:: WAYBEL_4 semantic presentation

definition
let c1 be non empty reflexive RelStr ;
canceled;
func c1 -waybelow -> Relation of a1 means :Def2: :: WAYBEL_4:def 2
for b1, b2 being Element of a1 holds
( [b1,b2] in a2 iff b1 << b2 );
existence
ex b1 being Relation of c1 st
for b2, b3 being Element of c1 holds
( [b2,b3] in b1 iff b2 << b3 )
proof end;
uniqueness
for b1, b2 being Relation of c1 holds
( ( for b3, b4 being Element of c1 holds
( [b3,b4] in b1 iff b3 << b4 ) ) & ( for b3, b4 being Element of c1 holds
( [b3,b4] in b2 iff b3 << b4 ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def1 WAYBEL_4:def 1 :
canceled;

:: deftheorem Def2 defines -waybelow WAYBEL_4:def 2 :
for b1 being non empty reflexive RelStr
for b2 being Relation of b1 holds
( b2 = b1 -waybelow iff for b3, b4 being Element of b1 holds
( [b3,b4] in b2 iff b3 << b4 ) );

definition
let c1 be RelStr ;
func IntRel c1 -> Relation of a1 equals :: WAYBEL_4:def 3
the InternalRel of a1;
coherence
the InternalRel of c1 is Relation of c1
;
correctness
;
end;

:: deftheorem Def3 defines IntRel WAYBEL_4:def 3 :
for b1 being RelStr holds IntRel b1 = the InternalRel of b1;

Lemma2: for b1 being RelStr
for b2, b3 being Element of b1 holds
( [b2,b3] in IntRel b1 iff b2 <= b3 )
by ORDERS_2:def 9;

definition
let c1 be RelStr ;
let c2 be Relation of c1;
attr a2 is auxiliary(i) means :Def4: :: WAYBEL_4:def 4
for b1, b2 being Element of a1 holds
( [b1,b2] in a2 implies b1 <= b2 );
attr a2 is auxiliary(ii) means :Def5: :: WAYBEL_4:def 5
for b1, b2, b3, b4 being Element of a1 holds
( b4 <= b1 & [b1,b2] in a2 & b2 <= b3 implies [b4,b3] in a2 );
end;

:: deftheorem Def4 defines auxiliary(i) WAYBEL_4:def 4 :
for b1 being RelStr
for b2 being Relation of b1 holds
( b2 is auxiliary(i) iff for b3, b4 being Element of b1 holds
( [b3,b4] in b2 implies b3 <= b4 ) );

:: deftheorem Def5 defines auxiliary(ii) WAYBEL_4:def 5 :
for b1 being RelStr
for b2 being Relation of b1 holds
( b2 is auxiliary(ii) iff for b3, b4, b5, b6 being Element of b1 holds
( b6 <= b3 & [b3,b4] in b2 & b4 <= b5 implies [b6,b5] in b2 ) );

definition
let c1 be non empty RelStr ;
let c2 be Relation of c1;
attr a2 is auxiliary(iii) means :Def6: :: WAYBEL_4:def 6
for b1, b2, b3 being Element of a1 holds
( [b1,b3] in a2 & [b2,b3] in a2 implies [(b1 "\/" b2),b3] in a2 );
attr a2 is auxiliary(iv) means :Def7: :: WAYBEL_4:def 7
for b1 being Element of a1 holds [(Bottom a1),b1] in a2;
end;

:: deftheorem Def6 defines auxiliary(iii) WAYBEL_4:def 6 :
for b1 being non empty RelStr
for b2 being Relation of b1 holds
( b2 is auxiliary(iii) iff for b3, b4, b5 being Element of b1 holds
( [b3,b5] in b2 & [b4,b5] in b2 implies [(b3 "\/" b4),b5] in b2 ) );

:: deftheorem Def7 defines auxiliary(iv) WAYBEL_4:def 7 :
for b1 being non empty RelStr
for b2 being Relation of b1 holds
( b2 is auxiliary(iv) iff for b3 being Element of b1 holds [(Bottom b1),b3] in b2 );

definition
let c1 be non empty RelStr ;
let c2 be Relation of c1;
attr a2 is auxiliary means :Def8: :: WAYBEL_4:def 8
( a2 is auxiliary(i) & a2 is auxiliary(ii) & a2 is auxiliary(iii) & a2 is auxiliary(iv) );
end;

:: deftheorem Def8 defines auxiliary WAYBEL_4:def 8 :
for b1 being non empty RelStr
for b2 being Relation of b1 holds
( b2 is auxiliary iff ( b2 is auxiliary(i) & b2 is auxiliary(ii) & b2 is auxiliary(iii) & b2 is auxiliary(iv) ) );

registration
let c1 be non empty RelStr ;
cluster auxiliary -> auxiliary(i) auxiliary(ii) auxiliary(iii) auxiliary(iv) Relation of the carrier of a1,the carrier of a1;
coherence
for b1 being Relation of c1 holds
( b1 is auxiliary implies ( b1 is auxiliary(i) & b1 is auxiliary(ii) & b1 is auxiliary(iii) & b1 is auxiliary(iv) ) )
by Def8;
cluster auxiliary(i) auxiliary(ii) auxiliary(iii) auxiliary(iv) -> auxiliary Relation of the carrier of a1,the carrier of a1;
coherence
for b1 being Relation of c1 holds
( b1 is auxiliary(i) & b1 is auxiliary(ii) & b1 is auxiliary(iii) & b1 is auxiliary(iv) implies b1 is auxiliary )
by Def8;
end;

registration
let c1 be transitive antisymmetric lower-bounded with_suprema RelStr ;
cluster auxiliary(i) auxiliary(ii) auxiliary(iii) auxiliary(iv) auxiliary Relation of the carrier of a1,the carrier of a1;
existence
ex b1 being Relation of c1 st b1 is auxiliary
proof end;
end;

theorem Th1: :: WAYBEL_4:1
for b1 being lower-bounded sup-Semilattice
for b2 being auxiliary(ii) auxiliary(iii) Relation of b1
for b3, b4, b5, b6 being Element of b1 holds
( [b3,b5] in b2 & [b4,b6] in b2 implies [(b3 "\/" b4),(b5 "\/" b6)] in b2 )
proof end;

Lemma9: for b1 being lower-bounded sup-Semilattice
for b2 being auxiliary(i) auxiliary(ii) Relation of b1 holds b2 is transitive
proof end;

registration
let c1 be lower-bounded sup-Semilattice;
cluster auxiliary(i) auxiliary(ii) -> transitive Relation of the carrier of a1,the carrier of a1;
coherence
for b1 being Relation of c1 holds
( b1 is auxiliary(i) & b1 is auxiliary(ii) implies b1 is transitive )
by Lemma9;
end;

registration
let c1 be RelStr ;
cluster IntRel a1 -> auxiliary(i) ;
coherence
IntRel c1 is auxiliary(i)
proof end;
end;

registration
let c1 be transitive RelStr ;
cluster IntRel a1 -> auxiliary(i) auxiliary(ii) ;
coherence
IntRel c1 is auxiliary(ii)
proof end;
end;

registration
let c1 be antisymmetric with_suprema RelStr ;
cluster IntRel a1 -> auxiliary(i) auxiliary(iii) ;
coherence
IntRel c1 is auxiliary(iii)
proof end;
end;

registration
let c1 be non empty antisymmetric lower-bounded RelStr ;
cluster IntRel a1 -> auxiliary(i) auxiliary(iv) ;
coherence
IntRel c1 is auxiliary(iv)
proof end;
end;

definition
let c1 be lower-bounded sup-Semilattice;
func Aux c1 -> set means :Def9: :: WAYBEL_4:def 9
for b1 being set holds
( b1 in a2 iff b1 is auxiliary Relation of a1 );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff b2 is auxiliary Relation of c1 )
proof end;
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff b3 is auxiliary Relation of c1 ) ) & ( for b3 being set holds
( b3 in b2 iff b3 is auxiliary Relation of c1 ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def9 defines Aux WAYBEL_4:def 9 :
for b1 being lower-bounded sup-Semilattice
for b2 being set holds
( b2 = Aux b1 iff for b3 being set holds
( b3 in b2 iff b3 is auxiliary Relation of b1 ) );

registration
let c1 be lower-bounded sup-Semilattice;
cluster Aux a1 -> non empty ;
coherence
not Aux c1 is empty
proof end;
end;

Lemma11: for b1 being lower-bounded sup-Semilattice
for b2 being auxiliary(i) Relation of b1
for b3, b4 being set holds
( [b3,b4] in b2 implies [b3,b4] in IntRel b1 )
proof end;

theorem Th2: :: WAYBEL_4:2
for b1 being lower-bounded sup-Semilattice
for b2 being auxiliary(i) Relation of b1 holds b2 c= IntRel b1
proof end;

theorem Th3: :: WAYBEL_4:3
for b1 being lower-bounded sup-Semilattice holds Top (InclPoset (Aux b1)) = IntRel b1
proof end;

registration
let c1 be lower-bounded sup-Semilattice;
cluster InclPoset (Aux a1) -> upper-bounded ;
coherence
InclPoset (Aux c1) is upper-bounded
proof end;
end;

definition
let c1 be non empty RelStr ;
func AuxBottom c1 -> Relation of a1 means :Def10: :: WAYBEL_4:def 10
for b1, b2 being Element of a1 holds
( [b1,b2] in a2 iff b1 = Bottom a1 );
existence
ex b1 being Relation of c1 st
for b2, b3 being Element of c1 holds
( [b2,b3] in b1 iff b2 = Bottom c1 )
proof end;
uniqueness
for b1, b2 being Relation of c1 holds
( ( for b3, b4 being Element of c1 holds
( [b3,b4] in b1 iff b3 = Bottom c1 ) ) & ( for b3, b4 being Element of c1 holds
( [b3,b4] in b2 iff b3 = Bottom c1 ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def10 defines AuxBottom WAYBEL_4:def 10 :
for b1 being non empty RelStr
for b2 being Relation of b1 holds
( b2 = AuxBottom b1 iff for b3, b4 being Element of b1 holds
( [b3,b4] in b2 iff b3 = Bottom b1 ) );

registration
let c1 be lower-bounded sup-Semilattice;
cluster AuxBottom a1 -> transitive auxiliary(i) auxiliary(ii) auxiliary(iii) auxiliary(iv) auxiliary ;
coherence
AuxBottom c1 is auxiliary
proof end;
end;

theorem Th4: :: WAYBEL_4:4
for b1 being lower-bounded sup-Semilattice
for b2 being auxiliary(iv) Relation of b1 holds AuxBottom b1 c= b2
proof end;

theorem Th5: :: WAYBEL_4:5
for b1 being lower-bounded sup-Semilattice holds Bottom (InclPoset (Aux b1)) = AuxBottom b1
proof end;

registration
let c1 be lower-bounded sup-Semilattice;
cluster InclPoset (Aux a1) -> lower-bounded upper-bounded ;
coherence
InclPoset (Aux c1) is lower-bounded
proof end;
end;

theorem Th6: :: WAYBEL_4:6
for b1 being lower-bounded sup-Semilattice
for b2, b3 being auxiliary(i) Relation of b1 holds
b2 /\ b3 is auxiliary(i) Relation of b1
proof end;

theorem Th7: :: WAYBEL_4:7
for b1 being lower-bounded sup-Semilattice
for b2, b3 being auxiliary(ii) Relation of b1 holds
b2 /\ b3 is auxiliary(ii) Relation of b1
proof end;

theorem Th8: :: WAYBEL_4:8
for b1 being lower-bounded sup-Semilattice
for b2, b3 being auxiliary(iii) Relation of b1 holds
b2 /\ b3 is auxiliary(iii) Relation of b1
proof end;

theorem Th9: :: WAYBEL_4:9
for b1 being lower-bounded sup-Semilattice
for b2, b3 being auxiliary(iv) Relation of b1 holds
b2 /\ b3 is auxiliary(iv) Relation of b1
proof end;

theorem Th10: :: WAYBEL_4:10
for b1 being lower-bounded sup-Semilattice
for b2, b3 being auxiliary Relation of b1 holds
b2 /\ b3 is auxiliary Relation of b1
proof end;

theorem Th11: :: WAYBEL_4:11
for b1 being lower-bounded sup-Semilattice
for b2 being non empty Subset of (InclPoset (Aux b1)) holds
meet b2 is auxiliary Relation of b1
proof end;

registration
let c1 be lower-bounded sup-Semilattice;
cluster InclPoset (Aux a1) -> lower-bounded upper-bounded with_infima ;
coherence
InclPoset (Aux c1) is with_infima
proof end;
end;

registration
let c1 be lower-bounded sup-Semilattice;
cluster InclPoset (Aux a1) -> lower-bounded upper-bounded with_infima complete ;
coherence
InclPoset (Aux c1) is complete
proof end;
end;

definition
let c1 be non empty RelStr ;
let c2 be Element of c1;
let c3 be Relation of the carrier of c1;
E21: { b1 where B is Element of c1 : [b1,c2] in c3 } c= the carrier of c1
proof end;
func c3 -below c2 -> Subset of a1 equals :: WAYBEL_4:def 11
{ b1 where B is Element of a1 : [b1,a2] in a3 } ;
correctness
coherence
{ b1 where B is Element of c1 : [b1,c2] in c3 } is Subset of c1
;
by E21;
E22: { b1 where B is Element of c1 : [c2,b1] in c3 } c= the carrier of c1
proof end;
func c3 -above c2 -> Subset of a1 equals :: WAYBEL_4:def 12
{ b1 where B is Element of a1 : [a2,b1] in a3 } ;
correctness
coherence
{ b1 where B is Element of c1 : [c2,b1] in c3 } is Subset of c1
;
by E22;
end;

:: deftheorem Def11 defines -below WAYBEL_4:def 11 :
for b1 being non empty RelStr
for b2 being Element of b1
for b3 being Relation of the carrier of b1 holds b3 -below b2 = { b4 where B is Element of b1 : [b4,b2] in b3 } ;

:: deftheorem Def12 defines -above WAYBEL_4:def 12 :
for b1 being non empty RelStr
for b2 being Element of b1
for b3 being Relation of the carrier of b1 holds b3 -above b2 = { b4 where B is Element of b1 : [b2,b4] in b3 } ;

theorem Th12: :: WAYBEL_4:12
for b1 being lower-bounded sup-Semilattice
for b2 being Element of b1
for b3 being auxiliary(i) Relation of b1 holds b3 -below b2 c= downarrow b2
proof end;

registration
let c1 be lower-bounded sup-Semilattice;
let c2 be Element of c1;
let c3 be auxiliary(iv) Relation of c1;
cluster a3 -below a2 -> non empty ;
coherence
not c3 -below c2 is empty
proof end;
end;

registration
let c1 be lower-bounded sup-Semilattice;
let c2 be Element of c1;
let c3 be auxiliary(ii) Relation of c1;
cluster a3 -below a2 -> lower ;
coherence
c3 -below c2 is lower
proof end;
end;

registration
let c1 be lower-bounded sup-Semilattice;
let c2 be Element of c1;
let c3 be auxiliary(iii) Relation of c1;
cluster a3 -below a2 -> directed ;
coherence
c3 -below c2 is directed
proof end;
end;

definition
let c1 be lower-bounded sup-Semilattice;
let c2 be auxiliary(ii) auxiliary(iii) auxiliary(iv) Relation of c1;
func c2 -below -> Function of a1,(InclPoset (Ids a1)) means :Def13: :: WAYBEL_4:def 13
for b1 being Element of a1 holds a3 . b1 = a2 -below b1;
existence
ex b1 being Function of c1,(InclPoset (Ids c1)) st
for b2 being Element of c1 holds b1 . b2 = c2 -below b2
proof end;
uniqueness
for b1, b2 being Function of c1,(InclPoset (Ids c1)) holds
( ( for b3 being Element of c1 holds b1 . b3 = c2 -below b3 ) & ( for b3 being Element of c1 holds b2 . b3 = c2 -below b3 ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def13 defines -below WAYBEL_4:def 13 :
for b1 being lower-bounded sup-Semilattice
for b2 being auxiliary(ii) auxiliary(iii) auxiliary(iv) Relation of b1
for b3 being Function of b1,(InclPoset (Ids b1)) holds
( b3 = b2 -below iff for b4 being Element of b1 holds b3 . b4 = b2 -below b4 );

theorem Th13: :: WAYBEL_4:13
for b1 being non empty RelStr
for b2 being Relation of b1
for b3 being set
for b4 being Element of b1 holds
( b3 in b2 -below b4 iff [b3,b4] in b2 )
proof end;

theorem Th14: :: WAYBEL_4:14
for b1 being set
for b2 being sup-Semilattice
for b3 being Relation of b2
for b4 being Element of b2 holds
( b1 in b3 -above b4 iff [b4,b1] in b3 )
proof end;

Lemma24: for b1 being lower-bounded with_suprema Poset
for b2 being Relation of b1
for b3 being set
for b4 being Element of b1 holds
( b3 in downarrow b4 iff [b3,b4] in the InternalRel of b1 )
proof end;

theorem Th15: :: WAYBEL_4:15
for b1 being lower-bounded sup-Semilattice
for b2 being auxiliary(i) Relation of b1
for b3 being Element of b1 holds
( b2 = the InternalRel of b1 implies b2 -below b3 = downarrow b3 )
proof end;

definition
let c1 be non empty Poset;
func MonSet c1 -> strict RelStr means :Def14: :: WAYBEL_4:def 14
for b1 being set holds
( not ( b1 in the carrier of a2 & ( for b2 being Function of a1,(InclPoset (Ids a1)) holds
not ( b1 = b2 & b2 is monotone & ( for b3 being Element of a1 holds b2 . b3 c= downarrow b3 ) ) ) ) & ( ex b2 being Function of a1,(InclPoset (Ids a1)) st
( b1 = b2 & b2 is monotone & ( for b3 being Element of a1 holds b2 . b3 c= downarrow b3 ) ) implies b1 in the carrier of a2 ) & ( for b2, b3 being set holds
( [b2,b3] in the InternalRel of a2 iff ex b4, b5 being Function of a1,(InclPoset (Ids a1)) st
( b2 = b4 & b3 = b5 & b2 in the carrier of a2 & b3 in the carrier of a2 & b4 <= b5 ) ) ) );
existence
ex b1 being strict RelStr st
for b2 being set holds
( not ( b2 in the carrier of b1 & ( for b3 being Function of c1,(InclPoset (Ids c1)) holds
not ( b2 = b3 & b3 is monotone & ( for b4 being Element of c1 holds b3 . b4 c= downarrow b4 ) ) ) ) & ( ex b3 being Function of c1,(InclPoset (Ids c1)) st
( b2 = b3 & b3 is monotone & ( for b4 being Element of c1 holds b3 . b4 c= downarrow b4 ) ) implies b2 in the carrier of b1 ) & ( for b3, b4 being set holds
( [b3,b4] in the InternalRel of b1 iff ex b5, b6 being Function of c1,(InclPoset (Ids c1)) st
( b3 = b5 & b4 = b6 & b3 in the carrier of b1 & b4 in the carrier of b1 & b5 <= b6 ) ) ) )
proof end;
uniqueness
for b1, b2 being strict RelStr holds
( ( for b3 being set holds
( not ( b3 in the carrier of b1 & ( for b4 being Function of c1,(InclPoset (Ids c1)) holds
not ( b3 = b4 & b4 is monotone & ( for b5 being Element of c1 holds b4 . b5 c= downarrow b5 ) ) ) ) & ( ex b4 being Function of c1,(InclPoset (Ids c1)) st
( b3 = b4 & b4 is monotone & ( for b5 being Element of c1 holds b4 . b5 c= downarrow b5 ) ) implies b3 in the carrier of b1 ) & ( for b4, b5 being set holds
( [b4,b5] in the InternalRel of b1 iff ex b6, b7 being Function of c1,(InclPoset (Ids c1)) st
( b4 = b6 & b5 = b7 & b4 in the carrier of b1 & b5 in the carrier of b1 & b6 <= b7 ) ) ) ) ) & ( for b3 being set holds
( not ( b3 in the carrier of b2 & ( for b4 being Function of c1,(InclPoset (Ids c1)) holds
not ( b3 = b4 & b4 is monotone & ( for b5 being Element of c1 holds b4 . b5 c= downarrow b5 ) ) ) ) & ( ex b4 being Function of c1,(InclPoset (Ids c1)) st
( b3 = b4 & b4 is monotone & ( for b5 being Element of c1 holds b4 . b5 c= downarrow b5 ) ) implies b3 in the carrier of b2 ) & ( for b4, b5 being set holds
( [b4,b5] in the InternalRel of b2 iff ex b6, b7 being Function of c1,(InclPoset (Ids c1)) st
( b4 = b6 & b5 = b7 & b4 in the carrier of b2 & b5 in the carrier of b2 & b6 <= b7 ) ) ) ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def14 defines MonSet WAYBEL_4:def 14 :
for b1 being non empty Poset
for b2 being strict RelStr holds
( b2 = MonSet b1 iff for b3 being set holds
( not ( b3 in the carrier of b2 & ( for b4 being Function of b1,(InclPoset (Ids b1)) holds
not ( b3 = b4 & b4 is monotone & ( for b5 being Element of b1 holds b4 . b5 c= downarrow b5 ) ) ) ) & ( ex b4 being Function of b1,(InclPoset (Ids b1)) st
( b3 = b4 & b4 is monotone & ( for b5 being Element of b1 holds b4 . b5 c= downarrow b5 ) ) implies b3 in the carrier of b2 ) & ( for b4, b5 being set holds
( [b4,b5] in the InternalRel of b2 iff ex b6, b7 being Function of b1,(InclPoset (Ids b1)) st
( b4 = b6 & b5 = b7 & b4 in the carrier of b2 & b5 in the carrier of b2 & b6 <= b7 ) ) ) ) );

theorem Th16: :: WAYBEL_4:16
for b1 being lower-bounded sup-Semilattice holds
MonSet b1 is full SubRelStr of (InclPoset (Ids b1)) |^ the carrier of b1
proof end;

theorem Th17: :: WAYBEL_4:17
for b1 being lower-bounded sup-Semilattice
for b2 being auxiliary(ii) Relation of b1
for b3, b4 being Element of b1 holds
( b3 <= b4 implies b2 -below b3 c= b2 -below b4 )
proof end;

registration
let c1 be lower-bounded sup-Semilattice;
let c2 be auxiliary(ii) auxiliary(iii) auxiliary(iv) Relation of c1;
cluster a2 -below -> monotone ;
coherence
c2 -below is monotone
proof end;
end;

theorem Th18: :: WAYBEL_4:18
for b1 being lower-bounded sup-Semilattice
for b2 being auxiliary Relation of b1 holds b2 -below in the carrier of (MonSet b1)
proof end;

registration
let c1 be lower-bounded sup-Semilattice;
cluster MonSet a1 -> non empty strict ;
coherence
not MonSet c1 is empty
proof end;
end;

theorem Th19: :: WAYBEL_4:19
for b1 being lower-bounded sup-Semilattice holds IdsMap b1 in the carrier of (MonSet b1)
proof end;

theorem Th20: :: WAYBEL_4:20
for b1 being lower-bounded sup-Semilattice
for b2 being auxiliary Relation of b1 holds b2 -below <= IdsMap b1
proof end;

theorem Th21: :: WAYBEL_4:21
for b1 being non empty lower-bounded Poset
for b2 being Ideal of b1 holds Bottom b1 in b2
proof end;

theorem Th22: :: WAYBEL_4:22
for b1 being non empty upper-bounded Poset
for b2 being Filter of b1 holds Top b1 in b2
proof end;

theorem Th23: :: WAYBEL_4:23
for b1 being non empty lower-bounded Poset holds downarrow (Bottom b1) = {(Bottom b1)}
proof end;

theorem Th24: :: WAYBEL_4:24
for b1 being non empty upper-bounded Poset holds uparrow (Top b1) = {(Top b1)}
proof end;

Lemma31: for b1 being lower-bounded sup-Semilattice
for b2 being Ideal of b1 holds {(Bottom b1)} c= b2
proof end;

theorem Th25: :: WAYBEL_4:25
for b1 being lower-bounded sup-Semilattice holds
the carrier of b1 --> {(Bottom b1)} is Function of b1,(InclPoset (Ids b1))
proof end;

Lemma33: for b1 being lower-bounded sup-Semilattice
for b2 being Function of b1,(InclPoset (Ids b1)) holds
( b2 = the carrier of b1 --> {(Bottom b1)} implies b2 is monotone )
proof end;

theorem Th26: :: WAYBEL_4:26
for b1 being lower-bounded sup-Semilattice holds the carrier of b1 --> {(Bottom b1)} in the carrier of (MonSet b1)
proof end;

theorem Th27: :: WAYBEL_4:27
for b1 being lower-bounded sup-Semilattice
for b2 being auxiliary Relation of b1 holds [(the carrier of b1 --> {(Bottom b1)}),(b2 -below )] in the InternalRel of (MonSet b1)
proof end;

Lemma35: for b1 being lower-bounded sup-Semilattice holds
ex b2 being Element of (MonSet b1) st b2 is_>=_than the carrier of (MonSet b1)
proof end;

registration
let c1 be lower-bounded sup-Semilattice;
cluster MonSet a1 -> non empty strict upper-bounded ;
coherence
MonSet c1 is upper-bounded
proof end;
end;

definition
let c1 be lower-bounded sup-Semilattice;
func Rel2Map c1 -> Function of (InclPoset (Aux a1)),(MonSet a1) means :Def15: :: WAYBEL_4:def 15
for b1 being auxiliary Relation of a1 holds a2 . b1 = b1 -below ;
existence
ex b1 being Function of (InclPoset (Aux c1)),(MonSet c1) st
for b2 being auxiliary Relation of c1 holds b1 . b2 = b2 -below
proof end;
uniqueness
for b1, b2 being Function of (InclPoset (Aux c1)),(MonSet c1) holds
( ( for b3 being auxiliary Relation of c1 holds b1 . b3 = b3 -below ) & ( for b3 being auxiliary Relation of c1 holds b2 . b3 = b3 -below ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def15 defines Rel2Map WAYBEL_4:def 15 :
for b1 being lower-bounded sup-Semilattice
for b2 being Function of (InclPoset (Aux b1)),(MonSet b1) holds
( b2 = Rel2Map b1 iff for b3 being auxiliary Relation of b1 holds b2 . b3 = b3 -below );

theorem Th28: :: WAYBEL_4:28
for b1 being lower-bounded sup-Semilattice
for b2, b3 being auxiliary Relation of b1 holds
( b2 c= b3 implies b2 -below <= b3 -below )
proof end;

theorem Th29: :: WAYBEL_4:29
for b1 being lower-bounded sup-Semilattice
for b2 being Element of b1
for b3, b4 being Relation of b1 holds
( b3 c= b4 implies b3 -below b2 c= b4 -below b2 )
proof end;

Lemma38: for b1 being lower-bounded sup-Semilattice holds Rel2Map b1 is monotone
proof end;

registration
let c1 be lower-bounded sup-Semilattice;
cluster Rel2Map a1 -> monotone ;
coherence
Rel2Map c1 is monotone
by Lemma38;
end;

definition
let c1 be lower-bounded sup-Semilattice;
func Map2Rel c1 -> Function of (MonSet a1),(InclPoset (Aux a1)) means :Def16: :: WAYBEL_4:def 16
for b1 being set holds
not ( b1 in the carrier of (MonSet a1) & ( for b2 being auxiliary Relation of a1 holds
not ( b2 = a2 . b1 & ( for b3, b4 being set holds
( [b3,b4] in b2 iff ex b5, b6 being Element of a1ex b7 being Function of a1,(InclPoset (Ids a1)) st
( b5 = b3 & b6 = b4 & b7 = b1 & b5 in b7 . b6 ) ) ) ) ) );
existence
ex b1 being Function of (MonSet c1),(InclPoset (Aux c1)) st
for b2 being set holds
not ( b2 in the carrier of (MonSet c1) & ( for b3 being auxiliary Relation of c1 holds
not ( b3 = b1 . b2 & ( for b4, b5 being set holds
( [b4,b5] in b3 iff ex b6, b7 being Element of c1ex b8 being Function of c1,(InclPoset (Ids c1)) st
( b6 = b4 & b7 = b5 & b8 = b2 & b6 in b8 . b7 ) ) ) ) ) )
proof end;
uniqueness
for b1, b2 being Function of (MonSet c1),(InclPoset (Aux c1)) holds
( ( for b3 being set holds
not ( b3 in the carrier of (MonSet c1) & ( for b4 being auxiliary Relation of c1 holds
not ( b4 = b1 . b3 & ( for b5, b6 being set holds
( [b5,b6] in b4 iff ex b7, b8 being Element of c1ex b9 being Function of c1,(InclPoset (Ids c1)) st
( b7 = b5 & b8 = b6 & b9 = b3 & b7 in b9 . b8 ) ) ) ) ) ) ) & ( for b3 being set holds
not ( b3 in the carrier of (MonSet c1) & ( for b4 being auxiliary Relation of c1 holds
not ( b4 = b2 . b3 & ( for b5, b6 being set holds
( [b5,b6] in b4 iff ex b7, b8 being Element of c1ex b9 being Function of c1,(InclPoset (Ids c1)) st
( b7 = b5 & b8 = b6 & b9 = b3 & b7 in b9 . b8 ) ) ) ) ) ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def16 defines Map2Rel WAYBEL_4:def 16 :
for b1 being lower-bounded sup-Semilattice
for b2 being Function of (MonSet b1),(InclPoset (Aux b1)) holds
( b2 = Map2Rel b1 iff for b3 being set holds
not ( b3 in the carrier of (MonSet b1) & ( for b4 being auxiliary Relation of b1 holds
not ( b4 = b2 . b3 & ( for b5, b6 being set holds
( [b5,b6] in b4 iff ex b7, b8 being Element of b1ex b9 being Function of b1,(InclPoset (Ids b1)) st
( b7 = b5 & b8 = b6 & b9 = b3 & b7 in b9 . b8 ) ) ) ) ) ) );

Lemma40: for b1 being lower-bounded sup-Semilattice holds Map2Rel b1 is monotone
proof end;

registration
let c1 be lower-bounded sup-Semilattice;
cluster Map2Rel a1 -> monotone ;
coherence
Map2Rel c1 is monotone
by Lemma40;
end;

theorem Th30: :: WAYBEL_4:30
for b1 being lower-bounded sup-Semilattice holds (Map2Rel b1) * (Rel2Map b1) = id (dom (Rel2Map b1))
proof end;

theorem Th31: :: WAYBEL_4:31
for b1 being lower-bounded sup-Semilattice holds (Rel2Map b1) * (Map2Rel b1) = id the carrier of (MonSet b1)
proof end;

registration
let c1 be lower-bounded sup-Semilattice;
cluster Rel2Map a1 -> V8 monotone ;
coherence
Rel2Map c1 is one-to-one
proof end;
end;

theorem Th32: :: WAYBEL_4:32
for b1 being lower-bounded sup-Semilattice holds (Rel2Map b1) " = Map2Rel b1
proof end;

theorem Th33: :: WAYBEL_4:33
for b1 being lower-bounded sup-Semilattice holds Rel2Map b1 is isomorphic
proof end;

theorem Th34: :: WAYBEL_4:34
for b1 being complete LATTICE
for b2 being Element of b1 holds meet { b3 where B is Ideal of b1 : b2 <= sup b3 } = waybelow b2
proof end;

definition
let c1 be Semilattice;
let c2 be Ideal of c1;
func DownMap c2 -> Function of a1,(InclPoset (Ids a1)) means :Def17: :: WAYBEL_4:def 17
for b1 being Element of a1 holds
( ( b1 <= sup a2 implies a3 . b1 = (downarrow b1) /\ a2 ) & ( not b1 <= sup a2 implies a3 . b1 = downarrow b1 ) );
existence
ex b1 being Function of c1,(InclPoset (Ids c1)) st
for b2 being Element of c1 holds
( ( b2 <= sup c2 implies b1 . b2 = (downarrow b2) /\ c2 ) & ( not b2 <= sup c2 implies b1 . b2 = downarrow b2 ) )
proof end;
uniqueness
for b1, b2 being Function of c1,(InclPoset (Ids c1)) holds
( ( for b3 being Element of c1 holds
( ( b3 <= sup c2 implies b1 . b3 = (downarrow b3) /\ c2 ) & ( not b3 <= sup c2 implies b1 . b3 = downarrow b3 ) ) ) & ( for b3 being Element of c1 holds
( ( b3 <= sup c2 implies b2 . b3 = (downarrow b3) /\ c2 ) & ( not b3 <= sup c2 implies b2 . b3 = downarrow b3 ) ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def17 defines DownMap WAYBEL_4:def 17 :
for b1 being Semilattice
for b2 being Ideal of b1
for b3 being Function of b1,(InclPoset (Ids b1)) holds
( b3 = DownMap b2 iff for b4 being Element of b1 holds
( ( b4 <= sup b2 implies b3 . b4 = (downarrow b4) /\ b2 ) & ( not b4 <= sup b2 implies b3 . b4 = downarrow b4 ) ) );

Lemma46: for b1 being Semilattice
for b2 being Ideal of b1 holds DownMap b2 is monotone
proof end;

Lemma47: for b1 being Semilattice
for b2 being Element of b1
for b3 being Ideal of b1 holds (DownMap b3) . b2 c= downarrow b2
proof end;

theorem Th35: :: WAYBEL_4:35
for b1 being Semilattice
for b2 being Ideal of b1 holds DownMap b2 in the carrier of (MonSet b1)
proof end;

theorem Th36: :: WAYBEL_4:36
for b1 being reflexive antisymmetric with_infima RelStr
for b2 being Element of b1
for b3 being non empty lower Subset of b1 holds {b2} "/\" b3 = (downarrow b2) /\ b3
proof end;

definition
let c1 be non empty RelStr ;
let c2 be Relation of c1;
attr a2 is approximating means :Def18: :: WAYBEL_4:def 18
for b1 being Element of a1 holds b1 = sup (a2 -below b1);
end;

:: deftheorem Def18 defines approximating WAYBEL_4:def 18 :
for b1 being non empty RelStr
for b2 being Relation of b1 holds
( b2 is approximating iff for b3 being Element of b1 holds b3 = sup (b2 -below b3) );

definition
let c1 be non empty Poset;
let c2 be Function of c1,(InclPoset (Ids c1));
attr a2 is approximating means :Def19: :: WAYBEL_4:def 19
for b1 being Element of a1 holds
ex b2 being Subset of a1 st
( b2 = a2 . b1 & b1 = sup b2 );
end;

:: deftheorem Def19 defines approximating WAYBEL_4:def 19 :
for b1 being non empty Poset
for b2 being Function of b1,(InclPoset (Ids b1)) holds
( b2 is approximating iff for b3 being Element of b1 holds
ex b4 being Subset of b1 st
( b4 = b2 . b3 & b3 = sup b4 ) );

theorem Th37: :: WAYBEL_4:37
for b1 being lower-bounded meet-continuous Semilattice
for b2 being Ideal of b1 holds DownMap b2 is approximating
proof end;

Lemma53: for b1 being complete LATTICE
for b2 being Element of b1
for b3 being directed Subset of b1 holds sup ({b2} "/\" b3) <= b2
proof end;

theorem Th38: :: WAYBEL_4:38
for b1 being lower-bounded continuous LATTICE holds b1 is meet-continuous
proof end;

registration
cluster lower-bounded continuous -> lower-bounded meet-continuous RelStr ;
coherence
for b1 being lower-bounded LATTICE holds
( b1 is continuous implies b1 is meet-continuous )
by Th38;
end;

theorem Th39: :: WAYBEL_4:39
for b1 being lower-bounded continuous LATTICE
for b2 being Ideal of b1 holds DownMap b2 is approximating by Th37;

registration
let c1 be non empty reflexive antisymmetric RelStr ;
cluster a1 -waybelow -> auxiliary(i) ;
coherence
c1 -waybelow is auxiliary(i)
proof end;
end;

registration
let c1 be non empty reflexive transitive RelStr ;
cluster a1 -waybelow -> auxiliary(ii) ;
coherence
c1 -waybelow is auxiliary(ii)
proof end;
end;

registration
let c1 be with_suprema Poset;
cluster a1 -waybelow -> auxiliary(i) auxiliary(ii) auxiliary(iii) ;
coherence
c1 -waybelow is auxiliary(iii)
proof end;
end;

registration
let c1 be non empty /\-complete Poset;
cluster a1 -waybelow -> auxiliary(i) auxiliary(ii) auxiliary(iii) ;
coherence
c1 -waybelow is auxiliary(iii)
proof end;
end;

registration
let c1 be non empty reflexive antisymmetric lower-bounded RelStr ;
cluster a1 -waybelow -> auxiliary(i) auxiliary(iv) ;
coherence
c1 -waybelow is auxiliary(iv)
proof end;
end;

theorem Th40: :: WAYBEL_4:40
for b1 being complete LATTICE
for b2 being Element of b1 holds (b1 -waybelow ) -below b2 = waybelow b2
proof end;

theorem Th41: :: WAYBEL_4:41
for b1 being LATTICE holds IntRel b1 is approximating
proof end;

Lemma57: for b1 being lower-bounded continuous LATTICE holds b1 -waybelow is approximating
proof end;

registration
let c1 be lower-bounded continuous LATTICE;
cluster a1 -waybelow -> transitive auxiliary(i) auxiliary(ii) auxiliary(iii) auxiliary(iv) auxiliary approximating ;
coherence
c1 -waybelow is approximating
by Lemma57;
end;

registration
let c1 be complete LATTICE;
cluster transitive auxiliary(i) auxiliary(ii) auxiliary(iii) auxiliary(iv) auxiliary approximating Relation of the carrier of a1,the carrier of a1;
existence
ex b1 being Relation of c1 st
( b1 is approximating & b1 is auxiliary )
proof end;
end;

definition
let c1 be complete LATTICE;
defpred S1[ set ] means a1 is auxiliary approximating Relation of c1;
func App c1 -> set means :Def20: :: WAYBEL_4:def 20
for b1 being set holds
( b1 in a2 iff b1 is auxiliary approximating Relation of a1 );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff b2 is auxiliary approximating Relation of c1 )
proof end;
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff b3 is auxiliary approximating Relation of c1 ) ) & ( for b3 being set holds
( b3 in b2 iff b3 is auxiliary approximating Relation of c1 ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def20 defines App WAYBEL_4:def 20 :
for b1 being complete LATTICE
for b2 being set holds
( b2 = App b1 iff for b3 being set holds
( b3 in b2 iff b3 is auxiliary approximating Relation of b1 ) );

registration
let c1 be complete LATTICE;
cluster App a1 -> non empty ;
coherence
not App c1 is empty
proof end;
end;

theorem Th42: :: WAYBEL_4:42
for b1 being complete LATTICE
for b2 being Function of b1,(InclPoset (Ids b1)) holds
not ( b2 is approximating & b2 in the carrier of (MonSet b1) & ( for b3 being auxiliary approximating Relation of b1 holds
not b3 = (Map2Rel b1) . b2 ) )
proof end;

theorem Th43: :: WAYBEL_4:43
for b1 being complete LATTICE
for b2 being Element of b1 holds meet { ((DownMap b3) . b2) where B is Ideal of b1 : verum } = waybelow b2
proof end;

theorem Th44: :: WAYBEL_4:44
for b1 being lower-bounded meet-continuous LATTICE
for b2 being Element of b1 holds meet { (b3 -below b2) where B is auxiliary Relation of b1 : b3 in App b1 } = waybelow b2
proof end;

theorem Th45: :: WAYBEL_4:45
for b1 being complete LATTICE holds
( b1 is continuous iff for b2 being auxiliary approximating Relation of b1 holds
( b1 -waybelow c= b2 & b1 -waybelow is approximating ) )
proof end;

theorem Th46: :: WAYBEL_4:46
for b1 being complete LATTICE holds
( b1 is continuous iff ( b1 is meet-continuous & ex b2 being auxiliary approximating Relation of b1 st
for b3 being auxiliary approximating Relation of b1 holds b2 c= b3 ) )
proof end;

definition
let c1 be RelStr ;
let c2 be Relation of c1;
attr a2 is satisfying_SI means :Def21: :: WAYBEL_4:def 21
for b1, b2 being Element of a1 holds
not ( [b1,b2] in a2 & b1 <> b2 & ( for b3 being Element of a1 holds
not ( [b1,b3] in a2 & [b3,b2] in a2 & b1 <> b3 ) ) );
end;

:: deftheorem Def21 defines satisfying_SI WAYBEL_4:def 21 :
for b1 being RelStr
for b2 being Relation of b1 holds
( b2 is satisfying_SI iff for b3, b4 being Element of b1 holds
not ( [b3,b4] in b2 & b3 <> b4 & ( for b5 being Element of b1 holds
not ( [b3,b5] in b2 & [b5,b4] in b2 & b3 <> b5 ) ) ) );

notation
let c1 be RelStr ;
let c2 be Relation of c1;
synonym c2 satisfies_SI for satisfying_SI c2;
end;

definition
let c1 be RelStr ;
let c2 be Relation of c1;
attr a2 is satisfying_INT means :Def22: :: WAYBEL_4:def 22
for b1, b2 being Element of a1 holds
not ( [b1,b2] in a2 & ( for b3 being Element of a1 holds
not ( [b1,b3] in a2 & [b3,b2] in a2 ) ) );
end;

:: deftheorem Def22 defines satisfying_INT WAYBEL_4:def 22 :
for b1 being RelStr
for b2 being Relation of b1 holds
( b2 is satisfying_INT iff for b3, b4 being Element of b1 holds
not ( [b3,b4] in b2 & ( for b5 being Element of b1 holds
not ( [b3,b5] in b2 & [b5,b4] in b2 ) ) ) );

notation
let c1 be RelStr ;
let c2 be Relation of c1;
synonym c2 satisfies_INT for satisfying_INT c2;
end;

theorem Th47: :: WAYBEL_4:47
canceled;

theorem Th48: :: WAYBEL_4:48
for b1 being RelStr
for b2 being Relation of b1 holds
( b2 satisfies_SI implies b2 satisfies_INT )
proof end;

registration
let c1 be non empty RelStr ;
cluster satisfying_SI -> satisfying_INT Relation of the carrier of a1,the carrier of a1;
coherence
for b1 being Relation of c1 holds
( b1 is satisfying_SI implies b1 is satisfying_INT )
by Th48;
end;

theorem Th49: :: WAYBEL_4:49
for b1 being complete LATTICE
for b2, b3 being Element of b1
for b4 being approximating Relation of b1 holds
not ( not b2 <= b3 & ( for b5 being Element of b1 holds
not ( [b5,b2] in b4 & not b5 <= b3 ) ) )
proof end;

theorem Th50: :: WAYBEL_4:50
for b1 being complete LATTICE
for b2, b3 being Element of b1
for b4 being auxiliary(i) auxiliary(iii) approximating Relation of b1 holds
not ( [b2,b3] in b4 & b2 <> b3 & ( for b5 being Element of b1 holds
not ( b2 <= b5 & [b5,b3] in b4 & b2 <> b5 ) ) )
proof end;

theorem Th51: :: WAYBEL_4:51
for b1 being complete LATTICE
for b2, b3 being Element of b1
for b4 being auxiliary approximating Relation of b1 holds
not ( b2 << b3 & b2 <> b3 & ( for b5 being Element of b1 holds
not ( [b2,b5] in b4 & [b5,b3] in b4 & b2 <> b5 ) ) )
proof end;

theorem Th52: :: WAYBEL_4:52
for b1 being lower-bounded continuous LATTICE holds b1 -waybelow satisfies_SI
proof end;

registration
let c1 be lower-bounded continuous LATTICE;
cluster a1 -waybelow -> transitive auxiliary(i) auxiliary(ii) auxiliary(iii) auxiliary(iv) auxiliary approximating satisfying_SI satisfying_INT ;
coherence
c1 -waybelow is satisfying_SI
by Th52;
end;

theorem Th53: :: WAYBEL_4:53
for b1 being lower-bounded continuous LATTICE
for b2, b3 being Element of b1 holds
not ( b2 << b3 & ( for b4 being Element of b1 holds
not ( b2 << b4 & b4 << b3 ) ) )
proof end;

theorem Th54: :: WAYBEL_4:54
for b1 being lower-bounded continuous LATTICE
for b2, b3 being Element of b1 holds
( b2 << b3 iff for b4 being non empty directed Subset of b1 holds
not ( b3 <= sup b4 & ( for b5 being Element of b1 holds
not ( b5 in b4 & b2 << b5 ) ) ) )
proof end;

definition
let c1 be RelStr ;
let c2 be Subset of c1;
let c3 be Relation of the carrier of c1;
pred c2 is_directed_wrt c3 means :Def23: :: WAYBEL_4:def 23
for b1, b2 being Element of a1 holds
not ( b1 in a2 & b2 in a2 & ( for b3 being Element of a1 holds
not ( b3 in a2 & [b1,b3] in a3 & [b2,b3] in a3 ) ) );
end;

:: deftheorem Def23 defines is_directed_wrt WAYBEL_4:def 23 :
for b1 being RelStr
for b2 being Subset of b1
for b3 being Relation of the carrier of b1 holds
( b2 is_directed_wrt b3 iff for b4, b5 being Element of b1 holds
not ( b4 in b2 & b5 in b2 & ( for b6 being Element of b1 holds
not ( b6 in b2 & [b4,b6] in b3 & [b5,b6] in b3 ) ) ) );

theorem Th55: :: WAYBEL_4:55
for b1 being RelStr
for b2 being Subset of b1 holds
( b2 is_directed_wrt the InternalRel of b1 implies b2 is directed )
proof end;

definition
let c1, c2 be set ;
let c3 be Relation;
pred c2 is_maximal_wrt c1,c3 means :Def24: :: WAYBEL_4:def 24
( a2 in a1 & ( for b1 being set holds
not ( b1 in a1 & b1 <> a2 & [a2,b1] in a3 ) ) );
end;

:: deftheorem Def24 defines is_maximal_wrt WAYBEL_4:def 24 :
for b1, b2 being set
for b3 being Relation holds
( b2 is_maximal_wrt b1,b3 iff ( b2 in b1 & ( for b4 being set holds
not ( b4 in b1 & b4 <> b2 & [b2,b4] in b3 ) ) ) );

definition
let c1 be RelStr ;
let c2 be set ;
let c3 be Element of c1;
pred c3 is_maximal_in c2 means :Def25: :: WAYBEL_4:def 25
a3 is_maximal_wrt a2,the InternalRel of a1;
end;

:: deftheorem Def25 defines is_maximal_in WAYBEL_4:def 25 :
for b1 being RelStr
for b2 being set
for b3 being Element of b1 holds
( b3 is_maximal_in b2 iff b3 is_maximal_wrt b2,the InternalRel of b1 );

theorem Th56: :: WAYBEL_4:56
for b1 being RelStr
for b2 being set
for b3 being Element of b1 holds
( b3 is_maximal_in b2 iff ( b3 in b2 & ( for b4 being Element of b1 holds
not ( b4 in b2 & b3 < b4 ) ) ) )
proof end;

definition
let c1, c2 be set ;
let c3 be Relation;
pred c2 is_minimal_wrt c1,c3 means :Def26: :: WAYBEL_4:def 26
( a2 in a1 & ( for b1 being set holds
not ( b1 in a1 & b1 <> a2 & [b1,a2] in a3 ) ) );
end;

:: deftheorem Def26 defines is_minimal_wrt WAYBEL_4:def 26 :
for b1, b2 being set
for b3 being Relation holds
( b2 is_minimal_wrt b1,b3 iff ( b2 in b1 & ( for b4 being set holds
not ( b4 in b1 & b4 <> b2 & [b4,b2] in b3 ) ) ) );

definition
let c1 be RelStr ;
let c2 be set ;
let c3 be Element of c1;
pred c3 is_minimal_in c2 means :Def27: :: WAYBEL_4:def 27
a3 is_minimal_wrt a2,the InternalRel of a1;
end;

:: deftheorem Def27 defines is_minimal_in WAYBEL_4:def 27 :
for b1 being RelStr
for b2 being set
for b3 being Element of b1 holds
( b3 is_minimal_in b2 iff b3 is_minimal_wrt b2,the InternalRel of b1 );

theorem Th57: :: WAYBEL_4:57
for b1 being RelStr
for b2 being set
for b3 being Element of b1 holds
( b3 is_minimal_in b2 iff ( b3 in b2 & ( for b4 being Element of b1 holds
not ( b4 in b2 & b3 > b4 ) ) ) )
proof end;

theorem Th58: :: WAYBEL_4:58
for b1 being complete LATTICE
for b2 being Relation of b1 holds
( b2 satisfies_SI implies for b3 being Element of b1 holds
( ex b4 being Element of b1 st b4 is_maximal_wrt b2 -below b3,b2 implies [b3,b3] in b2 ) )
proof end;

theorem Th59: :: WAYBEL_4:59
for b1 being complete LATTICE
for b2 being Relation of b1 holds
( ( for b3 being Element of b1 holds
( ex b4 being Element of b1 st b4 is_maximal_wrt b2 -below b3,b2 implies [b3,b3] in b2 ) ) implies b2 satisfies_SI )
proof end;

theorem Th60: :: WAYBEL_4:60
for b1 being complete LATTICE
for b2 being auxiliary(ii) auxiliary(iii) Relation of b1 holds
( b2 satisfies_INT implies for b3 being Element of b1 holds b2 -below b3 is_directed_wrt b2 )
proof end;

theorem Th61: :: WAYBEL_4:61
for b1 being complete LATTICE
for b2 being Relation of b1 holds
( ( for b3 being Element of b1 holds b2 -below b3 is_directed_wrt b2 ) implies b2 satisfies_INT )
proof end;

theorem Th62: :: WAYBEL_4:62
for b1 being complete LATTICE
for b2 being auxiliary(i) auxiliary(ii) auxiliary(iii) approximating Relation of b1 holds
( b2 satisfies_INT implies b2 satisfies_SI )
proof end;

registration
let c1 be complete LATTICE;
cluster auxiliary approximating satisfying_INT -> transitive auxiliary(i) auxiliary(ii) auxiliary(iii) auxiliary(iv) auxiliary approximating satisfying_SI satisfying_INT Relation of the carrier of a1,the carrier of a1;
coherence
for b1 being auxiliary approximating Relation of c1 holds
( b1 is satisfying_INT implies b1 is satisfying_SI )
by Th62;
end;