Article WAYBEL34, MML version 4.99.1005

:: WAYBEL34:exreg 1
registration
  let a1, a2 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  cluster Galois Connection of a1,a2;
end;

:: WAYBEL34:th 1
theorem
for b1, b2, b3, b4 being non empty RelStr
   st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b3,the InternalRel of b3#) &
      RelStr(#the carrier of b2,the InternalRel of b2#) = RelStr(#the carrier of b4,the InternalRel of b4#)
for b5 being Connection of b1,b2
for b6 being Connection of b3,b4
      st b5 = b6 & b5 is Galois(b1, b2)
   holds b6 is Galois(b3, b4);

:: WAYBEL34:funcnot 1 => WAYBEL34:func 1
definition
  let a1, a2 be reflexive transitive antisymmetric with_suprema with_infima RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  assume a1 is complete & a2 is complete & a3 is infs-preserving(a1, a2);
  func LowerAdj A3 -> Function-like quasi_total Relation of the carrier of a2,the carrier of a1 means
    [a3,it] is Galois(a1, a2);
end;

:: WAYBEL34:def 1
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
   st b1 is complete & b2 is complete & b3 is infs-preserving(b1, b2)
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1 holds
      b4 = LowerAdj b3
   iff
      [b3,b4] is Galois(b1, b2);

:: WAYBEL34:funcnot 2 => WAYBEL34:func 2
definition
  let a1, a2 be reflexive transitive antisymmetric with_suprema with_infima RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a2,the carrier of a1;
  assume a1 is complete & a2 is complete & a3 is sups-preserving(a2, a1);
  func UpperAdj A3 -> Function-like quasi_total Relation of the carrier of a1,the carrier of a2 means
    [it,a3] is Galois(a1, a2);
end;

:: WAYBEL34:def 2
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
   st b1 is complete & b2 is complete & b3 is sups-preserving(b2, b1)
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b4 = UpperAdj b3
   iff
      [b4,b3] is Galois(b1, b2);

:: WAYBEL34:funcreg 1
registration
  let a1, a2 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  let a3 be Function-like quasi_total infs-preserving Relation of the carrier of a1,the carrier of a2;
  cluster LowerAdj a3 -> Function-like quasi_total lower_adjoint;
end;

:: WAYBEL34:funcreg 2
registration
  let a1, a2 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  let a3 be Function-like quasi_total sups-preserving Relation of the carrier of a2,the carrier of a1;
  cluster UpperAdj a3 -> Function-like quasi_total upper_adjoint;
end;

:: WAYBEL34:th 2
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total infs-preserving Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b2 holds
   (LowerAdj b3) . b4 = "/\"(b3 " uparrow b4,b1);

:: WAYBEL34:th 3
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total sups-preserving Relation of the carrier of b2,the carrier of b1
for b4 being Element of the carrier of b1 holds
   (UpperAdj b3) . b4 = "\/"(b3 " downarrow b4,b2);

:: WAYBEL34:funcnot 3 => WAYBEL34:func 3
definition
  let a1, a2 be RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  func A3 opp -> Function-like quasi_total Relation of the carrier of a1 ~,the carrier of a2 ~ equals
    a3;
end;

:: WAYBEL34:def 3
theorem
for b1, b2 being RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
   b3 opp = b3;

:: WAYBEL34:funcreg 3
registration
  let a1, a2 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  let a3 be Function-like quasi_total infs-preserving Relation of the carrier of a1,the carrier of a2;
  cluster a3 opp -> Function-like quasi_total lower_adjoint;
end;

:: WAYBEL34:funcreg 4
registration
  let a1, a2 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  let a3 be Function-like quasi_total sups-preserving Relation of the carrier of a1,the carrier of a2;
  cluster a3 opp -> Function-like quasi_total upper_adjoint;
end;

:: WAYBEL34:th 4
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total infs-preserving Relation of the carrier of b1,the carrier of b2 holds
   LowerAdj b3 = UpperAdj (b3 opp);

:: WAYBEL34:th 5
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total sups-preserving Relation of the carrier of b1,the carrier of b2 holds
   LowerAdj (b3 opp) = UpperAdj b3;

:: WAYBEL34:th 6
theorem
for b1 being non empty RelStr holds
   [id b1,id b1] is Galois(b1, b1);

:: WAYBEL34:th 7
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
   LowerAdj id b1 = id b1 & UpperAdj id b1 = id b1;

:: WAYBEL34:th 8
theorem
for b1, b2, b3 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b4 being Function-like quasi_total infs-preserving Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total infs-preserving Relation of the carrier of b2,the carrier of b3 holds
   LowerAdj (b5 * b4) = (LowerAdj b4) * LowerAdj b5;

:: WAYBEL34:th 9
theorem
for b1, b2, b3 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b4 being Function-like quasi_total sups-preserving Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total sups-preserving Relation of the carrier of b2,the carrier of b3 holds
   UpperAdj (b5 * b4) = (UpperAdj b4) * UpperAdj b5;

:: WAYBEL34:th 10
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total infs-preserving Relation of the carrier of b1,the carrier of b2 holds
   UpperAdj LowerAdj b3 = b3;

:: WAYBEL34:th 11
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total sups-preserving Relation of the carrier of b1,the carrier of b2 holds
   LowerAdj UpperAdj b3 = b3;

:: WAYBEL34:th 12
theorem
for b1 being non empty AltCatStr
for b2, b3, b4 being set
      st b4 in (the Arrows of b1) .(b2,b3)
   holds ex b5, b6 being Element of the carrier of b1 st
      b5 = b2 & b6 = b3 & b4 in <^b5,b6^> & b4 is Element of <^b5,b6^>;

:: WAYBEL34:funcnot 4 => WAYBEL34:func 4
definition
  let a1 be non empty set;
  assume ex b1 being Element of a1 st
       b1 is not empty;
  func A1 -INF_category -> non empty transitive strict associative with_units lattice-wise AltCatStr means
    (for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr holds
           b1 is Element of the carrier of it
        iff
           b1 is strict & b1 is complete & the carrier of b1 in a1) &
     (for b1, b2 being Element of the carrier of it
     for b3 being Function-like quasi_total monotone Relation of the carrier of latt b1,the carrier of latt b2 holds
           b3 in <^b1,b2^>
        iff
           b3 is infs-preserving(latt b1, latt b2));
end;

:: WAYBEL34:def 4
theorem
for b1 being non empty set
   st ex b2 being Element of b1 st
        b2 is not empty
for b2 being non empty transitive strict associative with_units lattice-wise AltCatStr holds
      b2 = b1 -INF_category
   iff
      (for b3 being reflexive transitive antisymmetric with_suprema with_infima RelStr holds
             b3 is Element of the carrier of b2
          iff
             b3 is strict & b3 is complete & the carrier of b3 in b1) &
       (for b3, b4 being Element of the carrier of b2
       for b5 being Function-like quasi_total monotone Relation of the carrier of latt b3,the carrier of latt b4 holds
             b5 in <^b3,b4^>
          iff
             b5 is infs-preserving(latt b3, latt b4));

:: WAYBEL34:funcnot 5 => WAYBEL34:func 5
definition
  let a1 be non empty set;
  assume ex b1 being Element of a1 st
       b1 is not empty;
  func A1 -SUP_category -> non empty transitive strict associative with_units lattice-wise AltCatStr means
    (for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr holds
           b1 is Element of the carrier of it
        iff
           b1 is strict & b1 is complete & the carrier of b1 in a1) &
     (for b1, b2 being Element of the carrier of it
     for b3 being Function-like quasi_total monotone Relation of the carrier of latt b1,the carrier of latt b2 holds
           b3 in <^b1,b2^>
        iff
           b3 is sups-preserving(latt b1, latt b2));
end;

:: WAYBEL34:def 5
theorem
for b1 being non empty set
   st ex b2 being Element of b1 st
        b2 is not empty
for b2 being non empty transitive strict associative with_units lattice-wise AltCatStr holds
      b2 = b1 -SUP_category
   iff
      (for b3 being reflexive transitive antisymmetric with_suprema with_infima RelStr holds
             b3 is Element of the carrier of b2
          iff
             b3 is strict & b3 is complete & the carrier of b3 in b1) &
       (for b3, b4 being Element of the carrier of b2
       for b5 being Function-like quasi_total monotone Relation of the carrier of latt b3,the carrier of latt b4 holds
             b5 in <^b3,b4^>
          iff
             b5 is sups-preserving(latt b3, latt b4));

:: WAYBEL34:funcreg 5
registration
  let a1 be with_non-empty_element set;
  cluster a1 -INF_category -> non empty transitive strict associative with_units lattice-wise with_complete_lattices;
end;

:: WAYBEL34:funcreg 6
registration
  let a1 be with_non-empty_element set;
  cluster a1 -SUP_category -> non empty transitive strict associative with_units lattice-wise with_complete_lattices;
end;

:: WAYBEL34:th 13
theorem
for b1 being with_non-empty_element set
for b2 being reflexive transitive antisymmetric with_suprema with_infima RelStr holds
      b2 is Element of the carrier of b1 -INF_category
   iff
      b2 is strict & b2 is complete & the carrier of b2 in b1;

:: WAYBEL34:th 14
theorem
for b1 being with_non-empty_element set
for b2, b3 being Element of the carrier of b1 -INF_category
for b4 being set holds
      b4 in <^b2,b3^>
   iff
      b4 is Function-like quasi_total infs-preserving Relation of the carrier of latt b2,the carrier of latt b3;

:: WAYBEL34:th 15
theorem
for b1 being with_non-empty_element set
for b2 being reflexive transitive antisymmetric with_suprema with_infima RelStr holds
      b2 is Element of the carrier of b1 -SUP_category
   iff
      b2 is strict & b2 is complete & the carrier of b2 in b1;

:: WAYBEL34:th 16
theorem
for b1 being with_non-empty_element set
for b2, b3 being Element of the carrier of b1 -SUP_category
for b4 being set holds
      b4 in <^b2,b3^>
   iff
      b4 is Function-like quasi_total sups-preserving Relation of the carrier of latt b2,the carrier of latt b3;

:: WAYBEL34:th 17
theorem
for b1 being with_non-empty_element set holds
   the carrier of b1 -INF_category = the carrier of b1 -SUP_category;

:: WAYBEL34:funcnot 6 => WAYBEL34:func 6
definition
  let a1 be with_non-empty_element set;
  func A1 LowerAdj -> strict contravariant Functor of a1 -INF_category,a1 -SUP_category means
    (for b1 being Element of the carrier of a1 -INF_category holds
        it . b1 = latt b1) &
     (for b1, b2 being Element of the carrier of a1 -INF_category
        st <^b1,b2^> <> {}
     for b3 being Element of <^b1,b2^> holds
        it . b3 = LowerAdj @ b3);
end;

:: WAYBEL34:def 6
theorem
for b1 being with_non-empty_element set
for b2 being strict contravariant Functor of b1 -INF_category,b1 -SUP_category holds
      b2 = b1 LowerAdj
   iff
      (for b3 being Element of the carrier of b1 -INF_category holds
          b2 . b3 = latt b3) &
       (for b3, b4 being Element of the carrier of b1 -INF_category
          st <^b3,b4^> <> {}
       for b5 being Element of <^b3,b4^> holds
          b2 . b5 = LowerAdj @ b5);

:: WAYBEL34:funcnot 7 => WAYBEL34:func 7
definition
  let a1 be with_non-empty_element set;
  func A1 UpperAdj -> strict contravariant Functor of a1 -SUP_category,a1 -INF_category means
    (for b1 being Element of the carrier of a1 -SUP_category holds
        it . b1 = latt b1) &
     (for b1, b2 being Element of the carrier of a1 -SUP_category
        st <^b1,b2^> <> {}
     for b3 being Element of <^b1,b2^> holds
        it . b3 = UpperAdj @ b3);
end;

:: WAYBEL34:def 7
theorem
for b1 being with_non-empty_element set
for b2 being strict contravariant Functor of b1 -SUP_category,b1 -INF_category holds
      b2 = b1 UpperAdj
   iff
      (for b3 being Element of the carrier of b1 -SUP_category holds
          b2 . b3 = latt b3) &
       (for b3, b4 being Element of the carrier of b1 -SUP_category
          st <^b3,b4^> <> {}
       for b5 being Element of <^b3,b4^> holds
          b2 . b5 = UpperAdj @ b5);

:: WAYBEL34:funcreg 7
registration
  let a1 be with_non-empty_element set;
  cluster a1 LowerAdj -> strict contravariant bijective;
end;

:: WAYBEL34:funcreg 8
registration
  let a1 be with_non-empty_element set;
  cluster a1 UpperAdj -> strict contravariant bijective;
end;

:: WAYBEL34:th 18
theorem
for b1 being with_non-empty_element set holds
   b1 LowerAdj " = b1 UpperAdj & b1 UpperAdj " = b1 LowerAdj;

:: WAYBEL34:th 19
theorem
for b1 being with_non-empty_element set holds
   b1 LowerAdj * (b1 UpperAdj) = id (b1 -SUP_category) &
    b1 UpperAdj * (b1 LowerAdj) = id (b1 -INF_category);

:: WAYBEL34:th 20
theorem
for b1 being with_non-empty_element set holds
   b1 -INF_category,b1 -SUP_category are_anti-isomorphic;

:: WAYBEL34:th 23
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total infs-preserving Relation of the carrier of b1,the carrier of b2 holds
      b3 is directed-sups-preserving(b1, b2)
   iff
      for b4 being Scott TopAugmentation of b2
      for b5 being Scott TopAugmentation of b1
      for b6 being open Element of bool the carrier of b4 holds
         uparrow ((LowerAdj b3) .: b6) is open Element of bool the carrier of b5;

:: WAYBEL34:attrnot 1 => WAYBEL34:attr 1
definition
  let a1, a2 be non empty reflexive RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is waybelow-preserving means
    for b1, b2 being Element of the carrier of a1
          st b1 is_way_below b2
       holds a3 . b1 is_way_below a3 . b2;
end;

:: WAYBEL34:dfs 8
definiens
  let a1, a2 be non empty reflexive RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is waybelow-preserving
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1
          st b1 is_way_below b2
       holds a3 . b1 is_way_below a3 . b2;

:: WAYBEL34:def 8
theorem
for b1, b2 being non empty reflexive RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is waybelow-preserving(b1, b2)
   iff
      for b4, b5 being Element of the carrier of b1
            st b4 is_way_below b5
         holds b3 . b4 is_way_below b3 . b5;

:: WAYBEL34:th 24
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total infs-preserving Relation of the carrier of b1,the carrier of b2
      st b3 is directed-sups-preserving(b1, b2)
   holds LowerAdj b3 is waybelow-preserving(b2, b1);

:: WAYBEL34:th 25
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being reflexive transitive antisymmetric continuous with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total infs-preserving Relation of the carrier of b1,the carrier of b2
      st LowerAdj b3 is waybelow-preserving(b2, b1)
   holds b3 is directed-sups-preserving(b1, b2);

:: WAYBEL34:attrnot 2 => WAYBEL34:attr 2
definition
  let a1, a2 be TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is relatively_open means
    for b1 being open Element of bool the carrier of a1 holds
       a3 .: b1 is open Element of bool the carrier of a2 | rng a3;
end;

:: WAYBEL34:dfs 9
definiens
  let a1, a2 be TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is relatively_open
it is sufficient to prove
  thus for b1 being open Element of bool the carrier of a1 holds
       a3 .: b1 is open Element of bool the carrier of a2 | rng a3;

:: WAYBEL34:def 9
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is relatively_open(b1, b2)
   iff
      for b4 being open Element of bool the carrier of b1 holds
         b3 .: b4 is open Element of bool the carrier of b2 | rng b3;

:: WAYBEL34:th 26
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is relatively_open(b1, b2)
   iff
      corestr b3 is open(b1, Image b3);

:: WAYBEL34:th 27
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total infs-preserving Relation of the carrier of b1,the carrier of b2
for b4 being Scott TopAugmentation of b2
for b5 being Scott TopAugmentation of b1
for b6 being open Element of bool the carrier of b4 holds
   (LowerAdj b3) .: b6 = (rng LowerAdj b3) /\ uparrow ((LowerAdj b3) .: b6);

:: WAYBEL34:th 28
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total infs-preserving Relation of the carrier of b1,the carrier of b2
for b4 being Scott TopAugmentation of b2
for b5 being Scott TopAugmentation of b1
   st for b6 being open Element of bool the carrier of b4 holds
        uparrow ((LowerAdj b3) .: b6) is open Element of bool the carrier of b5
for b6 being Function-like quasi_total Relation of the carrier of b4,the carrier of b5
      st b6 = LowerAdj b3
   holds b6 is relatively_open(b4, b5);

:: WAYBEL34:funcreg 9
registration
  let a1, a2 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  let a3 be Function-like quasi_total sups-preserving Relation of the carrier of a1,the carrier of a2;
  cluster Image a3 -> strict full complete;
end;

:: WAYBEL34:th 29
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total infs-preserving Relation of the carrier of b1,the carrier of b2
for b4 being Scott TopAugmentation of b2
for b5 being Scott TopAugmentation of b1
for b6 being Scott TopAugmentation of Image LowerAdj b3
for b7 being Function-like quasi_total Relation of the carrier of b4,the carrier of b5
for b8 being Function-like quasi_total Relation of the carrier of b4,the carrier of b6
      st b7 = LowerAdj b3 & b8 = b7 & b7 is relatively_open(b4, b5)
   holds b8 is open(b4, b6);

:: WAYBEL34:th 32
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total infs-preserving Relation of the carrier of b1,the carrier of b2
   st b3 is one-to-one
for b4 being Scott TopAugmentation of b2
for b5 being Scott TopAugmentation of b1
for b6 being Function-like quasi_total Relation of the carrier of b4,the carrier of b5
      st b6 = LowerAdj b3
   holds    b3 is directed-sups-preserving(b1, b2)
   iff
      b6 is open(b4, b5);

:: WAYBEL34:funcreg 10
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  let a2 be Function-like quasi_total projection Relation of the carrier of a1,the carrier of a1;
  cluster Image a2 -> strict full complete;
end;

:: WAYBEL34:th 33
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Function-like quasi_total kernel Relation of the carrier of b1,the carrier of b1 holds
   corestr b2 is infs-preserving(b1, Image b2) & inclusion b2 is sups-preserving(Image b2, b1) & LowerAdj corestr b2 = inclusion b2 & UpperAdj inclusion b2 = corestr b2;

:: WAYBEL34:th 34
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Function-like quasi_total kernel Relation of the carrier of b1,the carrier of b1 holds
      b2 is directed-sups-preserving(b1, b1)
   iff
      corestr b2 is directed-sups-preserving(b1, Image b2);

:: WAYBEL34:th 35
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Function-like quasi_total kernel Relation of the carrier of b1,the carrier of b1 holds
      b2 is directed-sups-preserving(b1, b1)
   iff
      for b3 being Scott TopAugmentation of Image b2
      for b4 being Scott TopAugmentation of b1
      for b5 being Element of bool the carrier of b1
            st b5 is open Element of bool the carrier of b3
         holds uparrow b5 is open Element of bool the carrier of b4;

:: WAYBEL34:th 36
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty full sups-inheriting SubRelStr of b1
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
      st b5 = b3 & b6 = b4 & b3 is_way_below b4
   holds b5 is_way_below b6;

:: WAYBEL34:th 37
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Function-like quasi_total kernel Relation of the carrier of b1,the carrier of b1
   st b2 is directed-sups-preserving(b1, b1)
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of Image b2
      st b5 = b3 & b6 = b4
   holds    b3 is_way_below b4
   iff
      b5 is_way_below b6;

:: WAYBEL34:th 38
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Function-like quasi_total kernel Relation of the carrier of b1,the carrier of b1
      st Image b2 is continuous &
         (for b3, b4 being Element of the carrier of b1
         for b5, b6 being Element of the carrier of Image b2
               st b5 = b3 & b6 = b4
            holds    b3 is_way_below b4
            iff
               b5 is_way_below b6)
   holds b2 is directed-sups-preserving(b1, b1);

:: WAYBEL34:th 39
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Function-like quasi_total closure Relation of the carrier of b1,the carrier of b1 holds
   corestr b2 is sups-preserving(b1, Image b2) & inclusion b2 is infs-preserving(Image b2, b1) & UpperAdj corestr b2 = inclusion b2 & LowerAdj inclusion b2 = corestr b2;

:: WAYBEL34:th 40
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Function-like quasi_total closure Relation of the carrier of b1,the carrier of b1 holds
      Image b2 is directed-sups-inheriting(b1)
   iff
      inclusion b2 is directed-sups-preserving(Image b2, b1);

:: WAYBEL34:th 41
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Function-like quasi_total closure Relation of the carrier of b1,the carrier of b1 holds
      Image b2 is directed-sups-inheriting(b1)
   iff
      for b3 being Scott TopAugmentation of Image b2
      for b4 being Scott TopAugmentation of b1
      for b5 being Function-like quasi_total Relation of the carrier of b4,the carrier of b3
            st b5 = b2
         holds b5 is open(b4, b3);

:: WAYBEL34:th 42
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Function-like quasi_total closure Relation of the carrier of b1,the carrier of b1
      st Image b2 is directed-sups-inheriting(b1)
   holds corestr b2 is waybelow-preserving(b1, Image b2);

:: WAYBEL34:th 43
theorem
for b1 being reflexive transitive antisymmetric continuous with_suprema with_infima complete RelStr
for b2 being Function-like quasi_total closure Relation of the carrier of b1,the carrier of b1
      st corestr b2 is waybelow-preserving(b1, Image b2)
   holds Image b2 is directed-sups-inheriting(b1);

:: WAYBEL34:funcnot 8 => WAYBEL34:func 8
definition
  let a1 be non empty set;
  func A1 -INF(SC)_category -> non empty transitive strict id-inheriting SubCatStr of a1 -INF_category means
    (for b1 being Element of the carrier of a1 -INF_category holds
        b1 is Element of the carrier of it) &
     (for b1, b2 being Element of the carrier of a1 -INF_category
     for b3, b4 being Element of the carrier of it
        st b3 = b1 & b4 = b2 & <^b1,b2^> <> {}
     for b5 being Element of <^b1,b2^> holds
           b5 in <^b3,b4^>
        iff
           @ b5 is directed-sups-preserving(latt b1, latt b2));
end;

:: WAYBEL34:def 10
theorem
for b1 being non empty set
for b2 being non empty transitive strict id-inheriting SubCatStr of b1 -INF_category holds
      b2 = b1 -INF(SC)_category
   iff
      (for b3 being Element of the carrier of b1 -INF_category holds
          b3 is Element of the carrier of b2) &
       (for b3, b4 being Element of the carrier of b1 -INF_category
       for b5, b6 being Element of the carrier of b2
          st b5 = b3 & b6 = b4 & <^b3,b4^> <> {}
       for b7 being Element of <^b3,b4^> holds
             b7 in <^b5,b6^>
          iff
             @ b7 is directed-sups-preserving(latt b3, latt b4));

:: WAYBEL34:funcnot 9 => WAYBEL34:func 9
definition
  let a1 be with_non-empty_element set;
  func A1 -SUP(SO)_category -> non empty transitive strict id-inheriting SubCatStr of a1 -SUP_category means
    (for b1 being Element of the carrier of a1 -SUP_category holds
        b1 is Element of the carrier of it) &
     (for b1, b2 being Element of the carrier of a1 -SUP_category
     for b3, b4 being Element of the carrier of it
        st b3 = b1 & b4 = b2 & <^b1,b2^> <> {}
     for b5 being Element of <^b1,b2^> holds
           b5 in <^b3,b4^>
        iff
           UpperAdj @ b5 is directed-sups-preserving(latt b2, latt b1));
end;

:: WAYBEL34:def 11
theorem
for b1 being with_non-empty_element set
for b2 being non empty transitive strict id-inheriting SubCatStr of b1 -SUP_category holds
      b2 = b1 -SUP(SO)_category
   iff
      (for b3 being Element of the carrier of b1 -SUP_category holds
          b3 is Element of the carrier of b2) &
       (for b3, b4 being Element of the carrier of b1 -SUP_category
       for b5, b6 being Element of the carrier of b2
          st b5 = b3 & b6 = b4 & <^b3,b4^> <> {}
       for b7 being Element of <^b3,b4^> holds
             b7 in <^b5,b6^>
          iff
             UpperAdj @ b7 is directed-sups-preserving(latt b4, latt b3));

:: WAYBEL34:th 44
theorem
for b1 being non empty RelStr
for b2 being non empty reflexive antisymmetric RelStr
for b3 being Element of the carrier of b2
for b4 being non empty Element of bool the carrier of b1 holds
   b1 --> b3 preserves_sup_of b4 & b1 --> b3 preserves_inf_of b4;

:: WAYBEL34:th 45
theorem
for b1 being non empty RelStr
for b2 being non empty reflexive antisymmetric lower-bounded RelStr holds
   b1 --> Bottom b2 is sups-preserving(b1, b2);

:: WAYBEL34:th 46
theorem
for b1 being non empty RelStr
for b2 being non empty reflexive antisymmetric upper-bounded RelStr holds
   b1 --> Top b2 is infs-preserving(b1, b2);

:: WAYBEL34:funcreg 11
registration
  let a1 be non empty RelStr;
  let a2 be non empty reflexive antisymmetric upper-bounded RelStr;
  cluster a1 --> Top a2 -> Function-like quasi_total infs-preserving directed-sups-preserving;
end;

:: WAYBEL34:funcreg 12
registration
  let a1 be non empty RelStr;
  let a2 be non empty reflexive antisymmetric lower-bounded RelStr;
  cluster a1 --> Bottom a2 -> Function-like quasi_total sups-preserving filtered-infs-preserving;
end;

:: WAYBEL34:exreg 2
registration
  let a1 be non empty RelStr;
  let a2 be non empty reflexive antisymmetric upper-bounded RelStr;
  cluster Relation-like Function-like non empty quasi_total total infs-preserving directed-sups-preserving Relation of the carrier of a1,the carrier of a2;
end;

:: WAYBEL34:exreg 3
registration
  let a1 be non empty RelStr;
  let a2 be non empty reflexive antisymmetric lower-bounded RelStr;
  cluster Relation-like Function-like non empty quasi_total total sups-preserving filtered-infs-preserving Relation of the carrier of a1,the carrier of a2;
end;

:: WAYBEL34:th 47
theorem
for b1 being with_non-empty_element set
for b2 being reflexive transitive antisymmetric with_suprema with_infima RelStr holds
      b2 is Element of the carrier of b1 -INF(SC)_category
   iff
      b2 is strict & b2 is complete & the carrier of b2 in b1;

:: WAYBEL34:th 48
theorem
for b1 being with_non-empty_element set
for b2, b3 being Element of the carrier of b1 -INF(SC)_category
for b4 being set holds
      b4 in <^b2,b3^>
   iff
      b4 is Function-like quasi_total infs-preserving directed-sups-preserving Relation of the carrier of latt b2,the carrier of latt b3;

:: WAYBEL34:th 49
theorem
for b1 being with_non-empty_element set
for b2 being reflexive transitive antisymmetric with_suprema with_infima RelStr holds
      b2 is Element of the carrier of b1 -SUP(SO)_category
   iff
      b2 is strict & b2 is complete & the carrier of b2 in b1;

:: WAYBEL34:th 50
theorem
for b1 being with_non-empty_element set
for b2, b3 being Element of the carrier of b1 -SUP(SO)_category
for b4 being set holds
      b4 in <^b2,b3^>
   iff
      ex b5 being Function-like quasi_total sups-preserving Relation of the carrier of latt b2,the carrier of latt b3 st
         b5 = b4 & UpperAdj b5 is directed-sups-preserving(latt b3, latt b2);

:: WAYBEL34:th 51
theorem
for b1 being with_non-empty_element set holds
   b1 -INF(SC)_category = Intersect(b1 -INF_category,b1 -UPS_category);

:: WAYBEL34:funcnot 10 => WAYBEL34:func 10
definition
  let a1 be with_non-empty_element set;
  func A1 -CL_category -> non empty transitive strict full id-inheriting SubCatStr of a1 -INF(SC)_category means
    for b1 being Element of the carrier of a1 -INF(SC)_category holds
          b1 is Element of the carrier of it
       iff
          latt b1 is continuous;
end;

:: WAYBEL34:def 12
theorem
for b1 being with_non-empty_element set
for b2 being non empty transitive strict full id-inheriting SubCatStr of b1 -INF(SC)_category holds
      b2 = b1 -CL_category
   iff
      for b3 being Element of the carrier of b1 -INF(SC)_category holds
            b3 is Element of the carrier of b2
         iff
            latt b3 is continuous;

:: WAYBEL34:funcreg 13
registration
  let a1 be with_non-empty_element set;
  cluster a1 -CL_category -> non empty transitive strict full id-inheriting with_complete_lattices;
end;

:: WAYBEL34:th 52
theorem
for b1 being with_non-empty_element set
for b2 being reflexive transitive antisymmetric with_suprema with_infima RelStr
      st the carrier of b2 in b1
   holds    b2 is Element of the carrier of b1 -CL_category
   iff
      b2 is strict & b2 is complete & b2 is continuous;

:: WAYBEL34:th 53
theorem
for b1 being with_non-empty_element set
for b2, b3 being Element of the carrier of b1 -CL_category
for b4 being set holds
      b4 in <^b2,b3^>
   iff
      b4 is Function-like quasi_total infs-preserving directed-sups-preserving Relation of the carrier of latt b2,the carrier of latt b3;

:: WAYBEL34:funcnot 11 => WAYBEL34:func 11
definition
  let a1 be with_non-empty_element set;
  func A1 -CL-opp_category -> non empty transitive strict full id-inheriting SubCatStr of a1 -SUP(SO)_category means
    for b1 being Element of the carrier of a1 -SUP(SO)_category holds
          b1 is Element of the carrier of it
       iff
          latt b1 is continuous;
end;

:: WAYBEL34:def 13
theorem
for b1 being with_non-empty_element set
for b2 being non empty transitive strict full id-inheriting SubCatStr of b1 -SUP(SO)_category holds
      b2 = b1 -CL-opp_category
   iff
      for b3 being Element of the carrier of b1 -SUP(SO)_category holds
            b3 is Element of the carrier of b2
         iff
            latt b3 is continuous;

:: WAYBEL34:th 54
theorem
for b1 being with_non-empty_element set
for b2 being reflexive transitive antisymmetric with_suprema with_infima RelStr
      st the carrier of b2 in b1
   holds    b2 is Element of the carrier of b1 -CL-opp_category
   iff
      b2 is strict & b2 is complete & b2 is continuous;

:: WAYBEL34:th 55
theorem
for b1 being with_non-empty_element set
for b2, b3 being Element of the carrier of b1 -CL-opp_category
for b4 being set holds
      b4 in <^b2,b3^>
   iff
      ex b5 being Function-like quasi_total sups-preserving Relation of the carrier of latt b2,the carrier of latt b3 st
         b5 = b4 & UpperAdj b5 is directed-sups-preserving(latt b3, latt b2);

:: WAYBEL34:th 56
theorem
for b1 being with_non-empty_element set holds
   b1 -INF(SC)_category,b1 -SUP(SO)_category are_anti-isomorphic_under b1 LowerAdj;

:: WAYBEL34:th 57
theorem
for b1 being with_non-empty_element set holds
   b1 -SUP(SO)_category,b1 -INF(SC)_category are_anti-isomorphic_under b1 UpperAdj;

:: WAYBEL34:th 58
theorem
for b1 being with_non-empty_element set holds
   b1 -CL_category,b1 -CL-opp_category are_anti-isomorphic_under b1 LowerAdj;

:: WAYBEL34:th 59
theorem
for b1 being with_non-empty_element set holds
   b1 -CL-opp_category,b1 -CL_category are_anti-isomorphic_under b1 UpperAdj;

:: WAYBEL34:attrnot 3 => WAYBEL34:attr 3
definition
  let a1, a2 be non empty reflexive RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is compact-preserving means
    for b1 being Element of the carrier of a1
          st b1 is compact(a1)
       holds a3 . b1 is compact(a2);
end;

:: WAYBEL34:dfs 14
definiens
  let a1, a2 be non empty reflexive RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is compact-preserving
it is sufficient to prove
  thus for b1 being Element of the carrier of a1
          st b1 is compact(a1)
       holds a3 . b1 is compact(a2);

:: WAYBEL34:def 14
theorem
for b1, b2 being non empty reflexive RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is compact-preserving(b1, b2)
   iff
      for b4 being Element of the carrier of b1
            st b4 is compact(b1)
         holds b3 . b4 is compact(b2);

:: WAYBEL34:th 60
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total sups-preserving Relation of the carrier of b2,the carrier of b1
      st b3 is waybelow-preserving(b2, b1)
   holds b3 is compact-preserving(b2, b1);

:: WAYBEL34:th 61
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total sups-preserving Relation of the carrier of b2,the carrier of b1
      st b2 is algebraic & b3 is compact-preserving(b2, b1)
   holds b3 is waybelow-preserving(b2, b1);

:: WAYBEL34:th 62
theorem
for b1, b2, b3 being non empty RelStr
for b4 being Element of bool the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
      st b5 preserves_sup_of b4 & b6 preserves_sup_of b5 .: b4
   holds b6 * b5 preserves_sup_of b4;

:: WAYBEL34:attrnot 4 => WAYBEL34:attr 4
definition
  let a1, a2 be non empty RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is finite-sups-preserving means
    for b1 being finite Element of bool the carrier of a1 holds
       a3 preserves_sup_of b1;
end;

:: WAYBEL34:dfs 15
definiens
  let a1, a2 be non empty RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is finite-sups-preserving
it is sufficient to prove
  thus for b1 being finite Element of bool the carrier of a1 holds
       a3 preserves_sup_of b1;

:: WAYBEL34:def 15
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is finite-sups-preserving(b1, b2)
   iff
      for b4 being finite Element of bool the carrier of b1 holds
         b3 preserves_sup_of b4;

:: WAYBEL34:attrnot 5 => WAYBEL34:attr 5
definition
  let a1, a2 be non empty RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is bottom-preserving means
    a3 preserves_sup_of {} a1;
end;

:: WAYBEL34:dfs 16
definiens
  let a1, a2 be non empty RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is bottom-preserving
it is sufficient to prove
  thus a3 preserves_sup_of {} a1;

:: WAYBEL34:def 16
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is bottom-preserving(b1, b2)
   iff
      b3 preserves_sup_of {} b1;

:: WAYBEL34:th 63
theorem
for b1, b2, b3 being non empty RelStr
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
      st b4 is finite-sups-preserving(b1, b2) & b5 is finite-sups-preserving(b2, b3)
   holds b5 * b4 is finite-sups-preserving(b1, b3);

:: WAYBEL34:attrnot 6 => WAYBEL34:attr 5
definition
  let a1, a2 be non empty RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is bottom-preserving means
    a3 . Bottom a1 = Bottom a2;
end;

:: WAYBEL34:dfs 17
definiens
  let a1, a2 be non empty antisymmetric lower-bounded RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is bottom-preserving
it is sufficient to prove
  thus a3 . Bottom a1 = Bottom a2;

:: WAYBEL34:def 17
theorem
for b1, b2 being non empty antisymmetric lower-bounded RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is bottom-preserving(b1, b2)
   iff
      b3 . Bottom b1 = Bottom b2;

:: WAYBEL34:attrnot 7 => WAYBEL34:attr 6
definition
  let a1 be non empty RelStr;
  let a2 be SubRelStr of a1;
  attr a2 is finite-sups-inheriting means
    for b1 being finite Element of bool the carrier of a2
          st ex_sup_of b1,a1
       holds "\/"(b1,a1) in the carrier of a2;
end;

:: WAYBEL34:dfs 18
definiens
  let a1 be non empty RelStr;
  let a2 be SubRelStr of a1;
To prove
     a2 is finite-sups-inheriting
it is sufficient to prove
  thus for b1 being finite Element of bool the carrier of a2
          st ex_sup_of b1,a1
       holds "\/"(b1,a1) in the carrier of a2;

:: WAYBEL34:def 18
theorem
for b1 being non empty RelStr
for b2 being SubRelStr of b1 holds
      b2 is finite-sups-inheriting(b1)
   iff
      for b3 being finite Element of bool the carrier of b2
            st ex_sup_of b3,b1
         holds "\/"(b3,b1) in the carrier of b2;

:: WAYBEL34:attrnot 8 => WAYBEL34:attr 7
definition
  let a1 be non empty RelStr;
  let a2 be SubRelStr of a1;
  attr a2 is bottom-inheriting means
    Bottom a1 in the carrier of a2;
end;

:: WAYBEL34:dfs 19
definiens
  let a1 be non empty RelStr;
  let a2 be SubRelStr of a1;
To prove
     a2 is bottom-inheriting
it is sufficient to prove
  thus Bottom a1 in the carrier of a2;

:: WAYBEL34:def 19
theorem
for b1 being non empty RelStr
for b2 being SubRelStr of b1 holds
      b2 is bottom-inheriting(b1)
   iff
      Bottom b1 in the carrier of b2;

:: WAYBEL34:condreg 1
registration
  let a1, a2 be non empty RelStr;
  cluster Function-like quasi_total sups-preserving -> bottom-preserving (Relation of the carrier of a1,the carrier of a2);
end;

:: WAYBEL34:condreg 2
registration
  let a1 be non empty antisymmetric lower-bounded RelStr;
  cluster finite-sups-inheriting -> join-inheriting bottom-inheriting (SubRelStr of a1);
end;

:: WAYBEL34:condreg 3
registration
  let a1 be non empty RelStr;
  cluster sups-inheriting -> finite-sups-inheriting (SubRelStr of a1);
end;

:: WAYBEL34:exreg 4
registration
  let a1, a2 be non empty reflexive transitive antisymmetric lower-bounded RelStr;
  cluster Relation-like Function-like non empty quasi_total total sups-preserving Relation of the carrier of a1,the carrier of a2;
end;

:: WAYBEL34:condreg 4
registration
  let a1 be non empty antisymmetric lower-bounded RelStr;
  cluster full bottom-inheriting -> non empty lower-bounded (SubRelStr of a1);
end;

:: WAYBEL34:exreg 5
registration
  let a1 be non empty antisymmetric lower-bounded RelStr;
  cluster non empty full sups-inheriting finite-sups-inheriting bottom-inheriting SubRelStr of a1;
end;

:: WAYBEL34:th 64
theorem
for b1 being non empty antisymmetric lower-bounded RelStr
for b2 being non empty full bottom-inheriting SubRelStr of b1 holds
   Bottom b2 = Bottom b1;

:: WAYBEL34:condreg 5
registration
  let a1 be non empty reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  cluster full join-inheriting bottom-inheriting -> finite-sups-inheriting (SubRelStr of a1);
end;

:: WAYBEL34:th 65
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is finite-sups-preserving(b1, b2)
   holds b3 is join-preserving(b1, b2) & b3 is bottom-preserving(b1, b2);

:: WAYBEL34:th 66
theorem
for b1, b2 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is join-preserving(b1, b2) & b3 is bottom-preserving(b1, b2)
   holds b3 is finite-sups-preserving(b1, b2);

:: WAYBEL34:condreg 6
registration
  let a1, a2 be non empty RelStr;
  cluster Function-like quasi_total sups-preserving -> finite-sups-preserving (Relation of the carrier of a1,the carrier of a2);
end;

:: WAYBEL34:condreg 7
registration
  let a1, a2 be non empty RelStr;
  cluster Function-like quasi_total finite-sups-preserving -> join-preserving bottom-preserving (Relation of the carrier of a1,the carrier of a2);
end;

:: WAYBEL34:exreg 6
registration
  let a1 be non empty RelStr;
  let a2 be non empty reflexive antisymmetric lower-bounded RelStr;
  cluster Relation-like Function-like non empty quasi_total total sups-preserving finite-sups-preserving Relation of the carrier of a1,the carrier of a2;
end;

:: WAYBEL34:funcreg 14
registration
  let a1 be non empty reflexive transitive antisymmetric lower-bounded RelStr;
  cluster CompactSublatt a1 -> strict lower-bounded full;
end;

:: WAYBEL34:th 67
theorem
for b1 being RelStr
for b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being SubRelStr of b1
for b5 being SubRelStr of b2
      st b3 .: the carrier of b4 c= the carrier of b5
   holds b3 | the carrier of b4 is Function-like quasi_total Relation of the carrier of b4,the carrier of b5;

:: WAYBEL34:th 68
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b3 being Function-like quasi_total join-preserving Relation of the carrier of b1,the carrier of b2
for b4 being non empty full join-inheriting SubRelStr of b1
for b5 being non empty full join-inheriting SubRelStr of b2
for b6 being Function-like quasi_total Relation of the carrier of b4,the carrier of b5
      st b6 = b3 | the carrier of b4
   holds b6 is join-preserving(b4, b5);

:: WAYBEL34:th 69
theorem
for b1, b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total finite-sups-preserving Relation of the carrier of b1,the carrier of b2
for b4 being non empty full finite-sups-inheriting SubRelStr of b1
for b5 being non empty full finite-sups-inheriting SubRelStr of b2
for b6 being Function-like quasi_total Relation of the carrier of b4,the carrier of b5
      st b6 = b3 | the carrier of b4
   holds b6 is finite-sups-preserving(b4, b5);

:: WAYBEL34:funcreg 15
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  cluster CompactSublatt a1 -> strict full finite-sups-inheriting;
end;

:: WAYBEL34:th 70
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total sups-preserving Relation of the carrier of b2,the carrier of b1 holds
      b3 is compact-preserving(b2, b1)
   iff
      b3 | the carrier of CompactSublatt b2 is Function-like quasi_total finite-sups-preserving Relation of the carrier of CompactSublatt b2,the carrier of CompactSublatt b1;

:: WAYBEL34:th 71
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
   st b2 is algebraic
for b3 being Function-like quasi_total infs-preserving Relation of the carrier of b1,the carrier of b2 holds
      b3 is directed-sups-preserving(b1, b2)
   iff
      (LowerAdj b3) | the carrier of CompactSublatt b2 is Function-like quasi_total finite-sups-preserving Relation of the carrier of CompactSublatt b2,the carrier of CompactSublatt b1;