Article WAYBEL28, MML version 4.99.1005

:: WAYBEL28:th 1
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty transitive directed NetStr over b1 holds
   inf b2 <= lim_inf b2;

:: WAYBEL28:th 2
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 b1
      st for b4 being subnet of b2 holds
           b3 = lim_inf b4
   holds b3 = lim_inf b2 &
    (for b4 being subnet of b2 holds
       inf b4 <= b3);

:: WAYBEL28:th 3
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 b1
      st b2 in NetUniv b1 &
         (for b4 being subnet of b2
               st b4 in NetUniv b1
            holds b3 = lim_inf b4)
   holds b3 = lim_inf b2 &
    (for b4 being subnet of b2
          st b4 in NetUniv b1
       holds inf b4 <= b3);

:: WAYBEL28:attrnot 1 => WAYBEL28:attr 1
definition
  let a1 be non empty RelStr;
  let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
  attr a2 is greater_or_equal_to_id means
    for b1 being Element of the carrier of a1 holds
       b1 <= a2 . b1;
end;

:: WAYBEL28:dfs 1
definiens
  let a1 be non empty RelStr;
  let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is greater_or_equal_to_id
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       b1 <= a2 . b1;

:: WAYBEL28:def 1
theorem
for b1 being non empty RelStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
      b2 is greater_or_equal_to_id(b1)
   iff
      for b3 being Element of the carrier of b1 holds
         b3 <= b2 . b3;

:: WAYBEL28:th 4
theorem
for b1 being non empty reflexive RelStr holds
   id b1 is greater_or_equal_to_id(b1);

:: WAYBEL28:th 5
theorem
for b1 being non empty directed RelStr
for b2, b3 being Element of the carrier of b1 holds
ex b4 being Element of the carrier of b1 st
   b2 <= b4 & b3 <= b4;

:: WAYBEL28:th 6
theorem
for b1 being non empty directed RelStr holds
   ex b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 st
      b2 is greater_or_equal_to_id(b1);

:: WAYBEL28:exreg 1
registration
  let a1 be non empty directed RelStr;
  cluster non empty Relation-like Function-like total quasi_total greater_or_equal_to_id Relation of the carrier of a1,the carrier of a1;
end;

:: WAYBEL28:exreg 2
registration
  let a1 be non empty reflexive RelStr;
  cluster non empty Relation-like Function-like total quasi_total greater_or_equal_to_id Relation of the carrier of a1,the carrier of a1;
end;

:: WAYBEL28:funcnot 1 => WAYBEL28:func 1
definition
  let a1 be non empty 1-sorted;
  let a2 be non empty NetStr over a1;
  let a3 be Function-like quasi_total Relation of the carrier of a2,the carrier of a2;
  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 = (the mapping of a2) * a3;
end;

:: WAYBEL28:def 2
theorem
for b1 being non empty 1-sorted
for b2 being non empty NetStr over b1
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b2
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 = (the mapping of b2) * b3;

:: WAYBEL28:th 7
theorem
for b1 being non empty 1-sorted
for b2 being non empty NetStr over b1
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b2 holds
   the carrier of b2 * b3 = the carrier of b2;

:: WAYBEL28:th 8
theorem
for b1 being non empty 1-sorted
for b2 being non empty NetStr over b1
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b2 holds
   b2 * b3 = NetStr(#the carrier of b2,the InternalRel of b2,(the mapping of b2) * b3#);

:: WAYBEL28:th 9
theorem
for b1 being non empty 1-sorted
for b2 being non empty transitive directed RelStr
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1 holds
   NetStr(#the carrier of b2,the InternalRel of b2,b3#) is non empty transitive directed NetStr over b1;

:: WAYBEL28:funcreg 1
registration
  let a1 be non empty 1-sorted;
  let a2 be non empty transitive directed RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a2,the carrier of a1;
  cluster NetStr(#the carrier of a2,the InternalRel of a2,a3#) -> non empty transitive strict directed;
end;

:: WAYBEL28:th 10
theorem
for b1 being non empty 1-sorted
for b2 being non empty transitive directed NetStr over b1
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b2 holds
   b2 * b3 is non empty transitive directed NetStr over b1;

:: WAYBEL28:funcreg 2
registration
  let a1 be non empty 1-sorted;
  let a2 be non empty transitive directed NetStr over a1;
  let a3 be Function-like quasi_total Relation of the carrier of a2,the carrier of a2;
  cluster a2 * a3 -> non empty transitive strict directed;
end;

:: WAYBEL28:th 11
theorem
for b1 being non empty 1-sorted
for b2 being non empty transitive directed NetStr over b1
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b2
      st b2 in NetUniv b1
   holds b2 * b3 in NetUniv b1;

:: WAYBEL28:th 12
theorem
for b1 being non empty 1-sorted
for b2, b3 being non empty transitive directed NetStr over b1
      st NetStr(#the carrier of b2,the InternalRel of b2,the mapping of b2#) = NetStr(#the carrier of b3,the InternalRel of b3,the mapping of b3#)
   holds b3 is subnet of b2;

:: WAYBEL28:th 13
theorem
for b1 being non empty 1-sorted
for b2 being non empty transitive directed NetStr over b1
for b3 being Function-like quasi_total greater_or_equal_to_id Relation of the carrier of b2,the carrier of b2 holds
   b2 * b3 is subnet of b2;

:: WAYBEL28:funcnot 2 => WAYBEL28:func 2
definition
  let a1 be non empty 1-sorted;
  let a2 be non empty transitive directed NetStr over a1;
  let a3 be Function-like quasi_total greater_or_equal_to_id Relation of the carrier of a2,the carrier of a2;
  redefine func a2 * a3 -> strict subnet of a2;
end;

:: WAYBEL28:th 14
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 b1
      st b2 in NetUniv b1 &
         b3 = lim_inf b2 &
         (for b4 being subnet of b2
               st b4 in NetUniv b1
            holds inf b4 <= b3)
   holds b3 = lim_inf b2 &
    (for b4 being Function-like quasi_total greater_or_equal_to_id Relation of the carrier of b2,the carrier of b2 holds
       inf (b2 * b4) <= b3);

:: WAYBEL28:th 15
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 b1
   st b3 = lim_inf b2 &
      (for b4 being Function-like quasi_total greater_or_equal_to_id Relation of the carrier of b2,the carrier of b2 holds
         inf (b2 * b4) <= b3)
for b4 being subnet of b2 holds
   b3 = lim_inf b4;

:: WAYBEL28:funcnot 3 => WAYBEL28:func 3
definition
  let a1 be non empty RelStr;
  func lim_inf-Convergence A1 -> Convergence-Class of a1 means
    for b1 being non empty transitive directed NetStr over a1
       st b1 in NetUniv a1
    for b2 being Element of the carrier of a1 holds
          [b1,b2] in it
       iff
          for b3 being subnet of b1 holds
             b2 = lim_inf b3;
end;

:: WAYBEL28:def 3
theorem
for b1 being non empty RelStr
for b2 being Convergence-Class of b1 holds
      b2 = lim_inf-Convergence b1
   iff
      for b3 being non empty transitive directed NetStr over b1
         st b3 in NetUniv b1
      for b4 being Element of the carrier of b1 holds
            [b3,b4] in b2
         iff
            for b5 being subnet of b3 holds
               b4 = lim_inf b5;

:: WAYBEL28:th 16
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 b1
      st b2 in NetUniv b1
   holds    [b2,b3] in lim_inf-Convergence b1
   iff
      for b4 being subnet of b2
            st b4 in NetUniv b1
         holds b3 = lim_inf b4;

:: WAYBEL28:th 17
theorem
for b1 being non empty RelStr
for b2 being non empty transitive directed constant NetStr over b1
for b3 being subnet of b2 holds
   b3 is constant(b1) & the_value_of b2 = the_value_of b3;

:: WAYBEL28:funcnot 4 => WAYBEL28:func 4
definition
  let a1 be non empty RelStr;
  func xi A1 -> Element of bool bool the carrier of a1 equals
    the topology of ConvergenceSpace lim_inf-Convergence a1;
end;

:: WAYBEL28:def 4
theorem
for b1 being non empty RelStr holds
   xi b1 = the topology of ConvergenceSpace lim_inf-Convergence b1;

:: WAYBEL28:th 18
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
   lim_inf-Convergence b1 is (CONSTANTS)(b1);

:: WAYBEL28:th 19
theorem
for b1 being non empty RelStr holds
   lim_inf-Convergence b1 is (SUBNETS)(b1);

:: WAYBEL28:th 20
theorem
for b1 being reflexive transitive antisymmetric continuous with_suprema with_infima complete RelStr holds
   lim_inf-Convergence b1 is (DIVERGENCE)(b1);

:: WAYBEL28:th 21
theorem
for b1 being non empty RelStr
for b2, b3 being set
      st [b2,b3] in lim_inf-Convergence b1
   holds b2 in NetUniv b1;

:: WAYBEL28:th 22
theorem
for b1 being non empty 1-sorted
for b2, b3 being Convergence-Class of b1
      st b2 c= b3
   holds the topology of ConvergenceSpace b3 c= the topology of ConvergenceSpace b2;

:: WAYBEL28:th 23
theorem
for b1 being non empty reflexive RelStr holds
   lim_inf-Convergence b1 c= Scott-Convergence b1;

:: WAYBEL28:th 24
theorem
for b1, b2 being set
      st b1 c= b2
   holds b1 in the_universe_of b2;

:: WAYBEL28:th 25
theorem
for b1 being non empty reflexive transitive RelStr
for b2 being non empty directed Element of bool the carrier of b1 holds
   Net-Str b2 in NetUniv b1;

:: WAYBEL28:th 26
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty directed Element of bool the carrier of b1
for b3 being subnet of Net-Str b2 holds
   lim_inf b3 = "\/"(b2,b1);

:: WAYBEL28:th 27
theorem
for b1 being non empty reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty directed Element of bool the carrier of b1 holds
   [Net-Str b2,"\/"(b2,b1)] in lim_inf-Convergence b1;

:: WAYBEL28:th 28
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Element of bool the carrier of b1
      st b2 in xi b1
   holds b2 is property(S)(b1);

:: WAYBEL28:th 29
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of bool the carrier of b1
      st b2 in sigma b1
   holds b2 in xi b1;

:: WAYBEL28:th 30
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Element of bool the carrier of b1
      st b2 is upper(b1) & b2 in xi b1
   holds b2 in sigma b1;

:: WAYBEL28: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
      st b2 is lower(b1)
   holds    b2 ` in xi b1
   iff
      b2 is closed_under_directed_sups(b1);