:: Kuratowski - Zorn Lemma
:: by Wojciech A. Trybulec and Grzegorz Bancerek
::
:: Received September 19, 1989
:: Copyright (c) 1990 Association of Mizar Users


begin

definition
attr c1 is strict ;
struct RelStr -> 1-sorted ;
aggr RelStr(# carrier, InternalRel #) -> RelStr ;
sel InternalRel c1 -> Relation of the carrier of c1;
end;

registration
let X be non empty set ;
let R be Relation of X;
cluster RelStr(# X,R #) -> non empty ;
coherence
not RelStr(# X,R #) is empty ;
end;

definition
let A be RelStr ;
canceled;
canceled;
attr A is total means :Def3: :: ORDERS_2:def 3
 the InternalRel of A is total ;
attr A is reflexive means :Def4: :: ORDERS_2:def 4
 the InternalRel of A is_reflexive_in the carrier of A;
attr A is transitive means :Def5: :: ORDERS_2:def 5
 the InternalRel of A is_transitive_in the carrier of A;
attr A is antisymmetric means :Def6: :: ORDERS_2:def 6
 the InternalRel of A is_antisymmetric_in the carrier of A;
end;

:: deftheorem ORDERS_2:def 1 :
canceled;

:: deftheorem ORDERS_2:def 2 :
canceled;

:: deftheorem Def3 defines total ORDERS_2:def 3 :
for A being RelStr holds
( A is total iff the InternalRel of A is total );

:: deftheorem Def4 defines reflexive ORDERS_2:def 4 :
for A being RelStr holds
( A is reflexive iff the InternalRel of A is_reflexive_in the carrier of A );

:: deftheorem Def5 defines transitive ORDERS_2:def 5 :
for A being RelStr holds
( A is transitive iff the InternalRel of A is_transitive_in the carrier of A );

:: deftheorem Def6 defines antisymmetric ORDERS_2:def 6 :
for A being RelStr holds
( A is antisymmetric iff the InternalRel of A is_antisymmetric_in the carrier of A );

registration
cluster non empty strict total reflexive transitive antisymmetric RelStr ;
existence
ex b1 being RelStr st
( not b1 is empty & b1 is reflexive & b1 is transitive & b1 is antisymmetric & b1 is strict & b1 is total )
proof end;
end;

registration
cluster reflexive -> total RelStr ;
coherence
for b1 being RelStr st b1 is reflexive holds
b1 is total
proof end;
end;

definition
mode Poset is reflexive transitive antisymmetric RelStr ;
end;

registration
let A be total RelStr ;
cluster the InternalRel of A -> total ;
coherence
the InternalRel of A is total by Def3;
end;

registration
let A be reflexive RelStr ;
cluster the InternalRel of A -> reflexive ;
coherence
the InternalRel of A is reflexive
proof end;
end;

registration
let A be total antisymmetric RelStr ;
cluster the InternalRel of A -> antisymmetric ;
coherence
the InternalRel of A is antisymmetric
proof end;
end;

registration
let A be total transitive RelStr ;
cluster the InternalRel of A -> transitive ;
coherence
the InternalRel of A is transitive
proof end;
end;

registration
let X be set ;
let O be reflexive total Relation of X;
cluster RelStr(# X,O #) -> reflexive ;
coherence
RelStr(# X,O #) is reflexive
proof end;
end;

registration
let X be set ;
let O be transitive total Relation of X;
cluster RelStr(# X,O #) -> transitive ;
coherence
RelStr(# X,O #) is transitive
proof end;
end;

registration
let X be set ;
let O be antisymmetric total Relation of X;
cluster RelStr(# X,O #) -> antisymmetric ;
coherence
RelStr(# X,O #) is antisymmetric
proof end;
end;

Lm1: for x being set
for A being non empty Poset
for S being Subset of A st x in S holds
x is Element of A
;

definition
let A be RelStr ;
let a1, a2 be Element of A;
canceled;
canceled;
pred a1 <= a2 means :Def9: :: ORDERS_2:def 9
[a1,a2] in the InternalRel of A;
end;

:: deftheorem ORDERS_2:def 7 :
canceled;

:: deftheorem ORDERS_2:def 8 :
canceled;

:: deftheorem Def9 defines <= ORDERS_2:def 9 :
for A being RelStr
for a1, a2 being Element of A holds
( a1 <= a2 iff [a1,a2] in the InternalRel of A );

notation
let A be RelStr ;
let a1, a2 be Element of A;
synonym a2 >= a1 for a1 <= a2;
end;

definition
let A be RelStr ;
let a1, a2 be Element of A;
pred a1 < a2 means :Def10: :: ORDERS_2:def 10
( a1 <= a2 & a1 <> a2 );
irreflexivity
for a1 being Element of A holds
( not a1 <= a1 or not a1 <> a1 ) ;
end;

:: deftheorem Def10 defines < ORDERS_2:def 10 :
for A being RelStr
for a1, a2 being Element of A holds
( a1 < a2 iff ( a1 <= a2 & a1 <> a2 ) );

notation
let A be RelStr ;
let a1, a2 be Element of A;
synonym a2 > a1 for a1 < a2;
end;

theorem :: ORDERS_2:1
canceled;

theorem :: ORDERS_2:2
canceled;

theorem :: ORDERS_2:3
canceled;

theorem :: ORDERS_2:4
canceled;

theorem :: ORDERS_2:5
canceled;

theorem :: ORDERS_2:6
canceled;

theorem :: ORDERS_2:7
canceled;

theorem :: ORDERS_2:8
canceled;

theorem :: ORDERS_2:9
canceled;

theorem :: ORDERS_2:10
canceled;

theorem :: ORDERS_2:11
canceled;

theorem :: ORDERS_2:12
canceled;

theorem :: ORDERS_2:13
canceled;

theorem :: ORDERS_2:14
canceled;

theorem :: ORDERS_2:15
canceled;

theorem :: ORDERS_2:16
canceled;

theorem :: ORDERS_2:17
canceled;

theorem :: ORDERS_2:18
canceled;

theorem :: ORDERS_2:19
canceled;

theorem :: ORDERS_2:20
canceled;

theorem :: ORDERS_2:21
canceled;

theorem :: ORDERS_2:22
canceled;

theorem :: ORDERS_2:23
canceled;

theorem Th24: :: ORDERS_2:24
for A being non empty reflexive RelStr
for a being Element of A holds a <= a
proof end;

definition
let A be non empty reflexive RelStr ;
let a1, a2 be Element of A;
:: original: <=
redefine pred a1 <= a2;
reflexivity
for a1 being Element of A holds a1 <= a1 by Th24;
end;

theorem Th25: :: ORDERS_2:25
for A being antisymmetric RelStr
for a1, a2 being Element of A st a1 <= a2 & a2 <= a1 holds
a1 = a2
proof end;

theorem Th26: :: ORDERS_2:26
for A being transitive RelStr
for a1, a2, a3 being Element of A st a1 <= a2 & a2 <= a3 holds
a1 <= a3
proof end;

theorem :: ORDERS_2:27
canceled;

theorem Th28: :: ORDERS_2:28
for A being antisymmetric RelStr
for a1, a2 being Element of A holds
( not a1 < a2 or not a2 < a1 )
proof end;

theorem Th29: :: ORDERS_2:29
for A being transitive antisymmetric RelStr
for a1, a2, a3 being Element of A st a1 < a2 & a2 < a3 holds
a1 < a3
proof end;

theorem Th30: :: ORDERS_2:30
for A being antisymmetric RelStr
for a1, a2 being Element of A st a1 <= a2 holds
not a2 < a1
proof end;

theorem :: ORDERS_2:31
canceled;

theorem Th32: :: ORDERS_2:32
for A being transitive antisymmetric RelStr
for a1, a2, a3 being Element of A st ( ( a1 < a2 & a2 <= a3 ) or ( a1 <= a2 & a2 < a3 ) ) holds
a1 < a3
proof end;

definition
let A be RelStr ;
let IT be Subset of A;
attr IT is strongly_connected means :Def11: :: ORDERS_2:def 11
 the InternalRel of A is_strongly_connected_in IT;
end;

:: deftheorem Def11 defines strongly_connected ORDERS_2:def 11 :
for A being RelStr
for IT being Subset of A holds
( IT is strongly_connected iff the InternalRel of A is_strongly_connected_in IT );

registration
let A be RelStr ;
cluster empty -> strongly_connected Element of bool the carrier of A;
coherence
for b1 being Subset of A st b1 is empty holds
b1 is strongly_connected
proof end;
end;

registration
let A be RelStr ;
cluster strongly_connected Element of bool the carrier of A;
existence
ex b1 being Subset of A st b1 is strongly_connected
proof end;
end;

definition
let A be RelStr ;
mode Chain of A is strongly_connected Subset of A;
end;

theorem :: ORDERS_2:33
canceled;

theorem :: ORDERS_2:34
canceled;

theorem Th35: :: ORDERS_2:35
for A being non empty reflexive RelStr
for a being Element of A holds {a} is Chain of A
proof end;

theorem Th36: :: ORDERS_2:36
for A being non empty reflexive RelStr
for a1, a2 being Element of A holds
( {a1,a2} is Chain of A iff ( a1 <= a2 or a2 <= a1 ) )
proof end;

theorem :: ORDERS_2:37
for A being RelStr
for C being Chain of A
for S being Subset of A st S c= C holds
S is Chain of A
proof end;

theorem Th38: :: ORDERS_2:38
for A being reflexive RelStr
for a1, a2 being Element of A holds
( ( a1 <= a2 or a2 <= a1 ) iff ex C being Chain of A st
( a1 in C & a2 in C ) )
proof end;

theorem Th39: :: ORDERS_2:39
for A being reflexive antisymmetric RelStr
for a1, a2 being Element of A holds
( ex C being Chain of A st
( a1 in C & a2 in C ) iff ( a1 < a2 iff not a2 <= a1 ) )
proof end;

theorem Th40: :: ORDERS_2:40
for A being RelStr
for T being Subset of A st the InternalRel of A well_orders T holds
T is Chain of A
proof end;

definition
let A be non empty Poset;
let S be Subset of A;
func UpperCone S -> Subset of A equals :: ORDERS_2:def 12
 { a1 where a1 is Element of A : for a2 being Element of A st a2 in S holds
a2 < a1 } ;
coherence
{ a1 where a1 is Element of A : for a2 being Element of A st a2 in S holds
a2 < a1 } is Subset of A
proof end;
end;

:: deftheorem defines UpperCone ORDERS_2:def 12 :
for A being non empty Poset
for S being Subset of A holds UpperCone S = { a1 where a1 is Element of A : for a2 being Element of A st a2 in S holds
a2 < a1 } ;

definition
let A be non empty Poset;
let S be Subset of A;
func LowerCone S -> Subset of A equals :: ORDERS_2:def 13
 { a1 where a1 is Element of A : for a2 being Element of A st a2 in S holds
a1 < a2 } ;
coherence
{ a1 where a1 is Element of A : for a2 being Element of A st a2 in S holds
a1 < a2 } is Subset of A
proof end;
end;

:: deftheorem defines LowerCone ORDERS_2:def 13 :
for A being non empty Poset
for S being Subset of A holds LowerCone S = { a1 where a1 is Element of A : for a2 being Element of A st a2 in S holds
a1 < a2 } ;

theorem :: ORDERS_2:41
canceled;

theorem :: ORDERS_2:42
canceled;

theorem Th43: :: ORDERS_2:43
for A being non empty Poset holds UpperCone ({} A) = the carrier of A
proof end;

theorem :: ORDERS_2:44
for A being non empty Poset holds UpperCone ([#] A) = {}
proof end;

theorem :: ORDERS_2:45
for A being non empty Poset holds LowerCone ({} A) = the carrier of A
proof end;

theorem :: ORDERS_2:46
for A being non empty Poset holds LowerCone ([#] A) = {}
proof end;

theorem Th47: :: ORDERS_2:47
for A being non empty Poset
for a being Element of A
for S being Subset of A st a in S holds
not a in UpperCone S
proof end;

theorem :: ORDERS_2:48
for A being non empty Poset
for a being Element of A holds not a in UpperCone {a}
proof end;

theorem Th49: :: ORDERS_2:49
for A being non empty Poset
for a being Element of A
for S being Subset of A st a in S holds
not a in LowerCone S
proof end;

theorem Th50: :: ORDERS_2:50
for A being non empty Poset
for a being Element of A holds not a in LowerCone {a}
proof end;

theorem :: ORDERS_2:51
for A being non empty Poset
for c, a being Element of A holds
( c < a iff a in UpperCone {c} )
proof end;

theorem Th52: :: ORDERS_2:52
for A being non empty Poset
for a, c being Element of A holds
( a < c iff a in LowerCone {c} )
proof end;

definition
let A be non empty Poset;
let S be Subset of A;
let a be Element of A;
func InitSegm (S,a) -> Subset of A equals :: ORDERS_2:def 14
(LowerCone {a}) /\ S;
correctness
coherence
(LowerCone {a}) /\ S is Subset of A;
;
end;

:: deftheorem defines InitSegm ORDERS_2:def 14 :
for A being non empty Poset
for S being Subset of A
for a being Element of A holds InitSegm (S,a) = (LowerCone {a}) /\ S;

definition
let A be non empty Poset;
let S be Subset of A;
mode Initial_Segm of S -> Subset of A means :Def15: :: ORDERS_2:def 15
 ex a being Element of A st
( a in S & it = InitSegm (S,a) ) if S <> {}
otherwise it = {} ;
existence
( ( S <> {} implies ex b1 being Subset of A ex a being Element of A st
( a in S & b1 = InitSegm (S,a) ) ) & ( not S <> {} implies ex b1 being Subset of A st b1 = {} ) )
proof end;
consistency
for b1 being Subset of A holds verum ;
end;

:: deftheorem Def15 defines Initial_Segm ORDERS_2:def 15 :
for A being non empty Poset
for S, b3 being Subset of A holds
( ( S <> {} implies ( b3 is Initial_Segm of S iff ex a being Element of A st
( a in S & b3 = InitSegm (S,a) ) ) ) & ( not S <> {} implies ( b3 is Initial_Segm of S iff b3 = {} ) ) );

theorem :: ORDERS_2:53
canceled;

theorem :: ORDERS_2:54
canceled;

theorem :: ORDERS_2:55
canceled;

theorem :: ORDERS_2:56
canceled;

theorem Th57: :: ORDERS_2:57
for A being non empty Poset
for a, b being Element of A
for S being Subset of A holds
( a in InitSegm (S,b) iff ( a < b & a in S ) )
proof end;

theorem :: ORDERS_2:58
canceled;

theorem :: ORDERS_2:59
canceled;

theorem :: ORDERS_2:60
for A being non empty Poset
for a being Element of A holds InitSegm (({} A),a) = {} ;

theorem :: ORDERS_2:61
canceled;

theorem Th62: :: ORDERS_2:62
for A being non empty Poset
for a being Element of A
for S being Subset of A holds not a in InitSegm (S,a)
proof end;

theorem :: ORDERS_2:63
canceled;

theorem :: ORDERS_2:64
for A being non empty Poset
for a1, a2 being Element of A
for S being Subset of A st a1 < a2 holds
InitSegm (S,a1) c= InitSegm (S,a2)
proof end;

theorem Th65: :: ORDERS_2:65
for A being non empty Poset
for a being Element of A
for S, T being Subset of A st S c= T holds
InitSegm (S,a) c= InitSegm (T,a)
proof end;

theorem :: ORDERS_2:66
canceled;

theorem Th67: :: ORDERS_2:67
for A being non empty Poset
for S being Subset of A
for I being Initial_Segm of S holds I c= S
proof end;

theorem Th68: :: ORDERS_2:68
for A being non empty Poset
for S being Subset of A holds
( S <> {} iff not S is Initial_Segm of S )
proof end;

theorem Th69: :: ORDERS_2:69
for A being non empty Poset
for S, T being Subset of A st S <> {} & S is Initial_Segm of T holds
not T is Initial_Segm of S
proof end;

theorem Th70: :: ORDERS_2:70
for A being non empty Poset
for a1, a2 being Element of A
for S, T being Subset of A st a1 < a2 & a1 in S & a2 in T & T is Initial_Segm of S holds
a1 in T
proof end;

theorem Th71: :: ORDERS_2:71
for A being non empty Poset
for a being Element of A
for S, T being Subset of A st a in S & S is Initial_Segm of T holds
InitSegm (S,a) = InitSegm (T,a)
proof end;

theorem :: ORDERS_2:72
for A being non empty Poset
for S, T being Subset of A st S c= T & the InternalRel of A well_orders T & ( for a1, a2 being Element of A st a2 in S & a1 < a2 holds
a1 in S ) & not S = T holds
S is Initial_Segm of T
proof end;

theorem Th73: :: ORDERS_2:73
for A being non empty Poset
for S, T being Subset of A st S c= T & the InternalRel of A well_orders T & ( for a1, a2 being Element of A st a2 in S & a1 in T & a1 < a2 holds
a1 in S ) & not S = T holds
S is Initial_Segm of T
proof end;

definition
let A be non empty Poset;
let f be Choice_Function of BOOL the carrier of A;
mode Chain of f -> Chain of A means :Def16: :: ORDERS_2:def 16
( it <> {} & the InternalRel of A well_orders it & ( for a being Element of A st a in it holds
f . (UpperCone (InitSegm (it,a))) = a ) );
existence
ex b1 being Chain of A st
( b1 <> {} & the InternalRel of A well_orders b1 & ( for a being Element of A st a in b1 holds
f . (UpperCone (InitSegm (b1,a))) = a ) )
proof end;
end;

:: deftheorem Def16 defines Chain ORDERS_2:def 16 :
for A being non empty Poset
for f being Choice_Function of BOOL the carrier of A
for b3 being Chain of A holds
( b3 is Chain of f iff ( b3 <> {} & the InternalRel of A well_orders b3 & ( for a being Element of A st a in b3 holds
f . (UpperCone (InitSegm (b3,a))) = a ) ) );

theorem :: ORDERS_2:74
canceled;

theorem :: ORDERS_2:75
canceled;

theorem :: ORDERS_2:76
canceled;

theorem :: ORDERS_2:77
canceled;

theorem Th78: :: ORDERS_2:78
for A being non empty Poset
for f being Choice_Function of BOOL the carrier of A holds {(f . the carrier of A)} is Chain of f
proof end;

theorem Th79: :: ORDERS_2:79
for A being non empty Poset
for f being Choice_Function of BOOL the carrier of A
for fC being Chain of f holds f . the carrier of A in fC
proof end;

theorem Th80: :: ORDERS_2:80
for A being non empty Poset
for a, b being Element of A
for f being Choice_Function of BOOL the carrier of A
for fC being Chain of f st a in fC & b = f . the carrier of A holds
b <= a
proof end;

theorem :: ORDERS_2:81
for A being non empty Poset
for a being Element of A
for f being Choice_Function of BOOL the carrier of A
for fC being Chain of f st a = f . the carrier of A holds
InitSegm (fC,a) = {}
proof end;

theorem :: ORDERS_2:82
for A being non empty Poset
for f being Choice_Function of BOOL the carrier of A
for fC1, fC2 being Chain of f holds fC1 meets fC2
proof end;

theorem Th83: :: ORDERS_2:83
for A being non empty Poset
for f being Choice_Function of BOOL the carrier of A
for fC1, fC2 being Chain of f st fC1 <> fC2 holds
( fC1 is Initial_Segm of fC2 iff fC2 is not Initial_Segm of fC1 )
proof end;

theorem Th84: :: ORDERS_2:84
for A being non empty Poset
for f being Choice_Function of BOOL the carrier of A
for fC1, fC2 being Chain of f holds
( fC1 c< fC2 iff fC1 is Initial_Segm of fC2 )
proof end;

definition
let A be non empty Poset;
let f be Choice_Function of BOOL the carrier of A;
func Chains f -> set means :Def17: :: ORDERS_2:def 17
 for x being set holds
( x in it iff x is Chain of f );
existence
ex b1 being set st
for x being set holds
( x in b1 iff x is Chain of f )
proof end;
uniqueness
for b1, b2 being set st ( for x being set holds
( x in b1 iff x is Chain of f ) ) & ( for x being set holds
( x in b2 iff x is Chain of f ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def17 defines Chains ORDERS_2:def 17 :
for A being non empty Poset
for f being Choice_Function of BOOL the carrier of A
for b3 being set holds
( b3 = Chains f iff for x being set holds
( x in b3 iff x is Chain of f ) );

registration
let A be non empty Poset;
let f be Choice_Function of BOOL the carrier of A;
cluster Chains f -> non empty ;
coherence
not Chains f is empty
proof end;
end;

Lm2: for A being non empty Poset
for f being Choice_Function of BOOL the carrier of A holds union (Chains f) is Subset of A
proof end;

Lm3: for A being non empty Poset
for f being Choice_Function of BOOL the carrier of A holds union (Chains f) is Chain of A
proof end;

theorem :: ORDERS_2:85
canceled;

theorem :: ORDERS_2:86
canceled;

theorem Th87: :: ORDERS_2:87
for A being non empty Poset
for f being Choice_Function of BOOL the carrier of A holds union (Chains f) <> {}
proof end;

theorem Th88: :: ORDERS_2:88
for A being non empty Poset
for S being Subset of A
for f being Choice_Function of BOOL the carrier of A
for fC being Chain of f st fC <> union (Chains f) & S = union (Chains f) holds
fC is Initial_Segm of S
proof end;

theorem Th89: :: ORDERS_2:89
for A being non empty Poset
for f being Choice_Function of BOOL the carrier of A holds union (Chains f) is Chain of f
proof end;

begin

theorem :: ORDERS_2:90
canceled;

theorem :: ORDERS_2:91
canceled;

theorem :: ORDERS_2:92
canceled;

theorem :: ORDERS_2:93
canceled;

theorem :: ORDERS_2:94
canceled;

theorem :: ORDERS_2:95
canceled;

theorem :: ORDERS_2:96
canceled;

theorem :: ORDERS_2:97
canceled;

theorem :: ORDERS_2:98
canceled;

theorem :: ORDERS_2:99
canceled;

theorem :: ORDERS_2:100
canceled;

theorem :: ORDERS_2:101
canceled;

theorem :: ORDERS_2:102
canceled;

theorem :: ORDERS_2:103
canceled;

theorem :: ORDERS_2:104
canceled;

theorem :: ORDERS_2:105
canceled;

theorem :: ORDERS_2:106
canceled;

theorem Th107: :: ORDERS_2:107
for A being non empty Poset
for S being Subset of A holds field ( the InternalRel of A |_2 S) = S
proof end;

theorem :: ORDERS_2:108
for A being non empty Poset
for S being Subset of A st the InternalRel of A |_2 S is being_linear-order holds
S is Chain of A
proof end;

theorem :: ORDERS_2:109
for A being non empty Poset
for C being Chain of A holds the InternalRel of A |_2 C is being_linear-order
proof end;

theorem :: ORDERS_2:110
canceled;

theorem :: ORDERS_2:111
canceled;

theorem :: ORDERS_2:112
canceled;

theorem :: ORDERS_2:113
canceled;

theorem :: ORDERS_2:114
canceled;

theorem :: ORDERS_2:115
canceled;

theorem :: ORDERS_2:116
canceled;

theorem :: ORDERS_2:117
canceled;

theorem :: ORDERS_2:118
canceled;

theorem :: ORDERS_2:119
canceled;

theorem :: ORDERS_2:120
canceled;

theorem :: ORDERS_2:121
canceled;

theorem :: ORDERS_2:122
canceled;

theorem :: ORDERS_2:123
canceled;

theorem :: ORDERS_2:124
canceled;

theorem :: ORDERS_2:125
canceled;

theorem :: ORDERS_2:126
canceled;

theorem :: ORDERS_2:127
canceled;

theorem :: ORDERS_2:128
canceled;

theorem :: ORDERS_2:129
canceled;

theorem :: ORDERS_2:130
canceled;

theorem :: ORDERS_2:131
canceled;

theorem :: ORDERS_2:132
canceled;

theorem :: ORDERS_2:133
canceled;

theorem :: ORDERS_2:134
canceled;

theorem :: ORDERS_2:135
for A being non empty Poset
for S being Subset of A st the InternalRel of A linearly_orders S holds
S is Chain of A
proof end;

theorem :: ORDERS_2:136
for A being non empty Poset
for C being Chain of A holds the InternalRel of A linearly_orders C
proof end;

theorem :: ORDERS_2:137
canceled;

theorem :: ORDERS_2:138
canceled;

theorem :: ORDERS_2:139
canceled;

theorem :: ORDERS_2:140
canceled;

theorem :: ORDERS_2:141
canceled;

theorem :: ORDERS_2:142
canceled;

theorem :: ORDERS_2:143
canceled;

theorem :: ORDERS_2:144
canceled;

theorem :: ORDERS_2:145
canceled;

theorem :: ORDERS_2:146
canceled;

theorem :: ORDERS_2:147
canceled;

theorem :: ORDERS_2:148
canceled;

theorem :: ORDERS_2:149
canceled;

theorem :: ORDERS_2:150
canceled;

theorem :: ORDERS_2:151
canceled;

theorem :: ORDERS_2:152
canceled;

theorem :: ORDERS_2:153
canceled;

theorem :: ORDERS_2:154
canceled;

theorem :: ORDERS_2:155
canceled;

theorem :: ORDERS_2:156
canceled;

theorem :: ORDERS_2:157
canceled;

theorem :: ORDERS_2:158
for A being non empty Poset
for a being Element of A holds
( a is_minimal_in the InternalRel of A iff for b being Element of A holds not b < a )
proof end;

theorem :: ORDERS_2:159
for A being non empty Poset
for a being Element of A holds
( a is_maximal_in the InternalRel of A iff for b being Element of A holds not a < b )
proof end;

theorem :: ORDERS_2:160
for A being non empty Poset
for a being Element of A holds
( a is_superior_of the InternalRel of A iff for b being Element of A st a <> b holds
b < a )
proof end;

theorem :: ORDERS_2:161
for A being non empty Poset
for a being Element of A holds
( a is_inferior_of the InternalRel of A iff for b being Element of A st a <> b holds
a < b )
proof end;

Lm4: for X, Y being set
for R being Relation st R is_connected_in X & Y c= X holds
R is_connected_in Y
proof end;

Lm5: for X, Y being set
for R being Relation st R well_orders X & Y c= X holds
R well_orders Y
proof end;

theorem Th162: :: ORDERS_2:162
for A being non empty Poset st ( for C being Chain of A ex a being Element of A st
for b being Element of A st b in C holds
b <= a ) holds
ex a being Element of A st
for b being Element of A holds not a < b
proof end;

theorem :: ORDERS_2:163
for A being non empty Poset st ( for C being Chain of A ex a being Element of A st
for b being Element of A st b in C holds
a <= b ) holds
ex a being Element of A st
for b being Element of A holds not b < a
proof end;

registration
cluster empty strict RelStr ;
existence
ex b1 being RelStr st
( b1 is strict & b1 is empty )
proof end;
end;