Article WAYBEL_7, MML version 4.99.1005

:: WAYBEL_7:th 3
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2, b3 being set
      st b2 c= b3
   holds "\/"(b2,b1) <= "\/"(b3,b1) & "/\"(b3,b1) <= "/\"(b2,b1);

:: WAYBEL_7:th 4
theorem
for b1 being set holds
   the carrier of BoolePoset b1 = bool b1;

:: WAYBEL_7:th 5
theorem
for b1 being non empty antisymmetric bounded RelStr holds
      b1 is trivial
   iff
      Top b1 = Bottom b1;

:: WAYBEL_7:funcreg 1
registration
  let a1 be set;
  cluster BoolePoset a1 -> strict Boolean;
end;

:: WAYBEL_7:funcreg 2
registration
  let a1 be non empty set;
  cluster BoolePoset a1 -> non trivial strict;
end;

:: WAYBEL_7:th 8
theorem
for b1 being non empty reflexive transitive antisymmetric lower-bounded RelStr
for b2 being non empty filtered upper Element of bool the carrier of b1 holds
      b2 is proper(bool the carrier of b1)
   iff
      not Bottom b1 in b2;

:: WAYBEL_7:exreg 1
registration
  cluster non empty non trivial strict reflexive transitive antisymmetric Boolean non void with_suprema with_infima RelStr;
end;

:: WAYBEL_7:exreg 2
registration
  let a1 be set;
  cluster non empty finite Element of bool bool a1;
end;

:: WAYBEL_7:exreg 3
registration
  let a1 be 1-sorted;
  cluster non empty finite Element of bool bool the carrier of a1;
end;

:: WAYBEL_7:exreg 4
registration
  let a1 be non empty non trivial reflexive transitive antisymmetric upper-bounded RelStr;
  cluster non empty proper filtered upper Element of bool the carrier of a1;
end;

:: WAYBEL_7:th 9
theorem
for b1 being set
for b2 being Element of the carrier of BoolePoset b1 holds
   'not' b2 = b1 \ b2;

:: WAYBEL_7:th 10
theorem
for b1 being set
for b2 being Element of bool the carrier of BoolePoset b1 holds
      b2 is lower(BoolePoset b1)
   iff
      for b3, b4 being set
            st b3 c= b4 & b4 in b2
         holds b3 in b2;

:: WAYBEL_7:th 11
theorem
for b1 being set
for b2 being Element of bool the carrier of BoolePoset b1 holds
      b2 is upper(BoolePoset b1)
   iff
      for b3, b4 being set
            st b3 c= b4 & b4 c= b1 & b3 in b2
         holds b4 in b2;

:: WAYBEL_7:th 12
theorem
for b1 being set
for b2 being lower Element of bool the carrier of BoolePoset b1 holds
      b2 is directed(BoolePoset b1)
   iff
      for b3, b4 being set
            st b3 in b2 & b4 in b2
         holds b3 \/ b4 in b2;

:: WAYBEL_7:th 13
theorem
for b1 being set
for b2 being upper Element of bool the carrier of BoolePoset b1 holds
      b2 is filtered(BoolePoset b1)
   iff
      for b3, b4 being set
            st b3 in b2 & b4 in b2
         holds b3 /\ b4 in b2;

:: WAYBEL_7:th 14
theorem
for b1 being set
for b2 being non empty lower Element of bool the carrier of BoolePoset b1 holds
      b2 is directed(BoolePoset b1)
   iff
      for b3 being finite Element of bool bool b1
            st b3 c= b2
         holds union b3 in b2;

:: WAYBEL_7:th 15
theorem
for b1 being set
for b2 being non empty upper Element of bool the carrier of BoolePoset b1 holds
      b2 is filtered(BoolePoset b1)
   iff
      for b3 being finite Element of bool bool b1
            st b3 c= b2
         holds Intersect b3 in b2;

:: WAYBEL_7:attrnot 1 => WAYBEL_7:attr 1
definition
  let a1 be reflexive transitive antisymmetric with_infima RelStr;
  let a2 be non empty directed lower Element of bool the carrier of a1;
  attr a2 is prime means
    for b1, b2 being Element of the carrier of a1
          st b1 "/\" b2 in a2 & not b1 in a2
       holds b2 in a2;
end;

:: WAYBEL_7:dfs 1
definiens
  let a1 be reflexive transitive antisymmetric with_infima RelStr;
  let a2 be non empty directed lower Element of bool the carrier of a1;
To prove
     a2 is prime
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1
          st b1 "/\" b2 in a2 & not b1 in a2
       holds b2 in a2;

:: WAYBEL_7:def 1
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
      b2 is prime(b1)
   iff
      for b3, b4 being Element of the carrier of b1
            st b3 "/\" b4 in b2 & not b3 in b2
         holds b4 in b2;

:: WAYBEL_7:th 16
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
      b2 is prime(b1)
   iff
      for b3 being non empty finite Element of bool the carrier of b1
            st "/\"(b3,b1) in b2
         holds ex b4 being Element of the carrier of b1 st
            b4 in b3 & b4 in b2;

:: WAYBEL_7:exreg 5
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima RelStr;
  cluster non empty directed lower prime Element of bool the carrier of a1;
end;

:: WAYBEL_7:th 17
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima RelStr
   st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#)
for b3 being set
      st b3 is non empty directed lower prime Element of bool the carrier of b1
   holds b3 is non empty directed lower prime Element of bool the carrier of b2;

:: WAYBEL_7:attrnot 2 => WAYBEL_7:attr 2
definition
  let a1 be reflexive transitive antisymmetric with_suprema RelStr;
  let a2 be non empty filtered upper Element of bool the carrier of a1;
  attr a2 is prime means
    for b1, b2 being Element of the carrier of a1
          st b1 "\/" b2 in a2 & not b1 in a2
       holds b2 in a2;
end;

:: WAYBEL_7:dfs 2
definiens
  let a1 be reflexive transitive antisymmetric with_suprema RelStr;
  let a2 be non empty filtered upper Element of bool the carrier of a1;
To prove
     a2 is prime
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1
          st b1 "\/" b2 in a2 & not b1 in a2
       holds b2 in a2;

:: WAYBEL_7:def 2
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2 being non empty filtered upper Element of bool the carrier of b1 holds
      b2 is prime(b1)
   iff
      for b3, b4 being Element of the carrier of b1
            st b3 "\/" b4 in b2 & not b3 in b2
         holds b4 in b2;

:: WAYBEL_7:th 18
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2 being non empty filtered upper Element of bool the carrier of b1 holds
      b2 is prime(b1)
   iff
      for b3 being non empty finite Element of bool the carrier of b1
            st "\/"(b3,b1) in b2
         holds ex b4 being Element of the carrier of b1 st
            b4 in b3 & b4 in b2;

:: WAYBEL_7:exreg 6
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima RelStr;
  cluster non empty filtered upper prime Element of bool the carrier of a1;
end;

:: WAYBEL_7:th 19
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima RelStr
   st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#)
for b3 being set
      st b3 is non empty filtered upper prime Element of bool the carrier of b1
   holds b3 is non empty filtered upper prime Element of bool the carrier of b2;

:: WAYBEL_7:th 20
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2 being set holds
      b2 is non empty directed lower prime Element of bool the carrier of b1
   iff
      b2 is non empty filtered upper prime Element of bool the carrier of b1 ~;

:: WAYBEL_7:th 21
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2 being set holds
      b2 is non empty filtered upper prime Element of bool the carrier of b1
   iff
      b2 is non empty directed lower prime Element of bool the carrier of b1 ~;

:: WAYBEL_7:th 22
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
      b2 is prime(b1)
   iff
      (b2 ` is non empty filtered upper Element of bool the carrier of b1 or b2 ` = {});

:: WAYBEL_7:th 23
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2 being non empty directed lower Element of bool the carrier of b1 holds
      b2 is prime(b1)
   iff
      b2 in PRIME InclPoset Ids b1;

:: WAYBEL_7:th 24
theorem
for b1 being reflexive transitive antisymmetric Boolean with_suprema with_infima RelStr
for b2 being non empty filtered upper Element of bool the carrier of b1 holds
      b2 is prime(b1)
   iff
      for b3 being Element of the carrier of b1
            st not b3 in b2
         holds 'not' b3 in b2;

:: WAYBEL_7:th 25
theorem
for b1 being set
for b2 being non empty filtered upper Element of bool the carrier of BoolePoset b1 holds
      b2 is prime(BoolePoset b1)
   iff
      for b3 being Element of bool b1
            st not b3 in b2
         holds b1 \ b3 in b2;

:: WAYBEL_7:attrnot 3 => WAYBEL_7:attr 3
definition
  let a1 be non empty reflexive transitive antisymmetric RelStr;
  let a2 be non empty filtered upper Element of bool the carrier of a1;
  attr a2 is ultra means
    a2 is proper(bool the carrier of a1) &
     (for b1 being non empty filtered upper Element of bool the carrier of a1
           st a2 c= b1 & a2 <> b1
        holds b1 = the carrier of a1);
end;

:: WAYBEL_7:dfs 3
definiens
  let a1 be non empty reflexive transitive antisymmetric RelStr;
  let a2 be non empty filtered upper Element of bool the carrier of a1;
To prove
     a2 is ultra
it is sufficient to prove
  thus a2 is proper(bool the carrier of a1) &
     (for b1 being non empty filtered upper Element of bool the carrier of a1
           st a2 c= b1 & a2 <> b1
        holds b1 = the carrier of a1);

:: WAYBEL_7:def 3
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being non empty filtered upper Element of bool the carrier of b1 holds
      b2 is ultra(b1)
   iff
      b2 is proper(bool the carrier of b1) &
       (for b3 being non empty filtered upper Element of bool the carrier of b1
             st b2 c= b3 & b2 <> b3
          holds b3 = the carrier of b1);

:: WAYBEL_7:condreg 1
registration
  let a1 be non empty reflexive transitive antisymmetric RelStr;
  cluster non empty filtered upper ultra -> proper (Element of bool the carrier of a1);
end;

:: WAYBEL_7:th 26
theorem
for b1 being reflexive transitive antisymmetric Boolean with_suprema with_infima RelStr
for b2 being non empty filtered upper Element of bool the carrier of b1 holds
      b2 is proper(bool the carrier of b1) & b2 is prime(b1)
   iff
      b2 is ultra(b1);

:: WAYBEL_7:th 27
theorem
for b1 being reflexive transitive antisymmetric distributive with_suprema with_infima RelStr
for b2 being non empty directed lower Element of bool the carrier of b1
for b3 being non empty filtered upper Element of bool the carrier of b1
      st b2 misses b3
   holds ex b4 being non empty directed lower Element of bool the carrier of b1 st
      b4 is prime(b1) & b2 c= b4 & b4 misses b3;

:: WAYBEL_7:th 28
theorem
for b1 being reflexive transitive antisymmetric distributive with_suprema with_infima RelStr
for b2 being non empty directed lower Element of bool the carrier of b1
for b3 being Element of the carrier of b1
      st not b3 in b2
   holds ex b4 being non empty directed lower Element of bool the carrier of b1 st
      b4 is prime(b1) & b2 c= b4 & not b3 in b4;

:: WAYBEL_7:th 29
theorem
for b1 being reflexive transitive antisymmetric distributive with_suprema with_infima RelStr
for b2 being non empty directed lower Element of bool the carrier of b1
for b3 being non empty filtered upper Element of bool the carrier of b1
      st b2 misses b3
   holds ex b4 being non empty filtered upper Element of bool the carrier of b1 st
      b4 is prime(b1) & b3 c= b4 & b2 misses b4;

:: WAYBEL_7:th 30
theorem
for b1 being non trivial reflexive transitive antisymmetric Boolean with_suprema with_infima RelStr
for b2 being non empty proper filtered upper Element of bool the carrier of b1 holds
   ex b3 being non empty filtered upper Element of bool the carrier of b1 st
      b2 c= b3 & b3 is ultra(b1);

:: WAYBEL_7:prednot 1 => WAYBEL_7:pred 1
definition
  let a1 be TopSpace-like TopStruct;
  let a2, a3 be set;
  pred A3 is_a_cluster_point_of A2,A1 means
    for b1 being Element of bool the carrier of a1
       st b1 is open(a1) & a3 in b1
    for b2 being set
          st b2 in a2
       holds b1 meets b2;
end;

:: WAYBEL_7:dfs 4
definiens
  let a1 be TopSpace-like TopStruct;
  let a2, a3 be set;
To prove
     a3 is_a_cluster_point_of a2,a1
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
       st b1 is open(a1) & a3 in b1
    for b2 being set
          st b2 in a2
       holds b1 meets b2;

:: WAYBEL_7:def 4
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being set holds
   b3 is_a_cluster_point_of b2,b1
iff
   for b4 being Element of bool the carrier of b1
      st b4 is open(b1) & b3 in b4
   for b5 being set
         st b5 in b2
      holds b4 meets b5;

:: WAYBEL_7:prednot 2 => WAYBEL_7:pred 2
definition
  let a1 be TopSpace-like TopStruct;
  let a2, a3 be set;
  pred A3 is_a_convergence_point_of A2,A1 means
    for b1 being Element of bool the carrier of a1
          st b1 is open(a1) & a3 in b1
       holds b1 in a2;
end;

:: WAYBEL_7:dfs 5
definiens
  let a1 be TopSpace-like TopStruct;
  let a2, a3 be set;
To prove
     a3 is_a_convergence_point_of a2,a1
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st b1 is open(a1) & a3 in b1
       holds b1 in a2;

:: WAYBEL_7:def 5
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being set holds
   b3 is_a_convergence_point_of b2,b1
iff
   for b4 being Element of bool the carrier of b1
         st b4 is open(b1) & b3 in b4
      holds b4 in b2;

:: WAYBEL_7:exreg 7
registration
  let a1 be non empty set;
  cluster non empty filtered upper ultra Element of bool the carrier of BoolePoset a1;
end;

:: WAYBEL_7:th 31
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty filtered upper ultra Element of bool the carrier of BoolePoset the carrier of b1
for b3 being set holds
      b3 is_a_cluster_point_of b2,b1
   iff
      b3 is_a_convergence_point_of b2,b1;

:: WAYBEL_7:th 32
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of InclPoset the topology of b1
   st b2 is_way_below b3
for b4 being non empty proper filtered upper Element of bool the carrier of BoolePoset the carrier of b1
      st b2 in b4
   holds ex b5 being Element of the carrier of b1 st
      b5 in b3 & b5 is_a_cluster_point_of b4,b1;

:: WAYBEL_7:th 33
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of InclPoset the topology of b1
   st b2 is_way_below b3
for b4 being non empty filtered upper ultra Element of bool the carrier of BoolePoset the carrier of b1
      st b2 in b4
   holds ex b5 being Element of the carrier of b1 st
      b5 in b3 & b5 is_a_convergence_point_of b4,b1;

:: WAYBEL_7:th 34
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of InclPoset the topology of b1
      st b2 c= b3 &
         (for b4 being non empty filtered upper ultra Element of bool the carrier of BoolePoset the carrier of b1
               st b2 in b4
            holds ex b5 being Element of the carrier of b1 st
               b5 in b3 & b5 is_a_convergence_point_of b4,b1)
   holds b2 is_way_below b3;

:: WAYBEL_7:th 35
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being prebasis of b1
for b3, b4 being Element of the carrier of InclPoset the topology of b1
      st b3 c= b4
   holds    b3 is_way_below b4
   iff
      for b5 being Element of bool b2
            st b4 c= union b5
         holds ex b6 being finite Element of bool b5 st
            b3 c= union b6;

:: WAYBEL_7:th 36
theorem
for b1 being reflexive transitive antisymmetric distributive with_suprema with_infima complete RelStr
for b2, b3 being Element of the carrier of b1 holds
   b2 is_way_below b3
iff
   for b4 being non empty directed lower prime Element of bool the carrier of b1
         st b3 <= "\/"(b4,b1)
      holds b2 in b4;

:: WAYBEL_7:th 37
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2 being Element of the carrier of b1
      st b2 is prime(b1)
   holds downarrow b2 is prime(b1);

:: WAYBEL_7:attrnot 4 => WAYBEL_7:attr 4
definition
  let a1 be reflexive transitive antisymmetric with_suprema with_infima RelStr;
  let a2 be Element of the carrier of a1;
  attr a2 is pseudoprime means
    ex b1 being non empty directed lower prime Element of bool the carrier of a1 st
       a2 = "\/"(b1,a1);
end;

:: WAYBEL_7:dfs 6
definiens
  let a1 be reflexive transitive antisymmetric with_suprema with_infima RelStr;
  let a2 be Element of the carrier of a1;
To prove
     a2 is pseudoprime
it is sufficient to prove
  thus ex b1 being non empty directed lower prime Element of bool the carrier of a1 st
       a2 = "\/"(b1,a1);

:: WAYBEL_7:def 6
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2 being Element of the carrier of b1 holds
      b2 is pseudoprime(b1)
   iff
      ex b3 being non empty directed lower prime Element of bool the carrier of b1 st
         b2 = "\/"(b3,b1);

:: WAYBEL_7:th 38
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2 being Element of the carrier of b1
      st b2 is prime(b1)
   holds b2 is pseudoprime(b1);

:: WAYBEL_7:th 39
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima continuous RelStr
for b2 being Element of the carrier of b1
   st b2 is pseudoprime(b1)
for b3 being non empty finite Element of bool the carrier of b1
      st "/\"(b3,b1) is_way_below b2
   holds ex b4 being Element of the carrier of b1 st
      b4 in b3 & b4 <= b2;

:: WAYBEL_7:th 40
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima continuous RelStr
for b2 being Element of the carrier of b1
      st (b2 = Top b1 implies Top b1 is not compact(b1)) &
         (for b3 being non empty finite Element of bool the carrier of b1
               st "/\"(b3,b1) is_way_below b2
            holds ex b4 being Element of the carrier of b1 st
               b4 in b3 & b4 <= b2)
   holds uparrow fininfs ((downarrow b2) `) misses waybelow b2;

:: WAYBEL_7:th 41
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima continuous RelStr
      st Top b1 is compact(b1)
   holds (for b2 being non empty finite Element of bool the carrier of b1
          st "/\"(b2,b1) is_way_below Top b1
       holds ex b3 being Element of the carrier of b1 st
          b3 in b2 & b3 <= Top b1) &
    uparrow fininfs ((downarrow Top b1) `) meets waybelow Top b1;

:: WAYBEL_7:th 42
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima continuous RelStr
for b2 being Element of the carrier of b1
   st uparrow fininfs ((downarrow b2) `) misses waybelow b2
for b3 being non empty finite Element of bool the carrier of b1
      st "/\"(b3,b1) is_way_below b2
   holds ex b4 being Element of the carrier of b1 st
      b4 in b3 & b4 <= b2;

:: WAYBEL_7:th 43
theorem
for b1 being reflexive transitive antisymmetric distributive with_suprema with_infima continuous RelStr
for b2 being Element of the carrier of b1
      st uparrow fininfs ((downarrow b2) `) misses waybelow b2
   holds b2 is pseudoprime(b1);

:: WAYBEL_7:attrnot 5 => WAYBEL_7: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 multiplicative means
    for b1, b2, b3 being Element of the carrier of a1
          st [b1,b2] in a2 & [b1,b3] in a2
       holds [b1,b2 "/\" b3] in a2;
end;

:: WAYBEL_7: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 multiplicative
it is sufficient to prove
  thus for b1, b2, b3 being Element of the carrier of a1
          st [b1,b2] in a2 & [b1,b3] in a2
       holds [b1,b2 "/\" b3] in a2;

:: WAYBEL_7: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 multiplicative(b1)
   iff
      for b3, b4, b5 being Element of the carrier of b1
            st [b3,b4] in b2 & [b3,b5] in b2
         holds [b3,b4 "/\" b5] in b2;

:: WAYBEL_7:funcreg 3
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
  let a2 be auxiliary Relation of the carrier of a1,the carrier of a1;
  let a3 be Element of the carrier of a1;
  cluster a2 -above a3 -> upper;
end;

:: WAYBEL_7:th 44
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b2 being auxiliary Relation of the carrier of b1,the carrier of b1 holds
      b2 is multiplicative(b1)
   iff
      for b3 being Element of the carrier of b1 holds
         b2 -above b3 is filtered(b1);

:: WAYBEL_7:th 45
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b2 being auxiliary Relation of the carrier of b1,the carrier of b1 holds
      b2 is multiplicative(b1)
   iff
      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_7:th 46
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b2 being auxiliary Relation of the carrier of b1,the carrier of b1 holds
      b2 is multiplicative(b1)
   iff
      for b3 being full SubRelStr of [:b1,b1:]
            st the carrier of b3 = b2
         holds b3 is meet-inheriting([:b1,b1:]);

:: WAYBEL_7:th 47
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b2 being auxiliary Relation of the carrier of b1,the carrier of b1 holds
      b2 is multiplicative(b1)
   iff
      b2 -below is meet-preserving(b1, InclPoset Ids b1);

:: WAYBEL_7:th 48
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr
   st b1 -waybelow is multiplicative(b1)
for b2 being Element of the carrier of b1 holds
      b2 is pseudoprime(b1)
   iff
      for b3, b4 being Element of the carrier of b1
            st b3 "/\" b4 is_way_below b2 & not b3 <= b2
         holds b4 <= b2;

:: WAYBEL_7:th 49
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr
   st b1 -waybelow is multiplicative(b1)
for b2 being Element of the carrier of b1
      st b2 is pseudoprime(b1)
   holds b2 is prime(b1);

:: WAYBEL_7:th 50
theorem
for b1 being reflexive transitive antisymmetric lower-bounded distributive with_suprema with_infima continuous RelStr
      st for b2 being Element of the carrier of b1
              st b2 is pseudoprime(b1)
           holds b2 is prime(b1)
   holds b1 -waybelow is multiplicative(b1);