Article YELLOW_2, MML version 4.99.1005

:: YELLOW_2:th 1
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
      b3 c= downarrow b2
   iff
      b3 is_<=_than b2;

:: YELLOW_2:th 2
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
      b3 c= uparrow b2
   iff
      b2 is_<=_than b3;

:: YELLOW_2:th 3
theorem
for b1 being transitive antisymmetric with_suprema RelStr
for b2, b3 being set
      st ex_sup_of b2,b1 & ex_sup_of b3,b1
   holds ex_sup_of b2 \/ b3,b1 &
    "\/"(b2 \/ b3,b1) = ("\/"(b2,b1)) "\/" "\/"(b3,b1);

:: YELLOW_2:th 4
theorem
for b1 being transitive antisymmetric with_infima RelStr
for b2, b3 being set
      st ex_inf_of b2,b1 & ex_inf_of b3,b1
   holds ex_inf_of b2 \/ b3,b1 &
    "/\"(b2 \/ b3,b1) = ("/\"(b2,b1)) "/\" "/\"(b3,b1);

:: YELLOW_2:th 5
theorem
for b1 being Relation-like set
for b2, b3 being set
      st b2 c= b3
   holds b1 |_2 b2 c= b1 |_2 b3;

:: YELLOW_2:th 6
theorem
for b1 being RelStr
for b2, b3 being full SubRelStr of b1
      st the carrier of b2 c= the carrier of b3
   holds the InternalRel of b2 c= the InternalRel of b3;

:: YELLOW_2:th 7
theorem
for b1 being set
for b2 being non empty RelStr
for b3 being non empty SubRelStr of b2 holds
   (b1 is directed Element of bool the carrier of b3 implies b1 is directed Element of bool the carrier of b2) &
    (b1 is filtered Element of bool the carrier of b3 implies b1 is filtered Element of bool the carrier of b2);

:: YELLOW_2:th 8
theorem
for b1 being non empty RelStr
for b2, b3 being non empty full SubRelStr of b1
   st the carrier of b2 c= the carrier of b3
for b4 being Element of bool the carrier of b2 holds
   b4 is Element of bool the carrier of b3 &
    (for b5 being Element of bool the carrier of b3
          st b4 = b5
       holds (b4 is filtered(b2) implies b5 is filtered(b3)) & (b4 is directed(b2) implies b5 is directed(b3)));

:: YELLOW_2:prednot 1 => YELLOW_2:pred 1
definition
  let a1 be set;
  let a2 be RelStr;
  let a3, a4 be Function-like quasi_total Relation of a1,the carrier of a2;
  pred A3 <= A4 means
    for b1 being set
          st b1 in a1
       holds ex b2, b3 being Element of the carrier of a2 st
          b2 = a3 . b1 & b3 = a4 . b1 & b2 <= b3;
end;

:: YELLOW_2:dfs 1
definiens
  let a1 be set;
  let a2 be RelStr;
  let a3, a4 be Function-like quasi_total Relation of a1,the carrier of a2;
To prove
     a3 <= a4
it is sufficient to prove
  thus for b1 being set
          st b1 in a1
       holds ex b2, b3 being Element of the carrier of a2 st
          b2 = a3 . b1 & b3 = a4 . b1 & b2 <= b3;

:: YELLOW_2:def 1
theorem
for b1 being set
for b2 being RelStr
for b3, b4 being Function-like quasi_total Relation of b1,the carrier of b2 holds
   b3 <= b4
iff
   for b5 being set
         st b5 in b1
      holds ex b6, b7 being Element of the carrier of b2 st
         b6 = b3 . b5 & b7 = b4 . b5 & b6 <= b7;

:: YELLOW_2:prednot 2 => YELLOW_2:pred 1
notation
  let a1 be set;
  let a2 be RelStr;
  let a3, a4 be Function-like quasi_total Relation of a1,the carrier of a2;
  synonym a4 >= a3 for a3 <= a4;
end;

:: YELLOW_2:th 10
theorem
for b1, b2 being non empty RelStr
for b3, b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
   b3 <= b4
iff
   for b5 being Element of the carrier of b1 holds
      b3 . b5 <= b4 . b5;

:: YELLOW_2:funcnot 1 => YELLOW_2:func 1
definition
  let a1, a2 be non empty RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  func Image A3 -> strict full SubRelStr of a2 equals
    subrelstr rng a3;
end;

:: YELLOW_2:def 2
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
   Image b3 = subrelstr rng b3;

:: YELLOW_2:th 12
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of Image b3 holds
   ex b5 being Element of the carrier of b1 st
      b3 . b5 = b4;

:: YELLOW_2:funcreg 1
registration
  let a1 be non empty RelStr;
  let a2 be non empty Element of bool the carrier of a1;
  cluster subrelstr a2 -> non empty strict full;
end;

:: YELLOW_2:funcreg 2
registration
  let a1, a2 be non empty RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  cluster Image a3 -> non empty strict full;
end;

:: YELLOW_2:th 13
theorem
for b1 being non empty RelStr holds
   id b1 is monotone(b1, b1);

:: YELLOW_2:th 14
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 b2
for b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
      st b4 is monotone(b1, b2) & b5 is monotone(b2, b3)
   holds b5 * b4 is monotone(b1, b3);

:: YELLOW_2:th 15
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1
for b5 being Element of the carrier of b1
      st b3 is monotone(b1, b2) & b5 is_<=_than b4
   holds b3 . b5 is_<=_than b3 .: b4;

:: YELLOW_2:th 16
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1
for b5 being Element of the carrier of b1
      st b3 is monotone(b1, b2) & b4 is_<=_than b5
   holds b3 .: b4 is_<=_than b3 . b5;

:: YELLOW_2:th 17
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being directed Element of bool the carrier of b1
      st b3 is monotone(b1, b2)
   holds b3 .: b4 is directed(b2);

:: YELLOW_2:th 18
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
      st b2 is directed-sups-preserving(b1, b1)
   holds b2 is monotone(b1, b1);

:: YELLOW_2:th 19
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
      st b2 is filtered-infs-preserving(b1, b1)
   holds b2 is monotone(b1, b1);

:: YELLOW_2:th 20
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
   st b2 is idempotent
for b3 being Element of the carrier of b1 holds
   b2 . (b2 . b3) = b2 . b3;

:: YELLOW_2:th 21
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
      st b2 is idempotent
   holds rng b2 = {b3 where b3 is Element of the carrier of b1: b3 = b2 . b3};

:: YELLOW_2:th 22
theorem
for b1 being set
for b2 being non empty 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b2
      st b3 is idempotent & b1 c= rng b3
   holds b3 .: b1 = b1;

:: YELLOW_2:th 23
theorem
for b1 being non empty RelStr holds
   id b1 is idempotent;

:: YELLOW_2:th 24
theorem
for b1 being set
for b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Element of the carrier of b2
      st b3 in b1
   holds b3 <= "\/"(b1,b2) & "/\"(b1,b2) <= b3;

:: YELLOW_2:th 25
theorem
for b1 being non empty RelStr holds
      for b2 being set holds
         ex_sup_of b2,b1
   iff
      for b2 being set holds
         ex_inf_of b2,b1;

:: YELLOW_2:th 26
theorem
for b1 being non empty RelStr
      st for b2 being set holds
           ex_sup_of b2,b1
   holds b1 is complete;

:: YELLOW_2:th 27
theorem
for b1 being non empty RelStr
      st for b2 being set holds
           ex_inf_of b2,b1
   holds b1 is complete;

:: YELLOW_2:th 28
theorem
for b1 being non empty RelStr
   st for b2 being Element of bool the carrier of b1 holds
        ex_inf_of b2,b1
for b2 being set holds
   ex_inf_of b2,b1 &
    "/\"(b2,b1) = "/\"(b2 /\ the carrier of b1,b1);

:: YELLOW_2:th 29
theorem
for b1 being non empty RelStr
   st for b2 being Element of bool the carrier of b1 holds
        ex_sup_of b2,b1
for b2 being set holds
   ex_sup_of b2,b1 &
    "\/"(b2,b1) = "\/"(b2 /\ the carrier of b1,b1);

:: YELLOW_2:th 30
theorem
for b1 being non empty RelStr
      st for b2 being Element of bool the carrier of b1 holds
           ex_inf_of b2,b1
   holds b1 is complete;

:: YELLOW_2:condreg 1
registration
  cluster non empty reflexive transitive antisymmetric upper-bounded up-complete /\-complete -> complete (RelStr);
end;

:: YELLOW_2:th 31
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
   st b2 is monotone(b1, b1)
for b3 being Element of bool the carrier of b1
      st b3 = {b4 where b4 is Element of the carrier of b1: b4 = b2 . b4}
   holds subrelstr b3 is reflexive transitive antisymmetric with_suprema with_infima complete RelStr;

:: YELLOW_2:th 32
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty full infs-inheriting SubRelStr of b1 holds
   b2 is reflexive transitive antisymmetric with_suprema with_infima complete RelStr;

:: YELLOW_2:th 33
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty full sups-inheriting SubRelStr of b1 holds
   b2 is reflexive transitive antisymmetric with_suprema with_infima complete RelStr;

:: YELLOW_2:th 34
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is sups-preserving(b1, b2)
   holds Image b3 is sups-inheriting(b2);

:: YELLOW_2:th 35
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is infs-preserving(b1, b2)
   holds Image b3 is infs-inheriting(b2);

:: YELLOW_2:th 36
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st (b3 is sups-preserving(b1, b2) or b3 is infs-preserving(b1, b2))
   holds Image b3 is reflexive transitive antisymmetric with_suprema with_infima complete RelStr;

:: YELLOW_2:th 37
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
      st b2 is idempotent & b2 is directed-sups-preserving(b1, b1)
   holds Image b2 is directed-sups-inheriting(b1) &
    Image b2 is reflexive transitive antisymmetric with_suprema with_infima complete RelStr;

:: YELLOW_2:th 38
theorem
for b1 being RelStr
for b2 being Element of bool bool the carrier of b1
      st for b3 being Element of bool the carrier of b1
              st b3 in b2
           holds b3 is lower(b1)
   holds meet b2 is lower Element of bool the carrier of b1;

:: YELLOW_2:th 39
theorem
for b1 being RelStr
for b2 being Element of bool bool the carrier of b1
      st for b3 being Element of bool the carrier of b1
              st b3 in b2
           holds b3 is upper(b1)
   holds meet b2 is upper Element of bool the carrier of b1;

:: YELLOW_2:th 40
theorem
for b1 being antisymmetric with_suprema RelStr
for b2 being Element of bool bool the carrier of b1
      st for b3 being Element of bool the carrier of b1
              st b3 in b2
           holds b3 is lower(b1) & b3 is directed(b1)
   holds meet b2 is directed Element of bool the carrier of b1;

:: YELLOW_2:th 41
theorem
for b1 being antisymmetric with_infima RelStr
for b2 being Element of bool bool the carrier of b1
      st for b3 being Element of bool the carrier of b1
              st b3 in b2
           holds b3 is upper(b1) & b3 is filtered(b1)
   holds meet b2 is filtered Element of bool the carrier of b1;

:: YELLOW_2:th 42
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2, b3 being non empty directed lower Element of bool the carrier of b1 holds
b2 /\ b3 is non empty directed lower Element of bool the carrier of b1;

:: YELLOW_2:funcreg 3
registration
  let a1 be non empty reflexive transitive RelStr;
  cluster Ids a1 -> non empty;
end;

:: YELLOW_2:th 43
theorem
for b1 being set
for b2 being non empty reflexive transitive RelStr holds
      b1 is Element of the carrier of InclPoset Ids b2
   iff
      b1 is non empty directed lower Element of bool the carrier of b2;

:: YELLOW_2:th 44
theorem
for b1 being set
for b2 being non empty reflexive transitive RelStr
for b3 being Element of the carrier of InclPoset Ids b2
      st b1 in b3
   holds b1 is Element of the carrier of b2;

:: YELLOW_2:th 45
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2, b3 being Element of the carrier of InclPoset Ids b1 holds
b2 "/\" b3 = b2 /\ b3;

:: YELLOW_2:funcreg 4
registration
  let a1 be reflexive transitive antisymmetric with_infima RelStr;
  cluster InclPoset Ids a1 -> strict with_infima;
end;

:: YELLOW_2:th 46
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2, b3 being Element of the carrier of InclPoset Ids b1 holds
ex b4 being Element of bool the carrier of b1 st
   b4 = {b5 where b5 is Element of the carrier of b1: (not b5 in b2 & not b5 in b3 implies ex b6, b7 being Element of the carrier of b1 st
       b6 in b2 & b7 in b3 & b5 = b6 "\/" b7)} &
    ex_sup_of {b2,b3},InclPoset Ids b1 &
    b2 "\/" b3 = downarrow b4;

:: YELLOW_2:funcreg 5
registration
  let a1 be reflexive transitive antisymmetric with_suprema RelStr;
  cluster InclPoset Ids a1 -> strict with_suprema;
end;

:: YELLOW_2:th 47
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded RelStr
for b2 being non empty Element of bool Ids b1 holds
   meet b2 is non empty directed lower Element of bool the carrier of b1;

:: YELLOW_2:th 48
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded RelStr
for b2 being non empty Element of bool the carrier of InclPoset Ids b1 holds
   ex_inf_of b2,InclPoset Ids b1 & "/\"(b2,InclPoset Ids b1) = meet b2;

:: YELLOW_2:th 49
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr holds
   ex_inf_of {},InclPoset Ids b1 &
    "/\"({},InclPoset Ids b1) = [#] b1;

:: YELLOW_2:th 50
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded RelStr holds
   InclPoset Ids b1 is complete;

:: YELLOW_2:funcreg 6
registration
  let a1 be reflexive transitive antisymmetric with_suprema lower-bounded RelStr;
  cluster InclPoset Ids a1 -> strict complete;
end;

:: YELLOW_2:funcnot 2 => YELLOW_2:func 2
definition
  let a1 be non empty reflexive transitive antisymmetric RelStr;
  func SupMap A1 -> Function-like quasi_total Relation of the carrier of InclPoset Ids a1,the carrier of a1 means
    for b1 being non empty directed lower Element of bool the carrier of a1 holds
       it . b1 = "\/"(b1,a1);
end;

:: YELLOW_2:def 3
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being Function-like quasi_total Relation of the carrier of InclPoset Ids b1,the carrier of b1 holds
      b2 = SupMap b1
   iff
      for b3 being non empty directed lower Element of bool the carrier of b1 holds
         b2 . b3 = "\/"(b3,b1);

:: YELLOW_2:th 51
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr holds
   dom SupMap b1 = Ids b1 & rng SupMap b1 is Element of bool the carrier of b1;

:: YELLOW_2:th 52
theorem
for b1 being set
for b2 being non empty reflexive transitive antisymmetric RelStr holds
      b1 in dom SupMap b2
   iff
      b1 is non empty directed lower Element of bool the carrier of b2;

:: YELLOW_2:th 53
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr holds
   SupMap b1 is monotone(InclPoset Ids b1, b1);

:: YELLOW_2:funcreg 7
registration
  let a1 be non empty reflexive transitive antisymmetric up-complete RelStr;
  cluster SupMap a1 -> Function-like quasi_total monotone;
end;

:: YELLOW_2:funcnot 3 => YELLOW_2:func 3
definition
  let a1 be non empty reflexive transitive antisymmetric RelStr;
  func IdsMap A1 -> Function-like quasi_total Relation of the carrier of a1,the carrier of InclPoset Ids a1 means
    for b1 being Element of the carrier of a1 holds
       it . b1 = downarrow b1;
end;

:: YELLOW_2:def 4
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of InclPoset Ids b1 holds
      b2 = IdsMap b1
   iff
      for b3 being Element of the carrier of b1 holds
         b2 . b3 = downarrow b3;

:: YELLOW_2:th 54
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr holds
   IdsMap b1 is monotone(b1, InclPoset Ids b1);

:: YELLOW_2:funcreg 8
registration
  let a1 be non empty reflexive transitive antisymmetric RelStr;
  cluster IdsMap a1 -> Function-like quasi_total monotone;
end;

:: YELLOW_2:funcnot 4 => YELLOW_2:func 4
definition
  let a1 be non empty RelStr;
  let a2 be Relation-like set;
  func \\/(A2,A1) -> Element of the carrier of a1 equals
    "\/"(proj2 a2,a1);
end;

:: YELLOW_2:def 5
theorem
for b1 being non empty RelStr
for b2 being Relation-like set holds
   \\/(b2,b1) = "\/"(proj2 b2,b1);

:: YELLOW_2:funcnot 5 => YELLOW_2:func 5
definition
  let a1 be non empty RelStr;
  let a2 be Relation-like set;
  func //\(A2,A1) -> Element of the carrier of a1 equals
    "/\"(proj2 a2,a1);
end;

:: YELLOW_2:def 6
theorem
for b1 being non empty RelStr
for b2 being Relation-like set holds
   //\(b2,b1) = "/\"(proj2 b2,b1);

:: YELLOW_2:funcnot 6 => YELLOW_2:func 4
notation
  let a1 be set;
  let a2 be non empty RelStr;
  let a3 be Function-like quasi_total Relation of a1,the carrier of a2;
  synonym Sup a3 for \\/(a2,a1);
end;

:: YELLOW_2:funcnot 7 => YELLOW_2:func 5
notation
  let a1 be set;
  let a2 be non empty RelStr;
  let a3 be Function-like quasi_total Relation of a1,the carrier of a2;
  synonym Inf a3 for //\(a2,a1);
end;

:: YELLOW_2:th 55
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty set
for b3 being Element of b2
for b4 being Function-like quasi_total Relation of b2,the carrier of b1 holds
   b4 . b3 <= \\/(b4,b1) & //\(b4,b1) <= b4 . b3;

:: YELLOW_2:th 56
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Element of the carrier of b1
for b3 being non empty set
for b4 being Function-like quasi_total Relation of b3,the carrier of b1
      st for b5 being Element of b3 holds
           b4 . b5 <= b2
   holds \\/(b4,b1) <= b2;

:: YELLOW_2:th 57
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Element of the carrier of b1
for b3 being non empty set
for b4 being Function-like quasi_total Relation of b3,the carrier of b1
      st for b5 being Element of b3 holds
           b2 <= b4 . b5
   holds b2 <= //\(b4,b1);