Article WAYBEL22, MML version 4.99.1005

:: WAYBEL22:th 1
theorem
for b1 being reflexive transitive antisymmetric upper-bounded with_infima RelStr
for b2 being non empty directed Element of bool the carrier of InclPoset Filt b1 holds
   "\/"(b2,InclPoset Filt b1) = union b2;

:: WAYBEL22:th 2
theorem
for b1, b2, b3 being non empty reflexive transitive antisymmetric complete RelStr
for b4 being CLHomomorphism of b1,b2
for b5 being CLHomomorphism of b2,b3 holds
   b5 * b4 is CLHomomorphism of b1,b3;

:: WAYBEL22:th 3
theorem
for b1 being non empty RelStr holds
   id b1 is infs-preserving(b1, b1);

:: WAYBEL22:th 4
theorem
for b1 being non empty RelStr holds
   id b1 is directed-sups-preserving(b1, b1);

:: WAYBEL22:th 5
theorem
for b1 being non empty reflexive transitive antisymmetric complete RelStr holds
   id b1 is CLHomomorphism of b1,b1;

:: WAYBEL22:th 6
theorem
for b1 being non empty reflexive transitive antisymmetric upper-bounded with_infima RelStr holds
   InclPoset Filt b1 is non empty full infs-inheriting directed-sups-inheriting SubRelStr of BoolePoset the carrier of b1;

:: WAYBEL22:funcreg 1
registration
  let a1 be non empty reflexive transitive antisymmetric upper-bounded with_infima RelStr;
  cluster InclPoset Filt a1 -> strict continuous;
end;

:: WAYBEL22:condreg 1
registration
  let a1 be non empty reflexive transitive antisymmetric upper-bounded RelStr;
  cluster -> non empty (Element of the carrier of InclPoset Filt a1);
end;

:: WAYBEL22:prednot 1 => WAYBEL22:pred 1
definition
  let a1 be non empty reflexive transitive antisymmetric complete continuous RelStr;
  let a2 be set;
  pred A2 is_FreeGen_set_of A1 means
    for b1 being non empty reflexive transitive antisymmetric complete continuous RelStr
    for b2 being Function-like quasi_total Relation of a2,the carrier of b1 holds
       ex b3 being CLHomomorphism of a1,b1 st
          b3 | a2 = b2 &
           (for b4 being CLHomomorphism of a1,b1
                 st b4 | a2 = b2
              holds b4 = b3);
end;

:: WAYBEL22:dfs 1
definiens
  let a1 be non empty reflexive transitive antisymmetric complete continuous RelStr;
  let a2 be set;
To prove
     a2 is_FreeGen_set_of a1
it is sufficient to prove
  thus for b1 being non empty reflexive transitive antisymmetric complete continuous RelStr
    for b2 being Function-like quasi_total Relation of a2,the carrier of b1 holds
       ex b3 being CLHomomorphism of a1,b1 st
          b3 | a2 = b2 &
           (for b4 being CLHomomorphism of a1,b1
                 st b4 | a2 = b2
              holds b4 = b3);

:: WAYBEL22:def 1
theorem
for b1 being non empty reflexive transitive antisymmetric complete continuous RelStr
for b2 being set holds
      b2 is_FreeGen_set_of b1
   iff
      for b3 being non empty reflexive transitive antisymmetric complete continuous RelStr
      for b4 being Function-like quasi_total Relation of b2,the carrier of b3 holds
         ex b5 being CLHomomorphism of b1,b3 st
            b5 | b2 = b4 &
             (for b6 being CLHomomorphism of b1,b3
                   st b6 | b2 = b4
                holds b6 = b5);

:: WAYBEL22:th 7
theorem
for b1 being non empty reflexive transitive antisymmetric complete continuous RelStr
for b2 being set
      st b2 is_FreeGen_set_of b1
   holds b2 is Element of bool the carrier of b1;

:: WAYBEL22:th 8
theorem
for b1 being non empty reflexive transitive antisymmetric complete continuous RelStr
for b2 being set
   st b2 is_FreeGen_set_of b1
for b3 being CLHomomorphism of b1,b1
      st b3 | b2 = id b2
   holds b3 = id b1;

:: WAYBEL22:funcnot 1 => WAYBEL22:func 1
definition
  let a1 be set;
  func FixedUltraFilters A1 -> Element of bool bool the carrier of BoolePoset a1 equals
    {uparrow b1 where b1 is Element of the carrier of BoolePoset a1: ex b2 being Element of a1 st
       b1 = {b2}};
end;

:: WAYBEL22:def 2
theorem
for b1 being set holds
   FixedUltraFilters b1 = {uparrow b2 where b2 is Element of the carrier of BoolePoset b1: ex b3 being Element of b1 st
      b2 = {b3}};

:: WAYBEL22:th 9
theorem
for b1 being set holds
   FixedUltraFilters b1 c= Filt BoolePoset b1;

:: WAYBEL22:th 10
theorem
for b1 being set holds
   Card FixedUltraFilters b1 = Card b1;

:: WAYBEL22:th 11
theorem
for b1 being set
for b2 being non empty filtered upper Element of bool the carrier of BoolePoset b1 holds
   b2 = "\/"({"/\"({uparrow b4 where b4 is Element of the carrier of BoolePoset b1: ex b5 being Element of b1 st
      b4 = {b5} & b5 in b3},InclPoset Filt BoolePoset b1) where b3 is Element of bool b1: b3 in b2},InclPoset Filt BoolePoset b1);

:: WAYBEL22:funcnot 2 => WAYBEL22:func 2
definition
  let a1 be set;
  let a2 be non empty reflexive transitive antisymmetric complete continuous RelStr;
  let a3 be Function-like quasi_total Relation of FixedUltraFilters a1,the carrier of a2;
  func A3 -extension_to_hom -> Function-like quasi_total Relation of the carrier of InclPoset Filt BoolePoset a1,the carrier of a2 means
    for b1 being Element of the carrier of InclPoset Filt BoolePoset a1 holds
       it . b1 = "\/"({"/\"({a3 . uparrow b3 where b3 is Element of the carrier of BoolePoset a1: ex b4 being Element of a1 st
          b3 = {b4} & b4 in b2},a2) where b2 is Element of bool a1: b2 in b1},a2);
end;

:: WAYBEL22:def 3
theorem
for b1 being set
for b2 being non empty reflexive transitive antisymmetric complete continuous RelStr
for b3 being Function-like quasi_total Relation of FixedUltraFilters b1,the carrier of b2
for b4 being Function-like quasi_total Relation of the carrier of InclPoset Filt BoolePoset b1,the carrier of b2 holds
      b4 = b3 -extension_to_hom
   iff
      for b5 being Element of the carrier of InclPoset Filt BoolePoset b1 holds
         b4 . b5 = "\/"({"/\"({b3 . uparrow b7 where b7 is Element of the carrier of BoolePoset b1: ex b8 being Element of b1 st
            b7 = {b8} & b8 in b6},b2) where b6 is Element of bool b1: b6 in b5},b2);

:: WAYBEL22:th 12
theorem
for b1 being set
for b2 being non empty reflexive transitive antisymmetric complete continuous RelStr
for b3 being Function-like quasi_total Relation of FixedUltraFilters b1,the carrier of b2 holds
   b3 -extension_to_hom is monotone(InclPoset Filt BoolePoset b1, b2);

:: WAYBEL22:th 13
theorem
for b1 being set
for b2 being non empty reflexive transitive antisymmetric complete continuous RelStr
for b3 being Function-like quasi_total Relation of FixedUltraFilters b1,the carrier of b2 holds
   b3 -extension_to_hom . Top InclPoset Filt BoolePoset b1 = Top b2;

:: WAYBEL22:funcreg 2
registration
  let a1 be set;
  let a2 be non empty reflexive transitive antisymmetric complete continuous RelStr;
  let a3 be Function-like quasi_total Relation of FixedUltraFilters a1,the carrier of a2;
  cluster a3 -extension_to_hom -> Function-like quasi_total directed-sups-preserving;
end;

:: WAYBEL22:funcreg 3
registration
  let a1 be set;
  let a2 be non empty reflexive transitive antisymmetric complete continuous RelStr;
  let a3 be Function-like quasi_total Relation of FixedUltraFilters a1,the carrier of a2;
  cluster a3 -extension_to_hom -> Function-like quasi_total infs-preserving;
end;

:: WAYBEL22:th 14
theorem
for b1 being set
for b2 being non empty reflexive transitive antisymmetric complete continuous RelStr
for b3 being Function-like quasi_total Relation of FixedUltraFilters b1,the carrier of b2 holds
   b3 -extension_to_hom | FixedUltraFilters b1 = b3;

:: WAYBEL22:th 15
theorem
for b1 being set
for b2 being non empty reflexive transitive antisymmetric complete continuous RelStr
for b3 being Function-like quasi_total Relation of FixedUltraFilters b1,the carrier of b2
for b4 being CLHomomorphism of InclPoset Filt BoolePoset b1,b2
      st b4 | FixedUltraFilters b1 = b3
   holds b4 = b3 -extension_to_hom;

:: WAYBEL22:th 16
theorem
for b1 being set holds
   FixedUltraFilters b1 is_FreeGen_set_of InclPoset Filt BoolePoset b1;

:: WAYBEL22:th 17
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr
for b3, b4 being set
      st b3 is_FreeGen_set_of b1 & b4 is_FreeGen_set_of b2 & Card b3 = Card b4
   holds b1,b2 are_isomorphic;

:: WAYBEL22:th 18
theorem
for b1 being set
for b2 being reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr
for b3 being set
      st b3 is_FreeGen_set_of b2 & Card b3 = Card b1
   holds b2,InclPoset Filt BoolePoset b1 are_isomorphic;