Article WAYBEL_3, MML version 4.99.1005

:: WAYBEL_3:prednot 1 => WAYBEL_3:pred 1
definition
  let a1 be non empty reflexive RelStr;
  let a2, a3 be Element of the carrier of a1;
  pred A2 is_way_below A3 means
    for b1 being non empty directed Element of bool the carrier of a1
          st a3 <= "\/"(b1,a1)
       holds ex b2 being Element of the carrier of a1 st
          b2 in b1 & a2 <= b2;
end;

:: WAYBEL_3:dfs 1
definiens
  let a1 be non empty reflexive RelStr;
  let a2, a3 be Element of the carrier of a1;
To prove
     a2 is_way_below a3
it is sufficient to prove
  thus for b1 being non empty directed Element of bool the carrier of a1
          st a3 <= "\/"(b1,a1)
       holds ex b2 being Element of the carrier of a1 st
          b2 in b1 & a2 <= b2;

:: WAYBEL_3:def 1
theorem
for b1 being non empty reflexive 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 <= b5;

:: WAYBEL_3:prednot 2 => WAYBEL_3:pred 1
notation
  let a1 be non empty reflexive RelStr;
  let a2, a3 be Element of the carrier of a1;
  synonym a2 << a3 for a2 is_way_below a3;
end;

:: WAYBEL_3:prednot 3 => WAYBEL_3:pred 1
notation
  let a1 be non empty reflexive RelStr;
  let a2, a3 be Element of the carrier of a1;
  synonym a3 >> a2 for a2 is_way_below a3;
end;

:: WAYBEL_3:attrnot 1 => WAYBEL_3:attr 1
definition
  let a1 be non empty reflexive RelStr;
  let a2 be Element of the carrier of a1;
  attr a2 is compact means
    a2 is_way_below a2;
end;

:: WAYBEL_3:dfs 2
definiens
  let a1 be non empty reflexive RelStr;
  let a2 be Element of the carrier of a1;
To prove
     a2 is compact
it is sufficient to prove
  thus a2 is_way_below a2;

:: WAYBEL_3:def 2
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of the carrier of b1 holds
      b2 is compact(b1)
   iff
      b2 is_way_below b2;

:: WAYBEL_3:attrnot 2 => WAYBEL_3:attr 1
notation
  let a1 be non empty reflexive RelStr;
  let a2 be Element of the carrier of a1;
  synonym isolated_from_below for compact;
end;

:: WAYBEL_3:th 1
theorem
for b1 being non empty reflexive antisymmetric RelStr
for b2, b3 being Element of the carrier of b1
      st b2 is_way_below b3
   holds b2 <= b3;

:: WAYBEL_3:th 2
theorem
for b1 being non empty reflexive transitive RelStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2 <= b3 & b3 is_way_below b4 & b4 <= b5
   holds b2 is_way_below b5;

:: WAYBEL_3:th 3
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
   st (b1 is with_suprema or b1 is /\-complete)
for b2, b3, b4 being Element of the carrier of b1
      st b2 is_way_below b4 & b3 is_way_below b4
   holds ex_sup_of {b2,b3},b1 & b2 "\/" b3 is_way_below b4;

:: WAYBEL_3:th 4
theorem
for b1 being non empty reflexive antisymmetric lower-bounded RelStr
for b2 being Element of the carrier of b1 holds
   Bottom b1 is_way_below b2;

:: WAYBEL_3:th 5
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 is_way_below b3 & b3 is_way_below b4
   holds b2 is_way_below b4;

:: WAYBEL_3:th 6
theorem
for b1 being non empty reflexive antisymmetric RelStr
for b2, b3 being Element of the carrier of b1
      st b2 is_way_below b3 & b3 is_way_below b2
   holds b2 = b3;

:: WAYBEL_3:funcnot 1 => WAYBEL_3:func 1
definition
  let a1 be non empty reflexive RelStr;
  let a2 be Element of the carrier of a1;
  func waybelow A2 -> Element of bool the carrier of a1 equals
    {b1 where b1 is Element of the carrier of a1: b1 is_way_below a2};
end;

:: WAYBEL_3:def 3
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of the carrier of b1 holds
   waybelow b2 = {b3 where b3 is Element of the carrier of b1: b3 is_way_below b2};

:: WAYBEL_3:funcnot 2 => WAYBEL_3:func 2
definition
  let a1 be non empty reflexive RelStr;
  let a2 be Element of the carrier of a1;
  func wayabove A2 -> Element of bool the carrier of a1 equals
    {b1 where b1 is Element of the carrier of a1: a2 is_way_below b1};
end;

:: WAYBEL_3:def 4
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of the carrier of b1 holds
   wayabove b2 = {b3 where b3 is Element of the carrier of b1: b2 is_way_below b3};

:: WAYBEL_3:th 7
theorem
for b1 being non empty reflexive RelStr
for b2, b3 being Element of the carrier of b1 holds
   b2 in waybelow b3
iff
   b2 is_way_below b3;

:: WAYBEL_3:th 8
theorem
for b1 being non empty reflexive RelStr
for b2, b3 being Element of the carrier of b1 holds
   b2 in wayabove b3
iff
   b3 is_way_below b2;

:: WAYBEL_3:th 9
theorem
for b1 being non empty reflexive antisymmetric RelStr
for b2 being Element of the carrier of b1 holds
   waybelow b2 is_<=_than b2;

:: WAYBEL_3:th 10
theorem
for b1 being non empty reflexive antisymmetric RelStr
for b2 being Element of the carrier of b1 holds
   b2 is_<=_than wayabove b2;

:: WAYBEL_3:th 11
theorem
for b1 being non empty reflexive antisymmetric RelStr
for b2 being Element of the carrier of b1 holds
   waybelow b2 c= downarrow b2 & wayabove b2 c= uparrow b2;

:: WAYBEL_3:th 12
theorem
for b1 being non empty reflexive transitive RelStr
for b2, b3 being Element of the carrier of b1
      st b2 <= b3
   holds waybelow b2 c= waybelow b3 & wayabove b3 c= wayabove b2;

:: WAYBEL_3:funcreg 1
registration
  let a1 be non empty reflexive antisymmetric lower-bounded RelStr;
  let a2 be Element of the carrier of a1;
  cluster waybelow a2 -> non empty;
end;

:: WAYBEL_3:funcreg 2
registration
  let a1 be non empty reflexive transitive RelStr;
  let a2 be Element of the carrier of a1;
  cluster waybelow a2 -> lower;
end;

:: WAYBEL_3:funcreg 3
registration
  let a1 be non empty reflexive transitive RelStr;
  let a2 be Element of the carrier of a1;
  cluster wayabove a2 -> upper;
end;

:: WAYBEL_3:funcreg 4
registration
  let a1 be reflexive transitive antisymmetric with_suprema RelStr;
  let a2 be Element of the carrier of a1;
  cluster waybelow a2 -> directed;
end;

:: WAYBEL_3:funcreg 5
registration
  let a1 be non empty reflexive transitive antisymmetric /\-complete RelStr;
  let a2 be Element of the carrier of a1;
  cluster waybelow a2 -> directed;
end;

:: WAYBEL_3:condreg 1
registration
  let a1 be non empty connected RelStr;
  cluster -> directed filtered (Element of bool the carrier of a1);
end;

:: WAYBEL_3:condreg 2
registration
  cluster non empty reflexive transitive antisymmetric lower-bounded connected up-complete -> complete (RelStr);
end;

:: WAYBEL_3:exreg 1
registration
  cluster non empty total reflexive transitive antisymmetric complete connected RelStr;
end;

:: WAYBEL_3:th 13
theorem
for b1 being non empty reflexive transitive antisymmetric connected up-complete RelStr
for b2, b3 being Element of the carrier of b1
      st b2 < b3
   holds b2 is_way_below b3;

:: WAYBEL_3:th 14
theorem
for b1 being non empty reflexive antisymmetric RelStr
for b2, b3 being Element of the carrier of b1
      st b2 is not compact(b1) & b2 is_way_below b3
   holds b2 < b3;

:: WAYBEL_3:th 15
theorem
for b1 being non empty reflexive antisymmetric lower-bounded RelStr holds
   Bottom b1 is compact(b1);

:: WAYBEL_3:th 16
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr
for b2 being non empty finite directed Element of bool the carrier of b1 holds
   "\/"(b2,b1) in b2;

:: WAYBEL_3:th 17
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr
   st b1 is finite
for b2 being Element of the carrier of b1 holds
   b2 is compact(b1);

:: WAYBEL_3:th 18
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2, b3 being Element of the carrier of b1
   st b2 is_way_below b3
for b4 being Element of bool the carrier of b1
      st b3 <= "\/"(b4,b1)
   holds ex b5 being finite Element of bool the carrier of b1 st
      b5 c= b4 & b2 <= "\/"(b5,b1);

:: WAYBEL_3:th 19
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2, b3 being Element of the carrier of b1
      st for b4 being Element of bool the carrier of b1
              st b3 <= "\/"(b4,b1)
           holds ex b5 being finite Element of bool the carrier of b1 st
              b5 c= b4 & b2 <= "\/"(b5,b1)
   holds b2 is_way_below b3;

:: WAYBEL_3:th 20
theorem
for b1 being non empty reflexive transitive RelStr
for b2, b3 being Element of the carrier of b1
   st b2 is_way_below b3
for b4 being non empty directed lower Element of bool the carrier of b1
      st b3 <= "\/"(b4,b1)
   holds b2 in b4;

:: WAYBEL_3:th 21
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr
for b2, b3 being Element of the carrier of b1
      st for b4 being non empty directed lower Element of bool the carrier of b1
              st b3 <= "\/"(b4,b1)
           holds b2 in b4
   holds b2 is_way_below b3;

:: WAYBEL_3:th 22
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
   st b1 is meet-continuous
for b2, b3 being Element of the carrier of b1 holds
   b2 is_way_below b3
iff
   for b4 being non empty directed lower Element of bool the carrier of b1
         st b3 = "\/"(b4,b1)
      holds b2 in b4;

:: WAYBEL_3:th 23
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
      for b2 being Element of the carrier of b1 holds
         b2 is compact(b1)
   iff
      for b2 being non empty Element of bool the carrier of b1 holds
         ex b3 being Element of the carrier of b1 st
            b3 in b2 &
             (for b4 being Element of the carrier of b1
                   st b4 in b2
                holds not b3 < b4);

:: WAYBEL_3:attrnot 3 => WAYBEL_3:attr 2
definition
  let a1 be non empty reflexive RelStr;
  attr a1 is satisfying_axiom_of_approximation means
    for b1 being Element of the carrier of a1 holds
       b1 = "\/"(waybelow b1,a1);
end;

:: WAYBEL_3:dfs 5
definiens
  let a1 be non empty reflexive RelStr;
To prove
     a1 is satisfying_axiom_of_approximation
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       b1 = "\/"(waybelow b1,a1);

:: WAYBEL_3:def 5
theorem
for b1 being non empty reflexive RelStr holds
      b1 is satisfying_axiom_of_approximation
   iff
      for b2 being Element of the carrier of b1 holds
         b2 = "\/"(waybelow b2,b1);

:: WAYBEL_3:condreg 3
registration
  cluster non empty trivial reflexive -> satisfying_axiom_of_approximation (RelStr);
end;

:: WAYBEL_3:attrnot 4 => WAYBEL_3:attr 3
definition
  let a1 be non empty reflexive RelStr;
  attr a1 is continuous means
    (for b1 being Element of the carrier of a1 holds
        waybelow b1 is not empty & waybelow b1 is directed(a1)) &
     a1 is up-complete &
     a1 is satisfying_axiom_of_approximation;
end;

:: WAYBEL_3:dfs 6
definiens
  let a1 be non empty reflexive RelStr;
To prove
     a1 is continuous
it is sufficient to prove
  thus (for b1 being Element of the carrier of a1 holds
        waybelow b1 is not empty & waybelow b1 is directed(a1)) &
     a1 is up-complete &
     a1 is satisfying_axiom_of_approximation;

:: WAYBEL_3:def 6
theorem
for b1 being non empty reflexive RelStr holds
      b1 is continuous
   iff
      (for b2 being Element of the carrier of b1 holds
          waybelow b2 is not empty & waybelow b2 is directed(b1)) &
       b1 is up-complete &
       b1 is satisfying_axiom_of_approximation;

:: WAYBEL_3:condreg 4
registration
  cluster non empty reflexive continuous -> up-complete satisfying_axiom_of_approximation (RelStr);
end;

:: WAYBEL_3:condreg 5
registration
  cluster reflexive transitive antisymmetric lower-bounded with_suprema up-complete satisfying_axiom_of_approximation -> continuous (RelStr);
end;

:: WAYBEL_3:exreg 2
registration
  cluster non empty strict total reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr;
end;

:: WAYBEL_3:funcreg 6
registration
  let a1 be non empty reflexive continuous RelStr;
  let a2 be Element of the carrier of a1;
  cluster waybelow a2 -> non empty directed;
end;

:: WAYBEL_3:th 24
theorem
for b1 being reflexive transitive antisymmetric with_infima up-complete RelStr
      st for b2 being Element of the carrier of b1 holds
           waybelow b2 is not empty & waybelow b2 is directed(b1)
   holds    b1 is satisfying_axiom_of_approximation
   iff
      for b2, b3 being Element of the carrier of b1
            st not b2 <= b3
         holds ex b4 being Element of the carrier of b1 st
            b4 is_way_below b2 & not b4 <= b3;

:: WAYBEL_3:th 25
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima continuous RelStr
for b2, b3 being Element of the carrier of b1 holds
   b2 <= b3
iff
   waybelow b2 c= waybelow b3;

:: WAYBEL_3:condreg 6
registration
  cluster non empty reflexive transitive antisymmetric complete connected -> satisfying_axiom_of_approximation (RelStr);
end;

:: WAYBEL_3:th 26
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
      st for b2 being Element of the carrier of b1 holds
           b2 is compact(b1)
   holds b1 is satisfying_axiom_of_approximation;

:: WAYBEL_3:attrnot 5 => WAYBEL_3:attr 4
definition
  let a1 be Relation-like set;
  attr a1 is non-Empty means
    for b1 being 1-sorted
          st b1 in proj2 a1
       holds b1 is not empty;
end;

:: WAYBEL_3:dfs 7
definiens
  let a1 be Relation-like set;
To prove
     a1 is non-Empty
it is sufficient to prove
  thus for b1 being 1-sorted
          st b1 in proj2 a1
       holds b1 is not empty;

:: WAYBEL_3:def 7
theorem
for b1 being Relation-like set holds
      b1 is non-Empty
   iff
      for b2 being 1-sorted
            st b2 in proj2 b1
         holds b2 is not empty;

:: WAYBEL_3:attrnot 6 => WAYBEL_3:attr 5
definition
  let a1 be Relation-like set;
  attr a1 is reflexive-yielding means
    for b1 being RelStr
          st b1 in proj2 a1
       holds b1 is reflexive;
end;

:: WAYBEL_3:dfs 8
definiens
  let a1 be Relation-like set;
To prove
     a1 is reflexive-yielding
it is sufficient to prove
  thus for b1 being RelStr
          st b1 in proj2 a1
       holds b1 is reflexive;

:: WAYBEL_3:def 8
theorem
for b1 being Relation-like set holds
      b1 is reflexive-yielding
   iff
      for b2 being RelStr
            st b2 in proj2 b1
         holds b2 is reflexive;

:: WAYBEL_3:exreg 3
registration
  let a1 be set;
  cluster Relation-like Function-like RelStr-yielding non-Empty reflexive-yielding ManySortedSet of a1;
end;

:: WAYBEL_3:funcreg 7
registration
  let a1 be set;
  let a2 be RelStr-yielding non-Empty ManySortedSet of a1;
  cluster product a2 -> non empty strict;
end;

:: WAYBEL_3:funcnot 3 => WAYBEL_3:func 3
definition
  let a1 be non empty set;
  let a2 be RelStr-yielding ManySortedSet of a1;
  let a3 be Element of a1;
  redefine func a2 . a3 -> RelStr;
end;

:: WAYBEL_3:funcreg 8
registration
  let a1 be non empty set;
  let a2 be RelStr-yielding non-Empty ManySortedSet of a1;
  let a3 be Element of a1;
  cluster a2 . a3 -> non empty;
end;

:: WAYBEL_3:funcreg 9
registration
  let a1 be set;
  let a2 be RelStr-yielding non-Empty ManySortedSet of a1;
  cluster product a2 -> constituted-Functions strict;
end;

:: WAYBEL_3:funcnot 4 => WAYBEL_3:func 4
definition
  let a1 be non empty set;
  let a2 be RelStr-yielding non-Empty ManySortedSet of a1;
  let a3 be Element of the carrier of product a2;
  let a4 be Element of a1;
  redefine func a3 . a4 -> Element of the carrier of a2 . a4;
end;

:: WAYBEL_3:funcnot 5 => WAYBEL_3:func 5
definition
  let a1 be non empty set;
  let a2 be RelStr-yielding non-Empty ManySortedSet of a1;
  let a3 be Element of a1;
  let a4 be Element of bool the carrier of product a2;
  redefine func pi(a4,a3) -> Element of bool the carrier of a2 . a3;
end;

:: WAYBEL_3:th 27
theorem
for b1 being non empty set
for b2 being RelStr-yielding non-Empty ManySortedSet of b1
for b3 being Relation-like Function-like set holds
      b3 is Element of the carrier of product b2
   iff
      proj1 b3 = b1 &
       (for b4 being Element of b1 holds
          b3 . b4 is Element of the carrier of b2 . b4);

:: WAYBEL_3:th 28
theorem
for b1 being non empty set
for b2 being RelStr-yielding non-Empty ManySortedSet of b1
for b3, b4 being Element of the carrier of product b2 holds
   b3 <= b4
iff
   for b5 being Element of b1 holds
      b3 . b5 <= b4 . b5;

:: WAYBEL_3:funcreg 10
registration
  let a1 be non empty set;
  let a2 be RelStr-yielding reflexive-yielding ManySortedSet of a1;
  let a3 be Element of a1;
  cluster a2 . a3 -> reflexive;
end;

:: WAYBEL_3:funcreg 11
registration
  let a1 be non empty set;
  let a2 be RelStr-yielding non-Empty reflexive-yielding ManySortedSet of a1;
  cluster product a2 -> strict reflexive;
end;

:: WAYBEL_3:th 29
theorem
for b1 being non empty set
for b2 being RelStr-yielding non-Empty ManySortedSet of b1
      st for b3 being Element of b1 holds
           b2 . b3 is transitive
   holds product b2 is transitive;

:: WAYBEL_3:th 30
theorem
for b1 being non empty set
for b2 being RelStr-yielding non-Empty ManySortedSet of b1
      st for b3 being Element of b1 holds
           b2 . b3 is antisymmetric
   holds product b2 is antisymmetric;

:: WAYBEL_3:th 31
theorem
for b1 being non empty set
for b2 being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of b1
      st for b3 being Element of b1 holds
           b2 . b3 is reflexive transitive antisymmetric with_suprema with_infima complete RelStr
   holds product b2 is reflexive transitive antisymmetric with_suprema with_infima complete RelStr;

:: WAYBEL_3:th 32
theorem
for b1 being non empty set
for b2 being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of b1
   st for b3 being Element of b1 holds
        b2 . b3 is reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Element of bool the carrier of product b2
for b4 being Element of b1 holds
   ("\/"(b3,product b2)) . b4 = "\/"(pi(b3,b4),b2 . b4);

:: WAYBEL_3:th 33
theorem
for b1 being non empty set
for b2 being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of b1
   st for b3 being Element of b1 holds
        b2 . b3 is reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3, b4 being Element of the carrier of product b2 holds
   b3 is_way_below b4
iff
   (for b5 being Element of b1 holds
       b3 . b5 is_way_below b4 . b5) &
    (ex b5 being finite Element of bool b1 st
       for b6 being Element of b1
             st not b6 in b5
          holds b3 . b6 = Bottom (b2 . b6));

:: WAYBEL_3: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 is_way_below b3
for b4 being Element of bool bool the carrier of b1
      st b4 is open(b1) & b3 c= union b4
   holds ex b5 being finite Element of bool b4 st
      b2 c= union b5;

:: WAYBEL_3:th 35
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of InclPoset the topology of b1
      st for b4 being Element of bool bool the carrier of b1
              st b4 is open(b1) & b3 c= union b4
           holds ex b5 being finite Element of bool b4 st
              b2 c= union b5
   holds b2 is_way_below b3;

:: WAYBEL_3:th 36
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of InclPoset the topology of b1
for b3 being Element of bool the carrier of b1
      st b2 = b3
   holds    b2 is compact(InclPoset the topology of b1)
   iff
      b3 is compact(b1);

:: WAYBEL_3:th 37
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of InclPoset the topology of b1
      st b2 = the carrier of b1
   holds    b2 is compact(InclPoset the topology of b1)
   iff
      b1 is compact;

:: WAYBEL_3:attrnot 7 => WAYBEL_3:attr 6
definition
  let a1 be non empty TopSpace-like TopStruct;
  attr a1 is locally-compact means
    for b1 being Element of the carrier of a1
    for b2 being Element of bool the carrier of a1
          st b1 in b2 & b2 is open(a1)
       holds ex b3 being Element of bool the carrier of a1 st
          b1 in Int b3 & b3 c= b2 & b3 is compact(a1);
end;

:: WAYBEL_3:dfs 9
definiens
  let a1 be non empty TopSpace-like TopStruct;
To prove
     a1 is locally-compact
it is sufficient to prove
  thus for b1 being Element of the carrier of a1
    for b2 being Element of bool the carrier of a1
          st b1 in b2 & b2 is open(a1)
       holds ex b3 being Element of bool the carrier of a1 st
          b1 in Int b3 & b3 c= b2 & b3 is compact(a1);

:: WAYBEL_3:def 9
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is locally-compact
   iff
      for b2 being Element of the carrier of b1
      for b3 being Element of bool the carrier of b1
            st b2 in b3 & b3 is open(b1)
         holds ex b4 being Element of bool the carrier of b1 st
            b2 in Int b4 & b4 c= b3 & b4 is compact(b1);

:: WAYBEL_3:condreg 7
registration
  cluster non empty TopSpace-like compact being_T2 -> being_T3 being_T4 locally-compact (TopStruct);
end;

:: WAYBEL_3:th 38
theorem
for b1 being set holds
   1TopSp {b1} is being_T2;

:: WAYBEL_3:exreg 4
registration
  cluster non empty TopSpace-like compact being_T2 TopStruct;
end;

:: WAYBEL_3:th 39
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of InclPoset the topology of b1
      st ex b4 being Element of bool the carrier of b1 st
           b2 c= b4 & b4 c= b3 & b4 is compact(b1)
   holds b2 is_way_below b3;

:: WAYBEL_3:th 40
theorem
for b1 being non empty TopSpace-like TopStruct
   st b1 is locally-compact
for b2, b3 being Element of the carrier of InclPoset the topology of b1
      st b2 is_way_below b3
   holds ex b4 being Element of bool the carrier of b1 st
      b2 c= b4 & b4 c= b3 & b4 is compact(b1);

:: WAYBEL_3:th 41
theorem
for b1 being non empty TopSpace-like TopStruct
   st b1 is locally-compact & b1 is being_T2
for b2, b3 being Element of the carrier of InclPoset the topology of b1
      st b2 is_way_below b3
   holds ex b4 being Element of bool the carrier of b1 st
      b4 = b2 & Cl b4 c= b3 & Cl b4 is compact(b1);

:: WAYBEL_3:th 42
theorem
for b1 being non empty TopSpace-like TopStruct
      st b1 is being_T3 & InclPoset the topology of b1 is continuous
   holds b1 is locally-compact;

:: WAYBEL_3:th 43
theorem
for b1 being non empty TopSpace-like TopStruct
      st b1 is locally-compact
   holds InclPoset the topology of b1 is continuous;