Article WAYBEL21, MML version 4.99.1005

:: WAYBEL21:modenot 1 => WAYBEL21:mode 1
definition
  let a1, a2 be reflexive transitive antisymmetric with_infima RelStr;
  assume (a1 is upper-bounded implies a2 is upper-bounded);
  mode SemilatticeHomomorphism of A1,A2 -> Function-like quasi_total Relation of the carrier of a1,the carrier of a2 means
    for b1 being finite Element of bool the carrier of a1 holds
       it preserves_inf_of b1;
end;

:: WAYBEL21:dfs 1
definiens
  let a1, a2 be reflexive transitive antisymmetric with_infima RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is SemilatticeHomomorphism of a1,a2
it is sufficient to prove
thus (a1 is upper-bounded implies a2 is upper-bounded);
  thus for b1 being finite Element of bool the carrier of a1 holds
       a3 preserves_inf_of b1;

:: WAYBEL21:def 1
theorem
for b1, b2 being reflexive transitive antisymmetric with_infima RelStr
   st (b1 is upper-bounded implies b2 is upper-bounded)
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is SemilatticeHomomorphism of b1,b2
   iff
      for b4 being finite Element of bool the carrier of b1 holds
         b3 preserves_inf_of b4;

:: WAYBEL21:condreg 1
registration
  let a1, a2 be reflexive transitive antisymmetric with_infima RelStr;
  cluster Function-like quasi_total meet-preserving -> monotone (Relation of the carrier of a1,the carrier of a2);
end;

:: WAYBEL21:condreg 2
registration
  let a1 be reflexive transitive antisymmetric with_infima RelStr;
  let a2 be reflexive transitive antisymmetric upper-bounded with_infima RelStr;
  cluster -> meet-preserving (SemilatticeHomomorphism of a1,a2);
end;

:: WAYBEL21:th 1
theorem
for b1, b2 being reflexive transitive antisymmetric upper-bounded with_infima RelStr
for b3 being SemilatticeHomomorphism of b1,b2 holds
   b3 . Top b1 = Top b2;

:: WAYBEL21:th 2
theorem
for b1, b2 being reflexive transitive antisymmetric with_infima RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
   st b3 is meet-preserving(b1, b2)
for b4 being non empty finite Element of bool the carrier of b1 holds
   b3 preserves_inf_of b4;

:: WAYBEL21:th 3
theorem
for b1, b2 being reflexive transitive antisymmetric upper-bounded with_infima RelStr
for b3 being Function-like quasi_total meet-preserving Relation of the carrier of b1,the carrier of b2
      st b3 . Top b1 = Top b2
   holds b3 is SemilatticeHomomorphism of b1,b2;

:: WAYBEL21:th 4
theorem
for b1, b2 being reflexive transitive antisymmetric with_infima RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
   st b3 is meet-preserving(b1, b2) &
      (for b4 being non empty filtered Element of bool the carrier of b1 holds
         b3 preserves_inf_of b4)
for b4 being non empty Element of bool the carrier of b1 holds
   b3 preserves_inf_of b4;

:: WAYBEL21:th 5
theorem
for b1, b2 being reflexive transitive antisymmetric with_infima RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is infs-preserving(b1, b2)
   holds b3 is SemilatticeHomomorphism of b1,b2;

:: WAYBEL21:th 6
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 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 b3,the carrier of b4
      st b5 = b6
   holds (b5 is infs-preserving(b1, b2) implies b6 is infs-preserving(b3, b4)) & (b5 is directed-sups-preserving(b1, b2) implies b6 is directed-sups-preserving(b3, b4));

:: WAYBEL21:th 7
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 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 b3,the carrier of b4
      st b5 = b6
   holds (b5 is sups-preserving(b1, b2) implies b6 is sups-preserving(b3, b4)) & (b5 is filtered-infs-preserving(b1, b2) implies b6 is filtered-infs-preserving(b3, b4));

:: WAYBEL21:th 8
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty full infs-inheriting SubRelStr of b1 holds
   incl(b2,b1) is infs-preserving(b2, b1);

:: WAYBEL21:th 9
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty full sups-inheriting SubRelStr of b1 holds
   incl(b2,b1) is sups-preserving(b2, b1);

:: WAYBEL21:th 10
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr
for b2 being non empty full directed-sups-inheriting SubRelStr of b1 holds
   incl(b2,b1) is directed-sups-preserving(b2, b1);

:: WAYBEL21:th 11
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty full filtered-infs-inheriting SubRelStr of b1 holds
   incl(b2,b1) is filtered-infs-preserving(b2, b1);

:: WAYBEL21:th 12
theorem
for b1, b2, b3 being RelStr
for b4 being SubRelStr of b1
      st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#) &
         RelStr(#the carrier of b4,the InternalRel of b4#) = RelStr(#the carrier of b3,the InternalRel of b3#)
   holds b3 is SubRelStr of b2 & (b4 is full(b1) implies b3 is full SubRelStr of b2);

:: WAYBEL21:th 13
theorem
for b1 being non empty RelStr holds
   b1 is full infs-inheriting sups-inheriting SubRelStr of b1;

:: WAYBEL21:exreg 1
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  cluster non empty reflexive transitive antisymmetric full meet-inheriting infs-inheriting filtered-infs-inheriting directed-sups-inheriting with_infima complete SubRelStr of a1;
end;

:: WAYBEL21:th 14
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being non empty full SubRelStr of b1 holds
      b2 is meet-inheriting(b1)
   iff
      for b3 being non empty finite Element of bool the carrier of b2 holds
         "/\"(b3,b1) in the carrier of b2;

:: WAYBEL21:th 15
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2 being non empty full SubRelStr of b1 holds
      b2 is join-inheriting(b1)
   iff
      for b3 being non empty finite Element of bool the carrier of b2 holds
         "\/"(b3,b1) in the carrier of b2;

:: WAYBEL21:th 16
theorem
for b1 being reflexive transitive antisymmetric upper-bounded with_infima RelStr
for b2 being non empty full meet-inheriting SubRelStr of b1
      st Top b1 in the carrier of b2 & b2 is filtered-infs-inheriting(b1)
   holds b2 is infs-inheriting(b1);

:: WAYBEL21:th 17
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2 being non empty full join-inheriting SubRelStr of b1
      st Bottom b1 in the carrier of b2 & b2 is directed-sups-inheriting(b1)
   holds b2 is sups-inheriting(b1);

:: WAYBEL21:th 18
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty full SubRelStr of b1
      st b2 is infs-inheriting(b1)
   holds b2 is complete;

:: WAYBEL21:th 19
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty full SubRelStr of b1
      st b2 is sups-inheriting(b1)
   holds b2 is complete;

:: WAYBEL21:th 20
theorem
for b1, b2 being non empty RelStr
for b3 being non empty full SubRelStr of b1
for b4 being non empty full SubRelStr of b2
      st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#) &
         the carrier of b3 = the carrier of b4 &
         b3 is infs-inheriting(b1)
   holds b4 is infs-inheriting(b2);

:: WAYBEL21:th 21
theorem
for b1, b2 being non empty RelStr
for b3 being non empty full SubRelStr of b1
for b4 being non empty full SubRelStr of b2
      st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#) &
         the carrier of b3 = the carrier of b4 &
         b3 is sups-inheriting(b1)
   holds b4 is sups-inheriting(b2);

:: WAYBEL21:th 22
theorem
for b1, b2 being non empty RelStr
for b3 being non empty full SubRelStr of b1
for b4 being non empty full SubRelStr of b2
      st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#) &
         the carrier of b3 = the carrier of b4 &
         b3 is directed-sups-inheriting(b1)
   holds b4 is directed-sups-inheriting(b2);

:: WAYBEL21:th 23
theorem
for b1, b2 being non empty RelStr
for b3 being non empty full SubRelStr of b1
for b4 being non empty full SubRelStr of b2
      st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#) &
         the carrier of b3 = the carrier of b4 &
         b3 is filtered-infs-inheriting(b1)
   holds b4 is filtered-infs-inheriting(b2);

:: WAYBEL21:th 24
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being non empty transitive directed NetStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b4 is continuous(b1, b2)
   holds b4 .: Lim b3 c= Lim (b4 * b3);

:: WAYBEL21:attrnot 1 => WAYBEL_0:attr 9
definition
  let a1 be non empty RelStr;
  let a2 be non empty NetStr over a1;
  attr a2 is antitone means
    for b1, b2 being Element of the carrier of a2
          st b1 <= b2
       holds a2 . b2 <= a2 . b1;
end;

:: WAYBEL21:dfs 2
definiens
  let a1 be non empty RelStr;
  let a2 be non empty NetStr over a1;
To prove
     a2 is antitone
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a2
          st b1 <= b2
       holds a2 . b2 <= a2 . b1;

:: WAYBEL21:def 2
theorem
for b1 being non empty RelStr
for b2 being non empty NetStr over b1 holds
      b2 is antitone(b1)
   iff
      for b3, b4 being Element of the carrier of b2
            st b3 <= b4
         holds b2 . b4 <= b2 . b3;

:: WAYBEL21:funcreg 1
registration
  let a1 be non empty reflexive RelStr;
  let a2 be Element of the carrier of a1;
  cluster {a2} opp+id -> non empty transitive strict directed monotone antitone;
end;

:: WAYBEL21:exreg 2
registration
  let a1 be non empty reflexive RelStr;
  cluster non empty reflexive transitive strict directed monotone antitone NetStr over a1;
end;

:: WAYBEL21:funcreg 2
registration
  let a1 be non empty RelStr;
  let a2 be non empty Element of bool the carrier of a1;
  cluster a2 opp+id -> non empty strict antitone;
end;

:: WAYBEL21:funcreg 3
registration
  let a1, a2 be non empty reflexive RelStr;
  let a3 be Function-like quasi_total monotone Relation of the carrier of a1,the carrier of a2;
  let a4 be non empty antitone NetStr over a1;
  cluster a3 * a4 -> strict antitone;
end;

:: WAYBEL21:th 25
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty transitive directed NetStr over b1 holds
   {"/\"({b2 . b4 where b4 is Element of the carrier of b2: b3 <= b4},b1) where b3 is Element of the carrier of b2: TRUE} is non empty directed Element of bool the carrier of b1;

:: WAYBEL21:th 26
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being non empty reflexive transitive directed monotone NetStr over b1 holds
   {"/\"({b2 . b4 where b4 is Element of the carrier of b2: b3 <= b4},b1) where b3 is Element of the carrier of b2: TRUE} is non empty directed Element of bool the carrier of b1;

:: WAYBEL21:th 27
theorem
for b1 being non empty 1-sorted
for b2 being non empty NetStr over b1
for b3 being set
      st rng the mapping of b2 c= b3
   holds b2 is_eventually_in b3;

:: WAYBEL21:th 28
theorem
for b1 being non empty reflexive transitive antisymmetric /\-complete RelStr
for b2 being non empty filtered Element of bool the carrier of b1 holds
   lim_inf (b2 opp+id) = "/\"(b2,b1);

:: WAYBEL21:th 29
theorem
for b1, b2 being non empty reflexive transitive antisymmetric /\-complete RelStr
for b3 being non empty filtered Element of bool the carrier of b1
for b4 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b2 holds
   lim_inf (b4 * (b3 opp+id)) = "/\"(b4 .: b3,b2);

:: WAYBEL21:th 30
theorem
for b1, b2 being non empty TopSpace-like reflexive transitive antisymmetric TopRelStr
for b3 being non empty filtered Element of bool the carrier of b1
for b4 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b2
for b5 being non empty filtered Element of bool the carrier of b2
      st b5 = b4 .: b3
   holds b4 * (b3 opp+id) is subnet of b5 opp+id;

:: WAYBEL21:th 31
theorem
for b1, b2 being non empty TopSpace-like reflexive transitive antisymmetric TopRelStr
for b3 being non empty filtered Element of bool the carrier of b1
for b4 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b2
for b5 being non empty filtered Element of bool the carrier of b2
      st b5 = b4 .: b3
   holds Lim (b5 opp+id) c= Lim (b4 * (b3 opp+id));

:: WAYBEL21:th 32
theorem
for b1 being non empty reflexive RelStr
for b2 being non empty Element of bool the carrier of b1 holds
   the mapping of Net-Str b2 = id b2 & the carrier of Net-Str b2 = b2 & Net-Str b2 is full SubRelStr of b1;

:: WAYBEL21:th 33
theorem
for b1, b2 being non empty reflexive transitive antisymmetric up-complete RelStr
for b3 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b2
for b4 being non empty directed Element of bool the carrier of b1 holds
   lim_inf (b3 * Net-Str b4) = "\/"(b3 .: b4,b2);

:: WAYBEL21:th 34
theorem
for b1 being non empty reflexive RelStr
for b2 being non empty directed Element of bool the carrier of b1
for b3, b4 being Element of the carrier of Net-Str b2 holds
   b3 <= b4
iff
   (Net-Str b2) . b3 <= (Net-Str b2) . b4;

:: WAYBEL21:th 35
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr
for b2 being non empty directed Element of bool the carrier of b1 holds
   "\/"(b2,b1) in Lim Net-Str b2;

:: WAYBEL21:modenot 2 => WAYBEL21:mode 2
definition
  let a1 be non empty 1-sorted;
  let a2 be non empty transitive directed NetStr over a1;
  let a3 be non empty NetStr over a1;
  assume a3 is subnet of a2;
  mode Embedding of A3,A2 -> Function-like quasi_total Relation of the carrier of a3,the carrier of a2 means
    the mapping of a3 = (the mapping of a2) * it &
     (for b1 being Element of the carrier of a2 holds
        ex b2 being Element of the carrier of a3 st
           for b3 being Element of the carrier of a3
                 st b2 <= b3
              holds b1 <= it . b3);
end;

:: WAYBEL21:dfs 3
definiens
  let a1 be non empty 1-sorted;
  let a2 be non empty transitive directed NetStr over a1;
  let a3 be non empty NetStr over a1;
  let a4 be Function-like quasi_total Relation of the carrier of a3,the carrier of a2;
To prove
     a4 is Embedding of a3,a2
it is sufficient to prove
thus a3 is subnet of a2;
  thus the mapping of a3 = (the mapping of a2) * a4 &
     (for b1 being Element of the carrier of a2 holds
        ex b2 being Element of the carrier of a3 st
           for b3 being Element of the carrier of a3
                 st b2 <= b3
              holds b1 <= a4 . b3);

:: WAYBEL21:def 3
theorem
for b1 being non empty 1-sorted
for b2 being non empty transitive directed NetStr over b1
for b3 being non empty NetStr over b1
   st b3 is subnet of b2
for b4 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2 holds
      b4 is Embedding of b3,b2
   iff
      the mapping of b3 = (the mapping of b2) * b4 &
       (for b5 being Element of the carrier of b2 holds
          ex b6 being Element of the carrier of b3 st
             for b7 being Element of the carrier of b3
                   st b6 <= b7
                holds b5 <= b4 . b7);

:: WAYBEL21:th 36
theorem
for b1 being non empty 1-sorted
for b2 being non empty transitive directed NetStr over b1
for b3 being subnet of b2
for b4 being Embedding of b3,b2
for b5 being Element of the carrier of b3 holds
   b3 . b5 = b2 . (b4 . b5);

:: WAYBEL21:th 37
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 subnet of b2 holds
   lim_inf b2 <= lim_inf b3;

:: WAYBEL21:th 38
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 subnet of b2
for b4 being Embedding of b3,b2
      st for b5 being Element of the carrier of b2
        for b6 being Element of the carrier of b3
              st b4 . b6 <= b5
           holds ex b7 being Element of the carrier of b3 st
              b6 <= b7 & b3 . b7 <= b2 . b5
   holds lim_inf b2 = lim_inf b3;

:: WAYBEL21:th 39
theorem
for b1 being non empty RelStr
for b2 being non empty transitive directed NetStr over b1
for b3 being non empty full SubNetStr of b2
      st for b4 being Element of the carrier of b2 holds
           ex b5 being Element of the carrier of b2 st
              b4 <= b5 & b5 in the carrier of b3
   holds b3 is subnet of b2 & incl(b3,b2) is Embedding of b3,b2;

:: WAYBEL21:th 40
theorem
for b1 being non empty RelStr
for b2 being non empty transitive directed NetStr over b1
for b3 being Element of the carrier of b2 holds
   b2 | b3 is subnet of b2 & incl(b2 | b3,b2) is Embedding of b2 | b3,b2;

:: WAYBEL21:th 41
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 the carrier of b2 holds
   lim_inf (b2 | b3) = lim_inf b2;

:: WAYBEL21:th 42
theorem
for b1 being non empty RelStr
for b2 being non empty transitive directed NetStr over b1
for b3 being set
      st b2 is_eventually_in b3
   holds ex b4 being Element of the carrier of b2 st
      b2 . b4 in b3 & rng the mapping of b2 | b4 c= b3;

:: WAYBEL21:th 43
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr
for b2 being non empty transitive directed eventually-filtered NetStr over b1 holds
   rng the mapping of b2 is non empty filtered Element of bool the carrier of b1;

:: WAYBEL21:th 44
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr
for b2 being non empty transitive directed eventually-filtered NetStr over b1 holds
   Lim b2 = {inf b2};

:: WAYBEL21:th 45
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr
for b3 being Function-like quasi_total meet-preserving Relation of the carrier of b1,the carrier of b2 holds
      b3 is continuous(b1, b2)
   iff
      b3 is directed-sups-preserving(b1, b2) &
       (for b4 being non empty Element of bool the carrier of b1 holds
          b3 preserves_inf_of b4);

:: WAYBEL21:th 46
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr
for b3 being SemilatticeHomomorphism of b1,b2 holds
      b3 is continuous(b1, b2)
   iff
      b3 is infs-preserving(b1, b2) & b3 is directed-sups-preserving(b1, b2);

:: WAYBEL21:attrnot 2 => WAYBEL21:attr 1
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 lim_infs-preserving means
    for b1 being non empty transitive directed NetStr over a1 holds
       a3 . lim_inf b1 = lim_inf (a3 * b1);
end;

:: WAYBEL21:dfs 4
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 lim_infs-preserving
it is sufficient to prove
  thus for b1 being non empty transitive directed NetStr over a1 holds
       a3 . lim_inf b1 = lim_inf (a3 * b1);

:: WAYBEL21:def 4
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 lim_infs-preserving(b1, b2)
   iff
      for b4 being non empty transitive directed NetStr over b1 holds
         b3 . lim_inf b4 = lim_inf (b3 * b4);

:: WAYBEL21:th 47
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr
for b3 being SemilatticeHomomorphism of b1,b2 holds
      b3 is continuous(b1, b2)
   iff
      b3 is lim_infs-preserving(b1, b2);

:: WAYBEL21:th 48
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric continuous with_suprema with_infima complete Lawson TopRelStr
for b2 being non empty full meet-inheriting SubRelStr of b1
      st Top b1 in the carrier of b2 &
         (ex b3 being Element of bool the carrier of b1 st
            b3 = the carrier of b2 & b3 is closed(b1))
   holds b2 is infs-inheriting(b1);

:: WAYBEL21:th 49
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric continuous with_suprema with_infima complete Lawson TopRelStr
for b2 being non empty full SubRelStr of b1
      st ex b3 being Element of bool the carrier of b1 st
           b3 = the carrier of b2 & b3 is closed(b1)
   holds b2 is directed-sups-inheriting(b1);

:: WAYBEL21:th 50
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric continuous with_suprema with_infima complete Lawson TopRelStr
for b2 being non empty full infs-inheriting directed-sups-inheriting SubRelStr of b1 holds
   ex b3 being Element of bool the carrier of b1 st
      b3 = the carrier of b2 & b3 is closed(b1);

:: WAYBEL21:th 51
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric continuous with_suprema with_infima complete Lawson TopRelStr
for b2 being non empty full infs-inheriting directed-sups-inheriting SubRelStr of b1
for b3 being non empty transitive directed NetStr over b1
      st b3 is_eventually_in the carrier of b2
   holds lim_inf b3 in the carrier of b2;

:: WAYBEL21:th 52
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric continuous with_suprema with_infima complete Lawson TopRelStr
for b2 being non empty full meet-inheriting SubRelStr of b1
      st Top b1 in the carrier of b2 &
         (for b3 being non empty transitive directed NetStr over b1
               st rng the mapping of b3 c= the carrier of b2
            holds lim_inf b3 in the carrier of b2)
   holds b2 is infs-inheriting(b1);

:: WAYBEL21:th 53
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric continuous with_suprema with_infima complete Lawson TopRelStr
for b2 being non empty full SubRelStr of b1
      st for b3 being non empty transitive directed NetStr over b1
              st rng the mapping of b3 c= the carrier of b2
           holds lim_inf b3 in the carrier of b2
   holds b2 is directed-sups-inheriting(b1);

:: WAYBEL21:th 54
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric continuous with_suprema with_infima complete Lawson TopRelStr
for b2 being non empty full meet-inheriting SubRelStr of b1
for b3 being Element of bool the carrier of b1
      st b3 = the carrier of b2 & Top b1 in b3
   holds    b3 is closed(b1)
   iff
      for b4 being non empty transitive directed NetStr over b1
            st b4 is_eventually_in b3
         holds lim_inf b4 in b3;