:: ORDERS_2 semantic presentation

definition
attr a1 is strict;
struct RelStr -> 1-sorted ;
aggr RelStr(# carrier, InternalRel #) -> RelStr ;
sel InternalRel c1 -> Relation of the carrier of a1;
end;

registration
let X be non empty set ;
let R be Relation of X;
cluster RelStr(# a1,a2 #) -> non empty ;
coherence
not RelStr(# X,R #) is empty
proof end;
end;

definition
let A be RelStr ;
canceled;
canceled;
canceled;
attr a1 is reflexive means :Def4: :: ORDERS_2:def 4
the InternalRel of A is_reflexive_in the carrier of A;
attr a1 is transitive means :Def5: :: ORDERS_2:def 5
the InternalRel of A is_transitive_in the carrier of A;
attr a1 is antisymmetric means :Def6: :: ORDERS_2:def 6
the InternalRel of A is_antisymmetric_in the carrier of A;
end;

:: deftheorem Def1 ORDERS_2:def 1 :
canceled;

:: deftheorem Def2 ORDERS_2:def 2 :
canceled;

:: deftheorem Def3 ORDERS_2:def 3 :
canceled;

:: 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 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 )
proof end;
end;

definition
mode Poset is reflexive transitive antisymmetric RelStr ;
end;

registration
let A be Poset;
cluster the InternalRel of a1 -> reflexive antisymmetric transitive total ;
coherence
( the InternalRel of A is total & the InternalRel of A is reflexive & the InternalRel of A is antisymmetric & the InternalRel of A is transitive )
proof end;
end;

registration
let X be set ;
let O be Order of X;
cluster RelStr(# a1,a2 #) -> reflexive transitive antisymmetric ;
coherence
( RelStr(# X,O #) is reflexive & RelStr(# X,O #) is transitive & RelStr(# X,O #) is antisymmetric )
proof end;
end;

Lemma29: 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 be Element of A, a2 be Element of A;
canceled;
canceled;
pred c2 <= c3 means :Def9: :: ORDERS_2:def 9
[a1,a2] in the InternalRel of A;
end;

:: deftheorem Def7 ORDERS_2:def 7 :
canceled;

:: deftheorem Def8 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 be Element of A, a2 be Element of A;
synonym c3 >= c2 for c2 <= c3;
end;

definition
let A be RelStr ;
let a1 be Element of A, a2 be Element of A;
pred c2 < c3 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 be Element of A, a2 be Element of A;
synonym c3 > c2 for c2 < c3;
end;

theorem Th1: :: ORDERS_2:1
canceled;

theorem Th2: :: ORDERS_2:2
canceled;

theorem Th3: :: ORDERS_2:3
canceled;

theorem Th4: :: ORDERS_2:4
canceled;

theorem Th5: :: ORDERS_2:5
canceled;

theorem Th6: :: ORDERS_2:6
canceled;

theorem Th7: :: ORDERS_2:7
canceled;

theorem Th8: :: ORDERS_2:8
canceled;

theorem Th9: :: ORDERS_2:9
canceled;

theorem Th10: :: ORDERS_2:10
canceled;

theorem Th11: :: ORDERS_2:11
canceled;

theorem Th12: :: ORDERS_2:12
canceled;

theorem Th13: :: ORDERS_2:13
canceled;

theorem Th14: :: ORDERS_2:14
canceled;

theorem Th15: :: ORDERS_2:15
canceled;

theorem Th16: :: ORDERS_2:16
canceled;

theorem Th17: :: ORDERS_2:17
canceled;

theorem Th18: :: ORDERS_2:18
canceled;

theorem Th19: :: ORDERS_2:19
canceled;

theorem Th20: :: ORDERS_2:20
canceled;

theorem Th21: :: ORDERS_2:21
canceled;

theorem Th22: :: ORDERS_2:22
canceled;

theorem Th23: :: 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 be Element of A, a2 be Element of A;
redefine pred <= as c2 <= c3;
reflexivity
for a1 being Element of A holds a1 <= a1
by ;
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 Th27: :: 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 Th31: :: 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 a2 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 {} a1 -> strongly_connected ;
coherence
{} A is strongly_connected
proof end;
end;

registration
let A be RelStr ;
cluster strongly_connected Element of bool the carrier of a1;
existence
ex b1 being Subset of A st b1 is strongly_connected
proof end;
end;

definition
let A be RelStr ;
mode Chain of a1 is strongly_connected Subset of a1;
end;

theorem Th33: :: ORDERS_2:33
canceled;

theorem Th34: :: 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 Th37: :: 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 c2 -> Subset of a1 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 Def12 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 c2 -> Subset of a1 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 Def13 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 Th41: :: ORDERS_2:41
canceled;

theorem Th42: :: 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 Th44: :: ORDERS_2:44
for A being non empty Poset holds UpperCone ([#] A) = {}
proof end;

theorem Th45: :: ORDERS_2:45
for A being non empty Poset holds LowerCone ({} A) = the carrier of A
proof end;

theorem Th46: :: 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 Th48: :: 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 Th51: :: 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 c2,c3 -> Subset of a1 equals :: ORDERS_2:def 14
(LowerCone {a}) /\ S;
correctness
coherence
(LowerCone {a}) /\ S is Subset of A
;
;
end;

:: deftheorem Def14 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 c2 -> Subset of a1 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 Th53: :: ORDERS_2:53
canceled;

theorem Th54: :: ORDERS_2:54
canceled;

theorem Th55: :: ORDERS_2:55
canceled;

theorem Th56: :: ORDERS_2:56
for x being set
for A being non empty Poset
for a being Element of A
for S being Subset of A holds
( x in InitSegm S,a iff ( x in LowerCone {a} & x in S ) ) by XBOOLE_0:def 3;

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 Th58: :: ORDERS_2:58
canceled;

theorem Th59: :: ORDERS_2:59
canceled;

theorem Th60: :: ORDERS_2:60
for A being non empty Poset
for a being Element of A holds InitSegm ({} A),a = {} ;

theorem Th61: :: ORDERS_2:61
for A being non empty Poset
for a being Element of A
for S being Subset of A holds InitSegm S,a c= S by XBOOLE_1:17;

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 Th63: :: ORDERS_2:63
canceled;

theorem Th64: :: 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 Th66: :: 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 Th72: :: 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 c2 -> Chain of a1 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 Th74: :: ORDERS_2:74
canceled;

theorem Th75: :: ORDERS_2:75
canceled;

theorem Th76: :: ORDERS_2:76
canceled;

theorem Th77: :: 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 Th81: :: 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 Th82: :: 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 c2 -> 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 a2 -> non empty ;
coherence
not Chains f is empty
proof end;
end;

Lemma116: 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;

Lemma117: 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 Th85: :: ORDERS_2:85
canceled;

theorem Th86: :: 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;

theorem Th90: :: ORDERS_2:90
canceled;

theorem Th91: :: ORDERS_2:91
canceled;

theorem Th92: :: ORDERS_2:92
canceled;

theorem Th93: :: ORDERS_2:93
canceled;

theorem Th94: :: ORDERS_2:94
canceled;

theorem Th95: :: ORDERS_2:95
canceled;

theorem Th96: :: ORDERS_2:96
canceled;

theorem Th97: :: ORDERS_2:97
canceled;

theorem Th98: :: ORDERS_2:98
canceled;

theorem Th99: :: ORDERS_2:99
canceled;

theorem Th100: :: ORDERS_2:100
canceled;

theorem Th101: :: ORDERS_2:101
canceled;

theorem Th102: :: ORDERS_2:102
canceled;

theorem Th103: :: ORDERS_2:103
canceled;

theorem Th104: :: ORDERS_2:104
canceled;

theorem Th105: :: ORDERS_2:105
canceled;

theorem Th106: :: 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 Th108: :: ORDERS_2:108
for A being non empty Poset
for S being Subset of A st the InternalRel of A |_2 S is_linear-order holds
S is Chain of A
proof end;

theorem Th109: :: ORDERS_2:109
for A being non empty Poset
for C being Chain of A holds the InternalRel of A |_2 C is_linear-order
proof end;

theorem Th110: :: ORDERS_2:110
canceled;

theorem Th111: :: ORDERS_2:111
canceled;

theorem Th112: :: ORDERS_2:112
canceled;

theorem Th113: :: ORDERS_2:113
canceled;

theorem Th114: :: ORDERS_2:114
canceled;

theorem Th115: :: ORDERS_2:115
canceled;

theorem Th116: :: ORDERS_2:116
canceled;

theorem Th117: :: ORDERS_2:117
canceled;

theorem Th118: :: ORDERS_2:118
canceled;

theorem Th119: :: ORDERS_2:119
canceled;

theorem Th120: :: ORDERS_2:120
canceled;

theorem Th121: :: ORDERS_2:121
canceled;

theorem Th122: :: ORDERS_2:122
canceled;

theorem Th123: :: ORDERS_2:123
canceled;

theorem Th124: :: ORDERS_2:124
canceled;

theorem Th125: :: ORDERS_2:125
canceled;

theorem Th126: :: ORDERS_2:126
canceled;

theorem Th127: :: ORDERS_2:127
canceled;

theorem Th128: :: ORDERS_2:128
canceled;

theorem Th129: :: ORDERS_2:129
canceled;

theorem Th130: :: ORDERS_2:130
canceled;

theorem Th131: :: ORDERS_2:131
canceled;

theorem Th132: :: ORDERS_2:132
canceled;

theorem Th133: :: ORDERS_2:133
canceled;

theorem Th134: :: ORDERS_2:134
canceled;

theorem Th135: :: 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 Th136: :: 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 Th137: :: ORDERS_2:137
canceled;

theorem Th138: :: ORDERS_2:138
canceled;

theorem Th139: :: ORDERS_2:139
canceled;

theorem Th140: :: ORDERS_2:140
canceled;

theorem Th141: :: ORDERS_2:141
canceled;

theorem Th142: :: ORDERS_2:142
canceled;

theorem Th143: :: ORDERS_2:143
canceled;

theorem Th144: :: ORDERS_2:144
canceled;

theorem Th145: :: ORDERS_2:145
canceled;

theorem Th146: :: ORDERS_2:146
canceled;

theorem Th147: :: ORDERS_2:147
canceled;

theorem Th148: :: ORDERS_2:148
canceled;

theorem Th149: :: ORDERS_2:149
canceled;

theorem Th150: :: ORDERS_2:150
canceled;

theorem Th151: :: ORDERS_2:151
canceled;

theorem Th152: :: ORDERS_2:152
canceled;

theorem Th153: :: ORDERS_2:153
canceled;

theorem Th154: :: ORDERS_2:154
canceled;

theorem Th155: :: ORDERS_2:155
canceled;

theorem Th156: :: ORDERS_2:156
canceled;

theorem Th157: :: ORDERS_2:157
canceled;

theorem Th158: :: 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 Th159: :: 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 Th160: :: 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 Th161: :: 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;

Lemma124: 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;

Lemma125: 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 Th163: :: 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;