Article WAYBEL25, MML version 4.99.1005

:: WAYBEL25:th 1
theorem
for b1 being Element of the carrier of Sierpinski_Space
      st b1 = 0
   holds {b1} is closed(Sierpinski_Space);

:: WAYBEL25:th 2
theorem
for b1 being Element of the carrier of Sierpinski_Space
      st b1 = 1
   holds {b1} is closed(not Sierpinski_Space);

:: WAYBEL25:funcreg 1
registration
  cluster Sierpinski_Space -> strict non being_T1;
end;

:: WAYBEL25:condreg 1
registration
  cluster TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott -> discerning (TopRelStr);
end;

:: WAYBEL25:exreg 1
registration
  cluster non empty strict TopSpace-like discerning injective TopStruct;
end;

:: WAYBEL25:exreg 2
registration
  cluster non empty TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete strict Scott TopRelStr;
end;

:: WAYBEL25:th 3
theorem
for b1 being non empty set
for b2 being Scott TopAugmentation of product (b1 --> BoolePoset 1) holds
   the carrier of b2 = the carrier of product (b1 --> Sierpinski_Space);

:: WAYBEL25:th 4
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Scott TopAugmentation of b1
for b4 being Scott TopAugmentation of b2
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 & b5 is isomorphic(b1, b2)
   holds b6 is being_homeomorphism(b3, b4);

:: WAYBEL25:th 5
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Scott TopAugmentation of b1
for b4 being Scott TopAugmentation of b2
      st b1,b2 are_isomorphic
   holds b3,b4 are_homeomorphic;

:: WAYBEL25:th 6
theorem
for b1, b2 being non empty TopSpace-like TopStruct
      st b1 is injective & b1,b2 are_homeomorphic
   holds b2 is injective;

:: WAYBEL25:th 7
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Scott TopAugmentation of b1
for b4 being Scott TopAugmentation of b2
      st b1,b2 are_isomorphic & b3 is injective
   holds b4 is injective;

:: WAYBEL25:prednot 1 => WAYBEL18:pred 1
definition
  let a1, a2 be TopStruct;
  pred A1 is_Retract_of A2 means
    ex b1 being Function-like quasi_total continuous Relation of the carrier of a1,the carrier of a2 st
       ex b2 being Function-like quasi_total continuous Relation of the carrier of a2,the carrier of a1 st
          b2 * b1 = id a1;
end;

:: WAYBEL25:dfs 1
definiens
  let a1, a2 be non empty TopSpace-like TopStruct;
To prove
     a1 is_Retract_of a2
it is sufficient to prove
  thus ex b1 being Function-like quasi_total continuous Relation of the carrier of a1,the carrier of a2 st
       ex b2 being Function-like quasi_total continuous Relation of the carrier of a2,the carrier of a1 st
          b2 * b1 = id a1;

:: WAYBEL25:def 1
theorem
for b1, b2 being non empty TopSpace-like TopStruct holds
   b1 is_Retract_of b2
iff
   ex b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2 st
      ex b4 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b1 st
         b4 * b3 = id b1;

:: WAYBEL25:th 8
theorem
for b1, b2, b3, b4 being non empty TopSpace-like TopStruct
      st TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#) &
         TopStruct(#the carrier of b3,the topology of b3#) = TopStruct(#the carrier of b4,the topology of b4#) &
         b1 is_Retract_of b3
   holds b2 is_Retract_of b4;

:: WAYBEL25:th 9
theorem
for b1, b2, b3 being non empty TopStruct
      st b2,b3 are_homeomorphic & b2 is_Retract_of b1
   holds b3 is_Retract_of b1;

:: WAYBEL25:th 10
theorem
for b1, b2 being non empty TopSpace-like TopStruct
      st b2 is injective & b1 is_Retract_of b2
   holds b1 is injective;

:: WAYBEL25:th 11
theorem
for b1 being non empty TopSpace-like injective TopStruct
for b2 being non empty TopSpace-like TopStruct
      st b1 is SubSpace of b2
   holds b1 is_Retract_of b2;

:: WAYBEL25:th 12
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr
for b2 being Scott TopAugmentation of b1 holds
   b2 is injective;

:: WAYBEL25:condreg 2
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr;
  cluster Scott -> injective (TopAugmentation of a1);
end;

:: WAYBEL25:funcreg 2
registration
  let a1 be non empty TopSpace-like injective TopStruct;
  cluster TopStruct(#the carrier of a1,the topology of a1#) -> strict injective;
end;

:: WAYBEL25:funcnot 1 => WAYBEL25:func 1
definition
  let a1 be TopStruct;
  func Omega A1 -> strict TopRelStr means
    TopStruct(#the carrier of it,the topology of it#) = TopStruct(#the carrier of a1,the topology of a1#) &
     (for b1, b2 being Element of the carrier of it holds
        b1 <= b2
     iff
        ex b3 being Element of bool the carrier of a1 st
           b3 = {b2} & b1 in Cl b3);
end;

:: WAYBEL25:def 2
theorem
for b1 being TopStruct
for b2 being strict TopRelStr holds
      b2 = Omega b1
   iff
      TopStruct(#the carrier of b2,the topology of b2#) = TopStruct(#the carrier of b1,the topology of b1#) &
       (for b3, b4 being Element of the carrier of b2 holds
          b3 <= b4
       iff
          ex b5 being Element of bool the carrier of b1 st
             b5 = {b4} & b3 in Cl b5);

:: WAYBEL25:funcreg 3
registration
  let a1 be empty TopStruct;
  cluster Omega a1 -> empty strict;
end;

:: WAYBEL25:funcreg 4
registration
  let a1 be non empty TopStruct;
  cluster Omega a1 -> non empty strict;
end;

:: WAYBEL25:funcreg 5
registration
  let a1 be TopSpace-like TopStruct;
  cluster Omega a1 -> TopSpace-like strict;
end;

:: WAYBEL25:funcreg 6
registration
  let a1 be TopStruct;
  cluster Omega a1 -> reflexive strict;
end;

:: WAYBEL25:funcreg 7
registration
  let a1 be TopStruct;
  cluster Omega a1 -> transitive strict;
end;

:: WAYBEL25:funcreg 8
registration
  let a1 be non empty TopSpace-like discerning TopStruct;
  cluster Omega a1 -> antisymmetric strict;
end;

:: WAYBEL25:th 13
theorem
for b1, b2 being TopSpace-like TopStruct
      st TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#)
   holds Omega b1 = Omega b2;

:: WAYBEL25:th 14
theorem
for b1 being non empty set
for b2 being non empty TopSpace-like TopStruct holds
   RelStr(#the carrier of Omega product (b1 --> b2),the InternalRel of Omega product (b1 --> b2)#) = RelStr(#the carrier of product (b1 --> Omega b2),the InternalRel of product (b1 --> Omega b2)#);

:: WAYBEL25:th 15
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr holds
   Omega b1 = TopRelStr(#the carrier of b1,the InternalRel of b1,the topology of b1#);

:: WAYBEL25:th 16
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Scott TopAugmentation of b1 holds
   RelStr(#the carrier of Omega b2,the InternalRel of Omega b2#) = RelStr(#the carrier of b1,the InternalRel of b1#);

:: WAYBEL25:funcreg 9
registration
  let a1 be TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr;
  cluster Omega a1 -> complete strict;
end;

:: WAYBEL25:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1 holds
   Omega b2 is full SubRelStr of Omega b1;

:: WAYBEL25:th 18
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of the carrier of Omega b1,the carrier of Omega b2
      st b3 = b4 & b3 is being_homeomorphism(b1, b2)
   holds b4 is isomorphic(Omega b1, Omega b2);

:: WAYBEL25:th 19
theorem
for b1, b2 being TopSpace-like TopStruct
      st b1,b2 are_homeomorphic
   holds Omega b1,Omega b2 are_isomorphic;

:: WAYBEL25:th 20
theorem
for b1 being non empty TopSpace-like discerning injective TopStruct holds
   Omega b1 is reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr;

:: WAYBEL25:funcreg 10
registration
  let a1 be non empty TopSpace-like discerning injective TopStruct;
  cluster Omega a1 -> complete strict continuous;
end;

:: WAYBEL25:th 21
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total continuous Relation of the carrier of Omega b1,the carrier of Omega b2 holds
   b3 is monotone(Omega b1, Omega b2);

:: WAYBEL25:condreg 3
registration
  let a1, a2 be non empty TopSpace-like TopStruct;
  cluster Function-like quasi_total continuous -> monotone (Relation of the carrier of Omega a1,the carrier of Omega a2);
end;

:: WAYBEL25:th 22
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of Omega b1 holds
   Cl {b2} = downarrow b2;

:: WAYBEL25:funcreg 11
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of the carrier of Omega a1;
  cluster Cl {a2} -> non empty directed lower;
end;

:: WAYBEL25:funcreg 12
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of the carrier of Omega a1;
  cluster downarrow a2 -> closed;
end;

:: WAYBEL25:th 23
theorem
for b1 being TopSpace-like TopStruct
for b2 being open Element of bool the carrier of Omega b1 holds
   b2 is upper(Omega b1);

:: WAYBEL25:condreg 4
registration
  let a1 be TopSpace-like TopStruct;
  cluster open -> upper (Element of bool the carrier of Omega a1);
end;

:: WAYBEL25:funcnot 2 => WAYBEL25:func 2
definition
  let a1 be non empty set;
  let a2, a3 be non empty RelStr;
  let a4 be non empty transitive directed NetStr over a3;
  let a5 be Element of a1;
  assume the carrier of a3 c= the carrier of a2 |^ a1;
  func commute(A4,A5,A2) -> non empty transitive strict directed NetStr over a2 means
    RelStr(#the carrier of it,the InternalRel of it#) = RelStr(#the carrier of a4,the InternalRel of a4#) &
     the mapping of it = (commute the mapping of a4) . a5;
end;

:: WAYBEL25:def 3
theorem
for b1 being non empty set
for b2, b3 being non empty RelStr
for b4 being non empty transitive directed NetStr over b3
for b5 being Element of b1
   st the carrier of b3 c= the carrier of b2 |^ b1
for b6 being non empty transitive strict directed NetStr over b2 holds
      b6 = commute(b4,b5,b2)
   iff
      RelStr(#the carrier of b6,the InternalRel of b6#) = RelStr(#the carrier of b4,the InternalRel of b4#) &
       the mapping of b6 = (commute the mapping of b4) . b5;

:: WAYBEL25:th 24
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being non empty transitive directed NetStr over ContMaps(b1,Omega b2)
for b4 being Element of the carrier of b3
for b5 being Element of the carrier of b1 holds
   (the mapping of commute(b3,b5,Omega b2)) . b4 = ((the mapping of b3) . b4) . b5;

:: WAYBEL25:th 25
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being non empty transitive directed eventually-directed NetStr over ContMaps(b1,Omega b2)
for b4 being Element of the carrier of b1 holds
   commute(b3,b4,Omega b2) is eventually-directed(Omega b2);

:: WAYBEL25:funcreg 13
registration
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be non empty transitive directed eventually-directed NetStr over ContMaps(a1,Omega a2);
  let a4 be Element of the carrier of a1;
  cluster commute(a3,a4,Omega a2) -> non empty transitive strict directed eventually-directed;
end;

:: WAYBEL25:condreg 5
registration
  let a1, a2 be non empty TopSpace-like TopStruct;
  cluster non empty transitive directed -> Function-yielding (NetStr over ContMaps(a1,Omega a2));
end;

:: WAYBEL25:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like discerning TopStruct
for b3 being non empty transitive directed NetStr over ContMaps(b1,Omega b2)
      st for b4 being Element of the carrier of b1 holds
           ex_sup_of commute(b3,b4,Omega b2)
   holds ex_sup_of rng the mapping of b3,(Omega b2) |^ the carrier of b1;

:: WAYBEL25:attrnot 1 => WAYBEL25:attr 1
definition
  let a1 be non empty TopSpace-like TopStruct;
  attr a1 is monotone-convergence means
    for b1 being non empty directed Element of bool the carrier of Omega a1 holds
       ex_sup_of b1,Omega a1 &
        (for b2 being open Element of bool the carrier of a1
              st "\/"(b1,Omega a1) in b2
           holds b1 meets b2);
end;

:: WAYBEL25:dfs 4
definiens
  let a1 be non empty TopSpace-like TopStruct;
To prove
     a1 is monotone-convergence
it is sufficient to prove
  thus for b1 being non empty directed Element of bool the carrier of Omega a1 holds
       ex_sup_of b1,Omega a1 &
        (for b2 being open Element of bool the carrier of a1
              st "\/"(b1,Omega a1) in b2
           holds b1 meets b2);

:: WAYBEL25:def 4
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is monotone-convergence
   iff
      for b2 being non empty directed Element of bool the carrier of Omega b1 holds
         ex_sup_of b2,Omega b1 &
          (for b3 being open Element of bool the carrier of b1
                st "\/"(b2,Omega b1) in b3
             holds b2 meets b3);

:: WAYBEL25:th 27
theorem
for b1, b2 being non empty TopSpace-like TopStruct
      st TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#) &
         b1 is monotone-convergence
   holds b2 is monotone-convergence;

:: WAYBEL25:condreg 6
registration
  cluster non empty trivial TopSpace-like discerning -> monotone-convergence (TopStruct);
end;

:: WAYBEL25:exreg 3
registration
  cluster non empty trivial strict TopSpace-like TopStruct;
end;

:: WAYBEL25:th 28
theorem
for b1 being non empty TopSpace-like discerning monotone-convergence TopStruct
for b2 being non empty TopSpace-like discerning TopStruct
      st b1,b2 are_homeomorphic
   holds b2 is monotone-convergence;

:: WAYBEL25:th 29
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr holds
   b1 is monotone-convergence;

:: WAYBEL25:condreg 7
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  cluster Scott -> monotone-convergence (TopAugmentation of a1);
end;

:: WAYBEL25:funcreg 14
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  let a2 be Scott TopAugmentation of a1;
  cluster TopStruct(#the carrier of a2,the topology of a2#) -> strict monotone-convergence;
end;

:: WAYBEL25:th 30
theorem
for b1 being non empty TopSpace-like discerning monotone-convergence TopStruct holds
   Omega b1 is up-complete;

:: WAYBEL25:funcreg 15
registration
  let a1 be non empty TopSpace-like discerning monotone-convergence TopStruct;
  cluster Omega a1 -> up-complete strict;
end;

:: WAYBEL25:th 31
theorem
for b1 being non empty TopSpace-like monotone-convergence TopStruct
for b2 being non empty directed eventually-directed NetStr over Omega b1 holds
   ex_sup_of b2;

:: WAYBEL25:th 32
theorem
for b1 being non empty TopSpace-like monotone-convergence TopStruct
for b2 being non empty transitive directed eventually-directed NetStr over Omega b1 holds
   sup b2 in Lim b2;

:: WAYBEL25:th 33
theorem
for b1 being non empty TopSpace-like monotone-convergence TopStruct
for b2 being non empty transitive directed eventually-directed NetStr over Omega b1 holds
   b2 is convergent(Omega b1);

:: WAYBEL25:condreg 8
registration
  let a1 be non empty TopSpace-like monotone-convergence TopStruct;
  cluster non empty transitive directed eventually-directed -> convergent (NetStr over Omega a1);
end;

:: WAYBEL25:th 34
theorem
for b1 being non empty TopSpace-like TopStruct
      st for b2 being non empty transitive directed eventually-directed NetStr over Omega b1 holds
           ex_sup_of b2 & sup b2 in Lim b2
   holds b1 is monotone-convergence;

:: WAYBEL25:th 35
theorem
for b1 being non empty TopSpace-like monotone-convergence TopStruct
for b2 being non empty TopSpace-like discerning TopStruct
for b3 being Function-like quasi_total continuous Relation of the carrier of Omega b1,the carrier of Omega b2 holds
   b3 is directed-sups-preserving(Omega b1, Omega b2);

:: WAYBEL25:condreg 9
registration
  let a1 be non empty TopSpace-like monotone-convergence TopStruct;
  let a2 be non empty TopSpace-like discerning TopStruct;
  cluster Function-like quasi_total continuous -> directed-sups-preserving (Relation of the carrier of Omega a1,the carrier of Omega a2);
end;

:: WAYBEL25:th 36
theorem
for b1 being non empty TopSpace-like discerning monotone-convergence TopStruct
for b2 being non empty TopSpace-like discerning TopStruct
      st b2 is_Retract_of b1
   holds b2 is monotone-convergence;

:: WAYBEL25:th 37
theorem
for b1 being non empty TopSpace-like discerning injective TopStruct
for b2 being Scott TopAugmentation of Omega b1 holds
   TopStruct(#the carrier of b2,the topology of b2#) = TopStruct(#the carrier of b1,the topology of b1#);

:: WAYBEL25:th 38
theorem
for b1 being non empty TopSpace-like discerning injective TopStruct holds
   b1 is compact & b1 is locally-compact & b1 is sober;

:: WAYBEL25:th 39
theorem
for b1 being non empty TopSpace-like discerning injective TopStruct holds
   b1 is monotone-convergence;

:: WAYBEL25:condreg 10
registration
  cluster non empty TopSpace-like discerning injective -> monotone-convergence (TopStruct);
end;

:: WAYBEL25:th 40
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like discerning monotone-convergence TopStruct
for b3 being non empty transitive directed NetStr over ContMaps(b1,Omega b2)
for b4, b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of Omega b2
      st b4 = "\/"(rng the mapping of b3,(Omega b2) |^ the carrier of b1) &
         ex_sup_of rng the mapping of b3,(Omega b2) |^ the carrier of b1 &
         b5 in rng the mapping of b3
   holds b5 <= b4;

:: WAYBEL25:th 41
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like discerning monotone-convergence TopStruct
for b3 being non empty transitive directed NetStr over ContMaps(b1,Omega b2)
for b4 being Element of the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of Omega b2
      st (for b6 being Element of the carrier of b1 holds
            commute(b3,b6,Omega b2) is eventually-directed(Omega b2)) &
         b5 = "\/"(rng the mapping of b3,(Omega b2) |^ the carrier of b1)
   holds b5 . b4 = sup commute(b3,b4,Omega b2);

:: WAYBEL25:th 42
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like discerning monotone-convergence TopStruct
for b3 being non empty transitive directed NetStr over ContMaps(b1,Omega b2)
      st for b4 being Element of the carrier of b1 holds
           commute(b3,b4,Omega b2) is eventually-directed(Omega b2)
   holds "\/"(rng the mapping of b3,(Omega b2) |^ the carrier of b1) is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2;

:: WAYBEL25:th 43
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like discerning monotone-convergence TopStruct holds
   ContMaps(b1,Omega b2) is directed-sups-inheriting SubRelStr of (Omega b2) |^ the carrier of b1;