%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP049^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n027.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:08 AM UTC 2026
% Result : Theorem 3.03s 1.63s
% Output : Refutation 3.03s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 3
% Syntax : Number of formulae : 18 ( 12 unt; 0 typ; 0 def)
% Number of atoms : 52 ( 14 equ; 0 cnn)
% Maximal formula atoms : 2 ( 2 avg)
% Number of connectives : 82 ( 10 ~; 3 |; 0 &; 66 @)
% ( 0 <=>; 3 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 3 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 90 ( 87 usr; 4 con; 0-4 aty)
% Number of variables : 0 ( 0 ^; 0 !; 0 ?; 0 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
product_prod_nat_nat: $tType ).
thf(type_def_6,type,
a: $tType ).
thf(type_def_8,type,
nat: $tType ).
thf(type_def_9,type,
set_Pr1986765409at_nat: $tType ).
thf(type_def_10,type,
set_nat: $tType ).
thf(type_def_11,type,
set_Pr1746169692at_nat: $tType ).
thf(type_def_12,type,
set_Pr1287749686at_nat: $tType ).
thf(type_def_13,type,
set_Pr1490359111at_nat: $tType ).
thf(type_def_14,type,
set_set_nat: $tType ).
thf(type_def_15,type,
set_se1612935105at_nat: $tType ).
thf(type_def_16,type,
list_P559422087at_nat: $tType ).
thf(type_def_17,type,
list_nat: $tType ).
thf(type_def_18,type,
produc1695820582at_nat: $tType ).
thf(type_def_19,type,
produc1271302400at_nat: $tType ).
thf(type_def_20,type,
produc842455143at_nat: $tType ).
thf(func_def_0,type,
set_Pr1490359111at_nat: $tType ).
thf(func_def_1,type,
produc842455143at_nat: $tType ).
thf(func_def_2,type,
set_Pr1287749686at_nat: $tType ).
thf(func_def_3,type,
set_Pr1746169692at_nat: $tType ).
thf(func_def_4,type,
produc1271302400at_nat: $tType ).
thf(func_def_5,type,
produc1695820582at_nat: $tType ).
thf(func_def_6,type,
set_se1612935105at_nat: $tType ).
thf(func_def_7,type,
list_P559422087at_nat: $tType ).
thf(func_def_8,type,
set_Pr1986765409at_nat: $tType ).
thf(func_def_9,type,
product_prod_nat_nat: $tType ).
thf(func_def_10,type,
set_set_nat: $tType ).
thf(func_def_11,type,
list_nat: $tType ).
thf(func_def_12,type,
set_nat: $tType ).
thf(func_def_13,type,
nat: $tType ).
thf(func_def_14,type,
a: $tType ).
thf(func_def_15,type,
edmond1517640972ysis_a: ( product_prod_nat_nat > a ) > $o ).
thf(func_def_16,type,
edmond475474835dges_a: ( product_prod_nat_nat > a ) > nat > nat > set_Pr1986765409at_nat ).
thf(func_def_17,type,
finite_finite_nat: set_nat > $o ).
thf(func_def_18,type,
finite772653738at_nat: set_Pr1986765409at_nat > $o ).
thf(func_def_19,type,
finite277291581at_nat: set_Pr1746169692at_nat > $o ).
thf(func_def_20,type,
finite2000257047at_nat: set_Pr1287749686at_nat > $o ).
thf(func_def_21,type,
finite48957584at_nat: set_Pr1490359111at_nat > $o ).
thf(func_def_22,type,
finite2012248349et_nat: set_set_nat > $o ).
thf(func_def_23,type,
finite1457549322at_nat: set_se1612935105at_nat > $o ).
thf(func_def_24,type,
finite_Graph_a: ( product_prod_nat_nat > a ) > $o ).
thf(func_def_25,type,
e_a: ( product_prod_nat_nat > a ) > set_Pr1986765409at_nat ).
thf(func_def_26,type,
v_a: ( product_prod_nat_nat > a ) > set_nat ).
thf(func_def_27,type,
adjacent_nodes_a: ( product_prod_nat_nat > a ) > nat > set_nat ).
thf(func_def_28,type,
connected_a: ( product_prod_nat_nat > a ) > nat > nat > $o ).
thf(func_def_29,type,
incoming_a: ( product_prod_nat_nat > a ) > nat > set_Pr1986765409at_nat ).
thf(func_def_30,type,
incoming_a2: ( product_prod_nat_nat > a ) > set_nat > set_Pr1986765409at_nat ).
thf(func_def_31,type,
isPath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).
thf(func_def_32,type,
isShortestPath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).
thf(func_def_33,type,
isSimplePath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).
thf(func_def_34,type,
outgoing_a: ( product_prod_nat_nat > a ) > nat > set_Pr1986765409at_nat ).
thf(func_def_35,type,
outgoing_a2: ( product_prod_nat_nat > a ) > set_nat > set_Pr1986765409at_nat ).
thf(func_def_36,type,
reachableNodes_a: ( product_prod_nat_nat > a ) > nat > set_nat ).
thf(func_def_37,type,
set_nat2: list_nat > set_nat ).
thf(func_def_38,type,
set_Pr2131844118at_nat: list_P559422087at_nat > set_Pr1986765409at_nat ).
thf(func_def_39,type,
ord_le1578155910_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(func_def_40,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_41,type,
ord_le1039616028_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
thf(func_def_42,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_43,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(func_def_44,type,
ord_le841296385at_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > $o ).
thf(func_def_45,type,
ord_le148299708at_nat: set_Pr1746169692at_nat > set_Pr1746169692at_nat > $o ).
thf(func_def_46,type,
ord_le1837363350at_nat: set_Pr1287749686at_nat > set_Pr1287749686at_nat > $o ).
thf(func_def_47,type,
ord_le465611495at_nat: set_Pr1490359111at_nat > set_Pr1490359111at_nat > $o ).
thf(func_def_48,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(func_def_49,type,
produc1933845336at_nat: nat > product_prod_nat_nat > produc1695820582at_nat ).
thf(func_def_50,type,
produc947540346at_nat: product_prod_nat_nat > nat > produc1271302400at_nat ).
thf(func_def_51,type,
produc1168807639at_nat: product_prod_nat_nat > product_prod_nat_nat > produc842455143at_nat ).
thf(func_def_52,type,
produc45129834at_nat: set_nat > ( nat > set_nat ) > set_Pr1986765409at_nat ).
thf(func_def_53,type,
produc894163943at_nat: set_nat > ( nat > set_Pr1986765409at_nat ) > set_Pr1746169692at_nat ).
thf(func_def_54,type,
produc2055342601at_nat: set_Pr1986765409at_nat > ( product_prod_nat_nat > set_nat ) > set_Pr1287749686at_nat ).
thf(func_def_55,type,
produc2056081288at_nat: set_Pr1986765409at_nat > ( product_prod_nat_nat > set_Pr1986765409at_nat ) > set_Pr1490359111at_nat ).
thf(func_def_56,type,
product_swap_nat_nat: product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_57,type,
domain_nat_nat: set_Pr1986765409at_nat > set_nat ).
thf(func_def_58,type,
domain37934156at_nat: set_Pr1490359111at_nat > set_Pr1986765409at_nat ).
thf(func_def_59,type,
image_nat_nat: set_Pr1986765409at_nat > set_nat > set_nat ).
thf(func_def_60,type,
image_981520924at_nat: set_Pr1746169692at_nat > set_nat > set_Pr1986765409at_nat ).
thf(func_def_61,type,
image_2142699582at_nat: set_Pr1287749686at_nat > set_Pr1986765409at_nat > set_nat ).
thf(func_def_62,type,
image_1655366419at_nat: set_Pr1490359111at_nat > set_Pr1986765409at_nat > set_Pr1986765409at_nat ).
thf(func_def_63,type,
range_nat_nat: set_Pr1986765409at_nat > set_nat ).
thf(func_def_64,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_65,type,
collec7649004at_nat: ( product_prod_nat_nat > $o ) > set_Pr1986765409at_nat ).
thf(func_def_66,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(func_def_67,type,
collec1606769740at_nat: ( set_Pr1986765409at_nat > $o ) > set_se1612935105at_nat ).
thf(func_def_68,type,
transi1645084429cl_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat ).
thf(func_def_69,type,
transi1484096900at_nat: set_Pr1490359111at_nat > set_Pr1490359111at_nat ).
thf(func_def_70,type,
member_nat: nat > set_nat > $o ).
thf(func_def_71,type,
member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
thf(func_def_72,type,
member1293241661at_nat: produc1695820582at_nat > set_Pr1746169692at_nat > $o ).
thf(func_def_73,type,
member868723479at_nat: produc1271302400at_nat > set_Pr1287749686at_nat > $o ).
thf(func_def_74,type,
member2027625872at_nat: produc842455143at_nat > set_Pr1490359111at_nat > $o ).
thf(func_def_75,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(func_def_76,type,
member298845450at_nat: set_Pr1986765409at_nat > set_se1612935105at_nat > $o ).
thf(func_def_77,type,
c: product_prod_nat_nat > a ).
thf(func_def_78,type,
s: nat ).
thf(func_def_79,type,
t: nat ).
thf(func_def_94,type,
sK0: ( set_Pr1986765409at_nat > set_Pr1986765409at_nat ) > set_Pr1986765409at_nat ).
thf(func_def_95,type,
sK1: ( set_Pr1986765409at_nat > set_Pr1986765409at_nat ) > set_Pr1986765409at_nat ).
thf(func_def_96,type,
sK2: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > set_Pr1986765409at_nat > set_Pr1986765409at_nat > product_prod_nat_nat ).
thf(func_def_97,type,
sK3: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > set_Pr1986765409at_nat > set_Pr1986765409at_nat > product_prod_nat_nat ).
thf(func_def_98,type,
sK4: ( set_Pr1986765409at_nat > set_Pr1986765409at_nat ) > set_Pr1986765409at_nat ).
thf(func_def_99,type,
sK5: ( set_Pr1986765409at_nat > set_Pr1986765409at_nat ) > set_Pr1986765409at_nat ).
thf(func_def_100,type,
sK6: nat > ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat ).
thf(func_def_101,type,
sK7: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > product_prod_nat_nat ).
thf(func_def_102,type,
sK8: nat > nat > list_P559422087at_nat ).
thf(func_def_103,type,
sK9: ( set_Pr1986765409at_nat > set_Pr1986765409at_nat ) > set_Pr1986765409at_nat ).
thf(func_def_104,type,
sK10: ( set_Pr1986765409at_nat > set_Pr1986765409at_nat ) > set_Pr1986765409at_nat ).
thf(func_def_105,type,
sK11: ( product_prod_nat_nat > nat > $o ) > set_Pr1986765409at_nat > set_nat > nat ).
thf(func_def_106,type,
sK12: ( product_prod_nat_nat > nat > $o ) > set_Pr1986765409at_nat > set_nat > product_prod_nat_nat ).
thf(func_def_107,type,
sK13: set_Pr1986765409at_nat > list_P559422087at_nat ).
thf(func_def_108,type,
sK14: ( set_Pr1986765409at_nat > set_Pr1986765409at_nat ) > set_Pr1986765409at_nat ).
thf(func_def_109,type,
sK15: ( set_Pr1986765409at_nat > set_Pr1986765409at_nat ) > set_Pr1986765409at_nat ).
thf(func_def_110,type,
sK16: ( product_prod_nat_nat > $o ) > product_prod_nat_nat ).
thf(func_def_111,type,
sK17: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > product_prod_nat_nat ).
thf(func_def_112,type,
sK18: set_Pr1986765409at_nat > set_Pr1986765409at_nat > nat ).
thf(func_def_113,type,
sK19: set_Pr1986765409at_nat > set_Pr1986765409at_nat > nat ).
thf(func_def_114,type,
sK20: list_P559422087at_nat > set_Pr1986765409at_nat > product_prod_nat_nat ).
thf(func_def_116,type,
ph22:
!>[X0: $tType] : X0 ).
thf(f1195,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1194,f1172]) ).
thf(f1172,plain,
( ( finite772653738at_nat @ ( e_a @ c ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f741]) ).
thf(f741,plain,
( ( finite772653738at_nat @ ( e_a @ c ) )
= $true ),
inference(fool_elimination,[status(thm)],[f740]) ).
thf(f740,plain,
finite772653738at_nat @ ( e_a @ c ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,axiom,
finite772653738at_nat @ ( e_a @ c ),
file('/export/starexec/sandbox2/tmp/tmp.tqDOf96qQm/DTF2THF_12988.p',fact_3_finite__E) ).
thf(f1194,plain,
( ( finite772653738at_nat @ ( e_a @ c ) )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f1193]) ).
thf(f1193,plain,
( ( $true != $true )
| ( ( finite772653738at_nat @ ( e_a @ c ) )
!= $true ) ),
inference(superposition,[status(thm)],[f1165,f1122]) ).
thf(f1122,plain,
( ( ( finite772653738at_nat @ ( edmond475474835dges_a @ c @ s @ t ) )
= $true )
| ( ( finite772653738at_nat @ ( e_a @ c ) )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1041]) ).
thf(f1041,plain,
( ( ( finite772653738at_nat @ ( e_a @ c ) )
!= $true )
| ( ( finite772653738at_nat @ ( edmond475474835dges_a @ c @ s @ t ) )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f791]) ).
thf(f791,plain,
( ( ( finite772653738at_nat @ ( e_a @ c ) )
= $true )
=> ( ( finite772653738at_nat @ ( edmond475474835dges_a @ c @ s @ t ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f790]) ).
thf(f790,plain,
( ( finite772653738at_nat @ ( e_a @ c ) )
=> ( finite772653738at_nat @ ( edmond475474835dges_a @ c @ s @ t ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
( ( finite772653738at_nat @ ( e_a @ c ) )
=> ( finite772653738at_nat @ ( edmond475474835dges_a @ c @ s @ t ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.tqDOf96qQm/DTF2THF_12988.p',fact_1__092_060open_062finite_AE_A_092_060Longrightarrow_062_Afinite_AspEdges_092_060close_062) ).
thf(f1165,plain,
( ( finite772653738at_nat @ ( edmond475474835dges_a @ c @ s @ t ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1018]) ).
thf(f1018,plain,
( ( finite772653738at_nat @ ( edmond475474835dges_a @ c @ s @ t ) )
!= $true ),
inference(flattening,[status(thm)],[f737]) ).
thf(f737,plain,
( ( finite772653738at_nat @ ( edmond475474835dges_a @ c @ s @ t ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f736]) ).
thf(f736,plain,
~ ( finite772653738at_nat @ ( edmond475474835dges_a @ c @ s @ t ) ),
inference(rectify,[status(thm)],[f356]) ).
thf(f356,negated_conjecture,
~ ( finite772653738at_nat @ ( edmond475474835dges_a @ c @ s @ t ) ),
inference(negated_conjecture,[status(cth)],[f355]) ).
thf(f355,conjecture,
finite772653738at_nat @ ( edmond475474835dges_a @ c @ s @ t ),
file('/export/starexec/sandbox2/tmp/tmp.tqDOf96qQm/DTF2THF_12988.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP049^1 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n027.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 08:24:52 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.18/0.45 ---- Original DTF file ---
% 0.18/0.45 thf(spec,logic,$$dhol).
% 0.18/0.45 %------------------------------------------------------------------------------
% 0.18/0.45 % File : ITP049^1 : TPTP v9.2.1. Released v7.5.0.
% 0.18/0.45 % Domain : Interactive Theorem Proving
% 0.18/0.45 % Problem : Sledgehammer EdmondsKarp_Termination_Abstract problem prob_229__7588066_1
% 0.18/0.45 % Version : Especial.
% 0.18/0.45 % English :
% 0.18/0.45
% 0.18/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.18/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.18/0.45 % Source : [Des21]
% 0.18/0.45 % Names : EdmondsKarp_Termination_Abstract/prob_229__7588066_1 [Des21]
% 0.18/0.45
% 0.18/0.45 % Status : Theorem
% 0.18/0.45 % Rating : 0.33 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% 0.18/0.45 % Syntax : Number of formulae : 435 ( 102 unt; 80 typ; 0 def)
% 0.18/0.45 % Number of atoms : 1099 ( 200 equ; 0 cnn)
% 0.18/0.45 % Maximal formula atoms : 12 ( 3 avg)
% 0.18/0.45 % Number of connectives : 3642 ( 106 ~; 10 |; 89 &;2864 @)
% 0.18/0.45 % ( 0 <=>; 573 =>; 0 <=; 0 <~>)
% 0.18/0.45 % Maximal formula depth : 22 ( 8 avg)
% 0.18/0.45 % Number of types : 16 ( 15 usr)
% 0.18/0.45 % Number of type conns : 343 ( 343 >; 0 *; 0 +; 0 <<)
% 0.18/0.45 % Number of symbols : 66 ( 65 usr; 2 con; 0-4 aty)
% 0.18/0.45 % Number of variables : 1227 ( 124 ^;1026 !; 77 ?;1227 :)
% 0.18/0.45 % SPC : TH0_THM_EQU_NAR
% 0.18/0.45
% 0.18/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 15:30:52.673
% 0.18/0.45 %------------------------------------------------------------------------------
% 0.18/0.45 % Could-be-implicit typings (15)
% 0.18/0.45 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.18/0.45 set_Pr1490359111at_nat: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.18/0.45 produc842455143at_nat: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
% 0.18/0.45 set_Pr1287749686at_nat: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.18/0.45 set_Pr1746169692at_nat: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
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% 0.18/0.45 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.18/0.45 set_set_nat: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
% 0.18/0.45 list_nat: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 0.18/0.45 set_nat: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Nat__Onat,type,
% 0.18/0.45 nat: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_tf__a,type,
% 0.18/0.45 a: $tType ).
% 0.18/0.45
% 0.18/0.45 % Explicit typings (65)
% 0.18/0.45 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__pndqeoznpl_Oek__analysis_001tf__a,type,
% 0.18/0.45 edmond1517640972ysis_a: ( product_prod_nat_nat > a ) > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__pndqeoznpl_Oek__analysis__defs_OspEdges_001tf__a,type,
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% 0.18/0.45
% 0.18/0.45 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
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% 0.18/0.45 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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% 0.18/0.45 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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% 0.18/0.45 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.18/0.45 finite48957584at_nat: set_Pr1490359111at_nat > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.18/0.45 finite2012248349et_nat: set_set_nat > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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% 0.18/0.45
% 0.18/0.45 thf(sy_c_Graph_OFinite__Graph_001tf__a,type,
% 0.18/0.45 finite_Graph_a: ( product_prod_nat_nat > a ) > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Graph_OGraph_OE_001tf__a,type,
% 0.18/0.45 e_a: ( product_prod_nat_nat > a ) > set_Pr1986765409at_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Graph_OGraph_OV_001tf__a,type,
% 0.18/0.45 v_a: ( product_prod_nat_nat > a ) > set_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Graph_OGraph_Oadjacent__nodes_001tf__a,type,
% 0.18/0.45 adjacent_nodes_a: ( product_prod_nat_nat > a ) > nat > set_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Graph_OGraph_Oconnected_001tf__a,type,
% 0.18/0.45 connected_a: ( product_prod_nat_nat > a ) > nat > nat > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Graph_OGraph_Oincoming_001tf__a,type,
% 0.18/0.45 incoming_a: ( product_prod_nat_nat > a ) > nat > set_Pr1986765409at_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Graph_OGraph_Oincoming_H_001tf__a,type,
% 0.18/0.45 incoming_a2: ( product_prod_nat_nat > a ) > set_nat > set_Pr1986765409at_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Graph_OGraph_OisPath_001tf__a,type,
% 0.18/0.45 isPath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Graph_OGraph_OisShortestPath_001tf__a,type,
% 0.18/0.45 isShortestPath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Graph_OGraph_OisSimplePath_001tf__a,type,
% 0.18/0.45 isSimplePath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Graph_OGraph_Ooutgoing_001tf__a,type,
% 0.18/0.45 outgoing_a: ( product_prod_nat_nat > a ) > nat > set_Pr1986765409at_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Graph_OGraph_Ooutgoing_H_001tf__a,type,
% 0.18/0.45 outgoing_a2: ( product_prod_nat_nat > a ) > set_nat > set_Pr1986765409at_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Graph_OGraph_OreachableNodes_001tf__a,type,
% 0.18/0.45 reachableNodes_a: ( product_prod_nat_nat > a ) > nat > set_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
% 0.18/0.45 set_nat2: list_nat > set_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.18/0.45 set_Pr2131844118at_nat: list_P559422087at_nat > set_Pr1986765409at_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
% 0.18/0.45 ord_le1578155910_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.18/0.45 ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 0.18/0.45 ord_le1039616028_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.18/0.45 ord_less_eq_nat: nat > nat > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.18/0.45 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.18/0.45 ord_le841296385at_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.18/0.45 ord_le148299708at_nat: set_Pr1746169692at_nat > set_Pr1746169692at_nat > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
% 0.18/0.45 ord_le1837363350at_nat: set_Pr1287749686at_nat > set_Pr1287749686at_nat > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.18/0.45 ord_le465611495at_nat: set_Pr1490359111at_nat > set_Pr1490359111at_nat > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.18/0.45 product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.18/0.45 produc1933845336at_nat: nat > product_prod_nat_nat > produc1695820582at_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 0.18/0.45 produc947540346at_nat: product_prod_nat_nat > nat > produc1271302400at_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.18/0.45 produc1168807639at_nat: product_prod_nat_nat > product_prod_nat_nat > produc842455143at_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.18/0.45 produc45129834at_nat: set_nat > ( nat > set_nat ) > set_Pr1986765409at_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.18/0.45 produc894163943at_nat: set_nat > ( nat > set_Pr1986765409at_nat ) > set_Pr1746169692at_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Product__Type_OSigma_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 0.18/0.45 produc2055342601at_nat: set_Pr1986765409at_nat > ( product_prod_nat_nat > set_nat ) > set_Pr1287749686at_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Product__Type_OSigma_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.18/0.45 produc2056081288at_nat: set_Pr1986765409at_nat > ( product_prod_nat_nat > set_Pr1986765409at_nat ) > set_Pr1490359111at_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Product__Type_Oprod_Oswap_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.18/0.45 product_swap_nat_nat: product_prod_nat_nat > product_prod_nat_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Relation_ODomain_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.18/0.45 domain_nat_nat: set_Pr1986765409at_nat > set_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Relation_ODomain_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.18/0.45 domain37934156at_nat: set_Pr1490359111at_nat > set_Pr1986765409at_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Relation_OImage_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.18/0.45 image_nat_nat: set_Pr1986765409at_nat > set_nat > set_nat ).
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% 0.18/0.45 thf(sy_c_Relation_OImage_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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% 0.18/0.45 thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
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% 0.18/0.45 thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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% 0.18/0.45
% 0.18/0.45 thf(sy_c_Relation_ORange_001t__Nat__Onat_001t__Nat__Onat,type,
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% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 0.18/0.45 collect_nat: ( nat > $o ) > set_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.18/0.45 collec7649004at_nat: ( product_prod_nat_nat > $o ) > set_Pr1986765409at_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.18/0.45 collect_set_nat: ( set_nat > $o ) > set_set_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.18/0.45 collec1606769740at_nat: ( set_Pr1986765409at_nat > $o ) > set_se1612935105at_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
% 0.18/0.45 transi1645084429cl_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.18/0.45 transi1484096900at_nat: set_Pr1490359111at_nat > set_Pr1490359111at_nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Nat__Onat,type,
% 0.18/0.45 member_nat: nat > set_nat > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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% 0.18/0.45 thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
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% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.18/0.45 member2027625872at_nat: produc842455143at_nat > set_Pr1490359111at_nat > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.18/0.45 member_set_nat: set_nat > set_set_nat > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.18/0.45 member298845450at_nat: set_Pr1986765409at_nat > set_se1612935105at_nat > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_c,type,
% 0.18/0.45 c: product_prod_nat_nat > a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_s,type,
% 0.18/0.45 s: nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_t,type,
% 0.18/0.45 t: nat ).
% 0.18/0.45
% 0.18/0.45 % Relevant facts (354)
% 0.18/0.45 thf(fact_0_ek__analysis__axioms,axiom,
% 0.18/0.45 edmond1517640972ysis_a @ c ).
% 0.18/0.45
% 0.18/0.45 % ek_analysis_axioms
% 0.18/0.45 thf(fact_1__092_060open_062finite_AE_A_092_060Longrightarrow_062_Afinite_AspEdges_092_060close_062,axiom,
% 0.18/0.45 ( ( finite772653738at_nat @ ( e_a @ c ) )
% 0.18/0.45 => ( finite772653738at_nat @ ( edmond475474835dges_a @ c @ s @ t ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % \<open>finite E \<Longrightarrow> finite spEdges\<close>
% 0.18/0.45 thf(fact_2_ek__analysis__defs_OspEdges_Ocong,axiom,
% 0.18/0.45 edmond475474835dges_a = edmond475474835dges_a ).
% 0.18/0.45
% 0.18/0.45 % ek_analysis_defs.spEdges.cong
% 0.18/0.45 thf(fact_3_finite__E,axiom,
% 0.18/0.45 finite772653738at_nat @ ( e_a @ c ) ).
% 0.18/0.45
% 0.18/0.45 % finite_E
% 0.18/0.45 thf(fact_4_spEdges__ss__E,axiom,
% 0.18/0.45 ord_le841296385at_nat @ ( edmond475474835dges_a @ c @ s @ t ) @ ( e_a @ c ) ).
% 0.18/0.45
% 0.18/0.45 % spEdges_ss_E
% 0.18/0.45 thf(fact_5_Finite__Graph__axioms,axiom,
% 0.18/0.45 finite_Graph_a @ c ).
% 0.18/0.45
% 0.18/0.45 % Finite_Graph_axioms
% 0.18/0.45 thf(fact_6_Finite__Graph__EI,axiom,
% 0.18/0.45 ( ( finite772653738at_nat @ ( e_a @ c ) )
% 0.18/0.45 => ( finite_Graph_a @ c ) ) ).
% 0.18/0.45
% 0.18/0.45 % Finite_Graph_EI
% 0.18/0.45 thf(fact_7_connected__refl,axiom,
% 0.18/0.45 ! [V: nat] : ( connected_a @ c @ V @ V ) ).
% 0.18/0.45
% 0.18/0.45 % connected_refl
% 0.18/0.45 thf(fact_8_spEdges__def,axiom,
% 0.18/0.45 ( ( edmond475474835dges_a @ c @ s @ t )
% 0.18/0.45 = ( collec7649004at_nat
% 0.18/0.45 @ ^ [E: product_prod_nat_nat] :
% 0.18/0.45 ? [P: list_P559422087at_nat] :
% 0.18/0.45 ( ( member701585322at_nat @ E @ ( set_Pr2131844118at_nat @ P ) )
% 0.18/0.45 & ( isShortestPath_a @ c @ s @ P @ t ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % spEdges_def
% 0.18/0.45 thf(fact_9_finite__Range,axiom,
% 0.18/0.45 ! [R: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( finite772653738at_nat @ R )
% 0.18/0.45 => ( finite_finite_nat @ ( range_nat_nat @ R ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_Range
% 0.18/0.45 thf(fact_10_finite__Domain,axiom,
% 0.18/0.45 ! [R: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( finite772653738at_nat @ R )
% 0.18/0.45 => ( finite_finite_nat @ ( domain_nat_nat @ R ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_Domain
% 0.18/0.45 thf(fact_11_Efin__imp__Vfin,axiom,
% 0.18/0.45 ( ( finite772653738at_nat @ ( e_a @ c ) )
% 0.18/0.45 => ( finite_finite_nat @ ( v_a @ c ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Efin_imp_Vfin
% 0.18/0.45 thf(fact_12_connected__inV__iff,axiom,
% 0.18/0.45 ! [U: nat,V: nat] :
% 0.18/0.45 ( ( connected_a @ c @ U @ V )
% 0.18/0.45 => ( ( member_nat @ V @ ( v_a @ c ) )
% 0.18/0.45 = ( member_nat @ U @ ( v_a @ c ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % connected_inV_iff
% 0.18/0.45 thf(fact_13_obtain__shortest__path,axiom,
% 0.18/0.45 ! [U: nat,V: nat] :
% 0.18/0.45 ( ( connected_a @ c @ U @ V )
% 0.18/0.45 => ~ ! [P2: list_P559422087at_nat] :
% 0.18/0.45 ~ ( isShortestPath_a @ c @ U @ P2 @ V ) ) ).
% 0.18/0.45
% 0.18/0.45 % obtain_shortest_path
% 0.18/0.45 thf(fact_14_finite__Collect__conjI,axiom,
% 0.18/0.45 ! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 0.18/0.45 ( ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 0.18/0.45 | ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) ) )
% 0.18/0.45 => ( finite772653738at_nat
% 0.18/0.45 @ ( collec7649004at_nat
% 0.18/0.45 @ ^ [X: product_prod_nat_nat] :
% 0.18/0.45 ( ( P3 @ X )
% 0.18/0.45 & ( Q @ X ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_Collect_conjI
% 0.18/0.45 thf(fact_15_finite__Collect__conjI,axiom,
% 0.18/0.45 ! [P3: nat > $o,Q: nat > $o] :
% 0.18/0.45 ( ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 0.18/0.45 | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
% 0.18/0.45 => ( finite_finite_nat
% 0.18/0.45 @ ( collect_nat
% 0.18/0.45 @ ^ [X: nat] :
% 0.18/0.45 ( ( P3 @ X )
% 0.18/0.45 & ( Q @ X ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_Collect_conjI
% 0.18/0.45 thf(fact_16_finite__Collect__disjI,axiom,
% 0.18/0.45 ! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 0.18/0.45 ( ( finite772653738at_nat
% 0.18/0.45 @ ( collec7649004at_nat
% 0.18/0.45 @ ^ [X: product_prod_nat_nat] :
% 0.18/0.45 ( ( P3 @ X )
% 0.18/0.45 | ( Q @ X ) ) ) )
% 0.18/0.45 = ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 0.18/0.45 & ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_Collect_disjI
% 0.18/0.45 thf(fact_17_finite__Collect__disjI,axiom,
% 0.18/0.45 ! [P3: nat > $o,Q: nat > $o] :
% 0.18/0.45 ( ( finite_finite_nat
% 0.18/0.45 @ ( collect_nat
% 0.18/0.45 @ ^ [X: nat] :
% 0.18/0.45 ( ( P3 @ X )
% 0.18/0.45 | ( Q @ X ) ) ) )
% 0.18/0.45 = ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 0.18/0.45 & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_Collect_disjI
% 0.18/0.45 thf(fact_18_finite__Collect__bounded__ex,axiom,
% 0.18/0.45 ! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 0.18/0.45 ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 0.18/0.45 => ( ( finite772653738at_nat
% 0.18/0.45 @ ( collec7649004at_nat
% 0.18/0.45 @ ^ [X: product_prod_nat_nat] :
% 0.18/0.45 ? [Y: product_prod_nat_nat] :
% 0.18/0.45 ( ( P3 @ Y )
% 0.18/0.45 & ( Q @ X @ Y ) ) ) )
% 0.18/0.45 = ( ! [Y: product_prod_nat_nat] :
% 0.18/0.45 ( ( P3 @ Y )
% 0.18/0.45 => ( finite772653738at_nat
% 0.18/0.45 @ ( collec7649004at_nat
% 0.18/0.45 @ ^ [X: product_prod_nat_nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_Collect_bounded_ex
% 0.18/0.45 thf(fact_19_finite__Collect__bounded__ex,axiom,
% 0.18/0.45 ! [P3: product_prod_nat_nat > $o,Q: nat > product_prod_nat_nat > $o] :
% 0.18/0.45 ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 0.18/0.45 => ( ( finite_finite_nat
% 0.18/0.45 @ ( collect_nat
% 0.18/0.45 @ ^ [X: nat] :
% 0.18/0.45 ? [Y: product_prod_nat_nat] :
% 0.18/0.45 ( ( P3 @ Y )
% 0.18/0.45 & ( Q @ X @ Y ) ) ) )
% 0.18/0.45 = ( ! [Y: product_prod_nat_nat] :
% 0.18/0.45 ( ( P3 @ Y )
% 0.18/0.45 => ( finite_finite_nat
% 0.18/0.45 @ ( collect_nat
% 0.18/0.45 @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_Collect_bounded_ex
% 0.18/0.45 thf(fact_20_finite__Collect__bounded__ex,axiom,
% 0.18/0.45 ! [P3: nat > $o,Q: product_prod_nat_nat > nat > $o] :
% 0.18/0.45 ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 0.18/0.45 => ( ( finite772653738at_nat
% 0.18/0.45 @ ( collec7649004at_nat
% 0.18/0.45 @ ^ [X: product_prod_nat_nat] :
% 0.18/0.45 ? [Y: nat] :
% 0.18/0.45 ( ( P3 @ Y )
% 0.18/0.45 & ( Q @ X @ Y ) ) ) )
% 0.18/0.45 = ( ! [Y: nat] :
% 0.18/0.45 ( ( P3 @ Y )
% 0.18/0.45 => ( finite772653738at_nat
% 0.18/0.45 @ ( collec7649004at_nat
% 0.18/0.45 @ ^ [X: product_prod_nat_nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_Collect_bounded_ex
% 0.18/0.45 thf(fact_21_finite__Collect__bounded__ex,axiom,
% 0.18/0.45 ! [P3: nat > $o,Q: nat > nat > $o] :
% 0.18/0.45 ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 0.18/0.45 => ( ( finite_finite_nat
% 0.18/0.45 @ ( collect_nat
% 0.18/0.45 @ ^ [X: nat] :
% 0.18/0.45 ? [Y: nat] :
% 0.18/0.45 ( ( P3 @ Y )
% 0.18/0.45 & ( Q @ X @ Y ) ) ) )
% 0.18/0.45 = ( ! [Y: nat] :
% 0.18/0.45 ( ( P3 @ Y )
% 0.18/0.45 => ( finite_finite_nat
% 0.18/0.45 @ ( collect_nat
% 0.18/0.45 @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_Collect_bounded_ex
% 0.18/0.45 thf(fact_22_finite__Collect__subsets,axiom,
% 0.18/0.45 ! [A: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( finite772653738at_nat @ A )
% 0.18/0.45 => ( finite1457549322at_nat
% 0.18/0.45 @ ( collec1606769740at_nat
% 0.18/0.45 @ ^ [B: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ B @ A ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_Collect_subsets
% 0.18/0.45 thf(fact_23_finite__Collect__subsets,axiom,
% 0.18/0.45 ! [A: set_nat] :
% 0.18/0.45 ( ( finite_finite_nat @ A )
% 0.18/0.45 => ( finite2012248349et_nat
% 0.18/0.45 @ ( collect_set_nat
% 0.18/0.45 @ ^ [B: set_nat] : ( ord_less_eq_set_nat @ B @ A ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_Collect_subsets
% 0.18/0.45 thf(fact_24_finite__V,axiom,
% 0.18/0.45 finite_finite_nat @ ( v_a @ c ) ).
% 0.18/0.45
% 0.18/0.45 % finite_V
% 0.18/0.45 thf(fact_25_Vfin__imp__Efin,axiom,
% 0.18/0.45 ( ( finite_finite_nat @ ( v_a @ c ) )
% 0.18/0.45 => ( finite772653738at_nat @ ( e_a @ c ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Vfin_imp_Efin
% 0.18/0.45 thf(fact_26_Range__mono,axiom,
% 0.18/0.45 ! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( ord_le841296385at_nat @ R @ S )
% 0.18/0.45 => ( ord_less_eq_set_nat @ ( range_nat_nat @ R ) @ ( range_nat_nat @ S ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Range_mono
% 0.18/0.45 thf(fact_27_Domain__mono,axiom,
% 0.18/0.45 ! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( ord_le841296385at_nat @ R @ S )
% 0.18/0.45 => ( ord_less_eq_set_nat @ ( domain_nat_nat @ R ) @ ( domain_nat_nat @ S ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Domain_mono
% 0.18/0.45 thf(fact_28_finite__image__set,axiom,
% 0.18/0.45 ! [P3: product_prod_nat_nat > $o,F: product_prod_nat_nat > product_prod_nat_nat] :
% 0.18/0.45 ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 0.18/0.45 => ( finite772653738at_nat
% 0.18/0.45 @ ( collec7649004at_nat
% 0.18/0.45 @ ^ [Uu: product_prod_nat_nat] :
% 0.18/0.45 ? [X: product_prod_nat_nat] :
% 0.18/0.45 ( ( Uu
% 0.18/0.45 = ( F @ X ) )
% 0.18/0.45 & ( P3 @ X ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_image_set
% 0.18/0.45 thf(fact_29_finite__image__set,axiom,
% 0.18/0.45 ! [P3: product_prod_nat_nat > $o,F: product_prod_nat_nat > nat] :
% 0.18/0.45 ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 0.18/0.45 => ( finite_finite_nat
% 0.18/0.45 @ ( collect_nat
% 0.18/0.45 @ ^ [Uu: nat] :
% 0.18/0.45 ? [X: product_prod_nat_nat] :
% 0.18/0.45 ( ( Uu
% 0.18/0.45 = ( F @ X ) )
% 0.18/0.45 & ( P3 @ X ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_image_set
% 0.18/0.45 thf(fact_30_finite__image__set,axiom,
% 0.18/0.45 ! [P3: nat > $o,F: nat > product_prod_nat_nat] :
% 0.18/0.45 ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 0.18/0.45 => ( finite772653738at_nat
% 0.18/0.45 @ ( collec7649004at_nat
% 0.18/0.45 @ ^ [Uu: product_prod_nat_nat] :
% 0.18/0.45 ? [X: nat] :
% 0.18/0.45 ( ( Uu
% 0.18/0.45 = ( F @ X ) )
% 0.18/0.45 & ( P3 @ X ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_image_set
% 0.18/0.45 thf(fact_31_finite__image__set,axiom,
% 0.18/0.45 ! [P3: nat > $o,F: nat > nat] :
% 0.18/0.45 ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 0.18/0.45 => ( finite_finite_nat
% 0.18/0.45 @ ( collect_nat
% 0.18/0.45 @ ^ [Uu: nat] :
% 0.18/0.45 ? [X: nat] :
% 0.18/0.45 ( ( Uu
% 0.18/0.45 = ( F @ X ) )
% 0.18/0.45 & ( P3 @ X ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_image_set
% 0.18/0.45 thf(fact_32_finite__image__set2,axiom,
% 0.18/0.45 ! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o,F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 0.18/0.45 ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 0.18/0.45 => ( ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) )
% 0.18/0.45 => ( finite772653738at_nat
% 0.18/0.45 @ ( collec7649004at_nat
% 0.18/0.45 @ ^ [Uu: product_prod_nat_nat] :
% 0.18/0.45 ? [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 0.18/0.45 ( ( Uu
% 0.18/0.45 = ( F @ X @ Y ) )
% 0.18/0.45 & ( P3 @ X )
% 0.18/0.45 & ( Q @ Y ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_image_set2
% 0.18/0.45 thf(fact_33_finite__image__set2,axiom,
% 0.18/0.45 ! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o,F: product_prod_nat_nat > product_prod_nat_nat > nat] :
% 0.18/0.45 ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 0.18/0.45 => ( ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) )
% 0.18/0.45 => ( finite_finite_nat
% 0.18/0.45 @ ( collect_nat
% 0.18/0.45 @ ^ [Uu: nat] :
% 0.18/0.45 ? [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 0.18/0.45 ( ( Uu
% 0.18/0.45 = ( F @ X @ Y ) )
% 0.18/0.45 & ( P3 @ X )
% 0.18/0.45 & ( Q @ Y ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_image_set2
% 0.18/0.45 thf(fact_34_finite__image__set2,axiom,
% 0.18/0.45 ! [P3: product_prod_nat_nat > $o,Q: nat > $o,F: product_prod_nat_nat > nat > product_prod_nat_nat] :
% 0.18/0.45 ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 0.18/0.45 => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 0.18/0.45 => ( finite772653738at_nat
% 0.18/0.45 @ ( collec7649004at_nat
% 0.18/0.45 @ ^ [Uu: product_prod_nat_nat] :
% 0.18/0.45 ? [X: product_prod_nat_nat,Y: nat] :
% 0.18/0.45 ( ( Uu
% 0.18/0.45 = ( F @ X @ Y ) )
% 0.18/0.45 & ( P3 @ X )
% 0.18/0.45 & ( Q @ Y ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_image_set2
% 0.18/0.45 thf(fact_35_finite__image__set2,axiom,
% 0.18/0.45 ! [P3: product_prod_nat_nat > $o,Q: nat > $o,F: product_prod_nat_nat > nat > nat] :
% 0.18/0.45 ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 0.18/0.45 => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 0.18/0.45 => ( finite_finite_nat
% 0.18/0.45 @ ( collect_nat
% 0.18/0.45 @ ^ [Uu: nat] :
% 0.18/0.45 ? [X: product_prod_nat_nat,Y: nat] :
% 0.18/0.45 ( ( Uu
% 0.18/0.45 = ( F @ X @ Y ) )
% 0.18/0.45 & ( P3 @ X )
% 0.18/0.45 & ( Q @ Y ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_image_set2
% 0.18/0.45 thf(fact_36_finite__image__set2,axiom,
% 0.18/0.45 ! [P3: nat > $o,Q: product_prod_nat_nat > $o,F: nat > product_prod_nat_nat > product_prod_nat_nat] :
% 0.18/0.45 ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 0.18/0.45 => ( ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) )
% 0.18/0.45 => ( finite772653738at_nat
% 0.18/0.45 @ ( collec7649004at_nat
% 0.18/0.45 @ ^ [Uu: product_prod_nat_nat] :
% 0.18/0.45 ? [X: nat,Y: product_prod_nat_nat] :
% 0.18/0.45 ( ( Uu
% 0.18/0.45 = ( F @ X @ Y ) )
% 0.18/0.45 & ( P3 @ X )
% 0.18/0.45 & ( Q @ Y ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_image_set2
% 0.18/0.45 thf(fact_37_finite__image__set2,axiom,
% 0.18/0.45 ! [P3: nat > $o,Q: product_prod_nat_nat > $o,F: nat > product_prod_nat_nat > nat] :
% 0.18/0.45 ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 0.18/0.45 => ( ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) )
% 0.18/0.45 => ( finite_finite_nat
% 0.18/0.45 @ ( collect_nat
% 0.18/0.45 @ ^ [Uu: nat] :
% 0.18/0.45 ? [X: nat,Y: product_prod_nat_nat] :
% 0.18/0.45 ( ( Uu
% 0.18/0.45 = ( F @ X @ Y ) )
% 0.18/0.45 & ( P3 @ X )
% 0.18/0.45 & ( Q @ Y ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_image_set2
% 0.18/0.45 thf(fact_38_finite__image__set2,axiom,
% 0.18/0.45 ! [P3: nat > $o,Q: nat > $o,F: nat > nat > product_prod_nat_nat] :
% 0.18/0.45 ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 0.18/0.45 => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 0.18/0.45 => ( finite772653738at_nat
% 0.18/0.45 @ ( collec7649004at_nat
% 0.18/0.45 @ ^ [Uu: product_prod_nat_nat] :
% 0.18/0.45 ? [X: nat,Y: nat] :
% 0.18/0.45 ( ( Uu
% 0.18/0.45 = ( F @ X @ Y ) )
% 0.18/0.45 & ( P3 @ X )
% 0.18/0.45 & ( Q @ Y ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_image_set2
% 0.18/0.45 thf(fact_39_finite__image__set2,axiom,
% 0.18/0.45 ! [P3: nat > $o,Q: nat > $o,F: nat > nat > nat] :
% 0.18/0.45 ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 0.18/0.45 => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 0.18/0.45 => ( finite_finite_nat
% 0.18/0.45 @ ( collect_nat
% 0.18/0.45 @ ^ [Uu: nat] :
% 0.18/0.45 ? [X: nat,Y: nat] :
% 0.18/0.45 ( ( Uu
% 0.18/0.45 = ( F @ X @ Y ) )
% 0.18/0.45 & ( P3 @ X )
% 0.18/0.45 & ( Q @ Y ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_image_set2
% 0.18/0.45 thf(fact_40_not__finite__existsD,axiom,
% 0.18/0.45 ! [P3: product_prod_nat_nat > $o] :
% 0.18/0.45 ( ~ ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 0.18/0.45 => ? [X_1: product_prod_nat_nat] : ( P3 @ X_1 ) ) ).
% 0.18/0.45
% 0.18/0.45 % not_finite_existsD
% 0.18/0.45 thf(fact_41_not__finite__existsD,axiom,
% 0.18/0.45 ! [P3: nat > $o] :
% 0.18/0.45 ( ~ ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 0.18/0.45 => ? [X_1: nat] : ( P3 @ X_1 ) ) ).
% 0.18/0.45
% 0.18/0.45 % not_finite_existsD
% 0.18/0.45 thf(fact_42_pigeonhole__infinite__rel,axiom,
% 0.18/0.45 ! [A: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,R2: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 0.18/0.45 ( ~ ( finite772653738at_nat @ A )
% 0.18/0.45 => ( ( finite772653738at_nat @ B2 )
% 0.18/0.45 => ( ! [X2: product_prod_nat_nat] :
% 0.18/0.45 ( ( member701585322at_nat @ X2 @ A )
% 0.18/0.45 => ? [Xa: product_prod_nat_nat] :
% 0.18/0.45 ( ( member701585322at_nat @ Xa @ B2 )
% 0.18/0.45 & ( R2 @ X2 @ Xa ) ) )
% 0.18/0.45 => ? [X2: product_prod_nat_nat] :
% 0.18/0.45 ( ( member701585322at_nat @ X2 @ B2 )
% 0.18/0.45 & ~ ( finite772653738at_nat
% 0.18/0.45 @ ( collec7649004at_nat
% 0.18/0.45 @ ^ [A2: product_prod_nat_nat] :
% 0.18/0.45 ( ( member701585322at_nat @ A2 @ A )
% 0.18/0.45 & ( R2 @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % pigeonhole_infinite_rel
% 0.18/0.45 thf(fact_43_pigeonhole__infinite__rel,axiom,
% 0.18/0.45 ! [A: set_Pr1986765409at_nat,B2: set_nat,R2: product_prod_nat_nat > nat > $o] :
% 0.18/0.45 ( ~ ( finite772653738at_nat @ A )
% 0.18/0.45 => ( ( finite_finite_nat @ B2 )
% 0.18/0.45 => ( ! [X2: product_prod_nat_nat] :
% 0.18/0.45 ( ( member701585322at_nat @ X2 @ A )
% 0.18/0.45 => ? [Xa: nat] :
% 0.18/0.45 ( ( member_nat @ Xa @ B2 )
% 0.18/0.45 & ( R2 @ X2 @ Xa ) ) )
% 0.18/0.45 => ? [X2: nat] :
% 0.18/0.45 ( ( member_nat @ X2 @ B2 )
% 0.18/0.45 & ~ ( finite772653738at_nat
% 0.18/0.45 @ ( collec7649004at_nat
% 0.18/0.45 @ ^ [A2: product_prod_nat_nat] :
% 0.18/0.45 ( ( member701585322at_nat @ A2 @ A )
% 0.18/0.45 & ( R2 @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % pigeonhole_infinite_rel
% 0.18/0.45 thf(fact_44_pigeonhole__infinite__rel,axiom,
% 0.18/0.45 ! [A: set_nat,B2: set_Pr1986765409at_nat,R2: nat > product_prod_nat_nat > $o] :
% 0.18/0.45 ( ~ ( finite_finite_nat @ A )
% 0.18/0.45 => ( ( finite772653738at_nat @ B2 )
% 0.18/0.45 => ( ! [X2: nat] :
% 0.18/0.45 ( ( member_nat @ X2 @ A )
% 0.18/0.45 => ? [Xa: product_prod_nat_nat] :
% 0.18/0.45 ( ( member701585322at_nat @ Xa @ B2 )
% 0.18/0.45 & ( R2 @ X2 @ Xa ) ) )
% 0.18/0.45 => ? [X2: product_prod_nat_nat] :
% 0.18/0.45 ( ( member701585322at_nat @ X2 @ B2 )
% 0.18/0.45 & ~ ( finite_finite_nat
% 0.18/0.45 @ ( collect_nat
% 0.18/0.45 @ ^ [A2: nat] :
% 0.18/0.45 ( ( member_nat @ A2 @ A )
% 0.18/0.45 & ( R2 @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % pigeonhole_infinite_rel
% 0.18/0.45 thf(fact_45_pigeonhole__infinite__rel,axiom,
% 0.18/0.45 ! [A: set_nat,B2: set_nat,R2: nat > nat > $o] :
% 0.18/0.45 ( ~ ( finite_finite_nat @ A )
% 0.18/0.45 => ( ( finite_finite_nat @ B2 )
% 0.18/0.45 => ( ! [X2: nat] :
% 0.18/0.45 ( ( member_nat @ X2 @ A )
% 0.18/0.45 => ? [Xa: nat] :
% 0.18/0.45 ( ( member_nat @ Xa @ B2 )
% 0.18/0.45 & ( R2 @ X2 @ Xa ) ) )
% 0.18/0.45 => ? [X2: nat] :
% 0.18/0.45 ( ( member_nat @ X2 @ B2 )
% 0.18/0.45 & ~ ( finite_finite_nat
% 0.18/0.45 @ ( collect_nat
% 0.18/0.45 @ ^ [A2: nat] :
% 0.18/0.45 ( ( member_nat @ A2 @ A )
% 0.18/0.45 & ( R2 @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % pigeonhole_infinite_rel
% 0.18/0.45 thf(fact_46_ek__analysis__def,axiom,
% 0.18/0.45 edmond1517640972ysis_a = finite_Graph_a ).
% 0.18/0.45
% 0.18/0.45 % ek_analysis_def
% 0.18/0.45 thf(fact_47_ek__analysis_Ointro,axiom,
% 0.18/0.45 ! [C: product_prod_nat_nat > a] :
% 0.18/0.45 ( ( finite_Graph_a @ C )
% 0.18/0.45 => ( edmond1517640972ysis_a @ C ) ) ).
% 0.18/0.45
% 0.18/0.45 % ek_analysis.intro
% 0.18/0.45 thf(fact_48_ek__analysis_Oaxioms,axiom,
% 0.18/0.45 ! [C: product_prod_nat_nat > a] :
% 0.18/0.45 ( ( edmond1517640972ysis_a @ C )
% 0.18/0.45 => ( finite_Graph_a @ C ) ) ).
% 0.18/0.45
% 0.18/0.45 % ek_analysis.axioms
% 0.18/0.45 thf(fact_49_ek__analysis__defs_OspEdges__def,axiom,
% 0.18/0.45 ( edmond475474835dges_a
% 0.18/0.45 = ( ^ [C2: product_prod_nat_nat > a,S2: nat,T: nat] :
% 0.18/0.45 ( collec7649004at_nat
% 0.18/0.45 @ ^ [E: product_prod_nat_nat] :
% 0.18/0.45 ? [P: list_P559422087at_nat] :
% 0.18/0.45 ( ( member701585322at_nat @ E @ ( set_Pr2131844118at_nat @ P ) )
% 0.18/0.45 & ( isShortestPath_a @ C2 @ S2 @ P @ T ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ek_analysis_defs.spEdges_def
% 0.18/0.45 thf(fact_50_finite__has__minimal2,axiom,
% 0.18/0.45 ! [A: set_se1612935105at_nat,A3: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( finite1457549322at_nat @ A )
% 0.18/0.45 => ( ( member298845450at_nat @ A3 @ A )
% 0.18/0.45 => ? [X2: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( member298845450at_nat @ X2 @ A )
% 0.18/0.45 & ( ord_le841296385at_nat @ X2 @ A3 )
% 0.18/0.45 & ! [Xa: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( member298845450at_nat @ Xa @ A )
% 0.18/0.45 => ( ( ord_le841296385at_nat @ Xa @ X2 )
% 0.18/0.45 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_has_minimal2
% 0.18/0.45 thf(fact_51_finite__has__minimal2,axiom,
% 0.18/0.45 ! [A: set_set_nat,A3: set_nat] :
% 0.18/0.45 ( ( finite2012248349et_nat @ A )
% 0.18/0.45 => ( ( member_set_nat @ A3 @ A )
% 0.18/0.45 => ? [X2: set_nat] :
% 0.18/0.45 ( ( member_set_nat @ X2 @ A )
% 0.18/0.45 & ( ord_less_eq_set_nat @ X2 @ A3 )
% 0.18/0.45 & ! [Xa: set_nat] :
% 0.18/0.45 ( ( member_set_nat @ Xa @ A )
% 0.18/0.45 => ( ( ord_less_eq_set_nat @ Xa @ X2 )
% 0.18/0.45 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_has_minimal2
% 0.18/0.45 thf(fact_52_finite__has__minimal2,axiom,
% 0.18/0.45 ! [A: set_nat,A3: nat] :
% 0.18/0.45 ( ( finite_finite_nat @ A )
% 0.18/0.45 => ( ( member_nat @ A3 @ A )
% 0.18/0.45 => ? [X2: nat] :
% 0.18/0.45 ( ( member_nat @ X2 @ A )
% 0.18/0.45 & ( ord_less_eq_nat @ X2 @ A3 )
% 0.18/0.45 & ! [Xa: nat] :
% 0.18/0.45 ( ( member_nat @ Xa @ A )
% 0.18/0.45 => ( ( ord_less_eq_nat @ Xa @ X2 )
% 0.18/0.45 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_has_minimal2
% 0.18/0.45 thf(fact_53_finite__has__maximal2,axiom,
% 0.18/0.45 ! [A: set_se1612935105at_nat,A3: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( finite1457549322at_nat @ A )
% 0.18/0.45 => ( ( member298845450at_nat @ A3 @ A )
% 0.18/0.45 => ? [X2: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( member298845450at_nat @ X2 @ A )
% 0.18/0.45 & ( ord_le841296385at_nat @ A3 @ X2 )
% 0.18/0.45 & ! [Xa: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( member298845450at_nat @ Xa @ A )
% 0.18/0.45 => ( ( ord_le841296385at_nat @ X2 @ Xa )
% 0.18/0.45 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_has_maximal2
% 0.18/0.45 thf(fact_54_finite__has__maximal2,axiom,
% 0.18/0.45 ! [A: set_set_nat,A3: set_nat] :
% 0.18/0.45 ( ( finite2012248349et_nat @ A )
% 0.18/0.45 => ( ( member_set_nat @ A3 @ A )
% 0.18/0.45 => ? [X2: set_nat] :
% 0.18/0.45 ( ( member_set_nat @ X2 @ A )
% 0.18/0.45 & ( ord_less_eq_set_nat @ A3 @ X2 )
% 0.18/0.45 & ! [Xa: set_nat] :
% 0.18/0.45 ( ( member_set_nat @ Xa @ A )
% 0.18/0.45 => ( ( ord_less_eq_set_nat @ X2 @ Xa )
% 0.18/0.45 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_has_maximal2
% 0.18/0.45 thf(fact_55_finite__has__maximal2,axiom,
% 0.18/0.45 ! [A: set_nat,A3: nat] :
% 0.18/0.45 ( ( finite_finite_nat @ A )
% 0.18/0.45 => ( ( member_nat @ A3 @ A )
% 0.18/0.45 => ? [X2: nat] :
% 0.18/0.45 ( ( member_nat @ X2 @ A )
% 0.18/0.45 & ( ord_less_eq_nat @ A3 @ X2 )
% 0.18/0.45 & ! [Xa: nat] :
% 0.18/0.45 ( ( member_nat @ Xa @ A )
% 0.18/0.45 => ( ( ord_less_eq_nat @ X2 @ Xa )
% 0.18/0.45 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_has_maximal2
% 0.18/0.45 thf(fact_56_rev__finite__subset,axiom,
% 0.18/0.45 ! [B2: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( finite772653738at_nat @ B2 )
% 0.18/0.45 => ( ( ord_le841296385at_nat @ A @ B2 )
% 0.18/0.45 => ( finite772653738at_nat @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rev_finite_subset
% 0.18/0.45 thf(fact_57_rev__finite__subset,axiom,
% 0.18/0.45 ! [B2: set_nat,A: set_nat] :
% 0.18/0.45 ( ( finite_finite_nat @ B2 )
% 0.18/0.45 => ( ( ord_less_eq_set_nat @ A @ B2 )
% 0.18/0.45 => ( finite_finite_nat @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rev_finite_subset
% 0.18/0.45 thf(fact_58_infinite__super,axiom,
% 0.18/0.45 ! [S3: set_Pr1986765409at_nat,T2: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( ord_le841296385at_nat @ S3 @ T2 )
% 0.18/0.45 => ( ~ ( finite772653738at_nat @ S3 )
% 0.18/0.45 => ~ ( finite772653738at_nat @ T2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % infinite_super
% 0.18/0.45 thf(fact_59_infinite__super,axiom,
% 0.18/0.45 ! [S3: set_nat,T2: set_nat] :
% 0.18/0.45 ( ( ord_less_eq_set_nat @ S3 @ T2 )
% 0.18/0.45 => ( ~ ( finite_finite_nat @ S3 )
% 0.18/0.45 => ~ ( finite_finite_nat @ T2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % infinite_super
% 0.18/0.45 thf(fact_60_finite__subset,axiom,
% 0.18/0.45 ! [A: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( ord_le841296385at_nat @ A @ B2 )
% 0.18/0.45 => ( ( finite772653738at_nat @ B2 )
% 0.18/0.45 => ( finite772653738at_nat @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_subset
% 0.18/0.45 thf(fact_61_finite__subset,axiom,
% 0.18/0.45 ! [A: set_nat,B2: set_nat] :
% 0.18/0.45 ( ( ord_less_eq_set_nat @ A @ B2 )
% 0.18/0.45 => ( ( finite_finite_nat @ B2 )
% 0.18/0.45 => ( finite_finite_nat @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_subset
% 0.18/0.45 thf(fact_62_ek__analysis_OspEdges__ss__E,axiom,
% 0.18/0.45 ! [C: product_prod_nat_nat > a,S: nat,T3: nat] :
% 0.18/0.45 ( ( edmond1517640972ysis_a @ C )
% 0.18/0.45 => ( ord_le841296385at_nat @ ( edmond475474835dges_a @ C @ S @ T3 ) @ ( e_a @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ek_analysis.spEdges_ss_E
% 0.18/0.45 thf(fact_63_incoming_H__edges,axiom,
% 0.18/0.45 ! [U2: set_nat] : ( ord_le841296385at_nat @ ( incoming_a2 @ c @ U2 ) @ ( e_a @ c ) ) ).
% 0.18/0.45
% 0.18/0.45 % incoming'_edges
% 0.18/0.45 thf(fact_64_outgoing_H__edges,axiom,
% 0.18/0.45 ! [U2: set_nat] : ( ord_le841296385at_nat @ ( outgoing_a2 @ c @ U2 ) @ ( e_a @ c ) ) ).
% 0.18/0.45
% 0.18/0.45 % outgoing'_edges
% 0.18/0.45 thf(fact_65_reachableNodes__def,axiom,
% 0.18/0.45 ! [U: nat] :
% 0.18/0.45 ( ( reachableNodes_a @ c @ U )
% 0.18/0.45 = ( collect_nat @ ( connected_a @ c @ U ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % reachableNodes_def
% 0.18/0.45 thf(fact_66_List_Ofinite__set,axiom,
% 0.18/0.45 ! [Xs: list_P559422087at_nat] : ( finite772653738at_nat @ ( set_Pr2131844118at_nat @ Xs ) ) ).
% 0.18/0.45
% 0.18/0.45 % List.finite_set
% 0.18/0.45 thf(fact_67_List_Ofinite__set,axiom,
% 0.18/0.45 ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% 0.18/0.45
% 0.18/0.45 % List.finite_set
% 0.18/0.45 thf(fact_68_Graph_OEfin__imp__Vfin,axiom,
% 0.18/0.45 ! [C: product_prod_nat_nat > a] :
% 0.18/0.45 ( ( finite772653738at_nat @ ( e_a @ C ) )
% 0.18/0.45 => ( finite_finite_nat @ ( v_a @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Graph.Efin_imp_Vfin
% 0.18/0.45 thf(fact_69_Graph_OVfin__imp__Efin,axiom,
% 0.18/0.45 ! [C: product_prod_nat_nat > a] :
% 0.18/0.45 ( ( finite_finite_nat @ ( v_a @ C ) )
% 0.18/0.45 => ( finite772653738at_nat @ ( e_a @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Graph.Vfin_imp_Efin
% 0.18/0.45 thf(fact_70_mem__Collect__eq,axiom,
% 0.18/0.45 ! [A3: product_prod_nat_nat,P3: product_prod_nat_nat > $o] :
% 0.18/0.45 ( ( member701585322at_nat @ A3 @ ( collec7649004at_nat @ P3 ) )
% 0.18/0.45 = ( P3 @ A3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_Collect_eq
% 0.18/0.45 thf(fact_71_mem__Collect__eq,axiom,
% 0.18/0.45 ! [A3: nat,P3: nat > $o] :
% 0.18/0.45 ( ( member_nat @ A3 @ ( collect_nat @ P3 ) )
% 0.18/0.45 = ( P3 @ A3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_Collect_eq
% 0.18/0.45 thf(fact_72_Collect__mem__eq,axiom,
% 0.18/0.45 ! [A: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( collec7649004at_nat
% 0.18/0.45 @ ^ [X: product_prod_nat_nat] : ( member701585322at_nat @ X @ A ) )
% 0.18/0.45 = A ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_mem_eq
% 0.18/0.45 thf(fact_73_Collect__mem__eq,axiom,
% 0.18/0.45 ! [A: set_nat] :
% 0.18/0.45 ( ( collect_nat
% 0.18/0.45 @ ^ [X: nat] : ( member_nat @ X @ A ) )
% 0.18/0.45 = A ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_mem_eq
% 0.18/0.45 thf(fact_74_Collect__cong,axiom,
% 0.18/0.45 ! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 0.18/0.45 ( ! [X2: product_prod_nat_nat] :
% 0.18/0.45 ( ( P3 @ X2 )
% 0.18/0.45 = ( Q @ X2 ) )
% 0.18/0.45 => ( ( collec7649004at_nat @ P3 )
% 0.18/0.45 = ( collec7649004at_nat @ Q ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_cong
% 0.18/0.45 thf(fact_75_Collect__cong,axiom,
% 0.18/0.45 ! [P3: nat > $o,Q: nat > $o] :
% 0.18/0.45 ( ! [X2: nat] :
% 0.18/0.45 ( ( P3 @ X2 )
% 0.18/0.45 = ( Q @ X2 ) )
% 0.18/0.45 => ( ( collect_nat @ P3 )
% 0.18/0.45 = ( collect_nat @ Q ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_cong
% 0.18/0.45 thf(fact_76_isShortestPath__level__edge_I1_J,axiom,
% 0.18/0.45 ! [S: nat,P4: list_P559422087at_nat,T3: nat,U: nat,V: nat] :
% 0.18/0.45 ( ( isShortestPath_a @ c @ S @ P4 @ T3 )
% 0.18/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 0.18/0.45 => ( connected_a @ c @ S @ U ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % isShortestPath_level_edge(1)
% 0.18/0.45 thf(fact_77_isShortestPath__level__edge_I2_J,axiom,
% 0.18/0.45 ! [S: nat,P4: list_P559422087at_nat,T3: nat,U: nat,V: nat] :
% 0.18/0.45 ( ( isShortestPath_a @ c @ S @ P4 @ T3 )
% 0.18/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 0.18/0.45 => ( connected_a @ c @ U @ V ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % isShortestPath_level_edge(2)
% 0.18/0.45 thf(fact_78_isShortestPath__level__edge_I3_J,axiom,
% 0.18/0.45 ! [S: nat,P4: list_P559422087at_nat,T3: nat,U: nat,V: nat] :
% 0.18/0.45 ( ( isShortestPath_a @ c @ S @ P4 @ T3 )
% 0.18/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 0.18/0.45 => ( connected_a @ c @ V @ T3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % isShortestPath_level_edge(3)
% 0.18/0.45 thf(fact_79_incoming__edges,axiom,
% 0.18/0.45 ! [U: nat] : ( ord_le841296385at_nat @ ( incoming_a @ c @ U ) @ ( e_a @ c ) ) ).
% 0.18/0.45
% 0.18/0.45 % incoming_edges
% 0.18/0.45 thf(fact_80_reachable__ss__V,axiom,
% 0.18/0.45 ! [S: nat] :
% 0.18/0.45 ( ( member_nat @ S @ ( v_a @ c ) )
% 0.18/0.45 => ( ord_less_eq_set_nat @ ( reachableNodes_a @ c @ S ) @ ( v_a @ c ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % reachable_ss_V
% 0.18/0.45 thf(fact_81_connected__append__edge,axiom,
% 0.18/0.45 ! [U: nat,V: nat,W: nat] :
% 0.18/0.45 ( ( connected_a @ c @ U @ V )
% 0.18/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ V @ W ) @ ( e_a @ c ) )
% 0.18/0.45 => ( connected_a @ c @ U @ W ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % connected_append_edge
% 0.18/0.45 thf(fact_82_reachableNodes__append__edge,axiom,
% 0.18/0.45 ! [U: nat,S: nat,V: nat] :
% 0.18/0.45 ( ( member_nat @ U @ ( reachableNodes_a @ c @ S ) )
% 0.18/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( e_a @ c ) )
% 0.18/0.45 => ( member_nat @ V @ ( reachableNodes_a @ c @ S ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % reachableNodes_append_edge
% 0.18/0.45 thf(fact_83_V__def,axiom,
% 0.18/0.45 ( ( v_a @ c )
% 0.18/0.45 = ( collect_nat
% 0.18/0.45 @ ^ [U3: nat] :
% 0.18/0.45 ? [V2: nat] :
% 0.18/0.45 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V2 ) @ ( e_a @ c ) )
% 0.18/0.45 | ( member701585322at_nat @ ( product_Pair_nat_nat @ V2 @ U3 ) @ ( e_a @ c ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % V_def
% 0.18/0.45 thf(fact_84_incoming__def,axiom,
% 0.18/0.45 ! [V: nat] :
% 0.18/0.45 ( ( incoming_a @ c @ V )
% 0.18/0.45 = ( collec7649004at_nat
% 0.18/0.45 @ ^ [Uu: product_prod_nat_nat] :
% 0.18/0.45 ? [U3: nat] :
% 0.18/0.45 ( ( Uu
% 0.18/0.45 = ( product_Pair_nat_nat @ U3 @ V ) )
% 0.18/0.45 & ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V ) @ ( e_a @ c ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % incoming_def
% 0.18/0.45 thf(fact_85_outgoing_H__def,axiom,
% 0.18/0.45 ! [K: set_nat] :
% 0.18/0.45 ( ( outgoing_a2 @ c @ K )
% 0.18/0.45 = ( collec7649004at_nat
% 0.18/0.45 @ ^ [Uu: product_prod_nat_nat] :
% 0.18/0.45 ? [U3: nat,V2: nat] :
% 0.18/0.45 ( ( Uu
% 0.18/0.45 = ( product_Pair_nat_nat @ V2 @ U3 ) )
% 0.18/0.45 & ~ ( member_nat @ U3 @ K )
% 0.18/0.45 & ( member_nat @ V2 @ K )
% 0.18/0.45 & ( member701585322at_nat @ ( product_Pair_nat_nat @ V2 @ U3 ) @ ( e_a @ c ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % outgoing'_def
% 0.18/0.45 thf(fact_86_incoming_H__def,axiom,
% 0.18/0.45 ! [K: set_nat] :
% 0.18/0.45 ( ( incoming_a2 @ c @ K )
% 0.18/0.45 = ( collec7649004at_nat
% 0.18/0.45 @ ^ [Uu: product_prod_nat_nat] :
% 0.18/0.45 ? [U3: nat,V2: nat] :
% 0.18/0.45 ( ( Uu
% 0.18/0.45 = ( product_Pair_nat_nat @ U3 @ V2 ) )
% 0.18/0.45 & ~ ( member_nat @ U3 @ K )
% 0.18/0.45 & ( member_nat @ V2 @ K )
% 0.18/0.45 & ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V2 ) @ ( e_a @ c ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % incoming'_def
% 0.18/0.45 thf(fact_87_finite__Collect__le__nat,axiom,
% 0.18/0.45 ! [K: nat] :
% 0.18/0.45 ( finite_finite_nat
% 0.18/0.45 @ ( collect_nat
% 0.18/0.45 @ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_Collect_le_nat
% 0.18/0.45 thf(fact_88_finite__incoming,axiom,
% 0.18/0.45 ! [U: nat] :
% 0.18/0.45 ( ( finite_finite_nat @ ( v_a @ c ) )
% 0.18/0.45 => ( finite772653738at_nat @ ( incoming_a @ c @ U ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_incoming
% 0.18/0.45 thf(fact_89_finite__outgoing_H,axiom,
% 0.18/0.45 ! [U2: set_nat] :
% 0.18/0.45 ( ( finite_finite_nat @ ( v_a @ c ) )
% 0.18/0.45 => ( finite772653738at_nat @ ( outgoing_a2 @ c @ U2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_outgoing'
% 0.18/0.45 thf(fact_90_finite__incoming_H,axiom,
% 0.18/0.45 ! [U2: set_nat] :
% 0.18/0.45 ( ( finite_finite_nat @ ( v_a @ c ) )
% 0.18/0.45 => ( finite772653738at_nat @ ( incoming_a2 @ c @ U2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % finite_incoming'
% 0.18/0.45 thf(fact_91_adjacent__nodes__ss__V,axiom,
% 0.18/0.45 ! [U: nat] : ( ord_less_eq_set_nat @ ( adjacent_nodes_a @ c @ U ) @ ( v_a @ c ) ) ).
% 0.18/0.45
% 0.18/0.45 % adjacent_nodes_ss_V
% 0.18/0.45 thf(fact_92_Graph_Oincoming__def,axiom,
% 0.18/0.45 ( incoming_a
% 0.18/0.45 = ( ^ [C2: product_prod_nat_nat > a,V2: nat] :
% 0.18/0.45 ( collec7649004at_nat
% 0.18/0.45 @ ^ [Uu: product_prod_nat_nat] :
% 0.18/0.45 ? [U3: nat] :
% 0.18/0.45 ( ( Uu
% 0.18/0.45 = ( product_Pair_nat_nat @ U3 @ V2 ) )
% 0.18/0.45 & ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V2 ) @ ( e_a @ C2 ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Graph.incoming_def
% 0.18/0.45 thf(fact_93_Graph_Oincoming_H__def,axiom,
% 0.18/0.45 ( incoming_a2
% 0.18/0.45 = ( ^ [C2: product_prod_nat_nat > a,K2: set_nat] :
% 0.18/0.45 ( collec7649004at_nat
% 0.18/0.45 @ ^ [Uu: product_prod_nat_nat] :
% 0.18/0.45 ? [U3: nat,V2: nat] :
% 0.18/0.45 ( ( Uu
% 0.18/0.45 = ( product_Pair_nat_nat @ U3 @ V2 ) )
% 0.18/0.45 & ~ ( member_nat @ U3 @ K2 )
% 0.18/0.45 & ( member_nat @ V2 @ K2 )
% 0.18/0.45 & ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V2 ) @ ( e_a @ C2 ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Graph.incoming'_def
% 0.18/0.45 thf(fact_94_Graph_Ooutgoing_H__def,axiom,
% 0.18/0.45 ( outgoing_a2
% 0.18/0.45 = ( ^ [C2: product_prod_nat_nat > a,K2: set_nat] :
% 0.18/0.45 ( collec7649004at_nat
% 0.18/0.45 @ ^ [Uu: product_prod_nat_nat] :
% 0.18/0.45 ? [U3: nat,V2: nat] :
% 0.18/0.45 ( ( Uu
% 0.18/0.45 = ( product_Pair_nat_nat @ V2 @ U3 ) )
% 0.18/0.45 & ~ ( member_nat @ U3 @ K2 )
% 0.18/0.45 & ( member_nat @ V2 @ K2 )
% 0.18/0.45 & ( member701585322at_nat @ ( product_Pair_nat_nat @ V2 @ U3 ) @ ( e_a @ C2 ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Graph.outgoing'_def
% 0.18/0.45 thf(fact_95_Graph_Oreachable__ss__V,axiom,
% 0.18/0.45 ! [S: nat,C: product_prod_nat_nat > a] :
% 0.18/0.45 ( ( member_nat @ S @ ( v_a @ C ) )
% 0.18/0.45 => ( ord_less_eq_set_nat @ ( reachableNodes_a @ C @ S ) @ ( v_a @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Graph.reachable_ss_V
% 0.18/0.45 thf(fact_96_Graph_OreachableNodes__append__edge,axiom,
% 0.18/0.45 ! [U: nat,C: product_prod_nat_nat > a,S: nat,V: nat] :
% 0.18/0.45 ( ( member_nat @ U @ ( reachableNodes_a @ C @ S ) )
% 0.18/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( e_a @ C ) )
% 0.18/0.45 => ( member_nat @ V @ ( reachableNodes_a @ C @ S ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Graph.reachableNodes_append_edge
% 0.18/0.45 thf(fact_97_Graph_OreachableNodes_Ocong,axiom,
% 0.18/0.45 reachableNodes_a = reachableNodes_a ).
% 0.18/0.45
% 0.18/0.45 % Graph.reachableNodes.cong
% 0.18/0.45 thf(fact_98_Graph_Ooutgoing_H_Ocong,axiom,
% 0.18/0.45 outgoing_a2 = outgoing_a2 ).
% 0.18/0.45
% 0.18/0.45 % Graph.outgoing'.cong
% 0.18/0.45 thf(fact_99_Graph_Oincoming_H_Ocong,axiom,
% 0.18/0.45 incoming_a2 = incoming_a2 ).
% 0.18/0.45
% 0.18/0.45 % Graph.incoming'.cong
% 0.18/0.45 thf(fact_100_Graph_Oincoming_Ocong,axiom,
% 0.18/0.45 incoming_a = incoming_a ).
% 0.18/0.45
% 0.18/0.45 % Graph.incoming.cong
% 0.18/0.45 thf(fact_101_pred__equals__eq2,axiom,
% 0.18/0.45 ! [R2: set_Pr1986765409at_nat,S3: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( ( ^ [X: nat,Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) )
% 0.18/0.45 = ( ^ [X: nat,Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S3 ) ) )
% 0.18/0.45 = ( R2 = S3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % pred_equals_eq2
% 0.18/0.45 thf(fact_102_pred__subset__eq2,axiom,
% 0.18/0.45 ! [R2: set_Pr1986765409at_nat,S3: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( ord_le1578155910_nat_o
% 0.18/0.45 @ ^ [X: nat,Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
% 0.18/0.45 @ ^ [X: nat,Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S3 ) )
% 0.18/0.45 = ( ord_le841296385at_nat @ R2 @ S3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % pred_subset_eq2
% 0.18/0.45 thf(fact_103_Graph_OV__def,axiom,
% 0.18/0.45 ( v_a
% 0.18/0.45 = ( ^ [C2: product_prod_nat_nat > a] :
% 0.18/0.45 ( collect_nat
% 0.18/0.45 @ ^ [U3: nat] :
% 0.18/0.45 ? [V2: nat] :
% 0.18/0.45 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V2 ) @ ( e_a @ C2 ) )
% 0.18/0.45 | ( member701585322at_nat @ ( product_Pair_nat_nat @ V2 @ U3 ) @ ( e_a @ C2 ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Graph.V_def
% 0.18/0.45 thf(fact_104_subrelI,axiom,
% 0.18/0.45 ! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ! [X2: nat,Y2: nat] :
% 0.18/0.45 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
% 0.18/0.45 => ( member701585322at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ S ) )
% 0.18/0.45 => ( ord_le841296385at_nat @ R @ S ) ) ).
% 0.18/0.45
% 0.18/0.45 % subrelI
% 0.18/0.45 thf(fact_105_Domain_Oinducts,axiom,
% 0.18/0.45 ! [X3: nat,R: set_Pr1986765409at_nat,P3: nat > $o] :
% 0.18/0.45 ( ( member_nat @ X3 @ ( domain_nat_nat @ R ) )
% 0.18/0.45 => ( ! [A4: nat,B3: nat] :
% 0.18/0.45 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ R )
% 0.18/0.45 => ( P3 @ A4 ) )
% 0.18/0.45 => ( P3 @ X3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Domain.inducts
% 0.18/0.45 thf(fact_106_Domain_ODomainI,axiom,
% 0.18/0.45 ! [A3: nat,B4: nat,R: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ R )
% 0.18/0.45 => ( member_nat @ A3 @ ( domain_nat_nat @ R ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Domain.DomainI
% 0.18/0.45 thf(fact_107_Domain_Osimps,axiom,
% 0.18/0.45 ! [A3: nat,R: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( member_nat @ A3 @ ( domain_nat_nat @ R ) )
% 0.18/0.45 = ( ? [A2: nat,B5: nat] :
% 0.18/0.45 ( ( A3 = A2 )
% 0.18/0.45 & ( member701585322at_nat @ ( product_Pair_nat_nat @ A2 @ B5 ) @ R ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Domain.simps
% 0.18/0.45 thf(fact_108_Domain_Ocases,axiom,
% 0.18/0.45 ! [A3: nat,R: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( member_nat @ A3 @ ( domain_nat_nat @ R ) )
% 0.18/0.45 => ~ ! [B3: nat] :
% 0.18/0.45 ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ R ) ) ).
% 0.18/0.45
% 0.18/0.45 % Domain.cases
% 0.18/0.45 thf(fact_109_Domain__iff,axiom,
% 0.18/0.45 ! [A3: nat,R: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( member_nat @ A3 @ ( domain_nat_nat @ R ) )
% 0.18/0.45 = ( ? [Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ Y ) @ R ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Domain_iff
% 0.18/0.45 thf(fact_110_DomainE,axiom,
% 0.18/0.45 ! [A3: nat,R: set_Pr1986765409at_nat] :
% 0.18/0.45 ( ( member_nat @ A3 @ ( domain_nat_nat @ R ) )
% 0.18/0.45 => ~ ! [B3: nat] :
% 0.18/0.45 ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ R ) ) ).
% 0.18/0.45
% 0.18/0.45 % DomainE
% 0.18/0.45 thf(fact_111_Range_Oinducts,axiom,
% 0.18/0.45 ! [X3: nat,R: set_Pr1986765409at_nat,P3: nat > $o] :
% 0.18/0.45 ( ( member_nat @ X3 @ ( range_nat_nat @ R ) )
% 0.18/0.45 => ( ! [A4: nat,B3: nat] :
% 0.18/0.45 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ R )
% 0.18/0.45 => ( P3 @ B3 ) )
% 0.18/0.45 => ( P3 @ X3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Range.inducts
% 0.18/0.45 thf(fact_112_Range_Ointros,axiom,
% 0.18/0.45 ! [A3: nat,B4: nat,R: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ R )
% 0.18/0.46 => ( member_nat @ B4 @ ( range_nat_nat @ R ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Range.intros
% 0.18/0.46 thf(fact_113_Range_Osimps,axiom,
% 0.18/0.46 ! [A3: nat,R: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member_nat @ A3 @ ( range_nat_nat @ R ) )
% 0.18/0.46 = ( ? [A2: nat,B5: nat] :
% 0.18/0.46 ( ( A3 = B5 )
% 0.18/0.46 & ( member701585322at_nat @ ( product_Pair_nat_nat @ A2 @ B5 ) @ R ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Range.simps
% 0.18/0.46 thf(fact_114_Range_Ocases,axiom,
% 0.18/0.46 ! [A3: nat,R: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member_nat @ A3 @ ( range_nat_nat @ R ) )
% 0.18/0.46 => ~ ! [A4: nat] :
% 0.18/0.46 ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ A4 @ A3 ) @ R ) ) ).
% 0.18/0.46
% 0.18/0.46 % Range.cases
% 0.18/0.46 thf(fact_115_Range__iff,axiom,
% 0.18/0.46 ! [A3: nat,R: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member_nat @ A3 @ ( range_nat_nat @ R ) )
% 0.18/0.46 = ( ? [Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ A3 ) @ R ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Range_iff
% 0.18/0.46 thf(fact_116_RangeE,axiom,
% 0.18/0.46 ! [B4: nat,R: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member_nat @ B4 @ ( range_nat_nat @ R ) )
% 0.18/0.46 => ~ ! [A4: nat] :
% 0.18/0.46 ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ A4 @ B4 ) @ R ) ) ).
% 0.18/0.46
% 0.18/0.46 % RangeE
% 0.18/0.46 thf(fact_117_Graph_OreachableNodes__def,axiom,
% 0.18/0.46 ( reachableNodes_a
% 0.18/0.46 = ( ^ [C2: product_prod_nat_nat > a,U3: nat] : ( collect_nat @ ( connected_a @ C2 @ U3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.reachableNodes_def
% 0.18/0.46 thf(fact_118_pred__subset__eq,axiom,
% 0.18/0.46 ! [R2: set_Pr1986765409at_nat,S3: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le1039616028_nat_o
% 0.18/0.46 @ ^ [X: product_prod_nat_nat] : ( member701585322at_nat @ X @ R2 )
% 0.18/0.46 @ ^ [X: product_prod_nat_nat] : ( member701585322at_nat @ X @ S3 ) )
% 0.18/0.46 = ( ord_le841296385at_nat @ R2 @ S3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % pred_subset_eq
% 0.18/0.46 thf(fact_119_pred__subset__eq,axiom,
% 0.18/0.46 ! [R2: set_nat,S3: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_nat_o
% 0.18/0.46 @ ^ [X: nat] : ( member_nat @ X @ R2 )
% 0.18/0.46 @ ^ [X: nat] : ( member_nat @ X @ S3 ) )
% 0.18/0.46 = ( ord_less_eq_set_nat @ R2 @ S3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % pred_subset_eq
% 0.18/0.46 thf(fact_120_Graph_Oconnected__append__edge,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,U: nat,V: nat,W: nat] :
% 0.18/0.46 ( ( connected_a @ C @ U @ V )
% 0.18/0.46 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ V @ W ) @ ( e_a @ C ) )
% 0.18/0.46 => ( connected_a @ C @ U @ W ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.connected_append_edge
% 0.18/0.46 thf(fact_121_Graph_Oincoming__edges,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,U: nat] : ( ord_le841296385at_nat @ ( incoming_a @ C @ U ) @ ( e_a @ C ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.incoming_edges
% 0.18/0.46 thf(fact_122_Graph_OisShortestPath__level__edge_I1_J,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,S: nat,P4: list_P559422087at_nat,T3: nat,U: nat,V: nat] :
% 0.18/0.46 ( ( isShortestPath_a @ C @ S @ P4 @ T3 )
% 0.18/0.46 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 0.18/0.46 => ( connected_a @ C @ S @ U ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.isShortestPath_level_edge(1)
% 0.18/0.46 thf(fact_123_Graph_OisShortestPath__level__edge_I2_J,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,S: nat,P4: list_P559422087at_nat,T3: nat,U: nat,V: nat] :
% 0.18/0.46 ( ( isShortestPath_a @ C @ S @ P4 @ T3 )
% 0.18/0.46 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 0.18/0.46 => ( connected_a @ C @ U @ V ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.isShortestPath_level_edge(2)
% 0.18/0.46 thf(fact_124_Graph_OisShortestPath__level__edge_I3_J,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,S: nat,P4: list_P559422087at_nat,T3: nat,U: nat,V: nat] :
% 0.18/0.46 ( ( isShortestPath_a @ C @ S @ P4 @ T3 )
% 0.18/0.46 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 0.18/0.46 => ( connected_a @ C @ V @ T3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.isShortestPath_level_edge(3)
% 0.18/0.46 thf(fact_125_Graph_Ooutgoing_H__edges,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,U2: set_nat] : ( ord_le841296385at_nat @ ( outgoing_a2 @ C @ U2 ) @ ( e_a @ C ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.outgoing'_edges
% 0.18/0.46 thf(fact_126_Graph_Oincoming_H__edges,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,U2: set_nat] : ( ord_le841296385at_nat @ ( incoming_a2 @ C @ U2 ) @ ( e_a @ C ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.incoming'_edges
% 0.18/0.46 thf(fact_127_Domain__unfold,axiom,
% 0.18/0.46 ( domain_nat_nat
% 0.18/0.46 = ( ^ [R3: set_Pr1986765409at_nat] :
% 0.18/0.46 ( collect_nat
% 0.18/0.46 @ ^ [X: nat] :
% 0.18/0.46 ? [Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Domain_unfold
% 0.18/0.46 thf(fact_128_Graph_Ofinite__incoming,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,U: nat] :
% 0.18/0.46 ( ( finite_finite_nat @ ( v_a @ C ) )
% 0.18/0.46 => ( finite772653738at_nat @ ( incoming_a @ C @ U ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.finite_incoming
% 0.18/0.46 thf(fact_129_Graph_OE_Ocong,axiom,
% 0.18/0.46 e_a = e_a ).
% 0.18/0.46
% 0.18/0.46 % Graph.E.cong
% 0.18/0.46 thf(fact_130_Graph_OV_Ocong,axiom,
% 0.18/0.46 v_a = v_a ).
% 0.18/0.46
% 0.18/0.46 % Graph.V.cong
% 0.18/0.46 thf(fact_131_Graph_Oconnected__refl,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,V: nat] : ( connected_a @ C @ V @ V ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.connected_refl
% 0.18/0.46 thf(fact_132_Graph_Oconnected_Ocong,axiom,
% 0.18/0.46 connected_a = connected_a ).
% 0.18/0.46
% 0.18/0.46 % Graph.connected.cong
% 0.18/0.46 thf(fact_133_Graph_Ofinite__outgoing_H,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,U2: set_nat] :
% 0.18/0.46 ( ( finite_finite_nat @ ( v_a @ C ) )
% 0.18/0.46 => ( finite772653738at_nat @ ( outgoing_a2 @ C @ U2 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.finite_outgoing'
% 0.18/0.46 thf(fact_134_Graph_OisShortestPath_Ocong,axiom,
% 0.18/0.46 isShortestPath_a = isShortestPath_a ).
% 0.18/0.46
% 0.18/0.46 % Graph.isShortestPath.cong
% 0.18/0.46 thf(fact_135_Graph_Ofinite__incoming_H,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,U2: set_nat] :
% 0.18/0.46 ( ( finite_finite_nat @ ( v_a @ C ) )
% 0.18/0.46 => ( finite772653738at_nat @ ( incoming_a2 @ C @ U2 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.finite_incoming'
% 0.18/0.46 thf(fact_136_finite__list,axiom,
% 0.18/0.46 ! [A: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( finite772653738at_nat @ A )
% 0.18/0.46 => ? [Xs2: list_P559422087at_nat] :
% 0.18/0.46 ( ( set_Pr2131844118at_nat @ Xs2 )
% 0.18/0.46 = A ) ) ).
% 0.18/0.46
% 0.18/0.46 % finite_list
% 0.18/0.46 thf(fact_137_finite__list,axiom,
% 0.18/0.46 ! [A: set_nat] :
% 0.18/0.46 ( ( finite_finite_nat @ A )
% 0.18/0.46 => ? [Xs2: list_nat] :
% 0.18/0.46 ( ( set_nat2 @ Xs2 )
% 0.18/0.46 = A ) ) ).
% 0.18/0.46
% 0.18/0.46 % finite_list
% 0.18/0.46 thf(fact_138_subset__code_I1_J,axiom,
% 0.18/0.46 ! [Xs: list_P559422087at_nat,B2: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ ( set_Pr2131844118at_nat @ Xs ) @ B2 )
% 0.18/0.46 = ( ! [X: product_prod_nat_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ X @ ( set_Pr2131844118at_nat @ Xs ) )
% 0.18/0.46 => ( member701585322at_nat @ X @ B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % subset_code(1)
% 0.18/0.46 thf(fact_139_subset__code_I1_J,axiom,
% 0.18/0.46 ! [Xs: list_nat,B2: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
% 0.18/0.46 = ( ! [X: nat] :
% 0.18/0.46 ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.18/0.46 => ( member_nat @ X @ B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % subset_code(1)
% 0.18/0.46 thf(fact_140_Graph_Oconnected__inV__iff,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,U: nat,V: nat] :
% 0.18/0.46 ( ( connected_a @ C @ U @ V )
% 0.18/0.46 => ( ( member_nat @ V @ ( v_a @ C ) )
% 0.18/0.46 = ( member_nat @ U @ ( v_a @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.connected_inV_iff
% 0.18/0.46 thf(fact_141_Graph_Oobtain__shortest__path,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,U: nat,V: nat] :
% 0.18/0.46 ( ( connected_a @ C @ U @ V )
% 0.18/0.46 => ~ ! [P2: list_P559422087at_nat] :
% 0.18/0.46 ~ ( isShortestPath_a @ C @ U @ P2 @ V ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.obtain_shortest_path
% 0.18/0.46 thf(fact_142_Finite__Graph_Ofinite__V,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a] :
% 0.18/0.46 ( ( finite_Graph_a @ C )
% 0.18/0.46 => ( finite_finite_nat @ ( v_a @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Finite_Graph.finite_V
% 0.18/0.46 thf(fact_143_Finite__Graph_Ointro,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a] :
% 0.18/0.46 ( ( finite_finite_nat @ ( v_a @ C ) )
% 0.18/0.46 => ( finite_Graph_a @ C ) ) ).
% 0.18/0.46
% 0.18/0.46 % Finite_Graph.intro
% 0.18/0.46 thf(fact_144_Finite__Graph__def,axiom,
% 0.18/0.46 ( finite_Graph_a
% 0.18/0.46 = ( ^ [C2: product_prod_nat_nat > a] : ( finite_finite_nat @ ( v_a @ C2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Finite_Graph_def
% 0.18/0.46 thf(fact_145_Finite__Graph_Ofinite__E,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a] :
% 0.18/0.46 ( ( finite_Graph_a @ C )
% 0.18/0.46 => ( finite772653738at_nat @ ( e_a @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Finite_Graph.finite_E
% 0.18/0.46 thf(fact_146_Graph_OFinite__Graph__EI,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a] :
% 0.18/0.46 ( ( finite772653738at_nat @ ( e_a @ C ) )
% 0.18/0.46 => ( finite_Graph_a @ C ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.Finite_Graph_EI
% 0.18/0.46 thf(fact_147_outgoing__def,axiom,
% 0.18/0.46 ! [V: nat] :
% 0.18/0.46 ( ( outgoing_a @ c @ V )
% 0.18/0.46 = ( collec7649004at_nat
% 0.18/0.46 @ ^ [Uu: product_prod_nat_nat] :
% 0.18/0.46 ? [U3: nat] :
% 0.18/0.46 ( ( Uu
% 0.18/0.46 = ( product_Pair_nat_nat @ V @ U3 ) )
% 0.18/0.46 & ( member701585322at_nat @ ( product_Pair_nat_nat @ V @ U3 ) @ ( e_a @ c ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % outgoing_def
% 0.18/0.46 thf(fact_148_outgoing__edges,axiom,
% 0.18/0.46 ! [U: nat] : ( ord_le841296385at_nat @ ( outgoing_a @ c @ U ) @ ( e_a @ c ) ) ).
% 0.18/0.46
% 0.18/0.46 % outgoing_edges
% 0.18/0.46 thf(fact_149_E__ss__VxV,axiom,
% 0.18/0.46 ( ord_le841296385at_nat @ ( e_a @ c )
% 0.18/0.46 @ ( produc45129834at_nat @ ( v_a @ c )
% 0.18/0.46 @ ^ [Uu: nat] : ( v_a @ c ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % E_ss_VxV
% 0.18/0.46 thf(fact_150_connected__edgeRtc,axiom,
% 0.18/0.46 ! [U: nat,V: nat] :
% 0.18/0.46 ( ( connected_a @ c @ U @ V )
% 0.18/0.46 = ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi1645084429cl_nat @ ( e_a @ c ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % connected_edgeRtc
% 0.18/0.46 thf(fact_151_isSPath__no__selfloop,axiom,
% 0.18/0.46 ! [U: nat,P4: list_P559422087at_nat,V: nat,U1: nat] :
% 0.18/0.46 ( ( isSimplePath_a @ c @ U @ P4 @ V )
% 0.18/0.46 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ U1 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % isSPath_no_selfloop
% 0.18/0.46 thf(fact_152_isSPath__sg__incoming,axiom,
% 0.18/0.46 ! [U: nat,P4: list_P559422087at_nat,V: nat,U1: nat,V1: nat,U22: nat] :
% 0.18/0.46 ( ( isSimplePath_a @ c @ U @ P4 @ V )
% 0.18/0.46 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 0.18/0.46 => ( ( U1 != U22 )
% 0.18/0.46 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U22 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % isSPath_sg_incoming
% 0.18/0.46 thf(fact_153_isSPath__sg__outgoing,axiom,
% 0.18/0.46 ! [U: nat,P4: list_P559422087at_nat,V: nat,U1: nat,V1: nat,V22: nat] :
% 0.18/0.46 ( ( isSimplePath_a @ c @ U @ P4 @ V )
% 0.18/0.46 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 0.18/0.46 => ( ( V1 != V22 )
% 0.18/0.46 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V22 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % isSPath_sg_outgoing
% 0.18/0.46 thf(fact_154_reachableNodes__E__closed,axiom,
% 0.18/0.46 ! [S: nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ ( e_a @ c ) @ ( reachableNodes_a @ c @ S ) ) @ ( reachableNodes_a @ c @ S ) ) ).
% 0.18/0.46
% 0.18/0.46 % reachableNodes_E_closed
% 0.18/0.46 thf(fact_155_shortestPath__is__simple,axiom,
% 0.18/0.46 ! [S: nat,P4: list_P559422087at_nat,T3: nat] :
% 0.18/0.46 ( ( isShortestPath_a @ c @ S @ P4 @ T3 )
% 0.18/0.46 => ( isSimplePath_a @ c @ S @ P4 @ T3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % shortestPath_is_simple
% 0.18/0.46 thf(fact_156_ImageI,axiom,
% 0.18/0.46 ! [A3: product_prod_nat_nat,B4: product_prod_nat_nat,R: set_Pr1490359111at_nat,A: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ A3 @ B4 ) @ R )
% 0.18/0.46 => ( ( member701585322at_nat @ A3 @ A )
% 0.18/0.46 => ( member701585322at_nat @ B4 @ ( image_1655366419at_nat @ R @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ImageI
% 0.18/0.46 thf(fact_157_ImageI,axiom,
% 0.18/0.46 ! [A3: product_prod_nat_nat,B4: nat,R: set_Pr1287749686at_nat,A: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member868723479at_nat @ ( produc947540346at_nat @ A3 @ B4 ) @ R )
% 0.18/0.46 => ( ( member701585322at_nat @ A3 @ A )
% 0.18/0.46 => ( member_nat @ B4 @ ( image_2142699582at_nat @ R @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ImageI
% 0.18/0.46 thf(fact_158_ImageI,axiom,
% 0.18/0.46 ! [A3: nat,B4: product_prod_nat_nat,R: set_Pr1746169692at_nat,A: set_nat] :
% 0.18/0.46 ( ( member1293241661at_nat @ ( produc1933845336at_nat @ A3 @ B4 ) @ R )
% 0.18/0.46 => ( ( member_nat @ A3 @ A )
% 0.18/0.46 => ( member701585322at_nat @ B4 @ ( image_981520924at_nat @ R @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ImageI
% 0.18/0.46 thf(fact_159_ImageI,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,R: set_Pr1986765409at_nat,A: set_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ R )
% 0.18/0.46 => ( ( member_nat @ A3 @ A )
% 0.18/0.46 => ( member_nat @ B4 @ ( image_nat_nat @ R @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ImageI
% 0.18/0.46 thf(fact_160_finite__SigmaI,axiom,
% 0.18/0.46 ! [A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( finite772653738at_nat @ A )
% 0.18/0.46 => ( ! [A4: product_prod_nat_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ A4 @ A )
% 0.18/0.46 => ( finite772653738at_nat @ ( B2 @ A4 ) ) )
% 0.18/0.46 => ( finite48957584at_nat @ ( produc2056081288at_nat @ A @ B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % finite_SigmaI
% 0.18/0.46 thf(fact_161_finite__SigmaI,axiom,
% 0.18/0.46 ! [A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_nat] :
% 0.18/0.46 ( ( finite772653738at_nat @ A )
% 0.18/0.46 => ( ! [A4: product_prod_nat_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ A4 @ A )
% 0.18/0.46 => ( finite_finite_nat @ ( B2 @ A4 ) ) )
% 0.18/0.46 => ( finite2000257047at_nat @ ( produc2055342601at_nat @ A @ B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % finite_SigmaI
% 0.18/0.46 thf(fact_162_finite__SigmaI,axiom,
% 0.18/0.46 ! [A: set_nat,B2: nat > set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( finite_finite_nat @ A )
% 0.18/0.46 => ( ! [A4: nat] :
% 0.18/0.46 ( ( member_nat @ A4 @ A )
% 0.18/0.46 => ( finite772653738at_nat @ ( B2 @ A4 ) ) )
% 0.18/0.46 => ( finite277291581at_nat @ ( produc894163943at_nat @ A @ B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % finite_SigmaI
% 0.18/0.46 thf(fact_163_finite__SigmaI,axiom,
% 0.18/0.46 ! [A: set_nat,B2: nat > set_nat] :
% 0.18/0.46 ( ( finite_finite_nat @ A )
% 0.18/0.46 => ( ! [A4: nat] :
% 0.18/0.46 ( ( member_nat @ A4 @ A )
% 0.18/0.46 => ( finite_finite_nat @ ( B2 @ A4 ) ) )
% 0.18/0.46 => ( finite772653738at_nat @ ( produc45129834at_nat @ A @ B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % finite_SigmaI
% 0.18/0.46 thf(fact_164_adjacent__nodes__finite,axiom,
% 0.18/0.46 ! [U: nat] : ( finite_finite_nat @ ( adjacent_nodes_a @ c @ U ) ) ).
% 0.18/0.46
% 0.18/0.46 % adjacent_nodes_finite
% 0.18/0.46 thf(fact_165_finite__outgoing,axiom,
% 0.18/0.46 ! [U: nat] :
% 0.18/0.46 ( ( finite_finite_nat @ ( v_a @ c ) )
% 0.18/0.46 => ( finite772653738at_nat @ ( outgoing_a @ c @ U ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % finite_outgoing
% 0.18/0.46 thf(fact_166_Graph_Oadjacent__nodes_Ocong,axiom,
% 0.18/0.46 adjacent_nodes_a = adjacent_nodes_a ).
% 0.18/0.46
% 0.18/0.46 % Graph.adjacent_nodes.cong
% 0.18/0.46 thf(fact_167_Graph_OisSimplePath_Ocong,axiom,
% 0.18/0.46 isSimplePath_a = isSimplePath_a ).
% 0.18/0.46
% 0.18/0.46 % Graph.isSimplePath.cong
% 0.18/0.46 thf(fact_168_Graph_Ooutgoing_Ocong,axiom,
% 0.18/0.46 outgoing_a = outgoing_a ).
% 0.18/0.46
% 0.18/0.46 % Graph.outgoing.cong
% 0.18/0.46 thf(fact_169_rev__ImageI,axiom,
% 0.18/0.46 ! [A3: product_prod_nat_nat,A: set_Pr1986765409at_nat,B4: product_prod_nat_nat,R: set_Pr1490359111at_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ A3 @ A )
% 0.18/0.46 => ( ( member2027625872at_nat @ ( produc1168807639at_nat @ A3 @ B4 ) @ R )
% 0.18/0.46 => ( member701585322at_nat @ B4 @ ( image_1655366419at_nat @ R @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % rev_ImageI
% 0.18/0.46 thf(fact_170_rev__ImageI,axiom,
% 0.18/0.46 ! [A3: product_prod_nat_nat,A: set_Pr1986765409at_nat,B4: nat,R: set_Pr1287749686at_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ A3 @ A )
% 0.18/0.46 => ( ( member868723479at_nat @ ( produc947540346at_nat @ A3 @ B4 ) @ R )
% 0.18/0.46 => ( member_nat @ B4 @ ( image_2142699582at_nat @ R @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % rev_ImageI
% 0.18/0.46 thf(fact_171_rev__ImageI,axiom,
% 0.18/0.46 ! [A3: nat,A: set_nat,B4: product_prod_nat_nat,R: set_Pr1746169692at_nat] :
% 0.18/0.46 ( ( member_nat @ A3 @ A )
% 0.18/0.46 => ( ( member1293241661at_nat @ ( produc1933845336at_nat @ A3 @ B4 ) @ R )
% 0.18/0.46 => ( member701585322at_nat @ B4 @ ( image_981520924at_nat @ R @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % rev_ImageI
% 0.18/0.46 thf(fact_172_rev__ImageI,axiom,
% 0.18/0.46 ! [A3: nat,A: set_nat,B4: nat,R: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member_nat @ A3 @ A )
% 0.18/0.46 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ R )
% 0.18/0.46 => ( member_nat @ B4 @ ( image_nat_nat @ R @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % rev_ImageI
% 0.18/0.46 thf(fact_173_Image__iff,axiom,
% 0.18/0.46 ! [B4: nat,R: set_Pr1986765409at_nat,A: set_nat] :
% 0.18/0.46 ( ( member_nat @ B4 @ ( image_nat_nat @ R @ A ) )
% 0.18/0.46 = ( ? [X: nat] :
% 0.18/0.46 ( ( member_nat @ X @ A )
% 0.18/0.46 & ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ B4 ) @ R ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Image_iff
% 0.18/0.46 thf(fact_174_ImageE,axiom,
% 0.18/0.46 ! [B4: product_prod_nat_nat,R: set_Pr1490359111at_nat,A: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ B4 @ ( image_1655366419at_nat @ R @ A ) )
% 0.18/0.46 => ~ ! [X2: product_prod_nat_nat] :
% 0.18/0.46 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ X2 @ B4 ) @ R )
% 0.18/0.46 => ~ ( member701585322at_nat @ X2 @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ImageE
% 0.18/0.46 thf(fact_175_ImageE,axiom,
% 0.18/0.46 ! [B4: product_prod_nat_nat,R: set_Pr1746169692at_nat,A: set_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ B4 @ ( image_981520924at_nat @ R @ A ) )
% 0.18/0.46 => ~ ! [X2: nat] :
% 0.18/0.46 ( ( member1293241661at_nat @ ( produc1933845336at_nat @ X2 @ B4 ) @ R )
% 0.18/0.46 => ~ ( member_nat @ X2 @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ImageE
% 0.18/0.46 thf(fact_176_ImageE,axiom,
% 0.18/0.46 ! [B4: nat,R: set_Pr1287749686at_nat,A: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member_nat @ B4 @ ( image_2142699582at_nat @ R @ A ) )
% 0.18/0.46 => ~ ! [X2: product_prod_nat_nat] :
% 0.18/0.46 ( ( member868723479at_nat @ ( produc947540346at_nat @ X2 @ B4 ) @ R )
% 0.18/0.46 => ~ ( member701585322at_nat @ X2 @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ImageE
% 0.18/0.46 thf(fact_177_ImageE,axiom,
% 0.18/0.46 ! [B4: nat,R: set_Pr1986765409at_nat,A: set_nat] :
% 0.18/0.46 ( ( member_nat @ B4 @ ( image_nat_nat @ R @ A ) )
% 0.18/0.46 => ~ ! [X2: nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X2 @ B4 ) @ R )
% 0.18/0.46 => ~ ( member_nat @ X2 @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ImageE
% 0.18/0.46 thf(fact_178_Image__subset,axiom,
% 0.18/0.46 ! [R: set_Pr1986765409at_nat,A: set_nat,B2: set_nat,C3: set_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ R
% 0.18/0.46 @ ( produc45129834at_nat @ A
% 0.18/0.46 @ ^ [Uu: nat] : B2 ) )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( image_nat_nat @ R @ C3 ) @ B2 ) ) ).
% 0.18/0.46
% 0.18/0.46 % Image_subset
% 0.18/0.46 thf(fact_179_Graph_OshortestPath__is__simple,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,S: nat,P4: list_P559422087at_nat,T3: nat] :
% 0.18/0.46 ( ( isShortestPath_a @ C @ S @ P4 @ T3 )
% 0.18/0.46 => ( isSimplePath_a @ C @ S @ P4 @ T3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.shortestPath_is_simple
% 0.18/0.46 thf(fact_180_Image__mono,axiom,
% 0.18/0.46 ! [R4: set_Pr1490359111at_nat,R: set_Pr1490359111at_nat,A5: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le465611495at_nat @ R4 @ R )
% 0.18/0.46 => ( ( ord_le841296385at_nat @ A5 @ A )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( image_1655366419at_nat @ R4 @ A5 ) @ ( image_1655366419at_nat @ R @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Image_mono
% 0.18/0.46 thf(fact_181_Image__mono,axiom,
% 0.18/0.46 ! [R4: set_Pr1287749686at_nat,R: set_Pr1287749686at_nat,A5: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le1837363350at_nat @ R4 @ R )
% 0.18/0.46 => ( ( ord_le841296385at_nat @ A5 @ A )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( image_2142699582at_nat @ R4 @ A5 ) @ ( image_2142699582at_nat @ R @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Image_mono
% 0.18/0.46 thf(fact_182_Image__mono,axiom,
% 0.18/0.46 ! [R4: set_Pr1746169692at_nat,R: set_Pr1746169692at_nat,A5: set_nat,A: set_nat] :
% 0.18/0.46 ( ( ord_le148299708at_nat @ R4 @ R )
% 0.18/0.46 => ( ( ord_less_eq_set_nat @ A5 @ A )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( image_981520924at_nat @ R4 @ A5 ) @ ( image_981520924at_nat @ R @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Image_mono
% 0.18/0.46 thf(fact_183_Image__mono,axiom,
% 0.18/0.46 ! [R4: set_Pr1986765409at_nat,R: set_Pr1986765409at_nat,A5: set_nat,A: set_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ R4 @ R )
% 0.18/0.46 => ( ( ord_less_eq_set_nat @ A5 @ A )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( image_nat_nat @ R4 @ A5 ) @ ( image_nat_nat @ R @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Image_mono
% 0.18/0.46 thf(fact_184_Relation_Ofinite__Image,axiom,
% 0.18/0.46 ! [R2: set_Pr1986765409at_nat,A: set_nat] :
% 0.18/0.46 ( ( finite772653738at_nat @ R2 )
% 0.18/0.46 => ( finite_finite_nat @ ( image_nat_nat @ R2 @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Relation.finite_Image
% 0.18/0.46 thf(fact_185_Graph_OreachableNodes__E__closed,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,S: nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ ( e_a @ C ) @ ( reachableNodes_a @ C @ S ) ) @ ( reachableNodes_a @ C @ S ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.reachableNodes_E_closed
% 0.18/0.46 thf(fact_186_finite__cartesian__product,axiom,
% 0.18/0.46 ! [A: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( finite772653738at_nat @ A )
% 0.18/0.46 => ( ( finite772653738at_nat @ B2 )
% 0.18/0.46 => ( finite48957584at_nat
% 0.18/0.46 @ ( produc2056081288at_nat @ A
% 0.18/0.46 @ ^ [Uu: product_prod_nat_nat] : B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % finite_cartesian_product
% 0.18/0.46 thf(fact_187_finite__cartesian__product,axiom,
% 0.18/0.46 ! [A: set_Pr1986765409at_nat,B2: set_nat] :
% 0.18/0.46 ( ( finite772653738at_nat @ A )
% 0.18/0.46 => ( ( finite_finite_nat @ B2 )
% 0.18/0.46 => ( finite2000257047at_nat
% 0.18/0.46 @ ( produc2055342601at_nat @ A
% 0.18/0.46 @ ^ [Uu: product_prod_nat_nat] : B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % finite_cartesian_product
% 0.18/0.46 thf(fact_188_finite__cartesian__product,axiom,
% 0.18/0.46 ! [A: set_nat,B2: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( finite_finite_nat @ A )
% 0.18/0.46 => ( ( finite772653738at_nat @ B2 )
% 0.18/0.46 => ( finite277291581at_nat
% 0.18/0.46 @ ( produc894163943at_nat @ A
% 0.18/0.46 @ ^ [Uu: nat] : B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % finite_cartesian_product
% 0.18/0.46 thf(fact_189_finite__cartesian__product,axiom,
% 0.18/0.46 ! [A: set_nat,B2: set_nat] :
% 0.18/0.46 ( ( finite_finite_nat @ A )
% 0.18/0.46 => ( ( finite_finite_nat @ B2 )
% 0.18/0.46 => ( finite772653738at_nat
% 0.18/0.46 @ ( produc45129834at_nat @ A
% 0.18/0.46 @ ^ [Uu: nat] : B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % finite_cartesian_product
% 0.18/0.46 thf(fact_190_Graph_OisSPath__sg__outgoing,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,U: nat,P4: list_P559422087at_nat,V: nat,U1: nat,V1: nat,V22: nat] :
% 0.18/0.46 ( ( isSimplePath_a @ C @ U @ P4 @ V )
% 0.18/0.46 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 0.18/0.46 => ( ( V1 != V22 )
% 0.18/0.46 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V22 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.isSPath_sg_outgoing
% 0.18/0.46 thf(fact_191_Graph_OisSPath__sg__incoming,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,U: nat,P4: list_P559422087at_nat,V: nat,U1: nat,V1: nat,U22: nat] :
% 0.18/0.46 ( ( isSimplePath_a @ C @ U @ P4 @ V )
% 0.18/0.46 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 0.18/0.46 => ( ( U1 != U22 )
% 0.18/0.46 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U22 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.isSPath_sg_incoming
% 0.18/0.46 thf(fact_192_Graph_OisSPath__no__selfloop,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,U: nat,P4: list_P559422087at_nat,V: nat,U1: nat] :
% 0.18/0.46 ( ( isSimplePath_a @ C @ U @ P4 @ V )
% 0.18/0.46 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ U1 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.isSPath_no_selfloop
% 0.18/0.46 thf(fact_193_Graph_Oadjacent__nodes__ss__V,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,U: nat] : ( ord_less_eq_set_nat @ ( adjacent_nodes_a @ C @ U ) @ ( v_a @ C ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.adjacent_nodes_ss_V
% 0.18/0.46 thf(fact_194_Graph_Ooutgoing__edges,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,U: nat] : ( ord_le841296385at_nat @ ( outgoing_a @ C @ U ) @ ( e_a @ C ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.outgoing_edges
% 0.18/0.46 thf(fact_195_Graph_Ooutgoing__def,axiom,
% 0.18/0.46 ( outgoing_a
% 0.18/0.46 = ( ^ [C2: product_prod_nat_nat > a,V2: nat] :
% 0.18/0.46 ( collec7649004at_nat
% 0.18/0.46 @ ^ [Uu: product_prod_nat_nat] :
% 0.18/0.46 ? [U3: nat] :
% 0.18/0.46 ( ( Uu
% 0.18/0.46 = ( product_Pair_nat_nat @ V2 @ U3 ) )
% 0.18/0.46 & ( member701585322at_nat @ ( product_Pair_nat_nat @ V2 @ U3 ) @ ( e_a @ C2 ) ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.outgoing_def
% 0.18/0.46 thf(fact_196_Graph_Oconnected__edgeRtc,axiom,
% 0.18/0.46 ( connected_a
% 0.18/0.46 = ( ^ [C2: product_prod_nat_nat > a,U3: nat,V2: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V2 ) @ ( transi1645084429cl_nat @ ( e_a @ C2 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.connected_edgeRtc
% 0.18/0.46 thf(fact_197_Graph_OE__ss__VxV,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a] :
% 0.18/0.46 ( ord_le841296385at_nat @ ( e_a @ C )
% 0.18/0.46 @ ( produc45129834at_nat @ ( v_a @ C )
% 0.18/0.46 @ ^ [Uu: nat] : ( v_a @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.E_ss_VxV
% 0.18/0.46 thf(fact_198_Graph_Ofinite__outgoing,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,U: nat] :
% 0.18/0.46 ( ( finite_finite_nat @ ( v_a @ C ) )
% 0.18/0.46 => ( finite772653738at_nat @ ( outgoing_a @ C @ U ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.finite_outgoing
% 0.18/0.46 thf(fact_199_Finite__Graph_Oadjacent__nodes__finite,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,U: nat] :
% 0.18/0.46 ( ( finite_Graph_a @ C )
% 0.18/0.46 => ( finite_finite_nat @ ( adjacent_nodes_a @ C @ U ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Finite_Graph.adjacent_nodes_finite
% 0.18/0.46 thf(fact_200_isSPath__nt__parallel,axiom,
% 0.18/0.46 ! [S: nat,P4: list_P559422087at_nat,T3: nat,E2: product_prod_nat_nat] :
% 0.18/0.46 ( ( isSimplePath_a @ c @ S @ P4 @ T3 )
% 0.18/0.46 => ( ( member701585322at_nat @ E2 @ ( set_Pr2131844118at_nat @ P4 ) )
% 0.18/0.46 => ~ ( member701585322at_nat @ ( product_swap_nat_nat @ E2 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % isSPath_nt_parallel
% 0.18/0.46 thf(fact_201_mem__Sigma__iff,axiom,
% 0.18/0.46 ! [A3: product_prod_nat_nat,B4: product_prod_nat_nat,A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ A3 @ B4 ) @ ( produc2056081288at_nat @ A @ B2 ) )
% 0.18/0.46 = ( ( member701585322at_nat @ A3 @ A )
% 0.18/0.46 & ( member701585322at_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % mem_Sigma_iff
% 0.18/0.46 thf(fact_202_mem__Sigma__iff,axiom,
% 0.18/0.46 ! [A3: product_prod_nat_nat,B4: nat,A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_nat] :
% 0.18/0.46 ( ( member868723479at_nat @ ( produc947540346at_nat @ A3 @ B4 ) @ ( produc2055342601at_nat @ A @ B2 ) )
% 0.18/0.46 = ( ( member701585322at_nat @ A3 @ A )
% 0.18/0.46 & ( member_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % mem_Sigma_iff
% 0.18/0.46 thf(fact_203_mem__Sigma__iff,axiom,
% 0.18/0.46 ! [A3: nat,B4: product_prod_nat_nat,A: set_nat,B2: nat > set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member1293241661at_nat @ ( produc1933845336at_nat @ A3 @ B4 ) @ ( produc894163943at_nat @ A @ B2 ) )
% 0.18/0.46 = ( ( member_nat @ A3 @ A )
% 0.18/0.46 & ( member701585322at_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % mem_Sigma_iff
% 0.18/0.46 thf(fact_204_mem__Sigma__iff,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,A: set_nat,B2: nat > set_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( produc45129834at_nat @ A @ B2 ) )
% 0.18/0.46 = ( ( member_nat @ A3 @ A )
% 0.18/0.46 & ( member_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % mem_Sigma_iff
% 0.18/0.46 thf(fact_205_SigmaI,axiom,
% 0.18/0.46 ! [A3: product_prod_nat_nat,A: set_Pr1986765409at_nat,B4: product_prod_nat_nat,B2: product_prod_nat_nat > set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ A3 @ A )
% 0.18/0.46 => ( ( member701585322at_nat @ B4 @ ( B2 @ A3 ) )
% 0.18/0.46 => ( member2027625872at_nat @ ( produc1168807639at_nat @ A3 @ B4 ) @ ( produc2056081288at_nat @ A @ B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % SigmaI
% 0.18/0.46 thf(fact_206_SigmaI,axiom,
% 0.18/0.46 ! [A3: product_prod_nat_nat,A: set_Pr1986765409at_nat,B4: nat,B2: product_prod_nat_nat > set_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ A3 @ A )
% 0.18/0.46 => ( ( member_nat @ B4 @ ( B2 @ A3 ) )
% 0.18/0.46 => ( member868723479at_nat @ ( produc947540346at_nat @ A3 @ B4 ) @ ( produc2055342601at_nat @ A @ B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % SigmaI
% 0.18/0.46 thf(fact_207_SigmaI,axiom,
% 0.18/0.46 ! [A3: nat,A: set_nat,B4: product_prod_nat_nat,B2: nat > set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member_nat @ A3 @ A )
% 0.18/0.46 => ( ( member701585322at_nat @ B4 @ ( B2 @ A3 ) )
% 0.18/0.46 => ( member1293241661at_nat @ ( produc1933845336at_nat @ A3 @ B4 ) @ ( produc894163943at_nat @ A @ B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % SigmaI
% 0.18/0.46 thf(fact_208_SigmaI,axiom,
% 0.18/0.46 ! [A3: nat,A: set_nat,B4: nat,B2: nat > set_nat] :
% 0.18/0.46 ( ( member_nat @ A3 @ A )
% 0.18/0.46 => ( ( member_nat @ B4 @ ( B2 @ A3 ) )
% 0.18/0.46 => ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( produc45129834at_nat @ A @ B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % SigmaI
% 0.18/0.46 thf(fact_209_trancl__subset__Sigma__aux,axiom,
% 0.18/0.46 ! [A3: product_prod_nat_nat,B4: product_prod_nat_nat,R: set_Pr1490359111at_nat,A: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ A3 @ B4 ) @ ( transi1484096900at_nat @ R ) )
% 0.18/0.46 => ( ( ord_le465611495at_nat @ R
% 0.18/0.46 @ ( produc2056081288at_nat @ A
% 0.18/0.46 @ ^ [Uu: product_prod_nat_nat] : A ) )
% 0.18/0.46 => ( ( A3 = B4 )
% 0.18/0.46 | ( member701585322at_nat @ A3 @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % trancl_subset_Sigma_aux
% 0.18/0.46 thf(fact_210_trancl__subset__Sigma__aux,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,R: set_Pr1986765409at_nat,A: set_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( transi1645084429cl_nat @ R ) )
% 0.18/0.46 => ( ( ord_le841296385at_nat @ R
% 0.18/0.46 @ ( produc45129834at_nat @ A
% 0.18/0.46 @ ^ [Uu: nat] : A ) )
% 0.18/0.46 => ( ( A3 = B4 )
% 0.18/0.46 | ( member_nat @ A3 @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % trancl_subset_Sigma_aux
% 0.18/0.46 thf(fact_211_old_Oprod_Oinject,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,A6: nat,B6: nat] :
% 0.18/0.46 ( ( ( product_Pair_nat_nat @ A3 @ B4 )
% 0.18/0.46 = ( product_Pair_nat_nat @ A6 @ B6 ) )
% 0.18/0.46 = ( ( A3 = A6 )
% 0.18/0.46 & ( B4 = B6 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % old.prod.inject
% 0.18/0.46 thf(fact_212_prod_Oinject,axiom,
% 0.18/0.46 ! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
% 0.18/0.46 ( ( ( product_Pair_nat_nat @ X1 @ X22 )
% 0.18/0.46 = ( product_Pair_nat_nat @ Y1 @ Y22 ) )
% 0.18/0.46 = ( ( X1 = Y1 )
% 0.18/0.46 & ( X22 = Y22 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % prod.inject
% 0.18/0.46 thf(fact_213_r__into__rtrancl,axiom,
% 0.18/0.46 ! [P4: product_prod_nat_nat,R: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ P4 @ R )
% 0.18/0.46 => ( member701585322at_nat @ P4 @ ( transi1645084429cl_nat @ R ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % r_into_rtrancl
% 0.18/0.46 thf(fact_214_rtrancl__idemp,axiom,
% 0.18/0.46 ! [R: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( transi1645084429cl_nat @ ( transi1645084429cl_nat @ R ) )
% 0.18/0.46 = ( transi1645084429cl_nat @ R ) ) ).
% 0.18/0.46
% 0.18/0.46 % rtrancl_idemp
% 0.18/0.46 thf(fact_215_swap__swap,axiom,
% 0.18/0.46 ! [P4: product_prod_nat_nat] :
% 0.18/0.46 ( ( product_swap_nat_nat @ ( product_swap_nat_nat @ P4 ) )
% 0.18/0.46 = P4 ) ).
% 0.18/0.46
% 0.18/0.46 % swap_swap
% 0.18/0.46 thf(fact_216_swap__simp,axiom,
% 0.18/0.46 ! [X3: nat,Y3: nat] :
% 0.18/0.46 ( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) )
% 0.18/0.46 = ( product_Pair_nat_nat @ Y3 @ X3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % swap_simp
% 0.18/0.46 thf(fact_217_converse__rtrancl__induct2,axiom,
% 0.18/0.46 ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R: set_Pr1490359111at_nat,P3: nat > nat > $o] :
% 0.18/0.46 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi1484096900at_nat @ R ) )
% 0.18/0.46 => ( ( P3 @ Bx @ By )
% 0.18/0.46 => ( ! [A4: nat,B3: nat,Aa: nat,Ba: nat] :
% 0.18/0.46 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R )
% 0.18/0.46 => ( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ Aa @ Ba ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi1484096900at_nat @ R ) )
% 0.18/0.46 => ( ( P3 @ Aa @ Ba )
% 0.18/0.46 => ( P3 @ A4 @ B3 ) ) ) )
% 0.18/0.46 => ( P3 @ Ax @ Ay ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % converse_rtrancl_induct2
% 0.18/0.46 thf(fact_218_converse__rtranclE2,axiom,
% 0.18/0.46 ! [Xa2: nat,Xb: nat,Za: nat,Zb: nat,R: set_Pr1490359111at_nat] :
% 0.18/0.46 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xb ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi1484096900at_nat @ R ) )
% 0.18/0.46 => ( ( ( product_Pair_nat_nat @ Xa2 @ Xb )
% 0.18/0.46 != ( product_Pair_nat_nat @ Za @ Zb ) )
% 0.18/0.46 => ~ ! [A4: nat,B3: nat] :
% 0.18/0.46 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xb ) @ ( product_Pair_nat_nat @ A4 @ B3 ) ) @ R )
% 0.18/0.46 => ~ ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi1484096900at_nat @ R ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % converse_rtranclE2
% 0.18/0.46 thf(fact_219_rtrancl__induct2,axiom,
% 0.18/0.46 ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R: set_Pr1490359111at_nat,P3: nat > nat > $o] :
% 0.18/0.46 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi1484096900at_nat @ R ) )
% 0.18/0.46 => ( ( P3 @ Ax @ Ay )
% 0.18/0.46 => ( ! [A4: nat,B3: nat,Aa: nat,Ba: nat] :
% 0.18/0.46 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A4 @ B3 ) ) @ ( transi1484096900at_nat @ R ) )
% 0.18/0.46 => ( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R )
% 0.18/0.46 => ( ( P3 @ A4 @ B3 )
% 0.18/0.46 => ( P3 @ Aa @ Ba ) ) ) )
% 0.18/0.46 => ( P3 @ Bx @ By ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % rtrancl_induct2
% 0.18/0.46 thf(fact_220_old_Oprod_Oinducts,axiom,
% 0.18/0.46 ! [P3: product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
% 0.18/0.46 ( ! [A4: nat,B3: nat] : ( P3 @ ( product_Pair_nat_nat @ A4 @ B3 ) )
% 0.18/0.46 => ( P3 @ Prod ) ) ).
% 0.18/0.46
% 0.18/0.46 % old.prod.inducts
% 0.18/0.46 thf(fact_221_old_Oprod_Oexhaust,axiom,
% 0.18/0.46 ! [Y3: product_prod_nat_nat] :
% 0.18/0.46 ~ ! [A4: nat,B3: nat] :
% 0.18/0.46 ( Y3
% 0.18/0.46 != ( product_Pair_nat_nat @ A4 @ B3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % old.prod.exhaust
% 0.18/0.46 thf(fact_222_Pair__inject,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,A6: nat,B6: nat] :
% 0.18/0.46 ( ( ( product_Pair_nat_nat @ A3 @ B4 )
% 0.18/0.46 = ( product_Pair_nat_nat @ A6 @ B6 ) )
% 0.18/0.46 => ~ ( ( A3 = A6 )
% 0.18/0.46 => ( B4 != B6 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Pair_inject
% 0.18/0.46 thf(fact_223_prod__cases,axiom,
% 0.18/0.46 ! [P3: product_prod_nat_nat > $o,P4: product_prod_nat_nat] :
% 0.18/0.46 ( ! [A4: nat,B3: nat] : ( P3 @ ( product_Pair_nat_nat @ A4 @ B3 ) )
% 0.18/0.46 => ( P3 @ P4 ) ) ).
% 0.18/0.46
% 0.18/0.46 % prod_cases
% 0.18/0.46 thf(fact_224_surj__pair,axiom,
% 0.18/0.46 ! [P4: product_prod_nat_nat] :
% 0.18/0.46 ? [X2: nat,Y2: nat] :
% 0.18/0.46 ( P4
% 0.18/0.46 = ( product_Pair_nat_nat @ X2 @ Y2 ) ) ).
% 0.18/0.46
% 0.18/0.46 % surj_pair
% 0.18/0.46 thf(fact_225_Graph_OisSPath__nt__parallel,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat > a,S: nat,P4: list_P559422087at_nat,T3: nat,E2: product_prod_nat_nat] :
% 0.18/0.46 ( ( isSimplePath_a @ C @ S @ P4 @ T3 )
% 0.18/0.46 => ( ( member701585322at_nat @ E2 @ ( set_Pr2131844118at_nat @ P4 ) )
% 0.18/0.46 => ~ ( member701585322at_nat @ ( product_swap_nat_nat @ E2 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Graph.isSPath_nt_parallel
% 0.18/0.46 thf(fact_226_Sigma__cong,axiom,
% 0.18/0.46 ! [A: set_nat,B2: set_nat,C3: nat > set_nat,D: nat > set_nat] :
% 0.18/0.46 ( ( A = B2 )
% 0.18/0.46 => ( ! [X2: nat] :
% 0.18/0.46 ( ( member_nat @ X2 @ B2 )
% 0.18/0.46 => ( ( C3 @ X2 )
% 0.18/0.46 = ( D @ X2 ) ) )
% 0.18/0.46 => ( ( produc45129834at_nat @ A @ C3 )
% 0.18/0.46 = ( produc45129834at_nat @ B2 @ D ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Sigma_cong
% 0.18/0.46 thf(fact_227_Times__eq__cancel2,axiom,
% 0.18/0.46 ! [X3: nat,C3: set_nat,A: set_nat,B2: set_nat] :
% 0.18/0.46 ( ( member_nat @ X3 @ C3 )
% 0.18/0.46 => ( ( ( produc45129834at_nat @ A
% 0.18/0.46 @ ^ [Uu: nat] : C3 )
% 0.18/0.46 = ( produc45129834at_nat @ B2
% 0.18/0.46 @ ^ [Uu: nat] : C3 ) )
% 0.18/0.46 = ( A = B2 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Times_eq_cancel2
% 0.18/0.46 thf(fact_228_SigmaE,axiom,
% 0.18/0.46 ! [C: produc842455143at_nat,A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member2027625872at_nat @ C @ ( produc2056081288at_nat @ A @ B2 ) )
% 0.18/0.46 => ~ ! [X2: product_prod_nat_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ X2 @ A )
% 0.18/0.46 => ! [Y2: product_prod_nat_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ Y2 @ ( B2 @ X2 ) )
% 0.18/0.46 => ( C
% 0.18/0.46 != ( produc1168807639at_nat @ X2 @ Y2 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % SigmaE
% 0.18/0.46 thf(fact_229_SigmaE,axiom,
% 0.18/0.46 ! [C: produc1271302400at_nat,A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_nat] :
% 0.18/0.46 ( ( member868723479at_nat @ C @ ( produc2055342601at_nat @ A @ B2 ) )
% 0.18/0.46 => ~ ! [X2: product_prod_nat_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ X2 @ A )
% 0.18/0.46 => ! [Y2: nat] :
% 0.18/0.46 ( ( member_nat @ Y2 @ ( B2 @ X2 ) )
% 0.18/0.46 => ( C
% 0.18/0.46 != ( produc947540346at_nat @ X2 @ Y2 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % SigmaE
% 0.18/0.46 thf(fact_230_SigmaE,axiom,
% 0.18/0.46 ! [C: produc1695820582at_nat,A: set_nat,B2: nat > set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member1293241661at_nat @ C @ ( produc894163943at_nat @ A @ B2 ) )
% 0.18/0.46 => ~ ! [X2: nat] :
% 0.18/0.46 ( ( member_nat @ X2 @ A )
% 0.18/0.46 => ! [Y2: product_prod_nat_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ Y2 @ ( B2 @ X2 ) )
% 0.18/0.46 => ( C
% 0.18/0.46 != ( produc1933845336at_nat @ X2 @ Y2 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % SigmaE
% 0.18/0.46 thf(fact_231_SigmaE,axiom,
% 0.18/0.46 ! [C: product_prod_nat_nat,A: set_nat,B2: nat > set_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ C @ ( produc45129834at_nat @ A @ B2 ) )
% 0.18/0.46 => ~ ! [X2: nat] :
% 0.18/0.46 ( ( member_nat @ X2 @ A )
% 0.18/0.46 => ! [Y2: nat] :
% 0.18/0.46 ( ( member_nat @ Y2 @ ( B2 @ X2 ) )
% 0.18/0.46 => ( C
% 0.18/0.46 != ( product_Pair_nat_nat @ X2 @ Y2 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % SigmaE
% 0.18/0.46 thf(fact_232_SigmaD1,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,A: set_nat,B2: nat > set_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( produc45129834at_nat @ A @ B2 ) )
% 0.18/0.46 => ( member_nat @ A3 @ A ) ) ).
% 0.18/0.46
% 0.18/0.46 % SigmaD1
% 0.18/0.46 thf(fact_233_SigmaD2,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,A: set_nat,B2: nat > set_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( produc45129834at_nat @ A @ B2 ) )
% 0.18/0.46 => ( member_nat @ B4 @ ( B2 @ A3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % SigmaD2
% 0.18/0.46 thf(fact_234_SigmaE2,axiom,
% 0.18/0.46 ! [A3: product_prod_nat_nat,B4: product_prod_nat_nat,A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ A3 @ B4 ) @ ( produc2056081288at_nat @ A @ B2 ) )
% 0.18/0.46 => ~ ( ( member701585322at_nat @ A3 @ A )
% 0.18/0.46 => ~ ( member701585322at_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % SigmaE2
% 0.18/0.46 thf(fact_235_SigmaE2,axiom,
% 0.18/0.46 ! [A3: product_prod_nat_nat,B4: nat,A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_nat] :
% 0.18/0.46 ( ( member868723479at_nat @ ( produc947540346at_nat @ A3 @ B4 ) @ ( produc2055342601at_nat @ A @ B2 ) )
% 0.18/0.46 => ~ ( ( member701585322at_nat @ A3 @ A )
% 0.18/0.46 => ~ ( member_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % SigmaE2
% 0.18/0.46 thf(fact_236_SigmaE2,axiom,
% 0.18/0.46 ! [A3: nat,B4: product_prod_nat_nat,A: set_nat,B2: nat > set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member1293241661at_nat @ ( produc1933845336at_nat @ A3 @ B4 ) @ ( produc894163943at_nat @ A @ B2 ) )
% 0.18/0.46 => ~ ( ( member_nat @ A3 @ A )
% 0.18/0.46 => ~ ( member701585322at_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % SigmaE2
% 0.18/0.46 thf(fact_237_SigmaE2,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,A: set_nat,B2: nat > set_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( produc45129834at_nat @ A @ B2 ) )
% 0.18/0.46 => ~ ( ( member_nat @ A3 @ A )
% 0.18/0.46 => ~ ( member_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % SigmaE2
% 0.18/0.46 thf(fact_238_rtranclE,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,R: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( transi1645084429cl_nat @ R ) )
% 0.18/0.46 => ( ( A3 != B4 )
% 0.18/0.46 => ~ ! [Y2: nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ Y2 ) @ ( transi1645084429cl_nat @ R ) )
% 0.18/0.46 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Y2 @ B4 ) @ R ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % rtranclE
% 0.18/0.46 thf(fact_239_rtrancl_Ocases,axiom,
% 0.18/0.46 ! [A1: nat,A22: nat,R: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi1645084429cl_nat @ R ) )
% 0.18/0.46 => ( ( A22 != A1 )
% 0.18/0.46 => ~ ! [B3: nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A1 @ B3 ) @ ( transi1645084429cl_nat @ R ) )
% 0.18/0.46 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ R ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % rtrancl.cases
% 0.18/0.46 thf(fact_240_rtrancl_Osimps,axiom,
% 0.18/0.46 ! [A1: nat,A22: nat,R: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi1645084429cl_nat @ R ) )
% 0.18/0.46 = ( ? [A2: nat] :
% 0.18/0.46 ( ( A1 = A2 )
% 0.18/0.46 & ( A22 = A2 ) )
% 0.18/0.46 | ? [A2: nat,B5: nat,C2: nat] :
% 0.18/0.46 ( ( A1 = A2 )
% 0.18/0.46 & ( A22 = C2 )
% 0.18/0.46 & ( member701585322at_nat @ ( product_Pair_nat_nat @ A2 @ B5 ) @ ( transi1645084429cl_nat @ R ) )
% 0.18/0.46 & ( member701585322at_nat @ ( product_Pair_nat_nat @ B5 @ C2 ) @ R ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % rtrancl.simps
% 0.18/0.46 thf(fact_241_rtrancl__trans,axiom,
% 0.18/0.46 ! [X3: nat,Y3: nat,R: set_Pr1986765409at_nat,Z: nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( transi1645084429cl_nat @ R ) )
% 0.18/0.46 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ ( transi1645084429cl_nat @ R ) )
% 0.18/0.46 => ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Z ) @ ( transi1645084429cl_nat @ R ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % rtrancl_trans
% 0.18/0.46 thf(fact_242_rtrancl__induct,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,R: set_Pr1986765409at_nat,P3: nat > $o] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( transi1645084429cl_nat @ R ) )
% 0.18/0.46 => ( ( P3 @ A3 )
% 0.18/0.46 => ( ! [Y2: nat,Z2: nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ Y2 ) @ ( transi1645084429cl_nat @ R ) )
% 0.18/0.46 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Y2 @ Z2 ) @ R )
% 0.18/0.46 => ( ( P3 @ Y2 )
% 0.18/0.46 => ( P3 @ Z2 ) ) ) )
% 0.18/0.46 => ( P3 @ B4 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % rtrancl_induct
% 0.18/0.46 thf(fact_243_rtrancl_Oinducts,axiom,
% 0.18/0.46 ! [X1: nat,X22: nat,R: set_Pr1986765409at_nat,P3: nat > nat > $o] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) @ ( transi1645084429cl_nat @ R ) )
% 0.18/0.46 => ( ! [A4: nat] : ( P3 @ A4 @ A4 )
% 0.18/0.46 => ( ! [A4: nat,B3: nat,C4: nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( transi1645084429cl_nat @ R ) )
% 0.18/0.46 => ( ( P3 @ A4 @ B3 )
% 0.18/0.46 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ B3 @ C4 ) @ R )
% 0.18/0.46 => ( P3 @ A4 @ C4 ) ) ) )
% 0.18/0.46 => ( P3 @ X1 @ X22 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % rtrancl.inducts
% 0.18/0.46 thf(fact_244_converse__rtranclE,axiom,
% 0.18/0.46 ! [X3: nat,Z: nat,R: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Z ) @ ( transi1645084429cl_nat @ R ) )
% 0.18/0.46 => ( ( X3 != Z )
% 0.18/0.46 => ~ ! [Y2: nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R )
% 0.18/0.46 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Y2 @ Z ) @ ( transi1645084429cl_nat @ R ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % converse_rtranclE
% 0.18/0.46 thf(fact_245_rtrancl_Ortrancl__refl,axiom,
% 0.18/0.46 ! [A3: nat,R: set_Pr1986765409at_nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ A3 ) @ ( transi1645084429cl_nat @ R ) ) ).
% 0.18/0.46
% 0.18/0.46 % rtrancl.rtrancl_refl
% 0.18/0.46 thf(fact_246_converse__rtrancl__induct,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,R: set_Pr1986765409at_nat,P3: nat > $o] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( transi1645084429cl_nat @ R ) )
% 0.18/0.46 => ( ( P3 @ B4 )
% 0.18/0.46 => ( ! [Y2: nat,Z2: nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Y2 @ Z2 ) @ R )
% 0.18/0.46 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Z2 @ B4 ) @ ( transi1645084429cl_nat @ R ) )
% 0.18/0.46 => ( ( P3 @ Z2 )
% 0.18/0.46 => ( P3 @ Y2 ) ) ) )
% 0.18/0.46 => ( P3 @ A3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % converse_rtrancl_induct
% 0.18/0.46 thf(fact_247_rtrancl_Ortrancl__into__rtrancl,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,R: set_Pr1986765409at_nat,C: nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( transi1645084429cl_nat @ R ) )
% 0.18/0.46 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ B4 @ C ) @ R )
% 0.18/0.46 => ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ C ) @ ( transi1645084429cl_nat @ R ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % rtrancl.rtrancl_into_rtrancl
% 0.18/0.46 thf(fact_248_converse__rtrancl__into__rtrancl,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,R: set_Pr1986765409at_nat,C: nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ R )
% 0.18/0.46 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ B4 @ C ) @ ( transi1645084429cl_nat @ R ) )
% 0.18/0.46 => ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ C ) @ ( transi1645084429cl_nat @ R ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % converse_rtrancl_into_rtrancl
% 0.18/0.46 thf(fact_249_rtrancl__mono,axiom,
% 0.18/0.46 ! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ R @ S )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( transi1645084429cl_nat @ R ) @ ( transi1645084429cl_nat @ S ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % rtrancl_mono
% 0.18/0.46 thf(fact_250_rtrancl__subset,axiom,
% 0.18/0.46 ! [R2: set_Pr1986765409at_nat,S3: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ R2 @ S3 )
% 0.18/0.46 => ( ( ord_le841296385at_nat @ S3 @ ( transi1645084429cl_nat @ R2 ) )
% 0.18/0.46 => ( ( transi1645084429cl_nat @ S3 )
% 0.18/0.46 = ( transi1645084429cl_nat @ R2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % rtrancl_subset
% 0.18/0.46 thf(fact_251_rtrancl__subset__rtrancl,axiom,
% 0.18/0.46 ! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ R @ ( transi1645084429cl_nat @ S ) )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( transi1645084429cl_nat @ R ) @ ( transi1645084429cl_nat @ S ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % rtrancl_subset_rtrancl
% 0.18/0.46 thf(fact_252_Sigma__mono,axiom,
% 0.18/0.46 ! [A: set_Pr1986765409at_nat,C3: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_Pr1986765409at_nat,D: product_prod_nat_nat > set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ A @ C3 )
% 0.18/0.46 => ( ! [X2: product_prod_nat_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ X2 @ A )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
% 0.18/0.46 => ( ord_le465611495at_nat @ ( produc2056081288at_nat @ A @ B2 ) @ ( produc2056081288at_nat @ C3 @ D ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Sigma_mono
% 0.18/0.46 thf(fact_253_Sigma__mono,axiom,
% 0.18/0.46 ! [A: set_Pr1986765409at_nat,C3: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_nat,D: product_prod_nat_nat > set_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ A @ C3 )
% 0.18/0.46 => ( ! [X2: product_prod_nat_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ X2 @ A )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
% 0.18/0.46 => ( ord_le1837363350at_nat @ ( produc2055342601at_nat @ A @ B2 ) @ ( produc2055342601at_nat @ C3 @ D ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Sigma_mono
% 0.18/0.46 thf(fact_254_Sigma__mono,axiom,
% 0.18/0.46 ! [A: set_nat,C3: set_nat,B2: nat > set_Pr1986765409at_nat,D: nat > set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ A @ C3 )
% 0.18/0.46 => ( ! [X2: nat] :
% 0.18/0.46 ( ( member_nat @ X2 @ A )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
% 0.18/0.46 => ( ord_le148299708at_nat @ ( produc894163943at_nat @ A @ B2 ) @ ( produc894163943at_nat @ C3 @ D ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Sigma_mono
% 0.18/0.46 thf(fact_255_Sigma__mono,axiom,
% 0.18/0.46 ! [A: set_nat,C3: set_nat,B2: nat > set_nat,D: nat > set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ A @ C3 )
% 0.18/0.46 => ( ! [X2: nat] :
% 0.18/0.46 ( ( member_nat @ X2 @ A )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( produc45129834at_nat @ A @ B2 ) @ ( produc45129834at_nat @ C3 @ D ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Sigma_mono
% 0.18/0.46 thf(fact_256_Image__closed__trancl,axiom,
% 0.18/0.46 ! [R: set_Pr1490359111at_nat,X4: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ ( image_1655366419at_nat @ R @ X4 ) @ X4 )
% 0.18/0.46 => ( ( image_1655366419at_nat @ ( transi1484096900at_nat @ R ) @ X4 )
% 0.18/0.46 = X4 ) ) ).
% 0.18/0.46
% 0.18/0.46 % Image_closed_trancl
% 0.18/0.46 thf(fact_257_Image__closed__trancl,axiom,
% 0.18/0.46 ! [R: set_Pr1986765409at_nat,X4: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R @ X4 ) @ X4 )
% 0.18/0.46 => ( ( image_nat_nat @ ( transi1645084429cl_nat @ R ) @ X4 )
% 0.18/0.46 = X4 ) ) ).
% 0.18/0.46
% 0.18/0.46 % Image_closed_trancl
% 0.18/0.46 thf(fact_258_Not__Domain__rtrancl,axiom,
% 0.18/0.46 ! [X3: product_prod_nat_nat,R2: set_Pr1490359111at_nat,Y3: product_prod_nat_nat] :
% 0.18/0.46 ( ~ ( member701585322at_nat @ X3 @ ( domain37934156at_nat @ R2 ) )
% 0.18/0.46 => ( ( member2027625872at_nat @ ( produc1168807639at_nat @ X3 @ Y3 ) @ ( transi1484096900at_nat @ R2 ) )
% 0.18/0.46 = ( X3 = Y3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Not_Domain_rtrancl
% 0.18/0.46 thf(fact_259_Not__Domain__rtrancl,axiom,
% 0.18/0.46 ! [X3: nat,R2: set_Pr1986765409at_nat,Y3: nat] :
% 0.18/0.46 ( ~ ( member_nat @ X3 @ ( domain_nat_nat @ R2 ) )
% 0.18/0.46 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( transi1645084429cl_nat @ R2 ) )
% 0.18/0.46 = ( X3 = Y3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Not_Domain_rtrancl
% 0.18/0.46 thf(fact_260_Times__subset__cancel2,axiom,
% 0.18/0.46 ! [X3: product_prod_nat_nat,C3: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ X3 @ C3 )
% 0.18/0.46 => ( ( ord_le465611495at_nat
% 0.18/0.46 @ ( produc2056081288at_nat @ A
% 0.18/0.46 @ ^ [Uu: product_prod_nat_nat] : C3 )
% 0.18/0.46 @ ( produc2056081288at_nat @ B2
% 0.18/0.46 @ ^ [Uu: product_prod_nat_nat] : C3 ) )
% 0.18/0.46 = ( ord_le841296385at_nat @ A @ B2 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Times_subset_cancel2
% 0.18/0.46 thf(fact_261_Times__subset__cancel2,axiom,
% 0.18/0.46 ! [X3: nat,C3: set_nat,A: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( member_nat @ X3 @ C3 )
% 0.18/0.46 => ( ( ord_le1837363350at_nat
% 0.18/0.46 @ ( produc2055342601at_nat @ A
% 0.18/0.46 @ ^ [Uu: product_prod_nat_nat] : C3 )
% 0.18/0.46 @ ( produc2055342601at_nat @ B2
% 0.18/0.46 @ ^ [Uu: product_prod_nat_nat] : C3 ) )
% 0.18/0.46 = ( ord_le841296385at_nat @ A @ B2 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Times_subset_cancel2
% 0.18/0.46 thf(fact_262_Times__subset__cancel2,axiom,
% 0.18/0.46 ! [X3: product_prod_nat_nat,C3: set_Pr1986765409at_nat,A: set_nat,B2: set_nat] :
% 0.18/0.46 ( ( member701585322at_nat @ X3 @ C3 )
% 0.18/0.46 => ( ( ord_le148299708at_nat
% 0.18/0.46 @ ( produc894163943at_nat @ A
% 0.18/0.46 @ ^ [Uu: nat] : C3 )
% 0.18/0.46 @ ( produc894163943at_nat @ B2
% 0.18/0.46 @ ^ [Uu: nat] : C3 ) )
% 0.18/0.46 = ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Times_subset_cancel2
% 0.18/0.46 thf(fact_263_Times__subset__cancel2,axiom,
% 0.18/0.46 ! [X3: nat,C3: set_nat,A: set_nat,B2: set_nat] :
% 0.18/0.46 ( ( member_nat @ X3 @ C3 )
% 0.18/0.46 => ( ( ord_le841296385at_nat
% 0.18/0.46 @ ( produc45129834at_nat @ A
% 0.18/0.46 @ ^ [Uu: nat] : C3 )
% 0.18/0.46 @ ( produc45129834at_nat @ B2
% 0.18/0.46 @ ^ [Uu: nat] : C3 ) )
% 0.18/0.46 = ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Times_subset_cancel2
% 0.18/0.46 thf(fact_264_finite__rtrancl__Image,axiom,
% 0.18/0.46 ! [R2: set_Pr1490359111at_nat,A: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( finite48957584at_nat @ R2 )
% 0.18/0.46 => ( ( finite772653738at_nat @ A )
% 0.18/0.46 => ( finite772653738at_nat @ ( image_1655366419at_nat @ ( transi1484096900at_nat @ R2 ) @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % finite_rtrancl_Image
% 0.18/0.46 thf(fact_265_finite__rtrancl__Image,axiom,
% 0.18/0.46 ! [R2: set_Pr1986765409at_nat,A: set_nat] :
% 0.18/0.46 ( ( finite772653738at_nat @ R2 )
% 0.18/0.46 => ( ( finite_finite_nat @ A )
% 0.18/0.46 => ( finite_finite_nat @ ( image_nat_nat @ ( transi1645084429cl_nat @ R2 ) @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % finite_rtrancl_Image
% 0.18/0.46 thf(fact_266_infinite__cartesian__product,axiom,
% 0.18/0.46 ! [A: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ~ ( finite772653738at_nat @ A )
% 0.18/0.46 => ( ~ ( finite772653738at_nat @ B2 )
% 0.18/0.46 => ~ ( finite48957584at_nat
% 0.18/0.46 @ ( produc2056081288at_nat @ A
% 0.18/0.46 @ ^ [Uu: product_prod_nat_nat] : B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % infinite_cartesian_product
% 0.18/0.46 thf(fact_267_infinite__cartesian__product,axiom,
% 0.18/0.46 ! [A: set_Pr1986765409at_nat,B2: set_nat] :
% 0.18/0.46 ( ~ ( finite772653738at_nat @ A )
% 0.18/0.46 => ( ~ ( finite_finite_nat @ B2 )
% 0.18/0.46 => ~ ( finite2000257047at_nat
% 0.18/0.46 @ ( produc2055342601at_nat @ A
% 0.18/0.46 @ ^ [Uu: product_prod_nat_nat] : B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % infinite_cartesian_product
% 0.18/0.46 thf(fact_268_infinite__cartesian__product,axiom,
% 0.18/0.46 ! [A: set_nat,B2: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ~ ( finite_finite_nat @ A )
% 0.18/0.46 => ( ~ ( finite772653738at_nat @ B2 )
% 0.18/0.46 => ~ ( finite277291581at_nat
% 0.18/0.46 @ ( produc894163943at_nat @ A
% 0.18/0.46 @ ^ [Uu: nat] : B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % infinite_cartesian_product
% 0.18/0.46 thf(fact_269_infinite__cartesian__product,axiom,
% 0.18/0.46 ! [A: set_nat,B2: set_nat] :
% 0.18/0.46 ( ~ ( finite_finite_nat @ A )
% 0.18/0.46 => ( ~ ( finite_finite_nat @ B2 )
% 0.18/0.46 => ~ ( finite772653738at_nat
% 0.18/0.46 @ ( produc45129834at_nat @ A
% 0.18/0.46 @ ^ [Uu: nat] : B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % infinite_cartesian_product
% 0.18/0.46 thf(fact_270_order__refl,axiom,
% 0.18/0.46 ! [X3: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ X3 @ X3 ) ).
% 0.18/0.46
% 0.18/0.46 % order_refl
% 0.18/0.46 thf(fact_271_order__refl,axiom,
% 0.18/0.46 ! [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ X3 ) ).
% 0.18/0.46
% 0.18/0.46 % order_refl
% 0.18/0.46 thf(fact_272_order__refl,axiom,
% 0.18/0.46 ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% 0.18/0.46
% 0.18/0.46 % order_refl
% 0.18/0.46 thf(fact_273_order__subst1,axiom,
% 0.18/0.46 ! [A3: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ A3 @ ( F @ B4 ) )
% 0.18/0.46 => ( ( ord_le841296385at_nat @ B4 @ C )
% 0.18/0.46 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_le841296385at_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_subst1
% 0.18/0.46 thf(fact_274_order__subst1,axiom,
% 0.18/0.46 ! [A3: set_Pr1986765409at_nat,F: set_nat > set_Pr1986765409at_nat,B4: set_nat,C: set_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ A3 @ ( F @ B4 ) )
% 0.18/0.46 => ( ( ord_less_eq_set_nat @ B4 @ C )
% 0.18/0.46 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_le841296385at_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_subst1
% 0.18/0.46 thf(fact_275_order__subst1,axiom,
% 0.18/0.46 ! [A3: set_Pr1986765409at_nat,F: nat > set_Pr1986765409at_nat,B4: nat,C: nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ A3 @ ( F @ B4 ) )
% 0.18/0.46 => ( ( ord_less_eq_nat @ B4 @ C )
% 0.18/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_le841296385at_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_subst1
% 0.18/0.46 thf(fact_276_order__subst1,axiom,
% 0.18/0.46 ! [A3: set_nat,F: set_Pr1986765409at_nat > set_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ A3 @ ( F @ B4 ) )
% 0.18/0.46 => ( ( ord_le841296385at_nat @ B4 @ C )
% 0.18/0.46 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_subst1
% 0.18/0.46 thf(fact_277_order__subst1,axiom,
% 0.18/0.46 ! [A3: set_nat,F: set_nat > set_nat,B4: set_nat,C: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ A3 @ ( F @ B4 ) )
% 0.18/0.46 => ( ( ord_less_eq_set_nat @ B4 @ C )
% 0.18/0.46 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_subst1
% 0.18/0.46 thf(fact_278_order__subst1,axiom,
% 0.18/0.46 ! [A3: set_nat,F: nat > set_nat,B4: nat,C: nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ A3 @ ( F @ B4 ) )
% 0.18/0.46 => ( ( ord_less_eq_nat @ B4 @ C )
% 0.18/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_subst1
% 0.18/0.46 thf(fact_279_order__subst1,axiom,
% 0.18/0.46 ! [A3: nat,F: set_Pr1986765409at_nat > nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ A3 @ ( F @ B4 ) )
% 0.18/0.46 => ( ( ord_le841296385at_nat @ B4 @ C )
% 0.18/0.46 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_subst1
% 0.18/0.46 thf(fact_280_order__subst1,axiom,
% 0.18/0.46 ! [A3: nat,F: set_nat > nat,B4: set_nat,C: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ A3 @ ( F @ B4 ) )
% 0.18/0.46 => ( ( ord_less_eq_set_nat @ B4 @ C )
% 0.18/0.46 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_subst1
% 0.18/0.46 thf(fact_281_order__subst1,axiom,
% 0.18/0.46 ! [A3: nat,F: nat > nat,B4: nat,C: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ A3 @ ( F @ B4 ) )
% 0.18/0.46 => ( ( ord_less_eq_nat @ B4 @ C )
% 0.18/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_subst1
% 0.18/0.46 thf(fact_282_order__subst2,axiom,
% 0.18/0.46 ! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ord_le841296385at_nat @ ( F @ B4 ) @ C )
% 0.18/0.46 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_subst2
% 0.18/0.46 thf(fact_283_order__subst2,axiom,
% 0.18/0.46 ! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_nat,C: set_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ord_less_eq_set_nat @ ( F @ B4 ) @ C )
% 0.18/0.46 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_subst2
% 0.18/0.46 thf(fact_284_order__subst2,axiom,
% 0.18/0.46 ! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > nat,C: nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ord_less_eq_nat @ ( F @ B4 ) @ C )
% 0.18/0.46 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_subst2
% 0.18/0.46 thf(fact_285_order__subst2,axiom,
% 0.18/0.46 ! [A3: set_nat,B4: set_nat,F: set_nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ord_le841296385at_nat @ ( F @ B4 ) @ C )
% 0.18/0.46 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_subst2
% 0.18/0.46 thf(fact_286_order__subst2,axiom,
% 0.18/0.46 ! [A3: set_nat,B4: set_nat,F: set_nat > set_nat,C: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ord_less_eq_set_nat @ ( F @ B4 ) @ C )
% 0.18/0.46 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_subst2
% 0.18/0.46 thf(fact_287_order__subst2,axiom,
% 0.18/0.46 ! [A3: set_nat,B4: set_nat,F: set_nat > nat,C: nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ord_less_eq_nat @ ( F @ B4 ) @ C )
% 0.18/0.46 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_subst2
% 0.18/0.46 thf(fact_288_order__subst2,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,F: nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ord_le841296385at_nat @ ( F @ B4 ) @ C )
% 0.18/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_subst2
% 0.18/0.46 thf(fact_289_order__subst2,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,F: nat > set_nat,C: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ord_less_eq_set_nat @ ( F @ B4 ) @ C )
% 0.18/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_subst2
% 0.18/0.46 thf(fact_290_order__subst2,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,F: nat > nat,C: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ord_less_eq_nat @ ( F @ B4 ) @ C )
% 0.18/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_subst2
% 0.18/0.46 thf(fact_291_ord__eq__le__subst,axiom,
% 0.18/0.46 ! [A3: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( A3
% 0.18/0.46 = ( F @ B4 ) )
% 0.18/0.46 => ( ( ord_le841296385at_nat @ B4 @ C )
% 0.18/0.46 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_le841296385at_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_le_subst
% 0.18/0.46 thf(fact_292_ord__eq__le__subst,axiom,
% 0.18/0.46 ! [A3: set_nat,F: set_Pr1986765409at_nat > set_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( A3
% 0.18/0.46 = ( F @ B4 ) )
% 0.18/0.46 => ( ( ord_le841296385at_nat @ B4 @ C )
% 0.18/0.46 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_le_subst
% 0.18/0.46 thf(fact_293_ord__eq__le__subst,axiom,
% 0.18/0.46 ! [A3: nat,F: set_Pr1986765409at_nat > nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( A3
% 0.18/0.46 = ( F @ B4 ) )
% 0.18/0.46 => ( ( ord_le841296385at_nat @ B4 @ C )
% 0.18/0.46 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_le_subst
% 0.18/0.46 thf(fact_294_ord__eq__le__subst,axiom,
% 0.18/0.46 ! [A3: set_Pr1986765409at_nat,F: set_nat > set_Pr1986765409at_nat,B4: set_nat,C: set_nat] :
% 0.18/0.46 ( ( A3
% 0.18/0.46 = ( F @ B4 ) )
% 0.18/0.46 => ( ( ord_less_eq_set_nat @ B4 @ C )
% 0.18/0.46 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_le841296385at_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_le_subst
% 0.18/0.46 thf(fact_295_ord__eq__le__subst,axiom,
% 0.18/0.46 ! [A3: set_nat,F: set_nat > set_nat,B4: set_nat,C: set_nat] :
% 0.18/0.46 ( ( A3
% 0.18/0.46 = ( F @ B4 ) )
% 0.18/0.46 => ( ( ord_less_eq_set_nat @ B4 @ C )
% 0.18/0.46 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_le_subst
% 0.18/0.46 thf(fact_296_ord__eq__le__subst,axiom,
% 0.18/0.46 ! [A3: nat,F: set_nat > nat,B4: set_nat,C: set_nat] :
% 0.18/0.46 ( ( A3
% 0.18/0.46 = ( F @ B4 ) )
% 0.18/0.46 => ( ( ord_less_eq_set_nat @ B4 @ C )
% 0.18/0.46 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_le_subst
% 0.18/0.46 thf(fact_297_ord__eq__le__subst,axiom,
% 0.18/0.46 ! [A3: set_Pr1986765409at_nat,F: nat > set_Pr1986765409at_nat,B4: nat,C: nat] :
% 0.18/0.46 ( ( A3
% 0.18/0.46 = ( F @ B4 ) )
% 0.18/0.46 => ( ( ord_less_eq_nat @ B4 @ C )
% 0.18/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_le841296385at_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_le_subst
% 0.18/0.46 thf(fact_298_ord__eq__le__subst,axiom,
% 0.18/0.46 ! [A3: set_nat,F: nat > set_nat,B4: nat,C: nat] :
% 0.18/0.46 ( ( A3
% 0.18/0.46 = ( F @ B4 ) )
% 0.18/0.46 => ( ( ord_less_eq_nat @ B4 @ C )
% 0.18/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_le_subst
% 0.18/0.46 thf(fact_299_ord__eq__le__subst,axiom,
% 0.18/0.46 ! [A3: nat,F: nat > nat,B4: nat,C: nat] :
% 0.18/0.46 ( ( A3
% 0.18/0.46 = ( F @ B4 ) )
% 0.18/0.46 => ( ( ord_less_eq_nat @ B4 @ C )
% 0.18/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_le_subst
% 0.18/0.46 thf(fact_300_ord__le__eq__subst,axiom,
% 0.18/0.46 ! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ( F @ B4 )
% 0.18/0.46 = C )
% 0.18/0.46 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_le_eq_subst
% 0.18/0.46 thf(fact_301_ord__le__eq__subst,axiom,
% 0.18/0.46 ! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_nat,C: set_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ( F @ B4 )
% 0.18/0.46 = C )
% 0.18/0.46 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_le_eq_subst
% 0.18/0.46 thf(fact_302_ord__le__eq__subst,axiom,
% 0.18/0.46 ! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > nat,C: nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ( F @ B4 )
% 0.18/0.46 = C )
% 0.18/0.46 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_le_eq_subst
% 0.18/0.46 thf(fact_303_ord__le__eq__subst,axiom,
% 0.18/0.46 ! [A3: set_nat,B4: set_nat,F: set_nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ( F @ B4 )
% 0.18/0.46 = C )
% 0.18/0.46 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_le_eq_subst
% 0.18/0.46 thf(fact_304_ord__le__eq__subst,axiom,
% 0.18/0.46 ! [A3: set_nat,B4: set_nat,F: set_nat > set_nat,C: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ( F @ B4 )
% 0.18/0.46 = C )
% 0.18/0.46 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_le_eq_subst
% 0.18/0.46 thf(fact_305_ord__le__eq__subst,axiom,
% 0.18/0.46 ! [A3: set_nat,B4: set_nat,F: set_nat > nat,C: nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ( F @ B4 )
% 0.18/0.46 = C )
% 0.18/0.46 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_le_eq_subst
% 0.18/0.46 thf(fact_306_ord__le__eq__subst,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,F: nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ( F @ B4 )
% 0.18/0.46 = C )
% 0.18/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_le841296385at_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_le_eq_subst
% 0.18/0.46 thf(fact_307_ord__le__eq__subst,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,F: nat > set_nat,C: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ( F @ B4 )
% 0.18/0.46 = C )
% 0.18/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_le_eq_subst
% 0.18/0.46 thf(fact_308_ord__le__eq__subst,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,F: nat > nat,C: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ( F @ B4 )
% 0.18/0.46 = C )
% 0.18/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.18/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.18/0.46 => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_le_eq_subst
% 0.18/0.46 thf(fact_309_eq__iff,axiom,
% 0.18/0.46 ( ( ^ [Y4: set_Pr1986765409at_nat,Z3: set_Pr1986765409at_nat] : ( Y4 = Z3 ) )
% 0.18/0.46 = ( ^ [X: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ X @ Y )
% 0.18/0.46 & ( ord_le841296385at_nat @ Y @ X ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % eq_iff
% 0.18/0.46 thf(fact_310_eq__iff,axiom,
% 0.18/0.46 ( ( ^ [Y4: set_nat,Z3: set_nat] : ( Y4 = Z3 ) )
% 0.18/0.46 = ( ^ [X: set_nat,Y: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.18/0.46 & ( ord_less_eq_set_nat @ Y @ X ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % eq_iff
% 0.18/0.46 thf(fact_311_eq__iff,axiom,
% 0.18/0.46 ( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
% 0.18/0.46 = ( ^ [X: nat,Y: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ X @ Y )
% 0.18/0.46 & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % eq_iff
% 0.18/0.46 thf(fact_312_antisym,axiom,
% 0.18/0.46 ! [X3: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ X3 @ Y3 )
% 0.18/0.46 => ( ( ord_le841296385at_nat @ Y3 @ X3 )
% 0.18/0.46 => ( X3 = Y3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % antisym
% 0.18/0.46 thf(fact_313_antisym,axiom,
% 0.18/0.46 ! [X3: set_nat,Y3: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ X3 @ Y3 )
% 0.18/0.46 => ( ( ord_less_eq_set_nat @ Y3 @ X3 )
% 0.18/0.46 => ( X3 = Y3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % antisym
% 0.18/0.46 thf(fact_314_antisym,axiom,
% 0.18/0.46 ! [X3: nat,Y3: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.18/0.46 => ( ( ord_less_eq_nat @ Y3 @ X3 )
% 0.18/0.46 => ( X3 = Y3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % antisym
% 0.18/0.46 thf(fact_315_linear,axiom,
% 0.18/0.46 ! [X3: nat,Y3: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.18/0.46 | ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % linear
% 0.18/0.46 thf(fact_316_eq__refl,axiom,
% 0.18/0.46 ! [X3: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( X3 = Y3 )
% 0.18/0.46 => ( ord_le841296385at_nat @ X3 @ Y3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % eq_refl
% 0.18/0.46 thf(fact_317_eq__refl,axiom,
% 0.18/0.46 ! [X3: set_nat,Y3: set_nat] :
% 0.18/0.46 ( ( X3 = Y3 )
% 0.18/0.46 => ( ord_less_eq_set_nat @ X3 @ Y3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % eq_refl
% 0.18/0.46 thf(fact_318_eq__refl,axiom,
% 0.18/0.46 ! [X3: nat,Y3: nat] :
% 0.18/0.46 ( ( X3 = Y3 )
% 0.18/0.46 => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % eq_refl
% 0.18/0.46 thf(fact_319_le__cases,axiom,
% 0.18/0.46 ! [X3: nat,Y3: nat] :
% 0.18/0.46 ( ~ ( ord_less_eq_nat @ X3 @ Y3 )
% 0.18/0.46 => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % le_cases
% 0.18/0.46 thf(fact_320_order_Otrans,axiom,
% 0.18/0.46 ! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ord_le841296385at_nat @ B4 @ C )
% 0.18/0.46 => ( ord_le841296385at_nat @ A3 @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order.trans
% 0.18/0.46 thf(fact_321_order_Otrans,axiom,
% 0.18/0.46 ! [A3: set_nat,B4: set_nat,C: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ord_less_eq_set_nat @ B4 @ C )
% 0.18/0.46 => ( ord_less_eq_set_nat @ A3 @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order.trans
% 0.18/0.46 thf(fact_322_order_Otrans,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,C: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ord_less_eq_nat @ B4 @ C )
% 0.18/0.46 => ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order.trans
% 0.18/0.46 thf(fact_323_le__cases3,axiom,
% 0.18/0.46 ! [X3: nat,Y3: nat,Z: nat] :
% 0.18/0.46 ( ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.18/0.46 => ~ ( ord_less_eq_nat @ Y3 @ Z ) )
% 0.18/0.46 => ( ( ( ord_less_eq_nat @ Y3 @ X3 )
% 0.18/0.46 => ~ ( ord_less_eq_nat @ X3 @ Z ) )
% 0.18/0.46 => ( ( ( ord_less_eq_nat @ X3 @ Z )
% 0.18/0.46 => ~ ( ord_less_eq_nat @ Z @ Y3 ) )
% 0.18/0.46 => ( ( ( ord_less_eq_nat @ Z @ Y3 )
% 0.18/0.46 => ~ ( ord_less_eq_nat @ Y3 @ X3 ) )
% 0.18/0.46 => ( ( ( ord_less_eq_nat @ Y3 @ Z )
% 0.18/0.46 => ~ ( ord_less_eq_nat @ Z @ X3 ) )
% 0.18/0.46 => ~ ( ( ord_less_eq_nat @ Z @ X3 )
% 0.18/0.46 => ~ ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % le_cases3
% 0.18/0.46 thf(fact_324_antisym__conv,axiom,
% 0.18/0.46 ! [Y3: set_Pr1986765409at_nat,X3: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ Y3 @ X3 )
% 0.18/0.46 => ( ( ord_le841296385at_nat @ X3 @ Y3 )
% 0.18/0.46 = ( X3 = Y3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % antisym_conv
% 0.18/0.46 thf(fact_325_antisym__conv,axiom,
% 0.18/0.46 ! [Y3: set_nat,X3: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ Y3 @ X3 )
% 0.18/0.46 => ( ( ord_less_eq_set_nat @ X3 @ Y3 )
% 0.18/0.46 = ( X3 = Y3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % antisym_conv
% 0.18/0.46 thf(fact_326_antisym__conv,axiom,
% 0.18/0.46 ! [Y3: nat,X3: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ Y3 @ X3 )
% 0.18/0.46 => ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.18/0.46 = ( X3 = Y3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % antisym_conv
% 0.18/0.46 thf(fact_327_order__class_Oorder_Oeq__iff,axiom,
% 0.18/0.46 ( ( ^ [Y4: set_Pr1986765409at_nat,Z3: set_Pr1986765409at_nat] : ( Y4 = Z3 ) )
% 0.18/0.46 = ( ^ [A2: set_Pr1986765409at_nat,B5: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ A2 @ B5 )
% 0.18/0.46 & ( ord_le841296385at_nat @ B5 @ A2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_class.order.eq_iff
% 0.18/0.46 thf(fact_328_order__class_Oorder_Oeq__iff,axiom,
% 0.18/0.46 ( ( ^ [Y4: set_nat,Z3: set_nat] : ( Y4 = Z3 ) )
% 0.18/0.46 = ( ^ [A2: set_nat,B5: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 0.18/0.46 & ( ord_less_eq_set_nat @ B5 @ A2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_class.order.eq_iff
% 0.18/0.46 thf(fact_329_order__class_Oorder_Oeq__iff,axiom,
% 0.18/0.46 ( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
% 0.18/0.46 = ( ^ [A2: nat,B5: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ A2 @ B5 )
% 0.18/0.46 & ( ord_less_eq_nat @ B5 @ A2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_class.order.eq_iff
% 0.18/0.46 thf(fact_330_ord__eq__le__trans,axiom,
% 0.18/0.46 ! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( A3 = B4 )
% 0.18/0.46 => ( ( ord_le841296385at_nat @ B4 @ C )
% 0.18/0.46 => ( ord_le841296385at_nat @ A3 @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_le_trans
% 0.18/0.46 thf(fact_331_ord__eq__le__trans,axiom,
% 0.18/0.46 ! [A3: set_nat,B4: set_nat,C: set_nat] :
% 0.18/0.46 ( ( A3 = B4 )
% 0.18/0.46 => ( ( ord_less_eq_set_nat @ B4 @ C )
% 0.18/0.46 => ( ord_less_eq_set_nat @ A3 @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_le_trans
% 0.18/0.46 thf(fact_332_ord__eq__le__trans,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,C: nat] :
% 0.18/0.46 ( ( A3 = B4 )
% 0.18/0.46 => ( ( ord_less_eq_nat @ B4 @ C )
% 0.18/0.46 => ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_le_trans
% 0.18/0.46 thf(fact_333_ord__le__eq__trans,axiom,
% 0.18/0.46 ! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( B4 = C )
% 0.18/0.46 => ( ord_le841296385at_nat @ A3 @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_le_eq_trans
% 0.18/0.46 thf(fact_334_ord__le__eq__trans,axiom,
% 0.18/0.46 ! [A3: set_nat,B4: set_nat,C: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( B4 = C )
% 0.18/0.46 => ( ord_less_eq_set_nat @ A3 @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_le_eq_trans
% 0.18/0.46 thf(fact_335_ord__le__eq__trans,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat,C: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( B4 = C )
% 0.18/0.46 => ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_le_eq_trans
% 0.18/0.46 thf(fact_336_order__class_Oorder_Oantisym,axiom,
% 0.18/0.46 ! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ord_le841296385at_nat @ B4 @ A3 )
% 0.18/0.46 => ( A3 = B4 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_class.order.antisym
% 0.18/0.46 thf(fact_337_order__class_Oorder_Oantisym,axiom,
% 0.18/0.46 ! [A3: set_nat,B4: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ord_less_eq_set_nat @ B4 @ A3 )
% 0.18/0.46 => ( A3 = B4 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_class.order.antisym
% 0.18/0.46 thf(fact_338_order__class_Oorder_Oantisym,axiom,
% 0.18/0.46 ! [A3: nat,B4: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ A3 @ B4 )
% 0.18/0.46 => ( ( ord_less_eq_nat @ B4 @ A3 )
% 0.18/0.46 => ( A3 = B4 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_class.order.antisym
% 0.18/0.46 thf(fact_339_order__trans,axiom,
% 0.18/0.46 ! [X3: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat,Z: set_Pr1986765409at_nat] :
% 0.18/0.46 ( ( ord_le841296385at_nat @ X3 @ Y3 )
% 0.18/0.46 => ( ( ord_le841296385at_nat @ Y3 @ Z )
% 0.18/0.46 => ( ord_le841296385at_nat @ X3 @ Z ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_trans
% 0.18/0.46 thf(fact_340_order__trans,axiom,
% 0.18/0.46 ! [X3: set_nat,Y3: set_nat,Z: set_nat] :
% 0.18/0.46 ( ( ord_less_eq_set_nat @ X3 @ Y3 )
% 0.18/0.46 => ( ( ord_less_eq_set_nat @ Y3 @ Z )
% 0.18/0.46 => ( ord_less_eq_set_nat @ X3 @ Z ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_trans
% 0.18/0.46 thf(fact_341_order__trans,axiom,
% 0.18/0.46 ! [X3: nat,Y3: nat,Z: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.18/0.46 => ( ( ord_less_eq_nat @ Y3 @ Z )
% 0.18/0.46 => ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_trans
% 0.18/0.46 thf(fact_342_dual__order_Orefl,axiom,
% 0.18/0.46 ! [A3: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ A3 @ A3 ) ).
% 0.18/0.46
% 0.18/0.46 % dual_order.refl
% 0.18/0.46 thf(fact_343_dual__order_Orefl,axiom,
% 0.18/0.46 ! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% 0.18/0.46
% 0.18/0.46 % dual_order.refl
% 0.18/0.46 thf(fact_344_dual__order_Orefl,axiom,
% 0.18/0.46 ! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% 0.18/0.46
% 0.18/0.46 % dual_order.refl
% 0.18/0.46 thf(fact_345_linorder__wlog,axiom,
% 0.18/0.46 ! [P3: nat > nat > $o,A3: nat,B4: nat] :
% 0.18/0.46 ( ! [A4: nat,B3: nat] :
% 0.18/0.46 ( ( ord_less_eq_nat @ A4 @ B3 )
% 0.18/0.46 => ( P3 @ A4 @ B3 ) )
% 0.18/0.46 => ( ! [A4: nat,B3: nat] :
% 0.18/0.46 ( ( P3 @ B3 @ A4 )
% 0.18/0.46 => ( P3 @ A4 @ B3 ) )
% 0.18/0.46 => ( P3 @ A3 @ B4 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % linorder_wlog
% 0.18/0.46 thf(fact_346_rtc__isPath,axiom,
% 0.18/0.46 ! [U: nat,V: nat] :
% 0.18/0.46 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi1645084429cl_nat @ ( e_a @ c ) ) )
% 0.18/0.46 => ? [P2: list_P559422087at_nat] : ( isPath_a @ c @ U @ P2 @ V ) ) ).
% 0.18/0.46
% 0.18/0.46 % rtc_isPath
% 0.18/0.46 thf(fact_347_isSPath__pathLE,axiom,
% 0.18/0.46 ! [S: nat,P4: list_P559422087at_nat,T3: nat] :
% 0.18/0.46 ( ( isPath_a @ c @ S @ P4 @ T3 )
% 0.18/0.46 => ? [P5: list_P559422087at_nat] : ( isSimplePath_a @ c @ S @ P5 @ T3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % isSPath_pathLE
% 0.18/0.46 thf(fact_348_connected__def,axiom,
% 0.18/0.46 ! [U: nat,V: nat] :
% 0.18/0.46 ( ( connected_a @ c @ U @ V )
% 0.18/0.46 = ( ? [P: list_P559422087at_nat] : ( isPath_a @ c @ U @ P @ V ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % connected_def
% 0.18/0.46 thf(fact_349_shortestPath__is__path,axiom,
% 0.18/0.46 ! [U: nat,P4: list_P559422087at_nat,V: nat] :
% 0.18/0.46 ( ( isShortestPath_a @ c @ U @ P4 @ V )
% 0.18/0.46 => ( isPath_a @ c @ U @ P4 @ V ) ) ).
% 0.18/0.46
% 0.18/0.46 % shortestPath_is_path
% 0.18/0.46 thf(fact_350_isPath__ex__edge1,axiom,
% 0.18/0.46 ! [U: nat,P4: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
% 0.18/0.46 ( ( isPath_a @ c @ U @ P4 @ V )
% 0.18/0.46 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 0.18/0.46 => ( ( U1 != U )
% 0.18/0.46 => ? [U23: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ U23 @ U1 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % isPath_ex_edge1
% 0.18/0.46 thf(fact_351_isPath__ex__edge2,axiom,
% 0.18/0.46 ! [U: nat,P4: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
% 0.18/0.46 ( ( isPath_a @ c @ U @ P4 @ V )
% 0.18/0.46 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 0.18/0.46 => ( ( V1 != V )
% 0.18/0.46 => ? [V23: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ V1 @ V23 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % isPath_ex_edge2
% 0.18/0.46 thf(fact_352_isPath__edgeset,axiom,
% 0.18/0.46 ! [U: nat,P4: list_P559422087at_nat,V: nat,E2: product_prod_nat_nat] :
% 0.18/0.46 ( ( isPath_a @ c @ U @ P4 @ V )
% 0.18/0.46 => ( ( member701585322at_nat @ E2 @ ( set_Pr2131844118at_nat @ P4 ) )
% 0.18/0.46 => ( member701585322at_nat @ E2 @ ( e_a @ c ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % isPath_edgeset
% 0.18/0.46 thf(fact_353_isPath__rtc,axiom,
% 0.18/0.46 ! [U: nat,P4: list_P559422087at_nat,V: nat] :
% 0.18/0.46 ( ( isPath_a @ c @ U @ P4 @ V )
% 0.18/0.46 => ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi1645084429cl_nat @ ( e_a @ c ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % isPath_rtc
% 0.18/0.46
% 0.18/0.46 % Conjectures (1)
% 0.18/0.46 thf(conj_0,conjecture,
% 0.18/0.46 finite772653738at_nat @ ( edmond475474835dges_a @ c @ s @ t ) ).
% 0.18/0.46
% 0.18/0.46 %------------------------------------------------------------------------------
% 0.18/0.46 ------------------------
% 2.86/1.54 ---- Embedded in THF ---
% 2.86/1.54 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.86/1.54 %%% Generated on Tue Feb 24 08:24:53 EST 2026
% 2.86/1.54 %%% using '$$dhol' embedding, version 1.3.0.
% 2.86/1.54 %%% Logic specification used:
% 2.86/1.54 %%% thf(spec, logic, $$dhol).
% 2.86/1.54
% 2.86/1.54 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.tqDOf96qQm/DTF2DTF12988.p
% See solution above
% 2.86/1.54 ------------------------
% 2.86/1.58 ---- Cleaned THF ---
% 2.86/1.58 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 2.86/1.58 set_Pr1490359111at_nat: $tType ).
% 2.86/1.58
% 2.86/1.58 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.86/1.58 produc842455143at_nat: $tType ).
% 2.86/1.58
% 2.86/1.58 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
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% 2.86/1.58 ! [V: nat] : ( connected_a @ c @ V @ V ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_8_spEdges__def,axiom,
% 2.86/1.58 ( ( edmond475474835dges_a @ c @ s @ t )
% 2.86/1.58 = ( collec7649004at_nat
% 2.86/1.58 @ ^ [E: product_prod_nat_nat] :
% 2.86/1.58 ? [P: list_P559422087at_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ E @ ( set_Pr2131844118at_nat @ P ) )
% 2.86/1.58 & ( isShortestPath_a @ c @ s @ P @ t ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_9_finite__Range,axiom,
% 2.86/1.58 ! [R: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( finite772653738at_nat @ R )
% 2.86/1.58 => ( finite_finite_nat @ ( range_nat_nat @ R ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_10_finite__Domain,axiom,
% 2.86/1.58 ! [R: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( finite772653738at_nat @ R )
% 2.86/1.58 => ( finite_finite_nat @ ( domain_nat_nat @ R ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_11_Efin__imp__Vfin,axiom,
% 2.86/1.58 ( ( finite772653738at_nat @ ( e_a @ c ) )
% 2.86/1.58 => ( finite_finite_nat @ ( v_a @ c ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_12_connected__inV__iff,axiom,
% 2.86/1.58 ! [U: nat,V: nat] :
% 2.86/1.58 ( ( connected_a @ c @ U @ V )
% 2.86/1.58 => ( ( member_nat @ V @ ( v_a @ c ) )
% 2.86/1.58 = ( member_nat @ U @ ( v_a @ c ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_13_obtain__shortest__path,axiom,
% 2.86/1.58 ! [U: nat,V: nat] :
% 2.86/1.58 ( ( connected_a @ c @ U @ V )
% 2.86/1.58 => ~ ! [P2: list_P559422087at_nat] :
% 2.86/1.58 ~ ( isShortestPath_a @ c @ U @ P2 @ V ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_14_finite__Collect__conjI,axiom,
% 2.86/1.58 ! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 2.86/1.58 ( ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 2.86/1.58 | ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) ) )
% 2.86/1.58 => ( finite772653738at_nat
% 2.86/1.58 @ ( collec7649004at_nat
% 2.86/1.58 @ ^ [X: product_prod_nat_nat] :
% 2.86/1.58 ( ( P3 @ X )
% 2.86/1.58 & ( Q @ X ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_15_finite__Collect__conjI,axiom,
% 2.86/1.58 ! [P3: nat > $o,Q: nat > $o] :
% 2.86/1.58 ( ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 2.86/1.58 | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
% 2.86/1.58 => ( finite_finite_nat
% 2.86/1.58 @ ( collect_nat
% 2.86/1.58 @ ^ [X: nat] :
% 2.86/1.58 ( ( P3 @ X )
% 2.86/1.58 & ( Q @ X ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_16_finite__Collect__disjI,axiom,
% 2.86/1.58 ! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 2.86/1.58 ( ( finite772653738at_nat
% 2.86/1.58 @ ( collec7649004at_nat
% 2.86/1.58 @ ^ [X: product_prod_nat_nat] :
% 2.86/1.58 ( ( P3 @ X )
% 2.86/1.58 | ( Q @ X ) ) ) )
% 2.86/1.58 = ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 2.86/1.58 & ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_17_finite__Collect__disjI,axiom,
% 2.86/1.58 ! [P3: nat > $o,Q: nat > $o] :
% 2.86/1.58 ( ( finite_finite_nat
% 2.86/1.58 @ ( collect_nat
% 2.86/1.58 @ ^ [X: nat] :
% 2.86/1.58 ( ( P3 @ X )
% 2.86/1.58 | ( Q @ X ) ) ) )
% 2.86/1.58 = ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 2.86/1.58 & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_18_finite__Collect__bounded__ex,axiom,
% 2.86/1.58 ! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 2.86/1.58 ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 2.86/1.58 => ( ( finite772653738at_nat
% 2.86/1.58 @ ( collec7649004at_nat
% 2.86/1.58 @ ^ [X: product_prod_nat_nat] :
% 2.86/1.58 ? [Y: product_prod_nat_nat] :
% 2.86/1.58 ( ( P3 @ Y )
% 2.86/1.58 & ( Q @ X @ Y ) ) ) )
% 2.86/1.58 = ( ! [Y: product_prod_nat_nat] :
% 2.86/1.58 ( ( P3 @ Y )
% 2.86/1.58 => ( finite772653738at_nat
% 2.86/1.58 @ ( collec7649004at_nat
% 2.86/1.58 @ ^ [X: product_prod_nat_nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_19_finite__Collect__bounded__ex,axiom,
% 2.86/1.58 ! [P3: product_prod_nat_nat > $o,Q: nat > product_prod_nat_nat > $o] :
% 2.86/1.58 ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 2.86/1.58 => ( ( finite_finite_nat
% 2.86/1.58 @ ( collect_nat
% 2.86/1.58 @ ^ [X: nat] :
% 2.86/1.58 ? [Y: product_prod_nat_nat] :
% 2.86/1.58 ( ( P3 @ Y )
% 2.86/1.58 & ( Q @ X @ Y ) ) ) )
% 2.86/1.58 = ( ! [Y: product_prod_nat_nat] :
% 2.86/1.58 ( ( P3 @ Y )
% 2.86/1.58 => ( finite_finite_nat
% 2.86/1.58 @ ( collect_nat
% 2.86/1.58 @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_20_finite__Collect__bounded__ex,axiom,
% 2.86/1.58 ! [P3: nat > $o,Q: product_prod_nat_nat > nat > $o] :
% 2.86/1.58 ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 2.86/1.58 => ( ( finite772653738at_nat
% 2.86/1.58 @ ( collec7649004at_nat
% 2.86/1.58 @ ^ [X: product_prod_nat_nat] :
% 2.86/1.58 ? [Y: nat] :
% 2.86/1.58 ( ( P3 @ Y )
% 2.86/1.58 & ( Q @ X @ Y ) ) ) )
% 2.86/1.58 = ( ! [Y: nat] :
% 2.86/1.58 ( ( P3 @ Y )
% 2.86/1.58 => ( finite772653738at_nat
% 2.86/1.58 @ ( collec7649004at_nat
% 2.86/1.58 @ ^ [X: product_prod_nat_nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_21_finite__Collect__bounded__ex,axiom,
% 2.86/1.58 ! [P3: nat > $o,Q: nat > nat > $o] :
% 2.86/1.58 ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 2.86/1.58 => ( ( finite_finite_nat
% 2.86/1.58 @ ( collect_nat
% 2.86/1.58 @ ^ [X: nat] :
% 2.86/1.58 ? [Y: nat] :
% 2.86/1.58 ( ( P3 @ Y )
% 2.86/1.58 & ( Q @ X @ Y ) ) ) )
% 2.86/1.58 = ( ! [Y: nat] :
% 2.86/1.58 ( ( P3 @ Y )
% 2.86/1.58 => ( finite_finite_nat
% 2.86/1.58 @ ( collect_nat
% 2.86/1.58 @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_22_finite__Collect__subsets,axiom,
% 2.86/1.58 ! [A: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( finite772653738at_nat @ A )
% 2.86/1.58 => ( finite1457549322at_nat
% 2.86/1.58 @ ( collec1606769740at_nat
% 2.86/1.58 @ ^ [B: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ B @ A ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_23_finite__Collect__subsets,axiom,
% 2.86/1.58 ! [A: set_nat] :
% 2.86/1.58 ( ( finite_finite_nat @ A )
% 2.86/1.58 => ( finite2012248349et_nat
% 2.86/1.58 @ ( collect_set_nat
% 2.86/1.58 @ ^ [B: set_nat] : ( ord_less_eq_set_nat @ B @ A ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_24_finite__V,axiom,
% 2.86/1.58 finite_finite_nat @ ( v_a @ c ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_25_Vfin__imp__Efin,axiom,
% 2.86/1.58 ( ( finite_finite_nat @ ( v_a @ c ) )
% 2.86/1.58 => ( finite772653738at_nat @ ( e_a @ c ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_26_Range__mono,axiom,
% 2.86/1.58 ! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ R @ S )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( range_nat_nat @ R ) @ ( range_nat_nat @ S ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_27_Domain__mono,axiom,
% 2.86/1.58 ! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ R @ S )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( domain_nat_nat @ R ) @ ( domain_nat_nat @ S ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_28_finite__image__set,axiom,
% 2.86/1.58 ! [P3: product_prod_nat_nat > $o,F: product_prod_nat_nat > product_prod_nat_nat] :
% 2.86/1.58 ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 2.86/1.58 => ( finite772653738at_nat
% 2.86/1.58 @ ( collec7649004at_nat
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] :
% 2.86/1.58 ? [X: product_prod_nat_nat] :
% 2.86/1.58 ( ( Uu
% 2.86/1.58 = ( F @ X ) )
% 2.86/1.58 & ( P3 @ X ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_29_finite__image__set,axiom,
% 2.86/1.58 ! [P3: product_prod_nat_nat > $o,F: product_prod_nat_nat > nat] :
% 2.86/1.58 ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 2.86/1.58 => ( finite_finite_nat
% 2.86/1.58 @ ( collect_nat
% 2.86/1.58 @ ^ [Uu: nat] :
% 2.86/1.58 ? [X: product_prod_nat_nat] :
% 2.86/1.58 ( ( Uu
% 2.86/1.58 = ( F @ X ) )
% 2.86/1.58 & ( P3 @ X ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_30_finite__image__set,axiom,
% 2.86/1.58 ! [P3: nat > $o,F: nat > product_prod_nat_nat] :
% 2.86/1.58 ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 2.86/1.58 => ( finite772653738at_nat
% 2.86/1.58 @ ( collec7649004at_nat
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] :
% 2.86/1.58 ? [X: nat] :
% 2.86/1.58 ( ( Uu
% 2.86/1.58 = ( F @ X ) )
% 2.86/1.58 & ( P3 @ X ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_31_finite__image__set,axiom,
% 2.86/1.58 ! [P3: nat > $o,F: nat > nat] :
% 2.86/1.58 ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 2.86/1.58 => ( finite_finite_nat
% 2.86/1.58 @ ( collect_nat
% 2.86/1.58 @ ^ [Uu: nat] :
% 2.86/1.58 ? [X: nat] :
% 2.86/1.58 ( ( Uu
% 2.86/1.58 = ( F @ X ) )
% 2.86/1.58 & ( P3 @ X ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_32_finite__image__set2,axiom,
% 2.86/1.58 ! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o,F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 2.86/1.58 ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 2.86/1.58 => ( ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) )
% 2.86/1.58 => ( finite772653738at_nat
% 2.86/1.58 @ ( collec7649004at_nat
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] :
% 2.86/1.58 ? [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 2.86/1.58 ( ( Uu
% 2.86/1.58 = ( F @ X @ Y ) )
% 2.86/1.58 & ( P3 @ X )
% 2.86/1.58 & ( Q @ Y ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_33_finite__image__set2,axiom,
% 2.86/1.58 ! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o,F: product_prod_nat_nat > product_prod_nat_nat > nat] :
% 2.86/1.58 ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 2.86/1.58 => ( ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) )
% 2.86/1.58 => ( finite_finite_nat
% 2.86/1.58 @ ( collect_nat
% 2.86/1.58 @ ^ [Uu: nat] :
% 2.86/1.58 ? [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 2.86/1.58 ( ( Uu
% 2.86/1.58 = ( F @ X @ Y ) )
% 2.86/1.58 & ( P3 @ X )
% 2.86/1.58 & ( Q @ Y ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_34_finite__image__set2,axiom,
% 2.86/1.58 ! [P3: product_prod_nat_nat > $o,Q: nat > $o,F: product_prod_nat_nat > nat > product_prod_nat_nat] :
% 2.86/1.58 ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 2.86/1.58 => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 2.86/1.58 => ( finite772653738at_nat
% 2.86/1.58 @ ( collec7649004at_nat
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] :
% 2.86/1.58 ? [X: product_prod_nat_nat,Y: nat] :
% 2.86/1.58 ( ( Uu
% 2.86/1.58 = ( F @ X @ Y ) )
% 2.86/1.58 & ( P3 @ X )
% 2.86/1.58 & ( Q @ Y ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_35_finite__image__set2,axiom,
% 2.86/1.58 ! [P3: product_prod_nat_nat > $o,Q: nat > $o,F: product_prod_nat_nat > nat > nat] :
% 2.86/1.58 ( ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 2.86/1.58 => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 2.86/1.58 => ( finite_finite_nat
% 2.86/1.58 @ ( collect_nat
% 2.86/1.58 @ ^ [Uu: nat] :
% 2.86/1.58 ? [X: product_prod_nat_nat,Y: nat] :
% 2.86/1.58 ( ( Uu
% 2.86/1.58 = ( F @ X @ Y ) )
% 2.86/1.58 & ( P3 @ X )
% 2.86/1.58 & ( Q @ Y ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_36_finite__image__set2,axiom,
% 2.86/1.58 ! [P3: nat > $o,Q: product_prod_nat_nat > $o,F: nat > product_prod_nat_nat > product_prod_nat_nat] :
% 2.86/1.58 ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 2.86/1.58 => ( ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) )
% 2.86/1.58 => ( finite772653738at_nat
% 2.86/1.58 @ ( collec7649004at_nat
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] :
% 2.86/1.58 ? [X: nat,Y: product_prod_nat_nat] :
% 2.86/1.58 ( ( Uu
% 2.86/1.58 = ( F @ X @ Y ) )
% 2.86/1.58 & ( P3 @ X )
% 2.86/1.58 & ( Q @ Y ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_37_finite__image__set2,axiom,
% 2.86/1.58 ! [P3: nat > $o,Q: product_prod_nat_nat > $o,F: nat > product_prod_nat_nat > nat] :
% 2.86/1.58 ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 2.86/1.58 => ( ( finite772653738at_nat @ ( collec7649004at_nat @ Q ) )
% 2.86/1.58 => ( finite_finite_nat
% 2.86/1.58 @ ( collect_nat
% 2.86/1.58 @ ^ [Uu: nat] :
% 2.86/1.58 ? [X: nat,Y: product_prod_nat_nat] :
% 2.86/1.58 ( ( Uu
% 2.86/1.58 = ( F @ X @ Y ) )
% 2.86/1.58 & ( P3 @ X )
% 2.86/1.58 & ( Q @ Y ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_38_finite__image__set2,axiom,
% 2.86/1.58 ! [P3: nat > $o,Q: nat > $o,F: nat > nat > product_prod_nat_nat] :
% 2.86/1.58 ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 2.86/1.58 => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 2.86/1.58 => ( finite772653738at_nat
% 2.86/1.58 @ ( collec7649004at_nat
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] :
% 2.86/1.58 ? [X: nat,Y: nat] :
% 2.86/1.58 ( ( Uu
% 2.86/1.58 = ( F @ X @ Y ) )
% 2.86/1.58 & ( P3 @ X )
% 2.86/1.58 & ( Q @ Y ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_39_finite__image__set2,axiom,
% 2.86/1.58 ! [P3: nat > $o,Q: nat > $o,F: nat > nat > nat] :
% 2.86/1.58 ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 2.86/1.58 => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 2.86/1.58 => ( finite_finite_nat
% 2.86/1.58 @ ( collect_nat
% 2.86/1.58 @ ^ [Uu: nat] :
% 2.86/1.58 ? [X: nat,Y: nat] :
% 2.86/1.58 ( ( Uu
% 2.86/1.58 = ( F @ X @ Y ) )
% 2.86/1.58 & ( P3 @ X )
% 2.86/1.58 & ( Q @ Y ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_40_not__finite__existsD,axiom,
% 2.86/1.58 ! [P3: product_prod_nat_nat > $o] :
% 2.86/1.58 ( ~ ( finite772653738at_nat @ ( collec7649004at_nat @ P3 ) )
% 2.86/1.58 => ? [X_1: product_prod_nat_nat] : ( P3 @ X_1 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_41_not__finite__existsD,axiom,
% 2.86/1.58 ! [P3: nat > $o] :
% 2.86/1.58 ( ~ ( finite_finite_nat @ ( collect_nat @ P3 ) )
% 2.86/1.58 => ? [X_1: nat] : ( P3 @ X_1 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_42_pigeonhole__infinite__rel,axiom,
% 2.86/1.58 ! [A: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,R2: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 2.86/1.58 ( ~ ( finite772653738at_nat @ A )
% 2.86/1.58 => ( ( finite772653738at_nat @ B2 )
% 2.86/1.58 => ( ! [X2: product_prod_nat_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ X2 @ A )
% 2.86/1.58 => ? [Xa: product_prod_nat_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ Xa @ B2 )
% 2.86/1.58 & ( R2 @ X2 @ Xa ) ) )
% 2.86/1.58 => ? [X2: product_prod_nat_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ X2 @ B2 )
% 2.86/1.58 & ~ ( finite772653738at_nat
% 2.86/1.58 @ ( collec7649004at_nat
% 2.86/1.58 @ ^ [A2: product_prod_nat_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ A2 @ A )
% 2.86/1.58 & ( R2 @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_43_pigeonhole__infinite__rel,axiom,
% 2.86/1.58 ! [A: set_Pr1986765409at_nat,B2: set_nat,R2: product_prod_nat_nat > nat > $o] :
% 2.86/1.58 ( ~ ( finite772653738at_nat @ A )
% 2.86/1.58 => ( ( finite_finite_nat @ B2 )
% 2.86/1.58 => ( ! [X2: product_prod_nat_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ X2 @ A )
% 2.86/1.58 => ? [Xa: nat] :
% 2.86/1.58 ( ( member_nat @ Xa @ B2 )
% 2.86/1.58 & ( R2 @ X2 @ Xa ) ) )
% 2.86/1.58 => ? [X2: nat] :
% 2.86/1.58 ( ( member_nat @ X2 @ B2 )
% 2.86/1.58 & ~ ( finite772653738at_nat
% 2.86/1.58 @ ( collec7649004at_nat
% 2.86/1.58 @ ^ [A2: product_prod_nat_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ A2 @ A )
% 2.86/1.58 & ( R2 @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_44_pigeonhole__infinite__rel,axiom,
% 2.86/1.58 ! [A: set_nat,B2: set_Pr1986765409at_nat,R2: nat > product_prod_nat_nat > $o] :
% 2.86/1.58 ( ~ ( finite_finite_nat @ A )
% 2.86/1.58 => ( ( finite772653738at_nat @ B2 )
% 2.86/1.58 => ( ! [X2: nat] :
% 2.86/1.58 ( ( member_nat @ X2 @ A )
% 2.86/1.58 => ? [Xa: product_prod_nat_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ Xa @ B2 )
% 2.86/1.58 & ( R2 @ X2 @ Xa ) ) )
% 2.86/1.58 => ? [X2: product_prod_nat_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ X2 @ B2 )
% 2.86/1.58 & ~ ( finite_finite_nat
% 2.86/1.58 @ ( collect_nat
% 2.86/1.58 @ ^ [A2: nat] :
% 2.86/1.58 ( ( member_nat @ A2 @ A )
% 2.86/1.58 & ( R2 @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_45_pigeonhole__infinite__rel,axiom,
% 2.86/1.58 ! [A: set_nat,B2: set_nat,R2: nat > nat > $o] :
% 2.86/1.58 ( ~ ( finite_finite_nat @ A )
% 2.86/1.58 => ( ( finite_finite_nat @ B2 )
% 2.86/1.58 => ( ! [X2: nat] :
% 2.86/1.58 ( ( member_nat @ X2 @ A )
% 2.86/1.58 => ? [Xa: nat] :
% 2.86/1.58 ( ( member_nat @ Xa @ B2 )
% 2.86/1.58 & ( R2 @ X2 @ Xa ) ) )
% 2.86/1.58 => ? [X2: nat] :
% 2.86/1.58 ( ( member_nat @ X2 @ B2 )
% 2.86/1.58 & ~ ( finite_finite_nat
% 2.86/1.58 @ ( collect_nat
% 2.86/1.58 @ ^ [A2: nat] :
% 2.86/1.58 ( ( member_nat @ A2 @ A )
% 2.86/1.58 & ( R2 @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_46_ek__analysis__def,axiom,
% 2.86/1.58 edmond1517640972ysis_a = finite_Graph_a ).
% 2.86/1.58
% 2.86/1.58 thf(fact_47_ek__analysis_Ointro,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a] :
% 2.86/1.58 ( ( finite_Graph_a @ C )
% 2.86/1.58 => ( edmond1517640972ysis_a @ C ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_48_ek__analysis_Oaxioms,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a] :
% 2.86/1.58 ( ( edmond1517640972ysis_a @ C )
% 2.86/1.58 => ( finite_Graph_a @ C ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_49_ek__analysis__defs_OspEdges__def,axiom,
% 2.86/1.58 ( edmond475474835dges_a
% 2.86/1.58 = ( ^ [C2: product_prod_nat_nat > a,S2: nat,T: nat] :
% 2.86/1.58 ( collec7649004at_nat
% 2.86/1.58 @ ^ [E: product_prod_nat_nat] :
% 2.86/1.58 ? [P: list_P559422087at_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ E @ ( set_Pr2131844118at_nat @ P ) )
% 2.86/1.58 & ( isShortestPath_a @ C2 @ S2 @ P @ T ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_50_finite__has__minimal2,axiom,
% 2.86/1.58 ! [A: set_se1612935105at_nat,A3: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( finite1457549322at_nat @ A )
% 2.86/1.58 => ( ( member298845450at_nat @ A3 @ A )
% 2.86/1.58 => ? [X2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member298845450at_nat @ X2 @ A )
% 2.86/1.58 & ( ord_le841296385at_nat @ X2 @ A3 )
% 2.86/1.58 & ! [Xa: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member298845450at_nat @ Xa @ A )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ Xa @ X2 )
% 2.86/1.58 => ( X2 = Xa ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_51_finite__has__minimal2,axiom,
% 2.86/1.58 ! [A: set_set_nat,A3: set_nat] :
% 2.86/1.58 ( ( finite2012248349et_nat @ A )
% 2.86/1.58 => ( ( member_set_nat @ A3 @ A )
% 2.86/1.58 => ? [X2: set_nat] :
% 2.86/1.58 ( ( member_set_nat @ X2 @ A )
% 2.86/1.58 & ( ord_less_eq_set_nat @ X2 @ A3 )
% 2.86/1.58 & ! [Xa: set_nat] :
% 2.86/1.58 ( ( member_set_nat @ Xa @ A )
% 2.86/1.58 => ( ( ord_less_eq_set_nat @ Xa @ X2 )
% 2.86/1.58 => ( X2 = Xa ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_52_finite__has__minimal2,axiom,
% 2.86/1.58 ! [A: set_nat,A3: nat] :
% 2.86/1.58 ( ( finite_finite_nat @ A )
% 2.86/1.58 => ( ( member_nat @ A3 @ A )
% 2.86/1.58 => ? [X2: nat] :
% 2.86/1.58 ( ( member_nat @ X2 @ A )
% 2.86/1.58 & ( ord_less_eq_nat @ X2 @ A3 )
% 2.86/1.58 & ! [Xa: nat] :
% 2.86/1.58 ( ( member_nat @ Xa @ A )
% 2.86/1.58 => ( ( ord_less_eq_nat @ Xa @ X2 )
% 2.86/1.58 => ( X2 = Xa ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_53_finite__has__maximal2,axiom,
% 2.86/1.58 ! [A: set_se1612935105at_nat,A3: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( finite1457549322at_nat @ A )
% 2.86/1.58 => ( ( member298845450at_nat @ A3 @ A )
% 2.86/1.58 => ? [X2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member298845450at_nat @ X2 @ A )
% 2.86/1.58 & ( ord_le841296385at_nat @ A3 @ X2 )
% 2.86/1.58 & ! [Xa: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member298845450at_nat @ Xa @ A )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ X2 @ Xa )
% 2.86/1.58 => ( X2 = Xa ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_54_finite__has__maximal2,axiom,
% 2.86/1.58 ! [A: set_set_nat,A3: set_nat] :
% 2.86/1.58 ( ( finite2012248349et_nat @ A )
% 2.86/1.58 => ( ( member_set_nat @ A3 @ A )
% 2.86/1.58 => ? [X2: set_nat] :
% 2.86/1.58 ( ( member_set_nat @ X2 @ A )
% 2.86/1.58 & ( ord_less_eq_set_nat @ A3 @ X2 )
% 2.86/1.58 & ! [Xa: set_nat] :
% 2.86/1.58 ( ( member_set_nat @ Xa @ A )
% 2.86/1.58 => ( ( ord_less_eq_set_nat @ X2 @ Xa )
% 2.86/1.58 => ( X2 = Xa ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_55_finite__has__maximal2,axiom,
% 2.86/1.58 ! [A: set_nat,A3: nat] :
% 2.86/1.58 ( ( finite_finite_nat @ A )
% 2.86/1.58 => ( ( member_nat @ A3 @ A )
% 2.86/1.58 => ? [X2: nat] :
% 2.86/1.58 ( ( member_nat @ X2 @ A )
% 2.86/1.58 & ( ord_less_eq_nat @ A3 @ X2 )
% 2.86/1.58 & ! [Xa: nat] :
% 2.86/1.58 ( ( member_nat @ Xa @ A )
% 2.86/1.58 => ( ( ord_less_eq_nat @ X2 @ Xa )
% 2.86/1.58 => ( X2 = Xa ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_56_rev__finite__subset,axiom,
% 2.86/1.58 ! [B2: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( finite772653738at_nat @ B2 )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ A @ B2 )
% 2.86/1.58 => ( finite772653738at_nat @ A ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_57_rev__finite__subset,axiom,
% 2.86/1.58 ! [B2: set_nat,A: set_nat] :
% 2.86/1.58 ( ( finite_finite_nat @ B2 )
% 2.86/1.58 => ( ( ord_less_eq_set_nat @ A @ B2 )
% 2.86/1.58 => ( finite_finite_nat @ A ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_58_infinite__super,axiom,
% 2.86/1.58 ! [S3: set_Pr1986765409at_nat,T2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ S3 @ T2 )
% 2.86/1.58 => ( ~ ( finite772653738at_nat @ S3 )
% 2.86/1.58 => ~ ( finite772653738at_nat @ T2 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_59_infinite__super,axiom,
% 2.86/1.58 ! [S3: set_nat,T2: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ S3 @ T2 )
% 2.86/1.58 => ( ~ ( finite_finite_nat @ S3 )
% 2.86/1.58 => ~ ( finite_finite_nat @ T2 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_60_finite__subset,axiom,
% 2.86/1.58 ! [A: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ A @ B2 )
% 2.86/1.58 => ( ( finite772653738at_nat @ B2 )
% 2.86/1.58 => ( finite772653738at_nat @ A ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_61_finite__subset,axiom,
% 2.86/1.58 ! [A: set_nat,B2: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ A @ B2 )
% 2.86/1.58 => ( ( finite_finite_nat @ B2 )
% 2.86/1.58 => ( finite_finite_nat @ A ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_62_ek__analysis_OspEdges__ss__E,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,S: nat,T3: nat] :
% 2.86/1.58 ( ( edmond1517640972ysis_a @ C )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( edmond475474835dges_a @ C @ S @ T3 ) @ ( e_a @ C ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_63_incoming_H__edges,axiom,
% 2.86/1.58 ! [U2: set_nat] : ( ord_le841296385at_nat @ ( incoming_a2 @ c @ U2 ) @ ( e_a @ c ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_64_outgoing_H__edges,axiom,
% 2.86/1.58 ! [U2: set_nat] : ( ord_le841296385at_nat @ ( outgoing_a2 @ c @ U2 ) @ ( e_a @ c ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_65_reachableNodes__def,axiom,
% 2.86/1.58 ! [U: nat] :
% 2.86/1.58 ( ( reachableNodes_a @ c @ U )
% 2.86/1.58 = ( collect_nat @ ( connected_a @ c @ U ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_66_List_Ofinite__set,axiom,
% 2.86/1.58 ! [Xs: list_P559422087at_nat] : ( finite772653738at_nat @ ( set_Pr2131844118at_nat @ Xs ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_67_List_Ofinite__set,axiom,
% 2.86/1.58 ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_68_Graph_OEfin__imp__Vfin,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a] :
% 2.86/1.58 ( ( finite772653738at_nat @ ( e_a @ C ) )
% 2.86/1.58 => ( finite_finite_nat @ ( v_a @ C ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_69_Graph_OVfin__imp__Efin,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a] :
% 2.86/1.58 ( ( finite_finite_nat @ ( v_a @ C ) )
% 2.86/1.58 => ( finite772653738at_nat @ ( e_a @ C ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_70_mem__Collect__eq,axiom,
% 2.86/1.58 ! [A3: product_prod_nat_nat,P3: product_prod_nat_nat > $o] :
% 2.86/1.58 ( ( member701585322at_nat @ A3 @ ( collec7649004at_nat @ P3 ) )
% 2.86/1.58 = ( P3 @ A3 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_71_mem__Collect__eq,axiom,
% 2.86/1.58 ! [A3: nat,P3: nat > $o] :
% 2.86/1.58 ( ( member_nat @ A3 @ ( collect_nat @ P3 ) )
% 2.86/1.58 = ( P3 @ A3 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_72_Collect__mem__eq,axiom,
% 2.86/1.58 ! [A: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( collec7649004at_nat
% 2.86/1.58 @ ^ [X: product_prod_nat_nat] : ( member701585322at_nat @ X @ A ) )
% 2.86/1.58 = A ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_73_Collect__mem__eq,axiom,
% 2.86/1.58 ! [A: set_nat] :
% 2.86/1.58 ( ( collect_nat
% 2.86/1.58 @ ^ [X: nat] : ( member_nat @ X @ A ) )
% 2.86/1.58 = A ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_74_Collect__cong,axiom,
% 2.86/1.58 ! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 2.86/1.58 ( ! [X2: product_prod_nat_nat] :
% 2.86/1.58 ( ( P3 @ X2 )
% 2.86/1.58 = ( Q @ X2 ) )
% 2.86/1.58 => ( ( collec7649004at_nat @ P3 )
% 2.86/1.58 = ( collec7649004at_nat @ Q ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_75_Collect__cong,axiom,
% 2.86/1.58 ! [P3: nat > $o,Q: nat > $o] :
% 2.86/1.58 ( ! [X2: nat] :
% 2.86/1.58 ( ( P3 @ X2 )
% 2.86/1.58 = ( Q @ X2 ) )
% 2.86/1.58 => ( ( collect_nat @ P3 )
% 2.86/1.58 = ( collect_nat @ Q ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_76_isShortestPath__level__edge_I1_J,axiom,
% 2.86/1.58 ! [S: nat,P4: list_P559422087at_nat,T3: nat,U: nat,V: nat] :
% 2.86/1.58 ( ( isShortestPath_a @ c @ S @ P4 @ T3 )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 2.86/1.58 => ( connected_a @ c @ S @ U ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_77_isShortestPath__level__edge_I2_J,axiom,
% 2.86/1.58 ! [S: nat,P4: list_P559422087at_nat,T3: nat,U: nat,V: nat] :
% 2.86/1.58 ( ( isShortestPath_a @ c @ S @ P4 @ T3 )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 2.86/1.58 => ( connected_a @ c @ U @ V ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_78_isShortestPath__level__edge_I3_J,axiom,
% 2.86/1.58 ! [S: nat,P4: list_P559422087at_nat,T3: nat,U: nat,V: nat] :
% 2.86/1.58 ( ( isShortestPath_a @ c @ S @ P4 @ T3 )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 2.86/1.58 => ( connected_a @ c @ V @ T3 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_79_incoming__edges,axiom,
% 2.86/1.58 ! [U: nat] : ( ord_le841296385at_nat @ ( incoming_a @ c @ U ) @ ( e_a @ c ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_80_reachable__ss__V,axiom,
% 2.86/1.58 ! [S: nat] :
% 2.86/1.58 ( ( member_nat @ S @ ( v_a @ c ) )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( reachableNodes_a @ c @ S ) @ ( v_a @ c ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_81_connected__append__edge,axiom,
% 2.86/1.58 ! [U: nat,V: nat,W: nat] :
% 2.86/1.58 ( ( connected_a @ c @ U @ V )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ V @ W ) @ ( e_a @ c ) )
% 2.86/1.58 => ( connected_a @ c @ U @ W ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_82_reachableNodes__append__edge,axiom,
% 2.86/1.58 ! [U: nat,S: nat,V: nat] :
% 2.86/1.58 ( ( member_nat @ U @ ( reachableNodes_a @ c @ S ) )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( e_a @ c ) )
% 2.86/1.58 => ( member_nat @ V @ ( reachableNodes_a @ c @ S ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_83_V__def,axiom,
% 2.86/1.58 ( ( v_a @ c )
% 2.86/1.58 = ( collect_nat
% 2.86/1.58 @ ^ [U3: nat] :
% 2.86/1.58 ? [V2: nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V2 ) @ ( e_a @ c ) )
% 2.86/1.58 | ( member701585322at_nat @ ( product_Pair_nat_nat @ V2 @ U3 ) @ ( e_a @ c ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_84_incoming__def,axiom,
% 2.86/1.58 ! [V: nat] :
% 2.86/1.58 ( ( incoming_a @ c @ V )
% 2.86/1.58 = ( collec7649004at_nat
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] :
% 2.86/1.58 ? [U3: nat] :
% 2.86/1.58 ( ( Uu
% 2.86/1.58 = ( product_Pair_nat_nat @ U3 @ V ) )
% 2.86/1.58 & ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V ) @ ( e_a @ c ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_85_outgoing_H__def,axiom,
% 2.86/1.58 ! [K: set_nat] :
% 2.86/1.58 ( ( outgoing_a2 @ c @ K )
% 2.86/1.58 = ( collec7649004at_nat
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] :
% 2.86/1.58 ? [U3: nat,V2: nat] :
% 2.86/1.58 ( ( Uu
% 2.86/1.58 = ( product_Pair_nat_nat @ V2 @ U3 ) )
% 2.86/1.58 & ~ ( member_nat @ U3 @ K )
% 2.86/1.58 & ( member_nat @ V2 @ K )
% 2.86/1.58 & ( member701585322at_nat @ ( product_Pair_nat_nat @ V2 @ U3 ) @ ( e_a @ c ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_86_incoming_H__def,axiom,
% 2.86/1.58 ! [K: set_nat] :
% 2.86/1.58 ( ( incoming_a2 @ c @ K )
% 2.86/1.58 = ( collec7649004at_nat
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] :
% 2.86/1.58 ? [U3: nat,V2: nat] :
% 2.86/1.58 ( ( Uu
% 2.86/1.58 = ( product_Pair_nat_nat @ U3 @ V2 ) )
% 2.86/1.58 & ~ ( member_nat @ U3 @ K )
% 2.86/1.58 & ( member_nat @ V2 @ K )
% 2.86/1.58 & ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V2 ) @ ( e_a @ c ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_87_finite__Collect__le__nat,axiom,
% 2.86/1.58 ! [K: nat] :
% 2.86/1.58 ( finite_finite_nat
% 2.86/1.58 @ ( collect_nat
% 2.86/1.58 @ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_88_finite__incoming,axiom,
% 2.86/1.58 ! [U: nat] :
% 2.86/1.58 ( ( finite_finite_nat @ ( v_a @ c ) )
% 2.86/1.58 => ( finite772653738at_nat @ ( incoming_a @ c @ U ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_89_finite__outgoing_H,axiom,
% 2.86/1.58 ! [U2: set_nat] :
% 2.86/1.58 ( ( finite_finite_nat @ ( v_a @ c ) )
% 2.86/1.58 => ( finite772653738at_nat @ ( outgoing_a2 @ c @ U2 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_90_finite__incoming_H,axiom,
% 2.86/1.58 ! [U2: set_nat] :
% 2.86/1.58 ( ( finite_finite_nat @ ( v_a @ c ) )
% 2.86/1.58 => ( finite772653738at_nat @ ( incoming_a2 @ c @ U2 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_91_adjacent__nodes__ss__V,axiom,
% 2.86/1.58 ! [U: nat] : ( ord_less_eq_set_nat @ ( adjacent_nodes_a @ c @ U ) @ ( v_a @ c ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_92_Graph_Oincoming__def,axiom,
% 2.86/1.58 ( incoming_a
% 2.86/1.58 = ( ^ [C2: product_prod_nat_nat > a,V2: nat] :
% 2.86/1.58 ( collec7649004at_nat
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] :
% 2.86/1.58 ? [U3: nat] :
% 2.86/1.58 ( ( Uu
% 2.86/1.58 = ( product_Pair_nat_nat @ U3 @ V2 ) )
% 2.86/1.58 & ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V2 ) @ ( e_a @ C2 ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_93_Graph_Oincoming_H__def,axiom,
% 2.86/1.58 ( incoming_a2
% 2.86/1.58 = ( ^ [C2: product_prod_nat_nat > a,K2: set_nat] :
% 2.86/1.58 ( collec7649004at_nat
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] :
% 2.86/1.58 ? [U3: nat,V2: nat] :
% 2.86/1.58 ( ( Uu
% 2.86/1.58 = ( product_Pair_nat_nat @ U3 @ V2 ) )
% 2.86/1.58 & ~ ( member_nat @ U3 @ K2 )
% 2.86/1.58 & ( member_nat @ V2 @ K2 )
% 2.86/1.58 & ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V2 ) @ ( e_a @ C2 ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_94_Graph_Ooutgoing_H__def,axiom,
% 2.86/1.58 ( outgoing_a2
% 2.86/1.58 = ( ^ [C2: product_prod_nat_nat > a,K2: set_nat] :
% 2.86/1.58 ( collec7649004at_nat
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] :
% 2.86/1.58 ? [U3: nat,V2: nat] :
% 2.86/1.58 ( ( Uu
% 2.86/1.58 = ( product_Pair_nat_nat @ V2 @ U3 ) )
% 2.86/1.58 & ~ ( member_nat @ U3 @ K2 )
% 2.86/1.58 & ( member_nat @ V2 @ K2 )
% 2.86/1.58 & ( member701585322at_nat @ ( product_Pair_nat_nat @ V2 @ U3 ) @ ( e_a @ C2 ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_95_Graph_Oreachable__ss__V,axiom,
% 2.86/1.58 ! [S: nat,C: product_prod_nat_nat > a] :
% 2.86/1.58 ( ( member_nat @ S @ ( v_a @ C ) )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( reachableNodes_a @ C @ S ) @ ( v_a @ C ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_96_Graph_OreachableNodes__append__edge,axiom,
% 2.86/1.58 ! [U: nat,C: product_prod_nat_nat > a,S: nat,V: nat] :
% 2.86/1.58 ( ( member_nat @ U @ ( reachableNodes_a @ C @ S ) )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( e_a @ C ) )
% 2.86/1.58 => ( member_nat @ V @ ( reachableNodes_a @ C @ S ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_97_Graph_OreachableNodes_Ocong,axiom,
% 2.86/1.58 reachableNodes_a = reachableNodes_a ).
% 2.86/1.58
% 2.86/1.58 thf(fact_98_Graph_Ooutgoing_H_Ocong,axiom,
% 2.86/1.58 outgoing_a2 = outgoing_a2 ).
% 2.86/1.58
% 2.86/1.58 thf(fact_99_Graph_Oincoming_H_Ocong,axiom,
% 2.86/1.58 incoming_a2 = incoming_a2 ).
% 2.86/1.58
% 2.86/1.58 thf(fact_100_Graph_Oincoming_Ocong,axiom,
% 2.86/1.58 incoming_a = incoming_a ).
% 2.86/1.58
% 2.86/1.58 thf(fact_101_pred__equals__eq2,axiom,
% 2.86/1.58 ! [R2: set_Pr1986765409at_nat,S3: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ( ^ [X: nat,Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) )
% 2.86/1.58 = ( ^ [X: nat,Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S3 ) ) )
% 2.86/1.58 = ( R2 = S3 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_102_pred__subset__eq2,axiom,
% 2.86/1.58 ! [R2: set_Pr1986765409at_nat,S3: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le1578155910_nat_o
% 2.86/1.58 @ ^ [X: nat,Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
% 2.86/1.58 @ ^ [X: nat,Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S3 ) )
% 2.86/1.58 = ( ord_le841296385at_nat @ R2 @ S3 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_103_Graph_OV__def,axiom,
% 2.86/1.58 ( v_a
% 2.86/1.58 = ( ^ [C2: product_prod_nat_nat > a] :
% 2.86/1.58 ( collect_nat
% 2.86/1.58 @ ^ [U3: nat] :
% 2.86/1.58 ? [V2: nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V2 ) @ ( e_a @ C2 ) )
% 2.86/1.58 | ( member701585322at_nat @ ( product_Pair_nat_nat @ V2 @ U3 ) @ ( e_a @ C2 ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_104_subrelI,axiom,
% 2.86/1.58 ! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ! [X2: nat,Y2: nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
% 2.86/1.58 => ( member701585322at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ S ) )
% 2.86/1.58 => ( ord_le841296385at_nat @ R @ S ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_105_Domain_Oinducts,axiom,
% 2.86/1.58 ! [X3: nat,R: set_Pr1986765409at_nat,P3: nat > $o] :
% 2.86/1.58 ( ( member_nat @ X3 @ ( domain_nat_nat @ R ) )
% 2.86/1.58 => ( ! [A4: nat,B3: nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ R )
% 2.86/1.58 => ( P3 @ A4 ) )
% 2.86/1.58 => ( P3 @ X3 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_106_Domain_ODomainI,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,R: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ R )
% 2.86/1.58 => ( member_nat @ A3 @ ( domain_nat_nat @ R ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_107_Domain_Osimps,axiom,
% 2.86/1.58 ! [A3: nat,R: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member_nat @ A3 @ ( domain_nat_nat @ R ) )
% 2.86/1.58 = ( ? [A2: nat,B5: nat] :
% 2.86/1.58 ( ( A3 = A2 )
% 2.86/1.58 & ( member701585322at_nat @ ( product_Pair_nat_nat @ A2 @ B5 ) @ R ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_108_Domain_Ocases,axiom,
% 2.86/1.58 ! [A3: nat,R: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member_nat @ A3 @ ( domain_nat_nat @ R ) )
% 2.86/1.58 => ~ ! [B3: nat] :
% 2.86/1.58 ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ R ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_109_Domain__iff,axiom,
% 2.86/1.58 ! [A3: nat,R: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member_nat @ A3 @ ( domain_nat_nat @ R ) )
% 2.86/1.58 = ( ? [Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ Y ) @ R ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_110_DomainE,axiom,
% 2.86/1.58 ! [A3: nat,R: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member_nat @ A3 @ ( domain_nat_nat @ R ) )
% 2.86/1.58 => ~ ! [B3: nat] :
% 2.86/1.58 ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ R ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_111_Range_Oinducts,axiom,
% 2.86/1.58 ! [X3: nat,R: set_Pr1986765409at_nat,P3: nat > $o] :
% 2.86/1.58 ( ( member_nat @ X3 @ ( range_nat_nat @ R ) )
% 2.86/1.58 => ( ! [A4: nat,B3: nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ R )
% 2.86/1.58 => ( P3 @ B3 ) )
% 2.86/1.58 => ( P3 @ X3 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_112_Range_Ointros,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,R: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ R )
% 2.86/1.58 => ( member_nat @ B4 @ ( range_nat_nat @ R ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_113_Range_Osimps,axiom,
% 2.86/1.58 ! [A3: nat,R: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member_nat @ A3 @ ( range_nat_nat @ R ) )
% 2.86/1.58 = ( ? [A2: nat,B5: nat] :
% 2.86/1.58 ( ( A3 = B5 )
% 2.86/1.58 & ( member701585322at_nat @ ( product_Pair_nat_nat @ A2 @ B5 ) @ R ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_114_Range_Ocases,axiom,
% 2.86/1.58 ! [A3: nat,R: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member_nat @ A3 @ ( range_nat_nat @ R ) )
% 2.86/1.58 => ~ ! [A4: nat] :
% 2.86/1.58 ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ A4 @ A3 ) @ R ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_115_Range__iff,axiom,
% 2.86/1.58 ! [A3: nat,R: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member_nat @ A3 @ ( range_nat_nat @ R ) )
% 2.86/1.58 = ( ? [Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ A3 ) @ R ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_116_RangeE,axiom,
% 2.86/1.58 ! [B4: nat,R: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member_nat @ B4 @ ( range_nat_nat @ R ) )
% 2.86/1.58 => ~ ! [A4: nat] :
% 2.86/1.58 ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ A4 @ B4 ) @ R ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_117_Graph_OreachableNodes__def,axiom,
% 2.86/1.58 ( reachableNodes_a
% 2.86/1.58 = ( ^ [C2: product_prod_nat_nat > a,U3: nat] : ( collect_nat @ ( connected_a @ C2 @ U3 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_118_pred__subset__eq,axiom,
% 2.86/1.58 ! [R2: set_Pr1986765409at_nat,S3: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le1039616028_nat_o
% 2.86/1.58 @ ^ [X: product_prod_nat_nat] : ( member701585322at_nat @ X @ R2 )
% 2.86/1.58 @ ^ [X: product_prod_nat_nat] : ( member701585322at_nat @ X @ S3 ) )
% 2.86/1.58 = ( ord_le841296385at_nat @ R2 @ S3 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_119_pred__subset__eq,axiom,
% 2.86/1.58 ! [R2: set_nat,S3: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_nat_o
% 2.86/1.58 @ ^ [X: nat] : ( member_nat @ X @ R2 )
% 2.86/1.58 @ ^ [X: nat] : ( member_nat @ X @ S3 ) )
% 2.86/1.58 = ( ord_less_eq_set_nat @ R2 @ S3 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_120_Graph_Oconnected__append__edge,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,U: nat,V: nat,W: nat] :
% 2.86/1.58 ( ( connected_a @ C @ U @ V )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ V @ W ) @ ( e_a @ C ) )
% 2.86/1.58 => ( connected_a @ C @ U @ W ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_121_Graph_Oincoming__edges,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,U: nat] : ( ord_le841296385at_nat @ ( incoming_a @ C @ U ) @ ( e_a @ C ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_122_Graph_OisShortestPath__level__edge_I1_J,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,S: nat,P4: list_P559422087at_nat,T3: nat,U: nat,V: nat] :
% 2.86/1.58 ( ( isShortestPath_a @ C @ S @ P4 @ T3 )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 2.86/1.58 => ( connected_a @ C @ S @ U ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_123_Graph_OisShortestPath__level__edge_I2_J,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,S: nat,P4: list_P559422087at_nat,T3: nat,U: nat,V: nat] :
% 2.86/1.58 ( ( isShortestPath_a @ C @ S @ P4 @ T3 )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 2.86/1.58 => ( connected_a @ C @ U @ V ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_124_Graph_OisShortestPath__level__edge_I3_J,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,S: nat,P4: list_P559422087at_nat,T3: nat,U: nat,V: nat] :
% 2.86/1.58 ( ( isShortestPath_a @ C @ S @ P4 @ T3 )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 2.86/1.58 => ( connected_a @ C @ V @ T3 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_125_Graph_Ooutgoing_H__edges,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,U2: set_nat] : ( ord_le841296385at_nat @ ( outgoing_a2 @ C @ U2 ) @ ( e_a @ C ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_126_Graph_Oincoming_H__edges,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,U2: set_nat] : ( ord_le841296385at_nat @ ( incoming_a2 @ C @ U2 ) @ ( e_a @ C ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_127_Domain__unfold,axiom,
% 2.86/1.58 ( domain_nat_nat
% 2.86/1.58 = ( ^ [R3: set_Pr1986765409at_nat] :
% 2.86/1.58 ( collect_nat
% 2.86/1.58 @ ^ [X: nat] :
% 2.86/1.58 ? [Y: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_128_Graph_Ofinite__incoming,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,U: nat] :
% 2.86/1.58 ( ( finite_finite_nat @ ( v_a @ C ) )
% 2.86/1.58 => ( finite772653738at_nat @ ( incoming_a @ C @ U ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_129_Graph_OE_Ocong,axiom,
% 2.86/1.58 e_a = e_a ).
% 2.86/1.58
% 2.86/1.58 thf(fact_130_Graph_OV_Ocong,axiom,
% 2.86/1.58 v_a = v_a ).
% 2.86/1.58
% 2.86/1.58 thf(fact_131_Graph_Oconnected__refl,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,V: nat] : ( connected_a @ C @ V @ V ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_132_Graph_Oconnected_Ocong,axiom,
% 2.86/1.58 connected_a = connected_a ).
% 2.86/1.58
% 2.86/1.58 thf(fact_133_Graph_Ofinite__outgoing_H,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,U2: set_nat] :
% 2.86/1.58 ( ( finite_finite_nat @ ( v_a @ C ) )
% 2.86/1.58 => ( finite772653738at_nat @ ( outgoing_a2 @ C @ U2 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_134_Graph_OisShortestPath_Ocong,axiom,
% 2.86/1.58 isShortestPath_a = isShortestPath_a ).
% 2.86/1.58
% 2.86/1.58 thf(fact_135_Graph_Ofinite__incoming_H,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,U2: set_nat] :
% 2.86/1.58 ( ( finite_finite_nat @ ( v_a @ C ) )
% 2.86/1.58 => ( finite772653738at_nat @ ( incoming_a2 @ C @ U2 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_136_finite__list,axiom,
% 2.86/1.58 ! [A: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( finite772653738at_nat @ A )
% 2.86/1.58 => ? [Xs2: list_P559422087at_nat] :
% 2.86/1.58 ( ( set_Pr2131844118at_nat @ Xs2 )
% 2.86/1.58 = A ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_137_finite__list,axiom,
% 2.86/1.58 ! [A: set_nat] :
% 2.86/1.58 ( ( finite_finite_nat @ A )
% 2.86/1.58 => ? [Xs2: list_nat] :
% 2.86/1.58 ( ( set_nat2 @ Xs2 )
% 2.86/1.58 = A ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_138_subset__code_I1_J,axiom,
% 2.86/1.58 ! [Xs: list_P559422087at_nat,B2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ ( set_Pr2131844118at_nat @ Xs ) @ B2 )
% 2.86/1.58 = ( ! [X: product_prod_nat_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ X @ ( set_Pr2131844118at_nat @ Xs ) )
% 2.86/1.58 => ( member701585322at_nat @ X @ B2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_139_subset__code_I1_J,axiom,
% 2.86/1.58 ! [Xs: list_nat,B2: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
% 2.86/1.58 = ( ! [X: nat] :
% 2.86/1.58 ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 2.86/1.58 => ( member_nat @ X @ B2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_140_Graph_Oconnected__inV__iff,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,U: nat,V: nat] :
% 2.86/1.58 ( ( connected_a @ C @ U @ V )
% 2.86/1.58 => ( ( member_nat @ V @ ( v_a @ C ) )
% 2.86/1.58 = ( member_nat @ U @ ( v_a @ C ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_141_Graph_Oobtain__shortest__path,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,U: nat,V: nat] :
% 2.86/1.58 ( ( connected_a @ C @ U @ V )
% 2.86/1.58 => ~ ! [P2: list_P559422087at_nat] :
% 2.86/1.58 ~ ( isShortestPath_a @ C @ U @ P2 @ V ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_142_Finite__Graph_Ofinite__V,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a] :
% 2.86/1.58 ( ( finite_Graph_a @ C )
% 2.86/1.58 => ( finite_finite_nat @ ( v_a @ C ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_143_Finite__Graph_Ointro,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a] :
% 2.86/1.58 ( ( finite_finite_nat @ ( v_a @ C ) )
% 2.86/1.58 => ( finite_Graph_a @ C ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_144_Finite__Graph__def,axiom,
% 2.86/1.58 ( finite_Graph_a
% 2.86/1.58 = ( ^ [C2: product_prod_nat_nat > a] : ( finite_finite_nat @ ( v_a @ C2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_145_Finite__Graph_Ofinite__E,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a] :
% 2.86/1.58 ( ( finite_Graph_a @ C )
% 2.86/1.58 => ( finite772653738at_nat @ ( e_a @ C ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_146_Graph_OFinite__Graph__EI,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a] :
% 2.86/1.58 ( ( finite772653738at_nat @ ( e_a @ C ) )
% 2.86/1.58 => ( finite_Graph_a @ C ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_147_outgoing__def,axiom,
% 2.86/1.58 ! [V: nat] :
% 2.86/1.58 ( ( outgoing_a @ c @ V )
% 2.86/1.58 = ( collec7649004at_nat
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] :
% 2.86/1.58 ? [U3: nat] :
% 2.86/1.58 ( ( Uu
% 2.86/1.58 = ( product_Pair_nat_nat @ V @ U3 ) )
% 2.86/1.58 & ( member701585322at_nat @ ( product_Pair_nat_nat @ V @ U3 ) @ ( e_a @ c ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_148_outgoing__edges,axiom,
% 2.86/1.58 ! [U: nat] : ( ord_le841296385at_nat @ ( outgoing_a @ c @ U ) @ ( e_a @ c ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_149_E__ss__VxV,axiom,
% 2.86/1.58 ( ord_le841296385at_nat @ ( e_a @ c )
% 2.86/1.58 @ ( produc45129834at_nat @ ( v_a @ c )
% 2.86/1.58 @ ^ [Uu: nat] : ( v_a @ c ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_150_connected__edgeRtc,axiom,
% 2.86/1.58 ! [U: nat,V: nat] :
% 2.86/1.58 ( ( connected_a @ c @ U @ V )
% 2.86/1.58 = ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi1645084429cl_nat @ ( e_a @ c ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_151_isSPath__no__selfloop,axiom,
% 2.86/1.58 ! [U: nat,P4: list_P559422087at_nat,V: nat,U1: nat] :
% 2.86/1.58 ( ( isSimplePath_a @ c @ U @ P4 @ V )
% 2.86/1.58 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ U1 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_152_isSPath__sg__incoming,axiom,
% 2.86/1.58 ! [U: nat,P4: list_P559422087at_nat,V: nat,U1: nat,V1: nat,U22: nat] :
% 2.86/1.58 ( ( isSimplePath_a @ c @ U @ P4 @ V )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 2.86/1.58 => ( ( U1 != U22 )
% 2.86/1.58 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U22 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_153_isSPath__sg__outgoing,axiom,
% 2.86/1.58 ! [U: nat,P4: list_P559422087at_nat,V: nat,U1: nat,V1: nat,V22: nat] :
% 2.86/1.58 ( ( isSimplePath_a @ c @ U @ P4 @ V )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 2.86/1.58 => ( ( V1 != V22 )
% 2.86/1.58 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V22 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_154_reachableNodes__E__closed,axiom,
% 2.86/1.58 ! [S: nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ ( e_a @ c ) @ ( reachableNodes_a @ c @ S ) ) @ ( reachableNodes_a @ c @ S ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_155_shortestPath__is__simple,axiom,
% 2.86/1.58 ! [S: nat,P4: list_P559422087at_nat,T3: nat] :
% 2.86/1.58 ( ( isShortestPath_a @ c @ S @ P4 @ T3 )
% 2.86/1.58 => ( isSimplePath_a @ c @ S @ P4 @ T3 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_156_ImageI,axiom,
% 2.86/1.58 ! [A3: product_prod_nat_nat,B4: product_prod_nat_nat,R: set_Pr1490359111at_nat,A: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ A3 @ B4 ) @ R )
% 2.86/1.58 => ( ( member701585322at_nat @ A3 @ A )
% 2.86/1.58 => ( member701585322at_nat @ B4 @ ( image_1655366419at_nat @ R @ A ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_157_ImageI,axiom,
% 2.86/1.58 ! [A3: product_prod_nat_nat,B4: nat,R: set_Pr1287749686at_nat,A: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member868723479at_nat @ ( produc947540346at_nat @ A3 @ B4 ) @ R )
% 2.86/1.58 => ( ( member701585322at_nat @ A3 @ A )
% 2.86/1.58 => ( member_nat @ B4 @ ( image_2142699582at_nat @ R @ A ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_158_ImageI,axiom,
% 2.86/1.58 ! [A3: nat,B4: product_prod_nat_nat,R: set_Pr1746169692at_nat,A: set_nat] :
% 2.86/1.58 ( ( member1293241661at_nat @ ( produc1933845336at_nat @ A3 @ B4 ) @ R )
% 2.86/1.58 => ( ( member_nat @ A3 @ A )
% 2.86/1.58 => ( member701585322at_nat @ B4 @ ( image_981520924at_nat @ R @ A ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_159_ImageI,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,R: set_Pr1986765409at_nat,A: set_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ R )
% 2.86/1.58 => ( ( member_nat @ A3 @ A )
% 2.86/1.58 => ( member_nat @ B4 @ ( image_nat_nat @ R @ A ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_160_finite__SigmaI,axiom,
% 2.86/1.58 ! [A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( finite772653738at_nat @ A )
% 2.86/1.58 => ( ! [A4: product_prod_nat_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ A4 @ A )
% 2.86/1.58 => ( finite772653738at_nat @ ( B2 @ A4 ) ) )
% 2.86/1.58 => ( finite48957584at_nat @ ( produc2056081288at_nat @ A @ B2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_161_finite__SigmaI,axiom,
% 2.86/1.58 ! [A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_nat] :
% 2.86/1.58 ( ( finite772653738at_nat @ A )
% 2.86/1.58 => ( ! [A4: product_prod_nat_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ A4 @ A )
% 2.86/1.58 => ( finite_finite_nat @ ( B2 @ A4 ) ) )
% 2.86/1.58 => ( finite2000257047at_nat @ ( produc2055342601at_nat @ A @ B2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_162_finite__SigmaI,axiom,
% 2.86/1.58 ! [A: set_nat,B2: nat > set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( finite_finite_nat @ A )
% 2.86/1.58 => ( ! [A4: nat] :
% 2.86/1.58 ( ( member_nat @ A4 @ A )
% 2.86/1.58 => ( finite772653738at_nat @ ( B2 @ A4 ) ) )
% 2.86/1.58 => ( finite277291581at_nat @ ( produc894163943at_nat @ A @ B2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_163_finite__SigmaI,axiom,
% 2.86/1.58 ! [A: set_nat,B2: nat > set_nat] :
% 2.86/1.58 ( ( finite_finite_nat @ A )
% 2.86/1.58 => ( ! [A4: nat] :
% 2.86/1.58 ( ( member_nat @ A4 @ A )
% 2.86/1.58 => ( finite_finite_nat @ ( B2 @ A4 ) ) )
% 2.86/1.58 => ( finite772653738at_nat @ ( produc45129834at_nat @ A @ B2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_164_adjacent__nodes__finite,axiom,
% 2.86/1.58 ! [U: nat] : ( finite_finite_nat @ ( adjacent_nodes_a @ c @ U ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_165_finite__outgoing,axiom,
% 2.86/1.58 ! [U: nat] :
% 2.86/1.58 ( ( finite_finite_nat @ ( v_a @ c ) )
% 2.86/1.58 => ( finite772653738at_nat @ ( outgoing_a @ c @ U ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_166_Graph_Oadjacent__nodes_Ocong,axiom,
% 2.86/1.58 adjacent_nodes_a = adjacent_nodes_a ).
% 2.86/1.58
% 2.86/1.58 thf(fact_167_Graph_OisSimplePath_Ocong,axiom,
% 2.86/1.58 isSimplePath_a = isSimplePath_a ).
% 2.86/1.58
% 2.86/1.58 thf(fact_168_Graph_Ooutgoing_Ocong,axiom,
% 2.86/1.58 outgoing_a = outgoing_a ).
% 2.86/1.58
% 2.86/1.58 thf(fact_169_rev__ImageI,axiom,
% 2.86/1.58 ! [A3: product_prod_nat_nat,A: set_Pr1986765409at_nat,B4: product_prod_nat_nat,R: set_Pr1490359111at_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ A3 @ A )
% 2.86/1.58 => ( ( member2027625872at_nat @ ( produc1168807639at_nat @ A3 @ B4 ) @ R )
% 2.86/1.58 => ( member701585322at_nat @ B4 @ ( image_1655366419at_nat @ R @ A ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_170_rev__ImageI,axiom,
% 2.86/1.58 ! [A3: product_prod_nat_nat,A: set_Pr1986765409at_nat,B4: nat,R: set_Pr1287749686at_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ A3 @ A )
% 2.86/1.58 => ( ( member868723479at_nat @ ( produc947540346at_nat @ A3 @ B4 ) @ R )
% 2.86/1.58 => ( member_nat @ B4 @ ( image_2142699582at_nat @ R @ A ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_171_rev__ImageI,axiom,
% 2.86/1.58 ! [A3: nat,A: set_nat,B4: product_prod_nat_nat,R: set_Pr1746169692at_nat] :
% 2.86/1.58 ( ( member_nat @ A3 @ A )
% 2.86/1.58 => ( ( member1293241661at_nat @ ( produc1933845336at_nat @ A3 @ B4 ) @ R )
% 2.86/1.58 => ( member701585322at_nat @ B4 @ ( image_981520924at_nat @ R @ A ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_172_rev__ImageI,axiom,
% 2.86/1.58 ! [A3: nat,A: set_nat,B4: nat,R: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member_nat @ A3 @ A )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ R )
% 2.86/1.58 => ( member_nat @ B4 @ ( image_nat_nat @ R @ A ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_173_Image__iff,axiom,
% 2.86/1.58 ! [B4: nat,R: set_Pr1986765409at_nat,A: set_nat] :
% 2.86/1.58 ( ( member_nat @ B4 @ ( image_nat_nat @ R @ A ) )
% 2.86/1.58 = ( ? [X: nat] :
% 2.86/1.58 ( ( member_nat @ X @ A )
% 2.86/1.58 & ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ B4 ) @ R ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_174_ImageE,axiom,
% 2.86/1.58 ! [B4: product_prod_nat_nat,R: set_Pr1490359111at_nat,A: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ B4 @ ( image_1655366419at_nat @ R @ A ) )
% 2.86/1.58 => ~ ! [X2: product_prod_nat_nat] :
% 2.86/1.58 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ X2 @ B4 ) @ R )
% 2.86/1.58 => ~ ( member701585322at_nat @ X2 @ A ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_175_ImageE,axiom,
% 2.86/1.58 ! [B4: product_prod_nat_nat,R: set_Pr1746169692at_nat,A: set_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ B4 @ ( image_981520924at_nat @ R @ A ) )
% 2.86/1.58 => ~ ! [X2: nat] :
% 2.86/1.58 ( ( member1293241661at_nat @ ( produc1933845336at_nat @ X2 @ B4 ) @ R )
% 2.86/1.58 => ~ ( member_nat @ X2 @ A ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_176_ImageE,axiom,
% 2.86/1.58 ! [B4: nat,R: set_Pr1287749686at_nat,A: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member_nat @ B4 @ ( image_2142699582at_nat @ R @ A ) )
% 2.86/1.58 => ~ ! [X2: product_prod_nat_nat] :
% 2.86/1.58 ( ( member868723479at_nat @ ( produc947540346at_nat @ X2 @ B4 ) @ R )
% 2.86/1.58 => ~ ( member701585322at_nat @ X2 @ A ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_177_ImageE,axiom,
% 2.86/1.58 ! [B4: nat,R: set_Pr1986765409at_nat,A: set_nat] :
% 2.86/1.58 ( ( member_nat @ B4 @ ( image_nat_nat @ R @ A ) )
% 2.86/1.58 => ~ ! [X2: nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X2 @ B4 ) @ R )
% 2.86/1.58 => ~ ( member_nat @ X2 @ A ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_178_Image__subset,axiom,
% 2.86/1.58 ! [R: set_Pr1986765409at_nat,A: set_nat,B2: set_nat,C3: set_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ R
% 2.86/1.58 @ ( produc45129834at_nat @ A
% 2.86/1.58 @ ^ [Uu: nat] : B2 ) )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( image_nat_nat @ R @ C3 ) @ B2 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_179_Graph_OshortestPath__is__simple,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,S: nat,P4: list_P559422087at_nat,T3: nat] :
% 2.86/1.58 ( ( isShortestPath_a @ C @ S @ P4 @ T3 )
% 2.86/1.58 => ( isSimplePath_a @ C @ S @ P4 @ T3 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_180_Image__mono,axiom,
% 2.86/1.58 ! [R4: set_Pr1490359111at_nat,R: set_Pr1490359111at_nat,A5: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le465611495at_nat @ R4 @ R )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ A5 @ A )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( image_1655366419at_nat @ R4 @ A5 ) @ ( image_1655366419at_nat @ R @ A ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_181_Image__mono,axiom,
% 2.86/1.58 ! [R4: set_Pr1287749686at_nat,R: set_Pr1287749686at_nat,A5: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le1837363350at_nat @ R4 @ R )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ A5 @ A )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( image_2142699582at_nat @ R4 @ A5 ) @ ( image_2142699582at_nat @ R @ A ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_182_Image__mono,axiom,
% 2.86/1.58 ! [R4: set_Pr1746169692at_nat,R: set_Pr1746169692at_nat,A5: set_nat,A: set_nat] :
% 2.86/1.58 ( ( ord_le148299708at_nat @ R4 @ R )
% 2.86/1.58 => ( ( ord_less_eq_set_nat @ A5 @ A )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( image_981520924at_nat @ R4 @ A5 ) @ ( image_981520924at_nat @ R @ A ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_183_Image__mono,axiom,
% 2.86/1.58 ! [R4: set_Pr1986765409at_nat,R: set_Pr1986765409at_nat,A5: set_nat,A: set_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ R4 @ R )
% 2.86/1.58 => ( ( ord_less_eq_set_nat @ A5 @ A )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( image_nat_nat @ R4 @ A5 ) @ ( image_nat_nat @ R @ A ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_184_Relation_Ofinite__Image,axiom,
% 2.86/1.58 ! [R2: set_Pr1986765409at_nat,A: set_nat] :
% 2.86/1.58 ( ( finite772653738at_nat @ R2 )
% 2.86/1.58 => ( finite_finite_nat @ ( image_nat_nat @ R2 @ A ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_185_Graph_OreachableNodes__E__closed,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,S: nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ ( e_a @ C ) @ ( reachableNodes_a @ C @ S ) ) @ ( reachableNodes_a @ C @ S ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_186_finite__cartesian__product,axiom,
% 2.86/1.58 ! [A: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( finite772653738at_nat @ A )
% 2.86/1.58 => ( ( finite772653738at_nat @ B2 )
% 2.86/1.58 => ( finite48957584at_nat
% 2.86/1.58 @ ( produc2056081288at_nat @ A
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] : B2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_187_finite__cartesian__product,axiom,
% 2.86/1.58 ! [A: set_Pr1986765409at_nat,B2: set_nat] :
% 2.86/1.58 ( ( finite772653738at_nat @ A )
% 2.86/1.58 => ( ( finite_finite_nat @ B2 )
% 2.86/1.58 => ( finite2000257047at_nat
% 2.86/1.58 @ ( produc2055342601at_nat @ A
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] : B2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_188_finite__cartesian__product,axiom,
% 2.86/1.58 ! [A: set_nat,B2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( finite_finite_nat @ A )
% 2.86/1.58 => ( ( finite772653738at_nat @ B2 )
% 2.86/1.58 => ( finite277291581at_nat
% 2.86/1.58 @ ( produc894163943at_nat @ A
% 2.86/1.58 @ ^ [Uu: nat] : B2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_189_finite__cartesian__product,axiom,
% 2.86/1.58 ! [A: set_nat,B2: set_nat] :
% 2.86/1.58 ( ( finite_finite_nat @ A )
% 2.86/1.58 => ( ( finite_finite_nat @ B2 )
% 2.86/1.58 => ( finite772653738at_nat
% 2.86/1.58 @ ( produc45129834at_nat @ A
% 2.86/1.58 @ ^ [Uu: nat] : B2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_190_Graph_OisSPath__sg__outgoing,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,U: nat,P4: list_P559422087at_nat,V: nat,U1: nat,V1: nat,V22: nat] :
% 2.86/1.58 ( ( isSimplePath_a @ C @ U @ P4 @ V )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 2.86/1.58 => ( ( V1 != V22 )
% 2.86/1.58 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V22 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_191_Graph_OisSPath__sg__incoming,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,U: nat,P4: list_P559422087at_nat,V: nat,U1: nat,V1: nat,U22: nat] :
% 2.86/1.58 ( ( isSimplePath_a @ C @ U @ P4 @ V )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 2.86/1.58 => ( ( U1 != U22 )
% 2.86/1.58 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U22 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_192_Graph_OisSPath__no__selfloop,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,U: nat,P4: list_P559422087at_nat,V: nat,U1: nat] :
% 2.86/1.58 ( ( isSimplePath_a @ C @ U @ P4 @ V )
% 2.86/1.58 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ U1 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_193_Graph_Oadjacent__nodes__ss__V,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,U: nat] : ( ord_less_eq_set_nat @ ( adjacent_nodes_a @ C @ U ) @ ( v_a @ C ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_194_Graph_Ooutgoing__edges,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,U: nat] : ( ord_le841296385at_nat @ ( outgoing_a @ C @ U ) @ ( e_a @ C ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_195_Graph_Ooutgoing__def,axiom,
% 2.86/1.58 ( outgoing_a
% 2.86/1.58 = ( ^ [C2: product_prod_nat_nat > a,V2: nat] :
% 2.86/1.58 ( collec7649004at_nat
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] :
% 2.86/1.58 ? [U3: nat] :
% 2.86/1.58 ( ( Uu
% 2.86/1.58 = ( product_Pair_nat_nat @ V2 @ U3 ) )
% 2.86/1.58 & ( member701585322at_nat @ ( product_Pair_nat_nat @ V2 @ U3 ) @ ( e_a @ C2 ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_196_Graph_Oconnected__edgeRtc,axiom,
% 2.86/1.58 ( connected_a
% 2.86/1.58 = ( ^ [C2: product_prod_nat_nat > a,U3: nat,V2: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ U3 @ V2 ) @ ( transi1645084429cl_nat @ ( e_a @ C2 ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_197_Graph_OE__ss__VxV,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a] :
% 2.86/1.58 ( ord_le841296385at_nat @ ( e_a @ C )
% 2.86/1.58 @ ( produc45129834at_nat @ ( v_a @ C )
% 2.86/1.58 @ ^ [Uu: nat] : ( v_a @ C ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_198_Graph_Ofinite__outgoing,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,U: nat] :
% 2.86/1.58 ( ( finite_finite_nat @ ( v_a @ C ) )
% 2.86/1.58 => ( finite772653738at_nat @ ( outgoing_a @ C @ U ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_199_Finite__Graph_Oadjacent__nodes__finite,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,U: nat] :
% 2.86/1.58 ( ( finite_Graph_a @ C )
% 2.86/1.58 => ( finite_finite_nat @ ( adjacent_nodes_a @ C @ U ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_200_isSPath__nt__parallel,axiom,
% 2.86/1.58 ! [S: nat,P4: list_P559422087at_nat,T3: nat,E2: product_prod_nat_nat] :
% 2.86/1.58 ( ( isSimplePath_a @ c @ S @ P4 @ T3 )
% 2.86/1.58 => ( ( member701585322at_nat @ E2 @ ( set_Pr2131844118at_nat @ P4 ) )
% 2.86/1.58 => ~ ( member701585322at_nat @ ( product_swap_nat_nat @ E2 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_201_mem__Sigma__iff,axiom,
% 2.86/1.58 ! [A3: product_prod_nat_nat,B4: product_prod_nat_nat,A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ A3 @ B4 ) @ ( produc2056081288at_nat @ A @ B2 ) )
% 2.86/1.58 = ( ( member701585322at_nat @ A3 @ A )
% 2.86/1.58 & ( member701585322at_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_202_mem__Sigma__iff,axiom,
% 2.86/1.58 ! [A3: product_prod_nat_nat,B4: nat,A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_nat] :
% 2.86/1.58 ( ( member868723479at_nat @ ( produc947540346at_nat @ A3 @ B4 ) @ ( produc2055342601at_nat @ A @ B2 ) )
% 2.86/1.58 = ( ( member701585322at_nat @ A3 @ A )
% 2.86/1.58 & ( member_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_203_mem__Sigma__iff,axiom,
% 2.86/1.58 ! [A3: nat,B4: product_prod_nat_nat,A: set_nat,B2: nat > set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member1293241661at_nat @ ( produc1933845336at_nat @ A3 @ B4 ) @ ( produc894163943at_nat @ A @ B2 ) )
% 2.86/1.58 = ( ( member_nat @ A3 @ A )
% 2.86/1.58 & ( member701585322at_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_204_mem__Sigma__iff,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,A: set_nat,B2: nat > set_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( produc45129834at_nat @ A @ B2 ) )
% 2.86/1.58 = ( ( member_nat @ A3 @ A )
% 2.86/1.58 & ( member_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_205_SigmaI,axiom,
% 2.86/1.58 ! [A3: product_prod_nat_nat,A: set_Pr1986765409at_nat,B4: product_prod_nat_nat,B2: product_prod_nat_nat > set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ A3 @ A )
% 2.86/1.58 => ( ( member701585322at_nat @ B4 @ ( B2 @ A3 ) )
% 2.86/1.58 => ( member2027625872at_nat @ ( produc1168807639at_nat @ A3 @ B4 ) @ ( produc2056081288at_nat @ A @ B2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_206_SigmaI,axiom,
% 2.86/1.58 ! [A3: product_prod_nat_nat,A: set_Pr1986765409at_nat,B4: nat,B2: product_prod_nat_nat > set_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ A3 @ A )
% 2.86/1.58 => ( ( member_nat @ B4 @ ( B2 @ A3 ) )
% 2.86/1.58 => ( member868723479at_nat @ ( produc947540346at_nat @ A3 @ B4 ) @ ( produc2055342601at_nat @ A @ B2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_207_SigmaI,axiom,
% 2.86/1.58 ! [A3: nat,A: set_nat,B4: product_prod_nat_nat,B2: nat > set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member_nat @ A3 @ A )
% 2.86/1.58 => ( ( member701585322at_nat @ B4 @ ( B2 @ A3 ) )
% 2.86/1.58 => ( member1293241661at_nat @ ( produc1933845336at_nat @ A3 @ B4 ) @ ( produc894163943at_nat @ A @ B2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_208_SigmaI,axiom,
% 2.86/1.58 ! [A3: nat,A: set_nat,B4: nat,B2: nat > set_nat] :
% 2.86/1.58 ( ( member_nat @ A3 @ A )
% 2.86/1.58 => ( ( member_nat @ B4 @ ( B2 @ A3 ) )
% 2.86/1.58 => ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( produc45129834at_nat @ A @ B2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_209_trancl__subset__Sigma__aux,axiom,
% 2.86/1.58 ! [A3: product_prod_nat_nat,B4: product_prod_nat_nat,R: set_Pr1490359111at_nat,A: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ A3 @ B4 ) @ ( transi1484096900at_nat @ R ) )
% 2.86/1.58 => ( ( ord_le465611495at_nat @ R
% 2.86/1.58 @ ( produc2056081288at_nat @ A
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] : A ) )
% 2.86/1.58 => ( ( A3 = B4 )
% 2.86/1.58 | ( member701585322at_nat @ A3 @ A ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_210_trancl__subset__Sigma__aux,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,R: set_Pr1986765409at_nat,A: set_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( transi1645084429cl_nat @ R ) )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ R
% 2.86/1.58 @ ( produc45129834at_nat @ A
% 2.86/1.58 @ ^ [Uu: nat] : A ) )
% 2.86/1.58 => ( ( A3 = B4 )
% 2.86/1.58 | ( member_nat @ A3 @ A ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_211_old_Oprod_Oinject,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,A6: nat,B6: nat] :
% 2.86/1.58 ( ( ( product_Pair_nat_nat @ A3 @ B4 )
% 2.86/1.58 = ( product_Pair_nat_nat @ A6 @ B6 ) )
% 2.86/1.58 = ( ( A3 = A6 )
% 2.86/1.58 & ( B4 = B6 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_212_prod_Oinject,axiom,
% 2.86/1.58 ! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
% 2.86/1.58 ( ( ( product_Pair_nat_nat @ X1 @ X22 )
% 2.86/1.58 = ( product_Pair_nat_nat @ Y1 @ Y22 ) )
% 2.86/1.58 = ( ( X1 = Y1 )
% 2.86/1.58 & ( X22 = Y22 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_213_r__into__rtrancl,axiom,
% 2.86/1.58 ! [P4: product_prod_nat_nat,R: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ P4 @ R )
% 2.86/1.58 => ( member701585322at_nat @ P4 @ ( transi1645084429cl_nat @ R ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_214_rtrancl__idemp,axiom,
% 2.86/1.58 ! [R: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( transi1645084429cl_nat @ ( transi1645084429cl_nat @ R ) )
% 2.86/1.58 = ( transi1645084429cl_nat @ R ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_215_swap__swap,axiom,
% 2.86/1.58 ! [P4: product_prod_nat_nat] :
% 2.86/1.58 ( ( product_swap_nat_nat @ ( product_swap_nat_nat @ P4 ) )
% 2.86/1.58 = P4 ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_216_swap__simp,axiom,
% 2.86/1.58 ! [X3: nat,Y3: nat] :
% 2.86/1.58 ( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) )
% 2.86/1.58 = ( product_Pair_nat_nat @ Y3 @ X3 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_217_converse__rtrancl__induct2,axiom,
% 2.86/1.58 ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R: set_Pr1490359111at_nat,P3: nat > nat > $o] :
% 2.86/1.58 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi1484096900at_nat @ R ) )
% 2.86/1.58 => ( ( P3 @ Bx @ By )
% 2.86/1.58 => ( ! [A4: nat,B3: nat,Aa: nat,Ba: nat] :
% 2.86/1.58 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R )
% 2.86/1.58 => ( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ Aa @ Ba ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi1484096900at_nat @ R ) )
% 2.86/1.58 => ( ( P3 @ Aa @ Ba )
% 2.86/1.58 => ( P3 @ A4 @ B3 ) ) ) )
% 2.86/1.58 => ( P3 @ Ax @ Ay ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_218_converse__rtranclE2,axiom,
% 2.86/1.58 ! [Xa2: nat,Xb: nat,Za: nat,Zb: nat,R: set_Pr1490359111at_nat] :
% 2.86/1.58 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xb ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi1484096900at_nat @ R ) )
% 2.86/1.58 => ( ( ( product_Pair_nat_nat @ Xa2 @ Xb )
% 2.86/1.58 != ( product_Pair_nat_nat @ Za @ Zb ) )
% 2.86/1.58 => ~ ! [A4: nat,B3: nat] :
% 2.86/1.58 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xb ) @ ( product_Pair_nat_nat @ A4 @ B3 ) ) @ R )
% 2.86/1.58 => ~ ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi1484096900at_nat @ R ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_219_rtrancl__induct2,axiom,
% 2.86/1.58 ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R: set_Pr1490359111at_nat,P3: nat > nat > $o] :
% 2.86/1.58 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi1484096900at_nat @ R ) )
% 2.86/1.58 => ( ( P3 @ Ax @ Ay )
% 2.86/1.58 => ( ! [A4: nat,B3: nat,Aa: nat,Ba: nat] :
% 2.86/1.58 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A4 @ B3 ) ) @ ( transi1484096900at_nat @ R ) )
% 2.86/1.58 => ( ( member2027625872at_nat @ ( produc1168807639at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R )
% 2.86/1.58 => ( ( P3 @ A4 @ B3 )
% 2.86/1.58 => ( P3 @ Aa @ Ba ) ) ) )
% 2.86/1.58 => ( P3 @ Bx @ By ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_220_old_Oprod_Oinducts,axiom,
% 2.86/1.58 ! [P3: product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
% 2.86/1.58 ( ! [A4: nat,B3: nat] : ( P3 @ ( product_Pair_nat_nat @ A4 @ B3 ) )
% 2.86/1.58 => ( P3 @ Prod ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_221_old_Oprod_Oexhaust,axiom,
% 2.86/1.58 ! [Y3: product_prod_nat_nat] :
% 2.86/1.58 ~ ! [A4: nat,B3: nat] :
% 2.86/1.58 ( Y3
% 2.86/1.58 != ( product_Pair_nat_nat @ A4 @ B3 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_222_Pair__inject,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,A6: nat,B6: nat] :
% 2.86/1.58 ( ( ( product_Pair_nat_nat @ A3 @ B4 )
% 2.86/1.58 = ( product_Pair_nat_nat @ A6 @ B6 ) )
% 2.86/1.58 => ~ ( ( A3 = A6 )
% 2.86/1.58 => ( B4 != B6 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_223_prod__cases,axiom,
% 2.86/1.58 ! [P3: product_prod_nat_nat > $o,P4: product_prod_nat_nat] :
% 2.86/1.58 ( ! [A4: nat,B3: nat] : ( P3 @ ( product_Pair_nat_nat @ A4 @ B3 ) )
% 2.86/1.58 => ( P3 @ P4 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_224_surj__pair,axiom,
% 2.86/1.58 ! [P4: product_prod_nat_nat] :
% 2.86/1.58 ? [X2: nat,Y2: nat] :
% 2.86/1.58 ( P4
% 2.86/1.58 = ( product_Pair_nat_nat @ X2 @ Y2 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_225_Graph_OisSPath__nt__parallel,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat > a,S: nat,P4: list_P559422087at_nat,T3: nat,E2: product_prod_nat_nat] :
% 2.86/1.58 ( ( isSimplePath_a @ C @ S @ P4 @ T3 )
% 2.86/1.58 => ( ( member701585322at_nat @ E2 @ ( set_Pr2131844118at_nat @ P4 ) )
% 2.86/1.58 => ~ ( member701585322at_nat @ ( product_swap_nat_nat @ E2 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_226_Sigma__cong,axiom,
% 2.86/1.58 ! [A: set_nat,B2: set_nat,C3: nat > set_nat,D: nat > set_nat] :
% 2.86/1.58 ( ( A = B2 )
% 2.86/1.58 => ( ! [X2: nat] :
% 2.86/1.58 ( ( member_nat @ X2 @ B2 )
% 2.86/1.58 => ( ( C3 @ X2 )
% 2.86/1.58 = ( D @ X2 ) ) )
% 2.86/1.58 => ( ( produc45129834at_nat @ A @ C3 )
% 2.86/1.58 = ( produc45129834at_nat @ B2 @ D ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_227_Times__eq__cancel2,axiom,
% 2.86/1.58 ! [X3: nat,C3: set_nat,A: set_nat,B2: set_nat] :
% 2.86/1.58 ( ( member_nat @ X3 @ C3 )
% 2.86/1.58 => ( ( ( produc45129834at_nat @ A
% 2.86/1.58 @ ^ [Uu: nat] : C3 )
% 2.86/1.58 = ( produc45129834at_nat @ B2
% 2.86/1.58 @ ^ [Uu: nat] : C3 ) )
% 2.86/1.58 = ( A = B2 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_228_SigmaE,axiom,
% 2.86/1.58 ! [C: produc842455143at_nat,A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member2027625872at_nat @ C @ ( produc2056081288at_nat @ A @ B2 ) )
% 2.86/1.58 => ~ ! [X2: product_prod_nat_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ X2 @ A )
% 2.86/1.58 => ! [Y2: product_prod_nat_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ Y2 @ ( B2 @ X2 ) )
% 2.86/1.58 => ( C
% 2.86/1.58 != ( produc1168807639at_nat @ X2 @ Y2 ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_229_SigmaE,axiom,
% 2.86/1.58 ! [C: produc1271302400at_nat,A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_nat] :
% 2.86/1.58 ( ( member868723479at_nat @ C @ ( produc2055342601at_nat @ A @ B2 ) )
% 2.86/1.58 => ~ ! [X2: product_prod_nat_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ X2 @ A )
% 2.86/1.58 => ! [Y2: nat] :
% 2.86/1.58 ( ( member_nat @ Y2 @ ( B2 @ X2 ) )
% 2.86/1.58 => ( C
% 2.86/1.58 != ( produc947540346at_nat @ X2 @ Y2 ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_230_SigmaE,axiom,
% 2.86/1.58 ! [C: produc1695820582at_nat,A: set_nat,B2: nat > set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member1293241661at_nat @ C @ ( produc894163943at_nat @ A @ B2 ) )
% 2.86/1.58 => ~ ! [X2: nat] :
% 2.86/1.58 ( ( member_nat @ X2 @ A )
% 2.86/1.58 => ! [Y2: product_prod_nat_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ Y2 @ ( B2 @ X2 ) )
% 2.86/1.58 => ( C
% 2.86/1.58 != ( produc1933845336at_nat @ X2 @ Y2 ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_231_SigmaE,axiom,
% 2.86/1.58 ! [C: product_prod_nat_nat,A: set_nat,B2: nat > set_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ C @ ( produc45129834at_nat @ A @ B2 ) )
% 2.86/1.58 => ~ ! [X2: nat] :
% 2.86/1.58 ( ( member_nat @ X2 @ A )
% 2.86/1.58 => ! [Y2: nat] :
% 2.86/1.58 ( ( member_nat @ Y2 @ ( B2 @ X2 ) )
% 2.86/1.58 => ( C
% 2.86/1.58 != ( product_Pair_nat_nat @ X2 @ Y2 ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_232_SigmaD1,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,A: set_nat,B2: nat > set_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( produc45129834at_nat @ A @ B2 ) )
% 2.86/1.58 => ( member_nat @ A3 @ A ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_233_SigmaD2,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,A: set_nat,B2: nat > set_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( produc45129834at_nat @ A @ B2 ) )
% 2.86/1.58 => ( member_nat @ B4 @ ( B2 @ A3 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_234_SigmaE2,axiom,
% 2.86/1.58 ! [A3: product_prod_nat_nat,B4: product_prod_nat_nat,A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member2027625872at_nat @ ( produc1168807639at_nat @ A3 @ B4 ) @ ( produc2056081288at_nat @ A @ B2 ) )
% 2.86/1.58 => ~ ( ( member701585322at_nat @ A3 @ A )
% 2.86/1.58 => ~ ( member701585322at_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_235_SigmaE2,axiom,
% 2.86/1.58 ! [A3: product_prod_nat_nat,B4: nat,A: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_nat] :
% 2.86/1.58 ( ( member868723479at_nat @ ( produc947540346at_nat @ A3 @ B4 ) @ ( produc2055342601at_nat @ A @ B2 ) )
% 2.86/1.58 => ~ ( ( member701585322at_nat @ A3 @ A )
% 2.86/1.58 => ~ ( member_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_236_SigmaE2,axiom,
% 2.86/1.58 ! [A3: nat,B4: product_prod_nat_nat,A: set_nat,B2: nat > set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member1293241661at_nat @ ( produc1933845336at_nat @ A3 @ B4 ) @ ( produc894163943at_nat @ A @ B2 ) )
% 2.86/1.58 => ~ ( ( member_nat @ A3 @ A )
% 2.86/1.58 => ~ ( member701585322at_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_237_SigmaE2,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,A: set_nat,B2: nat > set_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( produc45129834at_nat @ A @ B2 ) )
% 2.86/1.58 => ~ ( ( member_nat @ A3 @ A )
% 2.86/1.58 => ~ ( member_nat @ B4 @ ( B2 @ A3 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_238_rtranclE,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,R: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( transi1645084429cl_nat @ R ) )
% 2.86/1.58 => ( ( A3 != B4 )
% 2.86/1.58 => ~ ! [Y2: nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ Y2 ) @ ( transi1645084429cl_nat @ R ) )
% 2.86/1.58 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Y2 @ B4 ) @ R ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_239_rtrancl_Ocases,axiom,
% 2.86/1.58 ! [A1: nat,A22: nat,R: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi1645084429cl_nat @ R ) )
% 2.86/1.58 => ( ( A22 != A1 )
% 2.86/1.58 => ~ ! [B3: nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A1 @ B3 ) @ ( transi1645084429cl_nat @ R ) )
% 2.86/1.58 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ R ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_240_rtrancl_Osimps,axiom,
% 2.86/1.58 ! [A1: nat,A22: nat,R: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi1645084429cl_nat @ R ) )
% 2.86/1.58 = ( ? [A2: nat] :
% 2.86/1.58 ( ( A1 = A2 )
% 2.86/1.58 & ( A22 = A2 ) )
% 2.86/1.58 | ? [A2: nat,B5: nat,C2: nat] :
% 2.86/1.58 ( ( A1 = A2 )
% 2.86/1.58 & ( A22 = C2 )
% 2.86/1.58 & ( member701585322at_nat @ ( product_Pair_nat_nat @ A2 @ B5 ) @ ( transi1645084429cl_nat @ R ) )
% 2.86/1.58 & ( member701585322at_nat @ ( product_Pair_nat_nat @ B5 @ C2 ) @ R ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_241_rtrancl__trans,axiom,
% 2.86/1.58 ! [X3: nat,Y3: nat,R: set_Pr1986765409at_nat,Z: nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( transi1645084429cl_nat @ R ) )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ ( transi1645084429cl_nat @ R ) )
% 2.86/1.58 => ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Z ) @ ( transi1645084429cl_nat @ R ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_242_rtrancl__induct,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,R: set_Pr1986765409at_nat,P3: nat > $o] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( transi1645084429cl_nat @ R ) )
% 2.86/1.58 => ( ( P3 @ A3 )
% 2.86/1.58 => ( ! [Y2: nat,Z2: nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ Y2 ) @ ( transi1645084429cl_nat @ R ) )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Y2 @ Z2 ) @ R )
% 2.86/1.58 => ( ( P3 @ Y2 )
% 2.86/1.58 => ( P3 @ Z2 ) ) ) )
% 2.86/1.58 => ( P3 @ B4 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_243_rtrancl_Oinducts,axiom,
% 2.86/1.58 ! [X1: nat,X22: nat,R: set_Pr1986765409at_nat,P3: nat > nat > $o] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) @ ( transi1645084429cl_nat @ R ) )
% 2.86/1.58 => ( ! [A4: nat] : ( P3 @ A4 @ A4 )
% 2.86/1.58 => ( ! [A4: nat,B3: nat,C4: nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( transi1645084429cl_nat @ R ) )
% 2.86/1.58 => ( ( P3 @ A4 @ B3 )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ B3 @ C4 ) @ R )
% 2.86/1.58 => ( P3 @ A4 @ C4 ) ) ) )
% 2.86/1.58 => ( P3 @ X1 @ X22 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_244_converse__rtranclE,axiom,
% 2.86/1.58 ! [X3: nat,Z: nat,R: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Z ) @ ( transi1645084429cl_nat @ R ) )
% 2.86/1.58 => ( ( X3 != Z )
% 2.86/1.58 => ~ ! [Y2: nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R )
% 2.86/1.58 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Y2 @ Z ) @ ( transi1645084429cl_nat @ R ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_245_rtrancl_Ortrancl__refl,axiom,
% 2.86/1.58 ! [A3: nat,R: set_Pr1986765409at_nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ A3 ) @ ( transi1645084429cl_nat @ R ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_246_converse__rtrancl__induct,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,R: set_Pr1986765409at_nat,P3: nat > $o] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( transi1645084429cl_nat @ R ) )
% 2.86/1.58 => ( ( P3 @ B4 )
% 2.86/1.58 => ( ! [Y2: nat,Z2: nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Y2 @ Z2 ) @ R )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Z2 @ B4 ) @ ( transi1645084429cl_nat @ R ) )
% 2.86/1.58 => ( ( P3 @ Z2 )
% 2.86/1.58 => ( P3 @ Y2 ) ) ) )
% 2.86/1.58 => ( P3 @ A3 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_247_rtrancl_Ortrancl__into__rtrancl,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,R: set_Pr1986765409at_nat,C: nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( transi1645084429cl_nat @ R ) )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ B4 @ C ) @ R )
% 2.86/1.58 => ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ C ) @ ( transi1645084429cl_nat @ R ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_248_converse__rtrancl__into__rtrancl,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,R: set_Pr1986765409at_nat,C: nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ R )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ B4 @ C ) @ ( transi1645084429cl_nat @ R ) )
% 2.86/1.58 => ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ C ) @ ( transi1645084429cl_nat @ R ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_249_rtrancl__mono,axiom,
% 2.86/1.58 ! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ R @ S )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( transi1645084429cl_nat @ R ) @ ( transi1645084429cl_nat @ S ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_250_rtrancl__subset,axiom,
% 2.86/1.58 ! [R2: set_Pr1986765409at_nat,S3: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ R2 @ S3 )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ S3 @ ( transi1645084429cl_nat @ R2 ) )
% 2.86/1.58 => ( ( transi1645084429cl_nat @ S3 )
% 2.86/1.58 = ( transi1645084429cl_nat @ R2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_251_rtrancl__subset__rtrancl,axiom,
% 2.86/1.58 ! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ R @ ( transi1645084429cl_nat @ S ) )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( transi1645084429cl_nat @ R ) @ ( transi1645084429cl_nat @ S ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_252_Sigma__mono,axiom,
% 2.86/1.58 ! [A: set_Pr1986765409at_nat,C3: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_Pr1986765409at_nat,D: product_prod_nat_nat > set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ A @ C3 )
% 2.86/1.58 => ( ! [X2: product_prod_nat_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ X2 @ A )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
% 2.86/1.58 => ( ord_le465611495at_nat @ ( produc2056081288at_nat @ A @ B2 ) @ ( produc2056081288at_nat @ C3 @ D ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_253_Sigma__mono,axiom,
% 2.86/1.58 ! [A: set_Pr1986765409at_nat,C3: set_Pr1986765409at_nat,B2: product_prod_nat_nat > set_nat,D: product_prod_nat_nat > set_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ A @ C3 )
% 2.86/1.58 => ( ! [X2: product_prod_nat_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ X2 @ A )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
% 2.86/1.58 => ( ord_le1837363350at_nat @ ( produc2055342601at_nat @ A @ B2 ) @ ( produc2055342601at_nat @ C3 @ D ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_254_Sigma__mono,axiom,
% 2.86/1.58 ! [A: set_nat,C3: set_nat,B2: nat > set_Pr1986765409at_nat,D: nat > set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ A @ C3 )
% 2.86/1.58 => ( ! [X2: nat] :
% 2.86/1.58 ( ( member_nat @ X2 @ A )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
% 2.86/1.58 => ( ord_le148299708at_nat @ ( produc894163943at_nat @ A @ B2 ) @ ( produc894163943at_nat @ C3 @ D ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_255_Sigma__mono,axiom,
% 2.86/1.58 ! [A: set_nat,C3: set_nat,B2: nat > set_nat,D: nat > set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ A @ C3 )
% 2.86/1.58 => ( ! [X2: nat] :
% 2.86/1.58 ( ( member_nat @ X2 @ A )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( produc45129834at_nat @ A @ B2 ) @ ( produc45129834at_nat @ C3 @ D ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_256_Image__closed__trancl,axiom,
% 2.86/1.58 ! [R: set_Pr1490359111at_nat,X4: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ ( image_1655366419at_nat @ R @ X4 ) @ X4 )
% 2.86/1.58 => ( ( image_1655366419at_nat @ ( transi1484096900at_nat @ R ) @ X4 )
% 2.86/1.58 = X4 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_257_Image__closed__trancl,axiom,
% 2.86/1.58 ! [R: set_Pr1986765409at_nat,X4: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R @ X4 ) @ X4 )
% 2.86/1.58 => ( ( image_nat_nat @ ( transi1645084429cl_nat @ R ) @ X4 )
% 2.86/1.58 = X4 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_258_Not__Domain__rtrancl,axiom,
% 2.86/1.58 ! [X3: product_prod_nat_nat,R2: set_Pr1490359111at_nat,Y3: product_prod_nat_nat] :
% 2.86/1.58 ( ~ ( member701585322at_nat @ X3 @ ( domain37934156at_nat @ R2 ) )
% 2.86/1.58 => ( ( member2027625872at_nat @ ( produc1168807639at_nat @ X3 @ Y3 ) @ ( transi1484096900at_nat @ R2 ) )
% 2.86/1.58 = ( X3 = Y3 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_259_Not__Domain__rtrancl,axiom,
% 2.86/1.58 ! [X3: nat,R2: set_Pr1986765409at_nat,Y3: nat] :
% 2.86/1.58 ( ~ ( member_nat @ X3 @ ( domain_nat_nat @ R2 ) )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( transi1645084429cl_nat @ R2 ) )
% 2.86/1.58 = ( X3 = Y3 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_260_Times__subset__cancel2,axiom,
% 2.86/1.58 ! [X3: product_prod_nat_nat,C3: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ X3 @ C3 )
% 2.86/1.58 => ( ( ord_le465611495at_nat
% 2.86/1.58 @ ( produc2056081288at_nat @ A
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] : C3 )
% 2.86/1.58 @ ( produc2056081288at_nat @ B2
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] : C3 ) )
% 2.86/1.58 = ( ord_le841296385at_nat @ A @ B2 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_261_Times__subset__cancel2,axiom,
% 2.86/1.58 ! [X3: nat,C3: set_nat,A: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( member_nat @ X3 @ C3 )
% 2.86/1.58 => ( ( ord_le1837363350at_nat
% 2.86/1.58 @ ( produc2055342601at_nat @ A
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] : C3 )
% 2.86/1.58 @ ( produc2055342601at_nat @ B2
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] : C3 ) )
% 2.86/1.58 = ( ord_le841296385at_nat @ A @ B2 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_262_Times__subset__cancel2,axiom,
% 2.86/1.58 ! [X3: product_prod_nat_nat,C3: set_Pr1986765409at_nat,A: set_nat,B2: set_nat] :
% 2.86/1.58 ( ( member701585322at_nat @ X3 @ C3 )
% 2.86/1.58 => ( ( ord_le148299708at_nat
% 2.86/1.58 @ ( produc894163943at_nat @ A
% 2.86/1.58 @ ^ [Uu: nat] : C3 )
% 2.86/1.58 @ ( produc894163943at_nat @ B2
% 2.86/1.58 @ ^ [Uu: nat] : C3 ) )
% 2.86/1.58 = ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_263_Times__subset__cancel2,axiom,
% 2.86/1.58 ! [X3: nat,C3: set_nat,A: set_nat,B2: set_nat] :
% 2.86/1.58 ( ( member_nat @ X3 @ C3 )
% 2.86/1.58 => ( ( ord_le841296385at_nat
% 2.86/1.58 @ ( produc45129834at_nat @ A
% 2.86/1.58 @ ^ [Uu: nat] : C3 )
% 2.86/1.58 @ ( produc45129834at_nat @ B2
% 2.86/1.58 @ ^ [Uu: nat] : C3 ) )
% 2.86/1.58 = ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_264_finite__rtrancl__Image,axiom,
% 2.86/1.58 ! [R2: set_Pr1490359111at_nat,A: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( finite48957584at_nat @ R2 )
% 2.86/1.58 => ( ( finite772653738at_nat @ A )
% 2.86/1.58 => ( finite772653738at_nat @ ( image_1655366419at_nat @ ( transi1484096900at_nat @ R2 ) @ A ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_265_finite__rtrancl__Image,axiom,
% 2.86/1.58 ! [R2: set_Pr1986765409at_nat,A: set_nat] :
% 2.86/1.58 ( ( finite772653738at_nat @ R2 )
% 2.86/1.58 => ( ( finite_finite_nat @ A )
% 2.86/1.58 => ( finite_finite_nat @ ( image_nat_nat @ ( transi1645084429cl_nat @ R2 ) @ A ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_266_infinite__cartesian__product,axiom,
% 2.86/1.58 ! [A: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ~ ( finite772653738at_nat @ A )
% 2.86/1.58 => ( ~ ( finite772653738at_nat @ B2 )
% 2.86/1.58 => ~ ( finite48957584at_nat
% 2.86/1.58 @ ( produc2056081288at_nat @ A
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] : B2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_267_infinite__cartesian__product,axiom,
% 2.86/1.58 ! [A: set_Pr1986765409at_nat,B2: set_nat] :
% 2.86/1.58 ( ~ ( finite772653738at_nat @ A )
% 2.86/1.58 => ( ~ ( finite_finite_nat @ B2 )
% 2.86/1.58 => ~ ( finite2000257047at_nat
% 2.86/1.58 @ ( produc2055342601at_nat @ A
% 2.86/1.58 @ ^ [Uu: product_prod_nat_nat] : B2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_268_infinite__cartesian__product,axiom,
% 2.86/1.58 ! [A: set_nat,B2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ~ ( finite_finite_nat @ A )
% 2.86/1.58 => ( ~ ( finite772653738at_nat @ B2 )
% 2.86/1.58 => ~ ( finite277291581at_nat
% 2.86/1.58 @ ( produc894163943at_nat @ A
% 2.86/1.58 @ ^ [Uu: nat] : B2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_269_infinite__cartesian__product,axiom,
% 2.86/1.58 ! [A: set_nat,B2: set_nat] :
% 2.86/1.58 ( ~ ( finite_finite_nat @ A )
% 2.86/1.58 => ( ~ ( finite_finite_nat @ B2 )
% 2.86/1.58 => ~ ( finite772653738at_nat
% 2.86/1.58 @ ( produc45129834at_nat @ A
% 2.86/1.58 @ ^ [Uu: nat] : B2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_270_order__refl,axiom,
% 2.86/1.58 ! [X3: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ X3 @ X3 ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_271_order__refl,axiom,
% 2.86/1.58 ! [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ X3 ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_272_order__refl,axiom,
% 2.86/1.58 ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_273_order__subst1,axiom,
% 2.86/1.58 ! [A3: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ A3 @ ( F @ B4 ) )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ B4 @ C )
% 2.86/1.58 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_le841296385at_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_274_order__subst1,axiom,
% 2.86/1.58 ! [A3: set_Pr1986765409at_nat,F: set_nat > set_Pr1986765409at_nat,B4: set_nat,C: set_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ A3 @ ( F @ B4 ) )
% 2.86/1.58 => ( ( ord_less_eq_set_nat @ B4 @ C )
% 2.86/1.58 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_le841296385at_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_275_order__subst1,axiom,
% 2.86/1.58 ! [A3: set_Pr1986765409at_nat,F: nat > set_Pr1986765409at_nat,B4: nat,C: nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ A3 @ ( F @ B4 ) )
% 2.86/1.58 => ( ( ord_less_eq_nat @ B4 @ C )
% 2.86/1.58 => ( ! [X2: nat,Y2: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_le841296385at_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_276_order__subst1,axiom,
% 2.86/1.58 ! [A3: set_nat,F: set_Pr1986765409at_nat > set_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ A3 @ ( F @ B4 ) )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ B4 @ C )
% 2.86/1.58 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_277_order__subst1,axiom,
% 2.86/1.58 ! [A3: set_nat,F: set_nat > set_nat,B4: set_nat,C: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ A3 @ ( F @ B4 ) )
% 2.86/1.58 => ( ( ord_less_eq_set_nat @ B4 @ C )
% 2.86/1.58 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_278_order__subst1,axiom,
% 2.86/1.58 ! [A3: set_nat,F: nat > set_nat,B4: nat,C: nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ A3 @ ( F @ B4 ) )
% 2.86/1.58 => ( ( ord_less_eq_nat @ B4 @ C )
% 2.86/1.58 => ( ! [X2: nat,Y2: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_279_order__subst1,axiom,
% 2.86/1.58 ! [A3: nat,F: set_Pr1986765409at_nat > nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ A3 @ ( F @ B4 ) )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ B4 @ C )
% 2.86/1.58 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_280_order__subst1,axiom,
% 2.86/1.58 ! [A3: nat,F: set_nat > nat,B4: set_nat,C: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ A3 @ ( F @ B4 ) )
% 2.86/1.58 => ( ( ord_less_eq_set_nat @ B4 @ C )
% 2.86/1.58 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_281_order__subst1,axiom,
% 2.86/1.58 ! [A3: nat,F: nat > nat,B4: nat,C: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ A3 @ ( F @ B4 ) )
% 2.86/1.58 => ( ( ord_less_eq_nat @ B4 @ C )
% 2.86/1.58 => ( ! [X2: nat,Y2: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_282_order__subst2,axiom,
% 2.86/1.58 ! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ ( F @ B4 ) @ C )
% 2.86/1.58 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_283_order__subst2,axiom,
% 2.86/1.58 ! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_nat,C: set_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ord_less_eq_set_nat @ ( F @ B4 ) @ C )
% 2.86/1.58 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_284_order__subst2,axiom,
% 2.86/1.58 ! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > nat,C: nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ord_less_eq_nat @ ( F @ B4 ) @ C )
% 2.86/1.58 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_285_order__subst2,axiom,
% 2.86/1.58 ! [A3: set_nat,B4: set_nat,F: set_nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ ( F @ B4 ) @ C )
% 2.86/1.58 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_286_order__subst2,axiom,
% 2.86/1.58 ! [A3: set_nat,B4: set_nat,F: set_nat > set_nat,C: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ord_less_eq_set_nat @ ( F @ B4 ) @ C )
% 2.86/1.58 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_287_order__subst2,axiom,
% 2.86/1.58 ! [A3: set_nat,B4: set_nat,F: set_nat > nat,C: nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ord_less_eq_nat @ ( F @ B4 ) @ C )
% 2.86/1.58 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_288_order__subst2,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,F: nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ ( F @ B4 ) @ C )
% 2.86/1.58 => ( ! [X2: nat,Y2: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_289_order__subst2,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,F: nat > set_nat,C: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ord_less_eq_set_nat @ ( F @ B4 ) @ C )
% 2.86/1.58 => ( ! [X2: nat,Y2: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_290_order__subst2,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,F: nat > nat,C: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ord_less_eq_nat @ ( F @ B4 ) @ C )
% 2.86/1.58 => ( ! [X2: nat,Y2: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_291_ord__eq__le__subst,axiom,
% 2.86/1.58 ! [A3: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( A3
% 2.86/1.58 = ( F @ B4 ) )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ B4 @ C )
% 2.86/1.58 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_le841296385at_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_292_ord__eq__le__subst,axiom,
% 2.86/1.58 ! [A3: set_nat,F: set_Pr1986765409at_nat > set_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( A3
% 2.86/1.58 = ( F @ B4 ) )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ B4 @ C )
% 2.86/1.58 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_293_ord__eq__le__subst,axiom,
% 2.86/1.58 ! [A3: nat,F: set_Pr1986765409at_nat > nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( A3
% 2.86/1.58 = ( F @ B4 ) )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ B4 @ C )
% 2.86/1.58 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_294_ord__eq__le__subst,axiom,
% 2.86/1.58 ! [A3: set_Pr1986765409at_nat,F: set_nat > set_Pr1986765409at_nat,B4: set_nat,C: set_nat] :
% 2.86/1.58 ( ( A3
% 2.86/1.58 = ( F @ B4 ) )
% 2.86/1.58 => ( ( ord_less_eq_set_nat @ B4 @ C )
% 2.86/1.58 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_le841296385at_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_295_ord__eq__le__subst,axiom,
% 2.86/1.58 ! [A3: set_nat,F: set_nat > set_nat,B4: set_nat,C: set_nat] :
% 2.86/1.58 ( ( A3
% 2.86/1.58 = ( F @ B4 ) )
% 2.86/1.58 => ( ( ord_less_eq_set_nat @ B4 @ C )
% 2.86/1.58 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_296_ord__eq__le__subst,axiom,
% 2.86/1.58 ! [A3: nat,F: set_nat > nat,B4: set_nat,C: set_nat] :
% 2.86/1.58 ( ( A3
% 2.86/1.58 = ( F @ B4 ) )
% 2.86/1.58 => ( ( ord_less_eq_set_nat @ B4 @ C )
% 2.86/1.58 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_297_ord__eq__le__subst,axiom,
% 2.86/1.58 ! [A3: set_Pr1986765409at_nat,F: nat > set_Pr1986765409at_nat,B4: nat,C: nat] :
% 2.86/1.58 ( ( A3
% 2.86/1.58 = ( F @ B4 ) )
% 2.86/1.58 => ( ( ord_less_eq_nat @ B4 @ C )
% 2.86/1.58 => ( ! [X2: nat,Y2: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_le841296385at_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_298_ord__eq__le__subst,axiom,
% 2.86/1.58 ! [A3: set_nat,F: nat > set_nat,B4: nat,C: nat] :
% 2.86/1.58 ( ( A3
% 2.86/1.58 = ( F @ B4 ) )
% 2.86/1.58 => ( ( ord_less_eq_nat @ B4 @ C )
% 2.86/1.58 => ( ! [X2: nat,Y2: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_299_ord__eq__le__subst,axiom,
% 2.86/1.58 ! [A3: nat,F: nat > nat,B4: nat,C: nat] :
% 2.86/1.58 ( ( A3
% 2.86/1.58 = ( F @ B4 ) )
% 2.86/1.58 => ( ( ord_less_eq_nat @ B4 @ C )
% 2.86/1.58 => ( ! [X2: nat,Y2: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_300_ord__le__eq__subst,axiom,
% 2.86/1.58 ! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ( F @ B4 )
% 2.86/1.58 = C )
% 2.86/1.58 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_301_ord__le__eq__subst,axiom,
% 2.86/1.58 ! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_nat,C: set_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ( F @ B4 )
% 2.86/1.58 = C )
% 2.86/1.58 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_302_ord__le__eq__subst,axiom,
% 2.86/1.58 ! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > nat,C: nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ( F @ B4 )
% 2.86/1.58 = C )
% 2.86/1.58 => ( ! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_303_ord__le__eq__subst,axiom,
% 2.86/1.58 ! [A3: set_nat,B4: set_nat,F: set_nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ( F @ B4 )
% 2.86/1.58 = C )
% 2.86/1.58 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_304_ord__le__eq__subst,axiom,
% 2.86/1.58 ! [A3: set_nat,B4: set_nat,F: set_nat > set_nat,C: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ( F @ B4 )
% 2.86/1.58 = C )
% 2.86/1.58 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_305_ord__le__eq__subst,axiom,
% 2.86/1.58 ! [A3: set_nat,B4: set_nat,F: set_nat > nat,C: nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ( F @ B4 )
% 2.86/1.58 = C )
% 2.86/1.58 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_306_ord__le__eq__subst,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,F: nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ( F @ B4 )
% 2.86/1.58 = C )
% 2.86/1.58 => ( ! [X2: nat,Y2: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_le841296385at_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_307_ord__le__eq__subst,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,F: nat > set_nat,C: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ( F @ B4 )
% 2.86/1.58 = C )
% 2.86/1.58 => ( ! [X2: nat,Y2: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_308_ord__le__eq__subst,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,F: nat > nat,C: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ( F @ B4 )
% 2.86/1.58 = C )
% 2.86/1.58 => ( ! [X2: nat,Y2: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.86/1.58 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.86/1.58 => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_309_eq__iff,axiom,
% 2.86/1.58 ( ( ^ [Y4: set_Pr1986765409at_nat,Z3: set_Pr1986765409at_nat] : ( Y4 = Z3 ) )
% 2.86/1.58 = ( ^ [X: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ X @ Y )
% 2.86/1.58 & ( ord_le841296385at_nat @ Y @ X ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_310_eq__iff,axiom,
% 2.86/1.58 ( ( ^ [Y4: set_nat,Z3: set_nat] : ( Y4 = Z3 ) )
% 2.86/1.58 = ( ^ [X: set_nat,Y: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ X @ Y )
% 2.86/1.58 & ( ord_less_eq_set_nat @ Y @ X ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_311_eq__iff,axiom,
% 2.86/1.58 ( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
% 2.86/1.58 = ( ^ [X: nat,Y: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ X @ Y )
% 2.86/1.58 & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_312_antisym,axiom,
% 2.86/1.58 ! [X3: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ X3 @ Y3 )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ Y3 @ X3 )
% 2.86/1.58 => ( X3 = Y3 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_313_antisym,axiom,
% 2.86/1.58 ! [X3: set_nat,Y3: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ X3 @ Y3 )
% 2.86/1.58 => ( ( ord_less_eq_set_nat @ Y3 @ X3 )
% 2.86/1.58 => ( X3 = Y3 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_314_antisym,axiom,
% 2.86/1.58 ! [X3: nat,Y3: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 2.86/1.58 => ( ( ord_less_eq_nat @ Y3 @ X3 )
% 2.86/1.58 => ( X3 = Y3 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_315_linear,axiom,
% 2.86/1.58 ! [X3: nat,Y3: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 2.86/1.58 | ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_316_eq__refl,axiom,
% 2.86/1.58 ! [X3: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( X3 = Y3 )
% 2.86/1.58 => ( ord_le841296385at_nat @ X3 @ Y3 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_317_eq__refl,axiom,
% 2.86/1.58 ! [X3: set_nat,Y3: set_nat] :
% 2.86/1.58 ( ( X3 = Y3 )
% 2.86/1.58 => ( ord_less_eq_set_nat @ X3 @ Y3 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_318_eq__refl,axiom,
% 2.86/1.58 ! [X3: nat,Y3: nat] :
% 2.86/1.58 ( ( X3 = Y3 )
% 2.86/1.58 => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_319_le__cases,axiom,
% 2.86/1.58 ! [X3: nat,Y3: nat] :
% 2.86/1.58 ( ~ ( ord_less_eq_nat @ X3 @ Y3 )
% 2.86/1.58 => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_320_order_Otrans,axiom,
% 2.86/1.58 ! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ B4 @ C )
% 2.86/1.58 => ( ord_le841296385at_nat @ A3 @ C ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_321_order_Otrans,axiom,
% 2.86/1.58 ! [A3: set_nat,B4: set_nat,C: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ord_less_eq_set_nat @ B4 @ C )
% 2.86/1.58 => ( ord_less_eq_set_nat @ A3 @ C ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_322_order_Otrans,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,C: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ord_less_eq_nat @ B4 @ C )
% 2.86/1.58 => ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_323_le__cases3,axiom,
% 2.86/1.58 ! [X3: nat,Y3: nat,Z: nat] :
% 2.86/1.58 ( ( ( ord_less_eq_nat @ X3 @ Y3 )
% 2.86/1.58 => ~ ( ord_less_eq_nat @ Y3 @ Z ) )
% 2.86/1.58 => ( ( ( ord_less_eq_nat @ Y3 @ X3 )
% 2.86/1.58 => ~ ( ord_less_eq_nat @ X3 @ Z ) )
% 2.86/1.58 => ( ( ( ord_less_eq_nat @ X3 @ Z )
% 2.86/1.58 => ~ ( ord_less_eq_nat @ Z @ Y3 ) )
% 2.86/1.58 => ( ( ( ord_less_eq_nat @ Z @ Y3 )
% 2.86/1.58 => ~ ( ord_less_eq_nat @ Y3 @ X3 ) )
% 2.86/1.58 => ( ( ( ord_less_eq_nat @ Y3 @ Z )
% 2.86/1.58 => ~ ( ord_less_eq_nat @ Z @ X3 ) )
% 2.86/1.58 => ~ ( ( ord_less_eq_nat @ Z @ X3 )
% 2.86/1.58 => ~ ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_324_antisym__conv,axiom,
% 2.86/1.58 ! [Y3: set_Pr1986765409at_nat,X3: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ Y3 @ X3 )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ X3 @ Y3 )
% 2.86/1.58 = ( X3 = Y3 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_325_antisym__conv,axiom,
% 2.86/1.58 ! [Y3: set_nat,X3: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ Y3 @ X3 )
% 2.86/1.58 => ( ( ord_less_eq_set_nat @ X3 @ Y3 )
% 2.86/1.58 = ( X3 = Y3 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_326_antisym__conv,axiom,
% 2.86/1.58 ! [Y3: nat,X3: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ Y3 @ X3 )
% 2.86/1.58 => ( ( ord_less_eq_nat @ X3 @ Y3 )
% 2.86/1.58 = ( X3 = Y3 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_327_order__class_Oorder_Oeq__iff,axiom,
% 2.86/1.58 ( ( ^ [Y4: set_Pr1986765409at_nat,Z3: set_Pr1986765409at_nat] : ( Y4 = Z3 ) )
% 2.86/1.58 = ( ^ [A2: set_Pr1986765409at_nat,B5: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ A2 @ B5 )
% 2.86/1.58 & ( ord_le841296385at_nat @ B5 @ A2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_328_order__class_Oorder_Oeq__iff,axiom,
% 2.86/1.58 ( ( ^ [Y4: set_nat,Z3: set_nat] : ( Y4 = Z3 ) )
% 2.86/1.58 = ( ^ [A2: set_nat,B5: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 2.86/1.58 & ( ord_less_eq_set_nat @ B5 @ A2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_329_order__class_Oorder_Oeq__iff,axiom,
% 2.86/1.58 ( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
% 2.86/1.58 = ( ^ [A2: nat,B5: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ A2 @ B5 )
% 2.86/1.58 & ( ord_less_eq_nat @ B5 @ A2 ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_330_ord__eq__le__trans,axiom,
% 2.86/1.58 ! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( A3 = B4 )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ B4 @ C )
% 2.86/1.58 => ( ord_le841296385at_nat @ A3 @ C ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_331_ord__eq__le__trans,axiom,
% 2.86/1.58 ! [A3: set_nat,B4: set_nat,C: set_nat] :
% 2.86/1.58 ( ( A3 = B4 )
% 2.86/1.58 => ( ( ord_less_eq_set_nat @ B4 @ C )
% 2.86/1.58 => ( ord_less_eq_set_nat @ A3 @ C ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_332_ord__eq__le__trans,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,C: nat] :
% 2.86/1.58 ( ( A3 = B4 )
% 2.86/1.58 => ( ( ord_less_eq_nat @ B4 @ C )
% 2.86/1.58 => ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_333_ord__le__eq__trans,axiom,
% 2.86/1.58 ! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( B4 = C )
% 2.86/1.58 => ( ord_le841296385at_nat @ A3 @ C ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_334_ord__le__eq__trans,axiom,
% 2.86/1.58 ! [A3: set_nat,B4: set_nat,C: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( B4 = C )
% 2.86/1.58 => ( ord_less_eq_set_nat @ A3 @ C ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_335_ord__le__eq__trans,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat,C: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( B4 = C )
% 2.86/1.58 => ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_336_order__class_Oorder_Oantisym,axiom,
% 2.86/1.58 ! [A3: set_Pr1986765409at_nat,B4: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ B4 @ A3 )
% 2.86/1.58 => ( A3 = B4 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_337_order__class_Oorder_Oantisym,axiom,
% 2.86/1.58 ! [A3: set_nat,B4: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ord_less_eq_set_nat @ B4 @ A3 )
% 2.86/1.58 => ( A3 = B4 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_338_order__class_Oorder_Oantisym,axiom,
% 2.86/1.58 ! [A3: nat,B4: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ A3 @ B4 )
% 2.86/1.58 => ( ( ord_less_eq_nat @ B4 @ A3 )
% 2.86/1.58 => ( A3 = B4 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_339_order__trans,axiom,
% 2.86/1.58 ! [X3: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat,Z: set_Pr1986765409at_nat] :
% 2.86/1.58 ( ( ord_le841296385at_nat @ X3 @ Y3 )
% 2.86/1.58 => ( ( ord_le841296385at_nat @ Y3 @ Z )
% 2.86/1.58 => ( ord_le841296385at_nat @ X3 @ Z ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_340_order__trans,axiom,
% 2.86/1.58 ! [X3: set_nat,Y3: set_nat,Z: set_nat] :
% 2.86/1.58 ( ( ord_less_eq_set_nat @ X3 @ Y3 )
% 2.86/1.58 => ( ( ord_less_eq_set_nat @ Y3 @ Z )
% 2.86/1.58 => ( ord_less_eq_set_nat @ X3 @ Z ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_341_order__trans,axiom,
% 2.86/1.58 ! [X3: nat,Y3: nat,Z: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 2.86/1.58 => ( ( ord_less_eq_nat @ Y3 @ Z )
% 2.86/1.58 => ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_342_dual__order_Orefl,axiom,
% 2.86/1.58 ! [A3: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ A3 @ A3 ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_343_dual__order_Orefl,axiom,
% 2.86/1.58 ! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_344_dual__order_Orefl,axiom,
% 2.86/1.58 ! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_345_linorder__wlog,axiom,
% 2.86/1.58 ! [P3: nat > nat > $o,A3: nat,B4: nat] :
% 2.86/1.58 ( ! [A4: nat,B3: nat] :
% 2.86/1.58 ( ( ord_less_eq_nat @ A4 @ B3 )
% 2.86/1.58 => ( P3 @ A4 @ B3 ) )
% 2.86/1.58 => ( ! [A4: nat,B3: nat] :
% 2.86/1.58 ( ( P3 @ B3 @ A4 )
% 2.86/1.58 => ( P3 @ A4 @ B3 ) )
% 2.86/1.58 => ( P3 @ A3 @ B4 ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_346_rtc__isPath,axiom,
% 2.86/1.58 ! [U: nat,V: nat] :
% 2.86/1.58 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi1645084429cl_nat @ ( e_a @ c ) ) )
% 2.86/1.58 => ? [P2: list_P559422087at_nat] : ( isPath_a @ c @ U @ P2 @ V ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_347_isSPath__pathLE,axiom,
% 2.86/1.58 ! [S: nat,P4: list_P559422087at_nat,T3: nat] :
% 2.86/1.58 ( ( isPath_a @ c @ S @ P4 @ T3 )
% 2.86/1.58 => ? [P5: list_P559422087at_nat] : ( isSimplePath_a @ c @ S @ P5 @ T3 ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_348_connected__def,axiom,
% 2.86/1.58 ! [U: nat,V: nat] :
% 2.86/1.58 ( ( connected_a @ c @ U @ V )
% 2.86/1.58 = ( ? [P: list_P559422087at_nat] : ( isPath_a @ c @ U @ P @ V ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_349_shortestPath__is__path,axiom,
% 2.86/1.58 ! [U: nat,P4: list_P559422087at_nat,V: nat] :
% 2.86/1.58 ( ( isShortestPath_a @ c @ U @ P4 @ V )
% 2.86/1.58 => ( isPath_a @ c @ U @ P4 @ V ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_350_isPath__ex__edge1,axiom,
% 2.86/1.58 ! [U: nat,P4: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
% 2.86/1.58 ( ( isPath_a @ c @ U @ P4 @ V )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 2.86/1.58 => ( ( U1 != U )
% 2.86/1.58 => ? [U23: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ U23 @ U1 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_351_isPath__ex__edge2,axiom,
% 2.86/1.58 ! [U: nat,P4: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
% 2.86/1.58 ( ( isPath_a @ c @ U @ P4 @ V )
% 2.86/1.58 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P4 ) )
% 2.86/1.58 => ( ( V1 != V )
% 2.86/1.58 => ? [V23: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ V1 @ V23 ) @ ( set_Pr2131844118at_nat @ P4 ) ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_352_isPath__edgeset,axiom,
% 2.86/1.58 ! [U: nat,P4: list_P559422087at_nat,V: nat,E2: product_prod_nat_nat] :
% 2.86/1.58 ( ( isPath_a @ c @ U @ P4 @ V )
% 2.86/1.58 => ( ( member701585322at_nat @ E2 @ ( set_Pr2131844118at_nat @ P4 ) )
% 2.86/1.58 => ( member701585322at_nat @ E2 @ ( e_a @ c ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(fact_353_isPath__rtc,axiom,
% 2.86/1.58 ! [U: nat,P4: list_P559422087at_nat,V: nat] :
% 2.86/1.58 ( ( isPath_a @ c @ U @ P4 @ V )
% 2.86/1.58 => ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi1645084429cl_nat @ ( e_a @ c ) ) ) ) ).
% 2.86/1.58
% 2.86/1.58 thf(conj_0,conjecture,
% 2.86/1.58 finite772653738at_nat @ ( edmond475474835dges_a @ c @ s @ t ) ).
% 2.86/1.58 ------------------------
% 3.03/1.61 % (13134)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_12988 for (2999ds/2Mi)
% 3.03/1.61 % (13131)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_12988 for (2999ds/183Mi)
% 3.03/1.61 % (13136)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_12988 for (2999ds/275Mi)
% 3.03/1.61 % (13132)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_12988 for (2999ds/4Mi)
% 3.03/1.61 % (13133)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_12988 for (2999ds/27Mi)
% 3.03/1.61 % (13135)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_12988 for (2999ds/2Mi)
% 3.03/1.61 % (13137)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_12988 for (2999ds/18Mi)
% 3.03/1.61 % (13134)Instruction limit reached!
% 3.03/1.61 % (13134)------------------------------
% 3.03/1.61 % (13134)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.61 % (13134)Termination reason: Unknown
% 3.03/1.61 % (13134)Termination phase: shuffling
% 3.03/1.61
% 3.03/1.61 % (13134)Memory used [KB]: 1535
% 3.03/1.61 % (13134)Time elapsed: 0.003 s
% 3.03/1.61 % (13134)Instructions burned: 4 (million)
% 3.03/1.61 % (13134)------------------------------
% 3.03/1.61 % (13134)------------------------------
% 3.03/1.61 % (13135)Instruction limit reached!
% 3.03/1.61 % (13135)------------------------------
% 3.03/1.61 % (13135)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.61 % (13135)Termination reason: Unknown
% 3.03/1.61 % (13135)Termination phase: shuffling
% 3.03/1.61
% 3.03/1.61 % (13135)Memory used [KB]: 1535
% 3.03/1.61 % (13135)Time elapsed: 0.003 s
% 3.03/1.61 % (13135)Instructions burned: 4 (million)
% 3.03/1.61 % (13135)------------------------------
% 3.03/1.61 % (13135)------------------------------
% 3.03/1.61 % (13132)Instruction limit reached!
% 3.03/1.61 % (13132)------------------------------
% 3.03/1.61 % (13132)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.61 % (13132)Termination reason: Unknown
% 3.03/1.61 % (13132)Termination phase: shuffling
% 3.03/1.61
% 3.03/1.61 % (13132)Memory used [KB]: 1535
% 3.03/1.61 % (13132)Time elapsed: 0.004 s
% 3.03/1.61 % (13132)Instructions burned: 5 (million)
% 3.03/1.61 % (13132)------------------------------
% 3.03/1.61 % (13132)------------------------------
% 3.03/1.62 % (13137)Instruction limit reached!
% 3.03/1.62 % (13137)------------------------------
% 3.03/1.62 % (13137)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.62 % (13137)Termination reason: Unknown
% 3.03/1.62 % (13137)Termination phase: shuffling
% 3.03/1.62
% 3.03/1.62 % (13137)Memory used [KB]: 1791
% 3.03/1.62 % (13137)Time elapsed: 0.008 s
% 3.03/1.62 % (13137)Instructions burned: 20 (million)
% 3.03/1.62 % (13137)------------------------------
% 3.03/1.62 % (13137)------------------------------
% 3.03/1.62 % (13133)Instruction limit reached!
% 3.03/1.62 % (13133)------------------------------
% 3.03/1.62 % (13133)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.62 % (13133)Termination reason: Unknown
% 3.03/1.62 % (13133)Termination phase: shuffling
% 3.03/1.62
% 3.03/1.62 % (13133)Memory used [KB]: 2046
% 3.03/1.62 % (13133)Time elapsed: 0.011 s
% 3.03/1.62 % (13133)Instructions burned: 29 (million)
% 3.03/1.62 % (13133)------------------------------
% 3.03/1.62 % (13133)------------------------------
% 3.03/1.62 % (13139)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_12988 for (2999ds/3Mi)
% 3.03/1.62 % (13140)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_12988 for (2999ds/37Mi)
% 3.03/1.62 % (13139)Instruction limit reached!
% 3.03/1.62 % (13139)------------------------------
% 3.03/1.62 % (13139)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.62 % (13139)Termination reason: Unknown
% 3.03/1.62 % (13139)Termination phase: shuffling
% 3.03/1.62
% 3.03/1.62 % (13139)Memory used [KB]: 1407
% 3.03/1.62 % (13139)Time elapsed: 0.002 s
% 3.03/1.62 % (13139)Instructions burned: 3 (million)
% 3.03/1.62 % (13139)------------------------------
% 3.03/1.62 % (13139)------------------------------
% 3.03/1.63 % (13144)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_12988 for (2999ds/3Mi)
% 3.03/1.63 % (13147)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_12988 for (2999ds/1041Mi)
% 3.03/1.63 % (13144)Instruction limit reached!
% 3.03/1.63 % (13144)------------------------------
% 3.03/1.63 % (13144)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.63 % (13144)Termination reason: Unknown
% 3.03/1.63 % (13144)Termination phase: shuffling
% 3.03/1.63
% 3.03/1.63 % (13144)Memory used [KB]: 1535
% 3.03/1.63 % (13144)Time elapsed: 0.003 s
% 3.03/1.63 % (13144)Instructions burned: 3 (million)
% 3.03/1.63 % (13144)------------------------------
% 3.03/1.63 % (13144)------------------------------
% 3.03/1.63 % (13141)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_12988 for (2999ds/15Mi)
% 3.03/1.63 % (13140)Instruction limit reached!
% 3.03/1.63 % (13140)------------------------------
% 3.03/1.63 % (13140)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.63 % (13140)Termination reason: Unknown
% 3.03/1.63 % (13140)Termination phase: shuffling
% 3.03/1.63
% 3.03/1.63 % (13140)Memory used [KB]: 2174
% 3.03/1.63 % (13140)Time elapsed: 0.011 s
% 3.03/1.63 % (13140)Instructions burned: 37 (million)
% 3.03/1.63 % (13140)------------------------------
% 3.03/1.63 % (13140)------------------------------
% 3.03/1.63 % (13148)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_12988 for (2999ds/7Mi)
% 3.03/1.63 % (13136)First to succeed.
% 3.03/1.63 % (13148)Instruction limit reached!
% 3.03/1.63 % (13148)------------------------------
% 3.03/1.63 % (13148)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.63 % (13148)Termination reason: Unknown
% 3.03/1.63 % (13148)Termination phase: shuffling
% 3.03/1.63
% 3.03/1.63 % (13148)Memory used [KB]: 1663
% 3.03/1.63 % (13148)Time elapsed: 0.004 s
% 3.03/1.63 % (13148)Instructions burned: 9 (million)
% 3.03/1.63 % (13148)------------------------------
% 3.03/1.63 % (13148)------------------------------
% 3.03/1.63 % (13136)Refutation found. Thanks to Tanya!
% 3.03/1.63 % SZS status Theorem for DTF2THF_12988
% 3.03/1.63 % SZS output start Proof for DTF2THF_12988
% 3.03/1.63 thf(type_def_5, type, product_prod_nat_nat: $tType).
% 3.03/1.63 thf(type_def_6, type, a: $tType).
% 3.03/1.63 thf(type_def_8, type, nat: $tType).
% 3.03/1.63 thf(type_def_9, type, set_Pr1986765409at_nat: $tType).
% 3.03/1.63 thf(type_def_10, type, set_nat: $tType).
% 3.03/1.63 thf(type_def_11, type, set_Pr1746169692at_nat: $tType).
% 3.03/1.63 thf(type_def_12, type, set_Pr1287749686at_nat: $tType).
% 3.03/1.63 thf(type_def_13, type, set_Pr1490359111at_nat: $tType).
% 3.03/1.63 thf(type_def_14, type, set_set_nat: $tType).
% 3.03/1.63 thf(type_def_15, type, set_se1612935105at_nat: $tType).
% 3.03/1.63 thf(type_def_16, type, list_P559422087at_nat: $tType).
% 3.03/1.63 thf(type_def_17, type, list_nat: $tType).
% 3.03/1.63 thf(type_def_18, type, produc1695820582at_nat: $tType).
% 3.03/1.63 thf(type_def_19, type, produc1271302400at_nat: $tType).
% 3.03/1.63 thf(type_def_20, type, produc842455143at_nat: $tType).
% 3.03/1.63 thf(func_def_0, type, set_Pr1490359111at_nat: $tType).
% 3.03/1.63 thf(func_def_1, type, produc842455143at_nat: $tType).
% 3.03/1.63 thf(func_def_2, type, set_Pr1287749686at_nat: $tType).
% 3.03/1.63 thf(func_def_3, type, set_Pr1746169692at_nat: $tType).
% 3.03/1.63 thf(func_def_4, type, produc1271302400at_nat: $tType).
% 3.03/1.63 thf(func_def_5, type, produc1695820582at_nat: $tType).
% 3.03/1.63 thf(func_def_6, type, set_se1612935105at_nat: $tType).
% 3.03/1.63 thf(func_def_7, type, list_P559422087at_nat: $tType).
% 3.03/1.63 thf(func_def_8, type, set_Pr1986765409at_nat: $tType).
% 3.03/1.63 thf(func_def_9, type, product_prod_nat_nat: $tType).
% 3.03/1.63 thf(func_def_10, type, set_set_nat: $tType).
% 3.03/1.63 thf(func_def_11, type, list_nat: $tType).
% 3.03/1.63 thf(func_def_12, type, set_nat: $tType).
% 3.03/1.63 thf(func_def_13, type, nat: $tType).
% 3.03/1.63 thf(func_def_14, type, a: $tType).
% 3.03/1.63 thf(func_def_15, type, edmond1517640972ysis_a: (product_prod_nat_nat > a) > $o).
% 3.03/1.63 thf(func_def_16, type, edmond475474835dges_a: (product_prod_nat_nat > a) > nat > nat > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_17, type, finite_finite_nat: set_nat > $o).
% 3.03/1.63 thf(func_def_18, type, finite772653738at_nat: set_Pr1986765409at_nat > $o).
% 3.03/1.63 thf(func_def_19, type, finite277291581at_nat: set_Pr1746169692at_nat > $o).
% 3.03/1.63 thf(func_def_20, type, finite2000257047at_nat: set_Pr1287749686at_nat > $o).
% 3.03/1.63 thf(func_def_21, type, finite48957584at_nat: set_Pr1490359111at_nat > $o).
% 3.03/1.63 thf(func_def_22, type, finite2012248349et_nat: set_set_nat > $o).
% 3.03/1.63 thf(func_def_23, type, finite1457549322at_nat: set_se1612935105at_nat > $o).
% 3.03/1.63 thf(func_def_24, type, finite_Graph_a: (product_prod_nat_nat > a) > $o).
% 3.03/1.63 thf(func_def_25, type, e_a: (product_prod_nat_nat > a) > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_26, type, v_a: (product_prod_nat_nat > a) > set_nat).
% 3.03/1.63 thf(func_def_27, type, adjacent_nodes_a: (product_prod_nat_nat > a) > nat > set_nat).
% 3.03/1.63 thf(func_def_28, type, connected_a: (product_prod_nat_nat > a) > nat > nat > $o).
% 3.03/1.63 thf(func_def_29, type, incoming_a: (product_prod_nat_nat > a) > nat > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_30, type, incoming_a2: (product_prod_nat_nat > a) > set_nat > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_31, type, isPath_a: (product_prod_nat_nat > a) > nat > list_P559422087at_nat > nat > $o).
% 3.03/1.63 thf(func_def_32, type, isShortestPath_a: (product_prod_nat_nat > a) > nat > list_P559422087at_nat > nat > $o).
% 3.03/1.63 thf(func_def_33, type, isSimplePath_a: (product_prod_nat_nat > a) > nat > list_P559422087at_nat > nat > $o).
% 3.03/1.63 thf(func_def_34, type, outgoing_a: (product_prod_nat_nat > a) > nat > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_35, type, outgoing_a2: (product_prod_nat_nat > a) > set_nat > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_36, type, reachableNodes_a: (product_prod_nat_nat > a) > nat > set_nat).
% 3.03/1.63 thf(func_def_37, type, set_nat2: list_nat > set_nat).
% 3.03/1.63 thf(func_def_38, type, set_Pr2131844118at_nat: list_P559422087at_nat > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_39, type, ord_le1578155910_nat_o: (nat > nat > $o) > (nat > nat > $o) > $o).
% 3.03/1.63 thf(func_def_40, type, ord_less_eq_nat_o: (nat > $o) > (nat > $o) > $o).
% 3.03/1.63 thf(func_def_41, type, ord_le1039616028_nat_o: (product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > $o).
% 3.03/1.63 thf(func_def_42, type, ord_less_eq_nat: nat > nat > $o).
% 3.03/1.63 thf(func_def_43, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 3.03/1.63 thf(func_def_44, type, ord_le841296385at_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > $o).
% 3.03/1.63 thf(func_def_45, type, ord_le148299708at_nat: set_Pr1746169692at_nat > set_Pr1746169692at_nat > $o).
% 3.03/1.63 thf(func_def_46, type, ord_le1837363350at_nat: set_Pr1287749686at_nat > set_Pr1287749686at_nat > $o).
% 3.03/1.63 thf(func_def_47, type, ord_le465611495at_nat: set_Pr1490359111at_nat > set_Pr1490359111at_nat > $o).
% 3.03/1.63 thf(func_def_48, type, product_Pair_nat_nat: nat > nat > product_prod_nat_nat).
% 3.03/1.63 thf(func_def_49, type, produc1933845336at_nat: nat > product_prod_nat_nat > produc1695820582at_nat).
% 3.03/1.63 thf(func_def_50, type, produc947540346at_nat: product_prod_nat_nat > nat > produc1271302400at_nat).
% 3.03/1.63 thf(func_def_51, type, produc1168807639at_nat: product_prod_nat_nat > product_prod_nat_nat > produc842455143at_nat).
% 3.03/1.63 thf(func_def_52, type, produc45129834at_nat: set_nat > (nat > set_nat) > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_53, type, produc894163943at_nat: set_nat > (nat > set_Pr1986765409at_nat) > set_Pr1746169692at_nat).
% 3.03/1.63 thf(func_def_54, type, produc2055342601at_nat: set_Pr1986765409at_nat > (product_prod_nat_nat > set_nat) > set_Pr1287749686at_nat).
% 3.03/1.63 thf(func_def_55, type, produc2056081288at_nat: set_Pr1986765409at_nat > (product_prod_nat_nat > set_Pr1986765409at_nat) > set_Pr1490359111at_nat).
% 3.03/1.63 thf(func_def_56, type, product_swap_nat_nat: product_prod_nat_nat > product_prod_nat_nat).
% 3.03/1.63 thf(func_def_57, type, domain_nat_nat: set_Pr1986765409at_nat > set_nat).
% 3.03/1.63 thf(func_def_58, type, domain37934156at_nat: set_Pr1490359111at_nat > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_59, type, image_nat_nat: set_Pr1986765409at_nat > set_nat > set_nat).
% 3.03/1.63 thf(func_def_60, type, image_981520924at_nat: set_Pr1746169692at_nat > set_nat > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_61, type, image_2142699582at_nat: set_Pr1287749686at_nat > set_Pr1986765409at_nat > set_nat).
% 3.03/1.63 thf(func_def_62, type, image_1655366419at_nat: set_Pr1490359111at_nat > set_Pr1986765409at_nat > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_63, type, range_nat_nat: set_Pr1986765409at_nat > set_nat).
% 3.03/1.63 thf(func_def_64, type, collect_nat: (nat > $o) > set_nat).
% 3.03/1.63 thf(func_def_65, type, collec7649004at_nat: (product_prod_nat_nat > $o) > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_66, type, collect_set_nat: (set_nat > $o) > set_set_nat).
% 3.03/1.63 thf(func_def_67, type, collec1606769740at_nat: (set_Pr1986765409at_nat > $o) > set_se1612935105at_nat).
% 3.03/1.63 thf(func_def_68, type, transi1645084429cl_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_69, type, transi1484096900at_nat: set_Pr1490359111at_nat > set_Pr1490359111at_nat).
% 3.03/1.63 thf(func_def_70, type, member_nat: nat > set_nat > $o).
% 3.03/1.63 thf(func_def_71, type, member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o).
% 3.03/1.63 thf(func_def_72, type, member1293241661at_nat: produc1695820582at_nat > set_Pr1746169692at_nat > $o).
% 3.03/1.63 thf(func_def_73, type, member868723479at_nat: produc1271302400at_nat > set_Pr1287749686at_nat > $o).
% 3.03/1.63 thf(func_def_74, type, member2027625872at_nat: produc842455143at_nat > set_Pr1490359111at_nat > $o).
% 3.03/1.63 thf(func_def_75, type, member_set_nat: set_nat > set_set_nat > $o).
% 3.03/1.63 thf(func_def_76, type, member298845450at_nat: set_Pr1986765409at_nat > set_se1612935105at_nat > $o).
% 3.03/1.63 thf(func_def_77, type, c: product_prod_nat_nat > a).
% 3.03/1.63 thf(func_def_78, type, s: nat).
% 3.03/1.63 thf(func_def_79, type, t: nat).
% 3.03/1.63 thf(func_def_94, type, sK0: (set_Pr1986765409at_nat > set_Pr1986765409at_nat) > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_95, type, sK1: (set_Pr1986765409at_nat > set_Pr1986765409at_nat) > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_96, type, sK2: (product_prod_nat_nat > product_prod_nat_nat > $o) > set_Pr1986765409at_nat > set_Pr1986765409at_nat > product_prod_nat_nat).
% 3.03/1.63 thf(func_def_97, type, sK3: (product_prod_nat_nat > product_prod_nat_nat > $o) > set_Pr1986765409at_nat > set_Pr1986765409at_nat > product_prod_nat_nat).
% 3.03/1.63 thf(func_def_98, type, sK4: (set_Pr1986765409at_nat > set_Pr1986765409at_nat) > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_99, type, sK5: (set_Pr1986765409at_nat > set_Pr1986765409at_nat) > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_100, type, sK6: nat > (product_prod_nat_nat > a) > nat > list_P559422087at_nat).
% 3.03/1.63 thf(func_def_101, type, sK7: (product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > product_prod_nat_nat).
% 3.03/1.63 thf(func_def_102, type, sK8: nat > nat > list_P559422087at_nat).
% 3.03/1.63 thf(func_def_103, type, sK9: (set_Pr1986765409at_nat > set_Pr1986765409at_nat) > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_104, type, sK10: (set_Pr1986765409at_nat > set_Pr1986765409at_nat) > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_105, type, sK11: (product_prod_nat_nat > nat > $o) > set_Pr1986765409at_nat > set_nat > nat).
% 3.03/1.63 thf(func_def_106, type, sK12: (product_prod_nat_nat > nat > $o) > set_Pr1986765409at_nat > set_nat > product_prod_nat_nat).
% 3.03/1.63 thf(func_def_107, type, sK13: set_Pr1986765409at_nat > list_P559422087at_nat).
% 3.03/1.63 thf(func_def_108, type, sK14: (set_Pr1986765409at_nat > set_Pr1986765409at_nat) > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_109, type, sK15: (set_Pr1986765409at_nat > set_Pr1986765409at_nat) > set_Pr1986765409at_nat).
% 3.03/1.63 thf(func_def_110, type, sK16: (product_prod_nat_nat > $o) > product_prod_nat_nat).
% 3.03/1.63 thf(func_def_111, type, sK17: (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > product_prod_nat_nat).
% 3.03/1.63 thf(func_def_112, type, sK18: set_Pr1986765409at_nat > set_Pr1986765409at_nat > nat).
% 3.03/1.63 thf(func_def_113, type, sK19: set_Pr1986765409at_nat > set_Pr1986765409at_nat > nat).
% 3.03/1.63 thf(func_def_114, type, sK20: list_P559422087at_nat > set_Pr1986765409at_nat > product_prod_nat_nat).
% 3.03/1.63 thf(func_def_116, type, ph22: !>[X0: $tType]:(X0)).
% 3.03/1.63 thf(f1195,plain,(
% 3.03/1.63 $false),
% 3.03/1.63 inference(subsumption_resolution,[status(thm)],[f1194,f1172])).
% 3.03/1.63 thf(f1172,plain,(
% 3.03/1.63 ((finite772653738at_nat @ (e_a @ c)) = $true)),
% 3.03/1.63 inference(cnf_transformation,[status(thm)],[f741])).
% 3.03/1.63 thf(f741,plain,(
% 3.03/1.63 ((finite772653738at_nat @ (e_a @ c)) = $true)),
% 3.03/1.63 inference(fool_elimination,[status(thm)],[f740])).
% 3.03/1.63 thf(f740,plain,(
% 3.03/1.63 (finite772653738at_nat @ (e_a @ c))),
% 3.03/1.63 inference(rectify,[status(thm)],[f4])).
% 3.03/1.63 thf(f4,axiom,(
% 3.03/1.63 (finite772653738at_nat @ (e_a @ c))),
% 3.03/1.63 file('/export/starexec/sandbox2/tmp/tmp.tqDOf96qQm/DTF2THF_12988.p',fact_3_finite__E)).
% 3.03/1.63 thf(f1194,plain,(
% 3.03/1.63 ((finite772653738at_nat @ (e_a @ c)) != $true)),
% 3.03/1.63 inference(trivial_inequality_removal,[status(thm)],[f1193])).
% 3.03/1.63 thf(f1193,plain,(
% 3.03/1.63 ($true != $true) | ((finite772653738at_nat @ (e_a @ c)) != $true)),
% 3.03/1.63 inference(superposition,[status(thm)],[f1165,f1122])).
% 3.03/1.63 thf(f1122,plain,(
% 3.03/1.63 ((finite772653738at_nat @ (edmond475474835dges_a @ c @ s @ t)) = $true) | ((finite772653738at_nat @ (e_a @ c)) != $true)),
% 3.03/1.63 inference(cnf_transformation,[status(thm)],[f1041])).
% 3.03/1.63 thf(f1041,plain,(
% 3.03/1.63 ((finite772653738at_nat @ (e_a @ c)) != $true) | ((finite772653738at_nat @ (edmond475474835dges_a @ c @ s @ t)) = $true)),
% 3.03/1.63 inference(ennf_transformation,[status(thm)],[f791])).
% 3.03/1.63 thf(f791,plain,(
% 3.03/1.63 ((finite772653738at_nat @ (e_a @ c)) = $true) => ((finite772653738at_nat @ (edmond475474835dges_a @ c @ s @ t)) = $true)),
% 3.03/1.63 inference(fool_elimination,[status(thm)],[f790])).
% 3.03/1.63 thf(f790,plain,(
% 3.03/1.63 (finite772653738at_nat @ (e_a @ c)) => (finite772653738at_nat @ (edmond475474835dges_a @ c @ s @ t))),
% 3.03/1.63 inference(rectify,[status(thm)],[f2])).
% 3.03/1.63 thf(f2,axiom,(
% 3.03/1.63 (finite772653738at_nat @ (e_a @ c)) => (finite772653738at_nat @ (edmond475474835dges_a @ c @ s @ t))),
% 3.03/1.63 file('/export/starexec/sandbox2/tmp/tmp.tqDOf96qQm/DTF2THF_12988.p',fact_1__092_060open_062finite_AE_A_092_060Longrightarrow_062_Afinite_AspEdges_092_060close_062)).
% 3.03/1.63 thf(f1165,plain,(
% 3.03/1.63 ((finite772653738at_nat @ (edmond475474835dges_a @ c @ s @ t)) != $true)),
% 3.03/1.63 inference(cnf_transformation,[status(thm)],[f1018])).
% 3.03/1.63 thf(f1018,plain,(
% 3.03/1.63 ((finite772653738at_nat @ (edmond475474835dges_a @ c @ s @ t)) != $true)),
% 3.03/1.63 inference(flattening,[status(thm)],[f737])).
% 3.03/1.63 thf(f737,plain,(
% 3.03/1.63 ~((finite772653738at_nat @ (edmond475474835dges_a @ c @ s @ t)) = $true)),
% 3.03/1.63 inference(fool_elimination,[status(thm)],[f736])).
% 3.03/1.63 thf(f736,plain,(
% 3.03/1.63 ~(finite772653738at_nat @ (edmond475474835dges_a @ c @ s @ t))),
% 3.03/1.63 inference(rectify,[status(thm)],[f356])).
% 3.03/1.63 thf(f356,negated_conjecture,(
% 3.03/1.63 ~(finite772653738at_nat @ (edmond475474835dges_a @ c @ s @ t))),
% 3.03/1.63 inference(negated_conjecture,[status(cth)],[f355])).
% 3.03/1.63 thf(f355,conjecture,(
% 3.03/1.63 (finite772653738at_nat @ (edmond475474835dges_a @ c @ s @ t))),
% 3.03/1.63 file('/export/starexec/sandbox2/tmp/tmp.tqDOf96qQm/DTF2THF_12988.p',conj_0)).
% 3.03/1.63 % SZS output end Proof for DTF2THF_12988
% 3.03/1.63 % (13136)------------------------------
% 3.03/1.63 % (13136)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.63 % (13136)Termination reason: Refutation
% 3.03/1.63
% 3.03/1.63 % (13136)Memory used [KB]: 6908
% 3.03/1.63 % (13136)Time elapsed: 0.025 s
% 3.03/1.63 % (13136)Instructions burned: 76 (million)
% 3.03/1.63 % (13136)------------------------------
% 3.03/1.63 % (13136)------------------------------
% 3.03/1.63 % (13130)Success in time 0.04 s
%------------------------------------------------------------------------------