Article WAYBEL_4, MML version 4.99.1005

:: WAYBEL_4:funcnot 1 => WAYBEL_4:func 1
definition
  let a1 be non empty reflexive RelStr;
  func A1 -waybelow -> Relation of the carrier of a1,the carrier of a1 means
    for b1, b2 being Element of the carrier of a1 holds
       [b1,b2] in it
    iff
       b1 is_way_below b2;
end;

:: WAYBEL_4:def 2
theorem
for b1 being non empty reflexive RelStr
for b2 being Relation of the carrier of b1,the carrier of b1 holds
      b2 = b1 -waybelow
   iff
      for b3, b4 being Element of the carrier of b1 holds
         [b3,b4] in b2
      iff
         b3 is_way_below b4;

:: WAYBEL_4:funcnot 2 => WAYBEL_4:func 2
definition
  let a1 be RelStr;
  func IntRel A1 -> Relation of the carrier of a1,the carrier of a1 equals
    the InternalRel of a1;
end;

:: WAYBEL_4:def 3
theorem
for b1 being RelStr holds
   IntRel b1 = the InternalRel of b1;

:: WAYBEL_4:attrnot 1 => WAYBEL_4:attr 1
definition
  let a1 be RelStr;
  let a2 be Relation of the carrier of a1,the carrier of a1;
  attr a2 is auxiliary(i) means
    for b1, b2 being Element of the carrier of a1
          st [b1,b2] in a2
       holds b1 <= b2;
end;

:: WAYBEL_4:dfs 3
definiens
  let a1 be RelStr;
  let a2 be Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is auxiliary(i)
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1
          st [b1,b2] in a2
       holds b1 <= b2;

:: WAYBEL_4:def 4
theorem
for b1 being RelStr
for b2 being Relation of the carrier of b1,the carrier of b1 holds
      b2 is auxiliary(i)(b1)
   iff
      for b3, b4 being Element of the carrier of b1
            st [b3,b4] in b2
         holds b3 <= b4;

:: WAYBEL_4:attrnot 2 => WAYBEL_4:attr 2
definition
  let a1 be RelStr;
  let a2 be Relation of the carrier of a1,the carrier of a1;
  attr a2 is auxiliary(ii) means
    for b1, b2, b3, b4 being Element of the carrier of a1
          st b4 <= b1 & [b1,b2] in a2 & b2 <= b3
       holds [b4,b3] in a2;
end;

:: WAYBEL_4:dfs 4
definiens
  let a1 be RelStr;
  let a2 be Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is auxiliary(ii)
it is sufficient to prove
  thus for b1, b2, b3, b4 being Element of the carrier of a1
          st b4 <= b1 & [b1,b2] in a2 & b2 <= b3
       holds [b4,b3] in a2;

:: WAYBEL_4:def 5
theorem
for b1 being RelStr
for b2 being Relation of the carrier of b1,the carrier of b1 holds
      b2 is auxiliary(ii)(b1)
   iff
      for b3, b4, b5, b6 being Element of the carrier of b1
            st b6 <= b3 & [b3,b4] in b2 & b4 <= b5
         holds [b6,b5] in b2;

:: WAYBEL_4:attrnot 3 => WAYBEL_4:attr 3
definition
  let a1 be non empty RelStr;
  let a2 be Relation of the carrier of a1,the carrier of a1;
  attr a2 is auxiliary(iii) means
    for b1, b2, b3 being Element of the carrier of a1
          st [b1,b3] in a2 & [b2,b3] in a2
       holds [b1 "\/" b2,b3] in a2;
end;

:: WAYBEL_4:dfs 5
definiens
  let a1 be non empty RelStr;
  let a2 be Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is auxiliary(iii)
it is sufficient to prove
  thus for b1, b2, b3 being Element of the carrier of a1
          st [b1,b3] in a2 & [b2,b3] in a2
       holds [b1 "\/" b2,b3] in a2;

:: WAYBEL_4:def 6
theorem
for b1 being non empty RelStr
for b2 being Relation of the carrier of b1,the carrier of b1 holds
      b2 is auxiliary(iii)(b1)
   iff
      for b3, b4, b5 being Element of the carrier of b1
            st [b3,b5] in b2 & [b4,b5] in b2
         holds [b3 "\/" b4,b5] in b2;

:: WAYBEL_4:attrnot 4 => WAYBEL_4:attr 4
definition
  let a1 be non empty RelStr;
  let a2 be Relation of the carrier of a1,the carrier of a1;
  attr a2 is auxiliary(iv) means
    for b1 being Element of the carrier of a1 holds
       [Bottom a1,b1] in a2;
end;

:: WAYBEL_4:dfs 6
definiens
  let a1 be non empty RelStr;
  let a2 be Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is auxiliary(iv)
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       [Bottom a1,b1] in a2;

:: WAYBEL_4:def 7
theorem
for b1 being non empty RelStr
for b2 being Relation of the carrier of b1,the carrier of b1 holds
      b2 is auxiliary(iv)(b1)
   iff
      for b3 being Element of the carrier of b1 holds
         [Bottom b1,b3] in b2;

:: WAYBEL_4:attrnot 5 => WAYBEL_4:attr 5
definition
  let a1 be non empty RelStr;
  let a2 be Relation of the carrier of a1,the carrier of a1;
  attr a2 is auxiliary means
    a2 is auxiliary(i)(a1) & a2 is auxiliary(ii)(a1) & a2 is auxiliary(iii)(a1) & a2 is auxiliary(iv)(a1);
end;

:: WAYBEL_4:dfs 7
definiens
  let a1 be non empty RelStr;
  let a2 be Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is auxiliary
it is sufficient to prove
  thus a2 is auxiliary(i)(a1) & a2 is auxiliary(ii)(a1) & a2 is auxiliary(iii)(a1) & a2 is auxiliary(iv)(a1);

:: WAYBEL_4:def 8
theorem
for b1 being non empty RelStr
for b2 being Relation of the carrier of b1,the carrier of b1 holds
      b2 is auxiliary(b1)
   iff
      b2 is auxiliary(i)(b1) & b2 is auxiliary(ii)(b1) & b2 is auxiliary(iii)(b1) & b2 is auxiliary(iv)(b1);

:: WAYBEL_4:condreg 1
registration
  let a1 be non empty RelStr;
  cluster auxiliary -> auxiliary(i) auxiliary(ii) auxiliary(iii) auxiliary(iv) (Relation of the carrier of a1,the carrier of a1);
end;

:: WAYBEL_4:condreg 2
registration
  let a1 be non empty RelStr;
  cluster auxiliary(i) auxiliary(ii) auxiliary(iii) auxiliary(iv) -> auxiliary (Relation of the carrier of a1,the carrier of a1);
end;

:: WAYBEL_4:exreg 1
registration
  let a1 be transitive antisymmetric lower-bounded with_suprema RelStr;
  cluster Relation-like auxiliary Relation of the carrier of a1,the carrier of a1;
end;

:: WAYBEL_4:th 1
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2 being auxiliary(ii) auxiliary(iii) Relation of the carrier of b1,the carrier of b1
for b3, b4, b5, b6 being Element of the carrier of b1
      st [b3,b5] in b2 & [b4,b6] in b2
   holds [b3 "\/" b4,b5 "\/" b6] in b2;

:: WAYBEL_4:condreg 3
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  cluster auxiliary(i) auxiliary(ii) -> transitive (Relation of the carrier of a1,the carrier of a1);
end;

:: WAYBEL_4:funcreg 1
registration
  let a1 be RelStr;
  cluster IntRel a1 -> auxiliary(i);
end;

:: WAYBEL_4:funcreg 2
registration
  let a1 be transitive RelStr;
  cluster IntRel a1 -> auxiliary(ii);
end;

:: WAYBEL_4:funcreg 3
registration
  let a1 be antisymmetric with_suprema RelStr;
  cluster IntRel a1 -> auxiliary(iii);
end;

:: WAYBEL_4:funcreg 4
registration
  let a1 be non empty antisymmetric lower-bounded RelStr;
  cluster IntRel a1 -> auxiliary(iv);
end;

:: WAYBEL_4:funcnot 3 => WAYBEL_4:func 3
definition
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  func Aux A1 -> set means
    for b1 being set holds
          b1 in it
       iff
          b1 is auxiliary Relation of the carrier of a1,the carrier of a1;
end;

:: WAYBEL_4:def 9
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2 being set holds
      b2 = Aux b1
   iff
      for b3 being set holds
            b3 in b2
         iff
            b3 is auxiliary Relation of the carrier of b1,the carrier of b1;

:: WAYBEL_4:funcreg 5
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  cluster Aux a1 -> non empty;
end;

:: WAYBEL_4:th 2
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2 being auxiliary(i) Relation of the carrier of b1,the carrier of b1 holds
   b2 c= IntRel b1;

:: WAYBEL_4:th 3
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr holds
   Top InclPoset Aux b1 = IntRel b1;

:: WAYBEL_4:funcreg 6
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  cluster InclPoset Aux a1 -> strict upper-bounded;
end;

:: WAYBEL_4:funcnot 4 => WAYBEL_4:func 4
definition
  let a1 be non empty RelStr;
  func AuxBottom A1 -> Relation of the carrier of a1,the carrier of a1 means
    for b1, b2 being Element of the carrier of a1 holds
       [b1,b2] in it
    iff
       b1 = Bottom a1;
end;

:: WAYBEL_4:def 10
theorem
for b1 being non empty RelStr
for b2 being Relation of the carrier of b1,the carrier of b1 holds
      b2 = AuxBottom b1
   iff
      for b3, b4 being Element of the carrier of b1 holds
         [b3,b4] in b2
      iff
         b3 = Bottom b1;

:: WAYBEL_4:funcreg 7
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  cluster AuxBottom a1 -> auxiliary;
end;

:: WAYBEL_4:th 4
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2 being auxiliary(iv) Relation of the carrier of b1,the carrier of b1 holds
   AuxBottom b1 c= b2;

:: WAYBEL_4:th 5
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr holds
   Bottom InclPoset Aux b1 = AuxBottom b1;

:: WAYBEL_4:funcreg 8
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  cluster InclPoset Aux a1 -> strict lower-bounded;
end;

:: WAYBEL_4:th 6
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2, b3 being auxiliary(i) Relation of the carrier of b1,the carrier of b1 holds
b2 /\ b3 is auxiliary(i) Relation of the carrier of b1,the carrier of b1;

:: WAYBEL_4:th 7
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2, b3 being auxiliary(ii) Relation of the carrier of b1,the carrier of b1 holds
b2 /\ b3 is auxiliary(ii) Relation of the carrier of b1,the carrier of b1;

:: WAYBEL_4:th 8
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2, b3 being auxiliary(iii) Relation of the carrier of b1,the carrier of b1 holds
b2 /\ b3 is auxiliary(iii) Relation of the carrier of b1,the carrier of b1;

:: WAYBEL_4:th 9
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2, b3 being auxiliary(iv) Relation of the carrier of b1,the carrier of b1 holds
b2 /\ b3 is auxiliary(iv) Relation of the carrier of b1,the carrier of b1;

:: WAYBEL_4:th 10
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2, b3 being auxiliary Relation of the carrier of b1,the carrier of b1 holds
b2 /\ b3 is auxiliary Relation of the carrier of b1,the carrier of b1;

:: WAYBEL_4:th 11
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2 being non empty Element of bool the carrier of InclPoset Aux b1 holds
   meet b2 is auxiliary Relation of the carrier of b1,the carrier of b1;

:: WAYBEL_4:funcreg 9
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  cluster InclPoset Aux a1 -> strict with_infima;
end;

:: WAYBEL_4:funcreg 10
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  cluster InclPoset Aux a1 -> strict complete;
end;

:: WAYBEL_4:funcnot 5 => WAYBEL_4:func 5
definition
  let a1 be non empty RelStr;
  let a2 be Element of the carrier of a1;
  let a3 be Relation of the carrier of a1,the carrier of a1;
  func A3 -below A2 -> Element of bool the carrier of a1 equals
    {b1 where b1 is Element of the carrier of a1: [b1,a2] in a3};
end;

:: WAYBEL_4:def 11
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1
for b3 being Relation of the carrier of b1,the carrier of b1 holds
   b3 -below b2 = {b4 where b4 is Element of the carrier of b1: [b4,b2] in b3};

:: WAYBEL_4:funcnot 6 => WAYBEL_4:func 6
definition
  let a1 be non empty RelStr;
  let a2 be Element of the carrier of a1;
  let a3 be Relation of the carrier of a1,the carrier of a1;
  func A3 -above A2 -> Element of bool the carrier of a1 equals
    {b1 where b1 is Element of the carrier of a1: [a2,b1] in a3};
end;

:: WAYBEL_4:def 12
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1
for b3 being Relation of the carrier of b1,the carrier of b1 holds
   b3 -above b2 = {b4 where b4 is Element of the carrier of b1: [b2,b4] in b3};

:: WAYBEL_4:th 12
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2 being Element of the carrier of b1
for b3 being auxiliary(i) Relation of the carrier of b1,the carrier of b1 holds
   b3 -below b2 c= downarrow b2;

:: WAYBEL_4:funcreg 11
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  let a2 be Element of the carrier of a1;
  let a3 be auxiliary(iv) Relation of the carrier of a1,the carrier of a1;
  cluster a3 -below a2 -> non empty;
end;

:: WAYBEL_4:funcreg 12
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  let a2 be Element of the carrier of a1;
  let a3 be auxiliary(ii) Relation of the carrier of a1,the carrier of a1;
  cluster a3 -below a2 -> lower;
end;

:: WAYBEL_4:funcreg 13
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  let a2 be Element of the carrier of a1;
  let a3 be auxiliary(iii) Relation of the carrier of a1,the carrier of a1;
  cluster a3 -below a2 -> directed;
end;

:: WAYBEL_4:funcnot 7 => WAYBEL_4:func 7
definition
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  let a2 be auxiliary(ii) auxiliary(iii) auxiliary(iv) Relation of the carrier of a1,the carrier of a1;
  func A2 -below -> 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 = a2 -below b1;
end;

:: WAYBEL_4:def 13
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2 being auxiliary(ii) auxiliary(iii) auxiliary(iv) Relation of the carrier of b1,the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of InclPoset Ids b1 holds
      b3 = b2 -below
   iff
      for b4 being Element of the carrier of b1 holds
         b3 . b4 = b2 -below b4;

:: WAYBEL_4:th 13
theorem
for b1 being non empty RelStr
for b2 being Relation of the carrier of b1,the carrier of b1
for b3 being set
for b4 being Element of the carrier of b1 holds
      b3 in b2 -below b4
   iff
      [b3,b4] in b2;

:: WAYBEL_4:th 14
theorem
for b1 being set
for b2 being reflexive transitive antisymmetric with_suprema RelStr
for b3 being Relation of the carrier of b2,the carrier of b2
for b4 being Element of the carrier of b2 holds
      b1 in b3 -above b4
   iff
      [b4,b1] in b3;

:: WAYBEL_4:th 15
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2 being auxiliary(i) Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
      st b2 = the InternalRel of b1
   holds b2 -below b3 = downarrow b3;

:: WAYBEL_4:funcnot 8 => WAYBEL_4:func 8
definition
  let a1 be non empty reflexive transitive antisymmetric RelStr;
  func MonSet A1 -> strict RelStr means
    for b1 being set holds
       (b1 in the carrier of it implies ex b2 being Function-like quasi_total Relation of the carrier of a1,the carrier of InclPoset Ids a1 st
           b1 = b2 &
            b2 is monotone(a1, InclPoset Ids a1) &
            (for b3 being Element of the carrier of a1 holds
               b2 . b3 c= downarrow b3)) &
        (for b2 being Function-like quasi_total Relation of the carrier of a1,the carrier of InclPoset Ids a1
              st b1 = b2 & b2 is monotone(a1, InclPoset Ids a1)
           holds ex b3 being Element of the carrier of a1 st
              not b2 . b3 c= downarrow b3 or b1 in the carrier of it) &
        (for b2, b3 being set holds
           [b2,b3] in the InternalRel of it
        iff
           ex b4, b5 being Function-like quasi_total Relation of the carrier of a1,the carrier of InclPoset Ids a1 st
              b2 = b4 & b3 = b5 & b2 in the carrier of it & b3 in the carrier of it & b4 <= b5);
end;

:: WAYBEL_4:def 14
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being strict RelStr holds
      b2 = MonSet b1
   iff
      for b3 being set holds
         (b3 in the carrier of b2 implies ex b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of InclPoset Ids b1 st
             b3 = b4 &
              b4 is monotone(b1, InclPoset Ids b1) &
              (for b5 being Element of the carrier of b1 holds
                 b4 . b5 c= downarrow b5)) &
          (for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of InclPoset Ids b1
                st b3 = b4 & b4 is monotone(b1, InclPoset Ids b1)
             holds ex b5 being Element of the carrier of b1 st
                not b4 . b5 c= downarrow b5 or b3 in the carrier of b2) &
          (for b4, b5 being set holds
             [b4,b5] in the InternalRel of b2
          iff
             ex b6, b7 being Function-like quasi_total Relation of the carrier of b1,the carrier of InclPoset Ids b1 st
                b4 = b6 & b5 = b7 & b4 in the carrier of b2 & b5 in the carrier of b2 & b6 <= b7);

:: WAYBEL_4:th 16
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr holds
   MonSet b1 is full SubRelStr of (InclPoset Ids b1) |^ the carrier of b1;

:: WAYBEL_4:th 17
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2 being auxiliary(ii) Relation of the carrier of b1,the carrier of b1
for b3, b4 being Element of the carrier of b1
      st b3 <= b4
   holds b2 -below b3 c= b2 -below b4;

:: WAYBEL_4:funcreg 14
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  let a2 be auxiliary(ii) auxiliary(iii) auxiliary(iv) Relation of the carrier of a1,the carrier of a1;
  cluster a2 -below -> Function-like quasi_total monotone;
end;

:: WAYBEL_4:th 18
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2 being auxiliary Relation of the carrier of b1,the carrier of b1 holds
   b2 -below in the carrier of MonSet b1;

:: WAYBEL_4:funcreg 15
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  cluster MonSet a1 -> non empty strict;
end;

:: WAYBEL_4:th 19
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr holds
   IdsMap b1 in the carrier of MonSet b1;

:: WAYBEL_4:th 20
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2 being auxiliary Relation of the carrier of b1,the carrier of b1 holds
   b2 -below <= IdsMap b1;

:: WAYBEL_4:th 21
theorem
for b1 being non empty reflexive transitive antisymmetric lower-bounded RelStr
for b2 being non empty directed lower Element of bool the carrier of b1 holds
   Bottom b1 in b2;

:: WAYBEL_4:th 22
theorem
for b1 being non empty reflexive transitive antisymmetric upper-bounded RelStr
for b2 being non empty filtered upper Element of bool the carrier of b1 holds
   Top b1 in b2;

:: WAYBEL_4:th 23
theorem
for b1 being non empty reflexive transitive antisymmetric lower-bounded RelStr holds
   downarrow Bottom b1 = {Bottom b1};

:: WAYBEL_4:th 24
theorem
for b1 being non empty reflexive transitive antisymmetric upper-bounded RelStr holds
   uparrow Top b1 = {Top b1};

:: WAYBEL_4:th 25
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr holds
   (the carrier of b1) --> {Bottom b1} is Function-like quasi_total Relation of the carrier of b1,the carrier of InclPoset Ids b1;

:: WAYBEL_4:th 26
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr holds
   (the carrier of b1) --> {Bottom b1} in the carrier of MonSet b1;

:: WAYBEL_4:th 27
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2 being auxiliary Relation of the carrier of b1,the carrier of b1 holds
   [(the carrier of b1) --> {Bottom b1},b2 -below] in the InternalRel of MonSet b1;

:: WAYBEL_4:funcreg 16
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  cluster MonSet a1 -> strict upper-bounded;
end;

:: WAYBEL_4:funcnot 9 => WAYBEL_4:func 9
definition
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  func Rel2Map A1 -> Function-like quasi_total Relation of the carrier of InclPoset Aux a1,the carrier of MonSet a1 means
    for b1 being auxiliary Relation of the carrier of a1,the carrier of a1 holds
       it . b1 = b1 -below;
end;

:: WAYBEL_4:def 15
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2 being Function-like quasi_total Relation of the carrier of InclPoset Aux b1,the carrier of MonSet b1 holds
      b2 = Rel2Map b1
   iff
      for b3 being auxiliary Relation of the carrier of b1,the carrier of b1 holds
         b2 . b3 = b3 -below;

:: WAYBEL_4:th 28
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2, b3 being auxiliary Relation of the carrier of b1,the carrier of b1
      st b2 c= b3
   holds b2 -below <= b3 -below;

:: WAYBEL_4:th 29
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2 being Element of the carrier of b1
for b3, b4 being Relation of the carrier of b1,the carrier of b1
      st b3 c= b4
   holds b3 -below b2 c= b4 -below b2;

:: WAYBEL_4:funcreg 17
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  cluster Rel2Map a1 -> Function-like quasi_total monotone;
end;

:: WAYBEL_4:funcnot 10 => WAYBEL_4:func 10
definition
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  func Map2Rel A1 -> Function-like quasi_total Relation of the carrier of MonSet a1,the carrier of InclPoset Aux a1 means
    for b1 being set
          st b1 in the carrier of MonSet a1
       holds ex b2 being auxiliary Relation of the carrier of a1,the carrier of a1 st
          b2 = it . b1 &
           (for b3, b4 being set holds
              [b3,b4] in b2
           iff
              ex b5, b6 being Element of the carrier of a1 st
                 ex b7 being Function-like quasi_total Relation of the carrier of a1,the carrier of InclPoset Ids a1 st
                    b5 = b3 & b6 = b4 & b7 = b1 & b5 in b7 . b6);
end;

:: WAYBEL_4:def 16
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2 being Function-like quasi_total Relation of the carrier of MonSet b1,the carrier of InclPoset Aux b1 holds
      b2 = Map2Rel b1
   iff
      for b3 being set
            st b3 in the carrier of MonSet b1
         holds ex b4 being auxiliary Relation of the carrier of b1,the carrier of b1 st
            b4 = b2 . b3 &
             (for b5, b6 being set holds
                [b5,b6] in b4
             iff
                ex b7, b8 being Element of the carrier of b1 st
                   ex b9 being Function-like quasi_total Relation of the carrier of b1,the carrier of InclPoset Ids b1 st
                      b7 = b5 & b8 = b6 & b9 = b3 & b7 in b9 . b8);

:: WAYBEL_4:funcreg 18
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  cluster Map2Rel a1 -> Function-like quasi_total monotone;
end;

:: WAYBEL_4:th 30
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr holds
   (Map2Rel b1) * Rel2Map b1 = id dom Rel2Map b1;

:: WAYBEL_4:th 31
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr holds
   (Rel2Map b1) * Map2Rel b1 = id the carrier of MonSet b1;

:: WAYBEL_4:funcreg 19
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  cluster Rel2Map a1 -> Function-like one-to-one quasi_total;
end;

:: WAYBEL_4:th 32
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr holds
   (Rel2Map b1) " = Map2Rel b1;

:: WAYBEL_4:th 33
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr holds
   Rel2Map b1 is isomorphic(InclPoset Aux b1, MonSet b1);

:: WAYBEL_4:th 34
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Element of the carrier of b1 holds
   meet {b3 where b3 is non empty directed lower Element of bool the carrier of b1: b2 <= "\/"(b3,b1)} = waybelow b2;

:: WAYBEL_4:funcnot 11 => WAYBEL_4:func 11
definition
  let a1 be reflexive transitive antisymmetric with_infima RelStr;
  let a2 be non empty directed lower Element of bool the carrier of a1;
  func DownMap A2 -> 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
       (b1 <= "\/"(a2,a1) implies it . b1 = (downarrow b1) /\ a2) &
        (b1 <= "\/"(a2,a1) or it . b1 = downarrow b1);
end;

:: WAYBEL_4:def 17
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being non empty directed lower Element of bool the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of InclPoset Ids b1 holds
      b3 = DownMap b2
   iff
      for b4 being Element of the carrier of b1 holds
         (b4 <= "\/"(b2,b1) implies b3 . b4 = (downarrow b4) /\ b2) &
          (b4 <= "\/"(b2,b1) or b3 . b4 = downarrow b4);

:: WAYBEL_4:th 35
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being non empty directed lower Element of bool the carrier of b1 holds
   DownMap b2 in the carrier of MonSet b1;

:: WAYBEL_4:th 36
theorem
for b1 being reflexive antisymmetric with_infima RelStr
for b2 being Element of the carrier of b1
for b3 being non empty lower Element of bool the carrier of b1 holds
   {b2} "/\" b3 = (downarrow b2) /\ b3;

:: WAYBEL_4:attrnot 6 => WAYBEL_4:attr 6
definition
  let a1 be non empty RelStr;
  let a2 be Relation of the carrier of a1,the carrier of a1;
  attr a2 is approximating means
    for b1 being Element of the carrier of a1 holds
       b1 = "\/"(a2 -below b1,a1);
end;

:: WAYBEL_4:dfs 17
definiens
  let a1 be non empty RelStr;
  let a2 be Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is approximating
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       b1 = "\/"(a2 -below b1,a1);

:: WAYBEL_4:def 18
theorem
for b1 being non empty RelStr
for b2 being Relation of the carrier of b1,the carrier of b1 holds
      b2 is approximating(b1)
   iff
      for b3 being Element of the carrier of b1 holds
         b3 = "\/"(b2 -below b3,b1);

:: WAYBEL_4:attrnot 7 => WAYBEL_4:attr 7
definition
  let a1 be non empty reflexive transitive antisymmetric RelStr;
  let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of InclPoset Ids a1;
  attr a2 is approximating means
    for b1 being Element of the carrier of a1 holds
       ex b2 being Element of bool the carrier of a1 st
          b2 = a2 . b1 & b1 = "\/"(b2,a1);
end;

:: WAYBEL_4:dfs 18
definiens
  let a1 be non empty reflexive transitive antisymmetric RelStr;
  let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of InclPoset Ids a1;
To prove
     a2 is approximating
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       ex b2 being Element of bool the carrier of a1 st
          b2 = a2 . b1 & b1 = "\/"(b2,a1);

:: WAYBEL_4:def 19
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 is approximating(b1)
   iff
      for b3 being Element of the carrier of b1 holds
         ex b4 being Element of bool the carrier of b1 st
            b4 = b2 . b3 & b3 = "\/"(b4,b1);

:: WAYBEL_4:th 37
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_infima meet-continuous RelStr
for b2 being non empty directed lower Element of bool the carrier of b1 holds
   DownMap b2 is approximating(b1);

:: WAYBEL_4:th 38
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr holds
   b1 is meet-continuous;

:: WAYBEL_4:condreg 4
registration
  cluster reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous -> meet-continuous (RelStr);
end;

:: WAYBEL_4:th 39
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr
for b2 being non empty directed lower Element of bool the carrier of b1 holds
   DownMap b2 is approximating(b1);

:: WAYBEL_4:funcreg 20
registration
  let a1 be non empty reflexive antisymmetric RelStr;
  cluster a1 -waybelow -> auxiliary(i);
end;

:: WAYBEL_4:funcreg 21
registration
  let a1 be non empty reflexive transitive RelStr;
  cluster a1 -waybelow -> auxiliary(ii);
end;

:: WAYBEL_4:funcreg 22
registration
  let a1 be reflexive transitive antisymmetric with_suprema RelStr;
  cluster a1 -waybelow -> auxiliary(iii);
end;

:: WAYBEL_4:funcreg 23
registration
  let a1 be non empty reflexive transitive antisymmetric /\-complete RelStr;
  cluster a1 -waybelow -> auxiliary(iii);
end;

:: WAYBEL_4:funcreg 24
registration
  let a1 be non empty reflexive antisymmetric lower-bounded RelStr;
  cluster a1 -waybelow -> auxiliary(iv);
end;

:: WAYBEL_4:th 40
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Element of the carrier of b1 holds
   b1 -waybelow -below b2 = waybelow b2;

:: WAYBEL_4:th 41
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr holds
   IntRel b1 is approximating(b1);

:: WAYBEL_4:funcreg 25
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr;
  cluster a1 -waybelow -> approximating;
end;

:: WAYBEL_4:exreg 2
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  cluster Relation-like auxiliary approximating Relation of the carrier of a1,the carrier of a1;
end;

:: WAYBEL_4:funcnot 12 => WAYBEL_4:func 12
definition
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  func App A1 -> set means
    for b1 being set holds
          b1 in it
       iff
          b1 is auxiliary approximating Relation of the carrier of a1,the carrier of a1;
end;

:: WAYBEL_4:def 20
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being set holds
      b2 = App b1
   iff
      for b3 being set holds
            b3 in b2
         iff
            b3 is auxiliary approximating Relation of the carrier of b1,the carrier of b1;

:: WAYBEL_4:funcreg 26
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  cluster App a1 -> non empty;
end;

:: WAYBEL_4:th 42
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 InclPoset Ids b1
      st b2 is approximating(b1) & b2 in the carrier of MonSet b1
   holds ex b3 being auxiliary approximating Relation of the carrier of b1,the carrier of b1 st
      b3 = (Map2Rel b1) . b2;

:: WAYBEL_4:th 43
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Element of the carrier of b1 holds
   meet {(DownMap b3) . b2 where b3 is non empty directed lower Element of bool the carrier of b1: TRUE} = waybelow b2;

:: WAYBEL_4:th 44
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima meet-continuous RelStr
for b2 being Element of the carrier of b1 holds
   meet {b3 -below b2 where b3 is auxiliary Relation of the carrier of b1,the carrier of b1: b3 in App b1} = waybelow b2;

:: WAYBEL_4:th 45
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
      b1 is continuous
   iff
      for b2 being auxiliary approximating Relation of the carrier of b1,the carrier of b1 holds
         b1 -waybelow c= b2 & b1 -waybelow is approximating(b1);

:: WAYBEL_4:th 46
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
      b1 is continuous
   iff
      b1 is meet-continuous &
       (ex b2 being auxiliary approximating Relation of the carrier of b1,the carrier of b1 st
          for b3 being auxiliary approximating Relation of the carrier of b1,the carrier of b1 holds
             b2 c= b3);

:: WAYBEL_4:attrnot 8 => WAYBEL_4:attr 8
definition
  let a1 be RelStr;
  let a2 be Relation of the carrier of a1,the carrier of a1;
  attr a2 is satisfying_SI means
    for b1, b2 being Element of the carrier of a1
          st [b1,b2] in a2 & b1 <> b2
       holds ex b3 being Element of the carrier of a1 st
          [b1,b3] in a2 & [b3,b2] in a2 & b1 <> b3;
end;

:: WAYBEL_4:dfs 20
definiens
  let a1 be RelStr;
  let a2 be Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is satisfying_SI
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1
          st [b1,b2] in a2 & b1 <> b2
       holds ex b3 being Element of the carrier of a1 st
          [b1,b3] in a2 & [b3,b2] in a2 & b1 <> b3;

:: WAYBEL_4:def 21
theorem
for b1 being RelStr
for b2 being Relation of the carrier of b1,the carrier of b1 holds
      b2 is satisfying_SI(b1)
   iff
      for b3, b4 being Element of the carrier of b1
            st [b3,b4] in b2 & b3 <> b4
         holds ex b5 being Element of the carrier of b1 st
            [b3,b5] in b2 & [b5,b4] in b2 & b3 <> b5;

:: WAYBEL_4:prednot 1 => WAYBEL_4:attr 8
notation
  let a1 be RelStr;
  let a2 be Relation of the carrier of a1,the carrier of a1;
  synonym a2 satisfies_SI for satisfying_SI;
end;

:: WAYBEL_4:attrnot 9 => WAYBEL_4:attr 9
definition
  let a1 be RelStr;
  let a2 be Relation of the carrier of a1,the carrier of a1;
  attr a2 is satisfying_INT means
    for b1, b2 being Element of the carrier of a1
          st [b1,b2] in a2
       holds ex b3 being Element of the carrier of a1 st
          [b1,b3] in a2 & [b3,b2] in a2;
end;

:: WAYBEL_4:dfs 21
definiens
  let a1 be RelStr;
  let a2 be Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is satisfying_INT
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1
          st [b1,b2] in a2
       holds ex b3 being Element of the carrier of a1 st
          [b1,b3] in a2 & [b3,b2] in a2;

:: WAYBEL_4:def 22
theorem
for b1 being RelStr
for b2 being Relation of the carrier of b1,the carrier of b1 holds
      b2 is satisfying_INT(b1)
   iff
      for b3, b4 being Element of the carrier of b1
            st [b3,b4] in b2
         holds ex b5 being Element of the carrier of b1 st
            [b3,b5] in b2 & [b5,b4] in b2;

:: WAYBEL_4:prednot 2 => WAYBEL_4:attr 9
notation
  let a1 be RelStr;
  let a2 be Relation of the carrier of a1,the carrier of a1;
  synonym a2 satisfies_INT for satisfying_INT;
end;

:: WAYBEL_4:th 48
theorem
for b1 being RelStr
for b2 being Relation of the carrier of b1,the carrier of b1
      st b2 is satisfying_SI(b1)
   holds b2 is satisfying_INT(b1);

:: WAYBEL_4:condreg 5
registration
  let a1 be non empty RelStr;
  cluster satisfying_SI -> satisfying_INT (Relation of the carrier of a1,the carrier of a1);
end;

:: WAYBEL_4:th 49
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2, b3 being Element of the carrier of b1
for b4 being approximating Relation of the carrier of b1,the carrier of b1
      st not b2 <= b3
   holds ex b5 being Element of the carrier of b1 st
      [b5,b2] in b4 & not b5 <= b3;

:: WAYBEL_4:th 50
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2, b3 being Element of the carrier of b1
for b4 being auxiliary(i) auxiliary(iii) approximating Relation of the carrier of b1,the carrier of b1
      st [b2,b3] in b4 & b2 <> b3
   holds ex b5 being Element of the carrier of b1 st
      b2 <= b5 & [b5,b3] in b4 & b2 <> b5;

:: WAYBEL_4:th 51
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2, b3 being Element of the carrier of b1
for b4 being auxiliary approximating Relation of the carrier of b1,the carrier of b1
      st b2 is_way_below b3 & b2 <> b3
   holds ex b5 being Element of the carrier of b1 st
      [b2,b5] in b4 & [b5,b3] in b4 & b2 <> b5;

:: WAYBEL_4:th 52
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr holds
   b1 -waybelow is satisfying_SI(b1);

:: WAYBEL_4:funcreg 27
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr;
  cluster a1 -waybelow -> satisfying_SI;
end;

:: WAYBEL_4:th 53
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr
for b2, b3 being Element of the carrier of b1
      st b2 is_way_below b3
   holds ex b4 being Element of the carrier of b1 st
      b2 is_way_below b4 & b4 is_way_below b3;

:: WAYBEL_4:th 54
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr
for b2, b3 being Element of the carrier of b1 holds
   b2 is_way_below b3
iff
   for b4 being non empty directed Element of bool the carrier of b1
         st b3 <= "\/"(b4,b1)
      holds ex b5 being Element of the carrier of b1 st
         b5 in b4 & b2 is_way_below b5;

:: WAYBEL_4:prednot 3 => WAYBEL_4:pred 1
definition
  let a1 be RelStr;
  let a2 be Element of bool the carrier of a1;
  let a3 be Relation of the carrier of a1,the carrier of a1;
  pred A2 is_directed_wrt A3 means
    for b1, b2 being Element of the carrier of a1
          st b1 in a2 & b2 in a2
       holds ex b3 being Element of the carrier of a1 st
          b3 in a2 & [b1,b3] in a3 & [b2,b3] in a3;
end;

:: WAYBEL_4:dfs 22
definiens
  let a1 be RelStr;
  let a2 be Element of bool the carrier of a1;
  let a3 be Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is_directed_wrt a3
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1
          st b1 in a2 & b2 in a2
       holds ex b3 being Element of the carrier of a1 st
          b3 in a2 & [b1,b3] in a3 & [b2,b3] in a3;

:: WAYBEL_4:def 23
theorem
for b1 being RelStr
for b2 being Element of bool the carrier of b1
for b3 being Relation of the carrier of b1,the carrier of b1 holds
      b2 is_directed_wrt b3
   iff
      for b4, b5 being Element of the carrier of b1
            st b4 in b2 & b5 in b2
         holds ex b6 being Element of the carrier of b1 st
            b6 in b2 & [b4,b6] in b3 & [b5,b6] in b3;

:: WAYBEL_4:th 55
theorem
for b1 being RelStr
for b2 being Element of bool the carrier of b1
      st b2 is_directed_wrt the InternalRel of b1
   holds b2 is directed(b1);

:: WAYBEL_4:prednot 4 => WAYBEL_4:pred 2
definition
  let a1, a2 be set;
  let a3 be Relation-like set;
  pred A2 is_maximal_wrt A1,A3 means
    a2 in a1 &
     (for b1 being set
           st b1 in a1 & b1 <> a2
        holds not [a2,b1] in a3);
end;

:: WAYBEL_4:dfs 23
definiens
  let a1, a2 be set;
  let a3 be Relation-like set;
To prove
     a2 is_maximal_wrt a1,a3
it is sufficient to prove
  thus a2 in a1 &
     (for b1 being set
           st b1 in a1 & b1 <> a2
        holds not [a2,b1] in a3);

:: WAYBEL_4:def 24
theorem
for b1, b2 being set
for b3 being Relation-like set holds
      b2 is_maximal_wrt b1,b3
   iff
      b2 in b1 &
       (for b4 being set
             st b4 in b1 & b4 <> b2
          holds not [b2,b4] in b3);

:: WAYBEL_4:prednot 5 => WAYBEL_4:pred 3
definition
  let a1 be RelStr;
  let a2 be set;
  let a3 be Element of the carrier of a1;
  pred A3 is_maximal_in A2 means
    a3 is_maximal_wrt a2,the InternalRel of a1;
end;

:: WAYBEL_4:dfs 24
definiens
  let a1 be RelStr;
  let a2 be set;
  let a3 be Element of the carrier of a1;
To prove
     a3 is_maximal_in a2
it is sufficient to prove
  thus a3 is_maximal_wrt a2,the InternalRel of a1;

:: WAYBEL_4:def 25
theorem
for b1 being RelStr
for b2 being set
for b3 being Element of the carrier of b1 holds
      b3 is_maximal_in b2
   iff
      b3 is_maximal_wrt b2,the InternalRel of b1;

:: WAYBEL_4:th 56
theorem
for b1 being RelStr
for b2 being set
for b3 being Element of the carrier of b1 holds
      b3 is_maximal_in b2
   iff
      b3 in b2 &
       (for b4 being Element of the carrier of b1
             st b4 in b2
          holds not b3 < b4);

:: WAYBEL_4:prednot 6 => WAYBEL_4:pred 4
definition
  let a1, a2 be set;
  let a3 be Relation-like set;
  pred A2 is_minimal_wrt A1,A3 means
    a2 in a1 &
     (for b1 being set
           st b1 in a1 & b1 <> a2
        holds not [b1,a2] in a3);
end;

:: WAYBEL_4:dfs 25
definiens
  let a1, a2 be set;
  let a3 be Relation-like set;
To prove
     a2 is_minimal_wrt a1,a3
it is sufficient to prove
  thus a2 in a1 &
     (for b1 being set
           st b1 in a1 & b1 <> a2
        holds not [b1,a2] in a3);

:: WAYBEL_4:def 26
theorem
for b1, b2 being set
for b3 being Relation-like set holds
      b2 is_minimal_wrt b1,b3
   iff
      b2 in b1 &
       (for b4 being set
             st b4 in b1 & b4 <> b2
          holds not [b4,b2] in b3);

:: WAYBEL_4:prednot 7 => WAYBEL_4:pred 5
definition
  let a1 be RelStr;
  let a2 be set;
  let a3 be Element of the carrier of a1;
  pred A3 is_minimal_in A2 means
    a3 is_minimal_wrt a2,the InternalRel of a1;
end;

:: WAYBEL_4:dfs 26
definiens
  let a1 be RelStr;
  let a2 be set;
  let a3 be Element of the carrier of a1;
To prove
     a3 is_minimal_in a2
it is sufficient to prove
  thus a3 is_minimal_wrt a2,the InternalRel of a1;

:: WAYBEL_4:def 27
theorem
for b1 being RelStr
for b2 being set
for b3 being Element of the carrier of b1 holds
      b3 is_minimal_in b2
   iff
      b3 is_minimal_wrt b2,the InternalRel of b1;

:: WAYBEL_4:th 57
theorem
for b1 being RelStr
for b2 being set
for b3 being Element of the carrier of b1 holds
      b3 is_minimal_in b2
   iff
      b3 in b2 &
       (for b4 being Element of the carrier of b1
             st b4 in b2
          holds not b4 < b3);

:: WAYBEL_4:th 58
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Relation of the carrier of b1,the carrier of b1
   st b2 is satisfying_SI(b1)
for b3 being Element of the carrier of b1
      st ex b4 being Element of the carrier of b1 st
           b4 is_maximal_wrt b2 -below b3,b2
   holds [b3,b3] in b2;

:: WAYBEL_4:th 59
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Relation of the carrier of b1,the carrier of b1
      st for b3 being Element of the carrier of b1
              st ex b4 being Element of the carrier of b1 st
                   b4 is_maximal_wrt b2 -below b3,b2
           holds [b3,b3] in b2
   holds b2 is satisfying_SI(b1);

:: WAYBEL_4:th 60
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being auxiliary(ii) auxiliary(iii) Relation of the carrier of b1,the carrier of b1
   st b2 is satisfying_INT(b1)
for b3 being Element of the carrier of b1 holds
   b2 -below b3 is_directed_wrt b2;

:: WAYBEL_4:th 61
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Relation of the carrier of b1,the carrier of b1
      st for b3 being Element of the carrier of b1 holds
           b2 -below b3 is_directed_wrt b2
   holds b2 is satisfying_INT(b1);

:: WAYBEL_4:th 62
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being auxiliary(i) auxiliary(ii) auxiliary(iii) approximating Relation of the carrier of b1,the carrier of b1
      st b2 is satisfying_INT(b1)
   holds b2 is satisfying_SI(b1);

:: WAYBEL_4:condreg 6
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  cluster auxiliary approximating satisfying_INT -> satisfying_SI (Relation of the carrier of a1,the carrier of a1);
end;