Article WAYBEL30, MML version 4.99.1005

:: WAYBEL30:th 1
theorem
for b1 being set
for b2 being non empty set holds
   b1 /\ union b2 = union {b1 /\ b3 where b3 is Element of b2: TRUE};

:: WAYBEL30:th 2
theorem
for b1 being non empty reflexive transitive RelStr
for b2 being non empty directed Element of bool the carrier of InclPoset Ids b1 holds
   union b2 is non empty directed lower Element of bool the carrier of b1;

:: WAYBEL30:funcreg 1
registration
  let a1 be non empty reflexive transitive RelStr;
  cluster InclPoset Ids a1 -> strict up-complete;
end;

:: WAYBEL30:th 3
theorem
for b1 being non empty reflexive transitive RelStr
for b2 being non empty directed Element of bool the carrier of InclPoset Ids b1 holds
   "\/"(b2,InclPoset Ids b1) = union b2;

:: WAYBEL30:th 4
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being non empty directed Element of bool the carrier of InclPoset Ids b1
for b3 being Element of the carrier of InclPoset Ids b1 holds
   "\/"({b3} "/\" b2,InclPoset Ids b1) = union {b3 /\ b4 where b4 is Element of b2: TRUE};

:: WAYBEL30:funcreg 2
registration
  let a1 be reflexive transitive antisymmetric with_infima RelStr;
  cluster InclPoset Ids a1 -> strict satisfying_MC;
end;

:: WAYBEL30:condreg 1
registration
  let a1 be non empty trivial RelStr;
  cluster -> trivial (TopAugmentation of a1);
end;

:: WAYBEL30:th 5
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Scott TopAugmentation of b2
      st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#)
   holds TopRelStr(#the carrier of b3,the InternalRel of b3,the topology of b3#) = TopRelStr(#the carrier of b1,the InternalRel of b1,the topology of b1#);

:: WAYBEL30:th 6
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr
for b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being TopSpace-like Lawson TopAugmentation of b2
      st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#)
   holds TopRelStr(#the carrier of b3,the InternalRel of b3,the topology of b3#) = TopRelStr(#the carrier of b1,the InternalRel of b1,the topology of b1#);

:: WAYBEL30:th 7
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr
for b2 being Scott TopAugmentation of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4 & b4 is closed(b2)
   holds b3 is closed(b1);

:: WAYBEL30:funcreg 3
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  cluster lambda a1 -> non empty;
end;

:: WAYBEL30:funcreg 4
registration
  let a1 be TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr;
  cluster InclPoset sigma a1 -> non trivial strict complete;
end;

:: WAYBEL30:funcreg 5
registration
  let a1 be TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr;
  cluster InclPoset sigma a1 -> non trivial strict complete;
end;

:: WAYBEL30:funcreg 6
registration
  let a1 be TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr;
  cluster InclPoset lambda a1 -> non trivial strict complete;
end;

:: WAYBEL30:th 8
theorem
for b1 being non empty reflexive RelStr holds
   sigma b1 c= {b2 \ uparrow b3 where b2 is Element of bool the carrier of b1, b3 is Element of bool the carrier of b1: b2 in sigma b1 & b3 is finite};

:: WAYBEL30:th 9
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr holds
   lambda b1 = the topology of b1;

:: WAYBEL30:th 10
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr holds
   sigma b1 c= lambda b1;

:: WAYBEL30:th 11
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
      st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#)
   holds lambda b1 = lambda b2;

:: WAYBEL30:th 12
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr
for b2 being Element of bool the carrier of b1 holds
      b2 in lambda b1
   iff
      b2 is open(b1);

:: WAYBEL30:condreg 2
registration
  cluster non empty trivial TopSpace-like reflexive -> Scott (TopRelStr);
end;

:: WAYBEL30:condreg 3
registration
  cluster trivial TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete -> Lawson (TopRelStr);
end;

:: WAYBEL30:exreg 1
registration
  cluster non empty TopSpace-like reflexive transitive antisymmetric lower-bounded upper-bounded bounded up-complete /\-complete with_suprema with_infima complete non void meet-continuous strict Scott continuous TopRelStr;
end;

:: WAYBEL30:exreg 2
registration
  cluster non empty TopSpace-like reflexive transitive antisymmetric lower-bounded upper-bounded bounded up-complete /\-complete with_suprema with_infima complete non void compact being_T2 strict Lawson continuous TopRelStr;
end;

:: WAYBEL30:sch 1
scheme WAYBEL30:sch 1
{F1 -> TopSpace-like reflexive transitive antisymmetric with_suprema with_infima Scott TopRelStr,
  F2 -> set,
  F3 -> set}:
{F3(b1) where b1 is Element of the carrier of F1(): b1 in {}} = {}


:: WAYBEL30:th 13
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima meet-continuous RelStr
for b2 being Element of bool the carrier of b1
      st b2 is property(S)(b1)
   holds uparrow b2 is property(S)(b1);

:: WAYBEL30:funcreg 7
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima meet-continuous RelStr;
  let a2 be property(S) Element of bool the carrier of a1;
  cluster uparrow a2 -> property(S);
end;

:: WAYBEL30:th 14
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete meet-continuous Lawson TopRelStr
for b2 being Scott TopAugmentation of b1
for b3 being Element of bool the carrier of b1
      st b3 in lambda b1
   holds uparrow b3 in sigma b2;

:: WAYBEL30:th 15
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete meet-continuous Lawson TopRelStr
for b2 being Scott TopAugmentation of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4 & b3 is open(b1)
   holds uparrow b4 is open(b2);

:: WAYBEL30:th 16
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete meet-continuous Lawson TopRelStr
for b2 being Scott TopAugmentation of b1
for b3 being Element of the carrier of b2
for b4 being Element of the carrier of b1
for b5 being Basis of b4
      st b3 = b4
   holds {uparrow b6 where b6 is Element of bool the carrier of b1: b6 in b5} is Basis of b3;

:: WAYBEL30:th 17
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete meet-continuous Lawson TopRelStr
for b2 being Scott TopAugmentation of b1
for b3 being upper Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4
   holds Int b3 = Int b4;

:: WAYBEL30:th 18
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete meet-continuous Lawson TopRelStr
for b2 being Scott TopAugmentation of b1
for b3 being lower Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4
   holds Cl b3 = Cl b4;

:: WAYBEL30:th 19
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being TopSpace-like Lawson TopAugmentation of b1
for b4 being TopSpace-like Lawson TopAugmentation of b2
      st InclPoset sigma b2 is continuous
   holds the topology of [:b3,b4:] = lambda [:b1,b2:];

:: WAYBEL30:th 20
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being TopSpace-like Lawson TopAugmentation of [:b1,b2:]
for b4 being TopSpace-like Lawson TopAugmentation of b1
for b5 being TopSpace-like Lawson TopAugmentation of b2
      st InclPoset sigma b2 is continuous
   holds TopStruct(#the carrier of b3,the topology of b3#) = [:b4,b5:];

:: WAYBEL30:th 21
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete meet-continuous Lawson TopRelStr
for b2 being Element of the carrier of b1 holds
   b2 "/\" is continuous(b1, b1);

:: WAYBEL30:funcreg 8
registration
  let a1 be TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete meet-continuous Lawson TopRelStr;
  let a2 be Element of the carrier of a1;
  cluster a2 "/\" -> Function-like quasi_total continuous;
end;

:: WAYBEL30:th 22
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete meet-continuous Lawson TopRelStr
      st InclPoset sigma b1 is continuous
   holds b1 is topological_semilattice;

:: WAYBEL30:th 23
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete meet-continuous Lawson TopRelStr
      st InclPoset sigma b1 is continuous
   holds    b1 is being_T2
   iff
      for b2 being Element of bool the carrier of [:b1,b1:]
            st b2 = the InternalRel of b1
         holds b2 is closed([:b1,b1:]);

:: WAYBEL30:funcnot 1 => WAYBEL30:func 1
definition
  let a1 be non empty reflexive RelStr;
  let a2 be Element of bool the carrier of a1;
  func A2 ^0 -> Element of bool the carrier of a1 equals
    {b1 where b1 is Element of the carrier of a1: for b2 being non empty directed Element of bool the carrier of a1
          st b1 <= "\/"(b2,a1)
       holds a2 meets b2};
end;

:: WAYBEL30:def 1
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of bool the carrier of b1 holds
   b2 ^0 = {b3 where b3 is Element of the carrier of b1: for b4 being non empty directed Element of bool the carrier of b1
         st b3 <= "\/"(b4,b1)
      holds b2 meets b4};

:: WAYBEL30:funcreg 9
registration
  let a1 be non empty reflexive antisymmetric RelStr;
  let a2 be empty Element of bool the carrier of a1;
  cluster a2 ^0 -> empty;
end;

:: WAYBEL30:th 24
theorem
for b1 being non empty reflexive RelStr
for b2, b3 being Element of bool the carrier of b1
      st b2 c= b3
   holds b2 ^0 c= b3 ^0;

:: WAYBEL30:th 25
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of the carrier of b1 holds
   (uparrow b2) ^0 = wayabove b2;

:: WAYBEL30:th 26
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima Scott TopRelStr
for b2 being upper Element of bool the carrier of b1 holds
   Int b2 c= b2 ^0;

:: WAYBEL30:th 27
theorem
for b1 being non empty reflexive RelStr
for b2, b3 being Element of bool the carrier of b1 holds
b2 ^0 \/ (b3 ^0) c= (b2 \/ b3) ^0;

:: WAYBEL30:th 28
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima meet-continuous RelStr
for b2, b3 being upper Element of bool the carrier of b1 holds
b2 ^0 \/ (b3 ^0) = (b2 \/ b3) ^0;

:: WAYBEL30:th 29
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima meet-continuous Scott TopRelStr
for b2 being finite Element of bool the carrier of b1 holds
   Int uparrow b2 c= union {wayabove b3 where b3 is Element of the carrier of b1: b3 in b2};

:: WAYBEL30:th 30
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr holds
      b1 is continuous
   iff
      b1 is meet-continuous & b1 is being_T2;

:: WAYBEL30:condreg 4
registration
  cluster TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson continuous -> being_T2 (TopRelStr);
end;

:: WAYBEL30:condreg 5
registration
  cluster TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete meet-continuous being_T2 Lawson -> continuous (TopRelStr);
end;

:: WAYBEL30:attrnot 1 => WAYBEL30:attr 1
definition
  let a1 be non empty TopRelStr;
  attr a1 is with_small_semilattices means
    for b1 being Element of the carrier of a1 holds
       ex b2 being basis of b1 st
          for b3 being Element of bool the carrier of a1
                st b3 in b2
             holds subrelstr b3 is meet-inheriting(a1);
end;

:: WAYBEL30:dfs 2
definiens
  let a1 be non empty TopRelStr;
To prove
     a1 is with_small_semilattices
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       ex b2 being basis of b1 st
          for b3 being Element of bool the carrier of a1
                st b3 in b2
             holds subrelstr b3 is meet-inheriting(a1);

:: WAYBEL30:def 2
theorem
for b1 being non empty TopRelStr holds
      b1 is with_small_semilattices
   iff
      for b2 being Element of the carrier of b1 holds
         ex b3 being basis of b2 st
            for b4 being Element of bool the carrier of b1
                  st b4 in b3
               holds subrelstr b4 is meet-inheriting(b1);

:: WAYBEL30:attrnot 2 => WAYBEL30:attr 2
definition
  let a1 be non empty TopRelStr;
  attr a1 is with_compact_semilattices means
    for b1 being Element of the carrier of a1 holds
       ex b2 being basis of b1 st
          for b3 being Element of bool the carrier of a1
                st b3 in b2
             holds subrelstr b3 is meet-inheriting(a1) & b3 is compact(a1);
end;

:: WAYBEL30:dfs 3
definiens
  let a1 be non empty TopRelStr;
To prove
     a1 is with_compact_semilattices
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       ex b2 being basis of b1 st
          for b3 being Element of bool the carrier of a1
                st b3 in b2
             holds subrelstr b3 is meet-inheriting(a1) & b3 is compact(a1);

:: WAYBEL30:def 3
theorem
for b1 being non empty TopRelStr holds
      b1 is with_compact_semilattices
   iff
      for b2 being Element of the carrier of b1 holds
         ex b3 being basis of b2 st
            for b4 being Element of bool the carrier of b1
                  st b4 in b3
               holds subrelstr b4 is meet-inheriting(b1) & b4 is compact(b1);

:: WAYBEL30:attrnot 3 => WAYBEL30:attr 3
definition
  let a1 be non empty TopRelStr;
  attr a1 is with_open_semilattices means
    for b1 being Element of the carrier of a1 holds
       ex b2 being Basis of b1 st
          for b3 being Element of bool the carrier of a1
                st b3 in b2
             holds subrelstr b3 is meet-inheriting(a1);
end;

:: WAYBEL30:dfs 4
definiens
  let a1 be non empty TopRelStr;
To prove
     a1 is with_open_semilattices
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       ex b2 being Basis of b1 st
          for b3 being Element of bool the carrier of a1
                st b3 in b2
             holds subrelstr b3 is meet-inheriting(a1);

:: WAYBEL30:def 4
theorem
for b1 being non empty TopRelStr holds
      b1 is with_open_semilattices
   iff
      for b2 being Element of the carrier of b1 holds
         ex b3 being Basis of b2 st
            for b4 being Element of bool the carrier of b1
                  st b4 in b3
               holds subrelstr b4 is meet-inheriting(b1);

:: WAYBEL30:condreg 6
registration
  cluster non empty TopSpace-like with_open_semilattices -> with_small_semilattices (TopRelStr);
end;

:: WAYBEL30:condreg 7
registration
  cluster non empty TopSpace-like with_compact_semilattices -> with_small_semilattices (TopRelStr);
end;

:: WAYBEL30:condreg 8
registration
  cluster non empty anti-discrete -> with_small_semilattices with_open_semilattices (TopRelStr);
end;

:: WAYBEL30:condreg 9
registration
  cluster non empty trivial TopSpace-like reflexive -> with_compact_semilattices (TopRelStr);
end;

:: WAYBEL30:exreg 3
registration
  cluster non empty trivial TopSpace-like reflexive transitive antisymmetric with_suprema with_infima non void strict lower TopRelStr;
end;

:: WAYBEL30:th 31
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_infima topological_semilattice TopRelStr
for b2 being Element of bool the carrier of b1
      st subrelstr b2 is meet-inheriting(b1)
   holds subrelstr Cl b2 is meet-inheriting(b1);

:: WAYBEL30:th 32
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete meet-continuous Lawson TopRelStr
for b2 being Scott TopAugmentation of b1 holds
      for b3 being Element of the carrier of b2 holds
         ex b4 being Basis of b3 st
            for b5 being Element of bool the carrier of b2
                  st b5 in b4
               holds b5 is non empty filtered upper Element of bool the carrier of b2
   iff
      b1 is with_open_semilattices;

:: WAYBEL30:th 33
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr
for b2 being Scott TopAugmentation of b1
for b3 being Element of the carrier of b1 holds
   {"/\"(b4,b2) where b4 is Element of bool the carrier of b2: b3 in b4 & b4 in sigma b2} c= {"/\"(b4,b1) where b4 is Element of bool the carrier of b1: b3 in b4 & b4 in lambda b1};

:: WAYBEL30:th 34
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete meet-continuous Lawson TopRelStr
for b2 being Scott TopAugmentation of b1
for b3 being Element of the carrier of b1 holds
   {"/\"(b4,b2) where b4 is Element of bool the carrier of b2: b3 in b4 & b4 in sigma b2} = {"/\"(b4,b1) where b4 is Element of bool the carrier of b1: b3 in b4 & b4 in lambda b1};

:: WAYBEL30:th 35
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete meet-continuous Lawson TopRelStr holds
      b1 is continuous
   iff
      b1 is with_open_semilattices & InclPoset sigma b1 is continuous;

:: WAYBEL30:condreg 10
registration
  cluster TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson continuous -> with_open_semilattices (TopRelStr);
end;

:: WAYBEL30:funcreg 10
registration
  let a1 be TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson continuous TopRelStr;
  cluster InclPoset sigma a1 -> strict continuous;
end;

:: WAYBEL30:th 36
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson continuous TopRelStr holds
   b1 is compact & b1 is being_T2 & b1 is topological_semilattice & b1 is with_open_semilattices;

:: WAYBEL30:th 37
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete being_T2 Lawson topological_semilattice with_open_semilattices TopRelStr holds
   b1 is with_compact_semilattices;

:: WAYBEL30:th 38
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete meet-continuous being_T2 Lawson TopRelStr
for b2 being Element of the carrier of b1 holds
   b2 = "\/"({"/\"(b3,b1) where b3 is Element of bool the carrier of b1: b2 in b3 & b3 in lambda b1},b1);

:: WAYBEL30:th 39
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete meet-continuous Lawson TopRelStr holds
      b1 is continuous
   iff
      for b2 being Element of the carrier of b1 holds
         b2 = "\/"({"/\"(b3,b1) where b3 is Element of bool the carrier of b1: b2 in b3 & b3 in lambda b1},b1);

:: WAYBEL30:th 40
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete meet-continuous Lawson TopRelStr holds
      b1 is algebraic
   iff
      b1 is with_open_semilattices & InclPoset sigma b1 is algebraic;

:: WAYBEL30:funcreg 11
registration
  let a1 be TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete algebraic meet-continuous Lawson TopRelStr;
  cluster InclPoset sigma a1 -> strict algebraic;
end;