Article WAYBEL29, MML version 4.99.1005

:: WAYBEL29:th 2
theorem
for b1, b2 being non empty set
for b3 being non empty RelStr
for b4 being non empty SubRelStr of b3 |^ [:b1,b2:]
for b5 being non empty SubRelStr of (b3 |^ b2) |^ b1
for b6 being Function-like quasi_total Relation of the carrier of b4,the carrier of b5
      st b6 is currying & b6 is one-to-one & b6 is onto(the carrier of b4, the carrier of b5)
   holds b6 /" is uncurrying;

:: WAYBEL29:th 3
theorem
for b1, b2 being non empty set
for b3 being non empty RelStr
for b4 being non empty SubRelStr of b3 |^ [:b1,b2:]
for b5 being non empty SubRelStr of (b3 |^ b2) |^ b1
for b6 being Function-like quasi_total Relation of the carrier of b5,the carrier of b4
      st b6 is uncurrying & b6 is one-to-one & b6 is onto(the carrier of b5, the carrier of b4)
   holds b6 /" is currying;

:: WAYBEL29:th 4
theorem
for b1, b2 being non empty set
for b3 being non empty reflexive transitive antisymmetric RelStr
for b4 being non empty full SubRelStr of b3 |^ [:b1,b2:]
for b5 being non empty full SubRelStr of (b3 |^ b2) |^ b1
for b6 being Function-like quasi_total Relation of the carrier of b4,the carrier of b5
      st b6 is currying & b6 is one-to-one & b6 is onto(the carrier of b4, the carrier of b5)
   holds b6 is isomorphic(b4, b5);

:: WAYBEL29:th 5
theorem
for b1, b2 being non empty set
for b3 being non empty reflexive transitive antisymmetric RelStr
for b4 being non empty full SubRelStr of b3 |^ [:b1,b2:]
for b5 being non empty full SubRelStr of (b3 |^ b2) |^ b1
for b6 being Function-like quasi_total Relation of the carrier of b5,the carrier of b4
      st b6 is uncurrying & b6 is one-to-one & b6 is onto(the carrier of b5, the carrier of b4)
   holds b6 is isomorphic(b5, b4);

:: WAYBEL29:th 6
theorem
for b1, b2, b3, b4 being RelStr
   st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#) &
      RelStr(#the carrier of b3,the InternalRel of b3#) = 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 b3
   st b5 is isomorphic(b1, b3)
for b6 being Function-like quasi_total Relation of the carrier of b2,the carrier of b4
      st b6 = b5
   holds b6 is isomorphic(b2, b4);

:: WAYBEL29:th 7
theorem
for b1, b2, b3 being RelStr
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
   st b4 is isomorphic(b1, b2)
for b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
   st b5 is isomorphic(b2, b3)
for b6 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
      st b6 = b5 * b4
   holds b6 is isomorphic(b1, b3);

:: WAYBEL29:th 10
theorem
for b1, b2, b3, b4 being TopSpace-like TopStruct
      st TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b3,the topology of b3#) &
         TopStruct(#the carrier of b2,the topology of b2#) = TopStruct(#the carrier of b4,the topology of b4#)
   holds [:b1,b2:] = [:b3,b4:];

:: WAYBEL29:th 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like reflexive transitive antisymmetric up-complete Scott TopRelStr
for b3 being non empty directed Element of bool the carrier of ContMaps(b1,b2) holds
   "\/"(b3,b2 |^ the carrier of b1) is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2;

:: WAYBEL29:th 12
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like reflexive transitive antisymmetric up-complete Scott TopRelStr holds
   ContMaps(b1,b2) is directed-sups-inheriting SubRelStr of b2 |^ the carrier of b1;

:: WAYBEL29:th 13
theorem
for b1, b2 being TopStruct
   st TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#)
for b3, b4 being non empty TopRelStr
      st TopRelStr(#the carrier of b3,the InternalRel of b3,the topology of b3#) = TopRelStr(#the carrier of b4,the InternalRel of b4,the topology of b4#)
   holds ContMaps(b1,b3) = ContMaps(b2,b4);

:: WAYBEL29:condreg 1
registration
  cluster TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott continuous -> discerning injective (TopRelStr);
end;

:: WAYBEL29:exreg 1
registration
  cluster non empty TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete non void Scott continuous TopRelStr;
end;

:: WAYBEL29:funcreg 1
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty TopSpace-like reflexive transitive antisymmetric up-complete Scott TopRelStr;
  cluster ContMaps(a1,a2) -> strict up-complete;
end;

:: WAYBEL29:th 14
theorem
for b1 being non empty set
for b2 being non-Empty Poset-yielding ManySortedSet of b1
      st for b3 being Element of b1 holds
           b2 . b3 is up-complete
   holds product b2 is up-complete;

:: WAYBEL29:th 15
theorem
for b1 being non empty set
for b2 being non-Empty Poset-yielding ManySortedSet of b1
   st for b3 being Element of b1 holds
        b2 . b3 is up-complete & b2 . b3 is lower-bounded
for b3, b4 being Element of the carrier of product b2 holds
   b3 is_way_below b4
iff
   (for b5 being Element of b1 holds
       b3 . b5 is_way_below b4 . b5) &
    (ex b5 being finite Element of bool b1 st
       for b6 being Element of b1
             st not b6 in b5
          holds b3 . b6 = Bottom (b2 . b6));

:: WAYBEL29:funcreg 2
registration
  let a1 be set;
  let a2 be non empty reflexive antisymmetric lower-bounded RelStr;
  cluster a2 |^ a1 -> strict lower-bounded;
end;

:: WAYBEL29:funcreg 3
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty TopSpace-like reflexive transitive antisymmetric lower-bounded TopRelStr;
  cluster ContMaps(a1,a2) -> strict lower-bounded;
end;

:: WAYBEL29:condreg 2
registration
  let a1 be non empty reflexive transitive antisymmetric up-complete RelStr;
  cluster -> up-complete (TopAugmentation of a1);
end;

:: WAYBEL29:condreg 3
registration
  let a1 be non empty reflexive transitive antisymmetric up-complete RelStr;
  cluster Scott -> TopSpace-like (TopAugmentation of a1);
end;

:: WAYBEL29:exreg 2
registration
  let a1 be non empty reflexive transitive antisymmetric up-complete RelStr;
  cluster non empty reflexive transitive antisymmetric up-complete non void strict Scott TopAugmentation of a1;
end;

:: WAYBEL29:th 17
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr
for b2, b3 being Scott TopAugmentation of b1 holds
the topology of b2 = the topology of b3;

:: WAYBEL29:th 18
theorem
for b1, b2 being non empty reflexive antisymmetric up-complete TopRelStr
      st TopRelStr(#the carrier of b1,the InternalRel of b1,the topology of b1#) = TopRelStr(#the carrier of b2,the InternalRel of b2,the topology of b2#) &
         b1 is Scott
   holds b2 is Scott;

:: WAYBEL29:funcnot 1 => WAYBEL29:func 1
definition
  let a1 be non empty reflexive transitive antisymmetric up-complete RelStr;
  func Sigma A1 -> strict Scott TopAugmentation of a1 means
    TRUE;
end;

:: WAYBEL29:def 1
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr
for b2 being strict Scott TopAugmentation of b1 holds
   (b2 = Sigma b1 implies TRUE) & b2 = Sigma b1;

:: WAYBEL29:th 19
theorem
for b1 being non empty TopSpace-like reflexive transitive antisymmetric up-complete Scott TopRelStr holds
   Sigma b1 = TopRelStr(#the carrier of b1,the InternalRel of b1,the topology of b1#);

:: WAYBEL29:th 20
theorem
for b1, b2 being non empty reflexive transitive antisymmetric up-complete RelStr
      st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#)
   holds Sigma b1 = Sigma b2;

:: WAYBEL29:funcnot 2 => WAYBEL29:func 2
definition
  let a1, a2 be non empty reflexive transitive antisymmetric up-complete RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  func Sigma A3 -> Function-like quasi_total Relation of the carrier of Sigma a1,the carrier of Sigma a2 equals
    a3;
end;

:: WAYBEL29:def 2
theorem
for b1, b2 being non empty reflexive transitive antisymmetric up-complete RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
   Sigma b3 = b3;

:: WAYBEL29:funcreg 4
registration
  let a1, a2 be non empty reflexive transitive antisymmetric up-complete RelStr;
  let a3 be Function-like quasi_total directed-sups-preserving Relation of the carrier of a1,the carrier of a2;
  cluster Sigma a3 -> Function-like quasi_total continuous;
end;

:: WAYBEL29:th 21
theorem
for b1, b2 being non empty reflexive transitive antisymmetric up-complete RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is isomorphic(b1, b2)
   iff
      Sigma b3 is isomorphic(Sigma b1, Sigma b2);

:: WAYBEL29:th 22
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr holds
   oContMaps(b1,b2) = ContMaps(b1,b2);

:: WAYBEL29:funcnot 3 => WAYBEL29:func 3
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  func Theta(A1,A2) -> Function-like quasi_total Relation of the carrier of InclPoset the topology of [:a1,a2:],the carrier of ContMaps(a1,Sigma InclPoset the topology of a2) means
    for b1 being open Element of bool the carrier of [:a1,a2:] holds
       it . b1 = (b1,the carrier of a1)*graph;
end;

:: WAYBEL29:def 3
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of InclPoset the topology of [:b1,b2:],the carrier of ContMaps(b1,Sigma InclPoset the topology of b2) holds
      b3 = Theta(b1,b2)
   iff
      for b4 being open Element of bool the carrier of [:b1,b2:] holds
         b3 . b4 = (b4,the carrier of b1)*graph;

:: WAYBEL29:funcnot 4 => WAYBEL29:func 4
definition
  let a1 be non empty TopSpace-like TopStruct;
  func alpha A1 -> Function-like quasi_total Relation of the carrier of oContMaps(a1,Sierpinski_Space),the carrier of InclPoset the topology of a1 means
    for b1 being Function-like quasi_total continuous Relation of the carrier of a1,the carrier of Sierpinski_Space holds
       it . b1 = b1 " {1};
end;

:: WAYBEL29:def 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of the carrier of oContMaps(b1,Sierpinski_Space),the carrier of InclPoset the topology of b1 holds
      b2 = alpha b1
   iff
      for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of Sierpinski_Space holds
         b2 . b3 = b3 " {1};

:: WAYBEL29:th 23
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being open Element of bool the carrier of b1 holds
   (alpha b1) /" . b2 = chi(b2,the carrier of b1);

:: WAYBEL29:funcreg 5
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster alpha a1 -> Function-like quasi_total isomorphic;
end;

:: WAYBEL29:funcreg 6
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster (alpha a1) /" -> Function-like quasi_total isomorphic;
end;

:: WAYBEL29:funcreg 7
registration
  let a1 be non empty TopSpace-like discerning injective TopStruct;
  cluster Omega a1 -> strict Scott;
end;

:: WAYBEL29:funcreg 8
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster oContMaps(a1,Sierpinski_Space) -> non empty strict complete;
end;

:: WAYBEL29:th 24
theorem
Omega Sierpinski_Space = Sigma BoolePoset 1;

:: WAYBEL29:funcreg 9
registration
  let a1 be non empty set;
  let a2 be non empty TopSpace-like discerning injective TopStruct;
  cluster product (a1 => a2) -> strict TopSpace-like injective;
end;

:: WAYBEL29:th 25
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr holds
   Omega product (b1 => Sigma b2) = Sigma product (b1 => b2);

:: WAYBEL29:th 26
theorem
for b1 being non empty set
for b2 being non empty TopSpace-like discerning injective TopStruct holds
   Omega product (b1 => b2) = Sigma product (b1 => Omega b2);

:: WAYBEL29:funcnot 5 => WAYBEL29:func 5
definition
  let a1 be non empty set;
  let a2, a3 be non empty TopSpace-like TopStruct;
  func commute(A2,A1,A3) -> Function-like quasi_total Relation of the carrier of oContMaps(a2,product (a1 => a3)),the carrier of (oContMaps(a2,a3)) |^ a1 means
    for b1 being Function-like quasi_total continuous Relation of the carrier of a2,the carrier of product (a1 => a3) holds
       it . b1 = commute b1;
end;

:: WAYBEL29:def 5
theorem
for b1 being non empty set
for b2, b3 being non empty TopSpace-like TopStruct
for b4 being Function-like quasi_total Relation of the carrier of oContMaps(b2,product (b1 => b3)),the carrier of (oContMaps(b2,b3)) |^ b1 holds
      b4 = commute(b2,b1,b3)
   iff
      for b5 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of product (b1 => b3) holds
         b4 . b5 = commute b5;

:: WAYBEL29:funcreg 10
registration
  let a1 be non empty set;
  let a2, a3 be non empty TopSpace-like TopStruct;
  cluster commute(a2,a1,a3) -> Function-like one-to-one quasi_total onto;
end;

:: WAYBEL29:funcreg 11
registration
  let a1 be non empty set;
  let a2 be non empty TopSpace-like TopStruct;
  cluster commute(a2,a1,Sierpinski_Space) -> Function-like quasi_total isomorphic;
end;

:: WAYBEL29:th 27
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Scott TopAugmentation of InclPoset the topology of b2
for b4, b5 being Element of the carrier of ContMaps(b1,b3)
      st b4 <= b5
   holds *graph b4 c= *graph b5;

:: WAYBEL29:th 28
theorem
for b1 being non empty TopSpace-like discerning TopStruct holds
      for b2 being non empty TopSpace-like TopStruct
      for b3 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott continuous TopRelStr
      for b4 being Scott TopAugmentation of ContMaps(b1,b3) holds
         ex b5 being Function-like quasi_total Relation of the carrier of ContMaps(b2,b4),the carrier of ContMaps([:b2,b1:],b3) st
            ex b6 being Function-like quasi_total Relation of the carrier of ContMaps([:b2,b1:],b3),the carrier of ContMaps(b2,b4) st
               b5 is uncurrying &
                b5 is one-to-one &
                b5 is onto(the carrier of ContMaps(b2,b4), the carrier of ContMaps([:b2,b1:],b3)) &
                b6 is currying &
                b6 is one-to-one &
                b6 is onto(the carrier of ContMaps([:b2,b1:],b3), the carrier of ContMaps(b2,b4))
   iff
      for b2 being non empty TopSpace-like TopStruct
      for b3 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott continuous TopRelStr
      for b4 being Scott TopAugmentation of ContMaps(b1,b3) holds
         ex b5 being Function-like quasi_total Relation of the carrier of ContMaps(b2,b4),the carrier of ContMaps([:b2,b1:],b3) st
            ex b6 being Function-like quasi_total Relation of the carrier of ContMaps([:b2,b1:],b3),the carrier of ContMaps(b2,b4) st
               b5 is uncurrying &
                b5 is isomorphic(ContMaps(b2,b4), ContMaps([:b2,b1:],b3)) &
                b6 is currying &
                b6 is isomorphic(ContMaps([:b2,b1:],b3), ContMaps(b2,b4));

:: WAYBEL29:th 29
theorem
for b1 being non empty TopSpace-like discerning TopStruct holds
      InclPoset the topology of b1 is continuous
   iff
      for b2 being non empty TopSpace-like TopStruct holds
         Theta(b2,b1) is isomorphic(InclPoset the topology of [:b2,b1:], ContMaps(b2,Sigma InclPoset the topology of b1));

:: WAYBEL29:th 30
theorem
for b1 being non empty TopSpace-like discerning TopStruct holds
      InclPoset the topology of b1 is continuous
   iff
      for b2 being non empty TopSpace-like TopStruct
      for b3 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of Sigma InclPoset the topology of b1 holds
         *graph b3 is open Element of bool the carrier of [:b2,b1:];

:: WAYBEL29:th 31
theorem
for b1 being non empty TopSpace-like discerning TopStruct holds
      InclPoset the topology of b1 is continuous
   iff
      {[b2,b3] where b2 is open Element of bool the carrier of b1, b3 is Element of the carrier of b1: b3 in b2} is open Element of bool the carrier of [:Sigma InclPoset the topology of b1,b1:];

:: WAYBEL29:th 32
theorem
for b1 being non empty TopSpace-like discerning TopStruct holds
      InclPoset the topology of b1 is continuous
   iff
      for b2 being Element of the carrier of b1
      for b3 being open a_neighborhood of b2 holds
         ex b4 being open Element of bool the carrier of Sigma InclPoset the topology of b1 st
            b3 in b4 & meet b4 is a_neighborhood of b2;

:: WAYBEL29:th 33
theorem
for b1, b2, b3 being non empty RelStr
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
   st b4 is isomorphic(b1, b3)
for b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
      st b5 = b4 & b5 is isomorphic(b2, b3)
   holds RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#);

:: WAYBEL29:th 34
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
      InclPoset sigma b1 is continuous
   iff
      for b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
         sigma [:b2,b1:] = the topology of [:Sigma b2,Sigma b1:];

:: WAYBEL29:th 35
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
      for b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
         sigma [:b2,b1:] = the topology of [:Sigma b2,Sigma b1:]
   iff
      for b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
         TopStruct(#the carrier of Sigma [:b2,b1:],the topology of Sigma [:b2,b1:]#) = [:Sigma b2,Sigma b1:];

:: WAYBEL29:th 36
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
      for b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
         sigma [:b2,b1:] = the topology of [:Sigma b2,Sigma b1:]
   iff
      for b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
         Sigma [:b2,b1:] = Omega [:Sigma b2,Sigma b1:];

:: WAYBEL29:th 37
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
      InclPoset sigma b1 is continuous
   iff
      for b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
         Sigma [:b2,b1:] = Omega [:Sigma b2,Sigma b1:];