Article WAYBEL13, MML version 4.99.1005

:: WAYBEL13:th 1
theorem
for b1 being non empty reflexive transitive RelStr
for b2, b3 being Element of the carrier of b1
      st b2 <= b3
   holds compactbelow b2 c= compactbelow b3;

:: WAYBEL13:th 2
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of the carrier of b1 holds
   compactbelow b2 is Element of bool the carrier of CompactSublatt b1;

:: WAYBEL13:th 3
theorem
for b1 being RelStr
for b2 being SubRelStr of b1
for b3 being Element of bool the carrier of b2 holds
   b3 is Element of bool the carrier of b1;

:: WAYBEL13:th 4
theorem
for b1 being non empty reflexive transitive with_suprema RelStr holds
   the carrier of b1 is non empty directed lower Element of bool the carrier of b1;

:: WAYBEL13:th 5
theorem
for b1 being non empty reflexive antisymmetric lower-bounded RelStr
for b2 being non empty reflexive antisymmetric RelStr
      st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#) &
         b1 is up-complete
   holds the carrier of CompactSublatt b1 = the carrier of CompactSublatt b2;

:: WAYBEL13:th 6
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded algebraic RelStr
for b2 being non empty full infs-inheriting directed-sups-inheriting SubRelStr of b1 holds
   b2 is algebraic;

:: WAYBEL13:th 7
theorem
for b1, b2 being set
for b3 being non empty full infs-inheriting directed-sups-inheriting SubRelStr of BoolePoset b1 holds
      b2 in the carrier of CompactSublatt b3
   iff
      ex b4 being Element of the carrier of BoolePoset b1 st
         b4 is finite &
          b2 = meet {b5 where b5 is Element of the carrier of b3: b4 c= b5} &
          b4 c= b2;

:: WAYBEL13:th 8
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded RelStr holds
   InclPoset Ids b1 is non empty full infs-inheriting directed-sups-inheriting SubRelStr of BoolePoset the carrier of b1;

:: WAYBEL13:exreg 1
registration
  let a1 be non empty reflexive transitive RelStr;
  cluster non empty directed lower principal Element of bool the carrier of a1;
end;

:: WAYBEL13:th 9
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded RelStr
for b2 being non empty directed Element of bool the carrier of InclPoset Ids b1 holds
   "\/"(b2,InclPoset Ids b1) = union b2;

:: WAYBEL13:th 10
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded RelStr holds
   InclPoset Ids b1 is algebraic;

:: WAYBEL13:th 11
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded RelStr
for b2 being Element of the carrier of InclPoset Ids b1 holds
      b2 is compact(InclPoset Ids b1)
   iff
      b2 is non empty directed lower principal Element of bool the carrier of b1;

:: WAYBEL13:th 12
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded RelStr
for b2 being Element of the carrier of InclPoset Ids b1 holds
      b2 is compact(InclPoset Ids b1)
   iff
      ex b3 being Element of the carrier of b1 st
         b2 = downarrow b3;

:: WAYBEL13:th 13
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded RelStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of CompactSublatt InclPoset Ids b1
      st for b3 being Element of the carrier of b1 holds
           b2 . b3 = downarrow b3
   holds b2 is isomorphic(b1, CompactSublatt InclPoset Ids b1);

:: WAYBEL13:th 14
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr holds
   InclPoset Ids b1 is arithmetic;

:: WAYBEL13:th 15
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded RelStr holds
   CompactSublatt b1 is reflexive transitive antisymmetric with_suprema lower-bounded RelStr;

:: WAYBEL13:th 16
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded algebraic RelStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of InclPoset Ids CompactSublatt b1
      st for b3 being Element of the carrier of b1 holds
           b2 . b3 = compactbelow b3
   holds b2 is isomorphic(b1, InclPoset Ids CompactSublatt b1);

:: WAYBEL13:th 17
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded algebraic RelStr
for b2 being Element of the carrier of b1 holds
      compactbelow b2 is non empty directed lower principal Element of bool the carrier of CompactSublatt b1
   iff
      b2 is compact(b1);

:: WAYBEL13:th 18
theorem
for b1, b2 being non empty RelStr
for b3 being Element of bool the carrier of b1
for b4 being Element of the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b5 is isomorphic(b1, b2)
   holds    b4 is_<=_than b3
   iff
      b5 . b4 is_<=_than b5 .: b3;

:: WAYBEL13:th 19
theorem
for b1, b2 being non empty RelStr
for b3 being Element of bool the carrier of b1
for b4 being Element of the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b5 is isomorphic(b1, b2)
   holds    b3 is_<=_than b4
   iff
      b5 .: b3 is_<=_than b5 . b4;

:: WAYBEL13:th 20
theorem
for b1, b2 being non empty antisymmetric RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is isomorphic(b1, b2)
   holds b3 is infs-preserving(b1, b2) & b3 is sups-preserving(b1, b2);

:: WAYBEL13:condreg 1
registration
  let a1, a2 be non empty antisymmetric RelStr;
  cluster Function-like quasi_total isomorphic -> infs-preserving sups-preserving (Relation of the carrier of a1,the carrier of a2);
end;

:: WAYBEL13:th 21
theorem
for b1, b2, b3 being non empty transitive antisymmetric RelStr
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b4 is infs-preserving(b1, b2) & b2 is full infs-inheriting SubRelStr of b3 & b3 is complete
   holds ex b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3 st
      b4 = b5 & b5 is infs-preserving(b1, b3);

:: WAYBEL13:th 22
theorem
for b1, b2, b3 being non empty transitive antisymmetric RelStr
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b4 is monotone(b1, b2) & b4 is directed-sups-preserving(b1, b2) & b2 is full directed-sups-inheriting SubRelStr of b3 & b3 is complete
   holds ex b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3 st
      b4 = b5 & b5 is directed-sups-preserving(b1, b3);

:: WAYBEL13:th 23
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded RelStr holds
   InclPoset Ids CompactSublatt b1 is non empty full infs-inheriting directed-sups-inheriting SubRelStr of BoolePoset the carrier of CompactSublatt b1;

:: WAYBEL13:th 24
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded algebraic RelStr holds
   ex b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of BoolePoset the carrier of CompactSublatt b1 st
      b2 is infs-preserving(b1, BoolePoset the carrier of CompactSublatt b1) &
       b2 is directed-sups-preserving(b1, BoolePoset the carrier of CompactSublatt b1) &
       b2 is one-to-one &
       (for b3 being Element of the carrier of b1 holds
          b2 . b3 = compactbelow b3);

:: WAYBEL13:th 25
theorem
for b1 being non empty set
for b2 being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of b1
      st for b3 being Element of b1 holds
           b2 . b3 is reflexive transitive antisymmetric with_suprema with_infima lower-bounded algebraic RelStr
   holds product b2 is reflexive transitive antisymmetric with_suprema with_infima lower-bounded algebraic RelStr;

:: WAYBEL13:th 26
theorem
for b1, b2 being non empty RelStr
      st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#)
   holds b1,b2 are_isomorphic;

:: WAYBEL13:th 27
theorem
for b1, b2 being non empty reflexive transitive antisymmetric up-complete RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
   st b3 is isomorphic(b1, b2)
for b4, b5 being Element of the carrier of b1 holds
   b4 is_way_below b5
iff
   b3 . b4 is_way_below b3 . b5;

:: WAYBEL13:th 28
theorem
for b1, b2 being non empty reflexive transitive antisymmetric up-complete RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
   st b3 is isomorphic(b1, b2)
for b4 being Element of the carrier of b1 holds
      b4 is compact(b1)
   iff
      b3 . b4 is compact(b2);

:: WAYBEL13:th 29
theorem
for b1, b2 being non empty reflexive transitive antisymmetric up-complete RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
   st b3 is isomorphic(b1, b2)
for b4 being Element of the carrier of b1 holds
   b3 .: compactbelow b4 = compactbelow (b3 . b4);

:: WAYBEL13:th 30
theorem
for b1, b2 being non empty reflexive transitive antisymmetric RelStr
      st b1,b2 are_isomorphic & b1 is up-complete
   holds b2 is up-complete;

:: WAYBEL13:th 31
theorem
for b1, b2 being non empty reflexive transitive antisymmetric RelStr
      st b1,b2 are_isomorphic & b1 is complete & b1 is satisfying_axiom_K
   holds b2 is satisfying_axiom_K;

:: WAYBEL13:th 32
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema RelStr
      st b1,b2 are_isomorphic & b1 is lower-bounded & b1 is algebraic
   holds b2 is algebraic;

:: WAYBEL13:th 33
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded continuous RelStr holds
   SupMap b1 is infs-preserving(InclPoset Ids b1, b1) & SupMap b1 is sups-preserving(InclPoset Ids b1, b1);

:: WAYBEL13:th 34
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr holds
   (b1 is algebraic implies ex b2 being non empty set st
       ex b3 being full SubRelStr of BoolePoset b2 st
          b3 is infs-inheriting(BoolePoset b2) & b3 is directed-sups-inheriting(BoolePoset b2) & b1,b3 are_isomorphic) &
    (for b2 being set
    for b3 being full SubRelStr of BoolePoset b2
          st b3 is infs-inheriting(BoolePoset b2) & b3 is directed-sups-inheriting(BoolePoset b2)
       holds not b1,b3 are_isomorphic or b1 is algebraic);

:: WAYBEL13:th 35
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr holds
   (b1 is algebraic implies ex b2 being non empty set st
       ex b3 being Function-like quasi_total closure Relation of the carrier of BoolePoset b2,the carrier of BoolePoset b2 st
          b3 is directed-sups-preserving(BoolePoset b2, BoolePoset b2) & b1,Image b3 are_isomorphic) &
    (for b2 being set
    for b3 being Function-like quasi_total closure Relation of the carrier of BoolePoset b2,the carrier of BoolePoset b2
          st b3 is directed-sups-preserving(BoolePoset b2, BoolePoset b2)
       holds not b1,Image b3 are_isomorphic or b1 is algebraic);