Article WAYBEL14, MML version 4.99.1005

:: WAYBEL14:th 1
theorem
for b1 being set
for b2 being finite Element of bool bool b1 holds
   ex b3 being finite Element of bool bool b1 st
      b3 c= b2 &
       union b3 = union b2 &
       (for b4 being Element of bool b1
             st b4 in b3
          holds not b4 c= union (b3 \ {b4}));

:: WAYBEL14:th 2
theorem
for b1 being 1-sorted
for b2 being Element of bool the carrier of b1 holds
      b2 ` = the carrier of b1
   iff
      b2 is empty;

:: WAYBEL14:th 3
theorem
for b1 being non empty transitive antisymmetric with_infima RelStr
for b2, b3 being Element of the carrier of b1 holds
downarrow (b2 "/\" b3) = (downarrow b2) /\ downarrow b3;

:: WAYBEL14:th 4
theorem
for b1 being non empty transitive antisymmetric with_suprema RelStr
for b2, b3 being Element of the carrier of b1 holds
uparrow (b2 "\/" b3) = (uparrow b2) /\ uparrow b3;

:: WAYBEL14:th 5
theorem
for b1 being non empty antisymmetric complete RelStr
for b2 being lower Element of bool the carrier of b1
      st "\/"(b2,b1) in b2
   holds b2 = downarrow "\/"(b2,b1);

:: WAYBEL14:th 6
theorem
for b1 being non empty antisymmetric complete RelStr
for b2 being upper Element of bool the carrier of b1
      st "/\"(b2,b1) in b2
   holds b2 = uparrow "/\"(b2,b1);

:: WAYBEL14:th 7
theorem
for b1 being non empty reflexive transitive RelStr
for b2, b3 being Element of the carrier of b1 holds
   b2 is_way_below b3
iff
   uparrow b3 c= wayabove b2;

:: WAYBEL14:th 8
theorem
for b1 being non empty reflexive transitive RelStr
for b2, b3 being Element of the carrier of b1 holds
   b2 is_way_below b3
iff
   downarrow b2 c= waybelow b3;

:: WAYBEL14:th 9
theorem
for b1 being non empty reflexive antisymmetric complete RelStr
for b2 being Element of the carrier of b1 holds
   "\/"(waybelow b2,b1) <= b2 & b2 <= "/\"(wayabove b2,b1);

:: WAYBEL14:th 10
theorem
for b1 being non empty antisymmetric lower-bounded RelStr holds
   uparrow Bottom b1 = the carrier of b1;

:: WAYBEL14:th 11
theorem
for b1 being non empty antisymmetric upper-bounded RelStr holds
   downarrow Top b1 = the carrier of b1;

:: WAYBEL14:th 12
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2, b3 being Element of the carrier of b1 holds
(wayabove b2) "\/" wayabove b3 c= uparrow (b2 "\/" b3);

:: WAYBEL14:th 13
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2, b3 being Element of the carrier of b1 holds
(waybelow b2) "/\" waybelow b3 c= downarrow (b2 "/\" b3);

:: WAYBEL14:th 14
theorem
for b1 being non empty reflexive transitive antisymmetric with_suprema RelStr
for b2 being Element of the carrier of b1 holds
      b2 is co-prime(b1)
   iff
      for b3, b4 being Element of the carrier of b1
            st b2 <= b3 "\/" b4 & not b2 <= b3
         holds b2 <= b4;

:: WAYBEL14:th 15
theorem
for b1 being non empty reflexive transitive antisymmetric complete RelStr
for b2 being non empty Element of bool the carrier of b1 holds
   downarrow "/\"(b2,b1) = meet {downarrow b3 where b3 is Element of the carrier of b1: b3 in b2};

:: WAYBEL14:th 16
theorem
for b1 being non empty reflexive transitive antisymmetric complete RelStr
for b2 being non empty Element of bool the carrier of b1 holds
   uparrow "\/"(b2,b1) = meet {uparrow b3 where b3 is Element of the carrier of b1: b3 in b2};

:: WAYBEL14:funcreg 1
registration
  let a1 be reflexive transitive antisymmetric with_suprema RelStr;
  let a2 be Element of the carrier of a1;
  cluster compactbelow a2 -> directed;
end;

:: WAYBEL14:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being irreducible Element of bool the carrier of b1
for b3 being Element of the carrier of InclPoset the topology of b1
      st b3 = b2 `
   holds b3 is prime(InclPoset the topology of b1);

:: WAYBEL14:th 18
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of InclPoset the topology of b1 holds
b2 "\/" b3 = b2 \/ b3 & b2 "/\" b3 = b2 /\ b3;

:: WAYBEL14:th 19
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of InclPoset the topology of b1 holds
      b2 is prime(InclPoset the topology of b1)
   iff
      for b3, b4 being Element of the carrier of InclPoset the topology of b1
            st b3 /\ b4 c= b2 & not b3 c= b2
         holds b4 c= b2;

:: WAYBEL14:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of InclPoset the topology of b1 holds
      b2 is co-prime(InclPoset the topology of b1)
   iff
      for b3, b4 being Element of the carrier of InclPoset the topology of b1
            st b2 c= b3 \/ b4 & not b2 c= b3
         holds b2 c= b4;

:: WAYBEL14:funcreg 2
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster InclPoset the topology of a1 -> strict distributive;
end;

:: WAYBEL14:th 21
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima TopRelStr
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
for b5 being Element of bool bool the carrier of b2
      st TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#) &
         b3 = b4 &
         b5 is Basis of b4
   holds b5 is Basis of b3;

:: WAYBEL14:th 22
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima TopRelStr
for b2 being Element of the carrier of b1
      st for b3 being Element of bool the carrier of b1
              st b3 is open(b1)
           holds b3 is upper(b1)
   holds uparrow b2 is compact(b1);

:: WAYBEL14:funcreg 3
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  cluster sigma a1 -> non empty;
end;

:: WAYBEL14:th 23
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr holds
   sigma b1 = the topology of b1;

:: WAYBEL14:th 24
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b2 being Element of bool the carrier of b1 holds
      b2 in sigma b1
   iff
      b2 is open(b1);

:: WAYBEL14:th 25
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b2 being Element of bool the carrier of InclPoset sigma b1
for b3 being filtered Element of bool the carrier of b1
      st b2 = {(downarrow b4) ` where b4 is Element of the carrier of b1: b4 in b3}
   holds b2 is directed(InclPoset sigma b1);

:: WAYBEL14:th 26
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
      st b3 is open(b1) & b2 in b3
   holds "/\"(b3,b1) is_way_below b2;

:: WAYBEL14:attrnot 1 => WAYBEL14:attr 1
definition
  let a1 be non empty reflexive RelStr;
  let a2 be Function-like quasi_total Relation of the carrier of [:a1,a1:],the carrier of a1;
  attr a2 is jointly_Scott-continuous means
    for b1 being non empty TopSpace-like TopStruct
          st TopStruct(#the carrier of b1,the topology of b1#) = ConvergenceSpace Scott-Convergence a1
       holds ex b2 being Function-like quasi_total Relation of the carrier of [:b1,b1:],the carrier of b1 st
          b2 = a2 & b2 is continuous([:b1,b1:], b1);
end;

:: WAYBEL14:dfs 1
definiens
  let a1 be non empty reflexive RelStr;
  let a2 be Function-like quasi_total Relation of the carrier of [:a1,a1:],the carrier of a1;
To prove
     a2 is jointly_Scott-continuous
it is sufficient to prove
  thus for b1 being non empty TopSpace-like TopStruct
          st TopStruct(#the carrier of b1,the topology of b1#) = ConvergenceSpace Scott-Convergence a1
       holds ex b2 being Function-like quasi_total Relation of the carrier of [:b1,b1:],the carrier of b1 st
          b2 = a2 & b2 is continuous([:b1,b1:], b1);

:: WAYBEL14:def 1
theorem
for b1 being non empty reflexive RelStr
for b2 being Function-like quasi_total Relation of the carrier of [:b1,b1:],the carrier of b1 holds
      b2 is jointly_Scott-continuous(b1)
   iff
      for b3 being non empty TopSpace-like TopStruct
            st TopStruct(#the carrier of b3,the topology of b3#) = ConvergenceSpace Scott-Convergence b1
         holds ex b4 being Function-like quasi_total Relation of the carrier of [:b3,b3:],the carrier of b3 st
            b4 = b2 & b4 is continuous([:b3,b3:], b3);

:: WAYBEL14:th 27
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of InclPoset sigma b1
      st b3 = b2
   holds    b3 is co-prime(InclPoset sigma b1)
   iff
      b2 is filtered(b1) & b2 is upper(b1);

:: WAYBEL14:th 28
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of InclPoset sigma b1
      st b3 = b2 &
         (ex b4 being Element of the carrier of b1 st
            b2 = (downarrow b4) `)
   holds b3 is prime(InclPoset sigma b1) & b3 <> the carrier of b1;

:: WAYBEL14:th 29
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of InclPoset sigma b1
      st b3 = b2 & sup_op b1 is jointly_Scott-continuous(b1) & b3 is prime(InclPoset sigma b1) & b3 <> the carrier of b1
   holds ex b4 being Element of the carrier of b1 st
      b2 = (downarrow b4) `;

:: WAYBEL14:th 30
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
      st b1 is continuous
   holds sup_op b1 is jointly_Scott-continuous(b1);

:: WAYBEL14:th 31
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
      st sup_op b1 is jointly_Scott-continuous(b1)
   holds b1 is sober;

:: WAYBEL14:th 32
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
      st b1 is continuous
   holds b1 is compact & b1 is locally-compact & b1 is sober & b1 is Baire;

:: WAYBEL14:th 33
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b2 being Element of bool the carrier of b1
      st b1 is continuous & b2 in sigma b1
   holds b2 = union {wayabove b3 where b3 is Element of the carrier of b1: b3 in b2};

:: WAYBEL14:th 34
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
      st for b2 being Element of bool the carrier of b1
              st b2 in sigma b1
           holds b2 = union {wayabove b3 where b3 is Element of the carrier of b1: b3 in b2}
   holds b1 is continuous;

:: WAYBEL14:th 35
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b2 being Element of the carrier of b1
      st b1 is continuous
   holds ex b3 being Basis of b2 st
      for b4 being Element of bool the carrier of b1
            st b4 in b3
         holds b4 is open(b1) & b4 is filtered(b1);

:: WAYBEL14:th 36
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
      st b1 is continuous
   holds InclPoset sigma b1 is continuous;

:: WAYBEL14:th 37
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b2 being Element of the carrier of b1
      st (for b3 being Element of the carrier of b1 holds
            ex b4 being Basis of b3 st
               for b5 being Element of bool the carrier of b1
                     st b5 in b4
                  holds b5 is open(b1) & b5 is filtered(b1)) &
         InclPoset sigma b1 is continuous
   holds b2 = "\/"({"/\"(b3,b1) where b3 is Element of bool the carrier of b1: b2 in b3 & b3 in sigma b1},b1);

:: WAYBEL14:th 38
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
      st for b2 being Element of the carrier of b1 holds
           b2 = "\/"({"/\"(b3,b1) where b3 is Element of bool the carrier of b1: b2 in b3 & b3 in sigma b1},b1)
   holds b1 is continuous;

:: WAYBEL14:th 39
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr holds
      for b2 being Element of the carrier of b1 holds
         ex b3 being Basis of b2 st
            for b4 being Element of bool the carrier of b1
                  st b4 in b3
               holds b4 is open(b1) & b4 is filtered(b1)
   iff
      for b2 being Element of the carrier of InclPoset sigma b1 holds
         ex b3 being Element of bool the carrier of InclPoset sigma b1 st
            b2 = "\/"(b3,InclPoset sigma b1) &
             (for b4 being Element of the carrier of InclPoset sigma b1
                   st b4 in b3
                holds b4 is co-prime(InclPoset sigma b1));

:: WAYBEL14:th 40
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr holds
      (for b2 being Element of the carrier of InclPoset sigma b1 holds
          ex b3 being Element of bool the carrier of InclPoset sigma b1 st
             b2 = "\/"(b3,InclPoset sigma b1) &
              (for b4 being Element of the carrier of InclPoset sigma b1
                    st b4 in b3
                 holds b4 is co-prime(InclPoset sigma b1))) &
       InclPoset sigma b1 is continuous
   iff
      InclPoset sigma b1 is completely-distributive;

:: WAYBEL14:th 41
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr holds
      InclPoset sigma b1 is completely-distributive
   iff
      InclPoset sigma b1 is continuous & (InclPoset sigma b1) ~ is continuous;

:: WAYBEL14:th 42
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
      st b1 is algebraic
   holds ex b2 being Basis of b1 st
      b2 = {uparrow b3 where b3 is Element of the carrier of b1: b3 in the carrier of CompactSublatt b1};

:: WAYBEL14:th 43
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
      st ex b2 being Basis of b1 st
           b2 = {uparrow b3 where b3 is Element of the carrier of b1: b3 in the carrier of CompactSublatt b1}
   holds InclPoset sigma b1 is algebraic &
    (for b2 being Element of the carrier of InclPoset sigma b1 holds
       ex b3 being Element of bool the carrier of InclPoset sigma b1 st
          b2 = "\/"(b3,InclPoset sigma b1) &
           (for b4 being Element of the carrier of InclPoset sigma b1
                 st b4 in b3
              holds b4 is co-prime(InclPoset sigma b1)));

:: WAYBEL14:th 44
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
      st InclPoset sigma b1 is algebraic &
         (for b2 being Element of the carrier of InclPoset sigma b1 holds
            ex b3 being Element of bool the carrier of InclPoset sigma b1 st
               b2 = "\/"(b3,InclPoset sigma b1) &
                (for b4 being Element of the carrier of InclPoset sigma b1
                      st b4 in b3
                   holds b4 is co-prime(InclPoset sigma b1)))
   holds ex b2 being Basis of b1 st
      b2 = {uparrow b3 where b3 is Element of the carrier of b1: b3 in the carrier of CompactSublatt b1};

:: WAYBEL14:th 45
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
      st ex b2 being Basis of b1 st
           b2 = {uparrow b3 where b3 is Element of the carrier of b1: b3 in the carrier of CompactSublatt b1}
   holds b1 is algebraic;