Article WAYBEL24, MML version 4.99.1005

:: WAYBEL24:th 1
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric up-complete with_suprema with_infima Scott TopRelStr
for b3 being Element of bool the carrier of SCMaps(b1,b2) holds
   "\/"(b3,SCMaps(b1,b2)) is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2;

:: WAYBEL24:condreg 1
registration
  let a1 be non empty RelStr;
  let a2 be non empty reflexive RelStr;
  cluster Function-like constant quasi_total -> monotone (Relation of the carrier of a1,the carrier of a2);
end;

:: WAYBEL24:funcreg 1
registration
  let a1 be non empty RelStr;
  let a2 be non empty reflexive RelStr;
  let a3 be Element of the carrier of a2;
  cluster a1 --> a3 -> Function-like quasi_total monotone;
end;

:: WAYBEL24:th 2
theorem
for b1 being non empty RelStr
for b2 being non empty reflexive antisymmetric lower-bounded RelStr holds
   Bottom MonMaps(b1,b2) = b1 --> Bottom b2;

:: WAYBEL24:th 3
theorem
for b1 being non empty RelStr
for b2 being non empty reflexive antisymmetric upper-bounded RelStr holds
   Top MonMaps(b1,b2) = b1 --> Top b2;

:: WAYBEL24:sch 1
scheme WAYBEL24:sch 1
{F1 -> non empty RelStr,
  F2 -> non empty RelStr,
  F3 -> Element of the carrier of F2(),
  F4 -> Relation-like Function-like set}:
F4() .: {F3(b1) where b1 is Element of the carrier of F1(): P1[b1]} = {F4() . F3(b1) where b1 is Element of the carrier of F1(): P1[b1]}
provided
   the carrier of F2() c= proj1 F4();


:: WAYBEL24:sch 2
scheme WAYBEL24:sch 2
{F1 -> reflexive transitive antisymmetric with_suprema with_infima RelStr,
  F2 -> set}:
{F2(b1) where b1 is Element of the carrier of F1(): P1[b1]} = {F2(b1) where b1 is Element of the carrier of F1(): P2[b1]}
provided
   for b1 being Element of the carrier of F1() holds
         P1[b1]
      iff
         P2[b1];


:: WAYBEL24:th 4
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1 holds
   b3 . b4 = "\/"(b3 .: downarrow b4,b2);

:: WAYBEL24:th 5
theorem
for b1, b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1 holds
   b3 . b4 = "\/"({b3 . b5 where b5 is Element of the carrier of b1: b5 <= b4},b2);

:: WAYBEL24:th 6
theorem
for b1 being RelStr
for b2 being non empty RelStr
for b3 being Element of bool the carrier of b2 |^ the carrier of b1 holds
   "\/"(b3,b2 |^ the carrier of b1) is Function-like quasi_total Relation of the carrier of b1,the carrier of b2;

:: WAYBEL24:funcnot 1 => WAYBEL24:func 1
definition
  let a1, a2, a3 be non empty RelStr;
  let a4 be Function-like quasi_total Relation of the carrier of [:a1,a2:],the carrier of a3;
  let a5 be Element of the carrier of a1;
  func Proj(A4,A5) -> Function-like quasi_total Relation of the carrier of a2,the carrier of a3 equals
    (curry a4) . a5;
end;

:: WAYBEL24:def 1
theorem
for b1, b2, b3 being non empty RelStr
for b4 being Function-like quasi_total Relation of the carrier of [:b1,b2:],the carrier of b3
for b5 being Element of the carrier of b1 holds
   Proj(b4,b5) = (curry b4) . b5;

:: WAYBEL24:th 7
theorem
for b1, b2, b3 being non empty RelStr
for b4 being Function-like quasi_total Relation of the carrier of [:b2,b3:],the carrier of b1
for b5 being Element of the carrier of b2
for b6 being Element of the carrier of b3 holds
   (Proj(b4,b5)) . b6 = b4 .(b5,b6);

:: WAYBEL24:funcnot 2 => WAYBEL24:func 2
definition
  let a1, a2, a3 be non empty RelStr;
  let a4 be Function-like quasi_total Relation of the carrier of [:a1,a2:],the carrier of a3;
  let a5 be Element of the carrier of a2;
  func Proj(A4,A5) -> Function-like quasi_total Relation of the carrier of a1,the carrier of a3 equals
    (curry' a4) . a5;
end;

:: WAYBEL24:def 2
theorem
for b1, b2, b3 being non empty RelStr
for b4 being Function-like quasi_total Relation of the carrier of [:b1,b2:],the carrier of b3
for b5 being Element of the carrier of b2 holds
   Proj(b4,b5) = (curry' b4) . b5;

:: WAYBEL24:th 8
theorem
for b1, b2, b3 being non empty RelStr
for b4 being Function-like quasi_total Relation of the carrier of [:b3,b2:],the carrier of b1
for b5 being Element of the carrier of b2
for b6 being Element of the carrier of b3 holds
   (Proj(b4,b5)) . b6 = b4 .(b6,b5);

:: WAYBEL24:th 9
theorem
for b1, b2, b3 being non empty RelStr
for b4 being Function-like quasi_total Relation of the carrier of [:b1,b2:],the carrier of b3
for b5 being Element of the carrier of b1
for b6 being Element of the carrier of b2 holds
   (Proj(b4,b5)) . b6 = (Proj(b4,b6)) . b5;

:: WAYBEL24:exreg 1
registration
  let a1 be non empty RelStr;
  let a2 be non empty reflexive RelStr;
  cluster non empty Relation-like Function-like quasi_total total antitone Relation of the carrier of a1,the carrier of a2;
end;

:: WAYBEL24:th 10
theorem
for b1, b2, b3 being non empty reflexive RelStr
for b4 being Function-like quasi_total Relation of the carrier of [:b1,b2:],the carrier of b3
for b5 being Element of the carrier of b1
for b6 being Element of the carrier of b2
      st b4 is monotone([:b1,b2:], b3)
   holds Proj(b4,b5) is monotone(b2, b3) & Proj(b4,b6) is monotone(b1, b3);

:: WAYBEL24:th 11
theorem
for b1, b2, b3 being non empty reflexive RelStr
for b4 being Function-like quasi_total Relation of the carrier of [:b1,b2:],the carrier of b3
for b5 being Element of the carrier of b1
for b6 being Element of the carrier of b2
      st b4 is antitone([:b1,b2:], b3)
   holds Proj(b4,b5) is antitone(b2, b3) & Proj(b4,b6) is antitone(b1, b3);

:: WAYBEL24:funcreg 2
registration
  let a1, a2, a3 be non empty reflexive RelStr;
  let a4 be Function-like quasi_total monotone Relation of the carrier of [:a1,a2:],the carrier of a3;
  let a5 be Element of the carrier of a1;
  cluster Proj(a4,a5) -> Function-like quasi_total monotone;
end;

:: WAYBEL24:funcreg 3
registration
  let a1, a2, a3 be non empty reflexive RelStr;
  let a4 be Function-like quasi_total monotone Relation of the carrier of [:a1,a2:],the carrier of a3;
  let a5 be Element of the carrier of a2;
  cluster Proj(a4,a5) -> Function-like quasi_total monotone;
end;

:: WAYBEL24:funcreg 4
registration
  let a1, a2, a3 be non empty reflexive RelStr;
  let a4 be Function-like quasi_total antitone Relation of the carrier of [:a1,a2:],the carrier of a3;
  let a5 be Element of the carrier of a1;
  cluster Proj(a4,a5) -> Function-like quasi_total antitone;
end;

:: WAYBEL24:funcreg 5
registration
  let a1, a2, a3 be non empty reflexive RelStr;
  let a4 be Function-like quasi_total antitone Relation of the carrier of [:a1,a2:],the carrier of a3;
  let a5 be Element of the carrier of a2;
  cluster Proj(a4,a5) -> Function-like quasi_total antitone;
end;

:: WAYBEL24:th 12
theorem
for b1, b2, b3 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b4 being Function-like quasi_total Relation of the carrier of [:b1,b2:],the carrier of b3
      st for b5 being Element of the carrier of b1
        for b6 being Element of the carrier of b2 holds
           Proj(b4,b5) is monotone(b2, b3) & Proj(b4,b6) is monotone(b1, b3)
   holds b4 is monotone([:b1,b2:], b3);

:: WAYBEL24:th 13
theorem
for b1, b2, b3 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b4 being Function-like quasi_total Relation of the carrier of [:b1,b2:],the carrier of b3
      st for b5 being Element of the carrier of b1
        for b6 being Element of the carrier of b2 holds
           Proj(b4,b5) is antitone(b2, b3) & Proj(b4,b6) is antitone(b1, b3)
   holds b4 is antitone([:b1,b2:], b3);

:: WAYBEL24:th 14
theorem
for b1, b2, b3 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b4 being Function-like quasi_total Relation of the carrier of [:b1,b2:],the carrier of b3
for b5 being Element of the carrier of b2
for b6 being Element of bool the carrier of b1 holds
   (Proj(b4,b5)) .: b6 = b4 .: [:b6,{b5}:];

:: WAYBEL24:th 15
theorem
for b1, b2, b3 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b4 being Function-like quasi_total Relation of the carrier of [:b1,b2:],the carrier of b3
for b5 being Element of the carrier of b1
for b6 being Element of bool the carrier of b2 holds
   (Proj(b4,b5)) .: b6 = b4 .: [:{b5},b6:];

:: WAYBEL24:th 16
theorem
for b1, b2, b3 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b4 being Function-like quasi_total Relation of the carrier of [:b1,b2:],the carrier of b3
for b5 being Element of the carrier of b1
for b6 being Element of the carrier of b2
      st b4 is directed-sups-preserving([:b1,b2:], b3)
   holds Proj(b4,b5) is directed-sups-preserving(b2, b3) & Proj(b4,b6) is directed-sups-preserving(b1, b3);

:: WAYBEL24:th 17
theorem
for b1, b2, b3 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b4 being Function-like quasi_total monotone Relation of the carrier of [:b1,b2:],the carrier of b3
for b5 being Element of the carrier of b1
for b6 being Element of the carrier of b2
for b7 being directed Element of bool the carrier of [:b1,b2:]
      st ex_sup_of b4 .: b7,b3 & b5 in proj1 b7 & b6 in proj2 b7
   holds b4 . [b5,b6] <= "\/"(b4 .: b7,b3);

:: WAYBEL24:th 18
theorem
for b1, b2, b3 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b4 being Function-like quasi_total Relation of the carrier of [:b1,b2:],the carrier of b3
      st for b5 being Element of the carrier of b1
        for b6 being Element of the carrier of b2 holds
           Proj(b4,b5) is directed-sups-preserving(b2, b3) & Proj(b4,b6) is directed-sups-preserving(b1, b3)
   holds b4 is directed-sups-preserving([:b1,b2:], b3);

:: WAYBEL24:th 19
theorem
for b1 being set
for b2 being non empty RelStr
for b3 being set holds
      b3 is Element of the carrier of b2 |^ b1
   iff
      b3 is Function-like quasi_total Relation of b1,the carrier of b2;

:: WAYBEL24:funcnot 3 => WAYBEL24:func 3
definition
  let a1 be TopStruct;
  let a2 be non empty TopRelStr;
  func ContMaps(A1,A2) -> strict RelStr means
    it is full SubRelStr of a2 |^ the carrier of a1 &
     (for b1 being set holds
           b1 in the carrier of it
        iff
           ex b2 being Function-like quasi_total Relation of the carrier of a1,the carrier of a2 st
              b1 = b2 & b2 is continuous(a1, a2));
end;

:: WAYBEL24:def 3
theorem
for b1 being TopStruct
for b2 being non empty TopRelStr
for b3 being strict RelStr holds
      b3 = ContMaps(b1,b2)
   iff
      b3 is full SubRelStr of b2 |^ the carrier of b1 &
       (for b4 being set holds
             b4 in the carrier of b3
          iff
             ex b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 st
                b4 = b5 & b5 is continuous(b1, b2));

:: WAYBEL24:funcreg 6
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty TopSpace-like TopRelStr;
  cluster ContMaps(a1,a2) -> non empty strict;
end;

:: WAYBEL24:funcreg 7
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty TopSpace-like TopRelStr;
  cluster ContMaps(a1,a2) -> constituted-Functions strict;
end;

:: WAYBEL24:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like reflexive TopRelStr
for b3, b4 being Element of the carrier of ContMaps(b1,b2) holds
   b3 <= b4
iff
   for b5 being Element of the carrier of b1 holds
      [b3 . b5,b4 . b5] in the InternalRel of b2;

:: WAYBEL24:th 21
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like reflexive TopRelStr
for b3 being set holds
      b3 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
   iff
      b3 is Element of the carrier of ContMaps(b1,b2);

:: WAYBEL24:funcreg 8
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty TopSpace-like reflexive TopRelStr;
  cluster ContMaps(a1,a2) -> strict reflexive;
end;

:: WAYBEL24:funcreg 9
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty TopSpace-like transitive TopRelStr;
  cluster ContMaps(a1,a2) -> strict transitive;
end;

:: WAYBEL24:funcreg 10
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty TopSpace-like antisymmetric TopRelStr;
  cluster ContMaps(a1,a2) -> strict antisymmetric;
end;

:: WAYBEL24:funcreg 11
registration
  let a1 be set;
  let a2 be non empty RelStr;
  cluster a2 |^ a1 -> constituted-Functions strict;
end;

:: WAYBEL24:th 22
theorem
for b1 being non empty 1-sorted
for b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3, b4, b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b6 being Element of the carrier of b1
      st b5 = "\/"({b3,b4},b2 |^ the carrier of b1)
   holds b5 . b6 = "\/"({b3 . b6,b4 . b6},b2);

:: WAYBEL24:th 23
theorem
for b1 being non empty set
for b2 being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of b1
   st for b3 being Element of b1 holds
        b2 . b3 is reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Element of bool the carrier of product b2
for b4 being Element of b1 holds
   ("/\"(b3,product b2)) . b4 = "/\"(pi(b3,b4),b2 . b4);

:: WAYBEL24:th 24
theorem
for b1 being non empty 1-sorted
for b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3, b4, b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b6 being Element of the carrier of b1
      st b5 = "/\"({b3,b4},b2 |^ the carrier of b1)
   holds b5 . b6 = "/\"({b3 . b6,b4 . b6},b2);

:: WAYBEL24:th 25
theorem
for b1 being non empty RelStr
for b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being non empty Element of bool the carrier of b2 |^ the carrier of b1
for b4 being Element of the carrier of b1 holds
   ("\/"(b3,b2 |^ the carrier of b1)) . b4 = "\/"({b5 . b4 where b5 is Element of the carrier of b2 |^ the carrier of b1: b5 in b3},b2);

:: WAYBEL24:th 26
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete TopRelStr
for b3 being non empty Element of bool the carrier of ContMaps(b1,b2)
for b4 being Element of the carrier of b1 holds
   ("\/"(b3,b2 |^ the carrier of b1)) . b4 = "\/"({b5 . b4 where b5 is Element of the carrier of b2 |^ the carrier of b1: b5 in b3},b2);

:: WAYBEL24:th 27
theorem
for b1 being non empty RelStr
for b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being non empty Element of bool the carrier of b2 |^ the carrier of b1
for b4 being non empty Element of bool the carrier of b1 holds
   ("\/"(b3,b2 |^ the carrier of b1)) .: b4 = {"\/"({b6 . b5 where b6 is Element of the carrier of b2 |^ the carrier of b1: b6 in b3},b2) where b5 is Element of the carrier of b1: b5 in b4};

:: WAYBEL24:th 28
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b3 being non empty Element of bool the carrier of ContMaps(b1,b2)
for b4 being non empty Element of bool the carrier of b1 holds
   ("\/"(b3,b2 |^ the carrier of b1)) .: b4 = {"\/"({b6 . b5 where b6 is Element of the carrier of b2 |^ the carrier of b1: b6 in b3},b2) where b5 is Element of the carrier of b1: b5 in b4};

:: WAYBEL24:sch 3
scheme WAYBEL24:sch 3
{F1 -> non empty TopRelStr,
  F2 -> set,
  F3 -> set}:
{F2(b1) where b1 is Element of the carrier of F1(): P1[b1]} = {F3(b1) where b1 is Element of the carrier of F1(): P1[b1]}
provided
   for b1 being Element of the carrier of F1()
         st P1[b1]
      holds F2(b1) = F3(b1);


:: WAYBEL24:sch 4
scheme WAYBEL24:sch 4
{F1 -> non empty RelStr,
  F2 -> set,
  F3 -> set}:
{F2(b1) where b1 is Element of the carrier of F1(): P1[b1]} = {F3(b1) where b1 is Element of the carrier of F1(): P1[b1]}
provided
   for b1 being Element of the carrier of F1()
         st P1[b1]
      holds F2(b1) = F3(b1);


:: WAYBEL24:sch 5
scheme WAYBEL24:sch 5
{F1 -> non empty TopRelStr,
  F2 -> non empty TopRelStr,
  F3 -> Element of the carrier of F2(),
  F4 -> Relation-like Function-like set}:
F4() .: {F3(b1) where b1 is Element of the carrier of F1(): P1[b1]} = {F4() . F3(b1) where b1 is Element of the carrier of F1(): P1[b1]}
provided
   the carrier of F2() c= proj1 F4();


:: WAYBEL24:th 29
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b3 being non empty Element of bool the carrier of ContMaps(b1,b2) holds
   "\/"(b3,b2 |^ the carrier of b1) is Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b2;

:: WAYBEL24:th 30
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b3 being non empty Element of bool the carrier of ContMaps(b1,b2)
for b4 being non empty directed Element of bool the carrier of b1 holds
   "\/"({"\/"({b5 . b6 where b6 is Element of the carrier of b1: b6 in b4},b2) where b5 is Element of the carrier of b2 |^ the carrier of b1: b5 in b3},b2) = "\/"({"\/"({b6 . b5 where b6 is Element of the carrier of b2 |^ the carrier of b1: b6 in b3},b2) where b5 is Element of the carrier of b1: b5 in b4},b2);

:: WAYBEL24:th 31
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b3 being non empty Element of bool the carrier of ContMaps(b1,b2)
for b4 being non empty directed Element of bool the carrier of b1 holds
   "\/"(("\/"(b3,b2 |^ the carrier of b1)) .: b4,b2) = ("\/"(b3,b2 |^ the carrier of b1)) . "\/"(b4,b1);

:: WAYBEL24:th 32
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b3 being non empty Element of bool the carrier of ContMaps(b1,b2) holds
   "\/"(b3,b2 |^ the carrier of b1) in the carrier of ContMaps(b1,b2);

:: WAYBEL24:th 33
theorem
for b1 being non empty RelStr
for b2 being non empty antisymmetric lower-bounded RelStr holds
   Bottom (b2 |^ the carrier of b1) = b1 --> Bottom b2;

:: WAYBEL24:th 34
theorem
for b1 being non empty RelStr
for b2 being non empty antisymmetric upper-bounded RelStr holds
   Top (b2 |^ the carrier of b1) = b1 --> Top b2;

:: WAYBEL24:funcreg 12
registration
  let a1 be non empty reflexive RelStr;
  let a2 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  let a3 be Element of the carrier of a2;
  cluster a1 --> a3 -> Function-like quasi_total directed-sups-preserving;
end;

:: WAYBEL24:th 35
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr holds
ContMaps(b1,b2) is sups-inheriting SubRelStr of b2 |^ the carrier of b1;

:: WAYBEL24:funcreg 13
registration
  let a1, a2 be TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr;
  cluster ContMaps(a1,a2) -> strict complete;
end;

:: WAYBEL24:th 36
theorem
for b1, b2 being non empty TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr holds
Bottom ContMaps(b1,b2) = b1 --> Bottom b2;

:: WAYBEL24:th 37
theorem
for b1, b2 being non empty TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr holds
Top ContMaps(b1,b2) = b1 --> Top b2;

:: WAYBEL24:th 38
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr holds
SCMaps(b1,b2) = ContMaps(b1,b2);