Article WAYBEL15, MML version 4.99.1005

:: WAYBEL15:th 1
theorem
for b1 being RelStr
for b2 being full SubRelStr of b1
for b3 being full SubRelStr of b2 holds
   b3 is full SubRelStr of b1;

:: WAYBEL15:th 2
theorem
for b1 being 1-sorted
for b2, b3 being non empty 1-sorted
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 onto(the carrier of b1, the carrier of b2) & b5 is onto(the carrier of b2, the carrier of b3)
   holds b5 * b4 is onto(the carrier of b1, the carrier of b3);

:: WAYBEL15:th 4
theorem
for b1 being set
for b2 being Element of the carrier of BoolePoset b1 holds
   uparrow b2 = {b3 where b3 is Element of bool b1: b2 c= b3};

:: WAYBEL15:th 5
theorem
for b1 being non empty antisymmetric upper-bounded RelStr
for b2 being Element of the carrier of b1
      st Top b1 <= b2
   holds b2 = Top b1;

:: WAYBEL15:th 6
theorem
for b1, b2 being non empty reflexive transitive antisymmetric RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
      st b3 is onto(the carrier of b1, the carrier of b2) & [b3,b4] is Galois(b1, b2)
   holds b2,Image b4 are_isomorphic;

:: WAYBEL15:th 7
theorem
for b1, b2, b3 being non empty reflexive transitive antisymmetric 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
for b6 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b7 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
      st [b4,b6] is Galois(b1, b2) & [b5,b7] is Galois(b2, b3)
   holds [b5 * b4,b6 * b7] is Galois(b1, b3);

:: WAYBEL15:th 8
theorem
for b1, b2 being non empty reflexive transitive antisymmetric RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
      st b4 = b3 " & b3 is isomorphic(b1, b2)
   holds [b3,b4] is Galois(b1, b2) & [b4,b3] is Galois(b2, b1);

:: WAYBEL15:th 9
theorem
for b1 being set holds
   BoolePoset b1 is arithmetic;

:: WAYBEL15:funcreg 1
registration
  let a1 be set;
  cluster BoolePoset a1 -> strict arithmetic;
end;

:: WAYBEL15:th 10
theorem
for b1, b2 being non empty reflexive transitive antisymmetric up-complete RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
   st b3 is isomorphic(b1, b2)
for b4 being Element of the carrier of b1 holds
   b3 .: waybelow b4 = waybelow (b3 . b4);

:: WAYBEL15:th 11
theorem
for b1, b2 being non empty reflexive transitive antisymmetric RelStr
      st b1,b2 are_isomorphic & b1 is continuous
   holds b2 is continuous;

:: WAYBEL15:th 12
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima RelStr
      st b1,b2 are_isomorphic & b1 is lower-bounded & b1 is arithmetic
   holds b2 is arithmetic;

:: WAYBEL15:th 13
theorem
for b1, b2, b3 being non empty reflexive transitive antisymmetric 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 directed-sups-preserving(b1, b2) & b5 is directed-sups-preserving(b2, b3)
   holds b5 * b4 is directed-sups-preserving(b1, b3);

:: WAYBEL15:th 14
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 Image b3 holds
   (inclusion b3) .: b4 = b4;

:: WAYBEL15:th 15
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of the carrier of BoolePoset b1,the carrier of BoolePoset b1
      st b2 is idempotent & b2 is directed-sups-preserving(BoolePoset b1, BoolePoset b1)
   holds inclusion b2 is directed-sups-preserving(Image b2, BoolePoset b1);

:: WAYBEL15:th 16
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr
for b2 being Function-like quasi_total kernel Relation of the carrier of b1,the carrier of b1
      st b2 is directed-sups-preserving(b1, b1)
   holds Image b2 is reflexive transitive antisymmetric with_suprema with_infima continuous RelStr;

:: WAYBEL15:th 17
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr
for b2 being Function-like quasi_total projection Relation of the carrier of b1,the carrier of b1
      st b2 is directed-sups-preserving(b1, b1)
   holds Image b2 is reflexive transitive antisymmetric with_suprema with_infima continuous RelStr;

:: WAYBEL15:th 18
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr holds
      b1 is continuous
   iff
      ex b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded arithmetic RelStr st
         ex b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1 st
            b3 is onto(the carrier of b2, the carrier of b1) & b3 is infs-preserving(b2, b1) & b3 is directed-sups-preserving(b2, b1);

:: WAYBEL15:th 19
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr holds
      b1 is continuous
   iff
      ex b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded algebraic RelStr st
         ex b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1 st
            b3 is onto(the carrier of b2, the carrier of b1) & b3 is infs-preserving(b2, b1) & b3 is directed-sups-preserving(b2, b1);

:: WAYBEL15:th 20
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr holds
   (b1 is continuous implies ex b2 being non empty set st
       ex b3 being Function-like quasi_total projection Relation of the carrier of BoolePoset b2,the carrier of BoolePoset b2 st
          b3 is directed-sups-preserving(BoolePoset b2, BoolePoset b2) & b1,Image b3 are_isomorphic) &
    (for b2 being set
    for b3 being Function-like quasi_total projection Relation of the carrier of BoolePoset b2,the carrier of BoolePoset b2
          st b3 is directed-sups-preserving(BoolePoset b2, BoolePoset b2)
       holds not b1,Image b3 are_isomorphic or b1 is continuous);

:: WAYBEL15:th 21
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1 holds
      b2 in PRIME (b1 ~)
   iff
      b2 is co-prime(b1);

:: WAYBEL15:attrnot 1 => WAYBEL15:attr 1
definition
  let a1 be non empty RelStr;
  let a2 be Element of the carrier of a1;
  attr a2 is atom means
    Bottom a1 < a2 &
     (for b1 being Element of the carrier of a1
           st Bottom a1 < b1 & b1 <= a2
        holds b1 = a2);
end;

:: WAYBEL15:dfs 1
definiens
  let a1 be non empty RelStr;
  let a2 be Element of the carrier of a1;
To prove
     a2 is atom
it is sufficient to prove
  thus Bottom a1 < a2 &
     (for b1 being Element of the carrier of a1
           st Bottom a1 < b1 & b1 <= a2
        holds b1 = a2);

:: WAYBEL15:def 1
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1 holds
      b2 is atom(b1)
   iff
      Bottom b1 < b2 &
       (for b3 being Element of the carrier of b1
             st Bottom b1 < b3 & b3 <= b2
          holds b3 = b2);

:: WAYBEL15:funcnot 1 => WAYBEL15:func 1
definition
  let a1 be non empty RelStr;
  func ATOM A1 -> Element of bool the carrier of a1 means
    for b1 being Element of the carrier of a1 holds
          b1 in it
       iff
          b1 is atom(a1);
end;

:: WAYBEL15:def 2
theorem
for b1 being non empty RelStr
for b2 being Element of bool the carrier of b1 holds
      b2 = ATOM b1
   iff
      for b3 being Element of the carrier of b1 holds
            b3 in b2
         iff
            b3 is atom(b1);

:: WAYBEL15:th 23
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima Boolean RelStr
for b2 being Element of the carrier of b1 holds
      b2 is atom(b1)
   iff
      b2 is co-prime(b1) & b2 <> Bottom b1;

:: WAYBEL15:condreg 1
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima Boolean RelStr;
  cluster atom -> co-prime (Element of the carrier of a1);
end;

:: WAYBEL15:th 24
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima Boolean RelStr holds
   ATOM b1 = (PRIME (b1 ~)) \ {Bottom b1};

:: WAYBEL15:th 25
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima Boolean RelStr
for b2, b3 being Element of the carrier of b1
      st b3 is atom(b1)
   holds    b3 <= b2
   iff
      not b3 <= 'not' b2;

:: WAYBEL15:th 26
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete Boolean RelStr
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
   b3 "/\" "\/"(b2,b1) = "\/"({b3 "/\" b4 where b4 is Element of the carrier of b1: b4 in b2},b1);

:: WAYBEL15:th 27
theorem
for b1 being non empty antisymmetric with_infima lower-bounded RelStr
for b2, b3 being Element of the carrier of b1
      st b2 is atom(b1) & b3 is atom(b1) & b2 <> b3
   holds b2 "/\" b3 = Bottom b1;

:: WAYBEL15:th 28
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete Boolean RelStr
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
      st b3 c= ATOM b1
   holds    b2 in b3
   iff
      b2 is atom(b1) & b2 <= "\/"(b3,b1);

:: WAYBEL15:th 29
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete Boolean RelStr
for b2, b3 being Element of bool the carrier of b1
      st b2 c= ATOM b1 & b3 c= ATOM b1
   holds    b2 c= b3
   iff
      "\/"(b2,b1) <= "\/"(b3,b1);

:: WAYBEL15:th 30
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima Boolean RelStr holds
      b1 is arithmetic
   iff
      ex b2 being set st
         b1,BoolePoset b2 are_isomorphic;

:: WAYBEL15:th 31
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima Boolean RelStr holds
      b1 is arithmetic
   iff
      b1 is algebraic;

:: WAYBEL15:th 32
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima Boolean RelStr holds
      b1 is arithmetic
   iff
      b1 is continuous;

:: WAYBEL15:th 33
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima Boolean RelStr holds
      b1 is arithmetic
   iff
      b1 is continuous & b1 ~ is continuous;

:: WAYBEL15:th 34
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima Boolean RelStr holds
      b1 is arithmetic
   iff
      b1 is completely-distributive;

:: WAYBEL15:th 35
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima Boolean RelStr holds
      b1 is arithmetic
   iff
      b1 is complete &
       (for b2 being Element of the carrier of b1 holds
          ex b3 being Element of bool the carrier of b1 st
             b3 c= ATOM b1 & b2 = "\/"(b3,b1));