Article WAYBEL11, MML version 4.99.1005

:: WAYBEL11:sch 1
scheme WAYBEL11:sch 1
{F1 -> non empty set,
  F2 -> non empty set,
  F3 -> set,
  F4 -> set}:
{F3(b1) where b1 is Element of F1(): P1[b1]} = {F4(b1, b2) where b1 is Element of F2(), b2 is Element of F1(): P1[b2]}
provided
   for b1 being Element of F2()
   for b2 being Element of F1() holds
      F3(b2) = F4(b1, b2);


:: WAYBEL11:th 1
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2, b3 being Element of bool the carrier of b1
      st b3 is_coarser_than b2
   holds "/\"(b2,b1) <= "/\"(b3,b1);

:: WAYBEL11:th 2
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2, b3 being Element of bool the carrier of b1
      st b2 is_finer_than b3
   holds "\/"(b2,b1) <= "\/"(b3,b1);

:: WAYBEL11:th 3
theorem
for b1 being RelStr
for b2 being upper Element of bool the carrier of b1
for b3 being directed Element of bool the carrier of b1 holds
   b2 /\ b3 is directed(b1);

:: WAYBEL11:exreg 1
registration
  let a1 be non empty reflexive RelStr;
  cluster non empty finite directed Element of bool the carrier of a1;
end;

:: WAYBEL11:th 4
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2 being non empty finite directed Element of bool the carrier of b1 holds
   "\/"(b2,b1) in b2;

:: WAYBEL11:exreg 2
registration
  cluster non empty trivial finite strict reflexive transitive antisymmetric with_suprema with_infima RelStr;
end;

:: WAYBEL11:condreg 1
registration
  let a1 be finite 1-sorted;
  cluster -> finite (Element of bool the carrier of a1);
end;

:: WAYBEL11:funcreg 1
registration
  let a1 be RelStr;
  cluster {} a1 -> lower upper;
end;

:: WAYBEL11:condreg 2
registration
  let a1 be non empty trivial RelStr;
  cluster -> upper (Element of bool the carrier of a1);
end;

:: WAYBEL11:th 5
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1
for b3 being upper Element of bool the carrier of b1
      st not b2 in b3
   holds b3 misses downarrow b2;

:: WAYBEL11:th 6
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1
for b3 being lower Element of bool the carrier of b1
      st b2 in b3
   holds downarrow b2 c= b3;

:: WAYBEL11:attrnot 1 => WAYBEL11:attr 1
definition
  let a1 be non empty reflexive RelStr;
  let a2 be Element of bool the carrier of a1;
  attr a2 is inaccessible_by_directed_joins means
    for b1 being non empty directed Element of bool the carrier of a1
          st "\/"(b1,a1) in a2
       holds b1 meets a2;
end;

:: WAYBEL11:dfs 1
definiens
  let a1 be non empty reflexive RelStr;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is inaccessible_by_directed_joins
it is sufficient to prove
  thus for b1 being non empty directed Element of bool the carrier of a1
          st "\/"(b1,a1) in a2
       holds b1 meets a2;

:: WAYBEL11:def 1
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of bool the carrier of b1 holds
      b2 is inaccessible_by_directed_joins(b1)
   iff
      for b3 being non empty directed Element of bool the carrier of b1
            st "\/"(b3,b1) in b2
         holds b3 meets b2;

:: WAYBEL11:attrnot 2 => WAYBEL11:attr 2
definition
  let a1 be non empty reflexive RelStr;
  let a2 be Element of bool the carrier of a1;
  attr a2 is closed_under_directed_sups means
    for b1 being non empty directed Element of bool the carrier of a1
          st b1 c= a2
       holds "\/"(b1,a1) in a2;
end;

:: WAYBEL11:dfs 2
definiens
  let a1 be non empty reflexive RelStr;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is closed_under_directed_sups
it is sufficient to prove
  thus for b1 being non empty directed Element of bool the carrier of a1
          st b1 c= a2
       holds "\/"(b1,a1) in a2;

:: WAYBEL11:def 2
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of bool the carrier of b1 holds
      b2 is closed_under_directed_sups(b1)
   iff
      for b3 being non empty directed Element of bool the carrier of b1
            st b3 c= b2
         holds "\/"(b3,b1) in b2;

:: WAYBEL11:attrnot 3 => WAYBEL11:attr 3
definition
  let a1 be non empty reflexive RelStr;
  let a2 be Element of bool the carrier of a1;
  attr a2 is property(S) means
    for b1 being non empty directed Element of bool the carrier of a1
          st "\/"(b1,a1) in a2
       holds ex b2 being Element of the carrier of a1 st
          b2 in b1 &
           (for b3 being Element of the carrier of a1
                 st b3 in b1 & b2 <= b3
              holds b3 in a2);
end;

:: WAYBEL11:dfs 3
definiens
  let a1 be non empty reflexive RelStr;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is property(S)
it is sufficient to prove
  thus for b1 being non empty directed Element of bool the carrier of a1
          st "\/"(b1,a1) in a2
       holds ex b2 being Element of the carrier of a1 st
          b2 in b1 &
           (for b3 being Element of the carrier of a1
                 st b3 in b1 & b2 <= b3
              holds b3 in a2);

:: WAYBEL11:def 3
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of bool the carrier of b1 holds
      b2 is property(S)(b1)
   iff
      for b3 being non empty directed Element of bool the carrier of b1
            st "\/"(b3,b1) in b2
         holds ex b4 being Element of the carrier of b1 st
            b4 in b3 &
             (for b5 being Element of the carrier of b1
                   st b5 in b3 & b4 <= b5
                holds b5 in b2);

:: WAYBEL11:attrnot 4 => WAYBEL11:attr 1
notation
  let a1 be non empty reflexive RelStr;
  let a2 be Element of bool the carrier of a1;
  synonym inaccessible for inaccessible_by_directed_joins;
end;

:: WAYBEL11:attrnot 5 => WAYBEL11:attr 2
notation
  let a1 be non empty reflexive RelStr;
  let a2 be Element of bool the carrier of a1;
  synonym directly_closed for closed_under_directed_sups;
end;

:: WAYBEL11:prednot 1 => WAYBEL11:attr 3
notation
  let a1 be non empty reflexive RelStr;
  let a2 be Element of bool the carrier of a1;
  synonym a2 has_the_property_(S) for property(S);
end;

:: WAYBEL11:funcreg 2
registration
  let a1 be non empty reflexive RelStr;
  cluster {} a1 -> closed_under_directed_sups property(S);
end;

:: WAYBEL11:exreg 3
registration
  let a1 be non empty reflexive RelStr;
  cluster closed_under_directed_sups property(S) Element of bool the carrier of a1;
end;

:: WAYBEL11:funcreg 3
registration
  let a1 be non empty reflexive RelStr;
  let a2 be property(S) Element of bool the carrier of a1;
  cluster a2 ` -> closed_under_directed_sups;
end;

:: WAYBEL11:attrnot 6 => WAYBEL11:attr 4
definition
  let a1 be non empty reflexive TopRelStr;
  attr a1 is Scott means
    for b1 being Element of bool the carrier of a1 holds
          b1 is open(a1)
       iff
          b1 is inaccessible_by_directed_joins(a1) & b1 is upper(a1);
end;

:: WAYBEL11:dfs 4
definiens
  let a1 be non empty reflexive TopRelStr;
To prove
     a1 is Scott
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1 holds
          b1 is open(a1)
       iff
          b1 is inaccessible_by_directed_joins(a1) & b1 is upper(a1);

:: WAYBEL11:def 4
theorem
for b1 being non empty reflexive TopRelStr holds
      b1 is Scott
   iff
      for b2 being Element of bool the carrier of b1 holds
            b2 is open(b1)
         iff
            b2 is inaccessible_by_directed_joins(b1) & b2 is upper(b1);

:: WAYBEL11:condreg 3
registration
  let a1 be non empty finite reflexive transitive antisymmetric with_suprema RelStr;
  cluster -> inaccessible_by_directed_joins (Element of bool the carrier of a1);
end;

:: WAYBEL11:attrnot 7 => WAYBEL11:attr 4
definition
  let a1 be non empty reflexive TopRelStr;
  attr a1 is Scott means
    for b1 being Element of bool the carrier of a1 holds
          b1 is open(a1)
       iff
          b1 is upper(a1);
end;

:: WAYBEL11:dfs 5
definiens
  let a1 be non empty finite reflexive transitive antisymmetric with_suprema TopRelStr;
To prove
     a1 is Scott
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1 holds
          b1 is open(a1)
       iff
          b1 is upper(a1);

:: WAYBEL11:def 5
theorem
for b1 being non empty finite reflexive transitive antisymmetric with_suprema TopRelStr holds
      b1 is Scott
   iff
      for b2 being Element of bool the carrier of b1 holds
            b2 is open(b1)
         iff
            b2 is upper(b1);

:: WAYBEL11:exreg 4
registration
  cluster non empty trivial TopSpace-like total reflexive transitive antisymmetric with_suprema with_infima complete strict Scott TopRelStr;
end;

:: WAYBEL11:funcreg 4
registration
  let a1 be non empty reflexive RelStr;
  cluster [#] a1 -> inaccessible_by_directed_joins closed_under_directed_sups;
end;

:: WAYBEL11:exreg 5
registration
  let a1 be non empty reflexive RelStr;
  cluster lower upper inaccessible_by_directed_joins closed_under_directed_sups Element of bool the carrier of a1;
end;

:: WAYBEL11:funcreg 5
registration
  let a1 be non empty reflexive RelStr;
  let a2 be inaccessible_by_directed_joins Element of bool the carrier of a1;
  cluster a2 ` -> closed_under_directed_sups;
end;

:: WAYBEL11:funcreg 6
registration
  let a1 be non empty reflexive RelStr;
  let a2 be closed_under_directed_sups Element of bool the carrier of a1;
  cluster a2 ` -> inaccessible_by_directed_joins;
end;

:: WAYBEL11:th 7
theorem
for b1 being non empty reflexive transitive up-complete Scott TopRelStr
for b2 being Element of bool the carrier of b1 holds
      b2 is closed(b1)
   iff
      b2 is closed_under_directed_sups(b1) & b2 is lower(b1);

:: WAYBEL11:th 8
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete TopRelStr
for b2 being Element of the carrier of b1 holds
   downarrow b2 is closed_under_directed_sups(b1);

:: WAYBEL11:th 9
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b2 being Element of the carrier of b1 holds
   Cl {b2} = downarrow b2;

:: WAYBEL11:th 10
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr holds
   b1 is non empty TopSpace-like discerning TopStruct;

:: WAYBEL11:th 11
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete Scott TopRelStr
for b2 being Element of the carrier of b1 holds
   downarrow b2 is closed(b1);

:: WAYBEL11:th 12
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric up-complete with_suprema with_infima Scott TopRelStr
for b2 being Element of the carrier of b1 holds
   (downarrow b2) ` is open(b1);

:: WAYBEL11:th 13
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric up-complete with_suprema with_infima Scott TopRelStr
for b2 being Element of the carrier of b1
for b3 being upper Element of bool the carrier of b1
      st not b2 in b3
   holds (downarrow b2) ` is a_neighborhood of b3;

:: WAYBEL11:th 14
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b2 being upper Element of bool the carrier of b1 holds
   ex b3 being Element of bool bool the carrier of b1 st
      b2 = meet b3 &
       (for b4 being Element of bool the carrier of b1
             st b4 in b3
          holds b4 is a_neighborhood of b2);

:: WAYBEL11:th 15
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima Scott TopRelStr
for b2 being Element of bool the carrier of b1 holds
      b2 is open(b1)
   iff
      b2 is upper(b1) & b2 is property(S)(b1);

:: WAYBEL11:condreg 4
registration
  let a1 be TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete TopRelStr;
  cluster lower -> property(S) (Element of bool the carrier of a1);
end;

:: WAYBEL11:th 16
theorem
for b1 being non empty reflexive transitive TopRelStr
      st the topology of b1 = {b2 where b2 is Element of bool the carrier of b1: b2 is property(S)(b1)}
   holds b1 is TopSpace-like;

:: WAYBEL11:funcnot 1 => WAYBEL11:func 1
definition
  let a1 be non empty RelStr;
  let a2 be non empty transitive directed NetStr over a1;
  func lim_inf A2 -> Element of the carrier of a1 equals
    "\/"({"/\"({a2 . b2 where b2 is Element of the carrier of a2: b1 <= b2},a1) where b1 is Element of the carrier of a2: TRUE},a1);
end;

:: WAYBEL11:def 6
theorem
for b1 being non empty RelStr
for b2 being non empty transitive directed NetStr over b1 holds
   lim_inf b2 = "\/"({"/\"({b2 . b4 where b4 is Element of the carrier of b2: b3 <= b4},b1) where b3 is Element of the carrier of b2: TRUE},b1);

:: WAYBEL11:prednot 2 => WAYBEL11:pred 1
definition
  let a1 be non empty reflexive RelStr;
  let a2 be non empty transitive directed NetStr over a1;
  let a3 be Element of the carrier of a1;
  pred A3 is_S-limit_of A2 means
    a3 <= lim_inf a2;
end;

:: WAYBEL11:dfs 7
definiens
  let a1 be non empty reflexive RelStr;
  let a2 be non empty transitive directed NetStr over a1;
  let a3 be Element of the carrier of a1;
To prove
     a3 is_S-limit_of a2
it is sufficient to prove
  thus a3 <= lim_inf a2;

:: WAYBEL11:def 7
theorem
for b1 being non empty reflexive RelStr
for b2 being non empty transitive directed NetStr over b1
for b3 being Element of the carrier of b1 holds
      b3 is_S-limit_of b2
   iff
      b3 <= lim_inf b2;

:: WAYBEL11:funcnot 2 => WAYBEL11:func 2
definition
  let a1 be non empty reflexive RelStr;
  func Scott-Convergence A1 -> Convergence-Class of a1 means
    for b1 being non empty transitive strict directed NetStr over a1
       st b1 in NetUniv a1
    for b2 being Element of the carrier of a1 holds
          [b1,b2] in it
       iff
          b2 is_S-limit_of b1;
end;

:: WAYBEL11:def 8
theorem
for b1 being non empty reflexive RelStr
for b2 being Convergence-Class of b1 holds
      b2 = Scott-Convergence b1
   iff
      for b3 being non empty transitive strict directed NetStr over b1
         st b3 in NetUniv b1
      for b4 being Element of the carrier of b1 holds
            [b3,b4] in b2
         iff
            b4 is_S-limit_of b3;

:: WAYBEL11:th 17
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty transitive directed NetStr over b1
for b3, b4 being Element of the carrier of b1
      st b3 is_S-limit_of b2 & b2 is_eventually_in downarrow b4
   holds b3 <= b4;

:: WAYBEL11:th 18
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty transitive directed NetStr over b1
for b3, b4 being Element of the carrier of b1
      st b2 is_eventually_in uparrow b4
   holds b4 <= lim_inf b2;

:: WAYBEL11:attrnot 8 => WAYBEL_0:attr 8
definition
  let a1 be non empty RelStr;
  let a2 be non empty NetStr over a1;
  attr a2 is monotone means
    for b1, b2 being Element of the carrier of a2
          st b1 <= b2
       holds a2 . b1 <= a2 . b2;
end;

:: WAYBEL11:dfs 9
definiens
  let a1 be non empty reflexive RelStr;
  let a2 be non empty NetStr over a1;
To prove
     a2 is monotone
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a2
          st b1 <= b2
       holds a2 . b1 <= a2 . b2;

:: WAYBEL11:def 9
theorem
for b1 being non empty reflexive RelStr
for b2 being non empty NetStr over b1 holds
      b2 is monotone(b1)
   iff
      for b3, b4 being Element of the carrier of b2
            st b3 <= b4
         holds b2 . b3 <= b2 . b4;

:: WAYBEL11:funcnot 3 => WAYBEL11:func 3
definition
  let a1 be non empty RelStr;
  let a2 be non empty set;
  let a3 be Function-like quasi_total Relation of a2,the carrier of a1;
  func Net-Str(A2,A3) -> non empty strict NetStr over a1 means
    the carrier of it = a2 &
     the mapping of it = a3 &
     (for b1, b2 being Element of the carrier of it holds
        b1 <= b2
     iff
        it . b1 <= it . b2);
end;

:: WAYBEL11:def 10
theorem
for b1 being non empty RelStr
for b2 being non empty set
for b3 being Function-like quasi_total Relation of b2,the carrier of b1
for b4 being non empty strict NetStr over b1 holds
      b4 = Net-Str(b2,b3)
   iff
      the carrier of b4 = b2 &
       the mapping of b4 = b3 &
       (for b5, b6 being Element of the carrier of b4 holds
          b5 <= b6
       iff
          b4 . b5 <= b4 . b6);

:: WAYBEL11:th 19
theorem
for b1 being non empty 1-sorted
for b2 being non empty NetStr over b1 holds
   rng the mapping of b2 = {b2 . b3 where b3 is Element of the carrier of b2: TRUE};

:: WAYBEL11:th 20
theorem
for b1 being non empty RelStr
for b2 being non empty set
for b3 being Function-like quasi_total Relation of b2,the carrier of b1
      st rng b3 is directed(b1)
   holds Net-Str(b2,b3) is directed;

:: WAYBEL11:funcreg 7
registration
  let a1 be non empty RelStr;
  let a2 be non empty set;
  let a3 be Function-like quasi_total Relation of a2,the carrier of a1;
  cluster Net-Str(a2,a3) -> non empty strict monotone;
end;

:: WAYBEL11:funcreg 8
registration
  let a1 be non empty transitive RelStr;
  let a2 be non empty set;
  let a3 be Function-like quasi_total Relation of a2,the carrier of a1;
  cluster Net-Str(a2,a3) -> non empty transitive strict;
end;

:: WAYBEL11:funcreg 9
registration
  let a1 be non empty reflexive RelStr;
  let a2 be non empty set;
  let a3 be Function-like quasi_total Relation of a2,the carrier of a1;
  cluster Net-Str(a2,a3) -> non empty reflexive strict;
end;

:: WAYBEL11:th 21
theorem
for b1 being non empty transitive RelStr
for b2 being non empty set
for b3 being Function-like quasi_total Relation of b2,the carrier of b1
      st b2 c= the carrier of b1 & Net-Str(b2,b3) is directed
   holds Net-Str(b2,b3) in NetUniv b1;

:: WAYBEL11:exreg 6
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima RelStr;
  cluster non empty reflexive transitive strict directed monotone NetStr over a1;
end;

:: WAYBEL11:th 22
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty reflexive transitive directed monotone NetStr over b1 holds
   lim_inf b2 = sup b2;

:: WAYBEL11:th 23
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty transitive directed constant NetStr over b1 holds
   the_value_of b2 = lim_inf b2;

:: WAYBEL11:th 24
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty transitive directed constant NetStr over b1 holds
   the_value_of b2 is_S-limit_of b2;

:: WAYBEL11:funcnot 4 => WAYBEL11:func 4
definition
  let a1 be non empty 1-sorted;
  let a2 be Element of the carrier of a1;
  func Net-Str A2 -> strict NetStr over a1 means
    the carrier of it = {a2} &
     the InternalRel of it = {[a2,a2]} &
     the mapping of it = id {a2};
end;

:: WAYBEL11:def 11
theorem
for b1 being non empty 1-sorted
for b2 being Element of the carrier of b1
for b3 being strict NetStr over b1 holds
      b3 = Net-Str b2
   iff
      the carrier of b3 = {b2} &
       the InternalRel of b3 = {[b2,b2]} &
       the mapping of b3 = id {b2};

:: WAYBEL11:funcreg 10
registration
  let a1 be non empty 1-sorted;
  let a2 be Element of the carrier of a1;
  cluster Net-Str a2 -> non empty strict;
end;

:: WAYBEL11:th 25
theorem
for b1 being non empty 1-sorted
for b2 being Element of the carrier of b1
for b3 being Element of the carrier of Net-Str b2 holds
   b3 = b2;

:: WAYBEL11:th 26
theorem
for b1 being non empty 1-sorted
for b2 being Element of the carrier of b1
for b3 being Element of the carrier of Net-Str b2 holds
   (Net-Str b2) . b3 = b2;

:: WAYBEL11:funcreg 11
registration
  let a1 be non empty 1-sorted;
  let a2 be Element of the carrier of a1;
  cluster Net-Str a2 -> reflexive transitive antisymmetric strict directed;
end;

:: WAYBEL11:th 27
theorem
for b1 being non empty 1-sorted
for b2 being Element of the carrier of b1
for b3 being set holds
      Net-Str b2 is_eventually_in b3
   iff
      b2 in b3;

:: WAYBEL11:th 28
theorem
for b1 being non empty reflexive antisymmetric RelStr
for b2 being Element of the carrier of b1 holds
   b2 = lim_inf Net-Str b2;

:: WAYBEL11:th 29
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of the carrier of b1 holds
   Net-Str b2 in NetUniv b1;

:: WAYBEL11:th 30
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty transitive directed NetStr over b1
for b3 being Element of bool the carrier of b1
      st b3 = {"/\"({b2 . b5 where b5 is Element of the carrier of b2: b4 <= b5},b1) where b4 is Element of the carrier of b2: TRUE}
   holds b3 is not empty & b3 is directed(b1);

:: WAYBEL11:th 31
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Element of bool the carrier of b1 holds
      b2 in the topology of ConvergenceSpace Scott-Convergence b1
   iff
      b2 is inaccessible_by_directed_joins(b1) & b2 is upper(b1);

:: WAYBEL11:th 32
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr holds
   TopStruct(#the carrier of b1,the topology of b1#) = ConvergenceSpace Scott-Convergence b1;

:: WAYBEL11:th 33
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete TopRelStr
   st TopStruct(#the carrier of b1,the topology of b1#) = ConvergenceSpace Scott-Convergence b1
for b2 being Element of bool the carrier of b1 holds
      b2 is open(b1)
   iff
      b2 is inaccessible_by_directed_joins(b1) & b2 is upper(b1);

:: WAYBEL11:th 34
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete TopRelStr
      st TopStruct(#the carrier of b1,the topology of b1#) = ConvergenceSpace Scott-Convergence b1
   holds b1 is Scott;

:: WAYBEL11:funcreg 12
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  cluster Scott-Convergence a1 -> (CONSTANTS);
end;

:: WAYBEL11:funcreg 13
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  cluster Scott-Convergence a1 -> (SUBNETS);
end;

:: WAYBEL11:th 35
theorem
for b1 being non empty 1-sorted
for b2 being non empty transitive directed NetStr over b1
for b3 being set
for b4 being subnet of b2
   st b4 = b2 " b3
for b5 being Element of the carrier of b4 holds
   b4 . b5 in b3;

:: WAYBEL11:funcnot 5 => WAYBEL11:func 5
definition
  let a1 be non empty reflexive RelStr;
  func sigma A1 -> Element of bool bool the carrier of a1 equals
    the topology of ConvergenceSpace Scott-Convergence a1;
end;

:: WAYBEL11:def 12
theorem
for b1 being non empty reflexive RelStr holds
   sigma b1 = the topology of ConvergenceSpace Scott-Convergence b1;

:: WAYBEL11:th 36
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete continuous Scott TopRelStr
for b2 being Element of the carrier of b1 holds
   wayabove b2 is open(b1);

:: WAYBEL11:th 37
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete TopRelStr
      st the topology of b1 = sigma b1
   holds b1 is Scott;

:: WAYBEL11:funcreg 14
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr;
  cluster Scott-Convergence a1 -> topological;
end;

:: WAYBEL11:th 38
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete continuous Scott TopRelStr
for b2 being Element of the carrier of b1
for b3 being non empty transitive directed NetStr over b1
      st b3 in NetUniv b1
   holds    b2 is_S-limit_of b3
   iff
      b2 in Lim b3;

:: WAYBEL11:th 39
theorem
for b1 being non empty reflexive transitive antisymmetric complete RelStr
      st Scott-Convergence b1 is (ITERATED_LIMITS)(b1)
   holds b1 is continuous;

:: WAYBEL11:th 40
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr holds
      b1 is continuous
   iff
      Convergence b1 = Scott-Convergence b1;

:: WAYBEL11:th 41
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b2 being upper Element of bool the carrier of b1
      st b2 is Open(b1)
   holds b2 is open(b1);

:: WAYBEL11:th 42
theorem
for b1 being non empty RelStr
for b2 being upper Element of bool the carrier of b1
for b3 being Element of the carrier of b1
      st b3 in b2
   holds uparrow b3 c= b2;

:: WAYBEL11:th 43
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete continuous Scott TopRelStr
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
      st b3 is open(b1) & b2 in b3
   holds ex b4 being Element of the carrier of b1 st
      b4 is_way_below b2 & b4 in b3;

:: WAYBEL11:th 44
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete continuous Scott TopRelStr
for b2 being Element of the carrier of b1 holds
   {wayabove b3 where b3 is Element of the carrier of b1: b3 is_way_below b2} is Basis of b2;

:: WAYBEL11:th 45
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete continuous Scott TopRelStr holds
   {wayabove b2 where b2 is Element of the carrier of b1: TRUE} is Basis of b1;

:: WAYBEL11:th 46
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete continuous Scott TopRelStr
for b2 being upper Element of bool the carrier of b1 holds
      b2 is open(b1)
   iff
      b2 is Open(b1);

:: WAYBEL11:th 47
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete continuous Scott TopRelStr
for b2 being Element of the carrier of b1 holds
   Int uparrow b2 = wayabove b2;

:: WAYBEL11:th 48
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete continuous Scott TopRelStr
for b2 being Element of bool the carrier of b1 holds
   Int b2 = union {wayabove b3 where b3 is Element of the carrier of b1: wayabove b3 c= b2};