Article WAYBEL26, MML version 4.99.1005

:: WAYBEL26:funcnot 1 => YELLOW_1:func 5
notation
  let a1 be set;
  let a2 be RelStr-yielding ManySortedSet of a1;
  synonym a1 -POS_prod a2 for product a2;
end;

:: WAYBEL26:funcreg 1
registration
  let a1 be set;
  let a2 be RelStr-yielding non-Empty ManySortedSet of a1;
  cluster product a2 -> constituted-Functions strict;
end;

:: WAYBEL26:funcnot 2 => WAYBEL18:func 3
notation
  let a1 be set;
  let a2 be non-Empty TopSpace-yielding ManySortedSet of a1;
  synonym a1 -TOP_prod a2 for product a2;
end;

:: WAYBEL26:funcnot 3 => WAYBEL26:func 1
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  func oContMaps(A1,A2) -> non empty strict RelStr equals
    ContMaps(a1,Omega a2);
end;

:: WAYBEL26:def 1
theorem
for b1, b2 being non empty TopSpace-like TopStruct holds
oContMaps(b1,b2) = ContMaps(b1,Omega b2);

:: WAYBEL26:funcreg 2
registration
  let a1, a2 be non empty TopSpace-like TopStruct;
  cluster oContMaps(a1,a2) -> non empty constituted-Functions strict reflexive transitive;
end;

:: WAYBEL26:funcreg 3
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty TopSpace-like discerning TopStruct;
  cluster oContMaps(a1,a2) -> non empty strict antisymmetric;
end;

:: WAYBEL26:th 1
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being set holds
      b3 is Element of the carrier of oContMaps(b1,b2)
   iff
      b3 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of Omega b2;

:: WAYBEL26:th 2
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being set holds
      b3 is Element of the carrier of oContMaps(b1,b2)
   iff
      b3 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2;

:: WAYBEL26:th 3
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being Element of the carrier of oContMaps(b1,b2)
for b5, b6 being Function-like quasi_total Relation of the carrier of b1,the carrier of Omega b2
      st b3 = b5 & b4 = b6
   holds    b3 <= b4
   iff
      b5 <= b6;

:: WAYBEL26:funcnot 4 => WAYBEL26:func 2
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Element of the carrier of a1;
  let a4 be Element of bool the carrier of oContMaps(a1,a2);
  redefine func pi(a4,a3) -> Element of bool the carrier of Omega a2;
end;

:: WAYBEL26:funcreg 4
registration
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be set;
  let a4 be non empty Element of bool the carrier of oContMaps(a1,a2);
  cluster pi(a4,a3) -> non empty;
end;

:: WAYBEL26:th 4
theorem
Omega Sierpinski_Space is TopAugmentation of BoolePoset 1;

:: WAYBEL26:th 5
theorem
for b1 being non empty TopSpace-like TopStruct holds
   ex b2 being Function-like quasi_total Relation of the carrier of InclPoset the topology of b1,the carrier of oContMaps(b1,Sierpinski_Space) st
      b2 is isomorphic(InclPoset the topology of b1, oContMaps(b1,Sierpinski_Space)) &
       (for b3 being open Element of bool the carrier of b1 holds
          b2 . b3 = chi(b3,the carrier of b1));

:: WAYBEL26:th 6
theorem
for b1 being non empty TopSpace-like TopStruct holds
   InclPoset the topology of b1,oContMaps(b1,Sierpinski_Space) are_isomorphic;

:: WAYBEL26:funcnot 5 => WAYBEL26:func 3
definition
  let a1, a2, a3 be non empty TopSpace-like TopStruct;
  let a4 be Function-like quasi_total continuous Relation of the carrier of a2,the carrier of a3;
  func oContMaps(A1,A4) -> Function-like quasi_total Relation of the carrier of oContMaps(a1,a2),the carrier of oContMaps(a1,a3) means
    for b1 being Function-like quasi_total continuous Relation of the carrier of a1,the carrier of a2 holds
       it . b1 = a4 * b1;
end;

:: WAYBEL26:def 2
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
for b4 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b3
for b5 being Function-like quasi_total Relation of the carrier of oContMaps(b1,b2),the carrier of oContMaps(b1,b3) holds
      b5 = oContMaps(b1,b4)
   iff
      for b6 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2 holds
         b5 . b6 = b4 * b6;

:: WAYBEL26:funcnot 6 => WAYBEL26:func 4
definition
  let a1, a2, a3 be non empty TopSpace-like TopStruct;
  let a4 be Function-like quasi_total continuous Relation of the carrier of a2,the carrier of a3;
  func oContMaps(A4,A1) -> Function-like quasi_total Relation of the carrier of oContMaps(a3,a1),the carrier of oContMaps(a2,a1) means
    for b1 being Function-like quasi_total continuous Relation of the carrier of a3,the carrier of a1 holds
       it . b1 = b1 * a4;
end;

:: WAYBEL26:def 3
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
for b4 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b3
for b5 being Function-like quasi_total Relation of the carrier of oContMaps(b3,b1),the carrier of oContMaps(b2,b1) holds
      b5 = oContMaps(b4,b1)
   iff
      for b6 being Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b1 holds
         b5 . b6 = b6 * b4;

:: WAYBEL26:th 8
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like discerning monotone-convergence TopStruct holds
   oContMaps(b1,b2) is up-complete;

:: WAYBEL26:th 9
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
for b4 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b3 holds
   oContMaps(b1,b4) is monotone(oContMaps(b1,b2), oContMaps(b1,b3));

:: WAYBEL26:th 10
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b2
      st b3 is idempotent
   holds oContMaps(b1,b3) is idempotent;

:: WAYBEL26:th 11
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
for b4 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b3 holds
   oContMaps(b4,b1) is monotone(oContMaps(b3,b1), oContMaps(b2,b1));

:: WAYBEL26:th 12
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b2
      st b3 is idempotent
   holds oContMaps(b3,b1) is idempotent;

:: WAYBEL26:th 13
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
for b4 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b3
for b5 being Element of the carrier of b1
for b6 being Element of bool the carrier of oContMaps(b1,b2) holds
   pi((oContMaps(b1,b4)) .: b6,b5) = b4 .: pi(b6,b5);

:: WAYBEL26:th 14
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty TopSpace-like discerning monotone-convergence TopStruct
for b4 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b3 holds
   oContMaps(b1,b4) is directed-sups-preserving(oContMaps(b1,b2), oContMaps(b1,b3));

:: WAYBEL26:th 15
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
for b4 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b3
for b5 being Element of the carrier of b2
for b6 being Element of bool the carrier of oContMaps(b3,b1) holds
   pi((oContMaps(b4,b1)) .: b6,b5) = pi(b6,b4 . b5);

:: WAYBEL26:th 16
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being non empty TopSpace-like discerning monotone-convergence TopStruct
for b4 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2 holds
   oContMaps(b4,b3) is directed-sups-preserving(oContMaps(b2,b3), oContMaps(b1,b3));

:: WAYBEL26:th 17
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being non empty SubSpace of b2 holds
   oContMaps(b1,b3) is full SubRelStr of oContMaps(b1,b2);

:: WAYBEL26:th 18
theorem
for b1 being non empty TopSpace-like discerning monotone-convergence TopStruct
for b2 being non empty SubSpace of b1
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
      st b3 is being_a_retraction(b1, b2)
   holds Omega b2 is directed-sups-inheriting SubRelStr of Omega b1;

:: WAYBEL26:th 19
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 SubSpace of b2
for b4 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b3
      st b4 is being_a_retraction(b2, b3)
   holds oContMaps(b1,b4) is_a_retraction_of oContMaps(b1,b2),oContMaps(b1,b3);

:: WAYBEL26:th 20
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 SubSpace of b2
      st b3 is_a_retract_of b2
   holds oContMaps(b1,b3) is_a_retract_of oContMaps(b1,b2);

:: WAYBEL26:th 21
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
for b4 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b3
      st b4 is being_homeomorphism(b2, b3)
   holds oContMaps(b1,b4) is isomorphic(oContMaps(b1,b2), oContMaps(b1,b3));

:: WAYBEL26:th 22
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
      st b2,b3 are_homeomorphic
   holds oContMaps(b1,b2),oContMaps(b1,b3) are_isomorphic;

:: WAYBEL26:th 23
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 SubSpace of b2
      st b3 is_a_retract_of b2 & oContMaps(b1,b2) is complete & oContMaps(b1,b2) is continuous
   holds oContMaps(b1,b3) is complete & oContMaps(b1,b3) is continuous;

:: WAYBEL26:th 24
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty TopSpace-like discerning monotone-convergence TopStruct
      st b2 is_Retract_of b3 & oContMaps(b1,b3) is complete & oContMaps(b1,b3) is continuous
   holds oContMaps(b1,b2) is complete & oContMaps(b1,b2) is continuous;

:: WAYBEL26:th 25
theorem
for b1 being non empty non trivial TopSpace-like discerning TopStruct
      st b1 is not being_T1
   holds Sierpinski_Space is_Retract_of b1;

:: WAYBEL26:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty non trivial TopSpace-like discerning TopStruct
      st oContMaps(b1,b2) is with_suprema
   holds b2 is not being_T1;

:: WAYBEL26:funcreg 5
registration
  cluster Sierpinski_Space -> non trivial strict monotone-convergence;
end;

:: WAYBEL26:exreg 1
registration
  cluster non empty non trivial TopSpace-like discerning injective monotone-convergence TopStruct;
end;

:: WAYBEL26:th 27
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty non trivial TopSpace-like discerning monotone-convergence TopStruct
      st oContMaps(b1,b2) is complete & oContMaps(b1,b2) is continuous
   holds InclPoset the topology of b1 is continuous;

:: WAYBEL26:th 28
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being non empty TopSpace-like discerning monotone-convergence TopStruct holds
   ex b4 being Function-like quasi_total directed-sups-preserving projection Relation of the carrier of oContMaps(b1,b3),the carrier of oContMaps(b1,b3) st
      (for b5 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b3 holds
          b4 . b5 = b1 --> (b5 . b2)) &
       (ex b5 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b1 st
          b5 = b1 --> b2 & b4 = oContMaps(b5,b3));

:: WAYBEL26:th 29
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like discerning monotone-convergence TopStruct
      st oContMaps(b1,b2) is complete & oContMaps(b1,b2) is continuous
   holds Omega b2 is complete & Omega b2 is continuous;

:: WAYBEL26:th 30
theorem
for b1 being non empty 1-sorted
for b2 being non empty set
for b3 being non-Empty TopSpace-yielding ManySortedSet of b2
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of product b3
for b5 being Element of b2 holds
   (commute b4) . b5 = (proj(b3,b5)) * b4;

:: WAYBEL26:th 31
theorem
for b1 being 1-sorted
for b2 being set holds
   Carrier (b2 --> b1) = b2 --> the carrier of b1;

:: WAYBEL26:th 32
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being non empty set
for b4 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of product (b3 --> b2) holds
   commute b4 is Function-like quasi_total Relation of b3,the carrier of oContMaps(b1,b2);

:: WAYBEL26:th 33
theorem
for b1, b2 being non empty TopSpace-like TopStruct holds
the carrier of oContMaps(b1,b2) c= Funcs(the carrier of b1,the carrier of b2);

:: WAYBEL26:th 34
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being non empty set
for b4 being Function-like quasi_total Relation of b3,the carrier of oContMaps(b1,b2) holds
   commute b4 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of product (b3 --> b2);

:: WAYBEL26:th 35
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty set holds
   ex b3 being Function-like quasi_total Relation of the carrier of oContMaps(b1,product (b2 --> Sierpinski_Space)),the carrier of product (b2 --> oContMaps(b1,Sierpinski_Space)) st
      b3 is isomorphic(oContMaps(b1,product (b2 --> Sierpinski_Space)), product (b2 --> oContMaps(b1,Sierpinski_Space))) &
       (for b4 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of product (b2 --> Sierpinski_Space) holds
          b3 . b4 = commute b4);

:: WAYBEL26:th 36
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty set holds
   oContMaps(b1,product (b2 --> Sierpinski_Space)),product (b2 --> oContMaps(b1,Sierpinski_Space)) are_isomorphic;

:: WAYBEL26:th 37
theorem
for b1 being non empty TopSpace-like TopStruct
   st InclPoset the topology of b1 is continuous
for b2 being non empty TopSpace-like discerning injective TopStruct holds
   oContMaps(b1,b2) is complete & oContMaps(b1,b2) is continuous;

:: WAYBEL26:exreg 2
registration
  cluster non empty non trivial TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr;
end;

:: WAYBEL26:th 38
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non trivial TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr holds
      oContMaps(b1,b2) is complete & oContMaps(b1,b2) is continuous
   iff
      InclPoset the topology of b1 is continuous & b2 is continuous;

:: WAYBEL26:funcreg 6
registration
  let a1 be Relation-like Function-like set;
  cluster Union disjoin a1 -> Relation-like;
end;

:: WAYBEL26:funcnot 7 => WAYBEL26:func 5
definition
  let a1 be Relation-like Function-like set;
  func *graph A1 -> Relation-like set equals
    (Union disjoin a1) ~;
end;

:: WAYBEL26:def 4
theorem
for b1 being Relation-like Function-like set holds
   *graph b1 = (Union disjoin b1) ~;

:: WAYBEL26:th 39
theorem
for b1, b2 being set
for b3 being Relation-like Function-like set holds
      [b1,b2] in *graph b3
   iff
      b1 in proj1 b3 & b2 in b3 . b1;

:: WAYBEL26:th 40
theorem
for b1 being finite set holds
   proj1 b1 is finite & proj2 b1 is finite;

:: WAYBEL26:th 41
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Scott TopAugmentation of InclPoset the topology of b2
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
      st *graph b4 is open Element of bool the carrier of [:b1,b2:]
   holds b4 is continuous(b1, b3);

:: WAYBEL26:funcnot 8 => WAYBEL26:func 6
definition
  let a1 be Relation-like set;
  let a2 be set;
  func (A1,A2)*graph -> Relation-like Function-like set means
    proj1 it = a2 &
     (for b1 being set
           st b1 in a2
        holds it . b1 = Im(a1,b1));
end;

:: WAYBEL26:def 5
theorem
for b1 being Relation-like set
for b2 being set
for b3 being Relation-like Function-like set holds
      b3 = (b1,b2)*graph
   iff
      proj1 b3 = b2 &
       (for b4 being set
             st b4 in b2
          holds b3 . b4 = Im(b1,b4));

:: WAYBEL26:th 42
theorem
for b1 being Relation-like set
for b2 being set
      st proj1 b1 c= b2
   holds *graph ((b1,b2)*graph) = b1;

:: WAYBEL26:condreg 1
registration
  let a1, a2 be TopSpace-like TopStruct;
  cluster -> Relation-like (Element of bool the carrier of [:a1,a2:]);
end;

:: WAYBEL26:condreg 2
registration
  let a1, a2 be TopSpace-like TopStruct;
  cluster -> Relation-like (Element of the topology of [:a1,a2:]);
end;

:: WAYBEL26:th 43
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being open Element of bool the carrier of [:b1,b2:]
for b4 being Element of the carrier of b1 holds
   Im(b3,b4) is open Element of bool the carrier of b2;

:: WAYBEL26:th 44
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Scott TopAugmentation of InclPoset the topology of b2
for b4 being open Element of bool the carrier of [:b1,b2:] holds
   (b4,the carrier of b1)*graph is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b3;

:: WAYBEL26:th 45
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 open Element of bool the carrier of [:b1,b2:]
   st b4 c= b5
for b6, b7 being Element of the carrier of oContMaps(b1,b3)
      st b6 = (b4,the carrier of b1)*graph & b7 = (b5,the carrier of b1)*graph
   holds b6 <= b7;

:: WAYBEL26:th 46
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Scott TopAugmentation of InclPoset the topology of b2 holds
   ex b4 being Function-like quasi_total Relation of the carrier of InclPoset the topology of [:b1,b2:],the carrier of oContMaps(b1,b3) st
      b4 is monotone(InclPoset the topology of [:b1,b2:], oContMaps(b1,b3)) &
       (for b5 being open Element of bool the carrier of [:b1,b2:] holds
          b4 . b5 = (b5,the carrier of b1)*graph);