( sup_sup_set_a @ X4 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_291_le__supE,axiom,
! [A: set_b,B2: set_b,X4: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ A @ B2 ) @ X4 )
=> ~ ( ( ord_less_eq_set_b @ A @ X4 )
=> ~ ( ord_less_eq_set_b @ B2 @ X4 ) ) ) ).
% le_supE
thf(fact_292_le__supE,axiom,
! [A: nat,B2: nat,X4: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X4 )
=> ~ ( ( ord_less_eq_nat @ A @ X4 )
=> ~ ( ord_less_eq_nat @ B2 @ X4 ) ) ) ).
% le_supE
thf(fact_293_le__supE,axiom,
! [A: set_a,B2: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ X4 )
=> ~ ( ( ord_less_eq_set_a @ A @ X4 )
=> ~ ( ord_less_eq_set_a @ B2 @ X4 ) ) ) ).
% le_supE
thf(fact_294_le__supI,axiom,
! [A: set_b,X4: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A @ X4 )
=> ( ( ord_less_eq_set_b @ B2 @ X4 )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ A @ B2 ) @ X4 ) ) ) ).
% le_supI
thf(fact_295_le__supI,axiom,
! [A: nat,X4: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ X4 )
=> ( ( ord_less_eq_nat @ B2 @ X4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X4 ) ) ) ).
% le_supI
thf(fact_296_le__supI,axiom,
! [A: set_a,X4: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ X4 )
=> ( ( ord_less_eq_set_a @ B2 @ X4 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ X4 ) ) ) ).
% le_supI
thf(fact_297_sup__ge1,axiom,
! [X4: set_b,Y2: set_b] : ( ord_less_eq_set_b @ X4 @ ( sup_sup_set_b @ X4 @ Y2 ) ) ).
% sup_ge1
thf(fact_298_sup__ge1,axiom,
! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ X4 @ Y2 ) ) ).
% sup_ge1
thf(fact_299_sup__ge1,axiom,
! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X4 @ ( sup_sup_set_a @ X4 @ Y2 ) ) ).
% sup_ge1
thf(fact_300_sup__ge2,axiom,
! [Y2: set_b,X4: set_b] : ( ord_less_eq_set_b @ Y2 @ ( sup_sup_set_b @ X4 @ Y2 ) ) ).
% sup_ge2
thf(fact_301_sup__ge2,axiom,
! [Y2: nat,X4: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X4 @ Y2 ) ) ).
% sup_ge2
thf(fact_302_sup__ge2,axiom,
! [Y2: set_a,X4: set_a] : ( ord_less_eq_set_a @ Y2 @ ( sup_sup_set_a @ X4 @ Y2 ) ) ).
% sup_ge2
thf(fact_303_le__supI1,axiom,
! [X4: set_b,A: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ X4 @ A )
=> ( ord_less_eq_set_b @ X4 @ ( sup_sup_set_b @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_304_le__supI1,axiom,
! [X4: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X4 @ A )
=> ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_305_le__supI1,axiom,
! [X4: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X4 @ A )
=> ( ord_less_eq_set_a @ X4 @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_306_le__supI2,axiom,
! [X4: set_b,B2: set_b,A: set_b] :
( ( ord_less_eq_set_b @ X4 @ B2 )
=> ( ord_less_eq_set_b @ X4 @ ( sup_sup_set_b @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_307_le__supI2,axiom,
! [X4: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ X4 @ B2 )
=> ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_308_le__supI2,axiom,
! [X4: set_a,B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X4 @ B2 )
=> ( ord_less_eq_set_a @ X4 @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_309_sup_Omono,axiom,
! [C2: set_b,A: set_b,D: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ C2 @ A )
=> ( ( ord_less_eq_set_b @ D @ B2 )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ C2 @ D ) @ ( sup_sup_set_b @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_310_sup_Omono,axiom,
! [C2: nat,A: nat,D: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ( ord_less_eq_nat @ D @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D ) @ ( sup_sup_nat @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_311_sup_Omono,axiom,
! [C2: set_a,A: set_a,D: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ D @ B2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C2 @ D ) @ ( sup_sup_set_a @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_312_sup__mono,axiom,
! [A: set_b,C2: set_b,B2: set_b,D: set_b] :
( ( ord_less_eq_set_b @ A @ C2 )
=> ( ( ord_less_eq_set_b @ B2 @ D )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ A @ B2 ) @ ( sup_sup_set_b @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_313_sup__mono,axiom,
! [A: nat,C2: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ ( sup_sup_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_314_sup__mono,axiom,
! [A: set_a,C2: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_315_sup__least,axiom,
! [Y2: set_b,X4: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ Y2 @ X4 )
=> ( ( ord_less_eq_set_b @ Z2 @ X4 )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ Y2 @ Z2 ) @ X4 ) ) ) ).
% sup_least
thf(fact_316_sup__least,axiom,
! [Y2: nat,X4: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y2 @ X4 )
=> ( ( ord_less_eq_nat @ Z2 @ X4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y2 @ Z2 ) @ X4 ) ) ) ).
% sup_least
thf(fact_317_sup__least,axiom,
! [Y2: set_a,X4: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X4 )
=> ( ( ord_less_eq_set_a @ Z2 @ X4 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y2 @ Z2 ) @ X4 ) ) ) ).
% sup_least
thf(fact_318_le__iff__sup,axiom,
( ord_less_eq_set_b
= ( ^ [X: set_b,Y: set_b] :
( ( sup_sup_set_b @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_319_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_320_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_321_sup_OorderE,axiom,
! [B2: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B2 @ A )
=> ( A
= ( sup_sup_set_b @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_322_sup_OorderE,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( A
= ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_323_sup_OorderE,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( A
= ( sup_sup_set_a @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_324_sup_OorderI,axiom,
! [A: set_b,B2: set_b] :
( ( A
= ( sup_sup_set_b @ A @ B2 ) )
=> ( ord_less_eq_set_b @ B2 @ A ) ) ).
% sup.orderI
thf(fact_325_sup_OorderI,axiom,
! [A: nat,B2: nat] :
( ( A
= ( sup_sup_nat @ A @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A ) ) ).
% sup.orderI
thf(fact_326_sup_OorderI,axiom,
! [A: set_a,B2: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B2 ) )
=> ( ord_less_eq_set_a @ B2 @ A ) ) ).
% sup.orderI
thf(fact_327_sup__unique,axiom,
! [F: set_b > set_b > set_b,X4: set_b,Y2: set_b] :
( ! [X3: set_b,Y4: set_b] : ( ord_less_eq_set_b @ X3 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: set_b,Y4: set_b] : ( ord_less_eq_set_b @ Y4 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: set_b,Y4: set_b,Z4: set_b] :
( ( ord_less_eq_set_b @ Y4 @ X3 )
=> ( ( ord_less_eq_set_b @ Z4 @ X3 )
=> ( ord_less_eq_set_b @ ( F @ Y4 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup_set_b @ X4 @ Y2 )
= ( F @ X4 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_328_sup__unique,axiom,
! [F: nat > nat > nat,X4: nat,Y2: nat] :
( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: nat,Y4: nat,Z4: nat] :
( ( ord_less_eq_nat @ Y4 @ X3 )
=> ( ( ord_less_eq_nat @ Z4 @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y4 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X4 @ Y2 )
= ( F @ X4 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_329_sup__unique,axiom,
! [F: set_a > set_a > set_a,X4: set_a,Y2: set_a] :
( ! [X3: set_a,Y4: set_a] : ( ord_less_eq_set_a @ X3 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: set_a,Y4: set_a] : ( ord_less_eq_set_a @ Y4 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: set_a,Y4: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X3 )
=> ( ( ord_less_eq_set_a @ Z4 @ X3 )
=> ( ord_less_eq_set_a @ ( F @ Y4 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup_set_a @ X4 @ Y2 )
= ( F @ X4 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_330_sup_Oabsorb1,axiom,
! [B2: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B2 @ A )
=> ( ( sup_sup_set_b @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_331_sup_Oabsorb1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( sup_sup_nat @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_332_sup_Oabsorb1,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( sup_sup_set_a @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_333_sup_Oabsorb2,axiom,
! [A: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A @ B2 )
=> ( ( sup_sup_set_b @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_334_sup_Oabsorb2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( sup_sup_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_335_sup_Oabsorb2,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( sup_sup_set_a @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_336_sup__absorb1,axiom,
! [Y2: set_b,X4: set_b] :
( ( ord_less_eq_set_b @ Y2 @ X4 )
=> ( ( sup_sup_set_b @ X4 @ Y2 )
= X4 ) ) ).
% sup_absorb1
thf(fact_337_sup__absorb1,axiom,
! [Y2: nat,X4: nat] :
( ( ord_less_eq_nat @ Y2 @ X4 )
=> ( ( sup_sup_nat @ X4 @ Y2 )
= X4 ) ) ).
% sup_absorb1
thf(fact_338_sup__absorb1,axiom,
! [Y2: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X4 )
=> ( ( sup_sup_set_a @ X4 @ Y2 )
= X4 ) ) ).
% sup_absorb1
thf(fact_339_sup__absorb2,axiom,
! [X4: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y2 )
=> ( ( sup_sup_set_b @ X4 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_340_sup__absorb2,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ( sup_sup_nat @ X4 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_341_sup__absorb2,axiom,
! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ( sup_sup_set_a @ X4 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_342_sup_OboundedE,axiom,
! [B2: set_b,C2: set_b,A: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ B2 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_set_b @ B2 @ A )
=> ~ ( ord_less_eq_set_b @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_343_sup_OboundedE,axiom,
! [B2: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_nat @ B2 @ A )
=> ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_344_sup_OboundedE,axiom,
! [B2: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B2 @ A )
=> ~ ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_345_sup_OboundedI,axiom,
! [B2: set_b,A: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ B2 @ A )
=> ( ( ord_less_eq_set_b @ C2 @ A )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ B2 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_346_sup_OboundedI,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_347_sup_OboundedI,axiom,
! [B2: set_a,A: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_348_sup_Oorder__iff,axiom,
( ord_less_eq_set_b
= ( ^ [B4: set_b,A4: set_b] :
( A4
= ( sup_sup_set_b @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_349_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_350_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_351_sup_Ocobounded1,axiom,
! [A: set_b,B2: set_b] : ( ord_less_eq_set_b @ A @ ( sup_sup_set_b @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_352_sup_Ocobounded1,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_353_sup_Ocobounded1,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_354_sup_Ocobounded2,axiom,
! [B2: set_b,A: set_b] : ( ord_less_eq_set_b @ B2 @ ( sup_sup_set_b @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_355_sup_Ocobounded2,axiom,
! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_356_sup_Ocobounded2,axiom,
! [B2: set_a,A: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_357_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_b
= ( ^ [B4: set_b,A4: set_b] :
( ( sup_sup_set_b @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_358_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_359_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_360_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B4: set_b] :
( ( sup_sup_set_b @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_361_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_362_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_363_sup_OcoboundedI1,axiom,
! [C2: set_b,A: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ C2 @ A )
=> ( ord_less_eq_set_b @ C2 @ ( sup_sup_set_b @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_364_sup_OcoboundedI1,axiom,
! [C2: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_365_sup_OcoboundedI1,axiom,
! [C2: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_366_sup_OcoboundedI2,axiom,
! [C2: set_b,B2: set_b,A: set_b] :
( ( ord_less_eq_set_b @ C2 @ B2 )
=> ( ord_less_eq_set_b @ C2 @ ( sup_sup_set_b @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_367_sup_OcoboundedI2,axiom,
! [C2: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_368_sup_OcoboundedI2,axiom,
! [C2: set_a,B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_369_Int__Collect__mono,axiom,
! [A2: set_set_a,B: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_370_Int__Collect__mono,axiom,
! [A2: set_list_a,B: set_list_a,P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ A2 @ B )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ ( collect_list_a @ P ) ) @ ( inf_inf_set_list_a @ B @ ( collect_list_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_371_Int__Collect__mono,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_372_Int__Collect__mono,axiom,
! [A2: set_b,B: set_b,P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ ( collect_b @ P ) ) @ ( inf_inf_set_b @ B @ ( collect_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_373_Int__Collect__mono,axiom,
! [A2: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_374_Int__greatest,axiom,
! [C: set_b,A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ C @ A2 )
=> ( ( ord_less_eq_set_b @ C @ B )
=> ( ord_less_eq_set_b @ C @ ( inf_inf_set_b @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_375_Int__greatest,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_376_Int__absorb2,axiom,
! [A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( inf_inf_set_b @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_377_Int__absorb2,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( inf_inf_set_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_378_Int__absorb1,axiom,
! [B: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B @ A2 )
=> ( ( inf_inf_set_b @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_379_Int__absorb1,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_380_Int__lower2,axiom,
! [A2: set_b,B: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_381_Int__lower2,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_382_Int__lower1,axiom,
! [A2: set_b,B: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_383_Int__lower1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_384_Int__mono,axiom,
! [A2: set_b,C: set_b,B: set_b,D2: set_b] :
( ( ord_less_eq_set_b @ A2 @ C )
=> ( ( ord_less_eq_set_b @ B @ D2 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B ) @ ( inf_inf_set_b @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_385_Int__mono,axiom,
! [A2: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_386_Un__mono,axiom,
! [A2: set_b,C: set_b,B: set_b,D2: set_b] :
( ( ord_less_eq_set_b @ A2 @ C )
=> ( ( ord_less_eq_set_b @ B @ D2 )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ A2 @ B ) @ ( sup_sup_set_b @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_387_Un__mono,axiom,
! [A2: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_388_Un__least,axiom,
! [A2: set_b,C: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A2 @ C )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_389_Un__least,axiom,
! [A2: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_390_Un__upper1,axiom,
! [A2: set_b,B: set_b] : ( ord_less_eq_set_b @ A2 @ ( sup_sup_set_b @ A2 @ B ) ) ).
% Un_upper1
thf(fact_391_Un__upper1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper1
thf(fact_392_Un__upper2,axiom,
! [B: set_b,A2: set_b] : ( ord_less_eq_set_b @ B @ ( sup_sup_set_b @ A2 @ B ) ) ).
% Un_upper2
thf(fact_393_Un__upper2,axiom,
! [B: set_a,A2: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper2
thf(fact_394_Un__absorb1,axiom,
! [A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( sup_sup_set_b @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_395_Un__absorb1,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( sup_sup_set_a @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_396_Un__absorb2,axiom,
! [B: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B @ A2 )
=> ( ( sup_sup_set_b @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_397_Un__absorb2,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_398_subset__UnE,axiom,
! [C: set_b,A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ C @ ( sup_sup_set_b @ A2 @ B ) )
=> ~ ! [A5: set_b] :
( ( ord_less_eq_set_b @ A5 @ A2 )
=> ! [B5: set_b] :
( ( ord_less_eq_set_b @ B5 @ B )
=> ( C
!= ( sup_sup_set_b @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_399_subset__UnE,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B ) )
=> ~ ! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ A2 )
=> ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ B )
=> ( C
!= ( sup_sup_set_a @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_400_subset__Un__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B3: set_b] :
( ( sup_sup_set_b @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_401_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( sup_sup_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_402_inf__set__def,axiom,
( inf_inf_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( collect_set_a
@ ( inf_inf_set_a_o
@ ^ [X: set_a] : ( member_set_a @ X @ A3 )
@ ^ [X: set_a] : ( member_set_a @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_403_inf__set__def,axiom,
( inf_inf_set_list_a
= ( ^ [A3: set_list_a,B3: set_list_a] :
( collect_list_a
@ ( inf_inf_list_a_o
@ ^ [X: list_a] : ( member_list_a @ X @ A3 )
@ ^ [X: list_a] : ( member_list_a @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_404_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_405_inf__set__def,axiom,
( inf_inf_set_b
= ( ^ [A3: set_b,B3: set_b] :
( collect_b
@ ( inf_inf_b_o
@ ^ [X: b] : ( member_b @ X @ A3 )
@ ^ [X: b] : ( member_b @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_406_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B3: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X: a] : ( member_a @ X @ A3 )
@ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_407_sup__set__def,axiom,
( sup_sup_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( collect_set_a
@ ( sup_sup_set_a_o
@ ^ [X: set_a] : ( member_set_a @ X @ A3 )
@ ^ [X: set_a] : ( member_set_a @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_408_sup__set__def,axiom,
( sup_sup_set_list_a
= ( ^ [A3: set_list_a,B3: set_list_a] :
( collect_list_a
@ ( sup_sup_list_a_o
@ ^ [X: list_a] : ( member_list_a @ X @ A3 )
@ ^ [X: list_a] : ( member_list_a @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_409_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_410_sup__set__def,axiom,
( sup_sup_set_b
= ( ^ [A3: set_b,B3: set_b] :
( collect_b
@ ( sup_sup_b_o
@ ^ [X: b] : ( member_b @ X @ A3 )
@ ^ [X: b] : ( member_b @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_411_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X: a] : ( member_a @ X @ A3 )
@ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_412_distrib__inf__le,axiom,
! [X4: set_b,Y2: set_b,Z2: set_b] : ( ord_less_eq_set_b @ ( sup_sup_set_b @ ( inf_inf_set_b @ X4 @ Y2 ) @ ( inf_inf_set_b @ X4 @ Z2 ) ) @ ( inf_inf_set_b @ X4 @ ( sup_sup_set_b @ Y2 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_413_distrib__inf__le,axiom,
! [X4: nat,Y2: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X4 @ Y2 ) @ ( inf_inf_nat @ X4 @ Z2 ) ) @ ( inf_inf_nat @ X4 @ ( sup_sup_nat @ Y2 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_414_distrib__inf__le,axiom,
! [X4: set_a,Y2: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X4 @ Y2 ) @ ( inf_inf_set_a @ X4 @ Z2 ) ) @ ( inf_inf_set_a @ X4 @ ( sup_sup_set_a @ Y2 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_415_distrib__sup__le,axiom,
! [X4: set_b,Y2: set_b,Z2: set_b] : ( ord_less_eq_set_b @ ( sup_sup_set_b @ X4 @ ( inf_inf_set_b @ Y2 @ Z2 ) ) @ ( inf_inf_set_b @ ( sup_sup_set_b @ X4 @ Y2 ) @ ( sup_sup_set_b @ X4 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_416_distrib__sup__le,axiom,
! [X4: nat,Y2: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X4 @ ( inf_inf_nat @ Y2 @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X4 @ Y2 ) @ ( sup_sup_nat @ X4 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_417_distrib__sup__le,axiom,
! [X4: set_a,Y2: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X4 @ ( inf_inf_set_a @ Y2 @ Z2 ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X4 @ Y2 ) @ ( sup_sup_set_a @ X4 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_418_Un__Int__assoc__eq,axiom,
! [A2: set_b,B: set_b,C: set_b] :
( ( ( sup_sup_set_b @ ( inf_inf_set_b @ A2 @ B ) @ C )
= ( inf_inf_set_b @ A2 @ ( sup_sup_set_b @ B @ C ) ) )
= ( ord_less_eq_set_b @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_419_Un__Int__assoc__eq,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) )
= ( ord_less_eq_set_a @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_420_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X: set_a] : ( member_set_a @ X @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_421_ex__in__conv,axiom,
! [A2: set_b] :
( ( ? [X: b] : ( member_b @ X @ A2 ) )
= ( A2 != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_422_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X: a] : ( member_a @ X @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_423_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y4: set_a] :
~ ( member_set_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_424_equals0I,axiom,
! [A2: set_b] :
( ! [Y4: b] :
~ ( member_b @ Y4 @ A2 )
=> ( A2 = bot_bot_set_b ) ) ).
% equals0I
thf(fact_425_equals0I,axiom,
! [A2: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_426_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_427_equals0D,axiom,
! [A2: set_b,A: b] :
( ( A2 = bot_bot_set_b )
=> ~ ( member_b @ A @ A2 ) ) ).
% equals0D
thf(fact_428_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_429_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_430_emptyE,axiom,
! [A: b] :
~ ( member_b @ A @ bot_bot_set_b ) ).
% emptyE
thf(fact_431_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_432_inf__left__commute,axiom,
! [X4: set_b,Y2: set_b,Z2: set_b] :
( ( inf_inf_set_b @ X4 @ ( inf_inf_set_b @ Y2 @ Z2 ) )
= ( inf_inf_set_b @ Y2 @ ( inf_inf_set_b @ X4 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_433_inf__left__commute,axiom,
! [X4: set_a,Y2: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y2 @ Z2 ) )
= ( inf_inf_set_a @ Y2 @ ( inf_inf_set_a @ X4 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_434_inf_Oleft__commute,axiom,
! [B2: set_b,A: set_b,C2: set_b] :
( ( inf_inf_set_b @ B2 @ ( inf_inf_set_b @ A @ C2 ) )
= ( inf_inf_set_b @ A @ ( inf_inf_set_b @ B2 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_435_inf_Oleft__commute,axiom,
! [B2: set_a,A: set_a,C2: set_a] :
( ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A @ C2 ) )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_436_inf__commute,axiom,
( inf_inf_set_b
= ( ^ [X: set_b,Y: set_b] : ( inf_inf_set_b @ Y @ X ) ) ) ).
% inf_commute
thf(fact_437_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X: set_a,Y: set_a] : ( inf_inf_set_a @ Y @ X ) ) ) ).
% inf_commute
thf(fact_438_inf_Ocommute,axiom,
( inf_inf_set_b
= ( ^ [A4: set_b,B4: set_b] : ( inf_inf_set_b @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_439_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_440_inf__assoc,axiom,
! [X4: set_b,Y2: set_b,Z2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ X4 @ Y2 ) @ Z2 )
= ( inf_inf_set_b @ X4 @ ( inf_inf_set_b @ Y2 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_441_inf__assoc,axiom,
! [X4: set_a,Y2: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y2 ) @ Z2 )
= ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y2 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_442_inf_Oassoc,axiom,
! [A: set_b,B2: set_b,C2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A @ B2 ) @ C2 )
= ( inf_inf_set_b @ A @ ( inf_inf_set_b @ B2 @ C2 ) ) ) ).
% inf.assoc
thf(fact_443_inf_Oassoc,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% inf.assoc
thf(fact_444_inf__sup__aci_I1_J,axiom,
( inf_inf_set_b
= ( ^ [X: set_b,Y: set_b] : ( inf_inf_set_b @ Y @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_445_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X: set_a,Y: set_a] : ( inf_inf_set_a @ Y @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_446_inf__sup__aci_I2_J,axiom,
! [X4: set_b,Y2: set_b,Z2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ X4 @ Y2 ) @ Z2 )
= ( inf_inf_set_b @ X4 @ ( inf_inf_set_b @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_447_inf__sup__aci_I2_J,axiom,
! [X4: set_a,Y2: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y2 ) @ Z2 )
= ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_448_inf__sup__aci_I3_J,axiom,
! [X4: set_b,Y2: set_b,Z2: set_b] :
( ( inf_inf_set_b @ X4 @ ( inf_inf_set_b @ Y2 @ Z2 ) )
= ( inf_inf_set_b @ Y2 @ ( inf_inf_set_b @ X4 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_449_inf__sup__aci_I3_J,axiom,
! [X4: set_a,Y2: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y2 @ Z2 ) )
= ( inf_inf_set_a @ Y2 @ ( inf_inf_set_a @ X4 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_450_inf__sup__aci_I4_J,axiom,
! [X4: set_b,Y2: set_b] :
( ( inf_inf_set_b @ X4 @ ( inf_inf_set_b @ X4 @ Y2 ) )
= ( inf_inf_set_b @ X4 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_451_inf__sup__aci_I4_J,axiom,
! [X4: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y2 ) )
= ( inf_inf_set_a @ X4 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_452_inf__sup__aci_I8_J,axiom,
! [X4: set_b,Y2: set_b] :
( ( sup_sup_set_b @ X4 @ ( sup_sup_set_b @ X4 @ Y2 ) )
= ( sup_sup_set_b @ X4 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_453_inf__sup__aci_I8_J,axiom,
! [X4: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X4 @ ( sup_sup_set_a @ X4 @ Y2 ) )
= ( sup_sup_set_a @ X4 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_454_inf__sup__aci_I7_J,axiom,
! [X4: set_b,Y2: set_b,Z2: set_b] :
( ( sup_sup_set_b @ X4 @ ( sup_sup_set_b @ Y2 @ Z2 ) )
= ( sup_sup_set_b @ Y2 @ ( sup_sup_set_b @ X4 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_455_inf__sup__aci_I7_J,axiom,
! [X4: set_a,Y2: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X4 @ ( sup_sup_set_a @ Y2 @ Z2 ) )
= ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X4 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_456_inf__sup__aci_I6_J,axiom,
! [X4: set_b,Y2: set_b,Z2: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ X4 @ Y2 ) @ Z2 )
= ( sup_sup_set_b @ X4 @ ( sup_sup_set_b @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_457_inf__sup__aci_I6_J,axiom,
! [X4: set_a,Y2: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X4 @ Y2 ) @ Z2 )
= ( sup_sup_set_a @ X4 @ ( sup_sup_set_a @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_458_inf__sup__aci_I5_J,axiom,
( sup_sup_set_b
= ( ^ [X: set_b,Y: set_b] : ( sup_sup_set_b @ Y @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_459_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_460_sup_Oassoc,axiom,
! [A: set_b,B2: set_b,C2: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ A @ B2 ) @ C2 )
= ( sup_sup_set_b @ A @ ( sup_sup_set_b @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_461_sup_Oassoc,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B2 ) @ C2 )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_462_sup__assoc,axiom,
! [X4: set_b,Y2: set_b,Z2: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ X4 @ Y2 ) @ Z2 )
= ( sup_sup_set_b @ X4 @ ( sup_sup_set_b @ Y2 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_463_sup__assoc,axiom,
! [X4: set_a,Y2: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X4 @ Y2 ) @ Z2 )
= ( sup_sup_set_a @ X4 @ ( sup_sup_set_a @ Y2 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_464_sup_Ocommute,axiom,
( sup_sup_set_b
= ( ^ [A4: set_b,B4: set_b] : ( sup_sup_set_b @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_465_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B4: set_a] : ( sup_sup_set_a @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_466_sup__commute,axiom,
( sup_sup_set_b
= ( ^ [X: set_b,Y: set_b] : ( sup_sup_set_b @ Y @ X ) ) ) ).
% sup_commute
thf(fact_467_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X ) ) ) ).
% sup_commute
thf(fact_468_sup_Oleft__commute,axiom,
! [B2: set_b,A: set_b,C2: set_b] :
( ( sup_sup_set_b @ B2 @ ( sup_sup_set_b @ A @ C2 ) )
= ( sup_sup_set_b @ A @ ( sup_sup_set_b @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_469_sup_Oleft__commute,axiom,
! [B2: set_a,A: set_a,C2: set_a] :
( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A @ C2 ) )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_470_sup__left__commute,axiom,
! [X4: set_b,Y2: set_b,Z2: set_b] :
( ( sup_sup_set_b @ X4 @ ( sup_sup_set_b @ Y2 @ Z2 ) )
= ( sup_sup_set_b @ Y2 @ ( sup_sup_set_b @ X4 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_471_sup__left__commute,axiom,
! [X4: set_a,Y2: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X4 @ ( sup_sup_set_a @ Y2 @ Z2 ) )
= ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X4 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_472_Int__left__commute,axiom,
! [A2: set_b,B: set_b,C: set_b] :
( ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ B @ C ) )
= ( inf_inf_set_b @ B @ ( inf_inf_set_b @ A2 @ C ) ) ) ).
% Int_left_commute
thf(fact_473_Int__left__commute,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A2 @ C ) ) ) ).
% Int_left_commute
thf(fact_474_Int__left__absorb,axiom,
! [A2: set_b,B: set_b] :
( ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ A2 @ B ) )
= ( inf_inf_set_b @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_475_Int__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_476_Int__commute,axiom,
( inf_inf_set_b
= ( ^ [A3: set_b,B3: set_b] : ( inf_inf_set_b @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_477_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_478_Int__absorb,axiom,
! [A2: set_b] :
( ( inf_inf_set_b @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_479_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_480_Int__assoc,axiom,
! [A2: set_b,B: set_b,C: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A2 @ B ) @ C )
= ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ B @ C ) ) ) ).
% Int_assoc
thf(fact_481_Int__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_assoc
thf(fact_482_IntD2,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ( member_set_a @ C2 @ B ) ) ).
% IntD2
thf(fact_483_IntD2,axiom,
! [C2: b,A2: set_b,B: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A2 @ B ) )
=> ( member_b @ C2 @ B ) ) ).
% IntD2
thf(fact_484_IntD2,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ B ) ) ).
% IntD2
thf(fact_485_IntD1,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ( member_set_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_486_IntD1,axiom,
! [C2: b,A2: set_b,B: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A2 @ B ) )
=> ( member_b @ C2 @ A2 ) ) ).
% IntD1
thf(fact_487_IntD1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_488_IntE,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ~ ( ( member_set_a @ C2 @ A2 )
=> ~ ( member_set_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_489_IntE,axiom,
! [C2: b,A2: set_b,B: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A2 @ B ) )
=> ~ ( ( member_b @ C2 @ A2 )
=> ~ ( member_b @ C2 @ B ) ) ) ).
% IntE
thf(fact_490_IntE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ~ ( member_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_491_UnE,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) )
=> ( ~ ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_492_UnE,axiom,
! [C2: b,A2: set_b,B: set_b] :
( ( member_b @ C2 @ ( sup_sup_set_b @ A2 @ B ) )
=> ( ~ ( member_b @ C2 @ A2 )
=> ( member_b @ C2 @ B ) ) ) ).
% UnE
thf(fact_493_UnE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
=> ( ~ ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_494_UnI1,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_495_UnI1,axiom,
! [C2: b,A2: set_b,B: set_b] :
( ( member_b @ C2 @ A2 )
=> ( member_b @ C2 @ ( sup_sup_set_b @ A2 @ B ) ) ) ).
% UnI1
thf(fact_496_UnI1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_497_UnI2,axiom,
! [C2: set_a,B: set_set_a,A2: set_set_a] :
( ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_498_UnI2,axiom,
! [C2: b,B: set_b,A2: set_b] :
( ( member_b @ C2 @ B )
=> ( member_b @ C2 @ ( sup_sup_set_b @ A2 @ B ) ) ) ).
% UnI2
thf(fact_499_UnI2,axiom,
! [C2: a,B: set_a,A2: set_a] :
( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_500_bex__Un,axiom,
! [A2: set_b,B: set_b,P: b > $o] :
( ( ? [X: b] :
( ( member_b @ X @ ( sup_sup_set_b @ A2 @ B ) )
& ( P @ X ) ) )
= ( ? [X: b] :
( ( member_b @ X @ A2 )
& ( P @ X ) )
| ? [X: b] :
( ( member_b @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_501_bex__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ? [X: a] :
( ( member_a @ X @ ( sup_sup_set_a @ A2 @ B ) )
& ( P @ X ) ) )
= ( ? [X: a] :
( ( member_a @ X @ A2 )
& ( P @ X ) )
| ? [X: a] :
( ( member_a @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_502_ball__Un,axiom,
! [A2: set_b,B: set_b,P: b > $o] :
( ( ! [X: b] :
( ( member_b @ X @ ( sup_sup_set_b @ A2 @ B ) )
=> ( P @ X ) ) )
= ( ! [X: b] :
( ( member_b @ X @ A2 )
=> ( P @ X ) )
& ! [X: b] :
( ( member_b @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_503_ball__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ! [X: a] :
( ( member_a @ X @ ( sup_sup_set_a @ A2 @ B ) )
=> ( P @ X ) ) )
= ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( P @ X ) )
& ! [X: a] :
( ( member_a @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_504_Un__assoc,axiom,
! [A2: set_b,B: set_b,C: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ A2 @ B ) @ C )
= ( sup_sup_set_b @ A2 @ ( sup_sup_set_b @ B @ C ) ) ) ).
% Un_assoc
thf(fact_505_Un__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_assoc
thf(fact_506_Un__absorb,axiom,
! [A2: set_b] :
( ( sup_sup_set_b @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_507_Un__absorb,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_508_Un__commute,axiom,
( sup_sup_set_b
= ( ^ [A3: set_b,B3: set_b] : ( sup_sup_set_b @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_509_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_510_Un__left__absorb,axiom,
! [A2: set_b,B: set_b] :
( ( sup_sup_set_b @ A2 @ ( sup_sup_set_b @ A2 @ B ) )
= ( sup_sup_set_b @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_511_Un__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_512_Un__left__commute,axiom,
! [A2: set_b,B: set_b,C: set_b] :
( ( sup_sup_set_b @ A2 @ ( sup_sup_set_b @ B @ C ) )
= ( sup_sup_set_b @ B @ ( sup_sup_set_b @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_513_Un__left__commute,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_514_Bex__def,axiom,
( bex_b
= ( ^ [A3: set_b,P2: b > $o] :
? [X: b] :
( ( member_b @ X @ A3 )
& ( P2 @ X ) ) ) ) ).
% Bex_def
thf(fact_515_Bex__def,axiom,
( bex_a
= ( ^ [A3: set_a,P2: a > $o] :
? [X: a] :
( ( member_a @ X @ A3 )
& ( P2 @ X ) ) ) ) ).
% Bex_def
thf(fact_516_Bex__def,axiom,
( bex_set_a
= ( ^ [A3: set_set_a,P2: set_a > $o] :
? [X: set_a] :
( ( member_set_a @ X @ A3 )
& ( P2 @ X ) ) ) ) ).
% Bex_def
thf(fact_517_empty__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a
@ ^ [X: list_a] : $false ) ) ).
% empty_def
thf(fact_518_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X: nat] : $false ) ) ).
% empty_def
thf(fact_519_empty__def,axiom,
( bot_bot_set_b
= ( collect_b
@ ^ [X: b] : $false ) ) ).
% empty_def
thf(fact_520_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X: a] : $false ) ) ).
% empty_def
thf(fact_521_Int__def,axiom,
( inf_inf_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( collect_set_a
@ ^ [X: set_a] :
( ( member_set_a @ X @ A3 )
& ( member_set_a @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_522_Int__def,axiom,
( inf_inf_set_list_a
= ( ^ [A3: set_list_a,B3: set_list_a] :
( collect_list_a
@ ^ [X: list_a] :
( ( member_list_a @ X @ A3 )
& ( member_list_a @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_523_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
& ( member_nat @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_524_Int__def,axiom,
( inf_inf_set_b
= ( ^ [A3: set_b,B3: set_b] :
( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A3 )
& ( member_b @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_525_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B3: set_a] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A3 )
& ( member_a @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_526_Int__Collect,axiom,
! [X4: set_a,A2: set_set_a,P: set_a > $o] :
( ( member_set_a @ X4 @ ( inf_inf_set_set_a @ A2 @ ( collect_set_a @ P ) ) )
= ( ( member_set_a @ X4 @ A2 )
& ( P @ X4 ) ) ) ).
% Int_Collect
thf(fact_527_Int__Collect,axiom,
! [X4: list_a,A2: set_list_a,P: list_a > $o] :
( ( member_list_a @ X4 @ ( inf_inf_set_list_a @ A2 @ ( collect_list_a @ P ) ) )
= ( ( member_list_a @ X4 @ A2 )
& ( P @ X4 ) ) ) ).
% Int_Collect
thf(fact_528_Int__Collect,axiom,
! [X4: nat,A2: set_nat,P: nat > $o] :
( ( member_nat @ X4 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X4 @ A2 )
& ( P @ X4 ) ) ) ).
% Int_Collect
thf(fact_529_Int__Collect,axiom,
! [X4: b,A2: set_b,P: b > $o] :
( ( member_b @ X4 @ ( inf_inf_set_b @ A2 @ ( collect_b @ P ) ) )
= ( ( member_b @ X4 @ A2 )
& ( P @ X4 ) ) ) ).
% Int_Collect
thf(fact_530_Int__Collect,axiom,
! [X4: a,A2: set_a,P: a > $o] :
( ( member_a @ X4 @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) )
= ( ( member_a @ X4 @ A2 )
& ( P @ X4 ) ) ) ).
% Int_Collect
thf(fact_531_Collect__conj__eq,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( collect_list_a
@ ^ [X: list_a] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_532_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_533_Collect__conj__eq,axiom,
! [P: b > $o,Q: b > $o] :
( ( collect_b
@ ^ [X: b] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_534_Collect__conj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_535_Un__def,axiom,
( sup_sup_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( collect_set_a
@ ^ [X: set_a] :
( ( member_set_a @ X @ A3 )
| ( member_set_a @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_536_Un__def,axiom,
( sup_sup_set_list_a
= ( ^ [A3: set_list_a,B3: set_list_a] :
( collect_list_a
@ ^ [X: list_a] :
( ( member_list_a @ X @ A3 )
| ( member_list_a @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_537_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
| ( member_nat @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_538_Un__def,axiom,
( sup_sup_set_b
= ( ^ [A3: set_b,B3: set_b] :
( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A3 )
| ( member_b @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_539_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A3 )
| ( member_a @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_540_Collect__disj__eq,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( collect_list_a
@ ^ [X: list_a] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_sup_set_list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_541_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_542_Collect__disj__eq,axiom,
! [P: b > $o,Q: b > $o] :
( ( collect_b
@ ^ [X: b] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_sup_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_543_Collect__disj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_sup_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_544_disjoint__iff__not__equal,axiom,
! [A2: set_b,B: set_b] :
( ( ( inf_inf_set_b @ A2 @ B )
= bot_bot_set_b )
= ( ! [X: b] :
( ( member_b @ X @ A2 )
=> ! [Y: b] :
( ( member_b @ Y @ B )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_545_disjoint__iff__not__equal,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ! [Y: a] :
( ( member_a @ Y @ B )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_546_Int__empty__right,axiom,
! [A2: set_b] :
( ( inf_inf_set_b @ A2 @ bot_bot_set_b )
= bot_bot_set_b ) ).
% Int_empty_right
thf(fact_547_Int__empty__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_548_Int__empty__left,axiom,
! [B: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ B )
= bot_bot_set_b ) ).
% Int_empty_left
thf(fact_549_Int__empty__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_550_disjoint__iff,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ! [X: set_a] :
( ( member_set_a @ X @ A2 )
=> ~ ( member_set_a @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_551_disjoint__iff,axiom,
! [A2: set_b,B: set_b] :
( ( ( inf_inf_set_b @ A2 @ B )
= bot_bot_set_b )
= ( ! [X: b] :
( ( member_b @ X @ A2 )
=> ~ ( member_b @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_552_disjoint__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ~ ( member_a @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_553_Int__emptyI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ~ ( member_set_a @ X3 @ B ) )
=> ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_554_Int__emptyI,axiom,
! [A2: set_b,B: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ~ ( member_b @ X3 @ B ) )
=> ( ( inf_inf_set_b @ A2 @ B )
= bot_bot_set_b ) ) ).
% Int_emptyI
thf(fact_555_Int__emptyI,axiom,
! [A2: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_556_Un__empty__left,axiom,
! [B: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ B )
= B ) ).
% Un_empty_left
thf(fact_557_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_558_Un__empty__right,axiom,
! [A2: set_b] :
( ( sup_sup_set_b @ A2 @ bot_bot_set_b )
= A2 ) ).
% Un_empty_right
thf(fact_559_Un__empty__right,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Un_empty_right
thf(fact_560_distrib__imp1,axiom,
! [X4: set_b,Y2: set_b,Z2: set_b] :
( ! [X3: set_b,Y4: set_b,Z4: set_b] :
( ( inf_inf_set_b @ X3 @ ( sup_sup_set_b @ Y4 @ Z4 ) )
= ( sup_sup_set_b @ ( inf_inf_set_b @ X3 @ Y4 ) @ ( inf_inf_set_b @ X3 @ Z4 ) ) )
=> ( ( sup_sup_set_b @ X4 @ ( inf_inf_set_b @ Y2 @ Z2 ) )
= ( inf_inf_set_b @ ( sup_sup_set_b @ X4 @ Y2 ) @ ( sup_sup_set_b @ X4 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_561_distrib__imp1,axiom,
! [X4: set_a,Y2: set_a,Z2: set_a] :
( ! [X3: set_a,Y4: set_a,Z4: set_a] :
( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y4 @ Z4 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y4 ) @ ( inf_inf_set_a @ X3 @ Z4 ) ) )
=> ( ( sup_sup_set_a @ X4 @ ( inf_inf_set_a @ Y2 @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X4 @ Y2 ) @ ( sup_sup_set_a @ X4 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_562_distrib__imp2,axiom,
! [X4: set_b,Y2: set_b,Z2: set_b] :
( ! [X3: set_b,Y4: set_b,Z4: set_b] :
( ( sup_sup_set_b @ X3 @ ( inf_inf_set_b @ Y4 @ Z4 ) )
= ( inf_inf_set_b @ ( sup_sup_set_b @ X3 @ Y4 ) @ ( sup_sup_set_b @ X3 @ Z4 ) ) )
=> ( ( inf_inf_set_b @ X4 @ ( sup_sup_set_b @ Y2 @ Z2 ) )
= ( sup_sup_set_b @ ( inf_inf_set_b @ X4 @ Y2 ) @ ( inf_inf_set_b @ X4 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_563_distrib__imp2,axiom,
! [X4: set_a,Y2: set_a,Z2: set_a] :
( ! [X3: set_a,Y4: set_a,Z4: set_a] :
( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y4 @ Z4 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y4 ) @ ( sup_sup_set_a @ X3 @ Z4 ) ) )
=> ( ( inf_inf_set_a @ X4 @ ( sup_sup_set_a @ Y2 @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X4 @ Y2 ) @ ( inf_inf_set_a @ X4 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_564_inf__sup__distrib1,axiom,
! [X4: set_b,Y2: set_b,Z2: set_b] :
( ( inf_inf_set_b @ X4 @ ( sup_sup_set_b @ Y2 @ Z2 ) )
= ( sup_sup_set_b @ ( inf_inf_set_b @ X4 @ Y2 ) @ ( inf_inf_set_b @ X4 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_565_inf__sup__distrib1,axiom,
! [X4: set_a,Y2: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X4 @ ( sup_sup_set_a @ Y2 @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X4 @ Y2 ) @ ( inf_inf_set_a @ X4 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_566_inf__sup__distrib2,axiom,
! [Y2: set_b,Z2: set_b,X4: set_b] :
( ( inf_inf_set_b @ ( sup_sup_set_b @ Y2 @ Z2 ) @ X4 )
= ( sup_sup_set_b @ ( inf_inf_set_b @ Y2 @ X4 ) @ ( inf_inf_set_b @ Z2 @ X4 ) ) ) ).
% inf_sup_distrib2
thf(fact_567_inf__sup__distrib2,axiom,
! [Y2: set_a,Z2: set_a,X4: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y2 @ Z2 ) @ X4 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y2 @ X4 ) @ ( inf_inf_set_a @ Z2 @ X4 ) ) ) ).
% inf_sup_distrib2
thf(fact_568_sup__inf__distrib1,axiom,
! [X4: set_b,Y2: set_b,Z2: set_b] :
( ( sup_sup_set_b @ X4 @ ( inf_inf_set_b @ Y2 @ Z2 ) )
= ( inf_inf_set_b @ ( sup_sup_set_b @ X4 @ Y2 ) @ ( sup_sup_set_b @ X4 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_569_sup__inf__distrib1,axiom,
! [X4: set_a,Y2: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X4 @ ( inf_inf_set_a @ Y2 @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X4 @ Y2 ) @ ( sup_sup_set_a @ X4 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_570_sup__inf__distrib2,axiom,
! [Y2: set_b,Z2: set_b,X4: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ Y2 @ Z2 ) @ X4 )
= ( inf_inf_set_b @ ( sup_sup_set_b @ Y2 @ X4 ) @ ( sup_sup_set_b @ Z2 @ X4 ) ) ) ).
% sup_inf_distrib2
thf(fact_571_sup__inf__distrib2,axiom,
! [Y2: set_a,Z2: set_a,X4: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y2 @ Z2 ) @ X4 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y2 @ X4 ) @ ( sup_sup_set_a @ Z2 @ X4 ) ) ) ).
% sup_inf_distrib2
thf(fact_572_Un__Int__crazy,axiom,
! [A2: set_b,B: set_b,C: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ ( inf_inf_set_b @ A2 @ B ) @ ( inf_inf_set_b @ B @ C ) ) @ ( inf_inf_set_b @ C @ A2 ) )
= ( inf_inf_set_b @ ( inf_inf_set_b @ ( sup_sup_set_b @ A2 @ B ) @ ( sup_sup_set_b @ B @ C ) ) @ ( sup_sup_set_b @ C @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_573_Un__Int__crazy,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ B @ C ) ) @ ( inf_inf_set_a @ C @ A2 ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ B @ C ) ) @ ( sup_sup_set_a @ C @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_574_Int__Un__distrib,axiom,
! [A2: set_b,B: set_b,C: set_b] :
( ( inf_inf_set_b @ A2 @ ( sup_sup_set_b @ B @ C ) )
= ( sup_sup_set_b @ ( inf_inf_set_b @ A2 @ B ) @ ( inf_inf_set_b @ A2 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_575_Int__Un__distrib,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ A2 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_576_Un__Int__distrib,axiom,
! [A2: set_b,B: set_b,C: set_b] :
( ( sup_sup_set_b @ A2 @ ( inf_inf_set_b @ B @ C ) )
= ( inf_inf_set_b @ ( sup_sup_set_b @ A2 @ B ) @ ( sup_sup_set_b @ A2 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_577_Un__Int__distrib,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ A2 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_578_Int__Un__distrib2,axiom,
! [B: set_b,C: set_b,A2: set_b] :
( ( inf_inf_set_b @ ( sup_sup_set_b @ B @ C ) @ A2 )
= ( sup_sup_set_b @ ( inf_inf_set_b @ B @ A2 ) @ ( inf_inf_set_b @ C @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_579_Int__Un__distrib2,axiom,
! [B: set_a,C: set_a,A2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B @ C ) @ A2 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B @ A2 ) @ ( inf_inf_set_a @ C @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_580_Un__Int__distrib2,axiom,
! [B: set_b,C: set_b,A2: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ B @ C ) @ A2 )
= ( inf_inf_set_b @ ( sup_sup_set_b @ B @ A2 ) @ ( sup_sup_set_b @ C @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_581_Un__Int__distrib2,axiom,
! [B: set_a,C: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B @ C ) @ A2 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B @ A2 ) @ ( sup_sup_set_a @ C @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_582_T__arcs__compl__fin_H,axiom,
! [Es: set_b] :
( ( graph_a_b @ g )
=> ( ( ord_less_eq_set_b @ Es @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( finite_finite_b
@ ( collect_b
@ ^ [E22: b] :
( ( member_b @ E22 @ ( pre_ar1395965042833527383t_unit @ g ) )
& ? [X: b] :
( ( member_b @ X @ Es )
& ( ( pre_he5236287464308401016t_unit @ g @ E22 )
= ( pre_ta4931606617599662728t_unit @ g @ X ) )
& ( ( pre_he5236287464308401016t_unit @ g @ X )
= ( pre_ta4931606617599662728t_unit @ g @ E22 ) ) ) ) ) ) ) ) ).
% T_arcs_compl_fin'
thf(fact_583_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_b] :
( ( inf_inf_set_b @ X4 @ bot_bot_set_b )
= bot_bot_set_b ) ).
% boolean_algebra.conj_zero_right
thf(fact_584_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_585_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ X4 )
= bot_bot_set_b ) ).
% boolean_algebra.conj_zero_left
thf(fact_586_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X4 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_587_arcs__del__vert,axiom,
! [U: a] :
( ( pre_ar1395965042833527383t_unit @ ( pre_del_vert_a_b @ t @ U ) )
= ( collect_b
@ ^ [A4: b] :
( ( member_b @ A4 @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( pre_ta4931606617599662728t_unit @ t @ A4 )
!= U )
& ( ( pre_he5236287464308401016t_unit @ t @ A4 )
!= U ) ) ) ) ).
% arcs_del_vert
thf(fact_588_G_Oarcs__del__vert,axiom,
! [U: a] :
( ( pre_ar1395965042833527383t_unit @ ( pre_del_vert_a_b @ g @ U ) )
= ( collect_b
@ ^ [A4: b] :
( ( member_b @ A4 @ ( pre_ar1395965042833527383t_unit @ g ) )
& ( ( pre_ta4931606617599662728t_unit @ g @ A4 )
!= U )
& ( ( pre_he5236287464308401016t_unit @ g @ A4 )
!= U ) ) ) ) ).
% G.arcs_del_vert
thf(fact_589_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_left
thf(fact_590_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_right
thf(fact_591_bidirected__digraphI,axiom,
! [Arev: b > b] :
( ! [A6: b] :
( ~ ( member_b @ A6 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( Arev @ A6 )
= A6 ) )
=> ( ! [A6: b] :
( ( member_b @ A6 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( Arev @ A6 )
!= A6 ) )
=> ( ! [A6: b] :
( ( member_b @ A6 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( Arev @ ( Arev @ A6 ) )
= A6 ) )
=> ( ! [A6: b] :
( ( member_b @ A6 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( pre_ta4931606617599662728t_unit @ t @ ( Arev @ A6 ) )
= ( pre_he5236287464308401016t_unit @ t @ A6 ) ) )
=> ( bidire6463457107099887885ph_a_b @ t @ Arev ) ) ) ) ) ).
% bidirected_digraphI
thf(fact_592_G_Obidirected__digraphI,axiom,
! [Arev: b > b] :
( ! [A6: b] :
( ~ ( member_b @ A6 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( Arev @ A6 )
= A6 ) )
=> ( ! [A6: b] :
( ( member_b @ A6 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( Arev @ A6 )
!= A6 ) )
=> ( ! [A6: b] :
( ( member_b @ A6 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( Arev @ ( Arev @ A6 ) )
= A6 ) )
=> ( ! [A6: b] :
( ( member_b @ A6 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( pre_ta4931606617599662728t_unit @ g @ ( Arev @ A6 ) )
= ( pre_he5236287464308401016t_unit @ g @ A6 ) ) )
=> ( bidire6463457107099887885ph_a_b @ g @ Arev ) ) ) ) ) ).
% G.bidirected_digraphI
thf(fact_593_unique__arc__set,axiom,
! [U: a,V: a] :
( ( ( collect_b
@ ^ [E3: b] :
( ( member_b @ E3 @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( pre_ta4931606617599662728t_unit @ t @ E3 )
= U )
& ( ( pre_he5236287464308401016t_unit @ t @ E3 )
= V ) ) )
= bot_bot_set_b )
| ? [E4: b] :
( ( collect_b
@ ^ [F2: b] :
( ( member_b @ F2 @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( pre_ta4931606617599662728t_unit @ t @ F2 )
= U )
& ( ( pre_he5236287464308401016t_unit @ t @ F2 )
= V ) ) )
= ( insert_b @ E4 @ bot_bot_set_b ) ) ) ).
% unique_arc_set
thf(fact_594_branching__points__def,axiom,
( ( graph_4596510882073158607ts_a_b @ t )
= ( collect_a
@ ^ [X: a] :
? [Y: b] :
( ( member_b @ Y @ ( pre_ar1395965042833527383t_unit @ t ) )
& ? [Z: b] :
( ( member_b @ Z @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( Y != Z )
& ( ( pre_ta4931606617599662728t_unit @ t @ Y )
= X )
& ( ( pre_ta4931606617599662728t_unit @ t @ Z )
= X ) ) ) ) ) ).
% branching_points_def
thf(fact_595_finite__arcs,axiom,
finite_finite_b @ ( pre_ar1395965042833527383t_unit @ g ) ).
% finite_arcs
thf(fact_596_G_Otail__del__vert,axiom,
! [U: a] :
( ( pre_ta4931606617599662728t_unit @ ( pre_del_vert_a_b @ g @ U ) )
= ( pre_ta4931606617599662728t_unit @ g ) ) ).
% G.tail_del_vert
thf(fact_597_G_Ohead__del__vert,axiom,
! [U: a] :
( ( pre_he5236287464308401016t_unit @ ( pre_del_vert_a_b @ g @ U ) )
= ( pre_he5236287464308401016t_unit @ g ) ) ).
% G.head_del_vert
thf(fact_598_tail__del__vert,axiom,
! [U: a] :
( ( pre_ta4931606617599662728t_unit @ ( pre_del_vert_a_b @ t @ U ) )
= ( pre_ta4931606617599662728t_unit @ t ) ) ).
% tail_del_vert
thf(fact_599_head__del__vert,axiom,
! [U: a] :
( ( pre_he5236287464308401016t_unit @ ( pre_del_vert_a_b @ t @ U ) )
= ( pre_he5236287464308401016t_unit @ t ) ) ).
% head_del_vert
thf(fact_600_G_Obranch__in__supergraph,axiom,
! [C: pre_pr7278220950009878019t_unit,X4: a] :
( ( shorte3657265928840388360ph_a_b @ C @ g )
=> ( ( member_a @ X4 @ ( graph_4596510882073158607ts_a_b @ C ) )
=> ( member_a @ X4 @ ( graph_4596510882073158607ts_a_b @ g ) ) ) ) ).
% G.branch_in_supergraph
thf(fact_601_branch__in__supergraph,axiom,
! [C: pre_pr7278220950009878019t_unit,X4: a] :
( ( shorte3657265928840388360ph_a_b @ C @ t )
=> ( ( member_a @ X4 @ ( graph_4596510882073158607ts_a_b @ C ) )
=> ( member_a @ X4 @ ( graph_4596510882073158607ts_a_b @ t ) ) ) ) ).
% branch_in_supergraph
thf(fact_602_graph__del__vert,axiom,
! [X4: a] : ( graph_a_b @ ( pre_del_vert_a_b @ g @ X4 ) ) ).
% graph_del_vert
thf(fact_603_add__right__cancel,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B2 = C2 ) ) ).
% add_right_cancel
thf(fact_604_add__left__cancel,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C2 ) )
= ( B2 = C2 ) ) ).
% add_left_cancel
thf(fact_605_insert__absorb2,axiom,
! [X4: b,A2: set_b] :
( ( insert_b @ X4 @ ( insert_b @ X4 @ A2 ) )
= ( insert_b @ X4 @ A2 ) ) ).
% insert_absorb2
thf(fact_606_insert__absorb2,axiom,
! [X4: a,A2: set_a] :
( ( insert_a @ X4 @ ( insert_a @ X4 @ A2 ) )
= ( insert_a @ X4 @ A2 ) ) ).
% insert_absorb2
thf(fact_607_insert__iff,axiom,
! [A: b,B2: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_608_insert__iff,axiom,
! [A: a,B2: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_609_insert__iff,axiom,
! [A: set_a,B2: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_set_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_610_insertCI,axiom,
! [A: b,B: set_b,B2: b] :
( ( ~ ( member_b @ A @ B )
=> ( A = B2 ) )
=> ( member_b @ A @ ( insert_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_611_insertCI,axiom,
! [A: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A @ B )
=> ( A = B2 ) )
=> ( member_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_612_insertCI,axiom,
! [A: set_a,B: set_set_a,B2: set_a] :
( ( ~ ( member_set_a @ A @ B )
=> ( A = B2 ) )
=> ( member_set_a @ A @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_613_G_Obranching__points__def,axiom,
( ( graph_4596510882073158607ts_a_b @ g )
= ( collect_a
@ ^ [X: a] :
? [Y: b] :
( ( member_b @ Y @ ( pre_ar1395965042833527383t_unit @ g ) )
& ? [Z: b] :
( ( member_b @ Z @ ( pre_ar1395965042833527383t_unit @ g ) )
& ( Y != Z )
& ( ( pre_ta4931606617599662728t_unit @ g @ Y )
= X )
& ( ( pre_ta4931606617599662728t_unit @ g @ Z )
= X ) ) ) ) ) ).
% G.branching_points_def
thf(fact_614_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_615_singletonI,axiom,
! [A: b] : ( member_b @ A @ ( insert_b @ A @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_616_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_617_insert__subset,axiom,
! [X4: set_a,A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X4 @ A2 ) @ B )
= ( ( member_set_a @ X4 @ B )
& ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_618_insert__subset,axiom,
! [X4: b,A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ ( insert_b @ X4 @ A2 ) @ B )
= ( ( member_b @ X4 @ B )
& ( ord_less_eq_set_b @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_619_insert__subset,axiom,
! [X4: a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X4 @ A2 ) @ B )
= ( ( member_a @ X4 @ B )
& ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_620_Int__insert__left__if0,axiom,
! [A: set_a,C: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C )
= ( inf_inf_set_set_a @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_621_Int__insert__left__if0,axiom,
! [A: b,C: set_b,B: set_b] :
( ~ ( member_b @ A @ C )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B ) @ C )
= ( inf_inf_set_b @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_622_Int__insert__left__if0,axiom,
! [A: a,C: set_a,B: set_a] :
( ~ ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_623_Int__insert__left__if1,axiom,
! [A: set_a,C: set_set_a,B: set_set_a] :
( ( member_set_a @ A @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_624_Int__insert__left__if1,axiom,
! [A: b,C: set_b,B: set_b] :
( ( member_b @ A @ C )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B ) @ C )
= ( insert_b @ A @ ( inf_inf_set_b @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_625_Int__insert__left__if1,axiom,
! [A: a,C: set_a,B: set_a] :
( ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_626_insert__inter__insert,axiom,
! [A: b,A2: set_b,B: set_b] :
( ( inf_inf_set_b @ ( insert_b @ A @ A2 ) @ ( insert_b @ A @ B ) )
= ( insert_b @ A @ ( inf_inf_set_b @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_627_insert__inter__insert,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_628_Int__insert__right__if0,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_629_Int__insert__right__if0,axiom,
! [A: b,A2: set_b,B: set_b] :
( ~ ( member_b @ A @ A2 )
=> ( ( inf_inf_set_b @ A2 @ ( insert_b @ A @ B ) )
= ( inf_inf_set_b @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_630_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_631_Int__insert__right__if1,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_632_Int__insert__right__if1,axiom,
! [A: b,A2: set_b,B: set_b] :
( ( member_b @ A @ A2 )
=> ( ( inf_inf_set_b @ A2 @ ( insert_b @ A @ B ) )
= ( insert_b @ A @ ( inf_inf_set_b @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_633_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_634_Un__insert__right,axiom,
! [A2: set_b,A: b,B: set_b] :
( ( sup_sup_set_b @ A2 @ ( insert_b @ A @ B ) )
= ( insert_b @ A @ ( sup_sup_set_b @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_635_Un__insert__right,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_636_Un__insert__left,axiom,
! [A: b,B: set_b,C: set_b] :
( ( sup_sup_set_b @ ( insert_b @ A @ B ) @ C )
= ( insert_b @ A @ ( sup_sup_set_b @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_637_Un__insert__left,axiom,
! [A: a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B ) @ C )
= ( insert_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_638_singleton__conv,axiom,
! [A: list_a] :
( ( collect_list_a
@ ^ [X: list_a] : ( X = A ) )
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% singleton_conv
thf(fact_639_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X: nat] : ( X = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_640_singleton__conv,axiom,
! [A: b] :
( ( collect_b
@ ^ [X: b] : ( X = A ) )
= ( insert_b @ A @ bot_bot_set_b ) ) ).
% singleton_conv
thf(fact_641_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X: a] : ( X = A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_642_singleton__conv2,axiom,
! [A: list_a] :
( ( collect_list_a
@ ( ^ [Y3: list_a,Z3: list_a] : ( Y3 = Z3 )
@ A ) )
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% singleton_conv2
thf(fact_643_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_644_singleton__conv2,axiom,
! [A: b] :
( ( collect_b
@ ( ^ [Y3: b,Z3: b] : ( Y3 = Z3 )
@ A ) )
= ( insert_b @ A @ bot_bot_set_b ) ) ).
% singleton_conv2
thf(fact_645_singleton__conv2,axiom,
! [A: a] :
( ( collect_a
@ ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 )
@ A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_646_singleton__insert__inj__eq_H,axiom,
! [A: b,A2: set_b,B2: b] :
( ( ( insert_b @ A @ A2 )
= ( insert_b @ B2 @ bot_bot_set_b ) )
= ( ( A = B2 )
& ( ord_less_eq_set_b @ A2 @ ( insert_b @ B2 @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_647_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B2: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_648_singleton__insert__inj__eq,axiom,
! [B2: b,A: b,A2: set_b] :
( ( ( insert_b @ B2 @ bot_bot_set_b )
= ( insert_b @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_b @ A2 @ ( insert_b @ B2 @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_649_singleton__insert__inj__eq,axiom,
! [B2: a,A: a,A2: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_650_disjoint__insert_I2_J,axiom,
! [A2: set_set_a,B2: set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) )
= ( ~ ( member_set_a @ B2 @ A2 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_651_disjoint__insert_I2_J,axiom,
! [A2: set_b,B2: b,B: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ A2 @ ( insert_b @ B2 @ B ) ) )
= ( ~ ( member_b @ B2 @ A2 )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_652_disjoint__insert_I2_J,axiom,
! [A2: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B2 @ B ) ) )
= ( ~ ( member_a @ B2 @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_653_disjoint__insert_I1_J,axiom,
! [B: set_set_a,A: set_a,A2: set_set_a] :
( ( ( inf_inf_set_set_a @ B @ ( insert_set_a @ A @ A2 ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B )
& ( ( inf_inf_set_set_a @ B @ A2 )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_654_disjoint__insert_I1_J,axiom,
! [B: set_b,A: b,A2: set_b] :
( ( ( inf_inf_set_b @ B @ ( insert_b @ A @ A2 ) )
= bot_bot_set_b )
= ( ~ ( member_b @ A @ B )
& ( ( inf_inf_set_b @ B @ A2 )
= bot_bot_set_b ) ) ) ).
% disjoint_insert(1)
thf(fact_655_disjoint__insert_I1_J,axiom,
! [B: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ B @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_656_insert__disjoint_I2_J,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B ) )
= ( ~ ( member_set_a @ A @ B )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_657_insert__disjoint_I2_J,axiom,
! [A: b,A2: set_b,B: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ ( insert_b @ A @ A2 ) @ B ) )
= ( ~ ( member_b @ A @ B )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_658_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B ) )
= ( ~ ( member_a @ A @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_659_insert__disjoint_I1_J,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B )
& ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_660_insert__disjoint_I1_J,axiom,
! [A: b,A2: set_b,B: set_b] :
( ( ( inf_inf_set_b @ ( insert_b @ A @ A2 ) @ B )
= bot_bot_set_b )
= ( ~ ( member_b @ A @ B )
& ( ( inf_inf_set_b @ A2 @ B )
= bot_bot_set_b ) ) ) ).
% insert_disjoint(1)
thf(fact_661_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_662_G_Ois__chain__def,axiom,
( ( graph_3890552050688490787in_a_b @ g )
= ( ( graph_4596510882073158607ts_a_b @ g )
= bot_bot_set_a ) ) ).
% G.is_chain_def
thf(fact_663_is__chain__def,axiom,
( ( graph_3890552050688490787in_a_b @ t )
= ( ( graph_4596510882073158607ts_a_b @ t )
= bot_bot_set_a ) ) ).
% is_chain_def
thf(fact_664_mk__disjoint__insert,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ? [B6: set_b] :
( ( A2
= ( insert_b @ A @ B6 ) )
& ~ ( member_b @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_665_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B6: set_a] :
( ( A2
= ( insert_a @ A @ B6 ) )
& ~ ( member_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_666_mk__disjoint__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ? [B6: set_set_a] :
( ( A2
= ( insert_set_a @ A @ B6 ) )
& ~ ( member_set_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_667_insert__commute,axiom,
! [X4: b,Y2: b,A2: set_b] :
( ( insert_b @ X4 @ ( insert_b @ Y2 @ A2 ) )
= ( insert_b @ Y2 @ ( insert_b @ X4 @ A2 ) ) ) ).
% insert_commute
thf(fact_668_insert__commute,axiom,
! [X4: a,Y2: a,A2: set_a] :
( ( insert_a @ X4 @ ( insert_a @ Y2 @ A2 ) )
= ( insert_a @ Y2 @ ( insert_a @ X4 @ A2 ) ) ) ).
% insert_commute
thf(fact_669_insert__eq__iff,axiom,
! [A: b,A2: set_b,B2: b,B: set_b] :
( ~ ( member_b @ A @ A2 )
=> ( ~ ( member_b @ B2 @ B )
=> ( ( ( insert_b @ A @ A2 )
= ( insert_b @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_b] :
( ( A2
= ( insert_b @ B2 @ C3 ) )
& ~ ( member_b @ B2 @ C3 )
& ( B
= ( insert_b @ A @ C3 ) )
& ~ ( member_b @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_670_insert__eq__iff,axiom,
! [A: a,A2: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_a] :
( ( A2
= ( insert_a @ B2 @ C3 ) )
& ~ ( member_a @ B2 @ C3 )
& ( B
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_671_insert__eq__iff,axiom,
! [A: set_a,A2: set_set_a,B2: set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ B2 @ B )
=> ( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_set_a] :
( ( A2
= ( insert_set_a @ B2 @ C3 ) )
& ~ ( member_set_a @ B2 @ C3 )
& ( B
= ( insert_set_a @ A @ C3 ) )
& ~ ( member_set_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_672_insert__absorb,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ( ( insert_b @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_673_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_674_insert__absorb,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( insert_set_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_675_insert__ident,axiom,
! [X4: b,A2: set_b,B: set_b] :
( ~ ( member_b @ X4 @ A2 )
=> ( ~ ( member_b @ X4 @ B )
=> ( ( ( insert_b @ X4 @ A2 )
= ( insert_b @ X4 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_676_insert__ident,axiom,
! [X4: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X4 @ A2 )
=> ( ~ ( member_a @ X4 @ B )
=> ( ( ( insert_a @ X4 @ A2 )
= ( insert_a @ X4 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_677_insert__ident,axiom,
! [X4: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X4 @ A2 )
=> ( ~ ( member_set_a @ X4 @ B )
=> ( ( ( insert_set_a @ X4 @ A2 )
= ( insert_set_a @ X4 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_678_Set_Oset__insert,axiom,
! [X4: b,A2: set_b] :
( ( member_b @ X4 @ A2 )
=> ~ ! [B6: set_b] :
( ( A2
= ( insert_b @ X4 @ B6 ) )
=> ( member_b @ X4 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_679_Set_Oset__insert,axiom,
! [X4: a,A2: set_a] :
( ( member_a @ X4 @ A2 )
=> ~ ! [B6: set_a] :
( ( A2
= ( insert_a @ X4 @ B6 ) )
=> ( member_a @ X4 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_680_Set_Oset__insert,axiom,
! [X4: set_a,A2: set_set_a] :
( ( member_set_a @ X4 @ A2 )
=> ~ ! [B6: set_set_a] :
( ( A2
= ( insert_set_a @ X4 @ B6 ) )
=> ( member_set_a @ X4 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_681_insertI2,axiom,
! [A: b,B: set_b,B2: b] :
( ( member_b @ A @ B )
=> ( member_b @ A @ ( insert_b @ B2 @ B ) ) ) ).
% insertI2
thf(fact_682_insertI2,axiom,
! [A: a,B: set_a,B2: a] :
( ( member_a @ A @ B )
=> ( member_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_683_insertI2,axiom,
! [A: set_a,B: set_set_a,B2: set_a] :
( ( member_set_a @ A @ B )
=> ( member_set_a @ A @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_684_insertI1,axiom,
! [A: b,B: set_b] : ( member_b @ A @ ( insert_b @ A @ B ) ) ).
% insertI1
thf(fact_685_insertI1,axiom,
! [A: a,B: set_a] : ( member_a @ A @ ( insert_a @ A @ B ) ) ).
% insertI1
thf(fact_686_insertI1,axiom,
! [A: set_a,B: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B ) ) ).
% insertI1
thf(fact_687_insertE,axiom,
! [A: b,B2: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_b @ A @ A2 ) ) ) ).
% insertE
thf(fact_688_insertE,axiom,
! [A: a,B2: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_689_insertE,axiom,
! [A: set_a,B2: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_set_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_690_insert__compr,axiom,
( insert_set_a
= ( ^ [A4: set_a,B3: set_set_a] :
( collect_set_a
@ ^ [X: set_a] :
( ( X = A4 )
| ( member_set_a @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_691_insert__compr,axiom,
( insert_b
= ( ^ [A4: b,B3: set_b] :
( collect_b
@ ^ [X: b] :
( ( X = A4 )
| ( member_b @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_692_insert__compr,axiom,
( insert_a
= ( ^ [A4: a,B3: set_a] :
( collect_a
@ ^ [X: a] :
( ( X = A4 )
| ( member_a @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_693_insert__compr,axiom,
( insert_list_a
= ( ^ [A4: list_a,B3: set_list_a] :
( collect_list_a
@ ^ [X: list_a] :
( ( X = A4 )
| ( member_list_a @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_694_insert__compr,axiom,
( insert_nat
= ( ^ [A4: nat,B3: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( X = A4 )
| ( member_nat @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_695_insert__Collect,axiom,
! [A: b,P: b > $o] :
( ( insert_b @ A @ ( collect_b @ P ) )
= ( collect_b
@ ^ [U2: b] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_696_insert__Collect,axiom,
! [A: a,P: a > $o] :
( ( insert_a @ A @ ( collect_a @ P ) )
= ( collect_a
@ ^ [U2: a] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_697_insert__Collect,axiom,
! [A: list_a,P: list_a > $o] :
( ( insert_list_a @ A @ ( collect_list_a @ P ) )
= ( collect_list_a
@ ^ [U2: list_a] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_698_insert__Collect,axiom,
! [A: nat,P: nat > $o] :
( ( insert_nat @ A @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U2: nat] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_699_bot__set__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a @ bot_bot_list_a_o ) ) ).
% bot_set_def
thf(fact_700_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_701_bot__set__def,axiom,
( bot_bot_set_b
= ( collect_b @ bot_bot_b_o ) ) ).
% bot_set_def
thf(fact_702_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_703_singletonD,axiom,
! [B2: set_a,A: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_704_singletonD,axiom,
! [B2: b,A: b] :
( ( member_b @ B2 @ ( insert_b @ A @ bot_bot_set_b ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_705_singletonD,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_706_singleton__iff,axiom,
! [B2: set_a,A: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_707_singleton__iff,axiom,
! [B2: b,A: b] :
( ( member_b @ B2 @ ( insert_b @ A @ bot_bot_set_b ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_708_singleton__iff,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_709_doubleton__eq__iff,axiom,
! [A: b,B2: b,C2: b,D: b] :
( ( ( insert_b @ A @ ( insert_b @ B2 @ bot_bot_set_b ) )
= ( insert_b @ C2 @ ( insert_b @ D @ bot_bot_set_b ) ) )
= ( ( ( A = C2 )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_710_doubleton__eq__iff,axiom,
! [A: a,B2: a,C2: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C2 )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_711_insert__not__empty,axiom,
! [A: b,A2: set_b] :
( ( insert_b @ A @ A2 )
!= bot_bot_set_b ) ).
% insert_not_empty
thf(fact_712_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_713_singleton__inject,axiom,
! [A: b,B2: b] :
( ( ( insert_b @ A @ bot_bot_set_b )
= ( insert_b @ B2 @ bot_bot_set_b ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_714_singleton__inject,axiom,
! [A: a,B2: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_715_insert__mono,axiom,
! [C: set_b,D2: set_b,A: b] :
( ( ord_less_eq_set_b @ C @ D2 )
=> ( ord_less_eq_set_b @ ( insert_b @ A @ C ) @ ( insert_b @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_716_insert__mono,axiom,
! [C: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_717_subset__insert,axiom,
! [X4: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X4 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X4 @ B ) )
= ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_718_subset__insert,axiom,
! [X4: b,A2: set_b,B: set_b] :
( ~ ( member_b @ X4 @ A2 )
=> ( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X4 @ B ) )
= ( ord_less_eq_set_b @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_719_subset__insert,axiom,
! [X4: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X4 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X4 @ B ) )
= ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_720_subset__insertI,axiom,
! [B: set_b,A: b] : ( ord_less_eq_set_b @ B @ ( insert_b @ A @ B ) ) ).
% subset_insertI
thf(fact_721_subset__insertI,axiom,
! [B: set_a,A: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A @ B ) ) ).
% subset_insertI
thf(fact_722_subset__insertI2,axiom,
! [A2: set_b,B: set_b,B2: b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ord_less_eq_set_b @ A2 @ ( insert_b @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_723_subset__insertI2,axiom,
! [A2: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_724_Int__insert__left,axiom,
! [A: set_a,C: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B @ C ) ) ) )
& ( ~ ( member_set_a @ A @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C )
= ( inf_inf_set_set_a @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_725_Int__insert__left,axiom,
! [A: b,C: set_b,B: set_b] :
( ( ( member_b @ A @ C )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B ) @ C )
= ( insert_b @ A @ ( inf_inf_set_b @ B @ C ) ) ) )
& ( ~ ( member_b @ A @ C )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B ) @ C )
= ( inf_inf_set_b @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_726_Int__insert__left,axiom,
! [A: a,C: set_a,B: set_a] :
( ( ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) )
& ( ~ ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( inf_inf_set_a @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_727_Int__insert__right,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B ) ) ) )
& ( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_728_Int__insert__right,axiom,
! [A: b,A2: set_b,B: set_b] :
( ( ( member_b @ A @ A2 )
=> ( ( inf_inf_set_b @ A2 @ ( insert_b @ A @ B ) )
= ( insert_b @ A @ ( inf_inf_set_b @ A2 @ B ) ) ) )
& ( ~ ( member_b @ A @ A2 )
=> ( ( inf_inf_set_b @ A2 @ ( insert_b @ A @ B ) )
= ( inf_inf_set_b @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_729_Int__insert__right,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_730_less__eq__set__def,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ord_less_eq_set_a_o
@ ^ [X: set_a] : ( member_set_a @ X @ A3 )
@ ^ [X: set_a] : ( member_set_a @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_731_less__eq__set__def,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B3: set_b] :
( ord_less_eq_b_o
@ ^ [X: b] : ( member_b @ X @ A3 )
@ ^ [X: b] : ( member_b @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_732_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ A3 )
@ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_733_Collect__conv__if,axiom,
! [P: list_a > $o,A: list_a] :
( ( ( P @ A )
=> ( ( collect_list_a
@ ^ [X: list_a] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_list_a
@ ^ [X: list_a] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_list_a ) ) ) ).
% Collect_conv_if
thf(fact_734_Collect__conv__if,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_735_Collect__conv__if,axiom,
! [P: b > $o,A: b] :
( ( ( P @ A )
=> ( ( collect_b
@ ^ [X: b] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_b @ A @ bot_bot_set_b ) ) )
& ( ~ ( P @ A )
=> ( ( collect_b
@ ^ [X: b] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_b ) ) ) ).
% Collect_conv_if
thf(fact_736_Collect__conv__if,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X: a] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X: a] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_737_Collect__conv__if2,axiom,
! [P: list_a > $o,A: list_a] :
( ( ( P @ A )
=> ( ( collect_list_a
@ ^ [X: list_a] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_list_a
@ ^ [X: list_a] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_list_a ) ) ) ).
% Collect_conv_if2
thf(fact_738_Collect__conv__if2,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_739_Collect__conv__if2,axiom,
! [P: b > $o,A: b] :
( ( ( P @ A )
=> ( ( collect_b
@ ^ [X: b] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_b @ A @ bot_bot_set_b ) ) )
& ( ~ ( P @ A )
=> ( ( collect_b
@ ^ [X: b] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_b ) ) ) ).
% Collect_conv_if2
thf(fact_740_Collect__conv__if2,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X: a] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X: a] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_741_insert__def,axiom,
( insert_list_a
= ( ^ [A4: list_a] :
( sup_sup_set_list_a
@ ( collect_list_a
@ ^ [X: list_a] : ( X = A4 ) ) ) ) ) ).
% insert_def
thf(fact_742_insert__def,axiom,
( insert_nat
= ( ^ [A4: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X: nat] : ( X = A4 ) ) ) ) ) ).
% insert_def
thf(fact_743_insert__def,axiom,
( insert_b
= ( ^ [A4: b] :
( sup_sup_set_b
@ ( collect_b
@ ^ [X: b] : ( X = A4 ) ) ) ) ) ).
% insert_def
thf(fact_744_insert__def,axiom,
( insert_a
= ( ^ [A4: a] :
( sup_sup_set_a
@ ( collect_a
@ ^ [X: a] : ( X = A4 ) ) ) ) ) ).
% insert_def
thf(fact_745_subset__singleton__iff,axiom,
! [X5: set_b,A: b] :
( ( ord_less_eq_set_b @ X5 @ ( insert_b @ A @ bot_bot_set_b ) )
= ( ( X5 = bot_bot_set_b )
| ( X5
= ( insert_b @ A @ bot_bot_set_b ) ) ) ) ).
% subset_singleton_iff
thf(fact_746_subset__singleton__iff,axiom,
! [X5: set_a,A: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_747_subset__singletonD,axiom,
! [A2: set_b,X4: b] :
( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X4 @ bot_bot_set_b ) )
=> ( ( A2 = bot_bot_set_b )
| ( A2
= ( insert_b @ X4 @ bot_bot_set_b ) ) ) ) ).
% subset_singletonD
thf(fact_748_subset__singletonD,axiom,
! [A2: set_a,X4: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X4 @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_749_singleton__Un__iff,axiom,
! [X4: b,A2: set_b,B: set_b] :
( ( ( insert_b @ X4 @ bot_bot_set_b )
= ( sup_sup_set_b @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_b )
& ( B
= ( insert_b @ X4 @ bot_bot_set_b ) ) )
| ( ( A2
= ( insert_b @ X4 @ bot_bot_set_b ) )
& ( B = bot_bot_set_b ) )
| ( ( A2
= ( insert_b @ X4 @ bot_bot_set_b ) )
& ( B
= ( insert_b @ X4 @ bot_bot_set_b ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_750_singleton__Un__iff,axiom,
! [X4: a,A2: set_a,B: set_a] :
( ( ( insert_a @ X4 @ bot_bot_set_a )
= ( sup_sup_set_a @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B
= ( insert_a @ X4 @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X4 @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X4 @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_751_Un__singleton__iff,axiom,
! [A2: set_b,B: set_b,X4: b] :
( ( ( sup_sup_set_b @ A2 @ B )
= ( insert_b @ X4 @ bot_bot_set_b ) )
= ( ( ( A2 = bot_bot_set_b )
& ( B
= ( insert_b @ X4 @ bot_bot_set_b ) ) )
| ( ( A2
= ( insert_b @ X4 @ bot_bot_set_b ) )
& ( B = bot_bot_set_b ) )
| ( ( A2
= ( insert_b @ X4 @ bot_bot_set_b ) )
& ( B
= ( insert_b @ X4 @ bot_bot_set_b ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_752_Un__singleton__iff,axiom,
! [A2: set_a,B: set_a,X4: a] :
( ( ( sup_sup_set_a @ A2 @ B )
= ( insert_a @ X4 @ bot_bot_set_a ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B
= ( insert_a @ X4 @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X4 @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X4 @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_753_insert__is__Un,axiom,
( insert_b
= ( ^ [A4: b] : ( sup_sup_set_b @ ( insert_b @ A4 @ bot_bot_set_b ) ) ) ) ).
% insert_is_Un
thf(fact_754_insert__is__Un,axiom,
( insert_a
= ( ^ [A4: a] : ( sup_sup_set_a @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_755_add__right__imp__eq,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B2 = C2 ) ) ).
% add_right_imp_eq
thf(fact_756_add__left__imp__eq,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B2 = C2 ) ) ).
% add_left_imp_eq
thf(fact_757_add_Oleft__commute,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% add.left_commute
thf(fact_758_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_759_add_Oassoc,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% add.assoc
thf(fact_760_group__cancel_Oadd2,axiom,
! [B: nat,K: nat,B2: nat,A: nat] :
( ( B
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_761_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_762_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_763_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_764_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_b,K: set_b,A: set_b,B2: set_b] :
( ( A2
= ( inf_inf_set_b @ K @ A ) )
=> ( ( inf_inf_set_b @ A2 @ B2 )
= ( inf_inf_set_b @ K @ ( inf_inf_set_b @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_765_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_a,K: set_a,A: set_a,B2: set_a] :
( ( A2
= ( inf_inf_set_a @ K @ A ) )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_766_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_b,K: set_b,B2: set_b,A: set_b] :
( ( B
= ( inf_inf_set_b @ K @ B2 ) )
=> ( ( inf_inf_set_b @ A @ B )
= ( inf_inf_set_b @ K @ ( inf_inf_set_b @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_767_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K: set_a,B2: set_a,A: set_a] :
( ( B
= ( inf_inf_set_a @ K @ B2 ) )
=> ( ( inf_inf_set_a @ A @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_768_boolean__algebra__cancel_Osup2,axiom,
! [B: set_b,K: set_b,B2: set_b,A: set_b] :
( ( B
= ( sup_sup_set_b @ K @ B2 ) )
=> ( ( sup_sup_set_b @ A @ B )
= ( sup_sup_set_b @ K @ ( sup_sup_set_b @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_769_boolean__algebra__cancel_Osup2,axiom,
! [B: set_a,K: set_a,B2: set_a,A: set_a] :
( ( B
= ( sup_sup_set_a @ K @ B2 ) )
=> ( ( sup_sup_set_a @ A @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_770_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_b,K: set_b,A: set_b,B2: set_b] :
( ( A2
= ( sup_sup_set_b @ K @ A ) )
=> ( ( sup_sup_set_b @ A2 @ B2 )
= ( sup_sup_set_b @ K @ ( sup_sup_set_b @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_771_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_a,K: set_a,A: set_a,B2: set_a] :
( ( A2
= ( sup_sup_set_a @ K @ A ) )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_772_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_773_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_774_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C4: nat] :
( B4
= ( plus_plus_nat @ A4 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_775_add__right__mono,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% add_right_mono
thf(fact_776_less__eqE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ~ ! [C5: nat] :
( B2
!= ( plus_plus_nat @ A @ C5 ) ) ) ).
% less_eqE
thf(fact_777_add__left__mono,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B2 ) ) ) ).
% add_left_mono
thf(fact_778_add__mono,axiom,
! [A: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_779_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_780_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_781_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_782_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_b] :
( ( sup_sup_set_b @ X4 @ bot_bot_set_b )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_783_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ X4 @ bot_bot_set_a )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_784_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: set_b,Z2: set_b,X4: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ Y2 @ Z2 ) @ X4 )
= ( inf_inf_set_b @ ( sup_sup_set_b @ Y2 @ X4 ) @ ( sup_sup_set_b @ Z2 @ X4 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_785_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: set_a,Z2: set_a,X4: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y2 @ Z2 ) @ X4 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y2 @ X4 ) @ ( sup_sup_set_a @ Z2 @ X4 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_786_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: set_b,Z2: set_b,X4: set_b] :
( ( inf_inf_set_b @ ( sup_sup_set_b @ Y2 @ Z2 ) @ X4 )
= ( sup_sup_set_b @ ( inf_inf_set_b @ Y2 @ X4 ) @ ( inf_inf_set_b @ Z2 @ X4 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_787_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: set_a,Z2: set_a,X4: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y2 @ Z2 ) @ X4 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y2 @ X4 ) @ ( inf_inf_set_a @ Z2 @ X4 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_788_boolean__algebra_Odisj__conj__distrib,axiom,
! [X4: set_b,Y2: set_b,Z2: set_b] :
( ( sup_sup_set_b @ X4 @ ( inf_inf_set_b @ Y2 @ Z2 ) )
= ( inf_inf_set_b @ ( sup_sup_set_b @ X4 @ Y2 ) @ ( sup_sup_set_b @ X4 @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_789_boolean__algebra_Odisj__conj__distrib,axiom,
! [X4: set_a,Y2: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X4 @ ( inf_inf_set_a @ Y2 @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X4 @ Y2 ) @ ( sup_sup_set_a @ X4 @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_790_boolean__algebra_Oconj__disj__distrib,axiom,
! [X4: set_b,Y2: set_b,Z2: set_b] :
( ( inf_inf_set_b @ X4 @ ( sup_sup_set_b @ Y2 @ Z2 ) )
= ( sup_sup_set_b @ ( inf_inf_set_b @ X4 @ Y2 ) @ ( inf_inf_set_b @ X4 @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_791_boolean__algebra_Oconj__disj__distrib,axiom,
! [X4: set_a,Y2: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X4 @ ( sup_sup_set_a @ Y2 @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X4 @ Y2 ) @ ( inf_inf_set_a @ X4 @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_792_last__branch__is__branch,axiom,
! [Y2: a] :
( ( member_a @ Y2 @ ( graph_1747835947655717337ts_a_b @ t ) )
=> ( member_a @ Y2 @ ( graph_4596510882073158607ts_a_b @ t ) ) ) ).
% last_branch_is_branch
thf(fact_793_G_Olast__branch__is__branch,axiom,
! [Y2: a] :
( ( member_a @ Y2 @ ( graph_1747835947655717337ts_a_b @ g ) )
=> ( member_a @ Y2 @ ( graph_4596510882073158607ts_a_b @ g ) ) ) ).
% G.last_branch_is_branch
thf(fact_794_T__arcs__compl__fin,axiom,
! [Es: set_b] :
( ( fin_digraph_a_b @ g )
=> ( ( ord_less_eq_set_b @ Es @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( finite_finite_b
@ ( collect_b
@ ^ [E22: b] :
( ( member_b @ E22 @ ( pre_ar1395965042833527383t_unit @ g ) )
& ? [X: b] :
( ( member_b @ X @ Es )
& ( ( pre_he5236287464308401016t_unit @ g @ E22 )
= ( pre_ta4931606617599662728t_unit @ g @ X ) )
& ( ( pre_he5236287464308401016t_unit @ g @ X )
= ( pre_ta4931606617599662728t_unit @ g @ E22 ) ) ) ) ) ) ) ) ).
% T_arcs_compl_fin
thf(fact_795_card__Un__disjoint,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( finite_finite_list_a @ B )
=> ( ( ( inf_inf_set_list_a @ A2 @ B )
= bot_bot_set_list_a )
=> ( ( finite_card_list_a @ ( sup_sup_set_list_a @ A2 @ B ) )
= ( plus_plus_nat @ ( finite_card_list_a @ A2 ) @ ( finite_card_list_a @ B ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_796_card__Un__disjoint,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_set_a @ B )
=> ( ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
=> ( ( finite_card_set_a @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( plus_plus_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_797_card__Un__disjoint,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
=> ( ( finite_card_nat @ ( sup_sup_set_nat @ A2 @ B ) )
= ( plus_plus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_798_card__Un__disjoint,axiom,
! [A2: set_b,B: set_b] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ B )
=> ( ( ( inf_inf_set_b @ A2 @ B )
= bot_bot_set_b )
=> ( ( finite_card_b @ ( sup_sup_set_b @ A2 @ B ) )
= ( plus_plus_nat @ ( finite_card_b @ A2 ) @ ( finite_card_b @ B ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_799_card__Un__disjoint,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
=> ( ( finite_card_a @ ( sup_sup_set_a @ A2 @ B ) )
= ( plus_plus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_800_finite__Collect__bex,axiom,
! [A2: set_nat,Q: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
? [Y: nat] :
( ( member_nat @ Y @ A2 )
& ( Q @ X @ Y ) ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Y: nat] : ( Q @ Y @ X ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_801_finite__Collect__bex,axiom,
! [A2: set_nat,Q: b > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_b
@ ( collect_b
@ ^ [X: b] :
? [Y: nat] :
( ( member_nat @ Y @ A2 )
& ( Q @ X @ Y ) ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( finite_finite_b
@ ( collect_b
@ ^ [Y: b] : ( Q @ Y @ X ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_802_finite__Collect__bex,axiom,
! [A2: set_nat,Q: a > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
? [Y: nat] :
( ( member_nat @ Y @ A2 )
& ( Q @ X @ Y ) ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( finite_finite_a
@ ( collect_a
@ ^ [Y: a] : ( Q @ Y @ X ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_803_finite__Collect__bex,axiom,
! [A2: set_b,Q: nat > b > $o] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
? [Y: b] :
( ( member_b @ Y @ A2 )
& ( Q @ X @ Y ) ) ) )
= ( ! [X: b] :
( ( member_b @ X @ A2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Y: nat] : ( Q @ Y @ X ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_804_finite__Collect__bex,axiom,
! [A2: set_b,Q: b > b > $o] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_b
@ ( collect_b
@ ^ [X: b] :
? [Y: b] :
( ( member_b @ Y @ A2 )
& ( Q @ X @ Y ) ) ) )
= ( ! [X: b] :
( ( member_b @ X @ A2 )
=> ( finite_finite_b
@ ( collect_b
@ ^ [Y: b] : ( Q @ Y @ X ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_805_finite__Collect__bex,axiom,
! [A2: set_b,Q: a > b > $o] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
? [Y: b] :
( ( member_b @ Y @ A2 )
& ( Q @ X @ Y ) ) ) )
= ( ! [X: b] :
( ( member_b @ X @ A2 )
=> ( finite_finite_a
@ ( collect_a
@ ^ [Y: a] : ( Q @ Y @ X ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_806_finite__Collect__bex,axiom,
! [A2: set_a,Q: nat > a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
? [Y: a] :
( ( member_a @ Y @ A2 )
& ( Q @ X @ Y ) ) ) )
= ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Y: nat] : ( Q @ Y @ X ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_807_finite__Collect__bex,axiom,
! [A2: set_a,Q: b > a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_b
@ ( collect_b
@ ^ [X: b] :
? [Y: a] :
( ( member_a @ Y @ A2 )
& ( Q @ X @ Y ) ) ) )
= ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( finite_finite_b
@ ( collect_b
@ ^ [Y: b] : ( Q @ Y @ X ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_808_finite__Collect__bex,axiom,
! [A2: set_a,Q: a > a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
? [Y: a] :
( ( member_a @ Y @ A2 )
& ( Q @ X @ Y ) ) ) )
= ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( finite_finite_a
@ ( collect_a
@ ^ [Y: a] : ( Q @ Y @ X ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_809_finite__Collect__bex,axiom,
! [A2: set_list_a,Q: nat > list_a > $o] :
( ( finite_finite_list_a @ A2 )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
? [Y: list_a] :
( ( member_list_a @ Y @ A2 )
& ( Q @ X @ Y ) ) ) )
= ( ! [X: list_a] :
( ( member_list_a @ X @ A2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Y: nat] : ( Q @ Y @ X ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_810_finite__Collect__subsets,axiom,
! [A2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( finite5282473924520328461list_a
@ ( collect_set_list_a
@ ^ [B3: set_list_a] : ( ord_le8861187494160871172list_a @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_811_finite__Collect__subsets,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite7209287970140883943_set_a
@ ( collect_set_set_a
@ ^ [B3: set_set_a] : ( ord_le3724670747650509150_set_a @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_812_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B3: set_nat] : ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_813_finite__Collect__subsets,axiom,
! [A2: set_b] :
( ( finite_finite_b @ A2 )
=> ( finite_finite_set_b
@ ( collect_set_b
@ ^ [B3: set_b] : ( ord_less_eq_set_b @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_814_finite__Collect__subsets,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B3: set_a] : ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_815_card__Un__Int,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( finite_finite_list_a @ B )
=> ( ( plus_plus_nat @ ( finite_card_list_a @ A2 ) @ ( finite_card_list_a @ B ) )
= ( plus_plus_nat @ ( finite_card_list_a @ ( sup_sup_set_list_a @ A2 @ B ) ) @ ( finite_card_list_a @ ( inf_inf_set_list_a @ A2 @ B ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_816_card__Un__Int,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_set_a @ B )
=> ( ( plus_plus_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B ) )
= ( plus_plus_nat @ ( finite_card_set_a @ ( sup_sup_set_set_a @ A2 @ B ) ) @ ( finite_card_set_a @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_817_card__Un__Int,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ( plus_plus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) )
= ( plus_plus_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A2 @ B ) ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_818_card__Un__Int,axiom,
! [A2: set_b,B: set_b] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ B )
=> ( ( plus_plus_nat @ ( finite_card_b @ A2 ) @ ( finite_card_b @ B ) )
= ( plus_plus_nat @ ( finite_card_b @ ( sup_sup_set_b @ A2 @ B ) ) @ ( finite_card_b @ ( inf_inf_set_b @ A2 @ B ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_819_card__Un__Int,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ( plus_plus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) )
= ( plus_plus_nat @ ( finite_card_a @ ( sup_sup_set_a @ A2 @ B ) ) @ ( finite_card_a @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_820_fin__digraph__axioms,axiom,
fin_digraph_a_b @ g ).
% fin_digraph_axioms
thf(fact_821_fin__digraph__del__vert,axiom,
! [U: a] : ( fin_digraph_a_b @ ( pre_del_vert_a_b @ g @ U ) ) ).
% fin_digraph_del_vert
thf(fact_822_finite__Collect__disjI,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X: list_a] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
& ( finite_finite_list_a @ ( collect_list_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_823_finite__Collect__disjI,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X: set_a] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_set_a @ ( collect_set_a @ P ) )
& ( finite_finite_set_a @ ( collect_set_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_824_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_825_finite__Collect__disjI,axiom,
! [P: b > $o,Q: b > $o] :
( ( finite_finite_b
@ ( collect_b
@ ^ [X: b] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_b @ ( collect_b @ P ) )
& ( finite_finite_b @ ( collect_b @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_826_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_827_finite__Collect__conjI,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
| ( finite_finite_list_a @ ( collect_list_a @ Q ) ) )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X: list_a] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_828_finite__Collect__conjI,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ( finite_finite_set_a @ ( collect_set_a @ P ) )
| ( finite_finite_set_a @ ( collect_set_a @ Q ) ) )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X: set_a] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_829_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_830_finite__Collect__conjI,axiom,
! [P: b > $o,Q: b > $o] :
( ( ( finite_finite_b @ ( collect_b @ P ) )
| ( finite_finite_b @ ( collect_b @ Q ) ) )
=> ( finite_finite_b
@ ( collect_b
@ ^ [X: b] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_831_finite__Collect__conjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_832_finite__insert,axiom,
! [A: list_a,A2: set_list_a] :
( ( finite_finite_list_a @ ( insert_list_a @ A @ A2 ) )
= ( finite_finite_list_a @ A2 ) ) ).
% finite_insert
thf(fact_833_finite__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( finite_finite_set_a @ ( insert_set_a @ A @ A2 ) )
= ( finite_finite_set_a @ A2 ) ) ).
% finite_insert
thf(fact_834_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_835_finite__insert,axiom,
! [A: b,A2: set_b] :
( ( finite_finite_b @ ( insert_b @ A @ A2 ) )
= ( finite_finite_b @ A2 ) ) ).
% finite_insert
thf(fact_836_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_837_finite__Int,axiom,
! [F3: set_list_a,G: set_list_a] :
( ( ( finite_finite_list_a @ F3 )
| ( finite_finite_list_a @ G ) )
=> ( finite_finite_list_a @ ( inf_inf_set_list_a @ F3 @ G ) ) ) ).
% finite_Int
thf(fact_838_finite__Int,axiom,
! [F3: set_set_a,G: set_set_a] :
( ( ( finite_finite_set_a @ F3 )
| ( finite_finite_set_a @ G ) )
=> ( finite_finite_set_a @ ( inf_inf_set_set_a @ F3 @ G ) ) ) ).
% finite_Int
thf(fact_839_finite__Int,axiom,
! [F3: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F3 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F3 @ G ) ) ) ).
% finite_Int
thf(fact_840_finite__Int,axiom,
! [F3: set_b,G: set_b] :
( ( ( finite_finite_b @ F3 )
| ( finite_finite_b @ G ) )
=> ( finite_finite_b @ ( inf_inf_set_b @ F3 @ G ) ) ) ).
% finite_Int
thf(fact_841_finite__Int,axiom,
! [F3: set_a,G: set_a] :
( ( ( finite_finite_a @ F3 )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F3 @ G ) ) ) ).
% finite_Int
thf(fact_842_finite__Un,axiom,
! [F3: set_list_a,G: set_list_a] :
( ( finite_finite_list_a @ ( sup_sup_set_list_a @ F3 @ G ) )
= ( ( finite_finite_list_a @ F3 )
& ( finite_finite_list_a @ G ) ) ) ).
% finite_Un
thf(fact_843_finite__Un,axiom,
! [F3: set_set_a,G: set_set_a] :
( ( finite_finite_set_a @ ( sup_sup_set_set_a @ F3 @ G ) )
= ( ( finite_finite_set_a @ F3 )
& ( finite_finite_set_a @ G ) ) ) ).
% finite_Un
thf(fact_844_finite__Un,axiom,
! [F3: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F3 @ G ) )
= ( ( finite_finite_nat @ F3 )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_845_finite__Un,axiom,
! [F3: set_b,G: set_b] :
( ( finite_finite_b @ ( sup_sup_set_b @ F3 @ G ) )
= ( ( finite_finite_b @ F3 )
& ( finite_finite_b @ G ) ) ) ).
% finite_Un
thf(fact_846_finite__Un,axiom,
! [F3: set_a,G: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F3 @ G ) )
= ( ( finite_finite_a @ F3 )
& ( finite_finite_a @ G ) ) ) ).
% finite_Un
thf(fact_847_fin__sp__costs__finite,axiom,
! [F: b > real] : ( finite7198162374296863863_ereal @ ( graph_7485366578106294827ts_a_b @ g @ F ) ) ).
% fin_sp_costs_finite
thf(fact_848_sp__costs__finite,axiom,
! [F: b > real] : ( finite7198162374296863863_ereal @ ( graph_1574344591923819902ts_a_b @ g @ F ) ) ).
% sp_costs_finite
thf(fact_849_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B )
& ( R @ X3 @ Xa2 ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_850_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_b,R: nat > b > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_b @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa2: b] :
( ( member_b @ Xa2 @ B )
& ( R @ X3 @ Xa2 ) ) )
=> ? [X3: b] :
( ( member_b @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_851_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_a,R: nat > a > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ B )
& ( R @ X3 @ Xa2 ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_852_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B: set_nat,R: b > nat > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B )
& ( R @ X3 @ Xa2 ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A4: b] :
( ( member_b @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_853_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B: set_b,R: b > b > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ B )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ? [Xa2: b] :
( ( member_b @ Xa2 @ B )
& ( R @ X3 @ Xa2 ) ) )
=> ? [X3: b] :
( ( member_b @ X3 @ B )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A4: b] :
( ( member_b @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_854_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B: set_a,R: b > a > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ B )
& ( R @ X3 @ Xa2 ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A4: b] :
( ( member_b @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_855_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B: set_nat,R: a > nat > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B )
& ( R @ X3 @ Xa2 ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_856_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B: set_b,R: a > b > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_b @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ? [Xa2: b] :
( ( member_b @ Xa2 @ B )
& ( R @ X3 @ Xa2 ) ) )
=> ? [X3: b] :
( ( member_b @ X3 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_857_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B: set_a,R: a > a > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ B )
& ( R @ X3 @ Xa2 ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_858_pigeonhole__infinite__rel,axiom,
! [A2: set_list_a,B: set_nat,R: list_a > nat > $o] :
( ~ ( finite_finite_list_a @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B )
& ( R @ X3 @ Xa2 ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_list_a
@ ( collect_list_a
@ ^ [A4: list_a] :
( ( member_list_a @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_859_not__finite__existsD,axiom,
! [P: list_a > $o] :
( ~ ( finite_finite_list_a @ ( collect_list_a @ P ) )
=> ? [X_1: list_a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_860_not__finite__existsD,axiom,
! [P: set_a > $o] :
( ~ ( finite_finite_set_a @ ( collect_set_a @ P ) )
=> ? [X_1: set_a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_861_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_862_not__finite__existsD,axiom,
! [P: b > $o] :
( ~ ( finite_finite_b @ ( collect_b @ P ) )
=> ? [X_1: b] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_863_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_864_finite__has__minimal2,axiom,
! [A2: set_set_b,A: set_b] :
( ( finite_finite_set_b @ A2 )
=> ( ( member_set_b @ A @ A2 )
=> ? [X3: set_b] :
( ( member_set_b @ X3 @ A2 )
& ( ord_less_eq_set_b @ X3 @ A )
& ! [Xa2: set_b] :
( ( member_set_b @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_b @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_865_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_866_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ X3 @ A )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_867_finite__has__maximal2,axiom,
! [A2: set_set_b,A: set_b] :
( ( finite_finite_set_b @ A2 )
=> ( ( member_set_b @ A @ A2 )
=> ? [X3: set_b] :
( ( member_set_b @ X3 @ A2 )
& ( ord_less_eq_set_b @ A @ X3 )
& ! [Xa2: set_b] :
( ( member_set_b @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_b @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_868_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_869_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ A @ X3 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_870_infinite__imp__nonempty,axiom,
! [S: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ( S != bot_bot_set_list_a ) ) ).
% infinite_imp_nonempty
thf(fact_871_infinite__imp__nonempty,axiom,
! [S: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ( S != bot_bot_set_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_872_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_873_infinite__imp__nonempty,axiom,
! [S: set_b] :
( ~ ( finite_finite_b @ S )
=> ( S != bot_bot_set_b ) ) ).
% infinite_imp_nonempty
thf(fact_874_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_875_finite_OemptyI,axiom,
finite_finite_list_a @ bot_bot_set_list_a ).
% finite.emptyI
thf(fact_876_finite_OemptyI,axiom,
finite_finite_set_a @ bot_bot_set_set_a ).
% finite.emptyI
thf(fact_877_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_878_finite_OemptyI,axiom,
finite_finite_b @ bot_bot_set_b ).
% finite.emptyI
thf(fact_879_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_880_rev__finite__subset,axiom,
! [B: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B )
=> ( finite_finite_list_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_881_rev__finite__subset,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_882_rev__finite__subset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_883_rev__finite__subset,axiom,
! [B: set_b,A2: set_b] :
( ( finite_finite_b @ B )
=> ( ( ord_less_eq_set_b @ A2 @ B )
=> ( finite_finite_b @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_884_rev__finite__subset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_885_infinite__super,axiom,
! [S: set_list_a,T: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ T )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ T ) ) ) ).
% infinite_super
thf(fact_886_infinite__super,axiom,
! [S: set_set_a,T: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S @ T )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ T ) ) ) ).
% infinite_super
thf(fact_887_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_888_infinite__super,axiom,
! [S: set_b,T: set_b] :
( ( ord_less_eq_set_b @ S @ T )
=> ( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ T ) ) ) ).
% infinite_super
thf(fact_889_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_890_finite__subset,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B )
=> ( ( finite_finite_list_a @ B )
=> ( finite_finite_list_a @ A2 ) ) ) ).
% finite_subset
thf(fact_891_finite__subset,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( finite_finite_set_a @ B )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% finite_subset
thf(fact_892_finite__subset,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_893_finite__subset,axiom,
! [A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( finite_finite_b @ B )
=> ( finite_finite_b @ A2 ) ) ) ).
% finite_subset
thf(fact_894_finite__subset,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_895_finite_OinsertI,axiom,
! [A2: set_list_a,A: list_a] :
( ( finite_finite_list_a @ A2 )
=> ( finite_finite_list_a @ ( insert_list_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_896_finite_OinsertI,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite_finite_set_a @ ( insert_set_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_897_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_898_finite_OinsertI,axiom,
! [A2: set_b,A: b] :
( ( finite_finite_b @ A2 )
=> ( finite_finite_b @ ( insert_b @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_899_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_900_finite__UnI,axiom,
! [F3: set_list_a,G: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ( finite_finite_list_a @ G )
=> ( finite_finite_list_a @ ( sup_sup_set_list_a @ F3 @ G ) ) ) ) ).
% finite_UnI
thf(fact_901_finite__UnI,axiom,
! [F3: set_set_a,G: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( finite_finite_set_a @ G )
=> ( finite_finite_set_a @ ( sup_sup_set_set_a @ F3 @ G ) ) ) ) ).
% finite_UnI
thf(fact_902_finite__UnI,axiom,
! [F3: set_nat,G: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F3 @ G ) ) ) ) ).
% finite_UnI
thf(fact_903_finite__UnI,axiom,
! [F3: set_b,G: set_b] :
( ( finite_finite_b @ F3 )
=> ( ( finite_finite_b @ G )
=> ( finite_finite_b @ ( sup_sup_set_b @ F3 @ G ) ) ) ) ).
% finite_UnI
thf(fact_904_finite__UnI,axiom,
! [F3: set_a,G: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( finite_finite_a @ G )
=> ( finite_finite_a @ ( sup_sup_set_a @ F3 @ G ) ) ) ) ).
% finite_UnI
thf(fact_905_Un__infinite,axiom,
! [S: set_list_a,T: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( sup_sup_set_list_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_906_Un__infinite,axiom,
! [S: set_set_a,T: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( sup_sup_set_set_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_907_Un__infinite,axiom,
! [S: set_nat,T: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_908_Un__infinite,axiom,
! [S: set_b,T: set_b] :
( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ ( sup_sup_set_b @ S @ T ) ) ) ).
% Un_infinite
thf(fact_909_Un__infinite,axiom,
! [S: set_a,T: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_910_infinite__Un,axiom,
! [S: set_list_a,T: set_list_a] :
( ( ~ ( finite_finite_list_a @ ( sup_sup_set_list_a @ S @ T ) ) )
= ( ~ ( finite_finite_list_a @ S )
| ~ ( finite_finite_list_a @ T ) ) ) ).
% infinite_Un
thf(fact_911_infinite__Un,axiom,
! [S: set_set_a,T: set_set_a] :
( ( ~ ( finite_finite_set_a @ ( sup_sup_set_set_a @ S @ T ) ) )
= ( ~ ( finite_finite_set_a @ S )
| ~ ( finite_finite_set_a @ T ) ) ) ).
% infinite_Un
thf(fact_912_infinite__Un,axiom,
! [S: set_nat,T: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_Un
thf(fact_913_infinite__Un,axiom,
! [S: set_b,T: set_b] :
( ( ~ ( finite_finite_b @ ( sup_sup_set_b @ S @ T ) ) )
= ( ~ ( finite_finite_b @ S )
| ~ ( finite_finite_b @ T ) ) ) ).
% infinite_Un
thf(fact_914_infinite__Un,axiom,
! [S: set_a,T: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T ) ) ) ).
% infinite_Un
thf(fact_915_card__insert__le,axiom,
! [A2: set_b,X4: b] : ( ord_less_eq_nat @ ( finite_card_b @ A2 ) @ ( finite_card_b @ ( insert_b @ X4 @ A2 ) ) ) ).
% card_insert_le
thf(fact_916_card__insert__le,axiom,
! [A2: set_a,X4: a] : ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ ( insert_a @ X4 @ A2 ) ) ) ).
% card_insert_le
thf(fact_917_card__insert__le,axiom,
! [A2: set_nat,X4: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( insert_nat @ X4 @ A2 ) ) ) ).
% card_insert_le
thf(fact_918_finite__has__maximal,axiom,
! [A2: set_set_b] :
( ( finite_finite_set_b @ A2 )
=> ( ( A2 != bot_bot_set_set_b )
=> ? [X3: set_b] :
( ( member_set_b @ X3 @ A2 )
& ! [Xa2: set_b] :
( ( member_set_b @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_b @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_919_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_920_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_921_finite__has__minimal,axiom,
! [A2: set_set_b] :
( ( finite_finite_set_b @ A2 )
=> ( ( A2 != bot_bot_set_set_b )
=> ? [X3: set_b] :
( ( member_set_b @ X3 @ A2 )
& ! [Xa2: set_b] :
( ( member_set_b @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_b @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_922_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_923_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_924_infinite__finite__induct,axiom,
! [P: set_list_a > $o,A2: set_list_a] :
( ! [A7: set_list_a] :
( ~ ( finite_finite_list_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X3: list_a,F4: set_list_a] :
( ( finite_finite_list_a @ F4 )
=> ( ~ ( member_list_a @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_list_a @ X3 @ F4 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_925_infinite__finite__induct,axiom,
! [P: set_set_a > $o,A2: set_set_a] :
( ! [A7: set_set_a] :
( ~ ( finite_finite_set_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X3: set_a,F4: set_set_a] :
( ( finite_finite_set_a @ F4 )
=> ( ~ ( member_set_a @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_set_a @ X3 @ F4 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_926_infinite__finite__induct,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ! [A7: set_nat] :
( ~ ( finite_finite_nat @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X3 @ F4 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_927_infinite__finite__induct,axiom,
! [P: set_b > $o,A2: set_b] :
( ! [A7: set_b] :
( ~ ( finite_finite_b @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_b )
=> ( ! [X3: b,F4: set_b] :
( ( finite_finite_b @ F4 )
=> ( ~ ( member_b @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_b @ X3 @ F4 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_928_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A7: set_a] :
( ~ ( finite_finite_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X3 @ F4 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_929_finite__ne__induct,axiom,
! [F3: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F3 )
=> ( ( F3 != bot_bot_set_list_a )
=> ( ! [X3: list_a] : ( P @ ( insert_list_a @ X3 @ bot_bot_set_list_a ) )
=> ( ! [X3: list_a,F4: set_list_a] :
( ( finite_finite_list_a @ F4 )
=> ( ( F4 != bot_bot_set_list_a )
=> ( ~ ( member_list_a @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_list_a @ X3 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_930_finite__ne__induct,axiom,
! [F3: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F3 )
=> ( ( F3 != bot_bot_set_set_a )
=> ( ! [X3: set_a] : ( P @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
=> ( ! [X3: set_a,F4: set_set_a] :
( ( finite_finite_set_a @ F4 )
=> ( ( F4 != bot_bot_set_set_a )
=> ( ~ ( member_set_a @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_set_a @ X3 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_931_finite__ne__induct,axiom,
! [F3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X3 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_932_finite__ne__induct,axiom,
! [F3: set_b,P: set_b > $o] :
( ( finite_finite_b @ F3 )
=> ( ( F3 != bot_bot_set_b )
=> ( ! [X3: b] : ( P @ ( insert_b @ X3 @ bot_bot_set_b ) )
=> ( ! [X3: b,F4: set_b] :
( ( finite_finite_b @ F4 )
=> ( ( F4 != bot_bot_set_b )
=> ( ~ ( member_b @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_b @ X3 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_933_finite__ne__induct,axiom,
! [F3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ! [X3: a] : ( P @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( F4 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X3 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_934_finite__induct,axiom,
! [F3: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F3 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X3: list_a,F4: set_list_a] :
( ( finite_finite_list_a @ F4 )
=> ( ~ ( member_list_a @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_list_a @ X3 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_935_finite__induct,axiom,
! [F3: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F3 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X3: set_a,F4: set_set_a] :
( ( finite_finite_set_a @ F4 )
=> ( ~ ( member_set_a @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_set_a @ X3 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_936_finite__induct,axiom,
! [F3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X3 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_937_finite__induct,axiom,
! [F3: set_b,P: set_b > $o] :
( ( finite_finite_b @ F3 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [X3: b,F4: set_b] :
( ( finite_finite_b @ F4 )
=> ( ~ ( member_b @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_b @ X3 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_938_finite__induct,axiom,
! [F3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X3 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_939_finite_Osimps,axiom,
( finite_finite_list_a
= ( ^ [A4: set_list_a] :
( ( A4 = bot_bot_set_list_a )
| ? [A3: set_list_a,B4: list_a] :
( ( A4
= ( insert_list_a @ B4 @ A3 ) )
& ( finite_finite_list_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_940_finite_Osimps,axiom,
( finite_finite_set_a
= ( ^ [A4: set_set_a] :
( ( A4 = bot_bot_set_set_a )
| ? [A3: set_set_a,B4: set_a] :
( ( A4
= ( insert_set_a @ B4 @ A3 ) )
& ( finite_finite_set_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_941_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A3: set_nat,B4: nat] :
( ( A4
= ( insert_nat @ B4 @ A3 ) )
& ( finite_finite_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_942_finite_Osimps,axiom,
( finite_finite_b
= ( ^ [A4: set_b] :
( ( A4 = bot_bot_set_b )
| ? [A3: set_b,B4: b] :
( ( A4
= ( insert_b @ B4 @ A3 ) )
& ( finite_finite_b @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_943_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A4: set_a] :
( ( A4 = bot_bot_set_a )
| ? [A3: set_a,B4: a] :
( ( A4
= ( insert_a @ B4 @ A3 ) )
& ( finite_finite_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_944_finite_Ocases,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( ( A != bot_bot_set_list_a )
=> ~ ! [A7: set_list_a] :
( ? [A6: list_a] :
( A
= ( insert_list_a @ A6 @ A7 ) )
=> ~ ( finite_finite_list_a @ A7 ) ) ) ) ).
% finite.cases
thf(fact_945_finite_Ocases,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ~ ! [A7: set_set_a] :
( ? [A6: set_a] :
( A
= ( insert_set_a @ A6 @ A7 ) )
=> ~ ( finite_finite_set_a @ A7 ) ) ) ) ).
% finite.cases
thf(fact_946_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A7: set_nat] :
( ? [A6: nat] :
( A
= ( insert_nat @ A6 @ A7 ) )
=> ~ ( finite_finite_nat @ A7 ) ) ) ) ).
% finite.cases
thf(fact_947_finite_Ocases,axiom,
! [A: set_b] :
( ( finite_finite_b @ A )
=> ( ( A != bot_bot_set_b )
=> ~ ! [A7: set_b] :
( ? [A6: b] :
( A
= ( insert_b @ A6 @ A7 ) )
=> ~ ( finite_finite_b @ A7 ) ) ) ) ).
% finite.cases
thf(fact_948_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A7: set_a] :
( ? [A6: a] :
( A
= ( insert_a @ A6 @ A7 ) )
=> ~ ( finite_finite_a @ A7 ) ) ) ) ).
% finite.cases
thf(fact_949_infinite__arbitrarily__large,axiom,
! [A2: set_list_a,N: nat] :
( ~ ( finite_finite_list_a @ A2 )
=> ? [B6: set_list_a] :
( ( finite_finite_list_a @ B6 )
& ( ( finite_card_list_a @ B6 )
= N )
& ( ord_le8861187494160871172list_a @ B6 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_950_infinite__arbitrarily__large,axiom,
! [A2: set_set_a,N: nat] :
( ~ ( finite_finite_set_a @ A2 )
=> ? [B6: set_set_a] :
( ( finite_finite_set_a @ B6 )
& ( ( finite_card_set_a @ B6 )
= N )
& ( ord_le3724670747650509150_set_a @ B6 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_951_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ( finite_card_nat @ B6 )
= N )
& ( ord_less_eq_set_nat @ B6 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_952_infinite__arbitrarily__large,axiom,
! [A2: set_b,N: nat] :
( ~ ( finite_finite_b @ A2 )
=> ? [B6: set_b] :
( ( finite_finite_b @ B6 )
& ( ( finite_card_b @ B6 )
= N )
& ( ord_less_eq_set_b @ B6 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_953_infinite__arbitrarily__large,axiom,
! [A2: set_a,N: nat] :
( ~ ( finite_finite_a @ A2 )
=> ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ( finite_card_a @ B6 )
= N )
& ( ord_less_eq_set_a @ B6 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_954_card__subset__eq,axiom,
! [B: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B )
=> ( ( ( finite_card_list_a @ A2 )
= ( finite_card_list_a @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_955_card__subset__eq,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ( finite_card_set_a @ A2 )
= ( finite_card_set_a @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_956_card__subset__eq,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_957_card__subset__eq,axiom,
! [B: set_b,A2: set_b] :
( ( finite_finite_b @ B )
=> ( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( ( finite_card_b @ A2 )
= ( finite_card_b @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_958_card__subset__eq,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_959_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_list_a,C: nat] :
( ! [G2: set_list_a] :
( ( ord_le8861187494160871172list_a @ G2 @ F3 )
=> ( ( finite_finite_list_a @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ G2 ) @ C ) ) )
=> ( ( finite_finite_list_a @ F3 )
& ( ord_less_eq_nat @ ( finite_card_list_a @ F3 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_960_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_set_a,C: nat] :
( ! [G2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ G2 @ F3 )
=> ( ( finite_finite_set_a @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ G2 ) @ C ) ) )
=> ( ( finite_finite_set_a @ F3 )
& ( ord_less_eq_nat @ ( finite_card_set_a @ F3 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_961_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_nat,C: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F3 )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C ) ) )
=> ( ( finite_finite_nat @ F3 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F3 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_962_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_b,C: nat] :
( ! [G2: set_b] :
( ( ord_less_eq_set_b @ G2 @ F3 )
=> ( ( finite_finite_b @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_b @ G2 ) @ C ) ) )
=> ( ( finite_finite_b @ F3 )
& ( ord_less_eq_nat @ ( finite_card_b @ F3 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_963_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_a,C: nat] :
( ! [G2: set_a] :
( ( ord_less_eq_set_a @ G2 @ F3 )
=> ( ( finite_finite_a @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ G2 ) @ C ) ) )
=> ( ( finite_finite_a @ F3 )
& ( ord_less_eq_nat @ ( finite_card_a @ F3 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_964_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_list_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_list_a @ S ) )
=> ~ ! [T3: set_list_a] :
( ( ord_le8861187494160871172list_a @ T3 @ S )
=> ( ( ( finite_card_list_a @ T3 )
= N )
=> ~ ( finite_finite_list_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_965_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ S ) )
=> ~ ! [T3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ T3 @ S )
=> ( ( ( finite_card_set_a @ T3 )
= N )
=> ~ ( finite_finite_set_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_966_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S )
=> ( ( ( finite_card_nat @ T3 )
= N )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_967_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_b] :
( ( ord_less_eq_nat @ N @ ( finite_card_b @ S ) )
=> ~ ! [T3: set_b] :
( ( ord_less_eq_set_b @ T3 @ S )
=> ( ( ( finite_card_b @ T3 )
= N )
=> ~ ( finite_finite_b @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_968_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
=> ~ ! [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ S )
=> ( ( ( finite_card_a @ T3 )
= N )
=> ~ ( finite_finite_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_969_exists__subset__between,axiom,
! [A2: set_list_a,N: nat,C: set_list_a] :
( ( ord_less_eq_nat @ ( finite_card_list_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_list_a @ C ) )
=> ( ( ord_le8861187494160871172list_a @ A2 @ C )
=> ( ( finite_finite_list_a @ C )
=> ? [B6: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B6 )
& ( ord_le8861187494160871172list_a @ B6 @ C )
& ( ( finite_card_list_a @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_970_exists__subset__between,axiom,
! [A2: set_set_a,N: nat,C: set_set_a] :
( ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ C ) )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ( finite_finite_set_a @ C )
=> ? [B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B6 )
& ( ord_le3724670747650509150_set_a @ B6 @ C )
& ( ( finite_card_set_a @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_971_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( finite_finite_nat @ C )
=> ? [B6: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B6 )
& ( ord_less_eq_set_nat @ B6 @ C )
& ( ( finite_card_nat @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_972_exists__subset__between,axiom,
! [A2: set_b,N: nat,C: set_b] :
( ( ord_less_eq_nat @ ( finite_card_b @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_b @ C ) )
=> ( ( ord_less_eq_set_b @ A2 @ C )
=> ( ( finite_finite_b @ C )
=> ? [B6: set_b] :
( ( ord_less_eq_set_b @ A2 @ B6 )
& ( ord_less_eq_set_b @ B6 @ C )
& ( ( finite_card_b @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_973_exists__subset__between,axiom,
! [A2: set_a,N: nat,C: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C ) )
=> ( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( finite_finite_a @ C )
=> ? [B6: set_a] :
( ( ord_less_eq_set_a @ A2 @ B6 )
& ( ord_less_eq_set_a @ B6 @ C )
& ( ( finite_card_a @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_974_card__seteq,axiom,
! [B: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_list_a @ B ) @ ( finite_card_list_a @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_975_card__seteq,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ B ) @ ( finite_card_set_a @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_976_card__seteq,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_977_card__seteq,axiom,
! [B: set_b,A2: set_b] :
( ( finite_finite_b @ B )
=> ( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_b @ B ) @ ( finite_card_b @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_978_card__seteq,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_979_card__mono,axiom,
! [B: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ A2 ) @ ( finite_card_list_a @ B ) ) ) ) ).
% card_mono
thf(fact_980_card__mono,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B ) ) ) ) ).
% card_mono
thf(fact_981_card__mono,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_982_card__mono,axiom,
! [B: set_b,A2: set_b] :
( ( finite_finite_b @ B )
=> ( ( ord_less_eq_set_b @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_b @ A2 ) @ ( finite_card_b @ B ) ) ) ) ).
% card_mono
thf(fact_983_card__mono,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ).
% card_mono
thf(fact_984_card__Un__le,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A2 @ B ) ) @ ( plus_plus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ).
% card_Un_le
thf(fact_985_card__Un__le,axiom,
! [A2: set_b,B: set_b] : ( ord_less_eq_nat @ ( finite_card_b @ ( sup_sup_set_b @ A2 @ B ) ) @ ( plus_plus_nat @ ( finite_card_b @ A2 ) @ ( finite_card_b @ B ) ) ) ).
% card_Un_le
thf(fact_986_card__Un__le,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_nat @ ( finite_card_a @ ( sup_sup_set_a @ A2 @ B ) ) @ ( plus_plus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ).
% card_Un_le
thf(fact_987_finite__subset__induct,axiom,
! [F3: set_list_a,A2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F3 )
=> ( ( ord_le8861187494160871172list_a @ F3 @ A2 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A6: list_a,F4: set_list_a] :
( ( finite_finite_list_a @ F4 )
=> ( ( member_list_a @ A6 @ A2 )
=> ( ~ ( member_list_a @ A6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_list_a @ A6 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_988_finite__subset__induct,axiom,
! [F3: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F3 )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A6: set_a,F4: set_set_a] :
( ( finite_finite_set_a @ F4 )
=> ( ( member_set_a @ A6 @ A2 )
=> ( ~ ( member_set_a @ A6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_set_a @ A6 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_989_finite__subset__induct,axiom,
! [F3: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A6 @ A2 )
=> ( ~ ( member_nat @ A6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ A6 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_990_finite__subset__induct,axiom,
! [F3: set_b,A2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F3 )
=> ( ( ord_less_eq_set_b @ F3 @ A2 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [A6: b,F4: set_b] :
( ( finite_finite_b @ F4 )
=> ( ( member_b @ A6 @ A2 )
=> ( ~ ( member_b @ A6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_b @ A6 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_991_finite__subset__induct,axiom,
! [F3: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A6 @ A2 )
=> ( ~ ( member_a @ A6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A6 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_992_finite__subset__induct_H,axiom,
! [F3: set_list_a,A2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F3 )
=> ( ( ord_le8861187494160871172list_a @ F3 @ A2 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A6: list_a,F4: set_list_a] :
( ( finite_finite_list_a @ F4 )
=> ( ( member_list_a @ A6 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ F4 @ A2 )
=> ( ~ ( member_list_a @ A6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_list_a @ A6 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_993_finite__subset__induct_H,axiom,
! [F3: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F3 )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A6: set_a,F4: set_set_a] :
( ( finite_finite_set_a @ F4 )
=> ( ( member_set_a @ A6 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ F4 @ A2 )
=> ( ~ ( member_set_a @ A6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_set_a @ A6 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_994_finite__subset__induct_H,axiom,
! [F3: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A6 @ A2 )
=> ( ( ord_less_eq_set_nat @ F4 @ A2 )
=> ( ~ ( member_nat @ A6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ A6 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_995_finite__subset__induct_H,axiom,
! [F3: set_b,A2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F3 )
=> ( ( ord_less_eq_set_b @ F3 @ A2 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [A6: b,F4: set_b] :
( ( finite_finite_b @ F4 )
=> ( ( member_b @ A6 @ A2 )
=> ( ( ord_less_eq_set_b @ F4 @ A2 )
=> ( ~ ( member_b @ A6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_b @ A6 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_996_finite__subset__induct_H,axiom,
! [F3: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A6 @ A2 )
=> ( ( ord_less_eq_set_a @ F4 @ A2 )
=> ( ~ ( member_a @ A6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A6 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_997_vpaths__finite,axiom,
( finite_finite_list_a
@ ( collect_list_a
@ ^ [P3: list_a] : ( vertex_vpath_a_b @ P3 @ g ) ) ) ).
% vpaths_finite
thf(fact_998_digraph__def,axiom,
( digraph_a_b
= ( ^ [G3: pre_pr7278220950009878019t_unit] :
( ( fin_digraph_a_b @ G3 )
& ( loopfree_digraph_a_b @ G3 )
& ( nomulti_digraph_a_b @ G3 ) ) ) ) ).
% digraph_def
thf(fact_999_digraph_Ointro,axiom,
! [G: pre_pr7278220950009878019t_unit] :
( ( fin_digraph_a_b @ G )
=> ( ( loopfree_digraph_a_b @ G )
=> ( ( nomulti_digraph_a_b @ G )
=> ( digraph_a_b @ G ) ) ) ) ).
% digraph.intro
thf(fact_1000_pre__digraph_Oarcs__del__vert,axiom,
! [G: pre_pr7278220950009878019t_unit,U: a] :
( ( pre_ar1395965042833527383t_unit @ ( pre_del_vert_a_b @ G @ U ) )
= ( collect_b
@ ^ [A4: b] :
( ( member_b @ A4 @ ( pre_ar1395965042833527383t_unit @ G ) )
& ( ( pre_ta4931606617599662728t_unit @ G @ A4 )
!= U )
& ( ( pre_he5236287464308401016t_unit @ G @ A4 )
!= U ) ) ) ) ).
% pre_digraph.arcs_del_vert
thf(fact_1001_order__refl,axiom,
! [X4: set_b] : ( ord_less_eq_set_b @ X4 @ X4 ) ).
% order_refl
thf(fact_1002_order__refl,axiom,
! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_1003_order__refl,axiom,
! [X4: set_a] : ( ord_less_eq_set_a @ X4 @ X4 ) ).
% order_refl
thf(fact_1004_dual__order_Orefl,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ A @ A ) ).
% dual_order.refl
thf(fact_1005_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_1006_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_1007_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1008_nle__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_1009_le__cases3,axiom,
! [X4: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X4 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X4 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X4 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X4 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1010_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_b,Z3: set_b] : ( Y3 = Z3 ) )
= ( ^ [X: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
& ( ord_less_eq_set_b @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1011_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1012_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z3: set_a] : ( Y3 = Z3 ) )
= ( ^ [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1013_ord__eq__le__trans,axiom,
! [A: set_b,B2: set_b,C2: set_b] :
( ( A = B2 )
=> ( ( ord_less_eq_set_b @ B2 @ C2 )
=> ( ord_less_eq_set_b @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_1014_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_1015_ord__eq__le__trans,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( A = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_1016_ord__le__eq__trans,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_1017_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1018_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1019_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1020_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1021_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1022_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1023_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1024_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1025_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1026_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1027_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1028_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1029_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1030_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1031_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1032_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1033_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1034_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1035_last__merging__points__def,axiom,
( ( graph_2659413520663303054ts_a_b @ t )
= ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( graph_2957805489637798020ts_a_b @ t ) )
& ~ ? [Y: a] :
( ( member_a @ Y @ ( graph_2957805489637798020ts_a_b @ t ) )
& ( Y != X )
& ( reachable_a_b @ t @ X @ Y ) ) ) ) ) ).
% last_merging_points_def
thf(fact_1036_G_Olast__merging__points__def,axiom,
( ( graph_2659413520663303054ts_a_b @ g )
= ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( graph_2957805489637798020ts_a_b @ g ) )
& ~ ? [Y: a] :
( ( member_a @ Y @ ( graph_2957805489637798020ts_a_b @ g ) )
& ( Y != X )
& ( reachable_a_b @ g @ X @ Y ) ) ) ) ) ).
% G.last_merging_points_def
thf(fact_1037_symmetric__reachable_H,axiom,
! [V: a,W: a] :
( ( reachable_a_b @ g @ V @ W )
=> ( reachable_a_b @ g @ W @ V ) ) ).
% symmetric_reachable'
thf(fact_1038_G_Oreachable__trans,axiom,
! [U: a,V: a,W: a] :
( ( reachable_a_b @ g @ U @ V )
=> ( ( reachable_a_b @ g @ V @ W )
=> ( reachable_a_b @ g @ U @ W ) ) ) ).
% G.reachable_trans
thf(fact_1039_reachable__trans,axiom,
! [U: a,V: a,W: a] :
( ( reachable_a_b @ t @ U @ V )
=> ( ( reachable_a_b @ t @ V @ W )
=> ( reachable_a_b @ t @ U @ W ) ) ) ).
% reachable_trans
thf(fact_1040_reachable__verts__finite,axiom,
! [U: a] : ( finite_finite_a @ ( collect_a @ ( reachable_a_b @ g @ U ) ) ) ).
% reachable_verts_finite
thf(fact_1041_reachable__subs,axiom,
! [R2: a] : ( ord_less_eq_set_a @ ( collect_a @ ( reachable_a_b @ t @ R2 ) ) @ ( collect_a @ ( reachable_a_b @ g @ R2 ) ) ) ).
% reachable_subs
thf(fact_1042_G_Olast__branch__alt,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( graph_1747835947655717337ts_a_b @ g ) )
=> ! [Z5: a] :
( ( ( reachable_a_b @ g @ X4 @ Z5 )
& ( Z5 != X4 ) )
=> ~ ( member_a @ Z5 @ ( graph_4596510882073158607ts_a_b @ g ) ) ) ) ).
% G.last_branch_alt
thf(fact_1043_last__branch__alt,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( graph_1747835947655717337ts_a_b @ t ) )
=> ! [Z5: a] :
( ( ( reachable_a_b @ t @ X4 @ Z5 )
& ( Z5 != X4 ) )
=> ~ ( member_a @ Z5 @ ( graph_4596510882073158607ts_a_b @ t ) ) ) ) ).
% last_branch_alt
thf(fact_1044_G_Olast__merge__alt,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( graph_2659413520663303054ts_a_b @ g ) )
=> ! [Z5: a] :
( ( ( reachable_a_b @ g @ X4 @ Z5 )
& ( Z5 != X4 ) )
=> ~ ( member_a @ Z5 @ ( graph_2957805489637798020ts_a_b @ g ) ) ) ) ).
% G.last_merge_alt
thf(fact_1045_last__merge__alt,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( graph_2659413520663303054ts_a_b @ t ) )
=> ! [Z5: a] :
( ( ( reachable_a_b @ t @ X4 @ Z5 )
& ( Z5 != X4 ) )
=> ~ ( member_a @ Z5 @ ( graph_2957805489637798020ts_a_b @ t ) ) ) ) ).
% last_merge_alt
thf(fact_1046_G_Olast__branching__points__def,axiom,
( ( graph_1747835947655717337ts_a_b @ g )
= ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( graph_4596510882073158607ts_a_b @ g ) )
& ~ ? [Y: a] :
( ( member_a @ Y @ ( graph_4596510882073158607ts_a_b @ g ) )
& ( Y != X )
& ( reachable_a_b @ g @ X @ Y ) ) ) ) ) ).
% G.last_branching_points_def
thf(fact_1047_last__branching__points__def,axiom,
( ( graph_1747835947655717337ts_a_b @ t )
= ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( graph_4596510882073158607ts_a_b @ t ) )
& ~ ? [Y: a] :
( ( member_a @ Y @ ( graph_4596510882073158607ts_a_b @ t ) )
& ( Y != X )
& ( reachable_a_b @ t @ X @ Y ) ) ) ) ) ).
% last_branching_points_def
thf(fact_1048_k__nh__reachable,axiom,
! [U: a,W: b > real,V: a,K: real] :
( ( member_a @ U @ ( graph_3921080825633621230od_a_b @ t @ W @ V @ K ) )
=> ( reachable_a_b @ t @ V @ U ) ) ).
% k_nh_reachable
thf(fact_1049_G_Ok__nh__reachable,axiom,
! [U: a,W: b > real,V: a,K: real] :
( ( member_a @ U @ ( graph_3921080825633621230od_a_b @ g @ W @ V @ K ) )
=> ( reachable_a_b @ g @ V @ U ) ) ).
% G.k_nh_reachable
thf(fact_1050_connected__minimal,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ~ ( reachable_a_b @ ( pre_del_arc_a_b @ t @ E ) @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( pre_he5236287464308401016t_unit @ t @ E ) ) ) ).
% connected_minimal
thf(fact_1051_scc__of__def,axiom,
! [U: a] :
( ( digrap2937667069914300949of_a_b @ t @ U )
= ( collect_a
@ ^ [V2: a] :
( ( reachable_a_b @ t @ U @ V2 )
& ( reachable_a_b @ t @ V2 @ U ) ) ) ) ).
% scc_of_def
thf(fact_1052_G_Odel__arc__commute,axiom,
! [B2: b,A: b] :
( ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ g @ B2 ) @ A )
= ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ g @ A ) @ B2 ) ) ).
% G.del_arc_commute
thf(fact_1053_del__arc__commute,axiom,
! [B2: b,A: b] :
( ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ t @ B2 ) @ A )
= ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ t @ A ) @ B2 ) ) ).
% del_arc_commute
thf(fact_1054_G_Oscc__of__eq,axiom,
! [U: a,V: a] :
( ( member_a @ U @ ( digrap2937667069914300949of_a_b @ g @ V ) )
=> ( ( digrap2937667069914300949of_a_b @ g @ U )
= ( digrap2937667069914300949of_a_b @ g @ V ) ) ) ).
% G.scc_of_eq
thf(fact_1055_scc__of__eq,axiom,
! [U: a,V: a] :
( ( member_a @ U @ ( digrap2937667069914300949of_a_b @ t @ V ) )
=> ( ( digrap2937667069914300949of_a_b @ t @ U )
= ( digrap2937667069914300949of_a_b @ t @ V ) ) ) ).
% scc_of_eq
thf(fact_1056_G_Osource__nmem__k__nh,axiom,
! [V: a,W: b > real,K: real] :
~ ( member_a @ V @ ( graph_3921080825633621230od_a_b @ g @ W @ V @ K ) ) ).
% G.source_nmem_k_nh
thf(fact_1057_source__nmem__k__nh,axiom,
! [V: a,W: b > real,K: real] :
~ ( member_a @ V @ ( graph_3921080825633621230od_a_b @ t @ W @ V @ K ) ) ).
% source_nmem_k_nh
thf(fact_1058_G_Odel__arc__in,axiom,
! [A: b] :
( ~ ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( pre_del_arc_a_b @ g @ A )
= g ) ) ).
% G.del_arc_in
thf(fact_1059_del__arc__in,axiom,
! [A: b] :
( ~ ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( pre_del_arc_a_b @ t @ A )
= t ) ) ).
% del_arc_in
thf(fact_1060_scc__ofI__reachable_H,axiom,
! [V: a,U: a] :
( ( reachable_a_b @ g @ V @ U )
=> ( member_a @ U @ ( digrap2937667069914300949of_a_b @ g @ V ) ) ) ).
% scc_ofI_reachable'
thf(fact_1061_scc__ofI__reachable,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ g @ U @ V )
=> ( member_a @ U @ ( digrap2937667069914300949of_a_b @ g @ V ) ) ) ).
% scc_ofI_reachable
thf(fact_1062_fin__digraph__del__arc,axiom,
! [A: b] : ( fin_digraph_a_b @ ( pre_del_arc_a_b @ g @ A ) ) ).
% fin_digraph_del_arc
thf(fact_1063_k__nh__finite,axiom,
! [W: b > real,V: a,K: real] : ( finite_finite_a @ ( graph_3921080825633621230od_a_b @ g @ W @ V @ K ) ) ).
% k_nh_finite
thf(fact_1064_G_Oscc__of__def,axiom,
! [U: a] :
( ( digrap2937667069914300949of_a_b @ g @ U )
= ( collect_a
@ ^ [V2: a] :
( ( reachable_a_b @ g @ U @ V2 )
& ( reachable_a_b @ g @ V2 @ U ) ) ) ) ).
% G.scc_of_def
thf(fact_1065_G_Odel__del__arc__collapse,axiom,
! [A: b] :
( ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ g @ A ) @ A )
= ( pre_del_arc_a_b @ g @ A ) ) ).
% G.del_del_arc_collapse
thf(fact_1066_del__del__arc__collapse,axiom,
! [A: b] :
( ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ t @ A ) @ A )
= ( pre_del_arc_a_b @ t @ A ) ) ).
% del_del_arc_collapse
thf(fact_1067_G_Ohead__del__arc,axiom,
! [A: b] :
( ( pre_he5236287464308401016t_unit @ ( pre_del_arc_a_b @ g @ A ) )
= ( pre_he5236287464308401016t_unit @ g ) ) ).
% G.head_del_arc
thf(fact_1068_G_Otail__del__arc,axiom,
! [A: b] :
( ( pre_ta4931606617599662728t_unit @ ( pre_del_arc_a_b @ g @ A ) )
= ( pre_ta4931606617599662728t_unit @ g ) ) ).
% G.tail_del_arc
thf(fact_1069_head__del__arc,axiom,
! [A: b] :
( ( pre_he5236287464308401016t_unit @ ( pre_del_arc_a_b @ t @ A ) )
= ( pre_he5236287464308401016t_unit @ t ) ) ).
% head_del_arc
thf(fact_1070_tail__del__arc,axiom,
! [A: b] :
( ( pre_ta4931606617599662728t_unit @ ( pre_del_arc_a_b @ t @ A ) )
= ( pre_ta4931606617599662728t_unit @ t ) ) ).
% tail_del_arc
thf(fact_1071_verts__finite__imp__arcs__finite,axiom,
( ( finite_finite_a @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( finite_finite_b @ ( pre_ar1395965042833527383t_unit @ t ) ) ) ).
% verts_finite_imp_arcs_finite
thf(fact_1072_in__sccs__verts__conv__reachable,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ t ) )
= ( ( S != bot_bot_set_a )
& ! [X: a] :
( ( member_a @ X @ S )
=> ! [Y: a] :
( ( member_a @ Y @ S )
=> ( reachable_a_b @ t @ X @ Y ) ) )
& ! [X: a] :
( ( member_a @ X @ S )
=> ! [V2: a] :
( ~ ( member_a @ V2 @ S )
=> ( ~ ( reachable_a_b @ t @ X @ V2 )
| ~ ( reachable_a_b @ t @ V2 @ X ) ) ) ) ) ) ).
% in_sccs_verts_conv_reachable
thf(fact_1073_G_Oin__sccs__verts__conv__reachable,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ g ) )
= ( ( S != bot_bot_set_a )
& ! [X: a] :
( ( member_a @ X @ S )
=> ! [Y: a] :
( ( member_a @ Y @ S )
=> ( reachable_a_b @ g @ X @ Y ) ) )
& ! [X: a] :
( ( member_a @ X @ S )
=> ! [V2: a] :
( ~ ( member_a @ V2 @ S )
=> ( ~ ( reachable_a_b @ g @ X @ V2 )
| ~ ( reachable_a_b @ g @ V2 @ X ) ) ) ) ) ) ).
% G.in_sccs_verts_conv_reachable
thf(fact_1074_finite__sccs__verts,axiom,
finite_finite_set_a @ ( digrap2871191568752656621ts_a_b @ g ) ).
% finite_sccs_verts
thf(fact_1075_G_Oreachable__in__verts_I2_J,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ g @ U @ V )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% G.reachable_in_verts(2)
thf(fact_1076_G_Oreachable__in__verts_I1_J,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ g @ U @ V )
=> ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% G.reachable_in_verts(1)
thf(fact_1077_reachable__in__verts_I2_J,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ t @ U @ V )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% reachable_in_verts(2)
thf(fact_1078_reachable__in__verts_I1_J,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ t @ U @ V )
=> ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% reachable_in_verts(1)
thf(fact_1079_non__empty,axiom,
( ( pre_ve642382030648772252t_unit @ t )
!= bot_bot_set_a ) ).
% non_empty
thf(fact_1080_G_Obranch__in__verts,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( graph_4596510882073158607ts_a_b @ g ) )
=> ( member_a @ X4 @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% G.branch_in_verts
thf(fact_1081_branch__in__verts,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( graph_4596510882073158607ts_a_b @ t ) )
=> ( member_a @ X4 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% branch_in_verts
thf(fact_1082_G_Omerge__in__verts,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( graph_2957805489637798020ts_a_b @ g ) )
=> ( member_a @ X4 @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% G.merge_in_verts
thf(fact_1083_G_Oin__scc__of__self,axiom,
! [U: a] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( member_a @ U @ ( digrap2937667069914300949of_a_b @ g @ U ) ) ) ).
% G.in_scc_of_self
thf(fact_1084_merge__in__verts,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( graph_2957805489637798020ts_a_b @ t ) )
=> ( member_a @ X4 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% merge_in_verts
thf(fact_1085_in__scc__of__self,axiom,
! [U: a] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( member_a @ U @ ( digrap2937667069914300949of_a_b @ t @ U ) ) ) ).
% in_scc_of_self
thf(fact_1086_G_Osccs__verts__disjoint,axiom,
! [S: set_a,T: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ g ) )
=> ( ( member_set_a @ T @ ( digrap2871191568752656621ts_a_b @ g ) )
=> ( ( S != T )
=> ( ( inf_inf_set_a @ S @ T )
= bot_bot_set_a ) ) ) ) ).
% G.sccs_verts_disjoint
thf(fact_1087_sccs__verts__disjoint,axiom,
! [S: set_a,T: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ t ) )
=> ( ( member_set_a @ T @ ( digrap2871191568752656621ts_a_b @ t ) )
=> ( ( S != T )
=> ( ( inf_inf_set_a @ S @ T )
= bot_bot_set_a ) ) ) ) ).
% sccs_verts_disjoint
thf(fact_1088_finite__verts,axiom,
finite_finite_a @ ( pre_ve642382030648772252t_unit @ g ) ).
% finite_verts
thf(fact_1089_G_Overts__add__vert,axiom,
! [U: a] :
( ( pre_ve642382030648772252t_unit @ ( pre_add_vert_a_b @ g @ U ) )
= ( insert_a @ U @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% G.verts_add_vert
thf(fact_1090_verts__add__vert,axiom,
! [U: a] :
( ( pre_ve642382030648772252t_unit @ ( pre_add_vert_a_b @ t @ U ) )
= ( insert_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% verts_add_vert
thf(fact_1091_G_Otail__in__verts,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( member_a @ ( pre_ta4931606617599662728t_unit @ g @ E ) @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% G.tail_in_verts
thf(fact_1092_G_Ohead__in__verts,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( member_a @ ( pre_he5236287464308401016t_unit @ g @ E ) @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% G.head_in_verts
thf(fact_1093_tail__in__verts,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% tail_in_verts
thf(fact_1094_head__in__verts,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member_a @ ( pre_he5236287464308401016t_unit @ t @ E ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% head_in_verts
thf(fact_1095_G_Oscc__of__empty__conv,axiom,
! [U: a] :
( ( ( digrap2937667069914300949of_a_b @ g @ U )
= bot_bot_set_a )
= ( ~ ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) ) ) ) ).
% G.scc_of_empty_conv
thf(fact_1096_scc__of__empty__conv,axiom,
! [U: a] :
( ( ( digrap2937667069914300949of_a_b @ t @ U )
= bot_bot_set_a )
= ( ~ ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% scc_of_empty_conv
thf(fact_1097_G_Osccs__verts__subsets,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ g ) )
=> ( ord_less_eq_set_a @ S @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% G.sccs_verts_subsets
thf(fact_1098_sccs__verts__subsets,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ t ) )
=> ( ord_less_eq_set_a @ S @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% sccs_verts_subsets
thf(fact_1099_G_Oscc__of__in__sccs__verts,axiom,
! [U: a] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( member_set_a @ ( digrap2937667069914300949of_a_b @ g @ U ) @ ( digrap2871191568752656621ts_a_b @ g ) ) ) ).
% G.scc_of_in_sccs_verts
thf(fact_1100_scc__of__in__sccs__verts,axiom,
! [U: a] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( member_set_a @ ( digrap2937667069914300949of_a_b @ t @ U ) @ ( digrap2871191568752656621ts_a_b @ t ) ) ) ).
% scc_of_in_sccs_verts
thf(fact_1101_G_Odel__vert__add__vert,axiom,
! [U: a] :
( ~ ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( pre_del_vert_a_b @ ( pre_add_vert_a_b @ g @ U ) @ U )
= g ) ) ).
% G.del_vert_add_vert
thf(fact_1102_del__vert__add__vert,axiom,
! [U: a] :
( ~ ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( pre_del_vert_a_b @ ( pre_add_vert_a_b @ t @ U ) @ U )
= t ) ) ).
% del_vert_add_vert
thf(fact_1103_verts__T__subset__G,axiom,
ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ t ) @ ( pre_ve642382030648772252t_unit @ g ) ).
% verts_T_subset_G
thf(fact_1104_G_Oreachable__refl,axiom,
! [V: a] :
( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( reachable_a_b @ g @ V @ V ) ) ).
% G.reachable_refl
thf(fact_1105_reachable__refl,axiom,
! [V: a] :
( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( reachable_a_b @ t @ V @ V ) ) ).
% reachable_refl
thf(fact_1106_G_Overts__del__arc,axiom,
! [A: b] :
( ( pre_ve642382030648772252t_unit @ ( pre_del_arc_a_b @ g @ A ) )
= ( pre_ve642382030648772252t_unit @ g ) ) ).
% G.verts_del_arc
thf(fact_1107_verts__del__arc,axiom,
! [A: b] :
( ( pre_ve642382030648772252t_unit @ ( pre_del_arc_a_b @ t @ A ) )
= ( pre_ve642382030648772252t_unit @ t ) ) ).
% verts_del_arc
thf(fact_1108_G_Ofin__digraphI,axiom,
( ( finite_finite_a @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( finite_finite_b @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( fin_digraph_a_b @ g ) ) ) ).
% G.fin_digraphI
thf(fact_1109_fin__digraphI,axiom,
( ( finite_finite_a @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( finite_finite_b @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( fin_digraph_a_b @ t ) ) ) ).
% fin_digraphI
thf(fact_1110_ex__leaf,axiom,
( ( finite_finite_a @ ( pre_ve642382030648772252t_unit @ t ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( pre_ve642382030648772252t_unit @ t ) )
& ( shorte1213025427933718126af_a_b @ t @ X3 ) ) ) ).
% ex_leaf
thf(fact_1111_subgraph__no__merge__chain,axiom,
! [C: pre_pr7278220950009878019t_unit] :
( ( shorte3657265928840388360ph_a_b @ C @ t )
=> ( ( ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ C )
@ ( minus_minus_set_a @ ( pre_ve642382030648772252t_unit @ t )
@ ( collect_a
@ ^ [X: a] :
? [Y: a] :
( ( member_a @ Y @ ( graph_2957805489637798020ts_a_b @ t ) )
& ( reachable_a_b @ t @ X @ Y ) ) ) ) )
=> ( graph_8150681439568091980in_a_b @ C ) ) ) ).
% subgraph_no_merge_chain
thf(fact_1112_G_Osubgraph__no__merge__chain,axiom,
! [C: pre_pr7278220950009878019t_unit] :
( ( shorte3657265928840388360ph_a_b @ C @ g )
=> ( ( ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ C )
@ ( minus_minus_set_a @ ( pre_ve642382030648772252t_unit @ g )
@ ( collect_a
@ ^ [X: a] :
? [Y: a] :
( ( member_a @ Y @ ( graph_2957805489637798020ts_a_b @ g ) )
& ( reachable_a_b @ g @ X @ Y ) ) ) ) )
=> ( graph_8150681439568091980in_a_b @ C ) ) ) ).
% G.subgraph_no_merge_chain
thf(fact_1113_subgraph__no__branch__chain,axiom,
! [C: pre_pr7278220950009878019t_unit] :
( ( shorte3657265928840388360ph_a_b @ C @ t )
=> ( ( ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ C )
@ ( minus_minus_set_a @ ( pre_ve642382030648772252t_unit @ t )
@ ( collect_a
@ ^ [X: a] :
? [Y: a] :
( ( member_a @ Y @ ( graph_4596510882073158607ts_a_b @ t ) )
& ( reachable_a_b @ t @ X @ Y ) ) ) ) )
=> ( graph_3890552050688490787in_a_b @ C ) ) ) ).
% subgraph_no_branch_chain
thf(fact_1114_G_Overts__del__vert,axiom,
! [U: a] :
( ( pre_ve642382030648772252t_unit @ ( pre_del_vert_a_b @ g @ U ) )
= ( minus_minus_set_a @ ( pre_ve642382030648772252t_unit @ g ) @ ( insert_a @ U @ bot_bot_set_a ) ) ) ).
% G.verts_del_vert
thf(fact_1115_verts__del__vert,axiom,
! [U: a] :
( ( pre_ve642382030648772252t_unit @ ( pre_del_vert_a_b @ t @ U ) )
= ( minus_minus_set_a @ ( pre_ve642382030648772252t_unit @ t ) @ ( insert_a @ U @ bot_bot_set_a ) ) ) ).
% verts_del_vert
thf(fact_1116_G_Osubgraph__no__branch__chain,axiom,
! [C: pre_pr7278220950009878019t_unit] :
( ( shorte3657265928840388360ph_a_b @ C @ g )
=> ( ( ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ C )
@ ( minus_minus_set_a @ ( pre_ve642382030648772252t_unit @ g )
@ ( collect_a
@ ^ [X: a] :
? [Y: a] :
( ( member_a @ Y @ ( graph_4596510882073158607ts_a_b @ g ) )
& ( reachable_a_b @ g @ X @ Y ) ) ) ) )
=> ( graph_3890552050688490787in_a_b @ C ) ) ) ).
% G.subgraph_no_branch_chain
thf(fact_1117_arcs__del__leaf,axiom,
! [E: b,V: a] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( ( pre_he5236287464308401016t_unit @ t @ E )
= V )
=> ( ( shorte1213025427933718126af_a_b @ t @ V )
=> ( ( pre_ar1395965042833527383t_unit @ ( pre_del_vert_a_b @ t @ V ) )
= ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ t ) @ ( insert_b @ E @ bot_bot_set_b ) ) ) ) ) ) ).
% arcs_del_leaf
thf(fact_1118_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1119_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1120_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1121_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1122_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1123_G_Oarcs__del__arc,axiom,
! [A: b] :
( ( pre_ar1395965042833527383t_unit @ ( pre_del_arc_a_b @ g @ A ) )
= ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ g ) @ ( insert_b @ A @ bot_bot_set_b ) ) ) ).
% G.arcs_del_arc
thf(fact_1124_arcs__del__arc,axiom,
! [A: b] :
( ( pre_ar1395965042833527383t_unit @ ( pre_del_arc_a_b @ t @ A ) )
= ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ t ) @ ( insert_b @ A @ bot_bot_set_b ) ) ) ).
% arcs_del_arc
thf(fact_1125_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1126_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1127_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1128_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1129_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1130_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1131_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1132_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1133_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1134_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1135_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1136_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1137_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1138_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1139_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1140_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1141_out__arcs__del__arc__iff,axiom,
! [A: b,U: a] :
( ( ( ( pre_ta4931606617599662728t_unit @ t @ A )
= U )
=> ( ( out_arcs_a_b @ ( pre_del_arc_a_b @ t @ A ) @ U )
= ( minus_minus_set_b @ ( out_arcs_a_b @ t @ U ) @ ( insert_b @ A @ bot_bot_set_b ) ) ) )
& ( ( ( pre_ta4931606617599662728t_unit @ t @ A )
!= U )
=> ( ( out_arcs_a_b @ ( pre_del_arc_a_b @ t @ A ) @ U )
= ( out_arcs_a_b @ t @ U ) ) ) ) ).
% out_arcs_del_arc_iff
thf(fact_1142_G_Oout__arcs__del__arc__iff,axiom,
! [A: b,U: a] :
( ( ( ( pre_ta4931606617599662728t_unit @ g @ A )
= U )
=> ( ( out_arcs_a_b @ ( pre_del_arc_a_b @ g @ A ) @ U )
= ( minus_minus_set_b @ ( out_arcs_a_b @ g @ U ) @ ( insert_b @ A @ bot_bot_set_b ) ) ) )
& ( ( ( pre_ta4931606617599662728t_unit @ g @ A )
!= U )
=> ( ( out_arcs_a_b @ ( pre_del_arc_a_b @ g @ A ) @ U )
= ( out_arcs_a_b @ g @ U ) ) ) ) ).
% G.out_arcs_del_arc_iff
thf(fact_1143_in__arcs__del__arc__iff,axiom,
! [A: b,U: a] :
( ( ( ( pre_he5236287464308401016t_unit @ t @ A )
= U )
=> ( ( in_arcs_a_b @ ( pre_del_arc_a_b @ t @ A ) @ U )
= ( minus_minus_set_b @ ( in_arcs_a_b @ t @ U ) @ ( insert_b @ A @ bot_bot_set_b ) ) ) )
& ( ( ( pre_he5236287464308401016t_unit @ t @ A )
!= U )
=> ( ( in_arcs_a_b @ ( pre_del_arc_a_b @ t @ A ) @ U )
= ( in_arcs_a_b @ t @ U ) ) ) ) ).
% in_arcs_del_arc_iff
thf(fact_1144_G_Onot__elem__no__in__arcs,axiom,
! [V: a] :
( ~ ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( in_arcs_a_b @ g @ V )
= bot_bot_set_b ) ) ).
% G.not_elem_no_in_arcs
thf(fact_1145_not__elem__no__in__arcs,axiom,
! [V: a] :
( ~ ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( in_arcs_a_b @ t @ V )
= bot_bot_set_b ) ) ).
% not_elem_no_in_arcs
thf(fact_1146_G_Onot__elem__no__out__arcs,axiom,
! [V: a] :
( ~ ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( out_arcs_a_b @ g @ V )
= bot_bot_set_b ) ) ).
% G.not_elem_no_out_arcs
thf(fact_1147_not__elem__no__out__arcs,axiom,
! [V: a] :
( ~ ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( out_arcs_a_b @ t @ V )
= bot_bot_set_b ) ) ).
% not_elem_no_out_arcs
thf(fact_1148_in__arcs__finite,axiom,
! [V: a] :
( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( finite_finite_b @ ( in_arcs_a_b @ t @ V ) ) ) ).
% in_arcs_finite
thf(fact_1149_G_Oleaf__def,axiom,
! [V: a] :
( ( shorte1213025427933718126af_a_b @ g @ V )
= ( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
& ( ( out_arcs_a_b @ g @ V )
= bot_bot_set_b ) ) ) ).
% G.leaf_def
thf(fact_1150_leaf__def,axiom,
! [V: a] :
( ( shorte1213025427933718126af_a_b @ t @ V )
= ( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) )
& ( ( out_arcs_a_b @ t @ V )
= bot_bot_set_b ) ) ) ).
% leaf_def
thf(fact_1151_G_Oarcs__del__vert2,axiom,
! [V: a] :
( ( pre_ar1395965042833527383t_unit @ ( pre_del_vert_a_b @ g @ V ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ g ) @ ( in_arcs_a_b @ g @ V ) ) @ ( out_arcs_a_b @ g @ V ) ) ) ).
% G.arcs_del_vert2
thf(fact_1152_arcs__del__vert2,axiom,
! [V: a] :
( ( pre_ar1395965042833527383t_unit @ ( pre_del_vert_a_b @ t @ V ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ t ) @ ( in_arcs_a_b @ t @ V ) ) @ ( out_arcs_a_b @ t @ V ) ) ) ).
% arcs_del_vert2
thf(fact_1153_G_Oin__arcs__del__arc__iff,axiom,
! [A: b,U: a] :
( ( ( ( pre_he5236287464308401016t_unit @ g @ A )
= U )
=> ( ( in_arcs_a_b @ ( pre_del_arc_a_b @ g @ A ) @ U )
= ( minus_minus_set_b @ ( in_arcs_a_b @ g @ U ) @ ( insert_b @ A @ bot_bot_set_b ) ) ) )
& ( ( ( pre_he5236287464308401016t_unit @ g @ A )
!= U )
=> ( ( in_arcs_a_b @ ( pre_del_arc_a_b @ g @ A ) @ U )
= ( in_arcs_a_b @ g @ U ) ) ) ) ).
% G.in_arcs_del_arc_iff
thf(fact_1154_finite__in__arcs,axiom,
! [V: a] : ( finite_finite_b @ ( in_arcs_a_b @ g @ V ) ) ).
% finite_in_arcs
thf(fact_1155_finite__out__arcs,axiom,
! [V: a] : ( finite_finite_b @ ( out_arcs_a_b @ g @ V ) ) ).
% finite_out_arcs
thf(fact_1156_verts__add__arc__conv,axiom,
! [A: b] :
( ( pre_ve642382030648772252t_unit @ ( pre_add_arc_a_b @ t @ A ) )
= ( sup_sup_set_a @ ( pre_ve642382030648772252t_unit @ t ) @ ( insert_a @ ( pre_ta4931606617599662728t_unit @ t @ A ) @ ( insert_a @ ( pre_he5236287464308401016t_unit @ t @ A ) @ bot_bot_set_a ) ) ) ) ).
% verts_add_arc_conv
thf(fact_1157_G_Overts__add__arc__conv,axiom,
! [A: b] :
( ( pre_ve642382030648772252t_unit @ ( pre_add_arc_a_b @ g @ A ) )
= ( sup_sup_set_a @ ( pre_ve642382030648772252t_unit @ g ) @ ( insert_a @ ( pre_ta4931606617599662728t_unit @ g @ A ) @ ( insert_a @ ( pre_he5236287464308401016t_unit @ g @ A ) @ bot_bot_set_a ) ) ) ) ).
% G.verts_add_arc_conv
thf(fact_1158_Suc__card__arcs__eq__card__verts,axiom,
( ( finite_finite_a @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( suc @ ( finite_card_b @ ( pre_ar1395965042833527383t_unit @ t ) ) )
= ( finite_card_a @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% Suc_card_arcs_eq_card_verts
thf(fact_1159_G_Oadd__arc__commute,axiom,
! [B2: b,A: b] :
( ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ g @ B2 ) @ A )
= ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ g @ A ) @ B2 ) ) ).
% G.add_arc_commute
thf(fact_1160_add__arc__commute,axiom,
! [B2: b,A: b] :
( ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ t @ B2 ) @ A )
= ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ t @ A ) @ B2 ) ) ).
% add_arc_commute
thf(fact_1161_G_Oadd__arc__in,axiom,
! [A: b] :
( ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( pre_add_arc_a_b @ g @ A )
= g ) ) ).
% G.add_arc_in
thf(fact_1162_add__arc__in,axiom,
! [A: b] :
( ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( pre_add_arc_a_b @ t @ A )
= t ) ) ).
% add_arc_in
thf(fact_1163_G_Oin__arcs__add__arc__iff,axiom,
! [A: b,U: a] :
( ( ( ( pre_he5236287464308401016t_unit @ g @ A )
= U )
=> ( ( in_arcs_a_b @ ( pre_add_arc_a_b @ g @ A ) @ U )
= ( insert_b @ A @ ( in_arcs_a_b @ g @ U ) ) ) )
& ( ( ( pre_he5236287464308401016t_unit @ g @ A )
!= U )
=> ( ( in_arcs_a_b @ ( pre_add_arc_a_b @ g @ A ) @ U )
= ( in_arcs_a_b @ g @ U ) ) ) ) ).
% G.in_arcs_add_arc_iff
thf(fact_1164_in__arcs__add__arc__iff,axiom,
! [A: b,U: a] :
( ( ( ( pre_he5236287464308401016t_unit @ t @ A )
= U )
=> ( ( in_arcs_a_b @ ( pre_add_arc_a_b @ t @ A ) @ U )
= ( insert_b @ A @ ( in_arcs_a_b @ t @ U ) ) ) )
& ( ( ( pre_he5236287464308401016t_unit @ t @ A )
!= U )
=> ( ( in_arcs_a_b @ ( pre_add_arc_a_b @ t @ A ) @ U )
= ( in_arcs_a_b @ t @ U ) ) ) ) ).
% in_arcs_add_arc_iff
thf(fact_1165_G_Oout__arcs__add__arc__iff,axiom,
! [A: b,U: a] :
( ( ( ( pre_ta4931606617599662728t_unit @ g @ A )
= U )
=> ( ( out_arcs_a_b @ ( pre_add_arc_a_b @ g @ A ) @ U )
= ( insert_b @ A @ ( out_arcs_a_b @ g @ U ) ) ) )
& ( ( ( pre_ta4931606617599662728t_unit @ g @ A )
!= U )
=> ( ( out_arcs_a_b @ ( pre_add_arc_a_b @ g @ A ) @ U )
= ( out_arcs_a_b @ g @ U ) ) ) ) ).
% G.out_arcs_add_arc_iff
thf(fact_1166_out__arcs__add__arc__iff,axiom,
! [A: b,U: a] :
( ( ( ( pre_ta4931606617599662728t_unit @ t @ A )
= U )
=> ( ( out_arcs_a_b @ ( pre_add_arc_a_b @ t @ A ) @ U )
= ( insert_b @ A @ ( out_arcs_a_b @ t @ U ) ) ) )
& ( ( ( pre_ta4931606617599662728t_unit @ t @ A )
!= U )
=> ( ( out_arcs_a_b @ ( pre_add_arc_a_b @ t @ A ) @ U )
= ( out_arcs_a_b @ t @ U ) ) ) ) ).
% out_arcs_add_arc_iff
thf(fact_1167_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1168_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1169_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_eq_nat @ I2 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_1170_G_Oadd__add__arc__collapse,axiom,
! [A: b] :
( ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ g @ A ) @ A )
= ( pre_add_arc_a_b @ g @ A ) ) ).
% G.add_add_arc_collapse
thf(fact_1171_add__add__arc__collapse,axiom,
! [A: b] :
( ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ t @ A ) @ A )
= ( pre_add_arc_a_b @ t @ A ) ) ).
% add_add_arc_collapse
thf(fact_1172_G_Otail__add__arc,axiom,
! [A: b] :
( ( pre_ta4931606617599662728t_unit @ ( pre_add_arc_a_b @ g @ A ) )
= ( pre_ta4931606617599662728t_unit @ g ) ) ).
% G.tail_add_arc
thf(fact_1173_G_Ohead__add__arc,axiom,
! [A: b] :
( ( pre_he5236287464308401016t_unit @ ( pre_add_arc_a_b @ g @ A ) )
= ( pre_he5236287464308401016t_unit @ g ) ) ).
% G.head_add_arc
thf(fact_1174_tail__add__arc,axiom,
! [A: b] :
( ( pre_ta4931606617599662728t_unit @ ( pre_add_arc_a_b @ t @ A ) )
= ( pre_ta4931606617599662728t_unit @ t ) ) ).
% tail_add_arc
thf(fact_1175_head__add__arc,axiom,
! [A: b] :
( ( pre_he5236287464308401016t_unit @ ( pre_add_arc_a_b @ t @ A ) )
= ( pre_he5236287464308401016t_unit @ t ) ) ).
% head_add_arc
thf(fact_1176_G_Oadd__del__arc__collapse,axiom,
! [A: b] :
( ( pre_add_arc_a_b @ ( pre_del_arc_a_b @ g @ A ) @ A )
= ( pre_add_arc_a_b @ g @ A ) ) ).
% G.add_del_arc_collapse
thf(fact_1177_add__del__arc__collapse,axiom,
! [A: b] :
( ( pre_add_arc_a_b @ ( pre_del_arc_a_b @ t @ A ) @ A )
= ( pre_add_arc_a_b @ t @ A ) ) ).
% add_del_arc_collapse
thf(fact_1178_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1179_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1180_G_Oarcs__add__arc,axiom,
! [A: b] :
( ( pre_ar1395965042833527383t_unit @ ( pre_add_arc_a_b @ g @ A ) )
= ( insert_b @ A @ ( pre_ar1395965042833527383t_unit @ g ) ) ) ).
% G.arcs_add_arc
thf(fact_1181_arcs__add__arc,axiom,
! [A: b] :
( ( pre_ar1395965042833527383t_unit @ ( pre_add_arc_a_b @ t @ A ) )
= ( insert_b @ A @ ( pre_ar1395965042833527383t_unit @ t ) ) ) ).
% arcs_add_arc
thf(fact_1182_G_Overts__add__arc,axiom,
! [A: b] :
( ( member_a @ ( pre_ta4931606617599662728t_unit @ g @ A ) @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( member_a @ ( pre_he5236287464308401016t_unit @ g @ A ) @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( pre_ve642382030648772252t_unit @ ( pre_add_arc_a_b @ g @ A ) )
= ( pre_ve642382030648772252t_unit @ g ) ) ) ) ).
% G.verts_add_arc
thf(fact_1183_verts__add__arc,axiom,
! [A: b] :
( ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ A ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( member_a @ ( pre_he5236287464308401016t_unit @ t @ A ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( pre_ve642382030648772252t_unit @ ( pre_add_arc_a_b @ t @ A ) )
= ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% verts_add_arc
thf(fact_1184_G_Odel__add__arc__collapse,axiom,
! [A: b] :
( ( member_a @ ( pre_ta4931606617599662728t_unit @ g @ A ) @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( member_a @ ( pre_he5236287464308401016t_unit @ g @ A ) @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( pre_del_arc_a_b @ ( pre_add_arc_a_b @ g @ A ) @ A )
= ( pre_del_arc_a_b @ g @ A ) ) ) ) ).
% G.del_add_arc_collapse
thf(fact_1185_del__add__arc__collapse,axiom,
! [A: b] :
( ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ A ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( member_a @ ( pre_he5236287464308401016t_unit @ t @ A ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( pre_del_arc_a_b @ ( pre_add_arc_a_b @ t @ A ) @ A )
= ( pre_del_arc_a_b @ t @ A ) ) ) ) ).
% del_add_arc_collapse
thf(fact_1186_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1187_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1188_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1189_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1190_Suc__le__D,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M3 )
=> ? [M4: nat] :
( M3
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1191_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1192_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1193_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1194_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1195_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1196_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y4: nat,Z4: nat] :
( ( R @ X3 @ Y4 )
=> ( ( R @ Y4 @ Z4 )
=> ( R @ X3 @ Z4 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1197_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1198_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1199_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1200_card__verts__le,axiom,
ord_less_eq_nat @ ( finite_card_a @ ( pre_ve642382030648772252t_unit @ t ) ) @ ( suc @ n ) ).
% card_verts_le
thf(fact_1201_nnvs__subs__k__nh,axiom,
! [W: b > real,U: a,N: nat,U3: set_a,K: real] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U3 )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ ( graph_3921080825633621230od_a_b @ g @ W @ U @ K ) ) )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ U3 @ ( insert_a @ U @ bot_bot_set_a ) ) @ ( graph_3921080825633621230od_a_b @ g @ W @ U @ K ) ) ) ) ).
% nnvs_subs_k_nh
thf(fact_1202_G_Onnvs__mem,axiom,
! [W: b > real,U: a,N: nat,U3: set_a] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U3 )
=> ( member_a @ U @ U3 ) ) ).
% G.nnvs_mem
thf(fact_1203_nnvs__mem,axiom,
! [W: b > real,U: a,N: nat,U3: set_a] :
( ( graph_3148032005746981223ts_a_b @ t @ W @ U @ N @ U3 )
=> ( member_a @ U @ U3 ) ) ).
% nnvs_mem
thf(fact_1204_G_Osource__mem__nnvs,axiom,
! [W: b > real,U: a,N: nat,U3: set_a] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U3 )
=> ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% G.source_mem_nnvs
thf(fact_1205_source__mem__nnvs,axiom,
! [W: b > real,U: a,N: nat,U3: set_a] :
( ( graph_3148032005746981223ts_a_b @ t @ W @ U @ N @ U3 )
=> ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% source_mem_nnvs
thf(fact_1206_G_Onnvs__finite,axiom,
! [W: b > real,U: a,N: nat,U3: set_a] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U3 )
=> ( finite_finite_a @ U3 ) ) ).
% G.nnvs_finite
thf(fact_1207_nnvs__finite,axiom,
! [W: b > real,U: a,N: nat,U3: set_a] :
( ( graph_3148032005746981223ts_a_b @ t @ W @ U @ N @ U3 )
=> ( finite_finite_a @ U3 ) ) ).
% nnvs_finite
thf(fact_1208_nnvs__subs__verts,axiom,
! [W: b > real,U: a,N: nat,U3: set_a] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U3 )
=> ( ord_less_eq_set_a @ U3 @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% nnvs_subs_verts
thf(fact_1209_G_Onnvs__card__le__n,axiom,
! [W: b > real,U: a,N: nat,U3: set_a] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U3 )
=> ( ord_less_eq_nat @ ( finite_card_a @ U3 ) @ ( suc @ N ) ) ) ).
% G.nnvs_card_le_n
thf(fact_1210_nnvs__card__le__n,axiom,
! [W: b > real,U: a,N: nat,U3: set_a] :
( ( graph_3148032005746981223ts_a_b @ t @ W @ U @ N @ U3 )
=> ( ord_less_eq_nat @ ( finite_card_a @ U3 ) @ ( suc @ N ) ) ) ).
% nnvs_card_le_n
thf(fact_1211_nnvs__imp__all__reachable,axiom,
! [W: b > real,U: a,N: nat,U3: set_a] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U3 )
=> ( ( ( suc @ N )
= ( finite_card_a @ ( collect_a @ ( reachable_a_b @ g @ U ) ) ) )
=> ( U3
= ( collect_a @ ( reachable_a_b @ g @ U ) ) ) ) ) ).
% nnvs_imp_all_reachable
thf(fact_1212_all__reachable__eq__nnvs,axiom,
! [U3: set_a,U: a,N: nat,W: b > real] :
( ( U3
= ( collect_a @ ( reachable_a_b @ g @ U ) ) )
=> ( ( ( finite_card_a @ U3 )
= ( suc @ N ) )
=> ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U3 ) ) ) ).
% all_reachable_eq_nnvs
thf(fact_1213_nnvs__imp__all__reachable__Suc,axiom,
! [W: b > real,U: a,N: nat,U3: set_a] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U3 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ ( collect_a @ ( reachable_a_b @ g @ U ) ) ) @ ( suc @ N ) )
=> ( U3
= ( collect_a @ ( reachable_a_b @ g @ U ) ) ) ) ) ).
% nnvs_imp_all_reachable_Suc
thf(fact_1214_all__reachable__eq__nnvs__Suc,axiom,
! [U: a,U3: set_a,N: nat,W: b > real] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( U3
= ( collect_a @ ( reachable_a_b @ g @ U ) ) )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ U3 ) @ ( suc @ N ) )
=> ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U3 ) ) ) ) ).
% all_reachable_eq_nnvs_Suc
thf(fact_1215_nnvs__imp__reachable,axiom,
! [W: b > real,U: a,N: nat,A2: set_a] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ A2 )
=> ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_a @ ( collect_a @ ( reachable_a_b @ g @ U ) ) ) )
=> ( ( ord_less_eq_set_a @ A2 @ ( collect_a @ ( reachable_a_b @ g @ U ) ) )
& ( ( finite_card_a @ A2 )
= ( suc @ N ) ) ) ) ) ).
% nnvs_imp_reachable
thf(fact_1216_reachable__subs__nnvs,axiom,
! [U: a,N: nat,W: b > real] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_a @ ( collect_a @ ( reachable_a_b @ g @ U ) ) ) )
=> ? [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ ( collect_a @ ( reachable_a_b @ g @ U ) ) )
& ( ( finite_card_a @ A7 )
= ( suc @ N ) )
& ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ A7 ) ) ) ) ).
% reachable_subs_nnvs
thf(fact_1217_con__Suc__card__arcs__eq__card__verts,axiom,
( ( graph_a_b @ g )
=> ( ( digrap8783888973171253482ed_a_b @ g )
=> ( ( ( suc @ n )
= ( finite_card_a @ ( pre_ve642382030648772252t_unit @ g ) ) )
=> ( ( suc @ ( finite_card_b @ ( pre_ar1395965042833527383t_unit @ t ) ) )
= ( finite_card_a @ ( pre_ve642382030648772252t_unit @ g ) ) ) ) ) ) ).
% con_Suc_card_arcs_eq_card_verts
thf(fact_1218_strongly__con__imp__card__verts__eq,axiom,
( ( fin_digraph_a_b @ g )
=> ( ( digrap8691851296217657702ed_a_b @ g )
=> ( ( ord_less_eq_nat @ ( suc @ n ) @ ( finite_card_a @ ( pre_ve642382030648772252t_unit @ g ) ) )
=> ( ( finite_card_a @ ( pre_ve642382030648772252t_unit @ t ) )
= ( suc @ n ) ) ) ) ) ).
% strongly_con_imp_card_verts_eq
thf(fact_1219_connected,axiom,
digrap8783888973171253482ed_a_b @ t ).
% connected
thf(fact_1220_G_Ostrongly__con__imp__reachable__eq__verts,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( digrap8691851296217657702ed_a_b @ g )
=> ( ( collect_a @ ( reachable_a_b @ g @ R2 ) )
= ( pre_ve642382030648772252t_unit @ g ) ) ) ) ).
% G.strongly_con_imp_reachable_eq_verts
thf(fact_1221_strongly__con__imp__reachable__eq__verts,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( digrap8691851296217657702ed_a_b @ t )
=> ( ( collect_a @ ( reachable_a_b @ t @ R2 ) )
= ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% strongly_con_imp_reachable_eq_verts
thf(fact_1222_connected__iff__reachable,axiom,
( ( digrap8783888973171253482ed_a_b @ g )
= ( ! [X: a] :
( ( member_a @ X @ ( pre_ve642382030648772252t_unit @ g ) )
=> ! [Y: a] :
( ( member_a @ Y @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( reachable_a_b @ g @ X @ Y ) ) )
& ( ( pre_ve642382030648772252t_unit @ g )
!= bot_bot_set_a ) ) ) ).
% connected_iff_reachable
thf(fact_1223_G_Overts__reachable__connected,axiom,
( ( ( pre_ve642382030648772252t_unit @ g )
!= bot_bot_set_a )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ! [Xa: a] :
( ( member_a @ Xa @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( reachable_a_b @ g @ X3 @ Xa ) ) )
=> ( digrap8783888973171253482ed_a_b @ g ) ) ) ).
% G.verts_reachable_connected
thf(fact_1224_verts__reachable__connected,axiom,
( ( ( pre_ve642382030648772252t_unit @ t )
!= bot_bot_set_a )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ! [Xa: a] :
( ( member_a @ Xa @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( reachable_a_b @ t @ X3 @ Xa ) ) )
=> ( digrap8783888973171253482ed_a_b @ t ) ) ) ).
% verts_reachable_connected
thf(fact_1225_G_Oconnected__arcs__empty,axiom,
( ( digrap8783888973171253482ed_a_b @ g )
=> ( ( ( pre_ar1395965042833527383t_unit @ g )
= bot_bot_set_b )
=> ( ( ( pre_ve642382030648772252t_unit @ g )
!= bot_bot_set_a )
=> ~ ! [V3: a] :
( ( pre_ve642382030648772252t_unit @ g )
!= ( insert_a @ V3 @ bot_bot_set_a ) ) ) ) ) ).
% G.connected_arcs_empty
thf(fact_1226_connected__arcs__empty,axiom,
( ( digrap8783888973171253482ed_a_b @ t )
=> ( ( ( pre_ar1395965042833527383t_unit @ t )
= bot_bot_set_b )
=> ( ( ( pre_ve642382030648772252t_unit @ t )
!= bot_bot_set_a )
=> ~ ! [V3: a] :
( ( pre_ve642382030648772252t_unit @ t )
!= ( insert_a @ V3 @ bot_bot_set_a ) ) ) ) ) ).
% connected_arcs_empty
thf(fact_1227_connected__verts__G__eq__T,axiom,
( ( graph_a_b @ g )
=> ( ( digrap8783888973171253482ed_a_b @ g )
=> ( ( ( suc @ n )
= ( finite_card_a @ ( pre_ve642382030648772252t_unit @ g ) ) )
=> ( ( pre_ve642382030648772252t_unit @ t )
= ( pre_ve642382030648772252t_unit @ g ) ) ) ) ) ).
% connected_verts_G_eq_T
thf(fact_1228_G_Ospanning__tree__imp__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5718416180170401981ee_a_b @ H @ g )
=> ( digrap8783888973171253482ed_a_b @ g ) ) ).
% G.spanning_tree_imp_connected
thf(fact_1229_spanning__tree__imp__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5718416180170401981ee_a_b @ H @ t )
=> ( digrap8783888973171253482ed_a_b @ t ) ) ).
% spanning_tree_imp_connected
thf(fact_1230_G_Oconnected__spanning__imp__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digraph_spanning_a_b @ H @ g )
=> ( ( digrap8783888973171253482ed_a_b @ H )
=> ( digrap8783888973171253482ed_a_b @ g ) ) ) ).
% G.connected_spanning_imp_connected
thf(fact_1231_connected__spanning__imp__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digraph_spanning_a_b @ H @ t )
=> ( ( digrap8783888973171253482ed_a_b @ H )
=> ( digrap8783888973171253482ed_a_b @ t ) ) ) ).
% connected_spanning_imp_connected
thf(fact_1232_G_Ostrongly__connected__spanning__imp__strongly__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digraph_spanning_a_b @ H @ g )
=> ( ( digrap8691851296217657702ed_a_b @ H )
=> ( digrap8691851296217657702ed_a_b @ g ) ) ) ).
% G.strongly_connected_spanning_imp_strongly_connected
thf(fact_1233_strongly__connected__spanning__imp__strongly__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digraph_spanning_a_b @ H @ t )
=> ( ( digrap8691851296217657702ed_a_b @ H )
=> ( digrap8691851296217657702ed_a_b @ t ) ) ) ).
% strongly_connected_spanning_imp_strongly_connected
thf(fact_1234_connected__verts,axiom,
( ( digrap8783888973171253482ed_a_b @ t )
=> ( ( ( pre_ar1395965042833527383t_unit @ t )
!= bot_bot_set_b )
=> ( ( pre_ve642382030648772252t_unit @ t )
= ( sup_sup_set_a @ ( image_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ ( pre_ar1395965042833527383t_unit @ t ) ) @ ( image_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ ( pre_ar1395965042833527383t_unit @ t ) ) ) ) ) ) ).
% connected_verts
thf(fact_1235_G_Oconnected__verts,axiom,
( ( digrap8783888973171253482ed_a_b @ g )
=> ( ( ( pre_ar1395965042833527383t_unit @ g )
!= bot_bot_set_b )
=> ( ( pre_ve642382030648772252t_unit @ g )
= ( sup_sup_set_a @ ( image_b_a @ ( pre_ta4931606617599662728t_unit @ g ) @ ( pre_ar1395965042833527383t_unit @ g ) ) @ ( image_b_a @ ( pre_he5236287464308401016t_unit @ g ) @ ( pre_ar1395965042833527383t_unit @ g ) ) ) ) ) ) ).
% G.connected_verts
thf(fact_1236_symmetric__connected__imp__strongly__connected,axiom,
( ( symmetric_a_b @ t )
=> ( ( digrap8783888973171253482ed_a_b @ t )
=> ( digrap8691851296217657702ed_a_b @ t ) ) ) ).
% symmetric_connected_imp_strongly_connected
thf(fact_1237_G_Osymmetric__connected__imp__strongly__connected,axiom,
( ( symmetric_a_b @ g )
=> ( ( digrap8783888973171253482ed_a_b @ g )
=> ( digrap8691851296217657702ed_a_b @ g ) ) ) ).
% G.symmetric_connected_imp_strongly_connected
thf(fact_1238_G_Osccs__verts__conv__scc__of,axiom,
( ( digrap2871191568752656621ts_a_b @ g )
= ( image_a_set_a @ ( digrap2937667069914300949of_a_b @ g ) @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% G.sccs_verts_conv_scc_of
thf(fact_1239_sccs__verts__conv__scc__of,axiom,
( ( digrap2871191568752656621ts_a_b @ t )
= ( image_a_set_a @ ( digrap2937667069914300949of_a_b @ t ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% sccs_verts_conv_scc_of
thf(fact_1240_sym__arcs,axiom,
symmetric_a_b @ g ).
% sym_arcs
thf(fact_1241_graphI,axiom,
( ( symmetric_a_b @ g )
=> ( graph_a_b @ g ) ) ).
% graphI
thf(fact_1242_zero__nnvs,axiom,
! [U: a,Uu: b > real] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( graph_3148032005746981223ts_a_b @ t @ Uu @ U @ zero_zero_nat @ ( insert_a @ U @ bot_bot_set_a ) ) ) ).
% zero_nnvs
thf(fact_1243_G_Ozero__nnvs,axiom,
! [U: a,Uu: b > real] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( graph_3148032005746981223ts_a_b @ g @ Uu @ U @ zero_zero_nat @ ( insert_a @ U @ bot_bot_set_a ) ) ) ).
% G.zero_nnvs
thf(fact_1244_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1245_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1246_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1247_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1248_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1249_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1250_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1251_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1252_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1253_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1254_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1255_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1256_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1257_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1258_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1259_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1260_sp__tree,axiom,
( ( fin_digraph_a_b @ g )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ ( collect_a @ ( reachable_a_b @ g @ source ) ) ) @ ( suc @ n ) )
=> ( ( pre_ve642382030648772252t_unit @ t )
= ( collect_a @ ( reachable_a_b @ g @ source ) ) ) ) ) ).
% sp_tree
thf(fact_1261_sp__tree2,axiom,
( ( fin_digraph_a_b @ g )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ ( pre_ve642382030648772252t_unit @ g ) ) @ ( suc @ n ) )
=> ( ( pre_ve642382030648772252t_unit @ t )
= ( collect_a @ ( reachable_a_b @ g @ source ) ) ) ) ) ).
% sp_tree2
thf(fact_1262_source__in__G,axiom,
member_a @ source @ ( pre_ve642382030648772252t_unit @ g ) ).
% source_in_G
thf(fact_1263_root__in__T,axiom,
member_a @ source @ ( pre_ve642382030648772252t_unit @ t ) ).
% root_in_T
thf(fact_1264_reachable__from__root,axiom,
! [V: a] :
( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( reachable_a_b @ t @ source @ V ) ) ).
% reachable_from_root
thf(fact_1265_in__arcs__root,axiom,
( ( in_arcs_a_b @ t @ source )
= bot_bot_set_b ) ).
% in_arcs_root
thf(fact_1266_ex__in__arc,axiom,
! [V: a] :
( ( V != source )
=> ( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) )
=> ? [E4: b] :
( ( in_arcs_a_b @ t @ V )
= ( insert_b @ E4 @ bot_bot_set_b ) ) ) ) ).
% ex_in_arc
thf(fact_1267_root__leaf__iff,axiom,
( ( shorte1213025427933718126af_a_b @ t @ source )
= ( ( pre_ve642382030648772252t_unit @ t )
= ( insert_a @ source @ bot_bot_set_a ) ) ) ).
% root_leaf_iff
thf(fact_1268_reachable__verts__G__eq__T,axiom,
( ( fin_digraph_a_b @ g )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( reachable_a_b @ g @ source @ X3 ) )
=> ( ( ( suc @ n )
= ( finite_card_a @ ( pre_ve642382030648772252t_unit @ g ) ) )
=> ( ( pre_ve642382030648772252t_unit @ t )
= ( pre_ve642382030648772252t_unit @ g ) ) ) ) ) ).
% reachable_verts_G_eq_T
thf(fact_1269_reachable__verts__G__subset__T,axiom,
( ( fin_digraph_a_b @ g )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( reachable_a_b @ g @ source @ X3 ) )
=> ( ( ( suc @ n )
= ( finite_card_a @ ( pre_ve642382030648772252t_unit @ g ) ) )
=> ( ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ g ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ) ).
% reachable_verts_G_subset_T
thf(fact_1270_partial,axiom,
graph_3148032005746981223ts_a_b @ g @ w @ source @ n @ ( pre_ve642382030648772252t_unit @ t ) ).
% partial
thf(fact_1271_k__nh__subs__nnvs,axiom,
! [W: b > real,U: a,N: nat,U3: set_a,K: real] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U3 )
=> ( ( ord_less_nat @ ( finite_card_a @ ( graph_3921080825633621230od_a_b @ g @ W @ U @ K ) ) @ ( finite_card_a @ U3 ) )
=> ( ord_less_eq_set_a @ ( graph_3921080825633621230od_a_b @ g @ W @ U @ K ) @ U3 ) ) ) ).
% k_nh_subs_nnvs
thf(fact_1272_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1273_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1274_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_1275_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
% Conjectures (1)
thf(conj_0,conjecture,
( ( finite_card_b
@ ( sup_sup_set_b
@ ( collect_b
@ ^ [E22: b] :
( ( member_b @ E22 @ ( pre_ar1395965042833527383t_unit @ g ) )
& ? [X: b] :
( ( member_b @ X @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( pre_he5236287464308401016t_unit @ g @ E22 )
= ( pre_ta4931606617599662728t_unit @ g @ X ) )
& ( ( pre_he5236287464308401016t_unit @ g @ X )
= ( pre_ta4931606617599662728t_unit @ g @ E22 ) ) ) ) )
@ ( pre_ar1395965042833527383t_unit @ t ) ) )
= ( plus_plus_nat
@ ( finite_card_b
@ ( collect_b
@ ^ [E22: b] :
( ( member_b @ E22 @ ( pre_ar1395965042833527383t_unit @ g ) )
& ? [X: b] :
( ( member_b @ X @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( pre_he5236287464308401016t_unit @ g @ E22 )
= ( pre_ta4931606617599662728t_unit @ g @ X ) )
& ( ( pre_he5236287464308401016t_unit @ g @ X )
= ( pre_ta4931606617599662728t_unit @ g @ E22 ) ) ) ) ) )
@ ( finite_card_b @ ( pre_ar1395965042833527383t_unit @ t ) ) ) ) ).
%------------------------------------------------------------------------------