Article WAYBEL32, MML version 4.99.1005

:: WAYBEL32:attrnot 1 => WAYBEL32:attr 1
definition
  let a1 be non empty TopRelStr;
  attr a1 is upper means
    {(downarrow b1) ` where b1 is Element of the carrier of a1: TRUE} is prebasis of a1;
end;

:: WAYBEL32:dfs 1
definiens
  let a1 be non empty TopRelStr;
To prove
     a1 is upper
it is sufficient to prove
  thus {(downarrow b1) ` where b1 is Element of the carrier of a1: TRUE} is prebasis of a1;

:: WAYBEL32:def 1
theorem
for b1 being non empty TopRelStr holds
      b1 is upper
   iff
      {(downarrow b2) ` where b2 is Element of the carrier of b1: TRUE} is prebasis of b1;

:: WAYBEL32:exreg 1
registration
  cluster non empty TopSpace-like reflexive transitive antisymmetric with_suprema with_infima up-complete strict Scott TopRelStr;
end;

:: WAYBEL32:attrnot 2 => WAYBEL32:attr 2
definition
  let a1 be non empty TopSpace-like reflexive TopRelStr;
  attr a1 is order_consistent means
    for b1 being Element of the carrier of a1 holds
       downarrow b1 = Cl {b1} &
        (for b2 being non empty transitive directed eventually-directed NetStr over a1
           st b1 = sup b2
        for b3 being a_neighborhood of b1 holds
           b2 is_eventually_in b3);
end;

:: WAYBEL32:dfs 2
definiens
  let a1 be non empty TopSpace-like reflexive TopRelStr;
To prove
     a1 is order_consistent
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       downarrow b1 = Cl {b1} &
        (for b2 being non empty transitive directed eventually-directed NetStr over a1
           st b1 = sup b2
        for b3 being a_neighborhood of b1 holds
           b2 is_eventually_in b3);

:: WAYBEL32:def 2
theorem
for b1 being non empty TopSpace-like reflexive TopRelStr holds
      b1 is order_consistent
   iff
      for b2 being Element of the carrier of b1 holds
         downarrow b2 = Cl {b2} &
          (for b3 being non empty transitive directed eventually-directed NetStr over b1
             st b2 = sup b3
          for b4 being a_neighborhood of b2 holds
             b3 is_eventually_in b4);

:: WAYBEL32:condreg 1
registration
  cluster non empty trivial TopSpace-like reflexive -> upper (TopRelStr);
end;

:: WAYBEL32:exreg 2
registration
  cluster non empty trivial TopSpace-like reflexive transitive antisymmetric with_suprema with_infima up-complete strict upper TopRelStr;
end;

:: WAYBEL32:th 1
theorem
for b1 being non empty TopSpace-like reflexive transitive antisymmetric up-complete upper TopRelStr
for b2 being Element of bool the carrier of b1
      st b2 is open(b1)
   holds b2 is upper(b1);

:: WAYBEL32:th 2
theorem
for b1 being non empty TopSpace-like reflexive transitive antisymmetric up-complete TopRelStr
      st b1 is upper
   holds b1 is order_consistent;

:: WAYBEL32:th 7
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr holds
   ex b2 being TopAugmentation of b1 st
      b2 is Scott;

:: WAYBEL32:th 8
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr
for b2 being TopAugmentation of b1
      st b2 is Scott
   holds b2 is TopSpace-like;

:: WAYBEL32:condreg 2
registration
  let a1 be non empty reflexive transitive antisymmetric up-complete RelStr;
  cluster Scott -> TopSpace-like (TopAugmentation of a1);
end;

:: WAYBEL32:exreg 3
registration
  let a1 be non empty reflexive transitive antisymmetric up-complete RelStr;
  cluster non empty TopSpace-like reflexive transitive antisymmetric up-complete Scott TopAugmentation of a1;
end;

:: WAYBEL32:th 9
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete Scott TopRelStr
for b2 being Element of the carrier of b1 holds
   Cl {b2} = downarrow b2;

:: WAYBEL32:th 10
theorem
for b1 being non empty TopSpace-like reflexive transitive antisymmetric up-complete Scott TopRelStr holds
   b1 is order_consistent;

:: WAYBEL32:th 11
theorem
for b1 being reflexive transitive antisymmetric 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);

:: WAYBEL32:th 12
theorem
for b1 being reflexive transitive antisymmetric with_infima /\-complete RelStr
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
      st b3 in b2
   holds "/\"(b2,b1) <= b3;

:: WAYBEL32:th 13
theorem
for b1 being reflexive transitive antisymmetric with_infima /\-complete RelStr
for b2 being non empty reflexive transitive directed monotone NetStr over b1 holds
   lim_inf b2 = sup b2;

:: WAYBEL32:th 14
theorem
for b1 being reflexive transitive antisymmetric 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);

:: WAYBEL32:th 15
theorem
for b1 being reflexive transitive antisymmetric with_infima up-complete /\-complete RelStr
for b2 being TopAugmentation of b1
      st the topology of b2 = sigma b1
   holds b2 is Scott;

:: WAYBEL32:exreg 4
registration
  let a1 be reflexive transitive antisymmetric with_infima up-complete /\-complete RelStr;
  cluster non empty TopSpace-like reflexive transitive antisymmetric up-complete strict Scott TopAugmentation of a1;
end;

:: WAYBEL32:th 16
theorem
for b1 being reflexive transitive antisymmetric with_infima up-complete /\-complete RelStr
for b2 being Scott TopAugmentation of b1 holds
   sigma b1 = the topology of b2;

:: WAYBEL32:th 17
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete Scott TopRelStr holds
   b1 is non empty TopSpace-like discerning TopStruct;

:: WAYBEL32:condreg 3
registration
  let a1 be non empty reflexive transitive antisymmetric up-complete RelStr;
  cluster -> up-complete (TopAugmentation of a1);
end;

:: WAYBEL32:th 18
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr
for b2 being Scott TopAugmentation of b1
for b3 being Element of the carrier of b2
for b4 being upper Element of bool the carrier of b2
      st not b3 in b4
   holds (downarrow b3) ` is a_neighborhood of b4;

:: WAYBEL32:th 19
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete TopRelStr
for b2 being Scott TopAugmentation of b1
for b3 being upper Element of bool the carrier of b2 holds
   ex b4 being Element of bool bool the carrier of b2 st
      b3 = meet b4 &
       (for b5 being Element of bool the carrier of b2
             st b5 in b4
          holds b5 is a_neighborhood of b3);

:: WAYBEL32:th 20
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete 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);

:: WAYBEL32:th 21
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete TopRelStr
for b2 being non empty directed Element of bool the carrier of b1
for b3 being Element of the carrier of b1
      st b3 in b2
   holds b3 <= "\/"(b2,b1);

:: WAYBEL32:condreg 4
registration
  let a1 be non empty reflexive transitive antisymmetric up-complete TopRelStr;
  cluster lower -> property(S) (Element of bool the carrier of a1);
end;

:: WAYBEL32:th 22
theorem
for b1 being non empty finite reflexive transitive antisymmetric up-complete RelStr
for b2 being Element of bool the carrier of b1 holds
   b2 is inaccessible_by_directed_joins(b1);

:: WAYBEL32:th 23
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete connected RelStr
for b2 being Scott TopAugmentation of b1
for b3 being Element of the carrier of b2 holds
   (downarrow b3) ` is open(b2);

:: WAYBEL32:th 24
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete connected RelStr
for b2 being Scott TopAugmentation of b1
for b3 being Element of bool the carrier of b2 holds
      b3 is open(b2)
   iff
      (b3 = the carrier of b2 or b3 in {(downarrow b4) ` where b4 is Element of the carrier of b2: TRUE});

:: WAYBEL32:exreg 5
registration
  let a1 be non empty reflexive transitive antisymmetric up-complete RelStr;
  cluster non empty TopSpace-like reflexive transitive antisymmetric up-complete order_consistent TopAugmentation of a1;
end;

:: WAYBEL32:exreg 6
registration
  cluster non empty TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete order_consistent TopRelStr;
end;

:: WAYBEL32:th 25
theorem
for b1 being non empty TopRelStr
for b2 being Element of bool the carrier of b1
      st (for b3 being Element of the carrier of b1 holds
            downarrow b3 = Cl {b3}) &
         b2 is open(b1)
   holds b2 is upper(b1);

:: WAYBEL32:th 26
theorem
for b1 being non empty TopRelStr
for b2 being Element of bool the carrier of b1
   st for b3 being Element of the carrier of b1 holds
        downarrow b3 = Cl {b3}
for b3 being Element of bool the carrier of b1
      st b3 is closed(b1)
   holds b3 is lower(b1);

:: WAYBEL32:funcnot 1 => WAYBEL32:func 1
definition
  let a1 be non empty 1-sorted;
  let a2 be non empty RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a2,the carrier of a1;
  func A2 *' A3 -> non empty strict NetStr over a1 means
    RelStr(#the carrier of it,the InternalRel of it#) = RelStr(#the carrier of a2,the InternalRel of a2#) &
     the mapping of it = a3;
end;

:: WAYBEL32:def 3
theorem
for b1 being non empty 1-sorted
for b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b4 being non empty strict NetStr over b1 holds
      b4 = b2 *' b3
   iff
      RelStr(#the carrier of b4,the InternalRel of b4#) = RelStr(#the carrier of b2,the InternalRel of b2#) &
       the mapping of b4 = b3;

:: WAYBEL32:funcreg 1
registration
  let a1 be non empty 1-sorted;
  let a2 be non empty transitive RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a2,the carrier of a1;
  cluster a2 *' a3 -> non empty transitive strict;
end;

:: WAYBEL32:funcreg 2
registration
  let a1 be non empty 1-sorted;
  let a2 be non empty directed RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a2,the carrier of a1;
  cluster a2 *' a3 -> non empty strict directed;
end;

:: WAYBEL32:funcnot 2 => WAYBEL32:func 2
definition
  let a1 be non empty RelStr;
  let a2 be non empty directed NetStr over a1;
  func inf_net A2 -> non empty strict directed NetStr over a1 means
    ex b1 being Function-like quasi_total Relation of the carrier of a2,the carrier of a1 st
       it = a2 *' b1 &
        (for b2 being Element of the carrier of a2 holds
           b1 . b2 = "/\"({a2 . b3 where b3 is Element of the carrier of a2: b2 <= b3},a1));
end;

:: WAYBEL32:def 4
theorem
for b1 being non empty RelStr
for b2 being non empty directed NetStr over b1
for b3 being non empty strict directed NetStr over b1 holds
      b3 = inf_net b2
   iff
      ex b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1 st
         b3 = b2 *' b4 &
          (for b5 being Element of the carrier of b2 holds
             b4 . b5 = "/\"({b2 . b6 where b6 is Element of the carrier of b2: b5 <= b6},b1));

:: WAYBEL32:funcreg 3
registration
  let a1 be non empty RelStr;
  let a2 be non empty transitive directed NetStr over a1;
  cluster inf_net a2 -> non empty transitive strict directed;
end;

:: WAYBEL32:funcreg 4
registration
  let a1 be non empty RelStr;
  let a2 be non empty transitive directed NetStr over a1;
  cluster inf_net a2 -> non empty strict directed;
end;

:: WAYBEL32:funcreg 5
registration
  let a1 be non empty reflexive antisymmetric /\-complete RelStr;
  let a2 be non empty transitive directed NetStr over a1;
  cluster inf_net a2 -> non empty strict directed monotone;
end;

:: WAYBEL32:funcreg 6
registration
  let a1 be non empty reflexive antisymmetric /\-complete RelStr;
  let a2 be non empty transitive directed NetStr over a1;
  cluster inf_net a2 -> non empty strict directed eventually-directed;
end;

:: WAYBEL32:th 28
theorem
for b1 being non empty RelStr
for b2 being non empty transitive directed NetStr over b1 holds
   rng the mapping of inf_net 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};

:: WAYBEL32:th 29
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima up-complete /\-complete RelStr
for b2 being non empty transitive directed NetStr over b1 holds
   sup inf_net b2 = lim_inf b2;

:: WAYBEL32:th 30
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima up-complete /\-complete RelStr
for b2 being non empty transitive directed NetStr over b1
for b3 being Element of the carrier of b2 holds
   sup inf_net b2 = lim_inf (b2 | b3);

:: WAYBEL32:th 31
theorem
for b1 being reflexive transitive antisymmetric with_infima /\-complete RelStr
for b2 being non empty transitive directed NetStr over b1
for b3 being upper Element of bool the carrier of b1
      st inf_net b2 is_eventually_in b3
   holds b2 is_eventually_in b3;

:: WAYBEL32:th 32
theorem
for b1 being reflexive transitive antisymmetric with_infima /\-complete RelStr
for b2 being non empty transitive directed NetStr over b1
for b3 being lower Element of bool the carrier of b1
      st b2 is_eventually_in b3
   holds inf_net b2 is_eventually_in b3;

:: WAYBEL32:th 33
theorem
for b1 being non empty TopSpace-like reflexive transitive antisymmetric with_suprema with_infima up-complete /\-complete order_consistent TopRelStr
for b2 being non empty transitive directed NetStr over b1
for b3 being Element of the carrier of b1
      st b3 <= lim_inf b2
   holds b3 is_a_cluster_point_of b2;

:: WAYBEL32:th 34
theorem
for b1 being non empty TopSpace-like reflexive transitive antisymmetric with_suprema with_infima up-complete /\-complete order_consistent TopRelStr
for b2 being non empty transitive directed eventually-directed NetStr over b1
for b3 being Element of the carrier of b1 holds
      b3 <= lim_inf b2
   iff
      b3 is_a_cluster_point_of b2;