TPTP Problem File: SLH0845^1.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_00523_022737__15033094_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1419 ( 559 unt; 139 typ; 0 def)
% Number of atoms : 3546 (1334 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 12220 ( 440 ~; 47 |; 309 &;10027 @)
% ( 0 <=>;1397 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 486 ( 486 >; 0 *; 0 +; 0 <<)
% Number of symbols : 129 ( 128 usr; 11 con; 0-5 aty)
% Number of variables : 3333 ( 131 ^;3122 !; 80 ?;3333 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:05:58.053
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__b,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (128)
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_List_Oappend_001tf__a,type,
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thf(sy_c_List_Oappend_001tf__b,type,
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thf(sy_c_List_Obutlast_001tf__a,type,
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thf(sy_c_List_Odistinct_001tf__a,type,
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thf(sy_c_List_Odistinct_001tf__b,type,
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thf(sy_c_List_Olast_001tf__a,type,
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thf(sy_c_List_Olast_001tf__b,type,
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thf(sy_c_List_Olist_OCons_001tf__a,type,
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thf(sy_c_Set_Oimage_001tf__b_001t__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J,type,
image_4434118323594779837t_unit: ( b > pre_pr7278220950009878019t_unit ) > set_b > set_pr5411798346947241657t_unit ).
thf(sy_c_Set_Oimage_001tf__b_001t__Set__Oset_Itf__a_J,type,
image_b_set_a: ( b > set_a ) > set_b > 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_Oimage_001tf__b_001tf__b,type,
image_b_b: ( b > b ) > set_b > set_b ).
thf(sy_c_Set_Oinsert_001t__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J,type,
insert6864688055023459379t_unit: pre_pr7278220950009878019t_unit > set_pr5411798346947241657t_unit > set_pr5411798346947241657t_unit ).
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_Owf__digraph_Omk__cycles__path_001tf__b,type,
shorte6374615165232202367path_b: nat > list_b > list_b ).
thf(sy_c_Shortest__Path__Tree_Odirected__tree_001tf__a_001tf__b,type,
shorte3810566709427824352ee_a_b: pre_pr7278220950009878019t_unit > a > $o ).
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_Stuff_ONOMATCH_001tf__a,type,
nOMATCH_a: a > a > $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_Vertex__Walk_Ovwalk_001tf__a_001tf__b,type,
vertex_vwalk_a_b: list_a > pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_member_001t__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J,type,
member6939884229742472986t_unit: pre_pr7278220950009878019t_unit > set_pr5411798346947241657t_unit > $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_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_U____,type,
u: set_a ).
thf(sy_v_source,type,
source: a ).
thf(sy_v_x____,type,
x: a ).
% Relevant facts (1279)
thf(fact_0__092_060open_062x_A_092_060notin_062_AU_092_060close_062,axiom,
~ ( member_a @ x @ u ) ).
% \<open>x \<notin> U\<close>
thf(fact_1_False,axiom,
( ( graph_2016941059203891550ts_a_b @ g @ source @ u )
= bot_bot_set_a ) ).
% False
thf(fact_2_assms_I1_J,axiom,
fin_digraph_a_b @ g ).
% assms(1)
thf(fact_3_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_4_G_Odisj__unvis__vis,axiom,
! [U: a,U2: set_a] :
( ( inf_inf_set_a @ ( graph_2016941059203891550ts_a_b @ g @ U @ U2 ) @ U2 )
= bot_bot_set_a ) ).
% G.disj_unvis_vis
thf(fact_5_source__in__G,axiom,
member_a @ source @ ( pre_ve642382030648772252t_unit @ g ) ).
% source_in_G
thf(fact_6_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_7_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_8_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_9_empty__iff,axiom,
! [C: pre_pr7278220950009878019t_unit] :
~ ( member6939884229742472986t_unit @ C @ bot_bo1839476491465656141t_unit ) ).
% empty_iff
thf(fact_10_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X: set_a] :
~ ( member_set_a @ X @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_11_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_12_all__not__in__conv,axiom,
! [A: set_b] :
( ( ! [X: b] :
~ ( member_b @ X @ A ) )
= ( A = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_13_all__not__in__conv,axiom,
! [A: set_pr5411798346947241657t_unit] :
( ( ! [X: pre_pr7278220950009878019t_unit] :
~ ( member6939884229742472986t_unit @ X @ A ) )
= ( A = bot_bo1839476491465656141t_unit ) ) ).
% all_not_in_conv
thf(fact_14_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X: set_a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_15_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_16_Collect__empty__eq,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( ! [X: b] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_17_Collect__empty__eq,axiom,
! [P: pre_pr7278220950009878019t_unit > $o] :
( ( ( collec8000012497822511960t_unit @ P )
= bot_bo1839476491465656141t_unit )
= ( ! [X: pre_pr7278220950009878019t_unit] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_18_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X: set_a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_19_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_20_empty__Collect__eq,axiom,
! [P: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P ) )
= ( ! [X: b] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_21_empty__Collect__eq,axiom,
! [P: pre_pr7278220950009878019t_unit > $o] :
( ( bot_bo1839476491465656141t_unit
= ( collec8000012497822511960t_unit @ P ) )
= ( ! [X: pre_pr7278220950009878019t_unit] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_22_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_23_x__reach,axiom,
reachable_a_b @ g @ source @ x ).
% x_reach
thf(fact_24_G_Oiapath__dist__ends,axiom,
! [U: a,P2: list_b,V: a] :
( ( pre_gen_iapath_a_b @ g @ ( verts3_a_b @ g ) @ U @ P2 @ V )
=> ( U != V ) ) ).
% G.iapath_dist_ends
thf(fact_25_G_Oinduced__subgraph__refl,axiom,
digrap5251062021860773499ph_a_b @ g @ g ).
% G.induced_subgraph_refl
thf(fact_26_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_27_assms_I2_J,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( reachable_a_b @ g @ source @ X2 ) ) ).
% assms(2)
thf(fact_28_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_29_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_30_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_31_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_32_IntI,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_33_IntI,axiom,
! [C: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C @ A )
=> ( ( member6939884229742472986t_unit @ C @ B )
=> ( member6939884229742472986t_unit @ C @ ( inf_in1092213268631476299t_unit @ A @ B ) ) ) ) ).
% IntI
thf(fact_34_IntI,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ A )
=> ( ( member_b @ C @ B )
=> ( member_b @ C @ ( inf_inf_set_b @ A @ B ) ) ) ) ).
% IntI
thf(fact_35_IntI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_36_Int__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
& ( member_set_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_37_Int__iff,axiom,
! [C: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C @ ( inf_in1092213268631476299t_unit @ A @ B ) )
= ( ( member6939884229742472986t_unit @ C @ A )
& ( member6939884229742472986t_unit @ C @ B ) ) ) ).
% Int_iff
thf(fact_38_Int__iff,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
= ( ( member_b @ C @ A )
& ( member_b @ C @ B ) ) ) ).
% Int_iff
thf(fact_39_Int__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_40_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_41_G_Oreachable__refl,axiom,
! [V: a] :
( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( reachable_a_b @ g @ V @ V ) ) ).
% G.reachable_refl
thf(fact_42_IntE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ~ ( ( member_set_a @ C @ A )
=> ~ ( member_set_a @ C @ B ) ) ) ).
% IntE
thf(fact_43_IntE,axiom,
! [C: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C @ ( inf_in1092213268631476299t_unit @ A @ B ) )
=> ~ ( ( member6939884229742472986t_unit @ C @ A )
=> ~ ( member6939884229742472986t_unit @ C @ B ) ) ) ).
% IntE
thf(fact_44_IntE,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
=> ~ ( ( member_b @ C @ A )
=> ~ ( member_b @ C @ B ) ) ) ).
% IntE
thf(fact_45_IntE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_46_IntD1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C @ A ) ) ).
% IntD1
thf(fact_47_IntD1,axiom,
! [C: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C @ ( inf_in1092213268631476299t_unit @ A @ B ) )
=> ( member6939884229742472986t_unit @ C @ A ) ) ).
% IntD1
thf(fact_48_IntD1,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
=> ( member_b @ C @ A ) ) ).
% IntD1
thf(fact_49_IntD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% IntD1
thf(fact_50_IntD2,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C @ B ) ) ).
% IntD2
thf(fact_51_IntD2,axiom,
! [C: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C @ ( inf_in1092213268631476299t_unit @ A @ B ) )
=> ( member6939884229742472986t_unit @ C @ B ) ) ).
% IntD2
thf(fact_52_IntD2,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
=> ( member_b @ C @ B ) ) ).
% IntD2
thf(fact_53_IntD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_54_Int__assoc,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_55_Int__absorb,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_56_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A2: set_a,B2: set_a] : ( inf_inf_set_a @ B2 @ A2 ) ) ) ).
% Int_commute
thf(fact_57_Int__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_58_Int__left__commute,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_59_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_60_bot__set__def,axiom,
( bot_bot_set_b
= ( collect_b @ bot_bot_b_o ) ) ).
% bot_set_def
thf(fact_61_bot__set__def,axiom,
( bot_bo1839476491465656141t_unit
= ( collec8000012497822511960t_unit @ bot_bo8537066411596906360unit_o ) ) ).
% bot_set_def
thf(fact_62_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_63_disjoint__iff__not__equal,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ! [Y: a] :
( ( member_a @ Y @ B )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_64_disjoint__iff__not__equal,axiom,
! [A: set_b,B: set_b] :
( ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b )
= ( ! [X: b] :
( ( member_b @ X @ A )
=> ! [Y: b] :
( ( member_b @ Y @ B )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_65_disjoint__iff__not__equal,axiom,
! [A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( ( inf_in1092213268631476299t_unit @ A @ B )
= bot_bo1839476491465656141t_unit )
= ( ! [X: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X @ A )
=> ! [Y: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ Y @ B )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_66_disjoint__iff__not__equal,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ! [Y: set_a] :
( ( member_set_a @ Y @ B )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_67_Int__empty__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_68_Int__empty__right,axiom,
! [A: set_b] :
( ( inf_inf_set_b @ A @ bot_bot_set_b )
= bot_bot_set_b ) ).
% Int_empty_right
thf(fact_69_Int__empty__right,axiom,
! [A: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ A @ bot_bo1839476491465656141t_unit )
= bot_bo1839476491465656141t_unit ) ).
% Int_empty_right
thf(fact_70_Int__empty__right,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ A @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% Int_empty_right
thf(fact_71_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_72_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_73_Int__empty__left,axiom,
! [B: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ bot_bo1839476491465656141t_unit @ B )
= bot_bo1839476491465656141t_unit ) ).
% Int_empty_left
thf(fact_74_Int__empty__left,axiom,
! [B: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ B )
= bot_bot_set_set_a ) ).
% Int_empty_left
thf(fact_75_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ~ ( member_a @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_76_disjoint__iff,axiom,
! [A: set_b,B: set_b] :
( ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b )
= ( ! [X: b] :
( ( member_b @ X @ A )
=> ~ ( member_b @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_77_disjoint__iff,axiom,
! [A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( ( inf_in1092213268631476299t_unit @ A @ B )
= bot_bo1839476491465656141t_unit )
= ( ! [X: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X @ A )
=> ~ ( member6939884229742472986t_unit @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_78_disjoint__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ~ ( member_set_a @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_79_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ X3 @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_80_Int__emptyI,axiom,
! [A: set_b,B: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ~ ( member_b @ X3 @ B ) )
=> ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b ) ) ).
% Int_emptyI
thf(fact_81_Int__emptyI,axiom,
! [A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ! [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ A )
=> ~ ( member6939884229742472986t_unit @ X3 @ B ) )
=> ( ( inf_in1092213268631476299t_unit @ A @ B )
= bot_bo1839476491465656141t_unit ) ) ).
% Int_emptyI
thf(fact_82_Int__emptyI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ~ ( member_set_a @ X3 @ B ) )
=> ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_83_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X: a] : ( member_a @ X @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_84_ex__in__conv,axiom,
! [A: set_b] :
( ( ? [X: b] : ( member_b @ X @ A ) )
= ( A != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_85_ex__in__conv,axiom,
! [A: set_pr5411798346947241657t_unit] :
( ( ? [X: pre_pr7278220950009878019t_unit] : ( member6939884229742472986t_unit @ X @ A ) )
= ( A != bot_bo1839476491465656141t_unit ) ) ).
% ex_in_conv
thf(fact_86_ex__in__conv,axiom,
! [A: set_set_a] :
( ( ? [X: set_a] : ( member_set_a @ X @ A ) )
= ( A != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_87_equals0I,axiom,
! [A: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_88_equals0I,axiom,
! [A: set_b] :
( ! [Y2: b] :
~ ( member_b @ Y2 @ A )
=> ( A = bot_bot_set_b ) ) ).
% equals0I
thf(fact_89_equals0I,axiom,
! [A: set_pr5411798346947241657t_unit] :
( ! [Y2: pre_pr7278220950009878019t_unit] :
~ ( member6939884229742472986t_unit @ Y2 @ A )
=> ( A = bot_bo1839476491465656141t_unit ) ) ).
% equals0I
thf(fact_90_equals0I,axiom,
! [A: set_set_a] :
( ! [Y2: set_a] :
~ ( member_set_a @ Y2 @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_91_equals0D,axiom,
! [A: set_a,A3: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A3 @ A ) ) ).
% equals0D
thf(fact_92_equals0D,axiom,
! [A: set_b,A3: b] :
( ( A = bot_bot_set_b )
=> ~ ( member_b @ A3 @ A ) ) ).
% equals0D
thf(fact_93_equals0D,axiom,
! [A: set_pr5411798346947241657t_unit,A3: pre_pr7278220950009878019t_unit] :
( ( A = bot_bo1839476491465656141t_unit )
=> ~ ( member6939884229742472986t_unit @ A3 @ A ) ) ).
% equals0D
thf(fact_94_equals0D,axiom,
! [A: set_set_a,A3: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A3 @ A ) ) ).
% equals0D
thf(fact_95_emptyE,axiom,
! [A3: a] :
~ ( member_a @ A3 @ bot_bot_set_a ) ).
% emptyE
thf(fact_96_emptyE,axiom,
! [A3: b] :
~ ( member_b @ A3 @ bot_bot_set_b ) ).
% emptyE
thf(fact_97_emptyE,axiom,
! [A3: pre_pr7278220950009878019t_unit] :
~ ( member6939884229742472986t_unit @ A3 @ bot_bo1839476491465656141t_unit ) ).
% emptyE
thf(fact_98_emptyE,axiom,
! [A3: set_a] :
~ ( member_set_a @ A3 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_99_x__def,axiom,
( ~ ( member_a @ x @ ( pre_ve642382030648772252t_unit @ t ) )
& ( member_a @ x @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% x_def
thf(fact_100_mem__Collect__eq,axiom,
! [A3: set_a,P: set_a > $o] :
( ( member_set_a @ A3 @ ( collect_set_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_101_mem__Collect__eq,axiom,
! [A3: pre_pr7278220950009878019t_unit,P: pre_pr7278220950009878019t_unit > $o] :
( ( member6939884229742472986t_unit @ A3 @ ( collec8000012497822511960t_unit @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_102_mem__Collect__eq,axiom,
! [A3: b,P: b > $o] :
( ( member_b @ A3 @ ( collect_b @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_103_mem__Collect__eq,axiom,
! [A3: a,P: a > $o] :
( ( member_a @ A3 @ ( collect_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_104_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X: set_a] : ( member_set_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_105_Collect__mem__eq,axiom,
! [A: set_pr5411798346947241657t_unit] :
( ( collec8000012497822511960t_unit
@ ^ [X: pre_pr7278220950009878019t_unit] : ( member6939884229742472986t_unit @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_106_Collect__mem__eq,axiom,
! [A: set_b] :
( ( collect_b
@ ^ [X: b] : ( member_b @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_107_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_108_Collect__cong,axiom,
! [P: pre_pr7278220950009878019t_unit > $o,Q: pre_pr7278220950009878019t_unit > $o] :
( ! [X3: pre_pr7278220950009878019t_unit] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec8000012497822511960t_unit @ P )
= ( collec8000012497822511960t_unit @ Q ) ) ) ).
% Collect_cong
thf(fact_109_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_110_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_111_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_112_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_113_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ X4 @ bot_bo1839476491465656141t_unit )
= bot_bo1839476491465656141t_unit ) ).
% boolean_algebra.conj_zero_right
thf(fact_114_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_115_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_116_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_117_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ bot_bo1839476491465656141t_unit @ X4 )
= bot_bo1839476491465656141t_unit ) ).
% boolean_algebra.conj_zero_left
thf(fact_118_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X4 )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_119_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_120_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_121_inf__bot__right,axiom,
! [X4: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ X4 @ bot_bo1839476491465656141t_unit )
= bot_bo1839476491465656141t_unit ) ).
% inf_bot_right
thf(fact_122_inf__bot__right,axiom,
! [X4: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% inf_bot_right
thf(fact_123_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_124_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_125_inf__bot__left,axiom,
! [X4: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ bot_bo1839476491465656141t_unit @ X4 )
= bot_bo1839476491465656141t_unit ) ).
% inf_bot_left
thf(fact_126_inf__bot__left,axiom,
! [X4: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X4 )
= bot_bot_set_set_a ) ).
% inf_bot_left
thf(fact_127_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_128_G_Oscc__disj,axiom,
! [C: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ g ) )
=> ( ( member6939884229742472986t_unit @ D @ ( digraph_pre_sccs_a_b @ g ) )
=> ( ( C != D )
=> ( ( inf_inf_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ D ) )
= bot_bot_set_a ) ) ) ) ).
% G.scc_disj
thf(fact_129_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_130_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_131_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_132_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_133_G_Oinduce__eq__iff__induced,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ H @ g )
=> ( ( digrap7873285959652527175ph_a_b @ g @ ( pre_ve642382030648772252t_unit @ H ) )
= H ) ) ).
% G.induce_eq_iff_induced
thf(fact_134_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_135_connected,axiom,
digrap8783888973171253482ed_a_b @ t ).
% connected
thf(fact_136_induced__subgraph__refl,axiom,
digrap5251062021860773499ph_a_b @ t @ t ).
% induced_subgraph_refl
thf(fact_137_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_138_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_139_fin__digraph__axioms,axiom,
fin_digraph_a_b @ t ).
% fin_digraph_axioms
thf(fact_140_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_141_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_142_non__empty,axiom,
( ( pre_ve642382030648772252t_unit @ t )
!= bot_bot_set_a ) ).
% non_empty
thf(fact_143_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_144_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_145_in__sccs__imp__induced,axiom,
! [C: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ t ) )
=> ( digrap5251062021860773499ph_a_b @ C @ t ) ) ).
% in_sccs_imp_induced
thf(fact_146_G_Oin__sccs__imp__induced,axiom,
! [C: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ g ) )
=> ( digrap5251062021860773499ph_a_b @ C @ g ) ) ).
% G.in_sccs_imp_induced
thf(fact_147_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_148__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_O_Ax_A_092_060notin_062_Averts_AT_A_092_060and_062_Ax_A_092_060in_062_Averts_AG_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X3: a] :
~ ( ~ ( member_a @ X3 @ ( pre_ve642382030648772252t_unit @ t ) )
& ( member_a @ X3 @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% \<open>\<And>thesis. (\<And>x. x \<notin> verts T \<and> x \<in> verts G \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_149_root__in__T,axiom,
member_a @ source @ ( pre_ve642382030648772252t_unit @ t ) ).
% root_in_T
thf(fact_150_inf__right__idem,axiom,
! [X4: set_a,Y3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ Y3 )
= ( inf_inf_set_a @ X4 @ Y3 ) ) ).
% inf_right_idem
thf(fact_151_inf_Oright__idem,axiom,
! [A3: set_a,B3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ B3 )
= ( inf_inf_set_a @ A3 @ B3 ) ) ).
% inf.right_idem
thf(fact_152_inf__left__idem,axiom,
! [X4: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y3 ) )
= ( inf_inf_set_a @ X4 @ Y3 ) ) ).
% inf_left_idem
thf(fact_153_inf_Oleft__idem,axiom,
! [A3: set_a,B3: set_a] :
( ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ A3 @ B3 ) )
= ( inf_inf_set_a @ A3 @ B3 ) ) ).
% inf.left_idem
thf(fact_154_inf__idem,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_155_inf_Oidem,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ A3 )
= A3 ) ).
% inf.idem
thf(fact_156_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_157_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_158_induce__eq__iff__induced,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ H @ t )
=> ( ( digrap7873285959652527175ph_a_b @ t @ ( pre_ve642382030648772252t_unit @ H ) )
= H ) ) ).
% induce_eq_iff_induced
thf(fact_159_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_160_in__verts__sccsD__sccs,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ t ) )
=> ( member6939884229742472986t_unit @ ( digrap7873285959652527175ph_a_b @ t @ S ) @ ( digraph_pre_sccs_a_b @ t ) ) ) ).
% in_verts_sccsD_sccs
thf(fact_161_in__sccs__verts__conv,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ t ) )
= ( member6939884229742472986t_unit @ ( digrap7873285959652527175ph_a_b @ t @ S ) @ ( digraph_pre_sccs_a_b @ t ) ) ) ).
% in_sccs_verts_conv
thf(fact_162_G_Oin__sccs__verts__conv,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ g ) )
= ( member6939884229742472986t_unit @ ( digrap7873285959652527175ph_a_b @ g @ S ) @ ( digraph_pre_sccs_a_b @ g ) ) ) ).
% G.in_sccs_verts_conv
thf(fact_163_G_Oin__verts__sccsD__sccs,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ g ) )
=> ( member6939884229742472986t_unit @ ( digrap7873285959652527175ph_a_b @ g @ S ) @ ( digraph_pre_sccs_a_b @ g ) ) ) ).
% G.in_verts_sccsD_sccs
thf(fact_164_disj__unvis__vis,axiom,
! [U: a,U2: set_a] :
( ( inf_inf_set_a @ ( graph_2016941059203891550ts_a_b @ t @ U @ U2 ) @ U2 )
= bot_bot_set_a ) ).
% disj_unvis_vis
thf(fact_165_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_166_reachable__from__root,axiom,
! [V: a] :
( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( reachable_a_b @ t @ source @ V ) ) ).
% reachable_from_root
thf(fact_167_iapath__dist__ends,axiom,
! [U: a,P2: list_b,V: a] :
( ( pre_gen_iapath_a_b @ t @ ( verts3_a_b @ t ) @ U @ P2 @ V )
=> ( U != V ) ) ).
% iapath_dist_ends
thf(fact_168_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_169_scc__disj,axiom,
! [C: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ t ) )
=> ( ( member6939884229742472986t_unit @ D @ ( digraph_pre_sccs_a_b @ t ) )
=> ( ( C != D )
=> ( ( inf_inf_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ D ) )
= bot_bot_set_a ) ) ) ) ).
% scc_disj
thf(fact_170__092_060open_062_092_060not_062_Averts_AG_A_092_060subseteq_062_Averts_AT_092_060close_062,axiom,
~ ( ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ g ) @ ( pre_ve642382030648772252t_unit @ t ) ) ).
% \<open>\<not> verts G \<subseteq> verts T\<close>
thf(fact_171_reachable__refl,axiom,
! [V: a] :
( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( reachable_a_b @ t @ V @ V ) ) ).
% reachable_refl
thf(fact_172_spanning__tree__imp__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5718416180170401981ee_a_b @ H @ t )
=> ( digrap8783888973171253482ed_a_b @ t ) ) ).
% spanning_tree_imp_connected
thf(fact_173_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_174_is__chain_H__def,axiom,
( ( graph_8150681439568091980in_a_b @ t )
= ( ( graph_2957805489637798020ts_a_b @ t )
= bot_bot_set_a ) ) ).
% is_chain'_def
thf(fact_175_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_176_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_177_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_178_last__merge__is__merge,axiom,
! [Y3: a] :
( ( member_a @ Y3 @ ( graph_2659413520663303054ts_a_b @ t ) )
=> ( member_a @ Y3 @ ( graph_2957805489637798020ts_a_b @ t ) ) ) ).
% last_merge_is_merge
thf(fact_179_G_Olast__merge__is__merge,axiom,
! [Y3: a] :
( ( member_a @ Y3 @ ( graph_2659413520663303054ts_a_b @ g ) )
=> ( member_a @ Y3 @ ( graph_2957805489637798020ts_a_b @ g ) ) ) ).
% G.last_merge_is_merge
thf(fact_180_last__merge__alt,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( graph_2659413520663303054ts_a_b @ t ) )
=> ! [Z: a] :
( ( ( reachable_a_b @ t @ X4 @ Z )
& ( Z != X4 ) )
=> ~ ( member_a @ Z @ ( graph_2957805489637798020ts_a_b @ t ) ) ) ) ).
% last_merge_alt
thf(fact_181_G_Olast__merge__alt,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( graph_2659413520663303054ts_a_b @ g ) )
=> ! [Z: a] :
( ( ( reachable_a_b @ g @ X4 @ Z )
& ( Z != X4 ) )
=> ~ ( member_a @ Z @ ( graph_2957805489637798020ts_a_b @ g ) ) ) ) ).
% G.last_merge_alt
thf(fact_182_inf__left__commute,axiom,
! [X4: set_a,Y3: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z2 ) )
= ( inf_inf_set_a @ Y3 @ ( inf_inf_set_a @ X4 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_183_inf_Oleft__commute,axiom,
! [B3: set_a,A3: set_a,C: set_a] :
( ( inf_inf_set_a @ B3 @ ( inf_inf_set_a @ A3 @ C ) )
= ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C ) ) ) ).
% inf.left_commute
thf(fact_184_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K: set_a,B3: set_a,A3: set_a] :
( ( B
= ( inf_inf_set_a @ K @ B3 ) )
=> ( ( inf_inf_set_a @ A3 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_185_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_a,K: set_a,A3: set_a,B3: set_a] :
( ( A
= ( inf_inf_set_a @ K @ A3 ) )
=> ( ( inf_inf_set_a @ A @ B3 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_186_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X: set_a,Y: set_a] : ( inf_inf_set_a @ Y @ X ) ) ) ).
% inf_commute
thf(fact_187_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_188_inf__assoc,axiom,
! [X4: set_a,Y3: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ Z2 )
= ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_189_inf_Oassoc,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ C )
= ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C ) ) ) ).
% inf.assoc
thf(fact_190_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_191_inf__sup__aci_I2_J,axiom,
! [X4: set_a,Y3: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ Z2 )
= ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_192_inf__sup__aci_I3_J,axiom,
! [X4: set_a,Y3: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z2 ) )
= ( inf_inf_set_a @ Y3 @ ( inf_inf_set_a @ X4 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_193_inf__sup__aci_I4_J,axiom,
! [X4: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y3 ) )
= ( inf_inf_set_a @ X4 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_194_loopfree_Oloopfree__digraph__axioms,axiom,
loopfree_digraph_a_b @ t ).
% loopfree.loopfree_digraph_axioms
thf(fact_195_nomulti_Onomulti__digraph__axioms,axiom,
nomulti_digraph_a_b @ t ).
% nomulti.nomulti_digraph_axioms
thf(fact_196_nearest__vert__reachable,axiom,
! [U: a,U2: set_a,W: b > real] :
( ( ( graph_2016941059203891550ts_a_b @ t @ U @ U2 )
!= bot_bot_set_a )
=> ( reachable_a_b @ t @ U @ ( graph_3614428260325061028rt_a_b @ t @ W @ U @ U2 ) ) ) ).
% nearest_vert_reachable
thf(fact_197_induce__subgraph__verts,axiom,
! [G: pre_pr7278220950009878019t_unit,Vs: set_a] :
( ( pre_ve642382030648772252t_unit @ ( digrap7873285959652527175ph_a_b @ G @ Vs ) )
= Vs ) ).
% induce_subgraph_verts
thf(fact_198_subgraph__axioms,axiom,
shorte3657265928840388360ph_a_b @ t @ g ).
% subgraph_axioms
thf(fact_199_G_Omerge__in__supergraph,axiom,
! [C2: pre_pr7278220950009878019t_unit,X4: a] :
( ( shorte3657265928840388360ph_a_b @ C2 @ g )
=> ( ( member_a @ X4 @ ( graph_2957805489637798020ts_a_b @ C2 ) )
=> ( member_a @ X4 @ ( graph_2957805489637798020ts_a_b @ g ) ) ) ) ).
% G.merge_in_supergraph
thf(fact_200_merge__in__supergraph,axiom,
! [C2: pre_pr7278220950009878019t_unit,X4: a] :
( ( shorte3657265928840388360ph_a_b @ C2 @ t )
=> ( ( member_a @ X4 @ ( graph_2957805489637798020ts_a_b @ C2 ) )
=> ( member_a @ X4 @ ( graph_2957805489637798020ts_a_b @ t ) ) ) ) ).
% merge_in_supergraph
thf(fact_201_some__unvis__vert_I1_J,axiom,
! [U: a,U2: set_a,X4: a,W: b > real] :
( ( ( graph_2016941059203891550ts_a_b @ t @ U @ U2 )
!= bot_bot_set_a )
=> ( ( X4
= ( graph_3614428260325061028rt_a_b @ t @ W @ U @ U2 ) )
=> ( member_a @ X4 @ ( graph_2016941059203891550ts_a_b @ t @ U @ U2 ) ) ) ) ).
% some_unvis_vert(1)
thf(fact_202_nearest__vert__not__mem,axiom,
! [U: a,U2: set_a,W: b > real] :
( ( ( graph_2016941059203891550ts_a_b @ t @ U @ U2 )
!= bot_bot_set_a )
=> ~ ( member_a @ ( graph_3614428260325061028rt_a_b @ t @ W @ U @ U2 ) @ U2 ) ) ).
% nearest_vert_not_mem
thf(fact_203_nearest__vert__unvis,axiom,
! [U: a,U2: set_a,W: b > real] :
( ( ( graph_2016941059203891550ts_a_b @ t @ U @ U2 )
!= bot_bot_set_a )
=> ( member_a @ ( graph_3614428260325061028rt_a_b @ t @ W @ U @ U2 ) @ ( graph_2016941059203891550ts_a_b @ t @ U @ U2 ) ) ) ).
% nearest_vert_unvis
thf(fact_204_G_Oreachable__induce__subgraphD,axiom,
! [S: set_a,U: a,V: a] :
( ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ g @ S ) @ U @ V )
=> ( ( ord_less_eq_set_a @ S @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( reachable_a_b @ g @ U @ V ) ) ) ).
% G.reachable_induce_subgraphD
thf(fact_205_dual__order_Orefl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_206_dual__order_Orefl,axiom,
! [A3: set_b] : ( ord_less_eq_set_b @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_207_order__refl,axiom,
! [X4: set_a] : ( ord_less_eq_set_a @ X4 @ X4 ) ).
% order_refl
thf(fact_208_order__refl,axiom,
! [X4: set_b] : ( ord_less_eq_set_b @ X4 @ X4 ) ).
% order_refl
thf(fact_209_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_210_subset__antisym,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_eq_set_b @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_211_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( member_set_a @ X3 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_212_subsetI,axiom,
! [A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ! [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ A )
=> ( member6939884229742472986t_unit @ X3 @ B ) )
=> ( ord_le8200006823705900825t_unit @ A @ B ) ) ).
% subsetI
thf(fact_213_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_214_subsetI,axiom,
! [A: set_b,B: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ( member_b @ X3 @ B ) )
=> ( ord_less_eq_set_b @ A @ B ) ) ).
% subsetI
thf(fact_215_in__sccs__subset__imp__eq,axiom,
! [C: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ t ) )
=> ( ( member6939884229742472986t_unit @ D @ ( digraph_pre_sccs_a_b @ t ) )
=> ( ( ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ D ) )
=> ( C = D ) ) ) ) ).
% in_sccs_subset_imp_eq
thf(fact_216_G_Oin__sccs__subset__imp__eq,axiom,
! [C: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ g ) )
=> ( ( member6939884229742472986t_unit @ D @ ( digraph_pre_sccs_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ D ) )
=> ( C = D ) ) ) ) ).
% G.in_sccs_subset_imp_eq
thf(fact_217_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_218_reachable__induce__ss,axiom,
! [S: set_a,U: a,V: a,T: set_a] :
( ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S ) @ U @ V )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ t @ T ) @ U @ V ) ) ) ).
% reachable_induce_ss
thf(fact_219_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_220_G_Oreachable__induce__ss,axiom,
! [S: set_a,U: a,V: a,T: set_a] :
( ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ g @ S ) @ U @ V )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ g @ T ) @ U @ V ) ) ) ).
% G.reachable_induce_ss
thf(fact_221_verts__T__subset__G,axiom,
ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ t ) @ ( pre_ve642382030648772252t_unit @ g ) ).
% verts_T_subset_G
thf(fact_222_reachable__induce__subgraphD,axiom,
! [S: set_a,U: a,V: a] :
( ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S ) @ U @ V )
=> ( ( ord_less_eq_set_a @ S @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( reachable_a_b @ t @ U @ V ) ) ) ).
% reachable_induce_subgraphD
thf(fact_223_le__inf__iff,axiom,
! [X4: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z2 ) )
= ( ( ord_less_eq_set_a @ X4 @ Y3 )
& ( ord_less_eq_set_a @ X4 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_224_le__inf__iff,axiom,
! [X4: set_b,Y3: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ X4 @ ( inf_inf_set_b @ Y3 @ Z2 ) )
= ( ( ord_less_eq_set_b @ X4 @ Y3 )
& ( ord_less_eq_set_b @ X4 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_225_inf_Obounded__iff,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C ) )
= ( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ A3 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_226_inf_Obounded__iff,axiom,
! [A3: set_b,B3: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A3 @ ( inf_inf_set_b @ B3 @ C ) )
= ( ( ord_less_eq_set_b @ A3 @ B3 )
& ( ord_less_eq_set_b @ A3 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_227_empty__subsetI,axiom,
! [A: set_pr5411798346947241657t_unit] : ( ord_le8200006823705900825t_unit @ bot_bo1839476491465656141t_unit @ A ) ).
% empty_subsetI
thf(fact_228_empty__subsetI,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% empty_subsetI
thf(fact_229_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_230_empty__subsetI,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A ) ).
% empty_subsetI
thf(fact_231_subset__empty,axiom,
! [A: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A @ bot_bo1839476491465656141t_unit )
= ( A = bot_bo1839476491465656141t_unit ) ) ).
% subset_empty
thf(fact_232_subset__empty,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_233_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_234_subset__empty,axiom,
! [A: set_b] :
( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
= ( A = bot_bot_set_b ) ) ).
% subset_empty
thf(fact_235_Int__subset__iff,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
= ( ( ord_less_eq_set_a @ C2 @ A )
& ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_236_Int__subset__iff,axiom,
! [C2: set_b,A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ C2 @ ( inf_inf_set_b @ A @ B ) )
= ( ( ord_less_eq_set_b @ C2 @ A )
& ( ord_less_eq_set_b @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_237_induced__induce,axiom,
! [Vs: set_a] :
( ( ord_less_eq_set_a @ Vs @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( digrap5251062021860773499ph_a_b @ ( digrap7873285959652527175ph_a_b @ t @ Vs ) @ t ) ) ).
% induced_induce
thf(fact_238_G_Oinduced__induce,axiom,
! [Vs: set_a] :
( ( ord_less_eq_set_a @ Vs @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( digrap5251062021860773499ph_a_b @ ( digrap7873285959652527175ph_a_b @ g @ Vs ) @ g ) ) ).
% G.induced_induce
thf(fact_239_order__antisym__conv,axiom,
! [Y3: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X4 )
=> ( ( ord_less_eq_set_a @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_240_order__antisym__conv,axiom,
! [Y3: set_b,X4: set_b] :
( ( ord_less_eq_set_b @ Y3 @ X4 )
=> ( ( ord_less_eq_set_b @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_241_ord__le__eq__subst,axiom,
! [A3: set_a,B3: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_242_ord__le__eq__subst,axiom,
! [A3: set_a,B3: set_a,F: set_a > set_b,C: set_b] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_243_ord__le__eq__subst,axiom,
! [A3: set_b,B3: set_b,F: set_b > set_a,C: set_a] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_244_ord__le__eq__subst,axiom,
! [A3: set_b,B3: set_b,F: set_b > set_b,C: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y2 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_245_ord__eq__le__subst,axiom,
! [A3: set_a,F: set_a > set_a,B3: set_a,C: set_a] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_246_ord__eq__le__subst,axiom,
! [A3: set_b,F: set_a > set_b,B3: set_a,C: set_a] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_b @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_247_ord__eq__le__subst,axiom,
! [A3: set_a,F: set_b > set_a,B3: set_b,C: set_b] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_b @ B3 @ C )
=> ( ! [X3: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_248_ord__eq__le__subst,axiom,
! [A3: set_b,F: set_b > set_b,B3: set_b,C: set_b] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_b @ B3 @ C )
=> ( ! [X3: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y2 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_b @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_249_order__eq__refl,axiom,
! [X4: set_a,Y3: set_a] :
( ( X4 = Y3 )
=> ( ord_less_eq_set_a @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_250_order__eq__refl,axiom,
! [X4: set_b,Y3: set_b] :
( ( X4 = Y3 )
=> ( ord_less_eq_set_b @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_251_order__subst2,axiom,
! [A3: set_a,B3: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_252_order__subst2,axiom,
! [A3: set_a,B3: set_a,F: set_a > set_b,C: set_b] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_b @ ( F @ B3 ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_253_order__subst2,axiom,
! [A3: set_b,B3: set_b,F: set_b > set_a,C: set_a] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
=> ( ! [X3: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_254_order__subst2,axiom,
! [A3: set_b,B3: set_b,F: set_b > set_b,C: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( ord_less_eq_set_b @ ( F @ B3 ) @ C )
=> ( ! [X3: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y2 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_255_order__subst1,axiom,
! [A3: set_a,F: set_a > set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_256_order__subst1,axiom,
! [A3: set_a,F: set_b > set_a,B3: set_b,C: set_b] :
( ( ord_less_eq_set_a @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_b @ B3 @ C )
=> ( ! [X3: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_257_order__subst1,axiom,
! [A3: set_b,F: set_a > set_b,B3: set_a,C: set_a] :
( ( ord_less_eq_set_b @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_b @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_258_order__subst1,axiom,
! [A3: set_b,F: set_b > set_b,B3: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_b @ B3 @ C )
=> ( ! [X3: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y2 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_b @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_259_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_260_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_b,Z3: set_b] : ( Y4 = Z3 ) )
= ( ^ [A4: set_b,B4: set_b] :
( ( ord_less_eq_set_b @ A4 @ B4 )
& ( ord_less_eq_set_b @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_261_Collect__mono__iff,axiom,
! [P: pre_pr7278220950009878019t_unit > $o,Q: pre_pr7278220950009878019t_unit > $o] :
( ( ord_le8200006823705900825t_unit @ ( collec8000012497822511960t_unit @ P ) @ ( collec8000012497822511960t_unit @ Q ) )
= ( ! [X: pre_pr7278220950009878019t_unit] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_262_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_263_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_264_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_265_set__eq__subset,axiom,
( ( ^ [Y4: set_b,Z3: set_b] : ( Y4 = Z3 ) )
= ( ^ [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
& ( ord_less_eq_set_b @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_266_antisym,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_267_antisym,axiom,
! [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( ord_less_eq_set_b @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_268_subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_269_subset__trans,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_eq_set_b @ B @ C2 )
=> ( ord_less_eq_set_b @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_270_Collect__mono,axiom,
! [P: pre_pr7278220950009878019t_unit > $o,Q: pre_pr7278220950009878019t_unit > $o] :
( ! [X3: pre_pr7278220950009878019t_unit] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8200006823705900825t_unit @ ( collec8000012497822511960t_unit @ P ) @ ( collec8000012497822511960t_unit @ Q ) ) ) ).
% Collect_mono
thf(fact_271_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_272_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_273_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_274_subset__refl,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ A @ A ) ).
% subset_refl
thf(fact_275_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A2: set_set_a,B2: set_set_a] :
! [T2: set_a] :
( ( member_set_a @ T2 @ A2 )
=> ( member_set_a @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_276_subset__iff,axiom,
( ord_le8200006823705900825t_unit
= ( ^ [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
! [T2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ T2 @ A2 )
=> ( member6939884229742472986t_unit @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_277_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B2: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A2 )
=> ( member_a @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_278_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A2: set_b,B2: set_b] :
! [T2: b] :
( ( member_b @ T2 @ A2 )
=> ( member_b @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_279_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_280_equalityD2,axiom,
! [A: set_b,B: set_b] :
( ( A = B )
=> ( ord_less_eq_set_b @ B @ A ) ) ).
% equalityD2
thf(fact_281_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_282_equalityD1,axiom,
! [A: set_b,B: set_b] :
( ( A = B )
=> ( ord_less_eq_set_b @ A @ B ) ) ).
% equalityD1
thf(fact_283_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A2: set_set_a,B2: set_set_a] :
! [X: set_a] :
( ( member_set_a @ X @ A2 )
=> ( member_set_a @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_284_subset__eq,axiom,
( ord_le8200006823705900825t_unit
= ( ^ [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
! [X: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X @ A2 )
=> ( member6939884229742472986t_unit @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_285_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B2: set_a] :
! [X: a] :
( ( member_a @ X @ A2 )
=> ( member_a @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_286_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A2: set_b,B2: set_b] :
! [X: b] :
( ( member_b @ X @ A2 )
=> ( member_b @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_287_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_288_equalityE,axiom,
! [A: set_b,B: set_b] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_b @ A @ B )
=> ~ ( ord_less_eq_set_b @ B @ A ) ) ) ).
% equalityE
thf(fact_289_subsetD,axiom,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_290_subsetD,axiom,
! [A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit,C: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ A @ B )
=> ( ( member6939884229742472986t_unit @ C @ A )
=> ( member6939884229742472986t_unit @ C @ B ) ) ) ).
% subsetD
thf(fact_291_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_292_subsetD,axiom,
! [A: set_b,B: set_b,C: b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( member_b @ C @ A )
=> ( member_b @ C @ B ) ) ) ).
% subsetD
thf(fact_293_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X4: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ X4 @ A )
=> ( member_set_a @ X4 @ B ) ) ) ).
% in_mono
thf(fact_294_in__mono,axiom,
! [A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit,X4: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ A @ B )
=> ( ( member6939884229742472986t_unit @ X4 @ A )
=> ( member6939884229742472986t_unit @ X4 @ B ) ) ) ).
% in_mono
thf(fact_295_in__mono,axiom,
! [A: set_a,B: set_a,X4: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X4 @ A )
=> ( member_a @ X4 @ B ) ) ) ).
% in_mono
thf(fact_296_in__mono,axiom,
! [A: set_b,B: set_b,X4: b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( member_b @ X4 @ A )
=> ( member_b @ X4 @ B ) ) ) ).
% in_mono
thf(fact_297_dual__order_Otrans,axiom,
! [B3: set_a,A3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ( ord_less_eq_set_a @ C @ B3 )
=> ( ord_less_eq_set_a @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_298_dual__order_Otrans,axiom,
! [B3: set_b,A3: set_b,C: set_b] :
( ( ord_less_eq_set_b @ B3 @ A3 )
=> ( ( ord_less_eq_set_b @ C @ B3 )
=> ( ord_less_eq_set_b @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_299_dual__order_Oantisym,axiom,
! [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_300_dual__order_Oantisym,axiom,
! [B3: set_b,A3: set_b] :
( ( ord_less_eq_set_b @ B3 @ A3 )
=> ( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_301_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_302_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_b,Z3: set_b] : ( Y4 = Z3 ) )
= ( ^ [A4: set_b,B4: set_b] :
( ( ord_less_eq_set_b @ B4 @ A4 )
& ( ord_less_eq_set_b @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_303_order__trans,axiom,
! [X4: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z2 )
=> ( ord_less_eq_set_a @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_304_order__trans,axiom,
! [X4: set_b,Y3: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y3 )
=> ( ( ord_less_eq_set_b @ Y3 @ Z2 )
=> ( ord_less_eq_set_b @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_305_order_Otrans,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_eq_set_a @ A3 @ C ) ) ) ).
% order.trans
thf(fact_306_order_Otrans,axiom,
! [A3: set_b,B3: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( ord_less_eq_set_b @ B3 @ C )
=> ( ord_less_eq_set_b @ A3 @ C ) ) ) ).
% order.trans
thf(fact_307_order__antisym,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_308_order__antisym,axiom,
! [X4: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y3 )
=> ( ( ord_less_eq_set_b @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_309_ord__le__eq__trans,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_a @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_310_ord__le__eq__trans,axiom,
! [A3: set_b,B3: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_b @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_311_ord__eq__le__trans,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( A3 = B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_eq_set_a @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_312_ord__eq__le__trans,axiom,
! [A3: set_b,B3: set_b,C: set_b] :
( ( A3 = B3 )
=> ( ( ord_less_eq_set_b @ B3 @ C )
=> ( ord_less_eq_set_b @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_313_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = 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_314_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_b,Z3: set_b] : ( Y4 = 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_315_bot_Oextremum__uniqueI,axiom,
! [A3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A3 @ bot_bo1839476491465656141t_unit )
=> ( A3 = bot_bo1839476491465656141t_unit ) ) ).
% bot.extremum_uniqueI
thf(fact_316_bot_Oextremum__uniqueI,axiom,
! [A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ bot_bot_set_set_a )
=> ( A3 = bot_bot_set_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_317_bot_Oextremum__uniqueI,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
=> ( A3 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_318_bot_Oextremum__uniqueI,axiom,
! [A3: set_b] :
( ( ord_less_eq_set_b @ A3 @ bot_bot_set_b )
=> ( A3 = bot_bot_set_b ) ) ).
% bot.extremum_uniqueI
thf(fact_319_bot_Oextremum__unique,axiom,
! [A3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A3 @ bot_bo1839476491465656141t_unit )
= ( A3 = bot_bo1839476491465656141t_unit ) ) ).
% bot.extremum_unique
thf(fact_320_bot_Oextremum__unique,axiom,
! [A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ bot_bot_set_set_a )
= ( A3 = bot_bot_set_set_a ) ) ).
% bot.extremum_unique
thf(fact_321_bot_Oextremum__unique,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_322_bot_Oextremum__unique,axiom,
! [A3: set_b] :
( ( ord_less_eq_set_b @ A3 @ bot_bot_set_b )
= ( A3 = bot_bot_set_b ) ) ).
% bot.extremum_unique
thf(fact_323_bot_Oextremum,axiom,
! [A3: set_pr5411798346947241657t_unit] : ( ord_le8200006823705900825t_unit @ bot_bo1839476491465656141t_unit @ A3 ) ).
% bot.extremum
thf(fact_324_bot_Oextremum,axiom,
! [A3: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A3 ) ).
% bot.extremum
thf(fact_325_bot_Oextremum,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% bot.extremum
thf(fact_326_bot_Oextremum,axiom,
! [A3: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A3 ) ).
% bot.extremum
thf(fact_327_inf__sup__ord_I2_J,axiom,
! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_328_inf__sup__ord_I2_J,axiom,
! [X4: set_b,Y3: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X4 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_329_inf__sup__ord_I1_J,axiom,
! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_330_inf__sup__ord_I1_J,axiom,
! [X4: set_b,Y3: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X4 @ Y3 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_331_inf__le1,axiom,
! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ X4 ) ).
% inf_le1
thf(fact_332_inf__le1,axiom,
! [X4: set_b,Y3: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X4 @ Y3 ) @ X4 ) ).
% inf_le1
thf(fact_333_inf__le2,axiom,
! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_334_inf__le2,axiom,
! [X4: set_b,Y3: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X4 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_335_le__infE,axiom,
! [X4: set_a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ A3 @ B3 ) )
=> ~ ( ( ord_less_eq_set_a @ X4 @ A3 )
=> ~ ( ord_less_eq_set_a @ X4 @ B3 ) ) ) ).
% le_infE
thf(fact_336_le__infE,axiom,
! [X4: set_b,A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ X4 @ ( inf_inf_set_b @ A3 @ B3 ) )
=> ~ ( ( ord_less_eq_set_b @ X4 @ A3 )
=> ~ ( ord_less_eq_set_b @ X4 @ B3 ) ) ) ).
% le_infE
thf(fact_337_le__infI,axiom,
! [X4: set_a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ X4 @ A3 )
=> ( ( ord_less_eq_set_a @ X4 @ B3 )
=> ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% le_infI
thf(fact_338_le__infI,axiom,
! [X4: set_b,A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ X4 @ A3 )
=> ( ( ord_less_eq_set_b @ X4 @ B3 )
=> ( ord_less_eq_set_b @ X4 @ ( inf_inf_set_b @ A3 @ B3 ) ) ) ) ).
% le_infI
thf(fact_339_inf__mono,axiom,
! [A3: set_a,C: set_a,B3: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A3 @ C )
=> ( ( ord_less_eq_set_a @ B3 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_340_inf__mono,axiom,
! [A3: set_b,C: set_b,B3: set_b,D: set_b] :
( ( ord_less_eq_set_b @ A3 @ C )
=> ( ( ord_less_eq_set_b @ B3 @ D )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A3 @ B3 ) @ ( inf_inf_set_b @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_341_le__infI1,axiom,
! [A3: set_a,X4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ X4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ X4 ) ) ).
% le_infI1
thf(fact_342_le__infI1,axiom,
! [A3: set_b,X4: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ X4 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A3 @ B3 ) @ X4 ) ) ).
% le_infI1
thf(fact_343_le__infI2,axiom,
! [B3: set_a,X4: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ X4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ X4 ) ) ).
% le_infI2
thf(fact_344_le__infI2,axiom,
! [B3: set_b,X4: set_b,A3: set_b] :
( ( ord_less_eq_set_b @ B3 @ X4 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A3 @ B3 ) @ X4 ) ) ).
% le_infI2
thf(fact_345_inf_OorderE,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( A3
= ( inf_inf_set_a @ A3 @ B3 ) ) ) ).
% inf.orderE
thf(fact_346_inf_OorderE,axiom,
! [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( A3
= ( inf_inf_set_b @ A3 @ B3 ) ) ) ).
% inf.orderE
thf(fact_347_inf_OorderI,axiom,
! [A3: set_a,B3: set_a] :
( ( A3
= ( inf_inf_set_a @ A3 @ B3 ) )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% inf.orderI
thf(fact_348_inf_OorderI,axiom,
! [A3: set_b,B3: set_b] :
( ( A3
= ( inf_inf_set_b @ A3 @ B3 ) )
=> ( ord_less_eq_set_b @ A3 @ B3 ) ) ).
% inf.orderI
thf(fact_349_inf__unique,axiom,
! [F: set_a > set_a > set_a,X4: set_a,Y3: set_a] :
( ! [X3: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: set_a,Y2: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_a @ X3 @ Z4 )
=> ( ord_less_eq_set_a @ X3 @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_inf_set_a @ X4 @ Y3 )
= ( F @ X4 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_350_inf__unique,axiom,
! [F: set_b > set_b > set_b,X4: set_b,Y3: set_b] :
( ! [X3: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: set_b,Y2: set_b,Z4: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y2 )
=> ( ( ord_less_eq_set_b @ X3 @ Z4 )
=> ( ord_less_eq_set_b @ X3 @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_inf_set_b @ X4 @ Y3 )
= ( F @ X4 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_351_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_352_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_353_inf_Oabsorb1,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( inf_inf_set_a @ A3 @ B3 )
= A3 ) ) ).
% inf.absorb1
thf(fact_354_inf_Oabsorb1,axiom,
! [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( inf_inf_set_b @ A3 @ B3 )
= A3 ) ) ).
% inf.absorb1
thf(fact_355_inf_Oabsorb2,axiom,
! [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_356_inf_Oabsorb2,axiom,
! [B3: set_b,A3: set_b] :
( ( ord_less_eq_set_b @ B3 @ A3 )
=> ( ( inf_inf_set_b @ A3 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_357_inf__absorb1,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( inf_inf_set_a @ X4 @ Y3 )
= X4 ) ) ).
% inf_absorb1
thf(fact_358_inf__absorb1,axiom,
! [X4: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y3 )
=> ( ( inf_inf_set_b @ X4 @ Y3 )
= X4 ) ) ).
% inf_absorb1
thf(fact_359_inf__absorb2,axiom,
! [Y3: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X4 )
=> ( ( inf_inf_set_a @ X4 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_360_inf__absorb2,axiom,
! [Y3: set_b,X4: set_b] :
( ( ord_less_eq_set_b @ Y3 @ X4 )
=> ( ( inf_inf_set_b @ X4 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_361_inf_OboundedE,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ~ ( ord_less_eq_set_a @ A3 @ C ) ) ) ).
% inf.boundedE
thf(fact_362_inf_OboundedE,axiom,
! [A3: set_b,B3: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A3 @ ( inf_inf_set_b @ B3 @ C ) )
=> ~ ( ( ord_less_eq_set_b @ A3 @ B3 )
=> ~ ( ord_less_eq_set_b @ A3 @ C ) ) ) ).
% inf.boundedE
thf(fact_363_inf_OboundedI,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ A3 @ C )
=> ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_364_inf_OboundedI,axiom,
! [A3: set_b,B3: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( ord_less_eq_set_b @ A3 @ C )
=> ( ord_less_eq_set_b @ A3 @ ( inf_inf_set_b @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_365_inf__greatest,axiom,
! [X4: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ord_less_eq_set_a @ X4 @ Z2 )
=> ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_366_inf__greatest,axiom,
! [X4: set_b,Y3: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y3 )
=> ( ( ord_less_eq_set_b @ X4 @ Z2 )
=> ( ord_less_eq_set_b @ X4 @ ( inf_inf_set_b @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_367_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_368_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_369_inf_Ocobounded1,axiom,
! [A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ A3 ) ).
% inf.cobounded1
thf(fact_370_inf_Ocobounded1,axiom,
! [A3: set_b,B3: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A3 @ B3 ) @ A3 ) ).
% inf.cobounded1
thf(fact_371_inf_Ocobounded2,axiom,
! [A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_372_inf_Ocobounded2,axiom,
! [A3: set_b,B3: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A3 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_373_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_374_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_375_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_376_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_377_inf_OcoboundedI1,axiom,
! [A3: set_a,C: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_378_inf_OcoboundedI1,axiom,
! [A3: set_b,C: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ C )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A3 @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_379_inf_OcoboundedI2,axiom,
! [B3: set_a,C: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_380_inf_OcoboundedI2,axiom,
! [B3: set_b,C: set_b,A3: set_b] :
( ( ord_less_eq_set_b @ B3 @ C )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A3 @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_381_Int__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_382_Int__mono,axiom,
! [A: set_b,C2: set_b,B: set_b,D2: set_b] :
( ( ord_less_eq_set_b @ A @ C2 )
=> ( ( ord_less_eq_set_b @ B @ D2 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ ( inf_inf_set_b @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_383_Int__lower1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_384_Int__lower1,axiom,
! [A: set_b,B: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_385_Int__lower2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_386_Int__lower2,axiom,
! [A: set_b,B: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_387_Int__absorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_388_Int__absorb1,axiom,
! [B: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B @ A )
=> ( ( inf_inf_set_b @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_389_Int__absorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_390_Int__absorb2,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( inf_inf_set_b @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_391_Int__greatest,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_392_Int__greatest,axiom,
! [C2: set_b,A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ C2 @ A )
=> ( ( ord_less_eq_set_b @ C2 @ B )
=> ( ord_less_eq_set_b @ C2 @ ( inf_inf_set_b @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_393_Int__Collect__mono,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_394_Int__Collect__mono,axiom,
! [A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit,P: pre_pr7278220950009878019t_unit > $o,Q: pre_pr7278220950009878019t_unit > $o] :
( ( ord_le8200006823705900825t_unit @ A @ B )
=> ( ! [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le8200006823705900825t_unit @ ( inf_in1092213268631476299t_unit @ A @ ( collec8000012497822511960t_unit @ P ) ) @ ( inf_in1092213268631476299t_unit @ B @ ( collec8000012497822511960t_unit @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_395_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_396_Int__Collect__mono,axiom,
! [A: set_b,B: set_b,P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ ( collect_b @ P ) ) @ ( inf_inf_set_b @ B @ ( collect_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_397_pre__digraph_Oin__sccs__subset__imp__eq,axiom,
! [C: pre_pr7278220950009878019t_unit,G: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ G ) )
=> ( ( member6939884229742472986t_unit @ D @ ( digraph_pre_sccs_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ D ) )
=> ( C = D ) ) ) ) ).
% pre_digraph.in_sccs_subset_imp_eq
thf(fact_398_pre__digraph_Osccs_Ocong,axiom,
digraph_pre_sccs_a_b = digraph_pre_sccs_a_b ).
% pre_digraph.sccs.cong
thf(fact_399_pre__digraph_Osccs__verts_Ocong,axiom,
digrap2871191568752656621ts_a_b = digrap2871191568752656621ts_a_b ).
% pre_digraph.sccs_verts.cong
thf(fact_400_pre__digraph_Oscc__of_Ocong,axiom,
digrap2937667069914300949of_a_b = digrap2937667069914300949of_a_b ).
% pre_digraph.scc_of.cong
thf(fact_401_induced__eq__verts__imp__eq,axiom,
! [G: pre_pr7278220950009878019t_unit,H: pre_pr7278220950009878019t_unit,G2: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ G @ H )
=> ( ( digrap5251062021860773499ph_a_b @ G2 @ H )
=> ( ( ( pre_ve642382030648772252t_unit @ G )
= ( pre_ve642382030648772252t_unit @ G2 ) )
=> ( G = G2 ) ) ) ) ).
% induced_eq_verts_imp_eq
thf(fact_402_pre__digraph_Oin__sccs__imp__induced,axiom,
! [C: pre_pr7278220950009878019t_unit,G: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ G ) )
=> ( digrap5251062021860773499ph_a_b @ C @ G ) ) ).
% pre_digraph.in_sccs_imp_induced
thf(fact_403_fin__digraph_Onearest__vert__reachable,axiom,
! [G: pre_pr7278220950009878019t_unit,U: a,U2: set_a,W: b > real] :
( ( fin_digraph_a_b @ G )
=> ( ( ( graph_2016941059203891550ts_a_b @ G @ U @ U2 )
!= bot_bot_set_a )
=> ( reachable_a_b @ G @ U @ ( graph_3614428260325061028rt_a_b @ G @ W @ U @ U2 ) ) ) ) ).
% fin_digraph.nearest_vert_reachable
thf(fact_404_fin__digraph_Onearest__vert__not__mem,axiom,
! [G: pre_pr7278220950009878019t_unit,U: a,U2: set_a,W: b > real] :
( ( fin_digraph_a_b @ G )
=> ( ( ( graph_2016941059203891550ts_a_b @ G @ U @ U2 )
!= bot_bot_set_a )
=> ~ ( member_a @ ( graph_3614428260325061028rt_a_b @ G @ W @ U @ U2 ) @ U2 ) ) ) ).
% fin_digraph.nearest_vert_not_mem
thf(fact_405_fin__digraph_Onearest__vert__unvis,axiom,
! [G: pre_pr7278220950009878019t_unit,U: a,U2: set_a,W: b > real] :
( ( fin_digraph_a_b @ G )
=> ( ( ( graph_2016941059203891550ts_a_b @ G @ U @ U2 )
!= bot_bot_set_a )
=> ( member_a @ ( graph_3614428260325061028rt_a_b @ G @ W @ U @ U2 ) @ ( graph_2016941059203891550ts_a_b @ G @ U @ U2 ) ) ) ) ).
% fin_digraph.nearest_vert_unvis
thf(fact_406_fin__digraph_Osome__unvis__vert_I1_J,axiom,
! [G: pre_pr7278220950009878019t_unit,U: a,U2: set_a,X4: a,W: b > real] :
( ( fin_digraph_a_b @ G )
=> ( ( ( graph_2016941059203891550ts_a_b @ G @ U @ U2 )
!= bot_bot_set_a )
=> ( ( X4
= ( graph_3614428260325061028rt_a_b @ G @ W @ U @ U2 ) )
=> ( member_a @ X4 @ ( graph_2016941059203891550ts_a_b @ G @ U @ U2 ) ) ) ) ) ).
% fin_digraph.some_unvis_vert(1)
thf(fact_407_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_408_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_409_G_Osccs__conv__sccs__verts,axiom,
( ( digraph_pre_sccs_a_b @ g )
= ( image_6801035452528096924t_unit @ ( digrap7873285959652527175ph_a_b @ g ) @ ( digrap2871191568752656621ts_a_b @ g ) ) ) ).
% G.sccs_conv_sccs_verts
thf(fact_410_sccs__conv__sccs__verts,axiom,
( ( digraph_pre_sccs_a_b @ t )
= ( image_6801035452528096924t_unit @ ( digrap7873285959652527175ph_a_b @ t ) @ ( digrap2871191568752656621ts_a_b @ t ) ) ) ).
% sccs_conv_sccs_verts
thf(fact_411_G_Oin__sccs__vertsI__sccs,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( image_7466199892558553556_set_a @ pre_ve642382030648772252t_unit @ ( digraph_pre_sccs_a_b @ g ) ) )
=> ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ g ) ) ) ).
% G.in_sccs_vertsI_sccs
thf(fact_412_image__eqI,axiom,
! [B3: a,F: a > a,X4: a,A: set_a] :
( ( B3
= ( F @ X4 ) )
=> ( ( member_a @ X4 @ A )
=> ( member_a @ B3 @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_413_image__eqI,axiom,
! [B3: b,F: a > b,X4: a,A: set_a] :
( ( B3
= ( F @ X4 ) )
=> ( ( member_a @ X4 @ A )
=> ( member_b @ B3 @ ( image_a_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_414_image__eqI,axiom,
! [B3: a,F: b > a,X4: b,A: set_b] :
( ( B3
= ( F @ X4 ) )
=> ( ( member_b @ X4 @ A )
=> ( member_a @ B3 @ ( image_b_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_415_image__eqI,axiom,
! [B3: b,F: b > b,X4: b,A: set_b] :
( ( B3
= ( F @ X4 ) )
=> ( ( member_b @ X4 @ A )
=> ( member_b @ B3 @ ( image_b_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_416_image__eqI,axiom,
! [B3: set_a,F: a > set_a,X4: a,A: set_a] :
( ( B3
= ( F @ X4 ) )
=> ( ( member_a @ X4 @ A )
=> ( member_set_a @ B3 @ ( image_a_set_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_417_image__eqI,axiom,
! [B3: a,F: set_a > a,X4: set_a,A: set_set_a] :
( ( B3
= ( F @ X4 ) )
=> ( ( member_set_a @ X4 @ A )
=> ( member_a @ B3 @ ( image_set_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_418_image__eqI,axiom,
! [B3: b,F: set_a > b,X4: set_a,A: set_set_a] :
( ( B3
= ( F @ X4 ) )
=> ( ( member_set_a @ X4 @ A )
=> ( member_b @ B3 @ ( image_set_a_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_419_image__eqI,axiom,
! [B3: set_a,F: b > set_a,X4: b,A: set_b] :
( ( B3
= ( F @ X4 ) )
=> ( ( member_b @ X4 @ A )
=> ( member_set_a @ B3 @ ( image_b_set_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_420_image__eqI,axiom,
! [B3: set_a,F: set_a > set_a,X4: set_a,A: set_set_a] :
( ( B3
= ( F @ X4 ) )
=> ( ( member_set_a @ X4 @ A )
=> ( member_set_a @ B3 @ ( image_set_a_set_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_421_image__eqI,axiom,
! [B3: pre_pr7278220950009878019t_unit,F: a > pre_pr7278220950009878019t_unit,X4: a,A: set_a] :
( ( B3
= ( F @ X4 ) )
=> ( ( member_a @ X4 @ A )
=> ( member6939884229742472986t_unit @ B3 @ ( image_5713294457175270716t_unit @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_422_in__sccs__vertsI__sccs,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( image_7466199892558553556_set_a @ pre_ve642382030648772252t_unit @ ( digraph_pre_sccs_a_b @ t ) ) )
=> ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ t ) ) ) ).
% in_sccs_vertsI_sccs
thf(fact_423_sccs__verts__conv,axiom,
( ( digrap2871191568752656621ts_a_b @ t )
= ( image_7466199892558553556_set_a @ pre_ve642382030648772252t_unit @ ( digraph_pre_sccs_a_b @ t ) ) ) ).
% sccs_verts_conv
thf(fact_424_G_Osccs__verts__conv,axiom,
( ( digrap2871191568752656621ts_a_b @ g )
= ( image_7466199892558553556_set_a @ pre_ve642382030648772252t_unit @ ( digraph_pre_sccs_a_b @ g ) ) ) ).
% G.sccs_verts_conv
thf(fact_425_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_426_image__empty,axiom,
! [F: a > b] :
( ( image_a_b @ F @ bot_bot_set_a )
= bot_bot_set_b ) ).
% image_empty
thf(fact_427_image__empty,axiom,
! [F: b > a] :
( ( image_b_a @ F @ bot_bot_set_b )
= bot_bot_set_a ) ).
% image_empty
thf(fact_428_image__empty,axiom,
! [F: b > b] :
( ( image_b_b @ F @ bot_bot_set_b )
= bot_bot_set_b ) ).
% image_empty
thf(fact_429_image__empty,axiom,
! [F: a > set_a] :
( ( image_a_set_a @ F @ bot_bot_set_a )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_430_image__empty,axiom,
! [F: b > set_a] :
( ( image_b_set_a @ F @ bot_bot_set_b )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_431_image__empty,axiom,
! [F: set_a > a] :
( ( image_set_a_a @ F @ bot_bot_set_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_432_image__empty,axiom,
! [F: set_a > b] :
( ( image_set_a_b @ F @ bot_bot_set_set_a )
= bot_bot_set_b ) ).
% image_empty
thf(fact_433_image__empty,axiom,
! [F: set_a > set_a] :
( ( image_set_a_set_a @ F @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_434_image__empty,axiom,
! [F: a > pre_pr7278220950009878019t_unit] :
( ( image_5713294457175270716t_unit @ F @ bot_bot_set_a )
= bot_bo1839476491465656141t_unit ) ).
% image_empty
thf(fact_435_empty__is__image,axiom,
! [F: a > a,A: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_436_empty__is__image,axiom,
! [F: b > a,A: set_b] :
( ( bot_bot_set_a
= ( image_b_a @ F @ A ) )
= ( A = bot_bot_set_b ) ) ).
% empty_is_image
thf(fact_437_empty__is__image,axiom,
! [F: a > b,A: set_a] :
( ( bot_bot_set_b
= ( image_a_b @ F @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_438_empty__is__image,axiom,
! [F: b > b,A: set_b] :
( ( bot_bot_set_b
= ( image_b_b @ F @ A ) )
= ( A = bot_bot_set_b ) ) ).
% empty_is_image
thf(fact_439_empty__is__image,axiom,
! [F: set_a > a,A: set_set_a] :
( ( bot_bot_set_a
= ( image_set_a_a @ F @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% empty_is_image
thf(fact_440_empty__is__image,axiom,
! [F: set_a > b,A: set_set_a] :
( ( bot_bot_set_b
= ( image_set_a_b @ F @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% empty_is_image
thf(fact_441_empty__is__image,axiom,
! [F: a > set_a,A: set_a] :
( ( bot_bot_set_set_a
= ( image_a_set_a @ F @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_442_empty__is__image,axiom,
! [F: b > set_a,A: set_b] :
( ( bot_bot_set_set_a
= ( image_b_set_a @ F @ A ) )
= ( A = bot_bot_set_b ) ) ).
% empty_is_image
thf(fact_443_empty__is__image,axiom,
! [F: set_a > set_a,A: set_set_a] :
( ( bot_bot_set_set_a
= ( image_set_a_set_a @ F @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% empty_is_image
thf(fact_444_empty__is__image,axiom,
! [F: pre_pr7278220950009878019t_unit > a,A: set_pr5411798346947241657t_unit] :
( ( bot_bot_set_a
= ( image_4969699134812999796unit_a @ F @ A ) )
= ( A = bot_bo1839476491465656141t_unit ) ) ).
% empty_is_image
thf(fact_445_image__is__empty,axiom,
! [F: a > a,A: set_a] :
( ( ( image_a_a @ F @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_446_image__is__empty,axiom,
! [F: b > a,A: set_b] :
( ( ( image_b_a @ F @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_b ) ) ).
% image_is_empty
thf(fact_447_image__is__empty,axiom,
! [F: a > b,A: set_a] :
( ( ( image_a_b @ F @ A )
= bot_bot_set_b )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_448_image__is__empty,axiom,
! [F: b > b,A: set_b] :
( ( ( image_b_b @ F @ A )
= bot_bot_set_b )
= ( A = bot_bot_set_b ) ) ).
% image_is_empty
thf(fact_449_image__is__empty,axiom,
! [F: set_a > a,A: set_set_a] :
( ( ( image_set_a_a @ F @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% image_is_empty
thf(fact_450_image__is__empty,axiom,
! [F: set_a > b,A: set_set_a] :
( ( ( image_set_a_b @ F @ A )
= bot_bot_set_b )
= ( A = bot_bot_set_set_a ) ) ).
% image_is_empty
thf(fact_451_image__is__empty,axiom,
! [F: a > set_a,A: set_a] :
( ( ( image_a_set_a @ F @ A )
= bot_bot_set_set_a )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_452_image__is__empty,axiom,
! [F: b > set_a,A: set_b] :
( ( ( image_b_set_a @ F @ A )
= bot_bot_set_set_a )
= ( A = bot_bot_set_b ) ) ).
% image_is_empty
thf(fact_453_image__is__empty,axiom,
! [F: set_a > set_a,A: set_set_a] :
( ( ( image_set_a_set_a @ F @ A )
= bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% image_is_empty
thf(fact_454_image__is__empty,axiom,
! [F: pre_pr7278220950009878019t_unit > a,A: set_pr5411798346947241657t_unit] :
( ( ( image_4969699134812999796unit_a @ F @ A )
= bot_bot_set_a )
= ( A = bot_bo1839476491465656141t_unit ) ) ).
% image_is_empty
thf(fact_455_imageI,axiom,
! [X4: a,A: set_a,F: a > a] :
( ( member_a @ X4 @ A )
=> ( member_a @ ( F @ X4 ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_456_imageI,axiom,
! [X4: a,A: set_a,F: a > b] :
( ( member_a @ X4 @ A )
=> ( member_b @ ( F @ X4 ) @ ( image_a_b @ F @ A ) ) ) ).
% imageI
thf(fact_457_imageI,axiom,
! [X4: b,A: set_b,F: b > a] :
( ( member_b @ X4 @ A )
=> ( member_a @ ( F @ X4 ) @ ( image_b_a @ F @ A ) ) ) ).
% imageI
thf(fact_458_imageI,axiom,
! [X4: b,A: set_b,F: b > b] :
( ( member_b @ X4 @ A )
=> ( member_b @ ( F @ X4 ) @ ( image_b_b @ F @ A ) ) ) ).
% imageI
thf(fact_459_imageI,axiom,
! [X4: a,A: set_a,F: a > set_a] :
( ( member_a @ X4 @ A )
=> ( member_set_a @ ( F @ X4 ) @ ( image_a_set_a @ F @ A ) ) ) ).
% imageI
thf(fact_460_imageI,axiom,
! [X4: set_a,A: set_set_a,F: set_a > a] :
( ( member_set_a @ X4 @ A )
=> ( member_a @ ( F @ X4 ) @ ( image_set_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_461_imageI,axiom,
! [X4: set_a,A: set_set_a,F: set_a > b] :
( ( member_set_a @ X4 @ A )
=> ( member_b @ ( F @ X4 ) @ ( image_set_a_b @ F @ A ) ) ) ).
% imageI
thf(fact_462_imageI,axiom,
! [X4: b,A: set_b,F: b > set_a] :
( ( member_b @ X4 @ A )
=> ( member_set_a @ ( F @ X4 ) @ ( image_b_set_a @ F @ A ) ) ) ).
% imageI
thf(fact_463_imageI,axiom,
! [X4: set_a,A: set_set_a,F: set_a > set_a] :
( ( member_set_a @ X4 @ A )
=> ( member_set_a @ ( F @ X4 ) @ ( image_set_a_set_a @ F @ A ) ) ) ).
% imageI
thf(fact_464_imageI,axiom,
! [X4: a,A: set_a,F: a > pre_pr7278220950009878019t_unit] :
( ( member_a @ X4 @ A )
=> ( member6939884229742472986t_unit @ ( F @ X4 ) @ ( image_5713294457175270716t_unit @ F @ A ) ) ) ).
% imageI
thf(fact_465_image__iff,axiom,
! [Z2: a,F: b > a,A: set_b] :
( ( member_a @ Z2 @ ( image_b_a @ F @ A ) )
= ( ? [X: b] :
( ( member_b @ X @ A )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_466_image__iff,axiom,
! [Z2: set_a,F: a > set_a,A: set_a] :
( ( member_set_a @ Z2 @ ( image_a_set_a @ F @ A ) )
= ( ? [X: a] :
( ( member_a @ X @ A )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_467_image__iff,axiom,
! [Z2: set_a,F: pre_pr7278220950009878019t_unit > set_a,A: set_pr5411798346947241657t_unit] :
( ( member_set_a @ Z2 @ ( image_7466199892558553556_set_a @ F @ A ) )
= ( ? [X: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X @ A )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_468_image__iff,axiom,
! [Z2: pre_pr7278220950009878019t_unit,F: set_a > pre_pr7278220950009878019t_unit,A: set_set_a] :
( ( member6939884229742472986t_unit @ Z2 @ ( image_6801035452528096924t_unit @ F @ A ) )
= ( ? [X: set_a] :
( ( member_set_a @ X @ A )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_469_bex__imageD,axiom,
! [F: a > set_a,A: set_a,P: set_a > $o] :
( ? [X2: set_a] :
( ( member_set_a @ X2 @ ( image_a_set_a @ F @ A ) )
& ( P @ X2 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_470_bex__imageD,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A: set_set_a,P: pre_pr7278220950009878019t_unit > $o] :
( ? [X2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X2 @ ( image_6801035452528096924t_unit @ F @ A ) )
& ( P @ X2 ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_471_bex__imageD,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A: set_pr5411798346947241657t_unit,P: set_a > $o] :
( ? [X2: set_a] :
( ( member_set_a @ X2 @ ( image_7466199892558553556_set_a @ F @ A ) )
& ( P @ X2 ) )
=> ? [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_472_bex__imageD,axiom,
! [F: b > a,A: set_b,P: a > $o] :
( ? [X2: a] :
( ( member_a @ X2 @ ( image_b_a @ F @ A ) )
& ( P @ X2 ) )
=> ? [X3: b] :
( ( member_b @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_473_image__cong,axiom,
! [M: set_a,N: set_a,F: a > set_a,G3: a > set_a] :
( ( M = N )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N )
=> ( ( F @ X3 )
= ( G3 @ X3 ) ) )
=> ( ( image_a_set_a @ F @ M )
= ( image_a_set_a @ G3 @ N ) ) ) ) ).
% image_cong
thf(fact_474_image__cong,axiom,
! [M: set_set_a,N: set_set_a,F: set_a > pre_pr7278220950009878019t_unit,G3: set_a > pre_pr7278220950009878019t_unit] :
( ( M = N )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ N )
=> ( ( F @ X3 )
= ( G3 @ X3 ) ) )
=> ( ( image_6801035452528096924t_unit @ F @ M )
= ( image_6801035452528096924t_unit @ G3 @ N ) ) ) ) ).
% image_cong
thf(fact_475_image__cong,axiom,
! [M: set_pr5411798346947241657t_unit,N: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > set_a,G3: pre_pr7278220950009878019t_unit > set_a] :
( ( M = N )
=> ( ! [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ N )
=> ( ( F @ X3 )
= ( G3 @ X3 ) ) )
=> ( ( image_7466199892558553556_set_a @ F @ M )
= ( image_7466199892558553556_set_a @ G3 @ N ) ) ) ) ).
% image_cong
thf(fact_476_image__cong,axiom,
! [M: set_b,N: set_b,F: b > a,G3: b > a] :
( ( M = N )
=> ( ! [X3: b] :
( ( member_b @ X3 @ N )
=> ( ( F @ X3 )
= ( G3 @ X3 ) ) )
=> ( ( image_b_a @ F @ M )
= ( image_b_a @ G3 @ N ) ) ) ) ).
% image_cong
thf(fact_477_ball__imageD,axiom,
! [F: a > set_a,A: set_a,P: set_a > $o] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( image_a_set_a @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( P @ ( F @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_478_ball__imageD,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A: set_set_a,P: pre_pr7278220950009878019t_unit > $o] :
( ! [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ ( image_6801035452528096924t_unit @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( P @ ( F @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_479_ball__imageD,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A: set_pr5411798346947241657t_unit,P: set_a > $o] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( image_7466199892558553556_set_a @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X2 @ A )
=> ( P @ ( F @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_480_ball__imageD,axiom,
! [F: b > a,A: set_b,P: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ ( image_b_a @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X2: b] :
( ( member_b @ X2 @ A )
=> ( P @ ( F @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_481_rev__image__eqI,axiom,
! [X4: a,A: set_a,B3: a,F: a > a] :
( ( member_a @ X4 @ A )
=> ( ( B3
= ( F @ X4 ) )
=> ( member_a @ B3 @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_482_rev__image__eqI,axiom,
! [X4: a,A: set_a,B3: b,F: a > b] :
( ( member_a @ X4 @ A )
=> ( ( B3
= ( F @ X4 ) )
=> ( member_b @ B3 @ ( image_a_b @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_483_rev__image__eqI,axiom,
! [X4: b,A: set_b,B3: a,F: b > a] :
( ( member_b @ X4 @ A )
=> ( ( B3
= ( F @ X4 ) )
=> ( member_a @ B3 @ ( image_b_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_484_rev__image__eqI,axiom,
! [X4: b,A: set_b,B3: b,F: b > b] :
( ( member_b @ X4 @ A )
=> ( ( B3
= ( F @ X4 ) )
=> ( member_b @ B3 @ ( image_b_b @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_485_rev__image__eqI,axiom,
! [X4: a,A: set_a,B3: set_a,F: a > set_a] :
( ( member_a @ X4 @ A )
=> ( ( B3
= ( F @ X4 ) )
=> ( member_set_a @ B3 @ ( image_a_set_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_486_rev__image__eqI,axiom,
! [X4: set_a,A: set_set_a,B3: a,F: set_a > a] :
( ( member_set_a @ X4 @ A )
=> ( ( B3
= ( F @ X4 ) )
=> ( member_a @ B3 @ ( image_set_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_487_rev__image__eqI,axiom,
! [X4: set_a,A: set_set_a,B3: b,F: set_a > b] :
( ( member_set_a @ X4 @ A )
=> ( ( B3
= ( F @ X4 ) )
=> ( member_b @ B3 @ ( image_set_a_b @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_488_rev__image__eqI,axiom,
! [X4: b,A: set_b,B3: set_a,F: b > set_a] :
( ( member_b @ X4 @ A )
=> ( ( B3
= ( F @ X4 ) )
=> ( member_set_a @ B3 @ ( image_b_set_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_489_rev__image__eqI,axiom,
! [X4: set_a,A: set_set_a,B3: set_a,F: set_a > set_a] :
( ( member_set_a @ X4 @ A )
=> ( ( B3
= ( F @ X4 ) )
=> ( member_set_a @ B3 @ ( image_set_a_set_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_490_rev__image__eqI,axiom,
! [X4: a,A: set_a,B3: pre_pr7278220950009878019t_unit,F: a > pre_pr7278220950009878019t_unit] :
( ( member_a @ X4 @ A )
=> ( ( B3
= ( F @ X4 ) )
=> ( member6939884229742472986t_unit @ B3 @ ( image_5713294457175270716t_unit @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_491_image__mono,axiom,
! [A: set_set_a,B: set_set_a,F: set_a > pre_pr7278220950009878019t_unit] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_le8200006823705900825t_unit @ ( image_6801035452528096924t_unit @ F @ A ) @ ( image_6801035452528096924t_unit @ F @ B ) ) ) ).
% image_mono
thf(fact_492_image__mono,axiom,
! [A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > set_a] :
( ( ord_le8200006823705900825t_unit @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_7466199892558553556_set_a @ F @ A ) @ ( image_7466199892558553556_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_493_image__mono,axiom,
! [A: set_a,B: set_a,F: a > set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_494_image__mono,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_495_image__mono,axiom,
! [A: set_a,B: set_a,F: a > b] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_b @ ( image_a_b @ F @ A ) @ ( image_a_b @ F @ B ) ) ) ).
% image_mono
thf(fact_496_image__mono,axiom,
! [A: set_b,B: set_b,F: b > a] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ord_less_eq_set_a @ ( image_b_a @ F @ A ) @ ( image_b_a @ F @ B ) ) ) ).
% image_mono
thf(fact_497_image__mono,axiom,
! [A: set_b,B: set_b,F: b > b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ord_less_eq_set_b @ ( image_b_b @ F @ A ) @ ( image_b_b @ F @ B ) ) ) ).
% image_mono
thf(fact_498_image__subsetI,axiom,
! [A: set_a,F: a > a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_499_image__subsetI,axiom,
! [A: set_b,F: b > a,B: set_a] :
( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_b_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_500_image__subsetI,axiom,
! [A: set_a,F: a > b,B: set_b] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_b @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_a_b @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_501_image__subsetI,axiom,
! [A: set_b,F: b > b,B: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ( member_b @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_b_b @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_502_image__subsetI,axiom,
! [A: set_a,F: a > set_a,B: set_set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_set_a @ ( F @ X3 ) @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_503_image__subsetI,axiom,
! [A: set_b,F: b > set_a,B: set_set_a] :
( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ( member_set_a @ ( F @ X3 ) @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( image_b_set_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_504_image__subsetI,axiom,
! [A: set_set_a,F: set_a > a,B: set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_505_image__subsetI,axiom,
! [A: set_set_a,F: set_a > b,B: set_b] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( member_b @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_set_a_b @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_506_image__subsetI,axiom,
! [A: set_set_a,F: set_a > set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( member_set_a @ ( F @ X3 ) @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_507_image__subsetI,axiom,
! [A: set_a,F: a > pre_pr7278220950009878019t_unit,B: set_pr5411798346947241657t_unit] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member6939884229742472986t_unit @ ( F @ X3 ) @ B ) )
=> ( ord_le8200006823705900825t_unit @ ( image_5713294457175270716t_unit @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_508_subset__imageE,axiom,
! [B: set_pr5411798346947241657t_unit,F: set_a > pre_pr7278220950009878019t_unit,A: set_set_a] :
( ( ord_le8200006823705900825t_unit @ B @ ( image_6801035452528096924t_unit @ F @ A ) )
=> ~ ! [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A )
=> ( B
!= ( image_6801035452528096924t_unit @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_509_subset__imageE,axiom,
! [B: set_set_a,F: pre_pr7278220950009878019t_unit > set_a,A: set_pr5411798346947241657t_unit] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_7466199892558553556_set_a @ F @ A ) )
=> ~ ! [C3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ C3 @ A )
=> ( B
!= ( image_7466199892558553556_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_510_subset__imageE,axiom,
! [B: set_set_a,F: a > set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_511_subset__imageE,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_512_subset__imageE,axiom,
! [B: set_a,F: b > a,A: set_b] :
( ( ord_less_eq_set_a @ B @ ( image_b_a @ F @ A ) )
=> ~ ! [C3: set_b] :
( ( ord_less_eq_set_b @ C3 @ A )
=> ( B
!= ( image_b_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_513_subset__imageE,axiom,
! [B: set_b,F: a > b,A: set_a] :
( ( ord_less_eq_set_b @ B @ ( image_a_b @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_b @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_514_subset__imageE,axiom,
! [B: set_b,F: b > b,A: set_b] :
( ( ord_less_eq_set_b @ B @ ( image_b_b @ F @ A ) )
=> ~ ! [C3: set_b] :
( ( ord_less_eq_set_b @ C3 @ A )
=> ( B
!= ( image_b_b @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_515_image__subset__iff,axiom,
! [F: a > set_a,A: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ B )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_set_a @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_516_image__subset__iff,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A: set_pr5411798346947241657t_unit,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_7466199892558553556_set_a @ F @ A ) @ B )
= ( ! [X: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X @ A )
=> ( member_set_a @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_517_image__subset__iff,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A: set_set_a,B: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ ( image_6801035452528096924t_unit @ F @ A ) @ B )
= ( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( member6939884229742472986t_unit @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_518_image__subset__iff,axiom,
! [F: b > a,A: set_b,B: set_a] :
( ( ord_less_eq_set_a @ ( image_b_a @ F @ A ) @ B )
= ( ! [X: b] :
( ( member_b @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_519_subset__image__iff,axiom,
! [B: set_pr5411798346947241657t_unit,F: set_a > pre_pr7278220950009878019t_unit,A: set_set_a] :
( ( ord_le8200006823705900825t_unit @ B @ ( image_6801035452528096924t_unit @ F @ A ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A )
& ( B
= ( image_6801035452528096924t_unit @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_520_subset__image__iff,axiom,
! [B: set_set_a,F: pre_pr7278220950009878019t_unit > set_a,A: set_pr5411798346947241657t_unit] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_7466199892558553556_set_a @ F @ A ) )
= ( ? [AA: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ AA @ A )
& ( B
= ( image_7466199892558553556_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_521_subset__image__iff,axiom,
! [B: set_set_a,F: a > set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_522_subset__image__iff,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_523_subset__image__iff,axiom,
! [B: set_a,F: b > a,A: set_b] :
( ( ord_less_eq_set_a @ B @ ( image_b_a @ F @ A ) )
= ( ? [AA: set_b] :
( ( ord_less_eq_set_b @ AA @ A )
& ( B
= ( image_b_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_524_subset__image__iff,axiom,
! [B: set_b,F: a > b,A: set_a] :
( ( ord_less_eq_set_b @ B @ ( image_a_b @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_525_subset__image__iff,axiom,
! [B: set_b,F: b > b,A: set_b] :
( ( ord_less_eq_set_b @ B @ ( image_b_b @ F @ A ) )
= ( ? [AA: set_b] :
( ( ord_less_eq_set_b @ AA @ A )
& ( B
= ( image_b_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_526_image__Int__subset,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A: set_set_a,B: set_set_a] : ( ord_le8200006823705900825t_unit @ ( image_6801035452528096924t_unit @ F @ ( inf_inf_set_set_a @ A @ B ) ) @ ( inf_in1092213268631476299t_unit @ ( image_6801035452528096924t_unit @ F @ A ) @ ( image_6801035452528096924t_unit @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_527_image__Int__subset,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] : ( ord_le3724670747650509150_set_a @ ( image_7466199892558553556_set_a @ F @ ( inf_in1092213268631476299t_unit @ A @ B ) ) @ ( inf_inf_set_set_a @ ( image_7466199892558553556_set_a @ F @ A ) @ ( image_7466199892558553556_set_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_528_image__Int__subset,axiom,
! [F: a > set_a,A: set_a,B: set_a] : ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_inf_set_set_a @ ( image_a_set_a @ F @ A ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_529_image__Int__subset,axiom,
! [F: b > a,A: set_b,B: set_b] : ( ord_less_eq_set_a @ ( image_b_a @ F @ ( inf_inf_set_b @ A @ B ) ) @ ( inf_inf_set_a @ ( image_b_a @ F @ A ) @ ( image_b_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_530_image__Int__subset,axiom,
! [F: a > a,A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_inf_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_531_image__Int__subset,axiom,
! [F: a > b,A: set_a,B: set_a] : ( ord_less_eq_set_b @ ( image_a_b @ F @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_inf_set_b @ ( image_a_b @ F @ A ) @ ( image_a_b @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_532_wf__digraph_Ounvisited__verts_Ocong,axiom,
graph_2016941059203891550ts_a_b = graph_2016941059203891550ts_a_b ).
% wf_digraph.unvisited_verts.cong
thf(fact_533_wf__digraph_Onearest__vert_Ocong,axiom,
graph_3614428260325061028rt_a_b = graph_3614428260325061028rt_a_b ).
% wf_digraph.nearest_vert.cong
thf(fact_534_wf__digraph_Ok__neighborhood_Ocong,axiom,
graph_3921080825633621230od_a_b = graph_3921080825633621230od_a_b ).
% wf_digraph.k_neighborhood.cong
thf(fact_535_G_Oeuler__imp__connected,axiom,
! [U: a,P2: list_b,V: a] :
( ( pre_euler_trail_a_b @ g @ U @ P2 @ V )
=> ( digrap8783888973171253482ed_a_b @ g ) ) ).
% G.euler_imp_connected
thf(fact_536_euler__imp__connected,axiom,
! [U: a,P2: list_b,V: a] :
( ( pre_euler_trail_a_b @ t @ U @ P2 @ V )
=> ( digrap8783888973171253482ed_a_b @ t ) ) ).
% euler_imp_connected
thf(fact_537_closed__euler2_I1_J,axiom,
! [U: a,P2: list_b] :
( ( pre_euler_trail_a_b @ t @ U @ P2 @ U )
=> ( digrap8783888973171253482ed_a_b @ t ) ) ).
% closed_euler2(1)
thf(fact_538_strict__sub,axiom,
ord_less_set_a @ ( pre_ve642382030648772252t_unit @ t ) @ ( pre_ve642382030648772252t_unit @ g ) ).
% strict_sub
thf(fact_539_unique__awalk,axiom,
! [V: a] :
( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) )
=> ? [X3: list_b] :
( ( arc_pre_awalk_a_b @ t @ source @ X3 @ V )
& ! [Y5: list_b] :
( ( arc_pre_awalk_a_b @ t @ source @ Y5 @ V )
=> ( Y5 = X3 ) ) ) ) ).
% unique_awalk
thf(fact_540_G_Osubgraph__induce__subgraphI,axiom,
! [V3: set_a] :
( ( ord_less_eq_set_a @ V3 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( digraph_subgraph_a_b @ ( digrap7873285959652527175ph_a_b @ g @ V3 ) @ g ) ) ).
% G.subgraph_induce_subgraphI
thf(fact_541_subgraph__induce__subgraphI,axiom,
! [V3: set_a] :
( ( ord_less_eq_set_a @ V3 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( digraph_subgraph_a_b @ ( digrap7873285959652527175ph_a_b @ t @ V3 ) @ t ) ) ).
% subgraph_induce_subgraphI
thf(fact_542_awalk__ends__eqD,axiom,
! [U: a,P2: list_b,V: a,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ U )
=> ( ( arc_pre_awalk_a_b @ t @ V @ P2 @ W )
=> ( V = W ) ) ) ).
% awalk_ends_eqD
thf(fact_543_unique__awalk__All,axiom,
! [U: a,V: a] :
( ? [P3: list_b] : ( arc_pre_awalk_a_b @ t @ U @ P3 @ V )
=> ? [X3: list_b] :
( ( arc_pre_awalk_a_b @ t @ U @ X3 @ V )
& ! [Y5: list_b] :
( ( arc_pre_awalk_a_b @ t @ U @ Y5 @ V )
=> ( Y5 = X3 ) ) ) ) ).
% unique_awalk_All
thf(fact_544_G_Oawalk__ends__eqD,axiom,
! [U: a,P2: list_b,V: a,W: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ U )
=> ( ( arc_pre_awalk_a_b @ g @ V @ P2 @ W )
=> ( V = W ) ) ) ).
% G.awalk_ends_eqD
thf(fact_545_subgraph__refl,axiom,
digraph_subgraph_a_b @ t @ t ).
% subgraph_refl
thf(fact_546_G_Osubgraph__refl,axiom,
digraph_subgraph_a_b @ g @ g ).
% G.subgraph_refl
thf(fact_547_awalk__hd__in__verts,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% awalk_hd_in_verts
thf(fact_548_awalk__last__in__verts,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% awalk_last_in_verts
thf(fact_549_G_Oawalk__last__in__verts,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% G.awalk_last_in_verts
thf(fact_550_G_Oawalk__hd__in__verts,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% G.awalk_hd_in_verts
thf(fact_551_reachable__awalk,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ t @ U @ V )
= ( ? [P4: list_b] : ( arc_pre_awalk_a_b @ t @ U @ P4 @ V ) ) ) ).
% reachable_awalk
thf(fact_552_reachable__awalkI,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( reachable_a_b @ t @ U @ V ) ) ).
% reachable_awalkI
thf(fact_553_G_Oreachable__awalkI,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( reachable_a_b @ g @ U @ V ) ) ).
% G.reachable_awalkI
thf(fact_554_G_Oreachable__awalk,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ g @ U @ V )
= ( ? [P4: list_b] : ( arc_pre_awalk_a_b @ g @ U @ P4 @ V ) ) ) ).
% G.reachable_awalk
thf(fact_555_subgraph__awalk__imp__awalk,axiom,
! [H: pre_pr7278220950009878019t_unit,U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ H @ U @ P2 @ V )
=> ( ( digraph_subgraph_a_b @ H @ t )
=> ( arc_pre_awalk_a_b @ t @ U @ P2 @ V ) ) ) ).
% subgraph_awalk_imp_awalk
thf(fact_556_G_Osubgraph__awalk__imp__awalk,axiom,
! [H: pre_pr7278220950009878019t_unit,U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ H @ U @ P2 @ V )
=> ( ( digraph_subgraph_a_b @ H @ g )
=> ( arc_pre_awalk_a_b @ g @ U @ P2 @ V ) ) ) ).
% G.subgraph_awalk_imp_awalk
thf(fact_557_reachable__mono,axiom,
! [H: pre_pr7278220950009878019t_unit,U: a,V: a] :
( ( reachable_a_b @ H @ U @ V )
=> ( ( digraph_subgraph_a_b @ H @ t )
=> ( reachable_a_b @ t @ U @ V ) ) ) ).
% reachable_mono
thf(fact_558_G_Oreachable__mono,axiom,
! [H: pre_pr7278220950009878019t_unit,U: a,V: a] :
( ( reachable_a_b @ H @ U @ V )
=> ( ( digraph_subgraph_a_b @ H @ g )
=> ( reachable_a_b @ g @ U @ V ) ) ) ).
% G.reachable_mono
thf(fact_559_awalk__sub__imp__awalk,axiom,
! [A3: a,P2: list_b,B3: a] :
( ( arc_pre_awalk_a_b @ t @ A3 @ P2 @ B3 )
=> ( arc_pre_awalk_a_b @ g @ A3 @ P2 @ B3 ) ) ).
% awalk_sub_imp_awalk
thf(fact_560_fin__digraph__subgraph,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H @ t )
=> ( fin_digraph_a_b @ H ) ) ).
% fin_digraph_subgraph
thf(fact_561_sub__G,axiom,
digraph_subgraph_a_b @ t @ g ).
% sub_G
thf(fact_562_psubsetI,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_563_psubsetI,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_b @ A @ B ) ) ) ).
% psubsetI
thf(fact_564_order__less__imp__not__less,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_set_a @ X4 @ Y3 )
=> ~ ( ord_less_set_a @ Y3 @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_565_order__less__imp__not__eq2,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_set_a @ X4 @ Y3 )
=> ( Y3 != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_566_order__less__imp__not__eq,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_set_a @ X4 @ Y3 )
=> ( X4 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_567_order__less__imp__triv,axiom,
! [X4: set_a,Y3: set_a,P: $o] :
( ( ord_less_set_a @ X4 @ Y3 )
=> ( ( ord_less_set_a @ Y3 @ X4 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_568_order__less__not__sym,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_set_a @ X4 @ Y3 )
=> ~ ( ord_less_set_a @ Y3 @ X4 ) ) ).
% order_less_not_sym
thf(fact_569_order__less__subst2,axiom,
! [A3: set_a,B3: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ( ord_less_set_a @ ( F @ B3 ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_set_a @ X3 @ Y2 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_570_order__less__subst1,axiom,
! [A3: set_a,F: set_a > set_a,B3: set_a,C: set_a] :
( ( ord_less_set_a @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_set_a @ B3 @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_set_a @ X3 @ Y2 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_571_order__less__irrefl,axiom,
! [X4: set_a] :
~ ( ord_less_set_a @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_572_ord__less__eq__subst,axiom,
! [A3: set_a,B3: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_set_a @ X3 @ Y2 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_573_ord__eq__less__subst,axiom,
! [A3: set_a,F: set_a > set_a,B3: set_a,C: set_a] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_set_a @ B3 @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_set_a @ X3 @ Y2 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_574_order__less__trans,axiom,
! [X4: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_set_a @ X4 @ Y3 )
=> ( ( ord_less_set_a @ Y3 @ Z2 )
=> ( ord_less_set_a @ X4 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_575_order__less__asym_H,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ~ ( ord_less_set_a @ B3 @ A3 ) ) ).
% order_less_asym'
thf(fact_576_order__less__asym,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_set_a @ X4 @ Y3 )
=> ~ ( ord_less_set_a @ Y3 @ X4 ) ) ).
% order_less_asym
thf(fact_577_psubset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% psubset_trans
thf(fact_578_psubsetD,axiom,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_579_psubsetD,axiom,
! [A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit,C: pre_pr7278220950009878019t_unit] :
( ( ord_le2693654750756130573t_unit @ A @ B )
=> ( ( member6939884229742472986t_unit @ C @ A )
=> ( member6939884229742472986t_unit @ C @ B ) ) ) ).
% psubsetD
thf(fact_580_psubsetD,axiom,
! [A: set_b,B: set_b,C: b] :
( ( ord_less_set_b @ A @ B )
=> ( ( member_b @ C @ A )
=> ( member_b @ C @ B ) ) ) ).
% psubsetD
thf(fact_581_psubsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_582_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: set_a,A3: set_a] :
( ( ord_less_set_a @ B3 @ A3 )
=> ( A3 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_583_order_Ostrict__implies__not__eq,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_584_dual__order_Ostrict__trans,axiom,
! [B3: set_a,A3: set_a,C: set_a] :
( ( ord_less_set_a @ B3 @ A3 )
=> ( ( ord_less_set_a @ C @ B3 )
=> ( ord_less_set_a @ C @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_585_order_Ostrict__trans,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ( ord_less_set_a @ B3 @ C )
=> ( ord_less_set_a @ A3 @ C ) ) ) ).
% order.strict_trans
thf(fact_586_dual__order_Oirrefl,axiom,
! [A3: set_a] :
~ ( ord_less_set_a @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_587_dual__order_Oasym,axiom,
! [B3: set_a,A3: set_a] :
( ( ord_less_set_a @ B3 @ A3 )
=> ~ ( ord_less_set_a @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_588_ord__less__eq__trans,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_set_a @ A3 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_589_ord__eq__less__trans,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( A3 = B3 )
=> ( ( ord_less_set_a @ B3 @ C )
=> ( ord_less_set_a @ A3 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_590_order_Oasym,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ~ ( ord_less_set_a @ B3 @ A3 ) ) ).
% order.asym
thf(fact_591_less__imp__neq,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_set_a @ X4 @ Y3 )
=> ( X4 != Y3 ) ) ).
% less_imp_neq
thf(fact_592_subgraph__trans,axiom,
! [G: pre_pr7278220950009878019t_unit,H: pre_pr7278220950009878019t_unit,I: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ G @ H )
=> ( ( digraph_subgraph_a_b @ H @ I )
=> ( digraph_subgraph_a_b @ G @ I ) ) ) ).
% subgraph_trans
thf(fact_593_pre__digraph_Oreachable__mono,axiom,
! [H: pre_pr7278220950009878019t_unit,U: a,V: a,G: pre_pr7278220950009878019t_unit] :
( ( reachable_a_b @ H @ U @ V )
=> ( ( digraph_subgraph_a_b @ H @ G )
=> ( reachable_a_b @ G @ U @ V ) ) ) ).
% pre_digraph.reachable_mono
thf(fact_594_order__le__imp__less__or__eq,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ord_less_set_a @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_595_order__le__imp__less__or__eq,axiom,
! [X4: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y3 )
=> ( ( ord_less_set_b @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_596_order__less__le__subst2,axiom,
! [A3: set_a,B3: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_set_a @ X3 @ Y2 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_597_order__less__le__subst2,axiom,
! [A3: set_a,B3: set_a,F: set_a > set_b,C: set_b] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_b @ ( F @ B3 ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_set_a @ X3 @ Y2 )
=> ( ord_less_set_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_b @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_598_order__less__le__subst1,axiom,
! [A3: set_a,F: set_a > set_a,B3: set_a,C: set_a] :
( ( ord_less_set_a @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_599_order__less__le__subst1,axiom,
! [A3: set_b,F: set_a > set_b,B3: set_a,C: set_a] :
( ( ord_less_set_b @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_b @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_600_order__less__le__subst1,axiom,
! [A3: set_a,F: set_b > set_a,B3: set_b,C: set_b] :
( ( ord_less_set_a @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_b @ B3 @ C )
=> ( ! [X3: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_601_order__less__le__subst1,axiom,
! [A3: set_b,F: set_b > set_b,B3: set_b,C: set_b] :
( ( ord_less_set_b @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_b @ B3 @ C )
=> ( ! [X3: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y2 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_b @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_602_order__le__less__subst2,axiom,
! [A3: set_a,B3: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_set_a @ ( F @ B3 ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_603_order__le__less__subst2,axiom,
! [A3: set_a,B3: set_a,F: set_a > set_b,C: set_b] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_set_b @ ( F @ B3 ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_b @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_604_order__le__less__subst2,axiom,
! [A3: set_b,B3: set_b,F: set_b > set_a,C: set_a] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( ord_less_set_a @ ( F @ B3 ) @ C )
=> ( ! [X3: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_605_order__le__less__subst2,axiom,
! [A3: set_b,B3: set_b,F: set_b > set_b,C: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( ord_less_set_b @ ( F @ B3 ) @ C )
=> ( ! [X3: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y2 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_b @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_606_order__le__less__subst1,axiom,
! [A3: set_a,F: set_a > set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_set_a @ B3 @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_set_a @ X3 @ Y2 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_607_order__le__less__subst1,axiom,
! [A3: set_b,F: set_a > set_b,B3: set_a,C: set_a] :
( ( ord_less_eq_set_b @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_set_a @ B3 @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_set_a @ X3 @ Y2 )
=> ( ord_less_set_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_b @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_608_order__less__le__trans,axiom,
! [X4: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_set_a @ X4 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z2 )
=> ( ord_less_set_a @ X4 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_609_order__less__le__trans,axiom,
! [X4: set_b,Y3: set_b,Z2: set_b] :
( ( ord_less_set_b @ X4 @ Y3 )
=> ( ( ord_less_eq_set_b @ Y3 @ Z2 )
=> ( ord_less_set_b @ X4 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_610_order__le__less__trans,axiom,
! [X4: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ord_less_set_a @ Y3 @ Z2 )
=> ( ord_less_set_a @ X4 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_611_order__le__less__trans,axiom,
! [X4: set_b,Y3: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y3 )
=> ( ( ord_less_set_b @ Y3 @ Z2 )
=> ( ord_less_set_b @ X4 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_612_order__neq__le__trans,axiom,
! [A3: set_a,B3: set_a] :
( ( A3 != B3 )
=> ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ord_less_set_a @ A3 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_613_order__neq__le__trans,axiom,
! [A3: set_b,B3: set_b] :
( ( A3 != B3 )
=> ( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ord_less_set_b @ A3 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_614_order__le__neq__trans,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_set_a @ A3 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_615_order__le__neq__trans,axiom,
! [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_set_b @ A3 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_616_order__less__imp__le,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ X4 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_617_order__less__imp__le,axiom,
! [X4: set_b,Y3: set_b] :
( ( ord_less_set_b @ X4 @ Y3 )
=> ( ord_less_eq_set_b @ X4 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_618_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_619_order__less__le,axiom,
( ord_less_set_b
= ( ^ [X: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_620_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_621_order__le__less,axiom,
( ord_less_eq_set_b
= ( ^ [X: set_b,Y: set_b] :
( ( ord_less_set_b @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_622_dual__order_Ostrict__implies__order,axiom,
! [B3: set_a,A3: set_a] :
( ( ord_less_set_a @ B3 @ A3 )
=> ( ord_less_eq_set_a @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_623_dual__order_Ostrict__implies__order,axiom,
! [B3: set_b,A3: set_b] :
( ( ord_less_set_b @ B3 @ A3 )
=> ( ord_less_eq_set_b @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_624_order_Ostrict__implies__order,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_625_order_Ostrict__implies__order,axiom,
! [A3: set_b,B3: set_b] :
( ( ord_less_set_b @ A3 @ B3 )
=> ( ord_less_eq_set_b @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_626_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ~ ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_627_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_b
= ( ^ [B4: set_b,A4: set_b] :
( ( ord_less_eq_set_b @ B4 @ A4 )
& ~ ( ord_less_eq_set_b @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_628_dual__order_Ostrict__trans2,axiom,
! [B3: set_a,A3: set_a,C: set_a] :
( ( ord_less_set_a @ B3 @ A3 )
=> ( ( ord_less_eq_set_a @ C @ B3 )
=> ( ord_less_set_a @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_629_dual__order_Ostrict__trans2,axiom,
! [B3: set_b,A3: set_b,C: set_b] :
( ( ord_less_set_b @ B3 @ A3 )
=> ( ( ord_less_eq_set_b @ C @ B3 )
=> ( ord_less_set_b @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_630_dual__order_Ostrict__trans1,axiom,
! [B3: set_a,A3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ( ord_less_set_a @ C @ B3 )
=> ( ord_less_set_a @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_631_dual__order_Ostrict__trans1,axiom,
! [B3: set_b,A3: set_b,C: set_b] :
( ( ord_less_eq_set_b @ B3 @ A3 )
=> ( ( ord_less_set_b @ C @ B3 )
=> ( ord_less_set_b @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_632_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_633_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_b
= ( ^ [B4: set_b,A4: set_b] :
( ( ord_less_eq_set_b @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_634_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_set_a @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_635_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_b
= ( ^ [B4: set_b,A4: set_b] :
( ( ord_less_set_b @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_636_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ~ ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_637_order_Ostrict__iff__not,axiom,
( ord_less_set_b
= ( ^ [A4: set_b,B4: set_b] :
( ( ord_less_eq_set_b @ A4 @ B4 )
& ~ ( ord_less_eq_set_b @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_638_order_Ostrict__trans2,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_set_a @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_639_order_Ostrict__trans2,axiom,
! [A3: set_b,B3: set_b,C: set_b] :
( ( ord_less_set_b @ A3 @ B3 )
=> ( ( ord_less_eq_set_b @ B3 @ C )
=> ( ord_less_set_b @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_640_order_Ostrict__trans1,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_set_a @ B3 @ C )
=> ( ord_less_set_a @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_641_order_Ostrict__trans1,axiom,
! [A3: set_b,B3: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( ord_less_set_b @ B3 @ C )
=> ( ord_less_set_b @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_642_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_643_order_Ostrict__iff__order,axiom,
( ord_less_set_b
= ( ^ [A4: set_b,B4: set_b] :
( ( ord_less_eq_set_b @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_644_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_set_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_645_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B4: set_b] :
( ( ord_less_set_b @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_646_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
& ~ ( ord_less_eq_set_a @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_647_less__le__not__le,axiom,
( ord_less_set_b
= ( ^ [X: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
& ~ ( ord_less_eq_set_b @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_648_antisym__conv2,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ~ ( ord_less_set_a @ X4 @ Y3 ) )
= ( X4 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_649_antisym__conv2,axiom,
! [X4: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y3 )
=> ( ( ~ ( ord_less_set_b @ X4 @ Y3 ) )
= ( X4 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_650_antisym__conv1,axiom,
! [X4: set_a,Y3: set_a] :
( ~ ( ord_less_set_a @ X4 @ Y3 )
=> ( ( ord_less_eq_set_a @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_651_antisym__conv1,axiom,
! [X4: set_b,Y3: set_b] :
( ~ ( ord_less_set_b @ X4 @ Y3 )
=> ( ( ord_less_eq_set_b @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_652_nless__le,axiom,
! [A3: set_a,B3: set_a] :
( ( ~ ( ord_less_set_a @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_set_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_653_nless__le,axiom,
! [A3: set_b,B3: set_b] :
( ( ~ ( ord_less_set_b @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_set_b @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_654_leD,axiom,
! [Y3: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X4 )
=> ~ ( ord_less_set_a @ X4 @ Y3 ) ) ).
% leD
thf(fact_655_leD,axiom,
! [Y3: set_b,X4: set_b] :
( ( ord_less_eq_set_b @ Y3 @ X4 )
=> ~ ( ord_less_set_b @ X4 @ Y3 ) ) ).
% leD
thf(fact_656_bot_Onot__eq__extremum,axiom,
! [A3: set_b] :
( ( A3 != bot_bot_set_b )
= ( ord_less_set_b @ bot_bot_set_b @ A3 ) ) ).
% bot.not_eq_extremum
thf(fact_657_bot_Onot__eq__extremum,axiom,
! [A3: set_pr5411798346947241657t_unit] :
( ( A3 != bot_bo1839476491465656141t_unit )
= ( ord_le2693654750756130573t_unit @ bot_bo1839476491465656141t_unit @ A3 ) ) ).
% bot.not_eq_extremum
thf(fact_658_bot_Onot__eq__extremum,axiom,
! [A3: set_set_a] :
( ( A3 != bot_bot_set_set_a )
= ( ord_less_set_set_a @ bot_bot_set_set_a @ A3 ) ) ).
% bot.not_eq_extremum
thf(fact_659_bot_Onot__eq__extremum,axiom,
! [A3: set_a] :
( ( A3 != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A3 ) ) ).
% bot.not_eq_extremum
thf(fact_660_bot_Oextremum__strict,axiom,
! [A3: set_b] :
~ ( ord_less_set_b @ A3 @ bot_bot_set_b ) ).
% bot.extremum_strict
thf(fact_661_bot_Oextremum__strict,axiom,
! [A3: set_pr5411798346947241657t_unit] :
~ ( ord_le2693654750756130573t_unit @ A3 @ bot_bo1839476491465656141t_unit ) ).
% bot.extremum_strict
thf(fact_662_bot_Oextremum__strict,axiom,
! [A3: set_set_a] :
~ ( ord_less_set_set_a @ A3 @ bot_bot_set_set_a ) ).
% bot.extremum_strict
thf(fact_663_bot_Oextremum__strict,axiom,
! [A3: set_a] :
~ ( ord_less_set_a @ A3 @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_664_not__psubset__empty,axiom,
! [A: set_b] :
~ ( ord_less_set_b @ A @ bot_bot_set_b ) ).
% not_psubset_empty
thf(fact_665_not__psubset__empty,axiom,
! [A: set_pr5411798346947241657t_unit] :
~ ( ord_le2693654750756130573t_unit @ A @ bot_bo1839476491465656141t_unit ) ).
% not_psubset_empty
thf(fact_666_not__psubset__empty,axiom,
! [A: set_set_a] :
~ ( ord_less_set_set_a @ A @ bot_bot_set_set_a ) ).
% not_psubset_empty
thf(fact_667_not__psubset__empty,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_668_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_669_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A2: set_b,B2: set_b] :
( ( ord_less_set_b @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_670_subset__psubset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_671_subset__psubset__trans,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_set_b @ B @ C2 )
=> ( ord_less_set_b @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_672_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_673_subset__not__subset__eq,axiom,
( ord_less_set_b
= ( ^ [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
& ~ ( ord_less_eq_set_b @ B2 @ A2 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_674_psubset__subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_675_psubset__subset__trans,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( ord_less_set_b @ A @ B )
=> ( ( ord_less_eq_set_b @ B @ C2 )
=> ( ord_less_set_b @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_676_psubset__imp__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_677_psubset__imp__subset,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_set_b @ A @ B )
=> ( ord_less_eq_set_b @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_678_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% psubset_eq
thf(fact_679_psubset__eq,axiom,
( ord_less_set_b
= ( ^ [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% psubset_eq
thf(fact_680_psubsetE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_681_psubsetE,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_set_b @ A @ B )
=> ~ ( ( ord_less_eq_set_b @ A @ B )
=> ( ord_less_eq_set_b @ B @ A ) ) ) ).
% psubsetE
thf(fact_682_fin__digraph_Ofin__digraph__subgraph,axiom,
! [G: pre_pr7278220950009878019t_unit,H: pre_pr7278220950009878019t_unit] :
( ( fin_digraph_a_b @ G )
=> ( ( digraph_subgraph_a_b @ H @ G )
=> ( fin_digraph_a_b @ H ) ) ) ).
% fin_digraph.fin_digraph_subgraph
thf(fact_683_less__infI1,axiom,
! [A3: set_a,X4: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ X4 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ X4 ) ) ).
% less_infI1
thf(fact_684_less__infI2,axiom,
! [B3: set_a,X4: set_a,A3: set_a] :
( ( ord_less_set_a @ B3 @ X4 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ X4 ) ) ).
% less_infI2
thf(fact_685_inf_Oabsorb3,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ( inf_inf_set_a @ A3 @ B3 )
= A3 ) ) ).
% inf.absorb3
thf(fact_686_inf_Oabsorb4,axiom,
! [B3: set_a,A3: set_a] :
( ( ord_less_set_a @ B3 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ B3 )
= B3 ) ) ).
% inf.absorb4
thf(fact_687_inf_Ostrict__boundedE,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( ord_less_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C ) )
=> ~ ( ( ord_less_set_a @ A3 @ B3 )
=> ~ ( ord_less_set_a @ A3 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_688_inf_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( A4
= ( inf_inf_set_a @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_689_inf_Ostrict__coboundedI1,axiom,
! [A3: set_a,C: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_690_inf_Ostrict__coboundedI2,axiom,
! [B3: set_a,C: set_a,A3: set_a] :
( ( ord_less_set_a @ B3 @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_691_induced__imp__subgraph,axiom,
! [H: pre_pr7278220950009878019t_unit,G: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ H @ G )
=> ( digraph_subgraph_a_b @ H @ G ) ) ).
% induced_imp_subgraph
thf(fact_692_subgraph__imp__subverts,axiom,
! [H: pre_pr7278220950009878019t_unit,G: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H @ G )
=> ( ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ H ) @ ( pre_ve642382030648772252t_unit @ G ) ) ) ).
% subgraph_imp_subverts
thf(fact_693_subgraph__induce__subgraphI2,axiom,
! [H: pre_pr7278220950009878019t_unit,G: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H @ G )
=> ( digraph_subgraph_a_b @ H @ ( digrap7873285959652527175ph_a_b @ G @ ( pre_ve642382030648772252t_unit @ H ) ) ) ) ).
% subgraph_induce_subgraphI2
thf(fact_694_spanning__def,axiom,
( digraph_spanning_a_b
= ( ^ [H2: pre_pr7278220950009878019t_unit,G4: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H2 @ G4 )
& ( ( pre_ve642382030648772252t_unit @ G4 )
= ( pre_ve642382030648772252t_unit @ H2 ) ) ) ) ) ).
% spanning_def
thf(fact_695_spanningE,axiom,
! [H: pre_pr7278220950009878019t_unit,G: pre_pr7278220950009878019t_unit] :
( ( digraph_spanning_a_b @ H @ G )
=> ( ( digraph_subgraph_a_b @ H @ G )
& ( ( pre_ve642382030648772252t_unit @ G )
= ( pre_ve642382030648772252t_unit @ H ) ) ) ) ).
% spanningE
thf(fact_696_G_Omk__cycles__path__awalk,axiom,
! [U: a,C: list_b,N2: nat] :
( ( arc_pre_awalk_a_b @ g @ U @ C @ U )
=> ( arc_pre_awalk_a_b @ g @ U @ ( shorte6374615165232202367path_b @ N2 @ C ) @ U ) ) ).
% G.mk_cycles_path_awalk
thf(fact_697_mk__cycles__path__awalk,axiom,
! [U: a,C: list_b,N2: nat] :
( ( arc_pre_awalk_a_b @ t @ U @ C @ U )
=> ( arc_pre_awalk_a_b @ t @ U @ ( shorte6374615165232202367path_b @ N2 @ C ) @ U ) ) ).
% mk_cycles_path_awalk
thf(fact_698_G_Oin__sccsE,axiom,
! [C: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ g ) )
=> ~ ( ( digrap5251062021860773499ph_a_b @ C @ g )
=> ( ( digrap8691851296217657702ed_a_b @ C )
=> ? [D3: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ D3 @ g )
& ( digrap8691851296217657702ed_a_b @ D3 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ D3 ) ) ) ) ) ) ).
% G.in_sccsE
thf(fact_699_in__sccsE,axiom,
! [C: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ t ) )
=> ~ ( ( digrap5251062021860773499ph_a_b @ C @ t )
=> ( ( digrap8691851296217657702ed_a_b @ C )
=> ? [D3: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ D3 @ t )
& ( digrap8691851296217657702ed_a_b @ D3 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ D3 ) ) ) ) ) ) ).
% in_sccsE
thf(fact_700_G_Osubgraph__cycle,axiom,
! [H: pre_pr7278220950009878019t_unit,P2: list_b] :
( ( digraph_subgraph_a_b @ H @ g )
=> ( ( arc_pre_cycle_a_b @ H @ P2 )
=> ( arc_pre_cycle_a_b @ g @ P2 ) ) ) ).
% G.subgraph_cycle
thf(fact_701_subgraph__cycle,axiom,
! [H: pre_pr7278220950009878019t_unit,P2: list_b] :
( ( digraph_subgraph_a_b @ H @ t )
=> ( ( arc_pre_cycle_a_b @ H @ P2 )
=> ( arc_pre_cycle_a_b @ t @ P2 ) ) ) ).
% subgraph_cycle
thf(fact_702_G_Ostrongly__connected__imp__induce__subgraph__strongly__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H @ g )
=> ( ( digrap8691851296217657702ed_a_b @ H )
=> ( digrap8691851296217657702ed_a_b @ ( digrap7873285959652527175ph_a_b @ g @ ( pre_ve642382030648772252t_unit @ H ) ) ) ) ) ).
% G.strongly_connected_imp_induce_subgraph_strongly_connected
thf(fact_703_cycle__free,axiom,
~ ? [X_1: list_b] : ( arc_pre_cycle_a_b @ t @ X_1 ) ).
% cycle_free
thf(fact_704_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_705_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_706_strongly__connected__imp__induce__subgraph__strongly__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H @ t )
=> ( ( digrap8691851296217657702ed_a_b @ H )
=> ( digrap8691851296217657702ed_a_b @ ( digrap7873285959652527175ph_a_b @ t @ ( pre_ve642382030648772252t_unit @ H ) ) ) ) ) ).
% strongly_connected_imp_induce_subgraph_strongly_connected
thf(fact_707_strongly__connectedI,axiom,
! [G: pre_pr7278220950009878019t_unit] :
( ( ( pre_ve642382030648772252t_unit @ G )
!= bot_bot_set_a )
=> ( ! [U3: a,V4: a] :
( ( member_a @ U3 @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( ( member_a @ V4 @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( reachable_a_b @ G @ U3 @ V4 ) ) )
=> ( digrap8691851296217657702ed_a_b @ G ) ) ) ).
% strongly_connectedI
thf(fact_708_in__sccsI,axiom,
! [C: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ C @ t )
=> ( ( digrap8691851296217657702ed_a_b @ C )
=> ( ~ ? [C4: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ C4 @ t )
& ( digrap8691851296217657702ed_a_b @ C4 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ C4 ) ) )
=> ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ t ) ) ) ) ) ).
% in_sccsI
thf(fact_709_G_Oin__sccsI,axiom,
! [C: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ C @ g )
=> ( ( digrap8691851296217657702ed_a_b @ C )
=> ( ~ ? [C4: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ C4 @ g )
& ( digrap8691851296217657702ed_a_b @ C4 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ C4 ) ) )
=> ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ g ) ) ) ) ) ).
% G.in_sccsI
thf(fact_710_strongly__connectedE,axiom,
! [G: pre_pr7278220950009878019t_unit] :
( ( digrap8691851296217657702ed_a_b @ G )
=> ! [U4: a,V5: a] :
( ( ( member_a @ U4 @ ( pre_ve642382030648772252t_unit @ G ) )
& ( member_a @ V5 @ ( pre_ve642382030648772252t_unit @ G ) ) )
=> ( reachable_a_b @ G @ U4 @ V5 ) ) ) ).
% strongly_connectedE
thf(fact_711_strongly__connected__def,axiom,
( digrap8691851296217657702ed_a_b
= ( ^ [G4: pre_pr7278220950009878019t_unit] :
( ( ( pre_ve642382030648772252t_unit @ G4 )
!= bot_bot_set_a )
& ! [X: a] :
( ( member_a @ X @ ( pre_ve642382030648772252t_unit @ G4 ) )
=> ! [Y: a] :
( ( member_a @ Y @ ( pre_ve642382030648772252t_unit @ G4 ) )
=> ( reachable_a_b @ G4 @ X @ Y ) ) ) ) ) ) ).
% strongly_connected_def
thf(fact_712_pre__digraph_Oin__sccsI,axiom,
! [C: pre_pr7278220950009878019t_unit,G: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ C @ G )
=> ( ( digrap8691851296217657702ed_a_b @ C )
=> ( ~ ? [C4: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ C4 @ G )
& ( digrap8691851296217657702ed_a_b @ C4 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ C4 ) ) )
=> ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ G ) ) ) ) ) ).
% pre_digraph.in_sccsI
thf(fact_713_pre__digraph_Oin__sccsE,axiom,
! [C: pre_pr7278220950009878019t_unit,G: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ G ) )
=> ~ ( ( digrap5251062021860773499ph_a_b @ C @ G )
=> ( ( digrap8691851296217657702ed_a_b @ C )
=> ? [D3: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ D3 @ G )
& ( digrap8691851296217657702ed_a_b @ D3 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ D3 ) ) ) ) ) ) ).
% pre_digraph.in_sccsE
thf(fact_714_G_Oclosed__w__imp__cycle,axiom,
! [P2: list_b] :
( ( arc_wf_closed_w_a_b @ g @ P2 )
=> ? [X_12: list_b] : ( arc_pre_cycle_a_b @ g @ X_12 ) ) ).
% G.closed_w_imp_cycle
thf(fact_715_closed__w__imp__cycle,axiom,
! [P2: list_b] :
( ( arc_wf_closed_w_a_b @ t @ P2 )
=> ? [X_12: list_b] : ( arc_pre_cycle_a_b @ t @ X_12 ) ) ).
% closed_w_imp_cycle
thf(fact_716_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_717_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_718_G_Oinduced__subgraph__altdef,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ H @ g )
= ( ( digraph_subgraph_a_b @ H @ g )
& ! [H3: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H3 @ g )
=> ( ( ( pre_ve642382030648772252t_unit @ H3 )
!= ( pre_ve642382030648772252t_unit @ H ) )
| ( ord_less_eq_set_b @ ( pre_ar1395965042833527383t_unit @ H3 ) @ ( pre_ar1395965042833527383t_unit @ H ) ) ) ) ) ) ).
% G.induced_subgraph_altdef
thf(fact_719_induced__subgraph__altdef,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ H @ t )
= ( ( digraph_subgraph_a_b @ H @ t )
& ! [H3: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H3 @ t )
=> ( ( ( pre_ve642382030648772252t_unit @ H3 )
!= ( pre_ve642382030648772252t_unit @ H ) )
| ( ord_less_eq_set_b @ ( pre_ar1395965042833527383t_unit @ H3 ) @ ( pre_ar1395965042833527383t_unit @ H ) ) ) ) ) ) ).
% induced_subgraph_altdef
thf(fact_720_induced__graph__imp__symmetric,axiom,
! [G: pre_pr7278220950009878019t_unit,H: pre_pr7278220950009878019t_unit] :
( ( symmetric_a_b @ G )
=> ( ( digrap5251062021860773499ph_a_b @ H @ G )
=> ( symmetric_a_b @ H ) ) ) ).
% induced_graph_imp_symmetric
thf(fact_721_induced__subgraphI_H,axiom,
! [H: pre_pr7278220950009878019t_unit,G: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H @ G )
=> ( ! [H4: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H4 @ G )
=> ( ( ( pre_ve642382030648772252t_unit @ H4 )
!= ( pre_ve642382030648772252t_unit @ H ) )
| ( ord_less_eq_set_b @ ( pre_ar1395965042833527383t_unit @ H4 ) @ ( pre_ar1395965042833527383t_unit @ H ) ) ) )
=> ( digrap5251062021860773499ph_a_b @ H @ G ) ) ) ).
% induced_subgraphI'
thf(fact_722_pre__digraph_Oinduced__subgraph__altdef,axiom,
( digrap5251062021860773499ph_a_b
= ( ^ [H2: pre_pr7278220950009878019t_unit,G4: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H2 @ G4 )
& ! [H3: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H3 @ G4 )
=> ( ( ( pre_ve642382030648772252t_unit @ H3 )
!= ( pre_ve642382030648772252t_unit @ H2 ) )
| ( ord_less_eq_set_b @ ( pre_ar1395965042833527383t_unit @ H3 ) @ ( pre_ar1395965042833527383t_unit @ H2 ) ) ) ) ) ) ) ).
% pre_digraph.induced_subgraph_altdef
thf(fact_723_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_724_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_725_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 )
=> ~ ! [V4: a] :
( ( pre_ve642382030648772252t_unit @ g )
!= ( insert_a @ V4 @ bot_bot_set_a ) ) ) ) ) ).
% G.connected_arcs_empty
thf(fact_726_connected__arcs__empty,axiom,
( ( digrap8783888973171253482ed_a_b @ t )
=> ( ( ( pre_ar1395965042833527383t_unit @ t )
= bot_bot_set_b )
=> ( ( ( pre_ve642382030648772252t_unit @ t )
!= bot_bot_set_a )
=> ~ ! [V4: a] :
( ( pre_ve642382030648772252t_unit @ t )
!= ( insert_a @ V4 @ bot_bot_set_a ) ) ) ) ) ).
% connected_arcs_empty
thf(fact_727_fin__digraph_Oclosed__euler2_I1_J,axiom,
! [G: pre_pr7278220950009878019t_unit,U: a,P2: list_b] :
( ( fin_digraph_a_b @ G )
=> ( ( pre_euler_trail_a_b @ G @ U @ P2 @ U )
=> ( digrap8783888973171253482ed_a_b @ G ) ) ) ).
% fin_digraph.closed_euler2(1)
thf(fact_728_insertCI,axiom,
! [A3: a,B: set_a,B3: a] :
( ( ~ ( member_a @ A3 @ B )
=> ( A3 = B3 ) )
=> ( member_a @ A3 @ ( insert_a @ B3 @ B ) ) ) ).
% insertCI
thf(fact_729_insertCI,axiom,
! [A3: set_a,B: set_set_a,B3: set_a] :
( ( ~ ( member_set_a @ A3 @ B )
=> ( A3 = B3 ) )
=> ( member_set_a @ A3 @ ( insert_set_a @ B3 @ B ) ) ) ).
% insertCI
thf(fact_730_insertCI,axiom,
! [A3: pre_pr7278220950009878019t_unit,B: set_pr5411798346947241657t_unit,B3: pre_pr7278220950009878019t_unit] :
( ( ~ ( member6939884229742472986t_unit @ A3 @ B )
=> ( A3 = B3 ) )
=> ( member6939884229742472986t_unit @ A3 @ ( insert6864688055023459379t_unit @ B3 @ B ) ) ) ).
% insertCI
thf(fact_731_insertCI,axiom,
! [A3: b,B: set_b,B3: b] :
( ( ~ ( member_b @ A3 @ B )
=> ( A3 = B3 ) )
=> ( member_b @ A3 @ ( insert_b @ B3 @ B ) ) ) ).
% insertCI
thf(fact_732_insert__iff,axiom,
! [A3: a,B3: a,A: set_a] :
( ( member_a @ A3 @ ( insert_a @ B3 @ A ) )
= ( ( A3 = B3 )
| ( member_a @ A3 @ A ) ) ) ).
% insert_iff
thf(fact_733_insert__iff,axiom,
! [A3: set_a,B3: set_a,A: set_set_a] :
( ( member_set_a @ A3 @ ( insert_set_a @ B3 @ A ) )
= ( ( A3 = B3 )
| ( member_set_a @ A3 @ A ) ) ) ).
% insert_iff
thf(fact_734_insert__iff,axiom,
! [A3: pre_pr7278220950009878019t_unit,B3: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A3 @ ( insert6864688055023459379t_unit @ B3 @ A ) )
= ( ( A3 = B3 )
| ( member6939884229742472986t_unit @ A3 @ A ) ) ) ).
% insert_iff
thf(fact_735_insert__iff,axiom,
! [A3: b,B3: b,A: set_b] :
( ( member_b @ A3 @ ( insert_b @ B3 @ A ) )
= ( ( A3 = B3 )
| ( member_b @ A3 @ A ) ) ) ).
% insert_iff
thf(fact_736_insert__absorb2,axiom,
! [X4: a,A: set_a] :
( ( insert_a @ X4 @ ( insert_a @ X4 @ A ) )
= ( insert_a @ X4 @ A ) ) ).
% insert_absorb2
thf(fact_737_insert__absorb2,axiom,
! [X4: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit] :
( ( insert6864688055023459379t_unit @ X4 @ ( insert6864688055023459379t_unit @ X4 @ A ) )
= ( insert6864688055023459379t_unit @ X4 @ A ) ) ).
% insert_absorb2
thf(fact_738_insert__absorb2,axiom,
! [X4: b,A: set_b] :
( ( insert_b @ X4 @ ( insert_b @ X4 @ A ) )
= ( insert_b @ X4 @ A ) ) ).
% insert_absorb2
thf(fact_739_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_740_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_741_image__insert,axiom,
! [F: a > a,A3: a,B: set_a] :
( ( image_a_a @ F @ ( insert_a @ A3 @ B ) )
= ( insert_a @ ( F @ A3 ) @ ( image_a_a @ F @ B ) ) ) ).
% image_insert
thf(fact_742_image__insert,axiom,
! [F: a > b,A3: a,B: set_a] :
( ( image_a_b @ F @ ( insert_a @ A3 @ B ) )
= ( insert_b @ ( F @ A3 ) @ ( image_a_b @ F @ B ) ) ) ).
% image_insert
thf(fact_743_image__insert,axiom,
! [F: b > a,A3: b,B: set_b] :
( ( image_b_a @ F @ ( insert_b @ A3 @ B ) )
= ( insert_a @ ( F @ A3 ) @ ( image_b_a @ F @ B ) ) ) ).
% image_insert
thf(fact_744_image__insert,axiom,
! [F: b > b,A3: b,B: set_b] :
( ( image_b_b @ F @ ( insert_b @ A3 @ B ) )
= ( insert_b @ ( F @ A3 ) @ ( image_b_b @ F @ B ) ) ) ).
% image_insert
thf(fact_745_image__insert,axiom,
! [F: a > set_a,A3: a,B: set_a] :
( ( image_a_set_a @ F @ ( insert_a @ A3 @ B ) )
= ( insert_set_a @ ( F @ A3 ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_insert
thf(fact_746_image__insert,axiom,
! [F: a > pre_pr7278220950009878019t_unit,A3: a,B: set_a] :
( ( image_5713294457175270716t_unit @ F @ ( insert_a @ A3 @ B ) )
= ( insert6864688055023459379t_unit @ ( F @ A3 ) @ ( image_5713294457175270716t_unit @ F @ B ) ) ) ).
% image_insert
thf(fact_747_image__insert,axiom,
! [F: pre_pr7278220950009878019t_unit > a,A3: pre_pr7278220950009878019t_unit,B: set_pr5411798346947241657t_unit] :
( ( image_4969699134812999796unit_a @ F @ ( insert6864688055023459379t_unit @ A3 @ B ) )
= ( insert_a @ ( F @ A3 ) @ ( image_4969699134812999796unit_a @ F @ B ) ) ) ).
% image_insert
thf(fact_748_image__insert,axiom,
! [F: pre_pr7278220950009878019t_unit > b,A3: pre_pr7278220950009878019t_unit,B: set_pr5411798346947241657t_unit] :
( ( image_4969699134812999797unit_b @ F @ ( insert6864688055023459379t_unit @ A3 @ B ) )
= ( insert_b @ ( F @ A3 ) @ ( image_4969699134812999797unit_b @ F @ B ) ) ) ).
% image_insert
thf(fact_749_image__insert,axiom,
! [F: b > pre_pr7278220950009878019t_unit,A3: b,B: set_b] :
( ( image_4434118323594779837t_unit @ F @ ( insert_b @ A3 @ B ) )
= ( insert6864688055023459379t_unit @ ( F @ A3 ) @ ( image_4434118323594779837t_unit @ F @ B ) ) ) ).
% image_insert
thf(fact_750_image__insert,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A3: set_a,B: set_set_a] :
( ( image_6801035452528096924t_unit @ F @ ( insert_set_a @ A3 @ B ) )
= ( insert6864688055023459379t_unit @ ( F @ A3 ) @ ( image_6801035452528096924t_unit @ F @ B ) ) ) ).
% image_insert
thf(fact_751_insert__image,axiom,
! [X4: a,A: set_a,F: a > a] :
( ( member_a @ X4 @ A )
=> ( ( insert_a @ ( F @ X4 ) @ ( image_a_a @ F @ A ) )
= ( image_a_a @ F @ A ) ) ) ).
% insert_image
thf(fact_752_insert__image,axiom,
! [X4: a,A: set_a,F: a > b] :
( ( member_a @ X4 @ A )
=> ( ( insert_b @ ( F @ X4 ) @ ( image_a_b @ F @ A ) )
= ( image_a_b @ F @ A ) ) ) ).
% insert_image
thf(fact_753_insert__image,axiom,
! [X4: b,A: set_b,F: b > a] :
( ( member_b @ X4 @ A )
=> ( ( insert_a @ ( F @ X4 ) @ ( image_b_a @ F @ A ) )
= ( image_b_a @ F @ A ) ) ) ).
% insert_image
thf(fact_754_insert__image,axiom,
! [X4: b,A: set_b,F: b > b] :
( ( member_b @ X4 @ A )
=> ( ( insert_b @ ( F @ X4 ) @ ( image_b_b @ F @ A ) )
= ( image_b_b @ F @ A ) ) ) ).
% insert_image
thf(fact_755_insert__image,axiom,
! [X4: a,A: set_a,F: a > set_a] :
( ( member_a @ X4 @ A )
=> ( ( insert_set_a @ ( F @ X4 ) @ ( image_a_set_a @ F @ A ) )
= ( image_a_set_a @ F @ A ) ) ) ).
% insert_image
thf(fact_756_insert__image,axiom,
! [X4: set_a,A: set_set_a,F: set_a > a] :
( ( member_set_a @ X4 @ A )
=> ( ( insert_a @ ( F @ X4 ) @ ( image_set_a_a @ F @ A ) )
= ( image_set_a_a @ F @ A ) ) ) ).
% insert_image
thf(fact_757_insert__image,axiom,
! [X4: set_a,A: set_set_a,F: set_a > b] :
( ( member_set_a @ X4 @ A )
=> ( ( insert_b @ ( F @ X4 ) @ ( image_set_a_b @ F @ A ) )
= ( image_set_a_b @ F @ A ) ) ) ).
% insert_image
thf(fact_758_insert__image,axiom,
! [X4: a,A: set_a,F: a > pre_pr7278220950009878019t_unit] :
( ( member_a @ X4 @ A )
=> ( ( insert6864688055023459379t_unit @ ( F @ X4 ) @ ( image_5713294457175270716t_unit @ F @ A ) )
= ( image_5713294457175270716t_unit @ F @ A ) ) ) ).
% insert_image
thf(fact_759_insert__image,axiom,
! [X4: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > a] :
( ( member6939884229742472986t_unit @ X4 @ A )
=> ( ( insert_a @ ( F @ X4 ) @ ( image_4969699134812999796unit_a @ F @ A ) )
= ( image_4969699134812999796unit_a @ F @ A ) ) ) ).
% insert_image
thf(fact_760_insert__image,axiom,
! [X4: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > b] :
( ( member6939884229742472986t_unit @ X4 @ A )
=> ( ( insert_b @ ( F @ X4 ) @ ( image_4969699134812999797unit_b @ F @ A ) )
= ( image_4969699134812999797unit_b @ F @ A ) ) ) ).
% insert_image
thf(fact_761_singletonI,axiom,
! [A3: a] : ( member_a @ A3 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_762_singletonI,axiom,
! [A3: b] : ( member_b @ A3 @ ( insert_b @ A3 @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_763_singletonI,axiom,
! [A3: pre_pr7278220950009878019t_unit] : ( member6939884229742472986t_unit @ A3 @ ( insert6864688055023459379t_unit @ A3 @ bot_bo1839476491465656141t_unit ) ) ).
% singletonI
thf(fact_764_singletonI,axiom,
! [A3: set_a] : ( member_set_a @ A3 @ ( insert_set_a @ A3 @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_765_insert__subset,axiom,
! [X4: set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X4 @ A ) @ B )
= ( ( member_set_a @ X4 @ B )
& ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_766_insert__subset,axiom,
! [X4: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ ( insert6864688055023459379t_unit @ X4 @ A ) @ B )
= ( ( member6939884229742472986t_unit @ X4 @ B )
& ( ord_le8200006823705900825t_unit @ A @ B ) ) ) ).
% insert_subset
thf(fact_767_insert__subset,axiom,
! [X4: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X4 @ A ) @ B )
= ( ( member_a @ X4 @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_768_insert__subset,axiom,
! [X4: b,A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ ( insert_b @ X4 @ A ) @ B )
= ( ( member_b @ X4 @ B )
& ( ord_less_eq_set_b @ A @ B ) ) ) ).
% insert_subset
thf(fact_769_Int__insert__right__if1,axiom,
! [A3: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ A3 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A3 @ B ) )
= ( insert_set_a @ A3 @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_770_Int__insert__right__if1,axiom,
! [A3: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A3 @ A )
=> ( ( inf_in1092213268631476299t_unit @ A @ ( insert6864688055023459379t_unit @ A3 @ B ) )
= ( insert6864688055023459379t_unit @ A3 @ ( inf_in1092213268631476299t_unit @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_771_Int__insert__right__if1,axiom,
! [A3: b,A: set_b,B: set_b] :
( ( member_b @ A3 @ A )
=> ( ( inf_inf_set_b @ A @ ( insert_b @ A3 @ B ) )
= ( insert_b @ A3 @ ( inf_inf_set_b @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_772_Int__insert__right__if1,axiom,
! [A3: a,A: set_a,B: set_a] :
( ( member_a @ A3 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A3 @ B ) )
= ( insert_a @ A3 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_773_Int__insert__right__if0,axiom,
! [A3: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A3 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A3 @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_774_Int__insert__right__if0,axiom,
! [A3: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ A3 @ A )
=> ( ( inf_in1092213268631476299t_unit @ A @ ( insert6864688055023459379t_unit @ A3 @ B ) )
= ( inf_in1092213268631476299t_unit @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_775_Int__insert__right__if0,axiom,
! [A3: b,A: set_b,B: set_b] :
( ~ ( member_b @ A3 @ A )
=> ( ( inf_inf_set_b @ A @ ( insert_b @ A3 @ B ) )
= ( inf_inf_set_b @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_776_Int__insert__right__if0,axiom,
! [A3: a,A: set_a,B: set_a] :
( ~ ( member_a @ A3 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A3 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_777_insert__inter__insert,axiom,
! [A3: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A3 @ A ) @ ( insert6864688055023459379t_unit @ A3 @ B ) )
= ( insert6864688055023459379t_unit @ A3 @ ( inf_in1092213268631476299t_unit @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_778_insert__inter__insert,axiom,
! [A3: b,A: set_b,B: set_b] :
( ( inf_inf_set_b @ ( insert_b @ A3 @ A ) @ ( insert_b @ A3 @ B ) )
= ( insert_b @ A3 @ ( inf_inf_set_b @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_779_insert__inter__insert,axiom,
! [A3: a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A3 @ A ) @ ( insert_a @ A3 @ B ) )
= ( insert_a @ A3 @ ( inf_inf_set_a @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_780_Int__insert__left__if1,axiom,
! [A3: set_a,C2: set_set_a,B: set_set_a] :
( ( member_set_a @ A3 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A3 @ B ) @ C2 )
= ( insert_set_a @ A3 @ ( inf_inf_set_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_781_Int__insert__left__if1,axiom,
! [A3: pre_pr7278220950009878019t_unit,C2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A3 @ C2 )
=> ( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A3 @ B ) @ C2 )
= ( insert6864688055023459379t_unit @ A3 @ ( inf_in1092213268631476299t_unit @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_782_Int__insert__left__if1,axiom,
! [A3: b,C2: set_b,B: set_b] :
( ( member_b @ A3 @ C2 )
=> ( ( inf_inf_set_b @ ( insert_b @ A3 @ B ) @ C2 )
= ( insert_b @ A3 @ ( inf_inf_set_b @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_783_Int__insert__left__if1,axiom,
! [A3: a,C2: set_a,B: set_a] :
( ( member_a @ A3 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A3 @ B ) @ C2 )
= ( insert_a @ A3 @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_784_Int__insert__left__if0,axiom,
! [A3: set_a,C2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A3 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A3 @ B ) @ C2 )
= ( inf_inf_set_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_785_Int__insert__left__if0,axiom,
! [A3: pre_pr7278220950009878019t_unit,C2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ A3 @ C2 )
=> ( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A3 @ B ) @ C2 )
= ( inf_in1092213268631476299t_unit @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_786_Int__insert__left__if0,axiom,
! [A3: b,C2: set_b,B: set_b] :
( ~ ( member_b @ A3 @ C2 )
=> ( ( inf_inf_set_b @ ( insert_b @ A3 @ B ) @ C2 )
= ( inf_inf_set_b @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_787_Int__insert__left__if0,axiom,
! [A3: a,C2: set_a,B: set_a] :
( ~ ( member_a @ A3 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A3 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_788_singleton__insert__inj__eq_H,axiom,
! [A3: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B3: pre_pr7278220950009878019t_unit] :
( ( ( insert6864688055023459379t_unit @ A3 @ A )
= ( insert6864688055023459379t_unit @ B3 @ bot_bo1839476491465656141t_unit ) )
= ( ( A3 = B3 )
& ( ord_le8200006823705900825t_unit @ A @ ( insert6864688055023459379t_unit @ B3 @ bot_bo1839476491465656141t_unit ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_789_singleton__insert__inj__eq_H,axiom,
! [A3: set_a,A: set_set_a,B3: set_a] :
( ( ( insert_set_a @ A3 @ A )
= ( insert_set_a @ B3 @ bot_bot_set_set_a ) )
= ( ( A3 = B3 )
& ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B3 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_790_singleton__insert__inj__eq_H,axiom,
! [A3: a,A: set_a,B3: a] :
( ( ( insert_a @ A3 @ A )
= ( insert_a @ B3 @ bot_bot_set_a ) )
= ( ( A3 = B3 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B3 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_791_singleton__insert__inj__eq_H,axiom,
! [A3: b,A: set_b,B3: b] :
( ( ( insert_b @ A3 @ A )
= ( insert_b @ B3 @ bot_bot_set_b ) )
= ( ( A3 = B3 )
& ( ord_less_eq_set_b @ A @ ( insert_b @ B3 @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_792_singleton__insert__inj__eq,axiom,
! [B3: pre_pr7278220950009878019t_unit,A3: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit] :
( ( ( insert6864688055023459379t_unit @ B3 @ bot_bo1839476491465656141t_unit )
= ( insert6864688055023459379t_unit @ A3 @ A ) )
= ( ( A3 = B3 )
& ( ord_le8200006823705900825t_unit @ A @ ( insert6864688055023459379t_unit @ B3 @ bot_bo1839476491465656141t_unit ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_793_singleton__insert__inj__eq,axiom,
! [B3: set_a,A3: set_a,A: set_set_a] :
( ( ( insert_set_a @ B3 @ bot_bot_set_set_a )
= ( insert_set_a @ A3 @ A ) )
= ( ( A3 = B3 )
& ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B3 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_794_singleton__insert__inj__eq,axiom,
! [B3: a,A3: a,A: set_a] :
( ( ( insert_a @ B3 @ bot_bot_set_a )
= ( insert_a @ A3 @ A ) )
= ( ( A3 = B3 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B3 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_795_singleton__insert__inj__eq,axiom,
! [B3: b,A3: b,A: set_b] :
( ( ( insert_b @ B3 @ bot_bot_set_b )
= ( insert_b @ A3 @ A ) )
= ( ( A3 = B3 )
& ( ord_less_eq_set_b @ A @ ( insert_b @ B3 @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_796_insert__disjoint_I1_J,axiom,
! [A3: a,A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A3 @ A ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A3 @ B )
& ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_797_insert__disjoint_I1_J,axiom,
! [A3: b,A: set_b,B: set_b] :
( ( ( inf_inf_set_b @ ( insert_b @ A3 @ A ) @ B )
= bot_bot_set_b )
= ( ~ ( member_b @ A3 @ B )
& ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b ) ) ) ).
% insert_disjoint(1)
thf(fact_798_insert__disjoint_I1_J,axiom,
! [A3: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A3 @ A ) @ B )
= bot_bo1839476491465656141t_unit )
= ( ~ ( member6939884229742472986t_unit @ A3 @ B )
& ( ( inf_in1092213268631476299t_unit @ A @ B )
= bot_bo1839476491465656141t_unit ) ) ) ).
% insert_disjoint(1)
thf(fact_799_insert__disjoint_I1_J,axiom,
! [A3: set_a,A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A3 @ A ) @ B )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A3 @ B )
& ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_800_insert__disjoint_I2_J,axiom,
! [A3: a,A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A3 @ A ) @ B ) )
= ( ~ ( member_a @ A3 @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_801_insert__disjoint_I2_J,axiom,
! [A3: b,A: set_b,B: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ ( insert_b @ A3 @ A ) @ B ) )
= ( ~ ( member_b @ A3 @ B )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_802_insert__disjoint_I2_J,axiom,
! [A3: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( bot_bo1839476491465656141t_unit
= ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A3 @ A ) @ B ) )
= ( ~ ( member6939884229742472986t_unit @ A3 @ B )
& ( bot_bo1839476491465656141t_unit
= ( inf_in1092213268631476299t_unit @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_803_insert__disjoint_I2_J,axiom,
! [A3: set_a,A: set_set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A3 @ A ) @ B ) )
= ( ~ ( member_set_a @ A3 @ B )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_804_disjoint__insert_I1_J,axiom,
! [B: set_a,A3: a,A: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A3 @ A ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A3 @ B )
& ( ( inf_inf_set_a @ B @ A )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_805_disjoint__insert_I1_J,axiom,
! [B: set_b,A3: b,A: set_b] :
( ( ( inf_inf_set_b @ B @ ( insert_b @ A3 @ A ) )
= bot_bot_set_b )
= ( ~ ( member_b @ A3 @ B )
& ( ( inf_inf_set_b @ B @ A )
= bot_bot_set_b ) ) ) ).
% disjoint_insert(1)
thf(fact_806_disjoint__insert_I1_J,axiom,
! [B: set_pr5411798346947241657t_unit,A3: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit] :
( ( ( inf_in1092213268631476299t_unit @ B @ ( insert6864688055023459379t_unit @ A3 @ A ) )
= bot_bo1839476491465656141t_unit )
= ( ~ ( member6939884229742472986t_unit @ A3 @ B )
& ( ( inf_in1092213268631476299t_unit @ B @ A )
= bot_bo1839476491465656141t_unit ) ) ) ).
% disjoint_insert(1)
thf(fact_807_disjoint__insert_I1_J,axiom,
! [B: set_set_a,A3: set_a,A: set_set_a] :
( ( ( inf_inf_set_set_a @ B @ ( insert_set_a @ A3 @ A ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A3 @ B )
& ( ( inf_inf_set_set_a @ B @ A )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_808_disjoint__insert_I2_J,axiom,
! [A: set_a,B3: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A @ ( insert_a @ B3 @ B ) ) )
= ( ~ ( member_a @ B3 @ A )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_809_disjoint__insert_I2_J,axiom,
! [A: set_b,B3: b,B: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ A @ ( insert_b @ B3 @ B ) ) )
= ( ~ ( member_b @ B3 @ A )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_810_disjoint__insert_I2_J,axiom,
! [A: set_pr5411798346947241657t_unit,B3: pre_pr7278220950009878019t_unit,B: set_pr5411798346947241657t_unit] :
( ( bot_bo1839476491465656141t_unit
= ( inf_in1092213268631476299t_unit @ A @ ( insert6864688055023459379t_unit @ B3 @ B ) ) )
= ( ~ ( member6939884229742472986t_unit @ B3 @ A )
& ( bot_bo1839476491465656141t_unit
= ( inf_in1092213268631476299t_unit @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_811_disjoint__insert_I2_J,axiom,
! [A: set_set_a,B3: set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ ( insert_set_a @ B3 @ B ) ) )
= ( ~ ( member_set_a @ B3 @ A )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_812_insert__mono,axiom,
! [C2: set_pr5411798346947241657t_unit,D2: set_pr5411798346947241657t_unit,A3: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ C2 @ D2 )
=> ( ord_le8200006823705900825t_unit @ ( insert6864688055023459379t_unit @ A3 @ C2 ) @ ( insert6864688055023459379t_unit @ A3 @ D2 ) ) ) ).
% insert_mono
thf(fact_813_insert__mono,axiom,
! [C2: set_a,D2: set_a,A3: a] :
( ( ord_less_eq_set_a @ C2 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A3 @ C2 ) @ ( insert_a @ A3 @ D2 ) ) ) ).
% insert_mono
thf(fact_814_insert__mono,axiom,
! [C2: set_b,D2: set_b,A3: b] :
( ( ord_less_eq_set_b @ C2 @ D2 )
=> ( ord_less_eq_set_b @ ( insert_b @ A3 @ C2 ) @ ( insert_b @ A3 @ D2 ) ) ) ).
% insert_mono
thf(fact_815_subset__insert,axiom,
! [X4: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X4 @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X4 @ B ) )
= ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_816_subset__insert,axiom,
! [X4: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ X4 @ A )
=> ( ( ord_le8200006823705900825t_unit @ A @ ( insert6864688055023459379t_unit @ X4 @ B ) )
= ( ord_le8200006823705900825t_unit @ A @ B ) ) ) ).
% subset_insert
thf(fact_817_subset__insert,axiom,
! [X4: a,A: set_a,B: set_a] :
( ~ ( member_a @ X4 @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X4 @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_818_subset__insert,axiom,
! [X4: b,A: set_b,B: set_b] :
( ~ ( member_b @ X4 @ A )
=> ( ( ord_less_eq_set_b @ A @ ( insert_b @ X4 @ B ) )
= ( ord_less_eq_set_b @ A @ B ) ) ) ).
% subset_insert
thf(fact_819_subset__insertI,axiom,
! [B: set_pr5411798346947241657t_unit,A3: pre_pr7278220950009878019t_unit] : ( ord_le8200006823705900825t_unit @ B @ ( insert6864688055023459379t_unit @ A3 @ B ) ) ).
% subset_insertI
thf(fact_820_subset__insertI,axiom,
! [B: set_a,A3: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A3 @ B ) ) ).
% subset_insertI
thf(fact_821_subset__insertI,axiom,
! [B: set_b,A3: b] : ( ord_less_eq_set_b @ B @ ( insert_b @ A3 @ B ) ) ).
% subset_insertI
thf(fact_822_subset__insertI2,axiom,
! [A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit,B3: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ A @ B )
=> ( ord_le8200006823705900825t_unit @ A @ ( insert6864688055023459379t_unit @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_823_subset__insertI2,axiom,
! [A: set_a,B: set_a,B3: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_824_subset__insertI2,axiom,
! [A: set_b,B: set_b,B3: b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ord_less_eq_set_b @ A @ ( insert_b @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_825_Int__insert__left,axiom,
! [A3: set_a,C2: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A3 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A3 @ B ) @ C2 )
= ( insert_set_a @ A3 @ ( inf_inf_set_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_set_a @ A3 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A3 @ B ) @ C2 )
= ( inf_inf_set_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_826_Int__insert__left,axiom,
! [A3: pre_pr7278220950009878019t_unit,C2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( ( member6939884229742472986t_unit @ A3 @ C2 )
=> ( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A3 @ B ) @ C2 )
= ( insert6864688055023459379t_unit @ A3 @ ( inf_in1092213268631476299t_unit @ B @ C2 ) ) ) )
& ( ~ ( member6939884229742472986t_unit @ A3 @ C2 )
=> ( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A3 @ B ) @ C2 )
= ( inf_in1092213268631476299t_unit @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_827_Int__insert__left,axiom,
! [A3: b,C2: set_b,B: set_b] :
( ( ( member_b @ A3 @ C2 )
=> ( ( inf_inf_set_b @ ( insert_b @ A3 @ B ) @ C2 )
= ( insert_b @ A3 @ ( inf_inf_set_b @ B @ C2 ) ) ) )
& ( ~ ( member_b @ A3 @ C2 )
=> ( ( inf_inf_set_b @ ( insert_b @ A3 @ B ) @ C2 )
= ( inf_inf_set_b @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_828_Int__insert__left,axiom,
! [A3: a,C2: set_a,B: set_a] :
( ( ( member_a @ A3 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A3 @ B ) @ C2 )
= ( insert_a @ A3 @ ( inf_inf_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_a @ A3 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A3 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_829_Int__insert__right,axiom,
! [A3: set_a,A: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A3 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A3 @ B ) )
= ( insert_set_a @ A3 @ ( inf_inf_set_set_a @ A @ B ) ) ) )
& ( ~ ( member_set_a @ A3 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A3 @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_830_Int__insert__right,axiom,
! [A3: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( ( member6939884229742472986t_unit @ A3 @ A )
=> ( ( inf_in1092213268631476299t_unit @ A @ ( insert6864688055023459379t_unit @ A3 @ B ) )
= ( insert6864688055023459379t_unit @ A3 @ ( inf_in1092213268631476299t_unit @ A @ B ) ) ) )
& ( ~ ( member6939884229742472986t_unit @ A3 @ A )
=> ( ( inf_in1092213268631476299t_unit @ A @ ( insert6864688055023459379t_unit @ A3 @ B ) )
= ( inf_in1092213268631476299t_unit @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_831_Int__insert__right,axiom,
! [A3: b,A: set_b,B: set_b] :
( ( ( member_b @ A3 @ A )
=> ( ( inf_inf_set_b @ A @ ( insert_b @ A3 @ B ) )
= ( insert_b @ A3 @ ( inf_inf_set_b @ A @ B ) ) ) )
& ( ~ ( member_b @ A3 @ A )
=> ( ( inf_inf_set_b @ A @ ( insert_b @ A3 @ B ) )
= ( inf_inf_set_b @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_832_Int__insert__right,axiom,
! [A3: a,A: set_a,B: set_a] :
( ( ( member_a @ A3 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A3 @ B ) )
= ( insert_a @ A3 @ ( inf_inf_set_a @ A @ B ) ) ) )
& ( ~ ( member_a @ A3 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A3 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_833_insertE,axiom,
! [A3: a,B3: a,A: set_a] :
( ( member_a @ A3 @ ( insert_a @ B3 @ A ) )
=> ( ( A3 != B3 )
=> ( member_a @ A3 @ A ) ) ) ).
% insertE
thf(fact_834_insertE,axiom,
! [A3: set_a,B3: set_a,A: set_set_a] :
( ( member_set_a @ A3 @ ( insert_set_a @ B3 @ A ) )
=> ( ( A3 != B3 )
=> ( member_set_a @ A3 @ A ) ) ) ).
% insertE
thf(fact_835_insertE,axiom,
! [A3: pre_pr7278220950009878019t_unit,B3: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A3 @ ( insert6864688055023459379t_unit @ B3 @ A ) )
=> ( ( A3 != B3 )
=> ( member6939884229742472986t_unit @ A3 @ A ) ) ) ).
% insertE
thf(fact_836_insertE,axiom,
! [A3: b,B3: b,A: set_b] :
( ( member_b @ A3 @ ( insert_b @ B3 @ A ) )
=> ( ( A3 != B3 )
=> ( member_b @ A3 @ A ) ) ) ).
% insertE
thf(fact_837_insertI1,axiom,
! [A3: a,B: set_a] : ( member_a @ A3 @ ( insert_a @ A3 @ B ) ) ).
% insertI1
thf(fact_838_insertI1,axiom,
! [A3: set_a,B: set_set_a] : ( member_set_a @ A3 @ ( insert_set_a @ A3 @ B ) ) ).
% insertI1
thf(fact_839_insertI1,axiom,
! [A3: pre_pr7278220950009878019t_unit,B: set_pr5411798346947241657t_unit] : ( member6939884229742472986t_unit @ A3 @ ( insert6864688055023459379t_unit @ A3 @ B ) ) ).
% insertI1
thf(fact_840_insertI1,axiom,
! [A3: b,B: set_b] : ( member_b @ A3 @ ( insert_b @ A3 @ B ) ) ).
% insertI1
thf(fact_841_insertI2,axiom,
! [A3: a,B: set_a,B3: a] :
( ( member_a @ A3 @ B )
=> ( member_a @ A3 @ ( insert_a @ B3 @ B ) ) ) ).
% insertI2
thf(fact_842_insertI2,axiom,
! [A3: set_a,B: set_set_a,B3: set_a] :
( ( member_set_a @ A3 @ B )
=> ( member_set_a @ A3 @ ( insert_set_a @ B3 @ B ) ) ) ).
% insertI2
thf(fact_843_insertI2,axiom,
! [A3: pre_pr7278220950009878019t_unit,B: set_pr5411798346947241657t_unit,B3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ A3 @ B )
=> ( member6939884229742472986t_unit @ A3 @ ( insert6864688055023459379t_unit @ B3 @ B ) ) ) ).
% insertI2
thf(fact_844_insertI2,axiom,
! [A3: b,B: set_b,B3: b] :
( ( member_b @ A3 @ B )
=> ( member_b @ A3 @ ( insert_b @ B3 @ B ) ) ) ).
% insertI2
thf(fact_845_Set_Oset__insert,axiom,
! [X4: a,A: set_a] :
( ( member_a @ X4 @ A )
=> ~ ! [B5: set_a] :
( ( A
= ( insert_a @ X4 @ B5 ) )
=> ( member_a @ X4 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_846_Set_Oset__insert,axiom,
! [X4: set_a,A: set_set_a] :
( ( member_set_a @ X4 @ A )
=> ~ ! [B5: set_set_a] :
( ( A
= ( insert_set_a @ X4 @ B5 ) )
=> ( member_set_a @ X4 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_847_Set_Oset__insert,axiom,
! [X4: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ X4 @ A )
=> ~ ! [B5: set_pr5411798346947241657t_unit] :
( ( A
= ( insert6864688055023459379t_unit @ X4 @ B5 ) )
=> ( member6939884229742472986t_unit @ X4 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_848_Set_Oset__insert,axiom,
! [X4: b,A: set_b] :
( ( member_b @ X4 @ A )
=> ~ ! [B5: set_b] :
( ( A
= ( insert_b @ X4 @ B5 ) )
=> ( member_b @ X4 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_849_insert__ident,axiom,
! [X4: a,A: set_a,B: set_a] :
( ~ ( member_a @ X4 @ A )
=> ( ~ ( member_a @ X4 @ B )
=> ( ( ( insert_a @ X4 @ A )
= ( insert_a @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_850_insert__ident,axiom,
! [X4: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X4 @ A )
=> ( ~ ( member_set_a @ X4 @ B )
=> ( ( ( insert_set_a @ X4 @ A )
= ( insert_set_a @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_851_insert__ident,axiom,
! [X4: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ X4 @ A )
=> ( ~ ( member6939884229742472986t_unit @ X4 @ B )
=> ( ( ( insert6864688055023459379t_unit @ X4 @ A )
= ( insert6864688055023459379t_unit @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_852_insert__ident,axiom,
! [X4: b,A: set_b,B: set_b] :
( ~ ( member_b @ X4 @ A )
=> ( ~ ( member_b @ X4 @ B )
=> ( ( ( insert_b @ X4 @ A )
= ( insert_b @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_853_insert__absorb,axiom,
! [A3: a,A: set_a] :
( ( member_a @ A3 @ A )
=> ( ( insert_a @ A3 @ A )
= A ) ) ).
% insert_absorb
thf(fact_854_insert__absorb,axiom,
! [A3: set_a,A: set_set_a] :
( ( member_set_a @ A3 @ A )
=> ( ( insert_set_a @ A3 @ A )
= A ) ) ).
% insert_absorb
thf(fact_855_insert__absorb,axiom,
! [A3: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A3 @ A )
=> ( ( insert6864688055023459379t_unit @ A3 @ A )
= A ) ) ).
% insert_absorb
thf(fact_856_insert__absorb,axiom,
! [A3: b,A: set_b] :
( ( member_b @ A3 @ A )
=> ( ( insert_b @ A3 @ A )
= A ) ) ).
% insert_absorb
thf(fact_857_insert__eq__iff,axiom,
! [A3: a,A: set_a,B3: a,B: set_a] :
( ~ ( member_a @ A3 @ A )
=> ( ~ ( member_a @ B3 @ B )
=> ( ( ( insert_a @ A3 @ A )
= ( insert_a @ B3 @ B ) )
= ( ( ( A3 = B3 )
=> ( A = B ) )
& ( ( A3 != B3 )
=> ? [C5: set_a] :
( ( A
= ( insert_a @ B3 @ C5 ) )
& ~ ( member_a @ B3 @ C5 )
& ( B
= ( insert_a @ A3 @ C5 ) )
& ~ ( member_a @ A3 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_858_insert__eq__iff,axiom,
! [A3: set_a,A: set_set_a,B3: set_a,B: set_set_a] :
( ~ ( member_set_a @ A3 @ A )
=> ( ~ ( member_set_a @ B3 @ B )
=> ( ( ( insert_set_a @ A3 @ A )
= ( insert_set_a @ B3 @ B ) )
= ( ( ( A3 = B3 )
=> ( A = B ) )
& ( ( A3 != B3 )
=> ? [C5: set_set_a] :
( ( A
= ( insert_set_a @ B3 @ C5 ) )
& ~ ( member_set_a @ B3 @ C5 )
& ( B
= ( insert_set_a @ A3 @ C5 ) )
& ~ ( member_set_a @ A3 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_859_insert__eq__iff,axiom,
! [A3: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B3: pre_pr7278220950009878019t_unit,B: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ A3 @ A )
=> ( ~ ( member6939884229742472986t_unit @ B3 @ B )
=> ( ( ( insert6864688055023459379t_unit @ A3 @ A )
= ( insert6864688055023459379t_unit @ B3 @ B ) )
= ( ( ( A3 = B3 )
=> ( A = B ) )
& ( ( A3 != B3 )
=> ? [C5: set_pr5411798346947241657t_unit] :
( ( A
= ( insert6864688055023459379t_unit @ B3 @ C5 ) )
& ~ ( member6939884229742472986t_unit @ B3 @ C5 )
& ( B
= ( insert6864688055023459379t_unit @ A3 @ C5 ) )
& ~ ( member6939884229742472986t_unit @ A3 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_860_insert__eq__iff,axiom,
! [A3: b,A: set_b,B3: b,B: set_b] :
( ~ ( member_b @ A3 @ A )
=> ( ~ ( member_b @ B3 @ B )
=> ( ( ( insert_b @ A3 @ A )
= ( insert_b @ B3 @ B ) )
= ( ( ( A3 = B3 )
=> ( A = B ) )
& ( ( A3 != B3 )
=> ? [C5: set_b] :
( ( A
= ( insert_b @ B3 @ C5 ) )
& ~ ( member_b @ B3 @ C5 )
& ( B
= ( insert_b @ A3 @ C5 ) )
& ~ ( member_b @ A3 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_861_insert__commute,axiom,
! [X4: a,Y3: a,A: set_a] :
( ( insert_a @ X4 @ ( insert_a @ Y3 @ A ) )
= ( insert_a @ Y3 @ ( insert_a @ X4 @ A ) ) ) ).
% insert_commute
thf(fact_862_insert__commute,axiom,
! [X4: pre_pr7278220950009878019t_unit,Y3: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit] :
( ( insert6864688055023459379t_unit @ X4 @ ( insert6864688055023459379t_unit @ Y3 @ A ) )
= ( insert6864688055023459379t_unit @ Y3 @ ( insert6864688055023459379t_unit @ X4 @ A ) ) ) ).
% insert_commute
thf(fact_863_insert__commute,axiom,
! [X4: b,Y3: b,A: set_b] :
( ( insert_b @ X4 @ ( insert_b @ Y3 @ A ) )
= ( insert_b @ Y3 @ ( insert_b @ X4 @ A ) ) ) ).
% insert_commute
thf(fact_864_mk__disjoint__insert,axiom,
! [A3: a,A: set_a] :
( ( member_a @ A3 @ A )
=> ? [B5: set_a] :
( ( A
= ( insert_a @ A3 @ B5 ) )
& ~ ( member_a @ A3 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_865_mk__disjoint__insert,axiom,
! [A3: set_a,A: set_set_a] :
( ( member_set_a @ A3 @ A )
=> ? [B5: set_set_a] :
( ( A
= ( insert_set_a @ A3 @ B5 ) )
& ~ ( member_set_a @ A3 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_866_mk__disjoint__insert,axiom,
! [A3: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A3 @ A )
=> ? [B5: set_pr5411798346947241657t_unit] :
( ( A
= ( insert6864688055023459379t_unit @ A3 @ B5 ) )
& ~ ( member6939884229742472986t_unit @ A3 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_867_mk__disjoint__insert,axiom,
! [A3: b,A: set_b] :
( ( member_b @ A3 @ A )
=> ? [B5: set_b] :
( ( A
= ( insert_b @ A3 @ B5 ) )
& ~ ( member_b @ A3 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_868_singleton__inject,axiom,
! [A3: a,B3: a] :
( ( ( insert_a @ A3 @ bot_bot_set_a )
= ( insert_a @ B3 @ bot_bot_set_a ) )
=> ( A3 = B3 ) ) ).
% singleton_inject
thf(fact_869_singleton__inject,axiom,
! [A3: b,B3: b] :
( ( ( insert_b @ A3 @ bot_bot_set_b )
= ( insert_b @ B3 @ bot_bot_set_b ) )
=> ( A3 = B3 ) ) ).
% singleton_inject
thf(fact_870_singleton__inject,axiom,
! [A3: pre_pr7278220950009878019t_unit,B3: pre_pr7278220950009878019t_unit] :
( ( ( insert6864688055023459379t_unit @ A3 @ bot_bo1839476491465656141t_unit )
= ( insert6864688055023459379t_unit @ B3 @ bot_bo1839476491465656141t_unit ) )
=> ( A3 = B3 ) ) ).
% singleton_inject
thf(fact_871_singleton__inject,axiom,
! [A3: set_a,B3: set_a] :
( ( ( insert_set_a @ A3 @ bot_bot_set_set_a )
= ( insert_set_a @ B3 @ bot_bot_set_set_a ) )
=> ( A3 = B3 ) ) ).
% singleton_inject
thf(fact_872_insert__not__empty,axiom,
! [A3: a,A: set_a] :
( ( insert_a @ A3 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_873_insert__not__empty,axiom,
! [A3: b,A: set_b] :
( ( insert_b @ A3 @ A )
!= bot_bot_set_b ) ).
% insert_not_empty
thf(fact_874_insert__not__empty,axiom,
! [A3: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit] :
( ( insert6864688055023459379t_unit @ A3 @ A )
!= bot_bo1839476491465656141t_unit ) ).
% insert_not_empty
thf(fact_875_insert__not__empty,axiom,
! [A3: set_a,A: set_set_a] :
( ( insert_set_a @ A3 @ A )
!= bot_bot_set_set_a ) ).
% insert_not_empty
thf(fact_876_doubleton__eq__iff,axiom,
! [A3: a,B3: a,C: a,D: a] :
( ( ( insert_a @ A3 @ ( insert_a @ B3 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A3 = C )
& ( B3 = D ) )
| ( ( A3 = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_877_doubleton__eq__iff,axiom,
! [A3: b,B3: b,C: b,D: b] :
( ( ( insert_b @ A3 @ ( insert_b @ B3 @ bot_bot_set_b ) )
= ( insert_b @ C @ ( insert_b @ D @ bot_bot_set_b ) ) )
= ( ( ( A3 = C )
& ( B3 = D ) )
| ( ( A3 = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_878_doubleton__eq__iff,axiom,
! [A3: pre_pr7278220950009878019t_unit,B3: pre_pr7278220950009878019t_unit,C: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( ( insert6864688055023459379t_unit @ A3 @ ( insert6864688055023459379t_unit @ B3 @ bot_bo1839476491465656141t_unit ) )
= ( insert6864688055023459379t_unit @ C @ ( insert6864688055023459379t_unit @ D @ bot_bo1839476491465656141t_unit ) ) )
= ( ( ( A3 = C )
& ( B3 = D ) )
| ( ( A3 = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_879_doubleton__eq__iff,axiom,
! [A3: set_a,B3: set_a,C: set_a,D: set_a] :
( ( ( insert_set_a @ A3 @ ( insert_set_a @ B3 @ bot_bot_set_set_a ) )
= ( insert_set_a @ C @ ( insert_set_a @ D @ bot_bot_set_set_a ) ) )
= ( ( ( A3 = C )
& ( B3 = D ) )
| ( ( A3 = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_880_singleton__iff,axiom,
! [B3: a,A3: a] :
( ( member_a @ B3 @ ( insert_a @ A3 @ bot_bot_set_a ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_881_singleton__iff,axiom,
! [B3: b,A3: b] :
( ( member_b @ B3 @ ( insert_b @ A3 @ bot_bot_set_b ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_882_singleton__iff,axiom,
! [B3: pre_pr7278220950009878019t_unit,A3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ B3 @ ( insert6864688055023459379t_unit @ A3 @ bot_bo1839476491465656141t_unit ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_883_singleton__iff,axiom,
! [B3: set_a,A3: set_a] :
( ( member_set_a @ B3 @ ( insert_set_a @ A3 @ bot_bot_set_set_a ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_884_singletonD,axiom,
! [B3: a,A3: a] :
( ( member_a @ B3 @ ( insert_a @ A3 @ bot_bot_set_a ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_885_singletonD,axiom,
! [B3: b,A3: b] :
( ( member_b @ B3 @ ( insert_b @ A3 @ bot_bot_set_b ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_886_singletonD,axiom,
! [B3: pre_pr7278220950009878019t_unit,A3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ B3 @ ( insert6864688055023459379t_unit @ A3 @ bot_bo1839476491465656141t_unit ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_887_singletonD,axiom,
! [B3: set_a,A3: set_a] :
( ( member_set_a @ B3 @ ( insert_set_a @ A3 @ bot_bot_set_set_a ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_888_subset__singleton__iff,axiom,
! [X5: set_pr5411798346947241657t_unit,A3: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ X5 @ ( insert6864688055023459379t_unit @ A3 @ bot_bo1839476491465656141t_unit ) )
= ( ( X5 = bot_bo1839476491465656141t_unit )
| ( X5
= ( insert6864688055023459379t_unit @ A3 @ bot_bo1839476491465656141t_unit ) ) ) ) ).
% subset_singleton_iff
thf(fact_889_subset__singleton__iff,axiom,
! [X5: set_set_a,A3: set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ ( insert_set_a @ A3 @ bot_bot_set_set_a ) )
= ( ( X5 = bot_bot_set_set_a )
| ( X5
= ( insert_set_a @ A3 @ bot_bot_set_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_890_subset__singleton__iff,axiom,
! [X5: set_a,A3: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A3 @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_891_subset__singleton__iff,axiom,
! [X5: set_b,A3: b] :
( ( ord_less_eq_set_b @ X5 @ ( insert_b @ A3 @ bot_bot_set_b ) )
= ( ( X5 = bot_bot_set_b )
| ( X5
= ( insert_b @ A3 @ bot_bot_set_b ) ) ) ) ).
% subset_singleton_iff
thf(fact_892_subset__singletonD,axiom,
! [A: set_pr5411798346947241657t_unit,X4: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ A @ ( insert6864688055023459379t_unit @ X4 @ bot_bo1839476491465656141t_unit ) )
=> ( ( A = bot_bo1839476491465656141t_unit )
| ( A
= ( insert6864688055023459379t_unit @ X4 @ bot_bo1839476491465656141t_unit ) ) ) ) ).
% subset_singletonD
thf(fact_893_subset__singletonD,axiom,
! [A: set_set_a,X4: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) )
=> ( ( A = bot_bot_set_set_a )
| ( A
= ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_894_subset__singletonD,axiom,
! [A: set_a,X4: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X4 @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_895_subset__singletonD,axiom,
! [A: set_b,X4: b] :
( ( ord_less_eq_set_b @ A @ ( insert_b @ X4 @ bot_bot_set_b ) )
=> ( ( A = bot_bot_set_b )
| ( A
= ( insert_b @ X4 @ bot_bot_set_b ) ) ) ) ).
% subset_singletonD
thf(fact_896_pre__digraph_Oeuler__trail_Ocong,axiom,
pre_euler_trail_a_b = pre_euler_trail_a_b ).
% pre_digraph.euler_trail.cong
thf(fact_897_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_898_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_899_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_900_G_Ounvis__insert,axiom,
! [U: a,X4: a,U2: set_a] :
( ( graph_2016941059203891550ts_a_b @ g @ U @ ( insert_a @ X4 @ U2 ) )
= ( minus_minus_set_a @ ( graph_2016941059203891550ts_a_b @ g @ U @ U2 ) @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).
% G.unvis_insert
thf(fact_901_unvis__insert,axiom,
! [U: a,X4: a,U2: set_a] :
( ( graph_2016941059203891550ts_a_b @ t @ U @ ( insert_a @ X4 @ U2 ) )
= ( minus_minus_set_a @ ( graph_2016941059203891550ts_a_b @ t @ U @ U2 ) @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).
% unvis_insert
thf(fact_902_subgraph__del__vert,axiom,
! [U: a] : ( digraph_subgraph_a_b @ ( pre_del_vert_a_b @ t @ U ) @ t ) ).
% subgraph_del_vert
thf(fact_903_G_Osubgraph__del__vert,axiom,
! [U: a] : ( digraph_subgraph_a_b @ ( pre_del_vert_a_b @ g @ U ) @ g ) ).
% G.subgraph_del_vert
thf(fact_904_strongly__connected__eq__iff,axiom,
( ( digrap8691851296217657702ed_a_b @ t )
= ( ( digraph_pre_sccs_a_b @ t )
= ( insert6864688055023459379t_unit @ t @ bot_bo1839476491465656141t_unit ) ) ) ).
% strongly_connected_eq_iff
thf(fact_905_G_Ostrongly__connected__eq__iff,axiom,
( ( digrap8691851296217657702ed_a_b @ g )
= ( ( digraph_pre_sccs_a_b @ g )
= ( insert6864688055023459379t_unit @ g @ bot_bo1839476491465656141t_unit ) ) ) ).
% G.strongly_connected_eq_iff
thf(fact_906_fin__digraph__del__vert,axiom,
! [U: a] : ( fin_digraph_a_b @ ( pre_del_vert_a_b @ t @ U ) ) ).
% fin_digraph_del_vert
thf(fact_907_Diff__idemp,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ).
% Diff_idemp
thf(fact_908_Diff__idemp,axiom,
! [A: set_b,B: set_b] :
( ( minus_minus_set_b @ ( minus_minus_set_b @ A @ B ) @ B )
= ( minus_minus_set_b @ A @ B ) ) ).
% Diff_idemp
thf(fact_909_Diff__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
& ~ ( member_set_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_910_Diff__iff,axiom,
! [C: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C @ ( minus_3777555517894451474t_unit @ A @ B ) )
= ( ( member6939884229742472986t_unit @ C @ A )
& ~ ( member6939884229742472986t_unit @ C @ B ) ) ) ).
% Diff_iff
thf(fact_911_Diff__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_912_Diff__iff,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A @ B ) )
= ( ( member_b @ C @ A )
& ~ ( member_b @ C @ B ) ) ) ).
% Diff_iff
thf(fact_913_DiffI,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( ~ ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_914_DiffI,axiom,
! [C: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C @ A )
=> ( ~ ( member6939884229742472986t_unit @ C @ B )
=> ( member6939884229742472986t_unit @ C @ ( minus_3777555517894451474t_unit @ A @ B ) ) ) ) ).
% DiffI
thf(fact_915_DiffI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_916_DiffI,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ A )
=> ( ~ ( member_b @ C @ B )
=> ( member_b @ C @ ( minus_minus_set_b @ A @ B ) ) ) ) ).
% DiffI
thf(fact_917_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_918_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_919_Diff__cancel,axiom,
! [A: set_pr5411798346947241657t_unit] :
( ( minus_3777555517894451474t_unit @ A @ A )
= bot_bo1839476491465656141t_unit ) ).
% Diff_cancel
thf(fact_920_Diff__cancel,axiom,
! [A: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ A )
= bot_bot_set_set_a ) ).
% Diff_cancel
thf(fact_921_Diff__cancel,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ A )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_922_Diff__cancel,axiom,
! [A: set_b] :
( ( minus_minus_set_b @ A @ A )
= bot_bot_set_b ) ).
% Diff_cancel
thf(fact_923_empty__Diff,axiom,
! [A: set_pr5411798346947241657t_unit] :
( ( minus_3777555517894451474t_unit @ bot_bo1839476491465656141t_unit @ A )
= bot_bo1839476491465656141t_unit ) ).
% empty_Diff
thf(fact_924_empty__Diff,axiom,
! [A: set_set_a] :
( ( minus_5736297505244876581_set_a @ bot_bot_set_set_a @ A )
= bot_bot_set_set_a ) ).
% empty_Diff
thf(fact_925_empty__Diff,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_926_empty__Diff,axiom,
! [A: set_b] :
( ( minus_minus_set_b @ bot_bot_set_b @ A )
= bot_bot_set_b ) ).
% empty_Diff
thf(fact_927_Diff__empty,axiom,
! [A: set_pr5411798346947241657t_unit] :
( ( minus_3777555517894451474t_unit @ A @ bot_bo1839476491465656141t_unit )
= A ) ).
% Diff_empty
thf(fact_928_Diff__empty,axiom,
! [A: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ bot_bot_set_set_a )
= A ) ).
% Diff_empty
thf(fact_929_Diff__empty,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ bot_bot_set_a )
= A ) ).
% Diff_empty
thf(fact_930_Diff__empty,axiom,
! [A: set_b] :
( ( minus_minus_set_b @ A @ bot_bot_set_b )
= A ) ).
% Diff_empty
thf(fact_931_insert__Diff1,axiom,
! [X4: set_a,B: set_set_a,A: set_set_a] :
( ( member_set_a @ X4 @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X4 @ A ) @ B )
= ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_932_insert__Diff1,axiom,
! [X4: pre_pr7278220950009878019t_unit,B: set_pr5411798346947241657t_unit,A: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ X4 @ B )
=> ( ( minus_3777555517894451474t_unit @ ( insert6864688055023459379t_unit @ X4 @ A ) @ B )
= ( minus_3777555517894451474t_unit @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_933_insert__Diff1,axiom,
! [X4: a,B: set_a,A: set_a] :
( ( member_a @ X4 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X4 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_934_insert__Diff1,axiom,
! [X4: b,B: set_b,A: set_b] :
( ( member_b @ X4 @ B )
=> ( ( minus_minus_set_b @ ( insert_b @ X4 @ A ) @ B )
= ( minus_minus_set_b @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_935_Diff__insert0,axiom,
! [X4: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X4 @ A )
=> ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X4 @ B ) )
= ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_936_Diff__insert0,axiom,
! [X4: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ X4 @ A )
=> ( ( minus_3777555517894451474t_unit @ A @ ( insert6864688055023459379t_unit @ X4 @ B ) )
= ( minus_3777555517894451474t_unit @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_937_Diff__insert0,axiom,
! [X4: a,A: set_a,B: set_a] :
( ~ ( member_a @ X4 @ A )
=> ( ( minus_minus_set_a @ A @ ( insert_a @ X4 @ B ) )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_938_Diff__insert0,axiom,
! [X4: b,A: set_b,B: set_b] :
( ~ ( member_b @ X4 @ A )
=> ( ( minus_minus_set_b @ A @ ( insert_b @ X4 @ B ) )
= ( minus_minus_set_b @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_939_Diff__eq__empty__iff,axiom,
! [A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( ( minus_3777555517894451474t_unit @ A @ B )
= bot_bo1839476491465656141t_unit )
= ( ord_le8200006823705900825t_unit @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_940_Diff__eq__empty__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_941_Diff__eq__empty__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_942_Diff__eq__empty__iff,axiom,
! [A: set_b,B: set_b] :
( ( ( minus_minus_set_b @ A @ B )
= bot_bot_set_b )
= ( ord_less_eq_set_b @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_943_insert__Diff__single,axiom,
! [A3: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit] :
( ( insert6864688055023459379t_unit @ A3 @ ( minus_3777555517894451474t_unit @ A @ ( insert6864688055023459379t_unit @ A3 @ bot_bo1839476491465656141t_unit ) ) )
= ( insert6864688055023459379t_unit @ A3 @ A ) ) ).
% insert_Diff_single
thf(fact_944_insert__Diff__single,axiom,
! [A3: set_a,A: set_set_a] :
( ( insert_set_a @ A3 @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ A3 @ bot_bot_set_set_a ) ) )
= ( insert_set_a @ A3 @ A ) ) ).
% insert_Diff_single
thf(fact_945_insert__Diff__single,axiom,
! [A3: a,A: set_a] :
( ( insert_a @ A3 @ ( minus_minus_set_a @ A @ ( insert_a @ A3 @ bot_bot_set_a ) ) )
= ( insert_a @ A3 @ A ) ) ).
% insert_Diff_single
thf(fact_946_insert__Diff__single,axiom,
! [A3: b,A: set_b] :
( ( insert_b @ A3 @ ( minus_minus_set_b @ A @ ( insert_b @ A3 @ bot_bot_set_b ) ) )
= ( insert_b @ A3 @ A ) ) ).
% insert_Diff_single
thf(fact_947_Diff__disjoint,axiom,
! [A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ A @ ( minus_3777555517894451474t_unit @ B @ A ) )
= bot_bo1839476491465656141t_unit ) ).
% Diff_disjoint
thf(fact_948_Diff__disjoint,axiom,
! [A: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( minus_5736297505244876581_set_a @ B @ A ) )
= bot_bot_set_set_a ) ).
% Diff_disjoint
thf(fact_949_Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_950_Diff__disjoint,axiom,
! [A: set_b,B: set_b] :
( ( inf_inf_set_b @ A @ ( minus_minus_set_b @ B @ A ) )
= bot_bot_set_b ) ).
% Diff_disjoint
thf(fact_951_insert__Diff__if,axiom,
! [X4: set_a,B: set_set_a,A: set_set_a] :
( ( ( member_set_a @ X4 @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X4 @ A ) @ B )
= ( minus_5736297505244876581_set_a @ A @ B ) ) )
& ( ~ ( member_set_a @ X4 @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X4 @ A ) @ B )
= ( insert_set_a @ X4 @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_952_insert__Diff__if,axiom,
! [X4: pre_pr7278220950009878019t_unit,B: set_pr5411798346947241657t_unit,A: set_pr5411798346947241657t_unit] :
( ( ( member6939884229742472986t_unit @ X4 @ B )
=> ( ( minus_3777555517894451474t_unit @ ( insert6864688055023459379t_unit @ X4 @ A ) @ B )
= ( minus_3777555517894451474t_unit @ A @ B ) ) )
& ( ~ ( member6939884229742472986t_unit @ X4 @ B )
=> ( ( minus_3777555517894451474t_unit @ ( insert6864688055023459379t_unit @ X4 @ A ) @ B )
= ( insert6864688055023459379t_unit @ X4 @ ( minus_3777555517894451474t_unit @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_953_insert__Diff__if,axiom,
! [X4: a,B: set_a,A: set_a] :
( ( ( member_a @ X4 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X4 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) )
& ( ~ ( member_a @ X4 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X4 @ A ) @ B )
= ( insert_a @ X4 @ ( minus_minus_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_954_insert__Diff__if,axiom,
! [X4: b,B: set_b,A: set_b] :
( ( ( member_b @ X4 @ B )
=> ( ( minus_minus_set_b @ ( insert_b @ X4 @ A ) @ B )
= ( minus_minus_set_b @ A @ B ) ) )
& ( ~ ( member_b @ X4 @ B )
=> ( ( minus_minus_set_b @ ( insert_b @ X4 @ A ) @ B )
= ( insert_b @ X4 @ ( minus_minus_set_b @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_955_DiffD2,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ~ ( member_set_a @ C @ B ) ) ).
% DiffD2
thf(fact_956_DiffD2,axiom,
! [C: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C @ ( minus_3777555517894451474t_unit @ A @ B ) )
=> ~ ( member6939884229742472986t_unit @ C @ B ) ) ).
% DiffD2
thf(fact_957_DiffD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( member_a @ C @ B ) ) ).
% DiffD2
thf(fact_958_DiffD2,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A @ B ) )
=> ~ ( member_b @ C @ B ) ) ).
% DiffD2
thf(fact_959_DiffD1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ( member_set_a @ C @ A ) ) ).
% DiffD1
thf(fact_960_DiffD1,axiom,
! [C: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C @ ( minus_3777555517894451474t_unit @ A @ B ) )
=> ( member6939884229742472986t_unit @ C @ A ) ) ).
% DiffD1
thf(fact_961_DiffD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% DiffD1
thf(fact_962_DiffD1,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A @ B ) )
=> ( member_b @ C @ A ) ) ).
% DiffD1
thf(fact_963_DiffE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ~ ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% DiffE
thf(fact_964_DiffE,axiom,
! [C: pre_pr7278220950009878019t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C @ ( minus_3777555517894451474t_unit @ A @ B ) )
=> ~ ( ( member6939884229742472986t_unit @ C @ A )
=> ( member6939884229742472986t_unit @ C @ B ) ) ) ).
% DiffE
thf(fact_965_DiffE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% DiffE
thf(fact_966_DiffE,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A @ B ) )
=> ~ ( ( member_b @ C @ A )
=> ( member_b @ C @ B ) ) ) ).
% DiffE
thf(fact_967_Diff__Int__distrib2,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B ) @ C2 )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_968_Diff__Int__distrib2,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( inf_inf_set_b @ ( minus_minus_set_b @ A @ B ) @ C2 )
= ( minus_minus_set_b @ ( inf_inf_set_b @ A @ C2 ) @ ( inf_inf_set_b @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_969_Diff__Int__distrib,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C2 @ A ) @ ( inf_inf_set_a @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_970_Diff__Int__distrib,axiom,
! [C2: set_b,A: set_b,B: set_b] :
( ( inf_inf_set_b @ C2 @ ( minus_minus_set_b @ A @ B ) )
= ( minus_minus_set_b @ ( inf_inf_set_b @ C2 @ A ) @ ( inf_inf_set_b @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_971_Diff__Diff__Int,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ A @ ( minus_minus_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_972_Diff__Diff__Int,axiom,
! [A: set_b,B: set_b] :
( ( minus_minus_set_b @ A @ ( minus_minus_set_b @ A @ B ) )
= ( inf_inf_set_b @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_973_Diff__Int2,axiom,
! [A: set_a,C2: set_a,B: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ ( inf_inf_set_a @ B @ C2 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_974_Diff__Int2,axiom,
! [A: set_b,C2: set_b,B: set_b] :
( ( minus_minus_set_b @ ( inf_inf_set_b @ A @ C2 ) @ ( inf_inf_set_b @ B @ C2 ) )
= ( minus_minus_set_b @ ( inf_inf_set_b @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_975_Int__Diff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_976_Int__Diff,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( minus_minus_set_b @ ( inf_inf_set_b @ A @ B ) @ C2 )
= ( inf_inf_set_b @ A @ ( minus_minus_set_b @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_977_Diff__mono,axiom,
! [A: set_a,C2: set_a,D2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ D2 @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_978_Diff__mono,axiom,
! [A: set_b,C2: set_b,D2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ C2 )
=> ( ( ord_less_eq_set_b @ D2 @ B )
=> ( ord_less_eq_set_b @ ( minus_minus_set_b @ A @ B ) @ ( minus_minus_set_b @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_979_tree__del__vert,axiom,
! [V: a] :
( ( V != source )
=> ( ( shorte1213025427933718126af_a_b @ t @ V )
=> ( shorte3810566709427824352ee_a_b @ ( pre_del_vert_a_b @ t @ V ) @ source ) ) ) ).
% tree_del_vert
thf(fact_980_ex__in__arc,axiom,
! [V: a] :
( ( V != source )
=> ( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) )
=> ? [E: b] :
( ( in_arcs_a_b @ t @ V )
= ( insert_b @ E @ bot_bot_set_b ) ) ) ) ).
% ex_in_arc
thf(fact_981_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_982_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_983_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_984_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_985_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_986_directed__tree__axioms,axiom,
shorte3810566709427824352ee_a_b @ t @ source ).
% directed_tree_axioms
thf(fact_987_in__arcs__root,axiom,
( ( in_arcs_a_b @ t @ source )
= bot_bot_set_b ) ).
% in_arcs_root
thf(fact_988_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_989_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_990_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_991_arcs__del__leaf,axiom,
! [E2: b,V: a] :
( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( ( pre_he5236287464308401016t_unit @ t @ E2 )
= 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 @ E2 @ bot_bot_set_b ) ) ) ) ) ) ).
% arcs_del_leaf
thf(fact_992_two__in__arcs__contr,axiom,
! [E1: b,E22: b] :
( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( member_b @ E22 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( E1 != E22 )
=> ( ( pre_he5236287464308401016t_unit @ t @ E1 )
!= ( pre_he5236287464308401016t_unit @ t @ E22 ) ) ) ) ) ).
% two_in_arcs_contr
thf(fact_993_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_994_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_995_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_996_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_997_headT__eq__headG,axiom,
( ( pre_he5236287464308401016t_unit @ t )
= ( pre_he5236287464308401016t_unit @ g ) ) ).
% headT_eq_headG
thf(fact_998_head__in__verts,axiom,
! [E2: b] :
( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member_a @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% head_in_verts
thf(fact_999_G_Ohead__in__verts,axiom,
! [E2: b] :
( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( member_a @ ( pre_he5236287464308401016t_unit @ g @ E2 ) @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% G.head_in_verts
thf(fact_1000_G_Oin__arcs__del__arc__iff,axiom,
! [A3: b,U: a] :
( ( ( ( pre_he5236287464308401016t_unit @ g @ A3 )
= U )
=> ( ( in_arcs_a_b @ ( pre_del_arc_a_b @ g @ A3 ) @ U )
= ( minus_minus_set_b @ ( in_arcs_a_b @ g @ U ) @ ( insert_b @ A3 @ bot_bot_set_b ) ) ) )
& ( ( ( pre_he5236287464308401016t_unit @ g @ A3 )
!= U )
=> ( ( in_arcs_a_b @ ( pre_del_arc_a_b @ g @ A3 ) @ U )
= ( in_arcs_a_b @ g @ U ) ) ) ) ).
% G.in_arcs_del_arc_iff
thf(fact_1001_in__arcs__del__arc__iff,axiom,
! [A3: b,U: a] :
( ( ( ( pre_he5236287464308401016t_unit @ t @ A3 )
= U )
=> ( ( in_arcs_a_b @ ( pre_del_arc_a_b @ t @ A3 ) @ U )
= ( minus_minus_set_b @ ( in_arcs_a_b @ t @ U ) @ ( insert_b @ A3 @ bot_bot_set_b ) ) ) )
& ( ( ( pre_he5236287464308401016t_unit @ t @ A3 )
!= U )
=> ( ( in_arcs_a_b @ ( pre_del_arc_a_b @ t @ A3 ) @ U )
= ( in_arcs_a_b @ t @ U ) ) ) ) ).
% in_arcs_del_arc_iff
thf(fact_1002_G_Oin__arcs__add__arc__iff,axiom,
! [A3: b,U: a] :
( ( ( ( pre_he5236287464308401016t_unit @ g @ A3 )
= U )
=> ( ( in_arcs_a_b @ ( pre_add_arc_a_b @ g @ A3 ) @ U )
= ( insert_b @ A3 @ ( in_arcs_a_b @ g @ U ) ) ) )
& ( ( ( pre_he5236287464308401016t_unit @ g @ A3 )
!= U )
=> ( ( in_arcs_a_b @ ( pre_add_arc_a_b @ g @ A3 ) @ U )
= ( in_arcs_a_b @ g @ U ) ) ) ) ).
% G.in_arcs_add_arc_iff
thf(fact_1003_in__arcs__add__arc__iff,axiom,
! [A3: b,U: a] :
( ( ( ( pre_he5236287464308401016t_unit @ t @ A3 )
= U )
=> ( ( in_arcs_a_b @ ( pre_add_arc_a_b @ t @ A3 ) @ U )
= ( insert_b @ A3 @ ( in_arcs_a_b @ t @ U ) ) ) )
& ( ( ( pre_he5236287464308401016t_unit @ t @ A3 )
!= U )
=> ( ( in_arcs_a_b @ ( pre_add_arc_a_b @ t @ A3 ) @ U )
= ( in_arcs_a_b @ t @ U ) ) ) ) ).
% in_arcs_add_arc_iff
thf(fact_1004_del__arc__commute,axiom,
! [B3: b,A3: b] :
( ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ t @ B3 ) @ A3 )
= ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ t @ A3 ) @ B3 ) ) ).
% del_arc_commute
thf(fact_1005_add__arc__commute,axiom,
! [B3: b,A3: b] :
( ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ t @ B3 ) @ A3 )
= ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ t @ A3 ) @ B3 ) ) ).
% add_arc_commute
thf(fact_1006_G_Odel__arc__commute,axiom,
! [B3: b,A3: b] :
( ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ g @ B3 ) @ A3 )
= ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ g @ A3 ) @ B3 ) ) ).
% G.del_arc_commute
thf(fact_1007_G_Oadd__arc__commute,axiom,
! [B3: b,A3: b] :
( ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ g @ B3 ) @ A3 )
= ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ g @ A3 ) @ B3 ) ) ).
% G.add_arc_commute
thf(fact_1008_del__arc__in,axiom,
! [A3: b] :
( ~ ( member_b @ A3 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( pre_del_arc_a_b @ t @ A3 )
= t ) ) ).
% del_arc_in
thf(fact_1009_add__arc__in,axiom,
! [A3: b] :
( ( member_b @ A3 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( pre_add_arc_a_b @ t @ A3 )
= t ) ) ).
% add_arc_in
thf(fact_1010_G_Odel__arc__in,axiom,
! [A3: b] :
( ~ ( member_b @ A3 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( pre_del_arc_a_b @ g @ A3 )
= g ) ) ).
% G.del_arc_in
thf(fact_1011_G_Oadd__arc__in,axiom,
! [A3: b] :
( ( member_b @ A3 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( pre_add_arc_a_b @ g @ A3 )
= g ) ) ).
% G.add_arc_in
thf(fact_1012_subgraph__del__arc,axiom,
! [A3: b] : ( digraph_subgraph_a_b @ ( pre_del_arc_a_b @ t @ A3 ) @ t ) ).
% subgraph_del_arc
thf(fact_1013_G_Osubgraph__del__arc,axiom,
! [A3: b] : ( digraph_subgraph_a_b @ ( pre_del_arc_a_b @ g @ A3 ) @ g ) ).
% G.subgraph_del_arc
thf(fact_1014_fin__digraph__del__arc,axiom,
! [A3: b] : ( fin_digraph_a_b @ ( pre_del_arc_a_b @ t @ A3 ) ) ).
% fin_digraph_del_arc
thf(fact_1015_del__del__arc__collapse,axiom,
! [A3: b] :
( ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ t @ A3 ) @ A3 )
= ( pre_del_arc_a_b @ t @ A3 ) ) ).
% del_del_arc_collapse
thf(fact_1016_add__add__arc__collapse,axiom,
! [A3: b] :
( ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ t @ A3 ) @ A3 )
= ( pre_add_arc_a_b @ t @ A3 ) ) ).
% add_add_arc_collapse
thf(fact_1017_G_Odel__del__arc__collapse,axiom,
! [A3: b] :
( ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ g @ A3 ) @ A3 )
= ( pre_del_arc_a_b @ g @ A3 ) ) ).
% G.del_del_arc_collapse
thf(fact_1018_G_Oadd__add__arc__collapse,axiom,
! [A3: b] :
( ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ g @ A3 ) @ A3 )
= ( pre_add_arc_a_b @ g @ A3 ) ) ).
% G.add_add_arc_collapse
thf(fact_1019_verts__del__arc,axiom,
! [A3: b] :
( ( pre_ve642382030648772252t_unit @ ( pre_del_arc_a_b @ t @ A3 ) )
= ( pre_ve642382030648772252t_unit @ t ) ) ).
% verts_del_arc
thf(fact_1020_G_Overts__del__arc,axiom,
! [A3: b] :
( ( pre_ve642382030648772252t_unit @ ( pre_del_arc_a_b @ g @ A3 ) )
= ( pre_ve642382030648772252t_unit @ g ) ) ).
% G.verts_del_arc
thf(fact_1021_head__del__arc,axiom,
! [A3: b] :
( ( pre_he5236287464308401016t_unit @ ( pre_del_arc_a_b @ t @ A3 ) )
= ( pre_he5236287464308401016t_unit @ t ) ) ).
% head_del_arc
thf(fact_1022_head__add__arc,axiom,
! [A3: b] :
( ( pre_he5236287464308401016t_unit @ ( pre_add_arc_a_b @ t @ A3 ) )
= ( pre_he5236287464308401016t_unit @ t ) ) ).
% head_add_arc
thf(fact_1023_G_Ohead__del__arc,axiom,
! [A3: b] :
( ( pre_he5236287464308401016t_unit @ ( pre_del_arc_a_b @ g @ A3 ) )
= ( pre_he5236287464308401016t_unit @ g ) ) ).
% G.head_del_arc
thf(fact_1024_G_Ohead__add__arc,axiom,
! [A3: b] :
( ( pre_he5236287464308401016t_unit @ ( pre_add_arc_a_b @ g @ A3 ) )
= ( pre_he5236287464308401016t_unit @ g ) ) ).
% G.head_add_arc
thf(fact_1025_add__del__arc__collapse,axiom,
! [A3: b] :
( ( pre_add_arc_a_b @ ( pre_del_arc_a_b @ t @ A3 ) @ A3 )
= ( pre_add_arc_a_b @ t @ A3 ) ) ).
% add_del_arc_collapse
thf(fact_1026_G_Oadd__del__arc__collapse,axiom,
! [A3: b] :
( ( pre_add_arc_a_b @ ( pre_del_arc_a_b @ g @ A3 ) @ A3 )
= ( pre_add_arc_a_b @ g @ A3 ) ) ).
% G.add_del_arc_collapse
thf(fact_1027_arcs__add__arc,axiom,
! [A3: b] :
( ( pre_ar1395965042833527383t_unit @ ( pre_add_arc_a_b @ t @ A3 ) )
= ( insert_b @ A3 @ ( pre_ar1395965042833527383t_unit @ t ) ) ) ).
% arcs_add_arc
thf(fact_1028_G_Oarcs__add__arc,axiom,
! [A3: b] :
( ( pre_ar1395965042833527383t_unit @ ( pre_add_arc_a_b @ g @ A3 ) )
= ( insert_b @ A3 @ ( pre_ar1395965042833527383t_unit @ g ) ) ) ).
% G.arcs_add_arc
thf(fact_1029_arcs__del__arc,axiom,
! [A3: b] :
( ( pre_ar1395965042833527383t_unit @ ( pre_del_arc_a_b @ t @ A3 ) )
= ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ t ) @ ( insert_b @ A3 @ bot_bot_set_b ) ) ) ).
% arcs_del_arc
thf(fact_1030_G_Oarcs__del__arc,axiom,
! [A3: b] :
( ( pre_ar1395965042833527383t_unit @ ( pre_del_arc_a_b @ g @ A3 ) )
= ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ g ) @ ( insert_b @ A3 @ bot_bot_set_b ) ) ) ).
% G.arcs_del_arc
thf(fact_1031_G_Oout__arcs__del__arc__iff,axiom,
! [A3: b,U: a] :
( ( ( ( pre_ta4931606617599662728t_unit @ g @ A3 )
= U )
=> ( ( out_arcs_a_b @ ( pre_del_arc_a_b @ g @ A3 ) @ U )
= ( minus_minus_set_b @ ( out_arcs_a_b @ g @ U ) @ ( insert_b @ A3 @ bot_bot_set_b ) ) ) )
& ( ( ( pre_ta4931606617599662728t_unit @ g @ A3 )
!= U )
=> ( ( out_arcs_a_b @ ( pre_del_arc_a_b @ g @ A3 ) @ U )
= ( out_arcs_a_b @ g @ U ) ) ) ) ).
% G.out_arcs_del_arc_iff
thf(fact_1032_out__arcs__del__arc__iff,axiom,
! [A3: b,U: a] :
( ( ( ( pre_ta4931606617599662728t_unit @ t @ A3 )
= U )
=> ( ( out_arcs_a_b @ ( pre_del_arc_a_b @ t @ A3 ) @ U )
= ( minus_minus_set_b @ ( out_arcs_a_b @ t @ U ) @ ( insert_b @ A3 @ bot_bot_set_b ) ) ) )
& ( ( ( pre_ta4931606617599662728t_unit @ t @ A3 )
!= U )
=> ( ( out_arcs_a_b @ ( pre_del_arc_a_b @ t @ A3 ) @ U )
= ( out_arcs_a_b @ t @ U ) ) ) ) ).
% out_arcs_del_arc_iff
thf(fact_1033_connected__minimal,axiom,
! [E2: b] :
( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ~ ( reachable_a_b @ ( pre_del_arc_a_b @ t @ E2 ) @ ( pre_ta4931606617599662728t_unit @ t @ E2 ) @ ( pre_he5236287464308401016t_unit @ t @ E2 ) ) ) ).
% connected_minimal
thf(fact_1034_G_Oout__arcs__add__arc__iff,axiom,
! [A3: b,U: a] :
( ( ( ( pre_ta4931606617599662728t_unit @ g @ A3 )
= U )
=> ( ( out_arcs_a_b @ ( pre_add_arc_a_b @ g @ A3 ) @ U )
= ( insert_b @ A3 @ ( out_arcs_a_b @ g @ U ) ) ) )
& ( ( ( pre_ta4931606617599662728t_unit @ g @ A3 )
!= U )
=> ( ( out_arcs_a_b @ ( pre_add_arc_a_b @ g @ A3 ) @ U )
= ( out_arcs_a_b @ g @ U ) ) ) ) ).
% G.out_arcs_add_arc_iff
thf(fact_1035_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_1036_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_1037_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_1038_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_1039_tailT__eq__tailG,axiom,
( ( pre_ta4931606617599662728t_unit @ t )
= ( pre_ta4931606617599662728t_unit @ g ) ) ).
% tailT_eq_tailG
thf(fact_1040_tail__in__verts,axiom,
! [E2: b] :
( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E2 ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% tail_in_verts
thf(fact_1041_G_Otail__in__verts,axiom,
! [E2: b] :
( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( member_a @ ( pre_ta4931606617599662728t_unit @ g @ E2 ) @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% G.tail_in_verts
thf(fact_1042_nomulti_Ono__multi__alt,axiom,
! [E1: b,E22: b] :
( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( member_b @ E22 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( E1 != E22 )
=> ( ( ( pre_he5236287464308401016t_unit @ t @ E1 )
!= ( pre_he5236287464308401016t_unit @ t @ E22 ) )
| ( ( pre_ta4931606617599662728t_unit @ t @ E1 )
!= ( pre_ta4931606617599662728t_unit @ t @ E22 ) ) ) ) ) ) ).
% nomulti.no_multi_alt
thf(fact_1043_loopfree_Ono__loops,axiom,
! [E2: b] :
( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( pre_ta4931606617599662728t_unit @ t @ E2 )
!= ( pre_he5236287464308401016t_unit @ t @ E2 ) ) ) ).
% loopfree.no_loops
thf(fact_1044_out__arcs__add__arc__iff,axiom,
! [A3: b,U: a] :
( ( ( ( pre_ta4931606617599662728t_unit @ t @ A3 )
= U )
=> ( ( out_arcs_a_b @ ( pre_add_arc_a_b @ t @ A3 ) @ U )
= ( insert_b @ A3 @ ( out_arcs_a_b @ t @ U ) ) ) )
& ( ( ( pre_ta4931606617599662728t_unit @ t @ A3 )
!= U )
=> ( ( out_arcs_a_b @ ( pre_add_arc_a_b @ t @ A3 ) @ U )
= ( out_arcs_a_b @ t @ U ) ) ) ) ).
% out_arcs_add_arc_iff
thf(fact_1045_tail__del__arc,axiom,
! [A3: b] :
( ( pre_ta4931606617599662728t_unit @ ( pre_del_arc_a_b @ t @ A3 ) )
= ( pre_ta4931606617599662728t_unit @ t ) ) ).
% tail_del_arc
thf(fact_1046_tail__add__arc,axiom,
! [A3: b] :
( ( pre_ta4931606617599662728t_unit @ ( pre_add_arc_a_b @ t @ A3 ) )
= ( pre_ta4931606617599662728t_unit @ t ) ) ).
% tail_add_arc
thf(fact_1047_G_Otail__del__arc,axiom,
! [A3: b] :
( ( pre_ta4931606617599662728t_unit @ ( pre_del_arc_a_b @ g @ A3 ) )
= ( pre_ta4931606617599662728t_unit @ g ) ) ).
% G.tail_del_arc
thf(fact_1048_G_Otail__add__arc,axiom,
! [A3: b] :
( ( pre_ta4931606617599662728t_unit @ ( pre_add_arc_a_b @ g @ A3 ) )
= ( pre_ta4931606617599662728t_unit @ g ) ) ).
% G.tail_add_arc
thf(fact_1049_verts__add__arc,axiom,
! [A3: b] :
( ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ A3 ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( member_a @ ( pre_he5236287464308401016t_unit @ t @ A3 ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( pre_ve642382030648772252t_unit @ ( pre_add_arc_a_b @ t @ A3 ) )
= ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% verts_add_arc
thf(fact_1050_G_Overts__add__arc,axiom,
! [A3: b] :
( ( member_a @ ( pre_ta4931606617599662728t_unit @ g @ A3 ) @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( member_a @ ( pre_he5236287464308401016t_unit @ g @ A3 ) @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( pre_ve642382030648772252t_unit @ ( pre_add_arc_a_b @ g @ A3 ) )
= ( pre_ve642382030648772252t_unit @ g ) ) ) ) ).
% G.verts_add_arc
thf(fact_1051_del__add__arc__collapse,axiom,
! [A3: b] :
( ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ A3 ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( member_a @ ( pre_he5236287464308401016t_unit @ t @ A3 ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( pre_del_arc_a_b @ ( pre_add_arc_a_b @ t @ A3 ) @ A3 )
= ( pre_del_arc_a_b @ t @ A3 ) ) ) ) ).
% del_add_arc_collapse
thf(fact_1052_G_Odel__add__arc__collapse,axiom,
! [A3: b] :
( ( member_a @ ( pre_ta4931606617599662728t_unit @ g @ A3 ) @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( member_a @ ( pre_he5236287464308401016t_unit @ g @ A3 ) @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( pre_del_arc_a_b @ ( pre_add_arc_a_b @ g @ A3 ) @ A3 )
= ( pre_del_arc_a_b @ g @ A3 ) ) ) ) ).
% G.del_add_arc_collapse
thf(fact_1053_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_1054_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_1055_G_Overts__add__arc__conv,axiom,
! [A3: b] :
( ( pre_ve642382030648772252t_unit @ ( pre_add_arc_a_b @ g @ A3 ) )
= ( sup_sup_set_a @ ( pre_ve642382030648772252t_unit @ g ) @ ( insert_a @ ( pre_ta4931606617599662728t_unit @ g @ A3 ) @ ( insert_a @ ( pre_he5236287464308401016t_unit @ g @ A3 ) @ bot_bot_set_a ) ) ) ) ).
% G.verts_add_arc_conv
thf(fact_1056_verts__add__arc__conv,axiom,
! [A3: b] :
( ( pre_ve642382030648772252t_unit @ ( pre_add_arc_a_b @ t @ A3 ) )
= ( sup_sup_set_a @ ( pre_ve642382030648772252t_unit @ t ) @ ( insert_a @ ( pre_ta4931606617599662728t_unit @ t @ A3 ) @ ( insert_a @ ( pre_he5236287464308401016t_unit @ t @ A3 ) @ bot_bot_set_a ) ) ) ) ).
% verts_add_arc_conv
thf(fact_1057_G_Obidirected__digraphI,axiom,
! [Arev: b > b] :
( ! [A5: b] :
( ~ ( member_b @ A5 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( Arev @ A5 )
= A5 ) )
=> ( ! [A5: b] :
( ( member_b @ A5 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( Arev @ A5 )
!= A5 ) )
=> ( ! [A5: b] :
( ( member_b @ A5 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( Arev @ ( Arev @ A5 ) )
= A5 ) )
=> ( ! [A5: b] :
( ( member_b @ A5 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( pre_ta4931606617599662728t_unit @ g @ ( Arev @ A5 ) )
= ( pre_he5236287464308401016t_unit @ g @ A5 ) ) )
=> ( bidire6463457107099887885ph_a_b @ g @ Arev ) ) ) ) ) ).
% G.bidirected_digraphI
thf(fact_1058_bidirected__digraphI,axiom,
! [Arev: b > b] :
( ! [A5: b] :
( ~ ( member_b @ A5 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( Arev @ A5 )
= A5 ) )
=> ( ! [A5: b] :
( ( member_b @ A5 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( Arev @ A5 )
!= A5 ) )
=> ( ! [A5: b] :
( ( member_b @ A5 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( Arev @ ( Arev @ A5 ) )
= A5 ) )
=> ( ! [A5: b] :
( ( member_b @ A5 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( pre_ta4931606617599662728t_unit @ t @ ( Arev @ A5 ) )
= ( pre_he5236287464308401016t_unit @ t @ A5 ) ) )
=> ( bidire6463457107099887885ph_a_b @ t @ Arev ) ) ) ) ) ).
% bidirected_digraphI
thf(fact_1059_G_Oawalk__Cons__iff,axiom,
! [U: a,E2: b,Es: list_b,W: a] :
( ( arc_pre_awalk_a_b @ g @ U @ ( cons_b @ E2 @ Es ) @ W )
= ( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ g ) )
& ( U
= ( pre_ta4931606617599662728t_unit @ g @ E2 ) )
& ( arc_pre_awalk_a_b @ g @ ( pre_he5236287464308401016t_unit @ g @ E2 ) @ Es @ W ) ) ) ).
% G.awalk_Cons_iff
thf(fact_1060_awalk__Cons__iff,axiom,
! [U: a,E2: b,Es: list_b,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ ( cons_b @ E2 @ Es ) @ W )
= ( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( U
= ( pre_ta4931606617599662728t_unit @ t @ E2 ) )
& ( arc_pre_awalk_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ Es @ W ) ) ) ).
% awalk_Cons_iff
thf(fact_1061_G_Oarc__implies__awalk,axiom,
! [E2: b] :
( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( arc_pre_awalk_a_b @ g @ ( pre_ta4931606617599662728t_unit @ g @ E2 ) @ ( cons_b @ E2 @ nil_b ) @ ( pre_he5236287464308401016t_unit @ g @ E2 ) ) ) ).
% G.arc_implies_awalk
thf(fact_1062_arc__implies__awalk,axiom,
! [E2: b] :
( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( arc_pre_awalk_a_b @ t @ ( pre_ta4931606617599662728t_unit @ t @ E2 ) @ ( cons_b @ E2 @ nil_b ) @ ( pre_he5236287464308401016t_unit @ t @ E2 ) ) ) ).
% arc_implies_awalk
thf(fact_1063_awalk__empty__ends,axiom,
! [U: a,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ nil_b @ V )
=> ( U = V ) ) ).
% awalk_empty_ends
thf(fact_1064_awalk__ends,axiom,
! [U: a,P2: list_b,V: a,U5: a,V6: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( arc_pre_awalk_a_b @ t @ U5 @ P2 @ V6 )
=> ( ( ( P2 != nil_b )
& ( U = U5 )
& ( V = V6 ) )
| ( ( P2 = nil_b )
& ( U = V )
& ( U5 = V6 ) ) ) ) ) ).
% awalk_ends
thf(fact_1065_G_Oawalk__ends,axiom,
! [U: a,P2: list_b,V: a,U5: a,V6: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( ( arc_pre_awalk_a_b @ g @ U5 @ P2 @ V6 )
=> ( ( ( P2 != nil_b )
& ( U = U5 )
& ( V = V6 ) )
| ( ( P2 = nil_b )
& ( U = V )
& ( U5 = V6 ) ) ) ) ) ).
% G.awalk_ends
thf(fact_1066_G_Oawalk__empty__ends,axiom,
! [U: a,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ nil_b @ V )
=> ( U = V ) ) ).
% G.awalk_empty_ends
thf(fact_1067_awalk__Nil__iff,axiom,
! [U: a,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ nil_b @ V )
= ( ( U = V )
& ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% awalk_Nil_iff
thf(fact_1068_G_Oawalk__Nil__iff,axiom,
! [U: a,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ nil_b @ V )
= ( ( U = V )
& ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) ) ) ) ).
% G.awalk_Nil_iff
thf(fact_1069_G_Ocas_Oelims_I3_J,axiom,
! [X4: a,Xa2: list_b,Xb: a] :
( ~ ( arc_pre_cas_a_b @ g @ X4 @ Xa2 @ Xb )
=> ( ( ( Xa2 = nil_b )
=> ( X4 = Xb ) )
=> ~ ! [E: b,Es2: list_b] :
( ( Xa2
= ( cons_b @ E @ Es2 ) )
=> ( ( ( pre_ta4931606617599662728t_unit @ g @ E )
= X4 )
& ( arc_pre_cas_a_b @ g @ ( pre_he5236287464308401016t_unit @ g @ E ) @ Es2 @ Xb ) ) ) ) ) ).
% G.cas.elims(3)
thf(fact_1070_G_Ocas_Oelims_I2_J,axiom,
! [X4: a,Xa2: list_b,Xb: a] :
( ( arc_pre_cas_a_b @ g @ X4 @ Xa2 @ Xb )
=> ( ( ( Xa2 = nil_b )
=> ( X4 != Xb ) )
=> ~ ! [E: b,Es2: list_b] :
( ( Xa2
= ( cons_b @ E @ Es2 ) )
=> ~ ( ( ( pre_ta4931606617599662728t_unit @ g @ E )
= X4 )
& ( arc_pre_cas_a_b @ g @ ( pre_he5236287464308401016t_unit @ g @ E ) @ Es2 @ Xb ) ) ) ) ) ).
% G.cas.elims(2)
thf(fact_1071_cas_Osimps_I1_J,axiom,
! [U: a,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ nil_b @ V )
= ( U = V ) ) ).
% cas.simps(1)
thf(fact_1072_cas__ends,axiom,
! [U: a,P2: list_b,V: a,U5: a,V6: a] :
( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( arc_pre_cas_a_b @ t @ U5 @ P2 @ V6 )
=> ( ( ( P2 != nil_b )
& ( U = U5 )
& ( V = V6 ) )
| ( ( P2 = nil_b )
& ( U = V )
& ( U5 = V6 ) ) ) ) ) ).
% cas_ends
thf(fact_1073_G_Ocas__ends,axiom,
! [U: a,P2: list_b,V: a,U5: a,V6: a] :
( ( arc_pre_cas_a_b @ g @ U @ P2 @ V )
=> ( ( arc_pre_cas_a_b @ g @ U5 @ P2 @ V6 )
=> ( ( ( P2 != nil_b )
& ( U = U5 )
& ( V = V6 ) )
| ( ( P2 = nil_b )
& ( U = V )
& ( U5 = V6 ) ) ) ) ) ).
% G.cas_ends
thf(fact_1074_G_Ocas_Osimps_I1_J,axiom,
! [U: a,V: a] :
( ( arc_pre_cas_a_b @ g @ U @ nil_b @ V )
= ( U = V ) ) ).
% G.cas.simps(1)
thf(fact_1075_cas_Osimps_I2_J,axiom,
! [U: a,E2: b,Es: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ ( cons_b @ E2 @ Es ) @ V )
= ( ( ( pre_ta4931606617599662728t_unit @ t @ E2 )
= U )
& ( arc_pre_cas_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ Es @ V ) ) ) ).
% cas.simps(2)
thf(fact_1076_G_Ocas_Osimps_I2_J,axiom,
! [U: a,E2: b,Es: list_b,V: a] :
( ( arc_pre_cas_a_b @ g @ U @ ( cons_b @ E2 @ Es ) @ V )
= ( ( ( pre_ta4931606617599662728t_unit @ g @ E2 )
= U )
& ( arc_pre_cas_a_b @ g @ ( pre_he5236287464308401016t_unit @ g @ E2 ) @ Es @ V ) ) ) ).
% G.cas.simps(2)
thf(fact_1077_cas_Oelims_I3_J,axiom,
! [X4: a,Xa2: list_b,Xb: a] :
( ~ ( arc_pre_cas_a_b @ t @ X4 @ Xa2 @ Xb )
=> ( ( ( Xa2 = nil_b )
=> ( X4 = Xb ) )
=> ~ ! [E: b,Es2: list_b] :
( ( Xa2
= ( cons_b @ E @ Es2 ) )
=> ( ( ( pre_ta4931606617599662728t_unit @ t @ E )
= X4 )
& ( arc_pre_cas_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E ) @ Es2 @ Xb ) ) ) ) ) ).
% cas.elims(3)
thf(fact_1078_cas_Oelims_I2_J,axiom,
! [X4: a,Xa2: list_b,Xb: a] :
( ( arc_pre_cas_a_b @ t @ X4 @ Xa2 @ Xb )
=> ( ( ( Xa2 = nil_b )
=> ( X4 != Xb ) )
=> ~ ! [E: b,Es2: list_b] :
( ( Xa2
= ( cons_b @ E @ Es2 ) )
=> ~ ( ( ( pre_ta4931606617599662728t_unit @ t @ E )
= X4 )
& ( arc_pre_cas_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E ) @ Es2 @ Xb ) ) ) ) ) ).
% cas.elims(2)
thf(fact_1079_cas_Oelims_I1_J,axiom,
! [X4: a,Xa2: list_b,Xb: a,Y3: $o] :
( ( ( arc_pre_cas_a_b @ t @ X4 @ Xa2 @ Xb )
= Y3 )
=> ( ( ( Xa2 = nil_b )
=> ( Y3
= ( X4 != Xb ) ) )
=> ~ ! [E: b,Es2: list_b] :
( ( Xa2
= ( cons_b @ E @ Es2 ) )
=> ( Y3
= ( ~ ( ( ( pre_ta4931606617599662728t_unit @ t @ E )
= X4 )
& ( arc_pre_cas_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E ) @ Es2 @ Xb ) ) ) ) ) ) ) ).
% cas.elims(1)
thf(fact_1080_G_Ocas_Oelims_I1_J,axiom,
! [X4: a,Xa2: list_b,Xb: a,Y3: $o] :
( ( ( arc_pre_cas_a_b @ g @ X4 @ Xa2 @ Xb )
= Y3 )
=> ( ( ( Xa2 = nil_b )
=> ( Y3
= ( X4 != Xb ) ) )
=> ~ ! [E: b,Es2: list_b] :
( ( Xa2
= ( cons_b @ E @ Es2 ) )
=> ( Y3
= ( ~ ( ( ( pre_ta4931606617599662728t_unit @ g @ E )
= X4 )
& ( arc_pre_cas_a_b @ g @ ( pre_he5236287464308401016t_unit @ g @ E ) @ Es2 @ Xb ) ) ) ) ) ) ) ).
% G.cas.elims(1)
thf(fact_1081_G_Oawalk__def,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
= ( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
& ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ g ) )
& ( arc_pre_cas_a_b @ g @ U @ P2 @ V ) ) ) ).
% G.awalk_def
thf(fact_1082_awalk__def,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
= ( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
& ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( arc_pre_cas_a_b @ t @ U @ P2 @ V ) ) ) ).
% awalk_def
thf(fact_1083_All__arcs__in__path,axiom,
! [E2: b] :
( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ? [P5: list_b,U3: a,V4: a] :
( ( arc_pre_awalk_a_b @ t @ U3 @ P5 @ V4 )
& ( member_b @ E2 @ ( set_b2 @ P5 ) ) ) ) ).
% All_arcs_in_path
thf(fact_1084_tail__and__head__eq__impl__cas,axiom,
! [X4: a,P2: list_b,Y3: a,G2: pre_pr7278220950009878019t_unit] :
( ( arc_pre_cas_a_b @ t @ X4 @ P2 @ Y3 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ ( set_b2 @ P2 ) )
=> ( ( pre_ta4931606617599662728t_unit @ t @ X3 )
= ( pre_ta4931606617599662728t_unit @ G2 @ X3 ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ ( set_b2 @ P2 ) )
=> ( ( pre_he5236287464308401016t_unit @ t @ X3 )
= ( pre_he5236287464308401016t_unit @ G2 @ X3 ) ) )
=> ( arc_pre_cas_a_b @ G2 @ X4 @ P2 @ Y3 ) ) ) ) ).
% tail_and_head_eq_impl_cas
thf(fact_1085_G_Otail__and__head__eq__impl__cas,axiom,
! [X4: a,P2: list_b,Y3: a,G2: pre_pr7278220950009878019t_unit] :
( ( arc_pre_cas_a_b @ g @ X4 @ P2 @ Y3 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ ( set_b2 @ P2 ) )
=> ( ( pre_ta4931606617599662728t_unit @ g @ X3 )
= ( pre_ta4931606617599662728t_unit @ G2 @ X3 ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ ( set_b2 @ P2 ) )
=> ( ( pre_he5236287464308401016t_unit @ g @ X3 )
= ( pre_he5236287464308401016t_unit @ G2 @ X3 ) ) )
=> ( arc_pre_cas_a_b @ G2 @ X4 @ P2 @ Y3 ) ) ) ) ).
% G.tail_and_head_eq_impl_cas
thf(fact_1086_G_Oawalk__connected,axiom,
! [U: a,P2: list_b,V: a] :
( ( digrap8783888973171253482ed_a_b @ g )
=> ( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( ( ( set_b2 @ P2 )
!= ( pre_ar1395965042833527383t_unit @ g ) )
=> ? [E: b] :
( ( member_b @ E @ ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ g ) @ ( set_b2 @ P2 ) ) )
& ( ( member_a @ ( pre_ta4931606617599662728t_unit @ g @ E ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) )
| ( member_a @ ( pre_he5236287464308401016t_unit @ g @ E ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) ) ) ) ) ) ) ).
% G.awalk_connected
thf(fact_1087_awalk__connected,axiom,
! [U: a,P2: list_b,V: a] :
( ( digrap8783888973171253482ed_a_b @ t )
=> ( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( ( set_b2 @ P2 )
!= ( pre_ar1395965042833527383t_unit @ t ) )
=> ? [E: b] :
( ( member_b @ E @ ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ t ) @ ( set_b2 @ P2 ) ) )
& ( ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
| ( member_a @ ( pre_he5236287464308401016t_unit @ t @ E ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ) ) ) ) ).
% awalk_connected
thf(fact_1088_awalk__verts__non__Nil,axiom,
! [U: a,P2: list_b] :
( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 )
!= nil_a ) ).
% awalk_verts_non_Nil
thf(fact_1089_G_Oawalk__verts__non__Nil,axiom,
! [U: a,P2: list_b] :
( ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 )
!= nil_a ) ).
% G.awalk_verts_non_Nil
thf(fact_1090_awalk__verts_Osimps_I1_J,axiom,
! [U: a] :
( ( arc_pr7493981781705774526ts_a_b @ t @ U @ nil_b )
= ( cons_a @ U @ nil_a ) ) ).
% awalk_verts.simps(1)
thf(fact_1091_awalk__verts__ne__eq,axiom,
! [P2: list_b,U: a,V: a] :
( ( P2 != nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 )
= ( arc_pr7493981781705774526ts_a_b @ t @ V @ P2 ) ) ) ).
% awalk_verts_ne_eq
thf(fact_1092_G_Oawalk__verts_Osimps_I1_J,axiom,
! [U: a] :
( ( arc_pr7493981781705774526ts_a_b @ g @ U @ nil_b )
= ( cons_a @ U @ nil_a ) ) ).
% G.awalk_verts.simps(1)
thf(fact_1093_G_Oawalk__verts__ne__eq,axiom,
! [P2: list_b,U: a,V: a] :
( ( P2 != nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 )
= ( arc_pr7493981781705774526ts_a_b @ g @ V @ P2 ) ) ) ).
% G.awalk_verts_ne_eq
thf(fact_1094_awalk__verts__induce,axiom,
! [S: set_a] :
( ( arc_pr7493981781705774526ts_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S ) )
= ( arc_pr7493981781705774526ts_a_b @ t ) ) ).
% awalk_verts_induce
thf(fact_1095_G_Oawalk__verts__induce,axiom,
! [S: set_a] :
( ( arc_pr7493981781705774526ts_a_b @ ( digrap7873285959652527175ph_a_b @ g @ S ) )
= ( arc_pr7493981781705774526ts_a_b @ g ) ) ).
% G.awalk_verts_induce
thf(fact_1096_hd__in__awalk__verts_I1_J,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( member_a @ U @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ).
% hd_in_awalk_verts(1)
thf(fact_1097_G_Ohd__in__awalk__verts_I1_J,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( member_a @ U @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) ) ) ).
% G.hd_in_awalk_verts(1)
thf(fact_1098_awalk__verts__reachable__to,axiom,
! [U: a,P2: list_b,V: a,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
=> ( reachable_a_b @ t @ W @ V ) ) ) ).
% awalk_verts_reachable_to
thf(fact_1099_awalk__verts__reachable__from,axiom,
! [U: a,P2: list_b,V: a,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
=> ( reachable_a_b @ t @ U @ W ) ) ) ).
% awalk_verts_reachable_from
thf(fact_1100_G_Oawalk__verts__reachable__from,axiom,
! [U: a,P2: list_b,V: a,W: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) )
=> ( reachable_a_b @ g @ U @ W ) ) ) ).
% G.awalk_verts_reachable_from
thf(fact_1101_G_Oawalk__verts__reachable__to,axiom,
! [U: a,P2: list_b,V: a,W: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) )
=> ( reachable_a_b @ g @ W @ V ) ) ) ).
% G.awalk_verts_reachable_to
thf(fact_1102_awalk__verts__arc1,axiom,
! [E2: b,P2: list_b,U: a] :
( ( member_b @ E2 @ ( set_b2 @ P2 ) )
=> ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E2 ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ).
% awalk_verts_arc1
thf(fact_1103_G_Oawalk__verts__arc1,axiom,
! [E2: b,P2: list_b,U: a] :
( ( member_b @ E2 @ ( set_b2 @ P2 ) )
=> ( member_a @ ( pre_ta4931606617599662728t_unit @ g @ E2 ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) ) ) ).
% G.awalk_verts_arc1
thf(fact_1104_awalk__verts_Osimps_I2_J,axiom,
! [U: a,E2: b,Es: list_b] :
( ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( cons_b @ E2 @ Es ) )
= ( cons_a @ ( pre_ta4931606617599662728t_unit @ t @ E2 ) @ ( arc_pr7493981781705774526ts_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ Es ) ) ) ).
% awalk_verts.simps(2)
thf(fact_1105_G_Oawalk__verts_Osimps_I2_J,axiom,
! [U: a,E2: b,Es: list_b] :
( ( arc_pr7493981781705774526ts_a_b @ g @ U @ ( cons_b @ E2 @ Es ) )
= ( cons_a @ ( pre_ta4931606617599662728t_unit @ g @ E2 ) @ ( arc_pr7493981781705774526ts_a_b @ g @ ( pre_he5236287464308401016t_unit @ g @ E2 ) @ Es ) ) ) ).
% G.awalk_verts.simps(2)
thf(fact_1106_awalk__del__vert,axiom,
! [U: a,P2: list_b,V: a,X4: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ~ ( member_a @ X4 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
=> ( arc_pre_awalk_a_b @ ( pre_del_vert_a_b @ t @ X4 ) @ U @ P2 @ V ) ) ) ).
% awalk_del_vert
thf(fact_1107_G_Oawalk__del__vert,axiom,
! [U: a,P2: list_b,V: a,X4: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( ~ ( member_a @ X4 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) )
=> ( arc_pre_awalk_a_b @ ( pre_del_vert_a_b @ g @ X4 ) @ U @ P2 @ V ) ) ) ).
% G.awalk_del_vert
thf(fact_1108_leaf__not__mem__awalk,axiom,
! [X4: a,U: a,P2: list_b,V: a] :
( ( shorte1213025427933718126af_a_b @ t @ X4 )
=> ( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( V != X4 )
=> ~ ( member_a @ X4 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ) ) ).
% leaf_not_mem_awalk
thf(fact_1109_awalk__verts__arc2,axiom,
! [U: a,P2: list_b,V: a,E2: b] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( member_b @ E2 @ ( set_b2 @ P2 ) )
=> ( member_a @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ) ).
% awalk_verts_arc2
thf(fact_1110_G_Oawalk__verts__arc2,axiom,
! [U: a,P2: list_b,V: a,E2: b] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( ( member_b @ E2 @ ( set_b2 @ P2 ) )
=> ( member_a @ ( pre_he5236287464308401016t_unit @ g @ E2 ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) ) ) ) ).
% G.awalk_verts_arc2
thf(fact_1111_awalk__induce,axiom,
! [U: a,P2: list_b,V: a,S: set_a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ S )
=> ( arc_pre_awalk_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S ) @ U @ P2 @ V ) ) ) ).
% awalk_induce
thf(fact_1112_G_Oawalk__induce,axiom,
! [U: a,P2: list_b,V: a,S: set_a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ S )
=> ( arc_pre_awalk_a_b @ ( digrap7873285959652527175ph_a_b @ g @ S ) @ U @ P2 @ V ) ) ) ).
% G.awalk_induce
thf(fact_1113_cas__induce,axiom,
! [U: a,P2: list_b,V: a,S: set_a] :
( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ S )
=> ( arc_pre_cas_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S ) @ U @ P2 @ V ) ) ) ).
% cas_induce
thf(fact_1114_G_Ocas__induce,axiom,
! [U: a,P2: list_b,V: a,S: set_a] :
( ( arc_pre_cas_a_b @ g @ U @ P2 @ V )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ S )
=> ( arc_pre_cas_a_b @ ( digrap7873285959652527175ph_a_b @ g @ S ) @ U @ P2 @ V ) ) ) ).
% G.cas_induce
thf(fact_1115_awalk__verts__in__verts,axiom,
! [U: a,P2: list_b,V: a] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( member_a @ V @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ) ).
% awalk_verts_in_verts
thf(fact_1116_G_Oawalk__verts__in__verts,axiom,
! [U: a,P2: list_b,V: a] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( member_a @ V @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) ) ) ) ) ).
% G.awalk_verts_in_verts
thf(fact_1117_awalk__verts__subset__if__p__sub,axiom,
! [U: a,P1: list_b,V: a,P22: list_b] :
( ( arc_pre_awalk_a_b @ t @ U @ P1 @ V )
=> ( ( arc_pre_awalk_a_b @ t @ U @ P22 @ V )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P1 ) @ ( set_b2 @ P22 ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P1 ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P22 ) ) ) ) ) ) ).
% awalk_verts_subset_if_p_sub
thf(fact_1118_G_Oawalk__verts__subset__if__p__sub,axiom,
! [U: a,P1: list_b,V: a,P22: list_b] :
( ( arc_pre_awalk_a_b @ g @ U @ P1 @ V )
=> ( ( arc_pre_awalk_a_b @ g @ U @ P22 @ V )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P1 ) @ ( set_b2 @ P22 ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P1 ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P22 ) ) ) ) ) ) ).
% G.awalk_verts_subset_if_p_sub
thf(fact_1119_awalk__vertex__props,axiom,
! [U: a,P2: list_b,V: a,P: a > $o,Q: a > $o] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( P2 != nil_b )
=> ( ! [W2: a] :
( ( member_a @ W2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
=> ( ( P @ W2 )
| ( Q @ W2 ) ) )
=> ( ( P @ U )
=> ( ( Q @ V )
=> ? [X3: b] :
( ( member_b @ X3 @ ( set_b2 @ P2 ) )
& ( P @ ( pre_ta4931606617599662728t_unit @ t @ X3 ) )
& ( Q @ ( pre_he5236287464308401016t_unit @ t @ X3 ) ) ) ) ) ) ) ) ).
% awalk_vertex_props
thf(fact_1120_G_Oawalk__vertex__props,axiom,
! [U: a,P2: list_b,V: a,P: a > $o,Q: a > $o] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( ( P2 != nil_b )
=> ( ! [W2: a] :
( ( member_a @ W2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) )
=> ( ( P @ W2 )
| ( Q @ W2 ) ) )
=> ( ( P @ U )
=> ( ( Q @ V )
=> ? [X3: b] :
( ( member_b @ X3 @ ( set_b2 @ P2 ) )
& ( P @ ( pre_ta4931606617599662728t_unit @ g @ X3 ) )
& ( Q @ ( pre_he5236287464308401016t_unit @ g @ X3 ) ) ) ) ) ) ) ) ).
% G.awalk_vertex_props
thf(fact_1121_awalkI,axiom,
! [U: a,P2: list_b,V: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( arc_pre_awalk_a_b @ t @ U @ P2 @ V ) ) ) ) ).
% awalkI
thf(fact_1122_G_OawalkI,axiom,
! [U: a,P2: list_b,V: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( arc_pre_cas_a_b @ g @ U @ P2 @ V )
=> ( arc_pre_awalk_a_b @ g @ U @ P2 @ V ) ) ) ) ).
% G.awalkI
thf(fact_1123_G_Otrail__connected,axiom,
! [U: a,P2: list_b,V: a] :
( ( digrap8783888973171253482ed_a_b @ g )
=> ( ( arc_pre_trail_a_b @ g @ U @ P2 @ V )
=> ( ( ( set_b2 @ P2 )
!= ( pre_ar1395965042833527383t_unit @ g ) )
=> ? [E: b] :
( ( member_b @ E @ ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ g ) @ ( set_b2 @ P2 ) ) )
& ( ( member_a @ ( pre_ta4931606617599662728t_unit @ g @ E ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) )
| ( member_a @ ( pre_he5236287464308401016t_unit @ g @ E ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) ) ) ) ) ) ) ).
% G.trail_connected
thf(fact_1124_trail__connected,axiom,
! [U: a,P2: list_b,V: a] :
( ( digrap8783888973171253482ed_a_b @ t )
=> ( ( arc_pre_trail_a_b @ t @ U @ P2 @ V )
=> ( ( ( set_b2 @ P2 )
!= ( pre_ar1395965042833527383t_unit @ t ) )
=> ? [E: b] :
( ( member_b @ E @ ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ t ) @ ( set_b2 @ P2 ) ) )
& ( ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
| ( member_a @ ( pre_he5236287464308401016t_unit @ t @ E ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ) ) ) ) ).
% trail_connected
thf(fact_1125_trail__Nil__iff,axiom,
! [U: a,V: a] :
( ( arc_pre_trail_a_b @ t @ U @ nil_b @ V )
= ( ( U = V )
& ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% trail_Nil_iff
thf(fact_1126_G_Otrail__Nil__iff,axiom,
! [U: a,V: a] :
( ( arc_pre_trail_a_b @ g @ U @ nil_b @ V )
= ( ( U = V )
& ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) ) ) ) ).
% G.trail_Nil_iff
thf(fact_1127_rotate__trailE_H,axiom,
! [U: a,P2: list_b,W: a] :
( ( arc_pre_trail_a_b @ t @ U @ P2 @ U )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
=> ~ ! [Q2: list_b] :
( ( arc_pre_trail_a_b @ t @ W @ Q2 @ W )
=> ( ( ( set_b2 @ Q2 )
= ( set_b2 @ P2 ) )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ W @ Q2 ) )
!= ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ) ) ) ).
% rotate_trailE'
thf(fact_1128_G_Orotate__trailE_H,axiom,
! [U: a,P2: list_b,W: a] :
( ( arc_pre_trail_a_b @ g @ U @ P2 @ U )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) )
=> ~ ! [Q2: list_b] :
( ( arc_pre_trail_a_b @ g @ W @ Q2 @ W )
=> ( ( ( set_b2 @ Q2 )
= ( set_b2 @ P2 ) )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ W @ Q2 ) )
!= ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) ) ) ) ) ) ).
% G.rotate_trailE'
thf(fact_1129_euler__trail__conv__connected,axiom,
! [U: a,P2: list_b,V: a] :
( ( digrap8783888973171253482ed_a_b @ t )
=> ( ( pre_euler_trail_a_b @ t @ U @ P2 @ V )
= ( ( arc_pre_trail_a_b @ t @ U @ P2 @ V )
& ( ( set_b2 @ P2 )
= ( pre_ar1395965042833527383t_unit @ t ) ) ) ) ) ).
% euler_trail_conv_connected
thf(fact_1130_G_Oeuler__trail__conv__connected,axiom,
! [U: a,P2: list_b,V: a] :
( ( digrap8783888973171253482ed_a_b @ g )
=> ( ( pre_euler_trail_a_b @ g @ U @ P2 @ V )
= ( ( arc_pre_trail_a_b @ g @ U @ P2 @ V )
& ( ( set_b2 @ P2 )
= ( pre_ar1395965042833527383t_unit @ g ) ) ) ) ) ).
% G.euler_trail_conv_connected
thf(fact_1131_trail__Cons__iff,axiom,
! [U: a,E2: b,Es: list_b,W: a] :
( ( arc_pre_trail_a_b @ t @ U @ ( cons_b @ E2 @ Es ) @ W )
= ( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( U
= ( pre_ta4931606617599662728t_unit @ t @ E2 ) )
& ~ ( member_b @ E2 @ ( set_b2 @ Es ) )
& ( arc_pre_trail_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ Es @ W ) ) ) ).
% trail_Cons_iff
thf(fact_1132_G_Otrail__Cons__iff,axiom,
! [U: a,E2: b,Es: list_b,W: a] :
( ( arc_pre_trail_a_b @ g @ U @ ( cons_b @ E2 @ Es ) @ W )
= ( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ g ) )
& ( U
= ( pre_ta4931606617599662728t_unit @ g @ E2 ) )
& ~ ( member_b @ E2 @ ( set_b2 @ Es ) )
& ( arc_pre_trail_a_b @ g @ ( pre_he5236287464308401016t_unit @ g @ E2 ) @ Es @ W ) ) ) ).
% G.trail_Cons_iff
thf(fact_1133_euler__trail__def,axiom,
! [U: a,P2: list_b,V: a] :
( ( pre_euler_trail_a_b @ t @ U @ P2 @ V )
= ( ( arc_pre_trail_a_b @ t @ U @ P2 @ V )
& ( ( set_b2 @ P2 )
= ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% euler_trail_def
thf(fact_1134_G_Oeuler__trail__def,axiom,
! [U: a,P2: list_b,V: a] :
( ( pre_euler_trail_a_b @ g @ U @ P2 @ V )
= ( ( arc_pre_trail_a_b @ g @ U @ P2 @ V )
& ( ( set_b2 @ P2 )
= ( pre_ar1395965042833527383t_unit @ g ) )
& ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) )
= ( pre_ve642382030648772252t_unit @ g ) ) ) ) ).
% G.euler_trail_def
thf(fact_1135_arc__balancedI__trail,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_trail_a_b @ t @ U @ P2 @ V )
=> ( pre_ar5931435604406180204ed_a_b @ t @ U @ ( set_b2 @ P2 ) @ V ) ) ).
% arc_balancedI_trail
thf(fact_1136_G_Oarc__balancedI__trail,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_trail_a_b @ g @ U @ P2 @ V )
=> ( pre_ar5931435604406180204ed_a_b @ g @ U @ ( set_b2 @ P2 ) @ V ) ) ).
% G.arc_balancedI_trail
thf(fact_1137_G_Oinner__verts__Cons,axiom,
! [U: a,E2: b,Es: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ ( cons_b @ E2 @ Es ) @ V )
=> ( ( ( Es != nil_b )
=> ( ( pre_inner_verts_a_b @ g @ ( cons_b @ E2 @ Es ) )
= ( cons_a @ ( pre_he5236287464308401016t_unit @ g @ E2 ) @ ( pre_inner_verts_a_b @ g @ Es ) ) ) )
& ( ( Es = nil_b )
=> ( ( pre_inner_verts_a_b @ g @ ( cons_b @ E2 @ Es ) )
= nil_a ) ) ) ) ).
% G.inner_verts_Cons
thf(fact_1138_inner__verts__Cons,axiom,
! [U: a,E2: b,Es: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ ( cons_b @ E2 @ Es ) @ V )
=> ( ( ( Es != nil_b )
=> ( ( pre_inner_verts_a_b @ t @ ( cons_b @ E2 @ Es ) )
= ( cons_a @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ ( pre_inner_verts_a_b @ t @ Es ) ) ) )
& ( ( Es = nil_b )
=> ( ( pre_inner_verts_a_b @ t @ ( cons_b @ E2 @ Es ) )
= nil_a ) ) ) ) ).
% inner_verts_Cons
thf(fact_1139_inner__verts__Nil,axiom,
( ( pre_inner_verts_a_b @ t @ nil_b )
= nil_a ) ).
% inner_verts_Nil
thf(fact_1140_G_Oinner__verts__Nil,axiom,
( ( pre_inner_verts_a_b @ g @ nil_b )
= nil_a ) ).
% G.inner_verts_Nil
thf(fact_1141_inner__verts__singleton,axiom,
! [X4: b] :
( ( pre_inner_verts_a_b @ t @ ( cons_b @ X4 @ nil_b ) )
= nil_a ) ).
% inner_verts_singleton
thf(fact_1142_G_Oinner__verts__singleton,axiom,
! [X4: b] :
( ( pre_inner_verts_a_b @ g @ ( cons_b @ X4 @ nil_b ) )
= nil_a ) ).
% G.inner_verts_singleton
thf(fact_1143_G_Oset__awalk__verts__cas,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_cas_a_b @ g @ U @ P2 @ V )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) )
= ( sup_sup_set_a @ ( sup_sup_set_a @ ( insert_a @ U @ bot_bot_set_a ) @ ( set_a2 @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ g ) @ P2 ) ) ) @ ( set_a2 @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ g ) @ P2 ) ) ) ) ) ).
% G.set_awalk_verts_cas
thf(fact_1144_set__awalk__verts__not__Nil,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( P2 != nil_b )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= ( sup_sup_set_a @ ( set_a2 @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P2 ) ) @ ( set_a2 @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ P2 ) ) ) ) ) ) ).
% set_awalk_verts_not_Nil
thf(fact_1145_G_Oset__awalk__verts__not__Nil,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( ( P2 != nil_b )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) )
= ( sup_sup_set_a @ ( set_a2 @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ g ) @ P2 ) ) @ ( set_a2 @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ g ) @ P2 ) ) ) ) ) ) ).
% G.set_awalk_verts_not_Nil
thf(fact_1146_set__awalk__verts__not__Nil__cas,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( P2 != nil_b )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= ( sup_sup_set_a @ ( set_a2 @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P2 ) ) @ ( set_a2 @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ P2 ) ) ) ) ) ) ).
% set_awalk_verts_not_Nil_cas
thf(fact_1147_G_Oset__awalk__verts__not__Nil__cas,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_cas_a_b @ g @ U @ P2 @ V )
=> ( ( P2 != nil_b )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) )
= ( sup_sup_set_a @ ( set_a2 @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ g ) @ P2 ) ) @ ( set_a2 @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ g ) @ P2 ) ) ) ) ) ) ).
% G.set_awalk_verts_not_Nil_cas
thf(fact_1148_set__awalk__verts,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= ( sup_sup_set_a @ ( sup_sup_set_a @ ( insert_a @ U @ bot_bot_set_a ) @ ( set_a2 @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P2 ) ) ) @ ( set_a2 @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ P2 ) ) ) ) ) ).
% set_awalk_verts
thf(fact_1149_G_Oset__awalk__verts,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) )
= ( sup_sup_set_a @ ( sup_sup_set_a @ ( insert_a @ U @ bot_bot_set_a ) @ ( set_a2 @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ g ) @ P2 ) ) ) @ ( set_a2 @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ g ) @ P2 ) ) ) ) ) ).
% G.set_awalk_verts
thf(fact_1150_set__awalk__verts__cas,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= ( sup_sup_set_a @ ( sup_sup_set_a @ ( insert_a @ U @ bot_bot_set_a ) @ ( set_a2 @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P2 ) ) ) @ ( set_a2 @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ P2 ) ) ) ) ) ).
% set_awalk_verts_cas
thf(fact_1151_G_Oawalk__verts__conv_H,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_cas_a_b @ g @ U @ P2 @ V )
=> ( ( ( P2 = nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 )
= ( cons_a @ U @ nil_a ) ) )
& ( ( P2 != nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 )
= ( cons_a @ ( pre_ta4931606617599662728t_unit @ g @ ( hd_b @ P2 ) ) @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ g ) @ P2 ) ) ) ) ) ) ).
% G.awalk_verts_conv'
thf(fact_1152_awalk__verts__conv_H,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( ( P2 = nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 )
= ( cons_a @ U @ nil_a ) ) )
& ( ( P2 != nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 )
= ( cons_a @ ( pre_ta4931606617599662728t_unit @ t @ ( hd_b @ P2 ) ) @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ P2 ) ) ) ) ) ) ).
% awalk_verts_conv'
thf(fact_1153_G_Oawalk__to__apath__verts__subset,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ ( arc_wf446166946845163101th_a_b @ g @ P2 ) ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) ) ) ).
% G.awalk_to_apath_verts_subset
thf(fact_1154_awalk__to__apath__verts__subset,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( arc_wf446166946845163101th_a_b @ t @ P2 ) ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ).
% awalk_to_apath_verts_subset
thf(fact_1155_awalk__to__apath__subset,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ord_less_eq_set_b @ ( set_b2 @ ( arc_wf446166946845163101th_a_b @ t @ P2 ) ) @ ( set_b2 @ P2 ) ) ) ).
% awalk_to_apath_subset
thf(fact_1156_G_Oawalk__to__apath__subset,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( ord_less_eq_set_b @ ( set_b2 @ ( arc_wf446166946845163101th_a_b @ g @ P2 ) ) @ ( set_b2 @ P2 ) ) ) ).
% G.awalk_to_apath_subset
thf(fact_1157_G_Ogen__iapath__def,axiom,
! [V3: set_a,U: a,P2: list_b,V: a] :
( ( pre_gen_iapath_a_b @ g @ V3 @ U @ P2 @ V )
= ( ( member_a @ U @ V3 )
& ( member_a @ V @ V3 )
& ( arc_pre_apath_a_b @ g @ U @ P2 @ V )
& ( ( inf_inf_set_a @ ( set_a2 @ ( pre_inner_verts_a_b @ g @ P2 ) ) @ V3 )
= bot_bot_set_a )
& ( P2 != nil_b ) ) ) ).
% G.gen_iapath_def
thf(fact_1158_gen__iapath__def,axiom,
! [V3: set_a,U: a,P2: list_b,V: a] :
( ( pre_gen_iapath_a_b @ t @ V3 @ U @ P2 @ V )
= ( ( member_a @ U @ V3 )
& ( member_a @ V @ V3 )
& ( arc_pre_apath_a_b @ t @ U @ P2 @ V )
& ( ( inf_inf_set_a @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ P2 ) ) @ V3 )
= bot_bot_set_a )
& ( P2 != nil_b ) ) ) ).
% gen_iapath_def
thf(fact_1159_apath__if__awalk,axiom,
! [R: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ R @ P2 @ V )
=> ( arc_pre_apath_a_b @ t @ R @ P2 @ V ) ) ).
% apath_if_awalk
thf(fact_1160_awalkI__apath,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_apath_a_b @ t @ U @ P2 @ V )
=> ( arc_pre_awalk_a_b @ t @ U @ P2 @ V ) ) ).
% awalkI_apath
thf(fact_1161_G_OawalkI__apath,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_apath_a_b @ g @ U @ P2 @ V )
=> ( arc_pre_awalk_a_b @ g @ U @ P2 @ V ) ) ).
% G.awalkI_apath
thf(fact_1162_reachable__apath,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ t @ U @ V )
= ( ? [P4: list_b] : ( arc_pre_apath_a_b @ t @ U @ P4 @ V ) ) ) ).
% reachable_apath
thf(fact_1163_G_Oreachable__apath,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ g @ U @ V )
= ( ? [P4: list_b] : ( arc_pre_apath_a_b @ g @ U @ P4 @ V ) ) ) ).
% G.reachable_apath
thf(fact_1164_apath__nonempty__ends,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_apath_a_b @ t @ U @ P2 @ V )
=> ( ( P2 != nil_b )
=> ( U != V ) ) ) ).
% apath_nonempty_ends
thf(fact_1165_apath__ends,axiom,
! [U: a,P2: list_b,V: a,U5: a,V6: a] :
( ( arc_pre_apath_a_b @ t @ U @ P2 @ V )
=> ( ( arc_pre_apath_a_b @ t @ U5 @ P2 @ V6 )
=> ( ( ( P2 != nil_b )
& ( U != V )
& ( U = U5 )
& ( V = V6 ) )
| ( ( P2 = nil_b )
& ( U = V )
& ( U5 = V6 ) ) ) ) ) ).
% apath_ends
thf(fact_1166_G_Oapath__ends,axiom,
! [U: a,P2: list_b,V: a,U5: a,V6: a] :
( ( arc_pre_apath_a_b @ g @ U @ P2 @ V )
=> ( ( arc_pre_apath_a_b @ g @ U5 @ P2 @ V6 )
=> ( ( ( P2 != nil_b )
& ( U != V )
& ( U = U5 )
& ( V = V6 ) )
| ( ( P2 = nil_b )
& ( U = V )
& ( U5 = V6 ) ) ) ) ) ).
% G.apath_ends
thf(fact_1167_G_Oapath__nonempty__ends,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_apath_a_b @ g @ U @ P2 @ V )
=> ( ( P2 != nil_b )
=> ( U != V ) ) ) ).
% G.apath_nonempty_ends
thf(fact_1168_subgraph__apath__imp__apath,axiom,
! [H: pre_pr7278220950009878019t_unit,U: a,P2: list_b,V: a] :
( ( arc_pre_apath_a_b @ H @ U @ P2 @ V )
=> ( ( digraph_subgraph_a_b @ H @ t )
=> ( arc_pre_apath_a_b @ t @ U @ P2 @ V ) ) ) ).
% subgraph_apath_imp_apath
thf(fact_1169_G_Osubgraph__apath__imp__apath,axiom,
! [H: pre_pr7278220950009878019t_unit,U: a,P2: list_b,V: a] :
( ( arc_pre_apath_a_b @ H @ U @ P2 @ V )
=> ( ( digraph_subgraph_a_b @ H @ g )
=> ( arc_pre_apath_a_b @ g @ U @ P2 @ V ) ) ) ).
% G.subgraph_apath_imp_apath
thf(fact_1170_hd__in__awalk__verts_I2_J,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_apath_a_b @ t @ U @ P2 @ V )
=> ( member_a @ U @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ).
% hd_in_awalk_verts(2)
thf(fact_1171_G_Ohd__in__awalk__verts_I2_J,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_apath_a_b @ g @ U @ P2 @ V )
=> ( member_a @ U @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) ) ) ).
% G.hd_in_awalk_verts(2)
thf(fact_1172_apath__Nil__iff,axiom,
! [U: a,V: a] :
( ( arc_pre_apath_a_b @ t @ U @ nil_b @ V )
= ( ( U = V )
& ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% apath_Nil_iff
thf(fact_1173_G_Oapath__Nil__iff,axiom,
! [U: a,V: a] :
( ( arc_pre_apath_a_b @ g @ U @ nil_b @ V )
= ( ( U = V )
& ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) ) ) ) ).
% G.apath_Nil_iff
thf(fact_1174_apath__awalk__to__apath,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( arc_pre_apath_a_b @ t @ U @ ( arc_wf446166946845163101th_a_b @ t @ P2 ) @ V ) ) ).
% apath_awalk_to_apath
thf(fact_1175_G_Oapath__awalk__to__apath,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( arc_pre_apath_a_b @ g @ U @ ( arc_wf446166946845163101th_a_b @ g @ P2 ) @ V ) ) ).
% G.apath_awalk_to_apath
thf(fact_1176_unique__apath__verts__in__awalk,axiom,
! [X4: a,U: a,P1: list_b,V: a,P22: list_b] :
( ( member_a @ X4 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P1 ) ) )
=> ( ( arc_pre_apath_a_b @ t @ U @ P1 @ V )
=> ( ( arc_pre_awalk_a_b @ t @ U @ P22 @ V )
=> ( ? [X2: list_b] :
( ( arc_pre_apath_a_b @ t @ U @ X2 @ V )
& ! [Y2: list_b] :
( ( arc_pre_apath_a_b @ t @ U @ Y2 @ V )
=> ( Y2 = X2 ) ) )
=> ( member_a @ X4 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P22 ) ) ) ) ) ) ) ).
% unique_apath_verts_in_awalk
thf(fact_1177_G_Ounique__apath__verts__in__awalk,axiom,
! [X4: a,U: a,P1: list_b,V: a,P22: list_b] :
( ( member_a @ X4 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P1 ) ) )
=> ( ( arc_pre_apath_a_b @ g @ U @ P1 @ V )
=> ( ( arc_pre_awalk_a_b @ g @ U @ P22 @ V )
=> ( ? [X2: list_b] :
( ( arc_pre_apath_a_b @ g @ U @ X2 @ V )
& ! [Y2: list_b] :
( ( arc_pre_apath_a_b @ g @ U @ Y2 @ V )
=> ( Y2 = X2 ) ) )
=> ( member_a @ X4 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P22 ) ) ) ) ) ) ) ).
% G.unique_apath_verts_in_awalk
thf(fact_1178_no__loops__in__apath,axiom,
! [U: a,P2: list_b,V: a,A3: b] :
( ( arc_pre_apath_a_b @ t @ U @ P2 @ V )
=> ( ( member_b @ A3 @ ( set_b2 @ P2 ) )
=> ( ( pre_ta4931606617599662728t_unit @ t @ A3 )
!= ( pre_he5236287464308401016t_unit @ t @ A3 ) ) ) ) ).
% no_loops_in_apath
thf(fact_1179_G_Ono__loops__in__apath,axiom,
! [U: a,P2: list_b,V: a,A3: b] :
( ( arc_pre_apath_a_b @ g @ U @ P2 @ V )
=> ( ( member_b @ A3 @ ( set_b2 @ P2 ) )
=> ( ( pre_ta4931606617599662728t_unit @ g @ A3 )
!= ( pre_he5236287464308401016t_unit @ g @ A3 ) ) ) ) ).
% G.no_loops_in_apath
thf(fact_1180_unique__apath__verts__sub__awalk,axiom,
! [U: a,P2: list_b,V: a,Q3: list_b] :
( ( arc_pre_apath_a_b @ t @ U @ P2 @ V )
=> ( ( arc_pre_awalk_a_b @ t @ U @ Q3 @ V )
=> ( ? [X2: list_b] :
( ( arc_pre_apath_a_b @ t @ U @ X2 @ V )
& ! [Y2: list_b] :
( ( arc_pre_apath_a_b @ t @ U @ Y2 @ V )
=> ( Y2 = X2 ) ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ Q3 ) ) ) ) ) ) ).
% unique_apath_verts_sub_awalk
thf(fact_1181_G_Ounique__apath__verts__sub__awalk,axiom,
! [U: a,P2: list_b,V: a,Q3: list_b] :
( ( arc_pre_apath_a_b @ g @ U @ P2 @ V )
=> ( ( arc_pre_awalk_a_b @ g @ U @ Q3 @ V )
=> ( ? [X2: list_b] :
( ( arc_pre_apath_a_b @ g @ U @ X2 @ V )
& ! [Y2: list_b] :
( ( arc_pre_apath_a_b @ g @ U @ Y2 @ V )
=> ( Y2 = X2 ) ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ Q3 ) ) ) ) ) ) ).
% G.unique_apath_verts_sub_awalk
thf(fact_1182_apath__Cons__iff,axiom,
! [U: a,E2: b,Es: list_b,W: a] :
( ( arc_pre_apath_a_b @ t @ U @ ( cons_b @ E2 @ Es ) @ W )
= ( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( pre_ta4931606617599662728t_unit @ t @ E2 )
= U )
& ( arc_pre_apath_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ Es @ W )
& ~ ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E2 ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ Es ) ) ) ) ) ).
% apath_Cons_iff
thf(fact_1183_G_Oapath__Cons__iff,axiom,
! [U: a,E2: b,Es: list_b,W: a] :
( ( arc_pre_apath_a_b @ g @ U @ ( cons_b @ E2 @ Es ) @ W )
= ( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ g ) )
& ( ( pre_ta4931606617599662728t_unit @ g @ E2 )
= U )
& ( arc_pre_apath_a_b @ g @ ( pre_he5236287464308401016t_unit @ g @ E2 ) @ Es @ W )
& ~ ( member_a @ ( pre_ta4931606617599662728t_unit @ g @ E2 ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ ( pre_he5236287464308401016t_unit @ g @ E2 ) @ Es ) ) ) ) ) ).
% G.apath_Cons_iff
thf(fact_1184_set__inner__verts,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_apath_a_b @ t @ U @ P2 @ V )
=> ( ( set_a2 @ ( pre_inner_verts_a_b @ t @ P2 ) )
= ( minus_minus_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( insert_a @ U @ ( insert_a @ V @ bot_bot_set_a ) ) ) ) ) ).
% set_inner_verts
thf(fact_1185_G_Oset__inner__verts,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_apath_a_b @ g @ U @ P2 @ V )
=> ( ( set_a2 @ ( pre_inner_verts_a_b @ g @ P2 ) )
= ( minus_minus_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ ( insert_a @ U @ ( insert_a @ V @ bot_bot_set_a ) ) ) ) ) ).
% G.set_inner_verts
thf(fact_1186_G_Ocas__simp,axiom,
! [Es: list_b,U: a,V: a] :
( ( Es != nil_b )
=> ( ( arc_pre_cas_a_b @ g @ U @ Es @ V )
= ( ( ( pre_ta4931606617599662728t_unit @ g @ ( hd_b @ Es ) )
= U )
& ( arc_pre_cas_a_b @ g @ ( pre_he5236287464308401016t_unit @ g @ ( hd_b @ Es ) ) @ ( tl_b @ Es ) @ V ) ) ) ) ).
% G.cas_simp
thf(fact_1187_cas__simp,axiom,
! [Es: list_b,U: a,V: a] :
( ( Es != nil_b )
=> ( ( arc_pre_cas_a_b @ t @ U @ Es @ V )
= ( ( ( pre_ta4931606617599662728t_unit @ t @ ( hd_b @ Es ) )
= U )
& ( arc_pre_cas_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ ( hd_b @ Es ) ) @ ( tl_b @ Es ) @ V ) ) ) ) ).
% cas_simp
thf(fact_1188_G_Oawlast__in__verts,axiom,
! [U: a,P2: list_b] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( member_a @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ g ) ) ) ) ).
% G.awlast_in_verts
thf(fact_1189_awlast__in__verts,axiom,
! [U: a,P2: list_b] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member_a @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% awlast_in_verts
thf(fact_1190_awlast__if__cas,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= V ) ) ).
% awlast_if_cas
thf(fact_1191_G_Oawlast__if__cas,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_cas_a_b @ g @ U @ P2 @ V )
=> ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) )
= V ) ) ).
% G.awlast_if_cas
thf(fact_1192_G_Oawalk__conv,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
= ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ g ) )
& ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ g ) )
& ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) )
= U )
& ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) )
= V )
& ( arc_pre_cas_a_b @ g @ U @ P2 @ V ) ) ) ).
% G.awalk_conv
thf(fact_1193_G_OawalkE_H,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ~ ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( arc_pre_cas_a_b @ g @ U @ P2 @ V )
=> ( ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) )
= U )
=> ( ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) )
= V )
=> ( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ~ ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) ) ) ) ) ) ) ) ) ).
% G.awalkE'
thf(fact_1194_awhd__if__cas,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= U ) ) ).
% awhd_if_cas
thf(fact_1195_G_Oawhd__if__cas,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_cas_a_b @ g @ U @ P2 @ V )
=> ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) )
= U ) ) ).
% G.awhd_if_cas
thf(fact_1196_awhd__in__verts,axiom,
! [U: a,P2: list_b] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member_a @ ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% awhd_in_verts
thf(fact_1197_G_Oawhd__in__verts,axiom,
! [U: a,P2: list_b] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( member_a @ ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ g ) ) ) ) ).
% G.awhd_in_verts
thf(fact_1198_awalk__conv,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
= ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ t ) )
& ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= U )
& ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= V )
& ( arc_pre_cas_a_b @ t @ U @ P2 @ V ) ) ) ).
% awalk_conv
thf(fact_1199_awalkE_H,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ~ ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= U )
=> ( ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= V )
=> ( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ~ ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ) ) ) ) ) ).
% awalkE'
thf(fact_1200_awalkE,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ~ ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= U )
=> ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
!= V ) ) ) ) ) ) ).
% awalkE
thf(fact_1201_G_OawalkE,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ~ ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( arc_pre_cas_a_b @ g @ U @ P2 @ V )
=> ( ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) )
= U )
=> ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) )
!= V ) ) ) ) ) ) ).
% G.awalkE
thf(fact_1202_awhd__of__awalk,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= U ) ) ).
% awhd_of_awalk
thf(fact_1203_G_Oawhd__of__awalk,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) )
= U ) ) ).
% G.awhd_of_awalk
thf(fact_1204_G_Oreachable__vpath__conv,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ g @ U @ V )
= ( ? [P4: list_a] :
( ( vertex_vpath_a_b @ P4 @ g )
& ( ( hd_a @ P4 )
= U )
& ( ( last_a @ P4 )
= V ) ) ) ) ).
% G.reachable_vpath_conv
thf(fact_1205_reachable__vpath__conv,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ t @ U @ V )
= ( ? [P4: list_a] :
( ( vertex_vpath_a_b @ P4 @ t )
& ( ( hd_a @ P4 )
= U )
& ( ( last_a @ P4 )
= V ) ) ) ) ).
% reachable_vpath_conv
thf(fact_1206_G_Oawlast__of__awalk,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( ( nOMATCH_a @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ V )
=> ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) )
= V ) ) ) ).
% G.awlast_of_awalk
thf(fact_1207_awlast__of__awalk,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( nOMATCH_a @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ V )
=> ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= V ) ) ) ).
% awlast_of_awalk
thf(fact_1208_G_Oset__awalk__verts__append__cas,axiom,
! [U: a,P2: list_b,V: a,Q3: list_b,W: a] :
( ( arc_pre_cas_a_b @ g @ U @ P2 @ V )
=> ( ( arc_pre_cas_a_b @ g @ V @ Q3 @ W )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ ( append_b @ P2 @ Q3 ) ) )
= ( sup_sup_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ V @ Q3 ) ) ) ) ) ) ).
% G.set_awalk_verts_append_cas
thf(fact_1209_awalk__appendI,axiom,
! [U: a,P2: list_b,V: a,Q3: list_b,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( arc_pre_awalk_a_b @ t @ V @ Q3 @ W )
=> ( arc_pre_awalk_a_b @ t @ U @ ( append_b @ P2 @ Q3 ) @ W ) ) ) ).
% awalk_appendI
thf(fact_1210_G_Oawalk__appendI,axiom,
! [U: a,P2: list_b,V: a,Q3: list_b,W: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( ( arc_pre_awalk_a_b @ g @ V @ Q3 @ W )
=> ( arc_pre_awalk_a_b @ g @ U @ ( append_b @ P2 @ Q3 ) @ W ) ) ) ).
% G.awalk_appendI
thf(fact_1211_awlast__append,axiom,
! [U: a,P2: list_b,Q3: list_b] :
( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P2 @ Q3 ) ) )
= ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ Q3 ) ) ) ).
% awlast_append
thf(fact_1212_G_Oawlast__append,axiom,
! [U: a,P2: list_b,Q3: list_b] :
( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ ( append_b @ P2 @ Q3 ) ) )
= ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ Q3 ) ) ) ).
% G.awlast_append
thf(fact_1213_awhd__append,axiom,
! [U: a,P2: list_b,Q3: list_b] :
( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P2 @ Q3 ) ) )
= ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ Q3 ) ) @ P2 ) ) ) ).
% awhd_append
thf(fact_1214_G_Oawhd__append,axiom,
! [U: a,P2: list_b,Q3: list_b] :
( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ ( append_b @ P2 @ Q3 ) ) )
= ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ g @ ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ Q3 ) ) @ P2 ) ) ) ).
% G.awhd_append
thf(fact_1215_in__set__inner__verts__appendI__l,axiom,
! [U: a,P2: list_b,Q3: list_b] :
( ( member_a @ U @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ P2 ) ) )
=> ( member_a @ U @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ ( append_b @ P2 @ Q3 ) ) ) ) ) ).
% in_set_inner_verts_appendI_l
thf(fact_1216_in__set__inner__verts__appendI__r,axiom,
! [U: a,Q3: list_b,P2: list_b] :
( ( member_a @ U @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ Q3 ) ) )
=> ( member_a @ U @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ ( append_b @ P2 @ Q3 ) ) ) ) ) ).
% in_set_inner_verts_appendI_r
thf(fact_1217_G_Oin__set__inner__verts__appendI__r,axiom,
! [U: a,Q3: list_b,P2: list_b] :
( ( member_a @ U @ ( set_a2 @ ( pre_inner_verts_a_b @ g @ Q3 ) ) )
=> ( member_a @ U @ ( set_a2 @ ( pre_inner_verts_a_b @ g @ ( append_b @ P2 @ Q3 ) ) ) ) ) ).
% G.in_set_inner_verts_appendI_r
thf(fact_1218_G_Oin__set__inner__verts__appendI__l,axiom,
! [U: a,P2: list_b,Q3: list_b] :
( ( member_a @ U @ ( set_a2 @ ( pre_inner_verts_a_b @ g @ P2 ) ) )
=> ( member_a @ U @ ( set_a2 @ ( pre_inner_verts_a_b @ g @ ( append_b @ P2 @ Q3 ) ) ) ) ) ).
% G.in_set_inner_verts_appendI_l
thf(fact_1219_rotate__awalkE,axiom,
! [U: a,P2: list_b,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ U )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
=> ~ ! [Q2: list_b,R2: list_b] :
( ( P2
= ( append_b @ Q2 @ R2 ) )
=> ( ( arc_pre_awalk_a_b @ t @ W @ ( append_b @ R2 @ Q2 ) @ W )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ W @ ( append_b @ R2 @ Q2 ) ) )
!= ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ) ) ) ).
% rotate_awalkE
thf(fact_1220_awalk__decomp,axiom,
! [U: a,P2: list_b,V: a,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
=> ? [Q2: list_b,R2: list_b] :
( ( P2
= ( append_b @ Q2 @ R2 ) )
& ( arc_pre_awalk_a_b @ t @ U @ Q2 @ W )
& ( arc_pre_awalk_a_b @ t @ W @ R2 @ V ) ) ) ) ).
% awalk_decomp
thf(fact_1221_G_Oawalk__decomp,axiom,
! [U: a,P2: list_b,V: a,W: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) )
=> ? [Q2: list_b,R2: list_b] :
( ( P2
= ( append_b @ Q2 @ R2 ) )
& ( arc_pre_awalk_a_b @ g @ U @ Q2 @ W )
& ( arc_pre_awalk_a_b @ g @ W @ R2 @ V ) ) ) ) ).
% G.awalk_decomp
thf(fact_1222_G_Orotate__awalkE,axiom,
! [U: a,P2: list_b,W: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ U )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) )
=> ~ ! [Q2: list_b,R2: list_b] :
( ( P2
= ( append_b @ Q2 @ R2 ) )
=> ( ( arc_pre_awalk_a_b @ g @ W @ ( append_b @ R2 @ Q2 ) @ W )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ W @ ( append_b @ R2 @ Q2 ) ) )
!= ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) ) ) ) ) ) ).
% G.rotate_awalkE
thf(fact_1223_rotate__trailE,axiom,
! [U: a,P2: list_b,W: a] :
( ( arc_pre_trail_a_b @ t @ U @ P2 @ U )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
=> ~ ! [Q2: list_b,R2: list_b] :
( ( P2
= ( append_b @ Q2 @ R2 ) )
=> ( ( arc_pre_trail_a_b @ t @ W @ ( append_b @ R2 @ Q2 ) @ W )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ W @ ( append_b @ R2 @ Q2 ) ) )
!= ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ) ) ) ).
% rotate_trailE
thf(fact_1224_G_Orotate__trailE,axiom,
! [U: a,P2: list_b,W: a] :
( ( arc_pre_trail_a_b @ g @ U @ P2 @ U )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) )
=> ~ ! [Q2: list_b,R2: list_b] :
( ( P2
= ( append_b @ Q2 @ R2 ) )
=> ( ( arc_pre_trail_a_b @ g @ W @ ( append_b @ R2 @ Q2 ) @ W )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ W @ ( append_b @ R2 @ Q2 ) ) )
!= ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) ) ) ) ) ) ).
% G.rotate_trailE
thf(fact_1225_set__awalk__verts__append,axiom,
! [U: a,P2: list_b,V: a,Q3: list_b,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( arc_pre_awalk_a_b @ t @ V @ Q3 @ W )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P2 @ Q3 ) ) )
= ( sup_sup_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ V @ Q3 ) ) ) ) ) ) ).
% set_awalk_verts_append
thf(fact_1226_G_Oset__awalk__verts__append,axiom,
! [U: a,P2: list_b,V: a,Q3: list_b,W: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( ( arc_pre_awalk_a_b @ g @ V @ Q3 @ W )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ ( append_b @ P2 @ Q3 ) ) )
= ( sup_sup_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ V @ Q3 ) ) ) ) ) ) ).
% G.set_awalk_verts_append
thf(fact_1227_awalk__verts__arc1__app,axiom,
! [E2: b,R: a,P1: list_b,P22: list_b] : ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E2 ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ R @ ( append_b @ P1 @ ( cons_b @ E2 @ P22 ) ) ) ) ) ).
% awalk_verts_arc1_app
thf(fact_1228_set__awalk__verts__append__cas,axiom,
! [U: a,P2: list_b,V: a,Q3: list_b,W: a] :
( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( arc_pre_cas_a_b @ t @ V @ Q3 @ W )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P2 @ Q3 ) ) )
= ( sup_sup_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ V @ Q3 ) ) ) ) ) ) ).
% set_awalk_verts_append_cas
thf(fact_1229_awalk__append__iff,axiom,
! [U: a,P2: list_b,Q3: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ ( append_b @ P2 @ Q3 ) @ V )
= ( ( arc_pre_awalk_a_b @ t @ U @ P2 @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
& ( arc_pre_awalk_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ Q3 @ V ) ) ) ).
% awalk_append_iff
thf(fact_1230_G_Oawalk__append__iff,axiom,
! [U: a,P2: list_b,Q3: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ ( append_b @ P2 @ Q3 ) @ V )
= ( ( arc_pre_awalk_a_b @ g @ U @ P2 @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) )
& ( arc_pre_awalk_a_b @ g @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ Q3 @ V ) ) ) ).
% G.awalk_append_iff
thf(fact_1231_cas__append__iff,axiom,
! [U: a,P2: list_b,Q3: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ ( append_b @ P2 @ Q3 ) @ V )
= ( ( arc_pre_cas_a_b @ t @ U @ P2 @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
& ( arc_pre_cas_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ Q3 @ V ) ) ) ).
% cas_append_iff
thf(fact_1232_G_Ocas__append__iff,axiom,
! [U: a,P2: list_b,Q3: list_b,V: a] :
( ( arc_pre_cas_a_b @ g @ U @ ( append_b @ P2 @ Q3 ) @ V )
= ( ( arc_pre_cas_a_b @ g @ U @ P2 @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) )
& ( arc_pre_cas_a_b @ g @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ Q3 @ V ) ) ) ).
% G.cas_append_iff
thf(fact_1233_G_Onot__distinct__if__head__eq__tail,axiom,
! [P2: b,U: a,E2: b,R: a,Ps: list_b,P22: list_b,V: a] :
( ( ( pre_ta4931606617599662728t_unit @ g @ P2 )
= U )
=> ( ( ( pre_he5236287464308401016t_unit @ g @ E2 )
= U )
=> ( ( arc_pre_awalk_a_b @ g @ R @ ( append_b @ Ps @ ( append_b @ ( cons_b @ P2 @ nil_b ) @ ( cons_b @ E2 @ P22 ) ) ) @ V )
=> ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ g @ R @ ( append_b @ Ps @ ( append_b @ ( cons_b @ P2 @ nil_b ) @ ( cons_b @ E2 @ P22 ) ) ) ) ) ) ) ) ).
% G.not_distinct_if_head_eq_tail
thf(fact_1234_not__distinct__if__head__eq__tail,axiom,
! [P2: b,U: a,E2: b,R: a,Ps: list_b,P22: list_b,V: a] :
( ( ( pre_ta4931606617599662728t_unit @ t @ P2 )
= U )
=> ( ( ( pre_he5236287464308401016t_unit @ t @ E2 )
= U )
=> ( ( arc_pre_awalk_a_b @ t @ R @ ( append_b @ Ps @ ( append_b @ ( cons_b @ P2 @ nil_b ) @ ( cons_b @ E2 @ P22 ) ) ) @ V )
=> ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ R @ ( append_b @ Ps @ ( append_b @ ( cons_b @ P2 @ nil_b ) @ ( cons_b @ E2 @ P22 ) ) ) ) ) ) ) ) ).
% not_distinct_if_head_eq_tail
thf(fact_1235_awalk__verts__append__distinct,axiom,
! [R: a,P1: list_b,P22: list_b] :
( ? [X_1: a] : ( arc_pre_awalk_a_b @ t @ R @ ( append_b @ P1 @ P22 ) @ X_1 )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ R @ ( append_b @ P1 @ P22 ) ) )
=> ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ R @ P1 ) ) ) ) ).
% awalk_verts_append_distinct
thf(fact_1236_G_Oawalk__verts__append__distinct,axiom,
! [R: a,P1: list_b,P22: list_b] :
( ? [X_1: a] : ( arc_pre_awalk_a_b @ g @ R @ ( append_b @ P1 @ P22 ) @ X_1 )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ g @ R @ ( append_b @ P1 @ P22 ) ) )
=> ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ g @ R @ P1 ) ) ) ) ).
% G.awalk_verts_append_distinct
thf(fact_1237_awalk__verts__append3,axiom,
! [U: a,P2: list_b,E2: b,Q3: list_b,R: a,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ ( append_b @ P2 @ ( cons_b @ E2 @ Q3 ) ) @ R )
=> ( ( arc_pre_awalk_a_b @ t @ V @ Q3 @ R )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P2 @ ( cons_b @ E2 @ Q3 ) ) )
= ( append_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) @ ( arc_pr7493981781705774526ts_a_b @ t @ V @ Q3 ) ) ) ) ) ).
% awalk_verts_append3
thf(fact_1238_G_Oawalk__verts__append3,axiom,
! [U: a,P2: list_b,E2: b,Q3: list_b,R: a,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ ( append_b @ P2 @ ( cons_b @ E2 @ Q3 ) ) @ R )
=> ( ( arc_pre_awalk_a_b @ g @ V @ Q3 @ R )
=> ( ( arc_pr7493981781705774526ts_a_b @ g @ U @ ( append_b @ P2 @ ( cons_b @ E2 @ Q3 ) ) )
= ( append_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) @ ( arc_pr7493981781705774526ts_a_b @ g @ V @ Q3 ) ) ) ) ) ).
% G.awalk_verts_append3
thf(fact_1239_apath__def,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_apath_a_b @ t @ U @ P2 @ V )
= ( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
& ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ).
% apath_def
thf(fact_1240_G_Oapath__def,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_apath_a_b @ g @ U @ P2 @ V )
= ( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
& ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) ) ) ).
% G.apath_def
thf(fact_1241_awalk__decomp__verts,axiom,
! [U: a,P2: list_b,V: a,Xs: list_a,Y3: a,Ys: list_a] :
( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 )
= ( append_a @ Xs @ ( cons_a @ Y3 @ Ys ) ) )
=> ~ ! [Q2: list_b] :
( ( arc_pre_cas_a_b @ t @ U @ Q2 @ Y3 )
=> ! [R2: list_b] :
( ( arc_pre_cas_a_b @ t @ Y3 @ R2 @ V )
=> ( ( P2
= ( append_b @ Q2 @ R2 ) )
=> ( ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ Q2 )
= ( append_a @ Xs @ ( cons_a @ Y3 @ nil_a ) ) )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ Y3 @ R2 )
!= ( cons_a @ Y3 @ Ys ) ) ) ) ) ) ) ) ).
% awalk_decomp_verts
thf(fact_1242_G_Oawalk__decomp__verts,axiom,
! [U: a,P2: list_b,V: a,Xs: list_a,Y3: a,Ys: list_a] :
( ( arc_pre_cas_a_b @ g @ U @ P2 @ V )
=> ( ( ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 )
= ( append_a @ Xs @ ( cons_a @ Y3 @ Ys ) ) )
=> ~ ! [Q2: list_b] :
( ( arc_pre_cas_a_b @ g @ U @ Q2 @ Y3 )
=> ! [R2: list_b] :
( ( arc_pre_cas_a_b @ g @ Y3 @ R2 @ V )
=> ( ( P2
= ( append_b @ Q2 @ R2 ) )
=> ( ( ( arc_pr7493981781705774526ts_a_b @ g @ U @ Q2 )
= ( append_a @ Xs @ ( cons_a @ Y3 @ nil_a ) ) )
=> ( ( arc_pr7493981781705774526ts_a_b @ g @ Y3 @ R2 )
!= ( cons_a @ Y3 @ Ys ) ) ) ) ) ) ) ) ).
% G.awalk_decomp_verts
thf(fact_1243_awalk__cyc__decompE_H,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
=> ~ ! [Q2: list_b,R2: list_b,S2: list_b] :
( ( P2
= ( append_b @ Q2 @ ( append_b @ R2 @ S2 ) ) )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ Q2 ) )
=> ( ? [W2: a] :
( ( arc_pre_awalk_a_b @ t @ U @ Q2 @ W2 )
& ( arc_pre_awalk_a_b @ t @ W2 @ R2 @ W2 )
& ( arc_pre_awalk_a_b @ t @ W2 @ S2 @ V ) )
=> ~ ( arc_wf_closed_w_a_b @ t @ R2 ) ) ) ) ) ) ).
% awalk_cyc_decompE'
thf(fact_1244_G_Oawalk__cyc__decompE_H,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) )
=> ~ ! [Q2: list_b,R2: list_b,S2: list_b] :
( ( P2
= ( append_b @ Q2 @ ( append_b @ R2 @ S2 ) ) )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ Q2 ) )
=> ( ? [W2: a] :
( ( arc_pre_awalk_a_b @ g @ U @ Q2 @ W2 )
& ( arc_pre_awalk_a_b @ g @ W2 @ R2 @ W2 )
& ( arc_pre_awalk_a_b @ g @ W2 @ S2 @ V ) )
=> ~ ( arc_wf_closed_w_a_b @ g @ R2 ) ) ) ) ) ) ).
% G.awalk_cyc_decompE'
thf(fact_1245_G_Oawalk__verts__conv,axiom,
! [P2: list_b,U: a] :
( ( ( P2 = nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 )
= ( cons_a @ U @ nil_a ) ) )
& ( ( P2 != nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 )
= ( append_a @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ g ) @ P2 ) @ ( cons_a @ ( pre_he5236287464308401016t_unit @ g @ ( last_b @ P2 ) ) @ nil_a ) ) ) ) ) ).
% G.awalk_verts_conv
thf(fact_1246_awalk__verts__conv,axiom,
! [P2: list_b,U: a] :
( ( ( P2 = nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 )
= ( cons_a @ U @ nil_a ) ) )
& ( ( P2 != nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 )
= ( append_a @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P2 ) @ ( cons_a @ ( pre_he5236287464308401016t_unit @ t @ ( last_b @ P2 ) ) @ nil_a ) ) ) ) ) ).
% awalk_verts_conv
thf(fact_1247_trail__def,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_trail_a_b @ t @ U @ P2 @ V )
= ( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
& ( distinct_b @ P2 ) ) ) ).
% trail_def
thf(fact_1248_G_Otrail__def,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_trail_a_b @ g @ U @ P2 @ V )
= ( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
& ( distinct_b @ P2 ) ) ) ).
% G.trail_def
thf(fact_1249_distinct__tl__verts__imp__distinct,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( distinct_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
=> ( distinct_b @ P2 ) ) ) ).
% distinct_tl_verts_imp_distinct
thf(fact_1250_distinct__verts__imp__distinct,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
=> ( distinct_b @ P2 ) ) ) ).
% distinct_verts_imp_distinct
thf(fact_1251_G_Odistinct__tl__verts__imp__distinct,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( ( distinct_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) )
=> ( distinct_b @ P2 ) ) ) ).
% G.distinct_tl_verts_imp_distinct
thf(fact_1252_G_Odistinct__verts__imp__distinct,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) )
=> ( distinct_b @ P2 ) ) ) ).
% G.distinct_verts_imp_distinct
thf(fact_1253_inner__verts__def,axiom,
! [P2: list_b] :
( ( pre_inner_verts_a_b @ t @ P2 )
= ( tl_a @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P2 ) ) ) ).
% inner_verts_def
thf(fact_1254_G_Oinner__verts__def,axiom,
! [P2: list_b] :
( ( pre_inner_verts_a_b @ g @ P2 )
= ( tl_a @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ g ) @ P2 ) ) ) ).
% G.inner_verts_def
thf(fact_1255_cycle__altdef,axiom,
! [P2: list_b] :
( ( arc_pre_cycle_a_b @ t @ P2 )
= ( ( arc_wf_closed_w_a_b @ t @ P2 )
& ? [U6: a] : ( distinct_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U6 @ P2 ) ) ) ) ) ).
% cycle_altdef
thf(fact_1256_G_Ocycle__altdef,axiom,
! [P2: list_b] :
( ( arc_pre_cycle_a_b @ g @ P2 )
= ( ( arc_wf_closed_w_a_b @ g @ P2 )
& ? [U6: a] : ( distinct_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U6 @ P2 ) ) ) ) ) ).
% G.cycle_altdef
thf(fact_1257_apath__decomp__disjoint,axiom,
! [U: a,P2: list_b,V: a,Q3: list_b,R: list_b,X4: a] :
( ( arc_pre_apath_a_b @ t @ U @ P2 @ V )
=> ( ( P2
= ( append_b @ Q3 @ R ) )
=> ( ( member_a @ X4 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ Q3 ) ) )
=> ~ ( member_a @ X4 @ ( set_a2 @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ Q3 ) ) @ R ) ) ) ) ) ) ) ).
% apath_decomp_disjoint
thf(fact_1258_G_Oapath__decomp__disjoint,axiom,
! [U: a,P2: list_b,V: a,Q3: list_b,R: list_b,X4: a] :
( ( arc_pre_apath_a_b @ g @ U @ P2 @ V )
=> ( ( P2
= ( append_b @ Q3 @ R ) )
=> ( ( member_a @ X4 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ Q3 ) ) )
=> ~ ( member_a @ X4 @ ( set_a2 @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ g @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ Q3 ) ) @ R ) ) ) ) ) ) ) ).
% G.apath_decomp_disjoint
thf(fact_1259_cycle__conv,axiom,
! [P2: list_b] :
( ( arc_pre_cycle_a_b @ t @ P2 )
= ( ? [U6: a] :
( ( arc_pre_awalk_a_b @ t @ U6 @ P2 @ U6 )
& ( distinct_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U6 @ P2 ) ) )
& ( distinct_b @ P2 )
& ( P2 != nil_b ) ) ) ) ).
% cycle_conv
thf(fact_1260_cycle__def,axiom,
! [P2: list_b] :
( ( arc_pre_cycle_a_b @ t @ P2 )
= ( ? [U6: a] :
( ( arc_pre_awalk_a_b @ t @ U6 @ P2 @ U6 )
& ( distinct_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U6 @ P2 ) ) )
& ( P2 != nil_b ) ) ) ) ).
% cycle_def
thf(fact_1261_awalk__verts__append,axiom,
! [U: a,P2: list_b,Q3: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ ( append_b @ P2 @ Q3 ) @ V )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P2 @ Q3 ) )
= ( append_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ Q3 ) ) ) ) ) ).
% awalk_verts_append
thf(fact_1262_G_Ocycle__conv,axiom,
! [P2: list_b] :
( ( arc_pre_cycle_a_b @ g @ P2 )
= ( ? [U6: a] :
( ( arc_pre_awalk_a_b @ g @ U6 @ P2 @ U6 )
& ( distinct_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U6 @ P2 ) ) )
& ( distinct_b @ P2 )
& ( P2 != nil_b ) ) ) ) ).
% G.cycle_conv
thf(fact_1263_G_Ocycle__def,axiom,
! [P2: list_b] :
( ( arc_pre_cycle_a_b @ g @ P2 )
= ( ? [U6: a] :
( ( arc_pre_awalk_a_b @ g @ U6 @ P2 @ U6 )
& ( distinct_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U6 @ P2 ) ) )
& ( P2 != nil_b ) ) ) ) ).
% G.cycle_def
thf(fact_1264_G_Oawalk__verts__append,axiom,
! [U: a,P2: list_b,Q3: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ ( append_b @ P2 @ Q3 ) @ V )
=> ( ( arc_pr7493981781705774526ts_a_b @ g @ U @ ( append_b @ P2 @ Q3 ) )
= ( append_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ g @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ Q3 ) ) ) ) ) ).
% G.awalk_verts_append
thf(fact_1265_awalk__verts__append__cas,axiom,
! [U: a,P2: list_b,Q3: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ ( append_b @ P2 @ Q3 ) @ V )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P2 @ Q3 ) )
= ( append_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ Q3 ) ) ) ) ) ).
% awalk_verts_append_cas
thf(fact_1266_G_Oawalk__verts__append__cas,axiom,
! [U: a,P2: list_b,Q3: list_b,V: a] :
( ( arc_pre_cas_a_b @ g @ U @ ( append_b @ P2 @ Q3 ) @ V )
=> ( ( arc_pr7493981781705774526ts_a_b @ g @ U @ ( append_b @ P2 @ Q3 ) )
= ( append_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ g @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ Q3 ) ) ) ) ) ).
% G.awalk_verts_append_cas
thf(fact_1267_apath__append__iff,axiom,
! [U: a,P2: list_b,Q3: list_b,V: a] :
( ( arc_pre_apath_a_b @ t @ U @ ( append_b @ P2 @ Q3 ) @ V )
= ( ( arc_pre_apath_a_b @ t @ U @ P2 @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
& ( arc_pre_apath_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ Q3 @ V )
& ( ( inf_inf_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( set_a2 @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ Q3 ) ) ) )
= bot_bot_set_a ) ) ) ).
% apath_append_iff
thf(fact_1268_G_Oapath__append__iff,axiom,
! [U: a,P2: list_b,Q3: list_b,V: a] :
( ( arc_pre_apath_a_b @ g @ U @ ( append_b @ P2 @ Q3 ) @ V )
= ( ( arc_pre_apath_a_b @ g @ U @ P2 @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) )
& ( arc_pre_apath_a_b @ g @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ Q3 @ V )
& ( ( inf_inf_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ ( set_a2 @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ g @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ Q3 ) ) ) )
= bot_bot_set_a ) ) ) ).
% G.apath_append_iff
thf(fact_1269_G_Oawalk__verts__append2,axiom,
! [U: a,P2: list_b,Q3: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ ( append_b @ P2 @ Q3 ) @ V )
=> ( ( arc_pr7493981781705774526ts_a_b @ g @ U @ ( append_b @ P2 @ Q3 ) )
= ( append_a @ ( butlast_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ ( arc_pr7493981781705774526ts_a_b @ g @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) @ Q3 ) ) ) ) ).
% G.awalk_verts_append2
thf(fact_1270_awalk__verts__append2,axiom,
! [U: a,P2: list_b,Q3: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ ( append_b @ P2 @ Q3 ) @ V )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P2 @ Q3 ) )
= ( append_a @ ( butlast_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( arc_pr7493981781705774526ts_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ Q3 ) ) ) ) ).
% awalk_verts_append2
thf(fact_1271_inner__verts__conv,axiom,
! [P2: list_b,U: a] :
( ( pre_inner_verts_a_b @ t @ P2 )
= ( butlast_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ).
% inner_verts_conv
thf(fact_1272_G_Oinner__verts__conv,axiom,
! [P2: list_b,U: a] :
( ( pre_inner_verts_a_b @ g @ P2 )
= ( butlast_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) ) ) ) ).
% G.inner_verts_conv
thf(fact_1273_G_Oreachable__vwalk__conv,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ g @ U @ V )
= ( ? [P4: list_a] :
( ( vertex_vwalk_a_b @ P4 @ g )
& ( ( hd_a @ P4 )
= U )
& ( ( last_a @ P4 )
= V ) ) ) ) ).
% G.reachable_vwalk_conv
thf(fact_1274_reachable__vwalk__conv,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ t @ U @ V )
= ( ? [P4: list_a] :
( ( vertex_vwalk_a_b @ P4 @ t )
& ( ( hd_a @ P4 )
= U )
& ( ( last_a @ P4 )
= V ) ) ) ) ).
% reachable_vwalk_conv
thf(fact_1275_awalk__imp__vwalk,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( vertex_vwalk_a_b @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) @ t ) ) ).
% awalk_imp_vwalk
thf(fact_1276_G_Oawalk__imp__vwalk,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U @ P2 @ V )
=> ( vertex_vwalk_a_b @ ( arc_pr7493981781705774526ts_a_b @ g @ U @ P2 ) @ g ) ) ).
% G.awalk_imp_vwalk
thf(fact_1277_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_1278_finite__verts,axiom,
finite_finite_a @ ( pre_ve642382030648772252t_unit @ t ) ).
% finite_verts
% Conjectures (1)
thf(conj_0,conjecture,
( ( graph_2016941059203891550ts_a_b @ g @ source @ u )
!= bot_bot_set_a ) ).
%------------------------------------------------------------------------------