Article WAYBEL23, MML version 4.99.1005

:: WAYBEL23:th 1
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being Element of the carrier of b1 holds
   compactbelow b2 = (waybelow b2) /\ the carrier of CompactSublatt b1;

:: WAYBEL23:funcnot 1 => WAYBEL23:func 1
definition
  let a1 be non empty reflexive transitive RelStr;
  let a2 be Element of bool the carrier of InclPoset Ids a1;
  redefine func union a2 -> Element of bool the carrier of a1;
end;

:: WAYBEL23:th 2
theorem
for b1 being non empty RelStr
for b2, b3 being Element of bool the carrier of b1
      st b2 c= b3
   holds finsups b2 c= finsups b3;

:: WAYBEL23:th 3
theorem
for b1 being non empty transitive RelStr
for b2 being non empty full sups-inheriting SubRelStr of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4
   holds finsups b3 c= finsups b4;

:: WAYBEL23:th 4
theorem
for b1 being non empty transitive antisymmetric complete RelStr
for b2 being non empty full sups-inheriting SubRelStr of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4
   holds finsups b3 = finsups b4;

:: WAYBEL23:th 5
theorem
for b1 being reflexive transitive antisymmetric with_suprema complete RelStr
for b2 being non empty full join-inheriting SubRelStr of b1
   st Bottom b1 in the carrier of b2
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4
   holds finsups b4 c= finsups b3;

:: WAYBEL23:th 6
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded RelStr
for b2 being Element of bool the carrier of InclPoset Ids b1 holds
   "\/"(b2,InclPoset Ids b1) = downarrow finsups union b2;

:: WAYBEL23:th 7
theorem
for b1 being reflexive transitive RelStr
for b2 being Element of bool the carrier of b1 holds
   downarrow downarrow b2 = downarrow b2;

:: WAYBEL23:th 8
theorem
for b1 being reflexive transitive RelStr
for b2 being Element of bool the carrier of b1 holds
   uparrow uparrow b2 = uparrow b2;

:: WAYBEL23:th 9
theorem
for b1 being non empty reflexive transitive RelStr
for b2 being Element of the carrier of b1 holds
   downarrow downarrow b2 = downarrow b2;

:: WAYBEL23:th 10
theorem
for b1 being non empty reflexive transitive RelStr
for b2 being Element of the carrier of b1 holds
   uparrow uparrow b2 = uparrow b2;

:: WAYBEL23:th 11
theorem
for b1 being non empty RelStr
for b2 being non empty SubRelStr of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4
   holds downarrow b4 c= downarrow b3;

:: WAYBEL23:th 12
theorem
for b1 being non empty RelStr
for b2 being non empty SubRelStr of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4
   holds uparrow b4 c= uparrow b3;

:: WAYBEL23:th 13
theorem
for b1 being non empty RelStr
for b2 being non empty SubRelStr of b1
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
      st b3 = b4
   holds downarrow b4 c= downarrow b3;

:: WAYBEL23:th 14
theorem
for b1 being non empty RelStr
for b2 being non empty SubRelStr of b1
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
      st b3 = b4
   holds uparrow b4 c= uparrow b3;

:: WAYBEL23:attrnot 1 => WAYBEL23:attr 1
definition
  let a1 be non empty RelStr;
  let a2 be Element of bool the carrier of a1;
  attr a2 is meet-closed means
    subrelstr a2 is meet-inheriting(a1);
end;

:: WAYBEL23:dfs 1
definiens
  let a1 be non empty RelStr;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is meet-closed
it is sufficient to prove
  thus subrelstr a2 is meet-inheriting(a1);

:: WAYBEL23:def 1
theorem
for b1 being non empty RelStr
for b2 being Element of bool the carrier of b1 holds
      b2 is meet-closed(b1)
   iff
      subrelstr b2 is meet-inheriting(b1);

:: WAYBEL23:attrnot 2 => WAYBEL23:attr 2
definition
  let a1 be non empty RelStr;
  let a2 be Element of bool the carrier of a1;
  attr a2 is join-closed means
    subrelstr a2 is join-inheriting(a1);
end;

:: WAYBEL23:dfs 2
definiens
  let a1 be non empty RelStr;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is join-closed
it is sufficient to prove
  thus subrelstr a2 is join-inheriting(a1);

:: WAYBEL23:def 2
theorem
for b1 being non empty RelStr
for b2 being Element of bool the carrier of b1 holds
      b2 is join-closed(b1)
   iff
      subrelstr b2 is join-inheriting(b1);

:: WAYBEL23:attrnot 3 => WAYBEL23:attr 3
definition
  let a1 be non empty RelStr;
  let a2 be Element of bool the carrier of a1;
  attr a2 is infs-closed means
    subrelstr a2 is infs-inheriting(a1);
end;

:: WAYBEL23:dfs 3
definiens
  let a1 be non empty RelStr;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is infs-closed
it is sufficient to prove
  thus subrelstr a2 is infs-inheriting(a1);

:: WAYBEL23:def 3
theorem
for b1 being non empty RelStr
for b2 being Element of bool the carrier of b1 holds
      b2 is infs-closed(b1)
   iff
      subrelstr b2 is infs-inheriting(b1);

:: WAYBEL23:attrnot 4 => WAYBEL23:attr 4
definition
  let a1 be non empty RelStr;
  let a2 be Element of bool the carrier of a1;
  attr a2 is sups-closed means
    subrelstr a2 is sups-inheriting(a1);
end;

:: WAYBEL23:dfs 4
definiens
  let a1 be non empty RelStr;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is sups-closed
it is sufficient to prove
  thus subrelstr a2 is sups-inheriting(a1);

:: WAYBEL23:def 4
theorem
for b1 being non empty RelStr
for b2 being Element of bool the carrier of b1 holds
      b2 is sups-closed(b1)
   iff
      subrelstr b2 is sups-inheriting(b1);

:: WAYBEL23:condreg 1
registration
  let a1 be non empty RelStr;
  cluster infs-closed -> meet-closed (Element of bool the carrier of a1);
end;

:: WAYBEL23:condreg 2
registration
  let a1 be non empty RelStr;
  cluster sups-closed -> join-closed (Element of bool the carrier of a1);
end;

:: WAYBEL23:exreg 1
registration
  let a1 be non empty RelStr;
  cluster non empty infs-closed sups-closed Element of bool the carrier of a1;
end;

:: WAYBEL23:th 15
theorem
for b1 being non empty RelStr
for b2 being Element of bool the carrier of b1 holds
      b2 is meet-closed(b1)
   iff
      for b3, b4 being Element of the carrier of b1
            st b3 in b2 & b4 in b2 & ex_inf_of {b3,b4},b1
         holds "/\"({b3,b4},b1) in b2;

:: WAYBEL23:th 16
theorem
for b1 being non empty RelStr
for b2 being Element of bool the carrier of b1 holds
      b2 is join-closed(b1)
   iff
      for b3, b4 being Element of the carrier of b1
            st b3 in b2 & b4 in b2 & ex_sup_of {b3,b4},b1
         holds "\/"({b3,b4},b1) in b2;

:: WAYBEL23:th 17
theorem
for b1 being antisymmetric with_infima RelStr
for b2 being Element of bool the carrier of b1 holds
      b2 is meet-closed(b1)
   iff
      for b3, b4 being Element of the carrier of b1
            st b3 in b2 & b4 in b2
         holds "/\"({b3,b4},b1) in b2;

:: WAYBEL23:th 18
theorem
for b1 being antisymmetric with_suprema RelStr
for b2 being Element of bool the carrier of b1 holds
      b2 is join-closed(b1)
   iff
      for b3, b4 being Element of the carrier of b1
            st b3 in b2 & b4 in b2
         holds "\/"({b3,b4},b1) in b2;

:: WAYBEL23:th 19
theorem
for b1 being non empty RelStr
for b2 being Element of bool the carrier of b1 holds
      b2 is infs-closed(b1)
   iff
      for b3 being Element of bool b2
            st ex_inf_of b3,b1
         holds "/\"(b3,b1) in b2;

:: WAYBEL23:th 20
theorem
for b1 being non empty RelStr
for b2 being Element of bool the carrier of b1 holds
      b2 is sups-closed(b1)
   iff
      for b3 being Element of bool b2
            st ex_sup_of b3,b1
         holds "\/"(b3,b1) in b2;

:: WAYBEL23:th 21
theorem
for b1 being non empty transitive RelStr
for b2 being non empty infs-closed Element of bool the carrier of b1
for b3 being Element of bool b2
      st ex_inf_of b3,b1
   holds ex_inf_of b3,subrelstr b2 & "/\"(b3,subrelstr b2) = "/\"(b3,b1);

:: WAYBEL23:th 22
theorem
for b1 being non empty transitive RelStr
for b2 being non empty sups-closed Element of bool the carrier of b1
for b3 being Element of bool b2
      st ex_sup_of b3,b1
   holds ex_sup_of b3,subrelstr b2 & "\/"(b3,subrelstr b2) = "\/"(b3,b1);

:: WAYBEL23:th 23
theorem
for b1 being non empty transitive RelStr
for b2 being non empty meet-closed Element of bool the carrier of b1
for b3, b4 being Element of b2
      st ex_inf_of {b3,b4},b1
   holds ex_inf_of {b3,b4},subrelstr b2 &
    "/\"({b3,b4},subrelstr b2) = "/\"({b3,b4},b1);

:: WAYBEL23:th 24
theorem
for b1 being non empty transitive RelStr
for b2 being non empty join-closed Element of bool the carrier of b1
for b3, b4 being Element of b2
      st ex_sup_of {b3,b4},b1
   holds ex_sup_of {b3,b4},subrelstr b2 &
    "\/"({b3,b4},subrelstr b2) = "\/"({b3,b4},b1);

:: WAYBEL23:th 25
theorem
for b1 being transitive antisymmetric with_infima RelStr
for b2 being non empty meet-closed Element of bool the carrier of b1 holds
   subrelstr b2 is with_infima;

:: WAYBEL23:th 26
theorem
for b1 being transitive antisymmetric with_suprema RelStr
for b2 being non empty join-closed Element of bool the carrier of b1 holds
   subrelstr b2 is with_suprema;

:: WAYBEL23:funcreg 1
registration
  let a1 be transitive antisymmetric with_infima RelStr;
  let a2 be non empty meet-closed Element of bool the carrier of a1;
  cluster subrelstr a2 -> strict with_infima full;
end;

:: WAYBEL23:funcreg 2
registration
  let a1 be transitive antisymmetric with_suprema RelStr;
  let a2 be non empty join-closed Element of bool the carrier of a1;
  cluster subrelstr a2 -> strict with_suprema full;
end;

:: WAYBEL23:th 27
theorem
for b1 being non empty transitive antisymmetric complete RelStr
for b2 being non empty infs-closed Element of bool the carrier of b1
for b3 being Element of bool b2 holds
   "/\"(b3,subrelstr b2) = "/\"(b3,b1);

:: WAYBEL23:th 28
theorem
for b1 being non empty transitive antisymmetric complete RelStr
for b2 being non empty sups-closed Element of bool the carrier of b1
for b3 being Element of bool b2 holds
   "\/"(b3,subrelstr b2) = "\/"(b3,b1);

:: WAYBEL23:th 29
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being meet-closed Element of bool the carrier of b1 holds
   b2 is filtered(b1);

:: WAYBEL23:th 30
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2 being join-closed Element of bool the carrier of b1 holds
   b2 is directed(b1);

:: WAYBEL23:condreg 3
registration
  let a1 be reflexive transitive antisymmetric with_infima RelStr;
  cluster meet-closed -> filtered (Element of bool the carrier of a1);
end;

:: WAYBEL23:condreg 4
registration
  let a1 be reflexive transitive antisymmetric with_suprema RelStr;
  cluster join-closed -> directed (Element of bool the carrier of a1);
end;

:: WAYBEL23:th 31
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being non empty upper Element of bool the carrier of b1 holds
      b2 is non empty filtered upper Element of bool the carrier of b1
   iff
      b2 is meet-closed(b1);

:: WAYBEL23:th 32
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2 being non empty lower Element of bool the carrier of b1 holds
      b2 is non empty directed lower Element of bool the carrier of b1
   iff
      b2 is join-closed(b1);

:: WAYBEL23:th 33
theorem
for b1 being non empty RelStr
for b2, b3 being join-closed Element of bool the carrier of b1 holds
b2 /\ b3 is join-closed(b1);

:: WAYBEL23:th 34
theorem
for b1 being non empty RelStr
for b2, b3 being meet-closed Element of bool the carrier of b1 holds
b2 /\ b3 is meet-closed(b1);

:: WAYBEL23:th 35
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2 being Element of the carrier of b1 holds
   downarrow b2 is join-closed(b1);

:: WAYBEL23:th 36
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being Element of the carrier of b1 holds
   downarrow b2 is meet-closed(b1);

:: WAYBEL23:th 37
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2 being Element of the carrier of b1 holds
   uparrow b2 is join-closed(b1);

:: WAYBEL23:th 38
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being Element of the carrier of b1 holds
   uparrow b2 is meet-closed(b1);

:: WAYBEL23:funcreg 3
registration
  let a1 be reflexive transitive antisymmetric with_suprema RelStr;
  let a2 be Element of the carrier of a1;
  cluster downarrow a2 -> join-closed;
end;

:: WAYBEL23:funcreg 4
registration
  let a1 be reflexive transitive antisymmetric with_suprema RelStr;
  let a2 be Element of the carrier of a1;
  cluster uparrow a2 -> join-closed;
end;

:: WAYBEL23:funcreg 5
registration
  let a1 be reflexive transitive antisymmetric with_infima RelStr;
  let a2 be Element of the carrier of a1;
  cluster downarrow a2 -> meet-closed;
end;

:: WAYBEL23:funcreg 6
registration
  let a1 be reflexive transitive antisymmetric with_infima RelStr;
  let a2 be Element of the carrier of a1;
  cluster uparrow a2 -> meet-closed;
end;

:: WAYBEL23:th 39
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2 being Element of the carrier of b1 holds
   waybelow b2 is join-closed(b1);

:: WAYBEL23:th 40
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being Element of the carrier of b1 holds
   waybelow b2 is meet-closed(b1);

:: WAYBEL23:th 41
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2 being Element of the carrier of b1 holds
   wayabove b2 is join-closed(b1);

:: WAYBEL23:funcreg 7
registration
  let a1 be reflexive transitive antisymmetric with_suprema RelStr;
  let a2 be Element of the carrier of a1;
  cluster waybelow a2 -> join-closed;
end;

:: WAYBEL23:funcreg 8
registration
  let a1 be reflexive transitive antisymmetric with_suprema RelStr;
  let a2 be Element of the carrier of a1;
  cluster wayabove a2 -> join-closed;
end;

:: WAYBEL23:funcreg 9
registration
  let a1 be reflexive transitive antisymmetric with_infima RelStr;
  let a2 be Element of the carrier of a1;
  cluster waybelow a2 -> meet-closed;
end;

:: WAYBEL23:funcnot 2 => WAYBEL23:func 2
definition
  let a1 be TopStruct;
  func weight A1 -> cardinal set equals
    meet {Card b1 where b1 is Basis of a1: TRUE};
end;

:: WAYBEL23:def 5
theorem
for b1 being TopStruct holds
   weight b1 = meet {Card b2 where b2 is Basis of b1: TRUE};

:: WAYBEL23:attrnot 5 => WAYBEL23:attr 5
definition
  let a1 be TopStruct;
  attr a1 is second-countable means
    weight a1 c= omega;
end;

:: WAYBEL23:dfs 6
definiens
  let a1 be TopStruct;
To prove
     a1 is second-countable
it is sufficient to prove
  thus weight a1 c= omega;

:: WAYBEL23:def 6
theorem
for b1 being TopStruct holds
      b1 is second-countable
   iff
      weight b1 c= omega;

:: WAYBEL23:modenot 1 => WAYBEL23:mode 1
definition
  let a1 be reflexive transitive antisymmetric with_suprema continuous RelStr;
  mode CLbasis of A1 -> Element of bool the carrier of a1 means
    it is join-closed(a1) &
     (for b1 being Element of the carrier of a1 holds
        b1 = "\/"((waybelow b1) /\ it,a1));
end;

:: WAYBEL23:dfs 7
definiens
  let a1 be reflexive transitive antisymmetric with_suprema continuous RelStr;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is CLbasis of a1
it is sufficient to prove
  thus a2 is join-closed(a1) &
     (for b1 being Element of the carrier of a1 holds
        b1 = "\/"((waybelow b1) /\ a2,a1));

:: WAYBEL23:def 7
theorem
for b1 being reflexive transitive antisymmetric with_suprema continuous RelStr
for b2 being Element of bool the carrier of b1 holds
      b2 is CLbasis of b1
   iff
      b2 is join-closed(b1) &
       (for b3 being Element of the carrier of b1 holds
          b3 = "\/"((waybelow b3) /\ b2,b1));

:: WAYBEL23:attrnot 6 => WAYBEL23:attr 6
definition
  let a1 be non empty RelStr;
  let a2 be Element of bool the carrier of a1;
  attr a2 is with_bottom means
    Bottom a1 in a2;
end;

:: WAYBEL23:dfs 8
definiens
  let a1 be non empty RelStr;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is with_bottom
it is sufficient to prove
  thus Bottom a1 in a2;

:: WAYBEL23:def 8
theorem
for b1 being non empty RelStr
for b2 being Element of bool the carrier of b1 holds
      b2 is with_bottom(b1)
   iff
      Bottom b1 in b2;

:: WAYBEL23:attrnot 7 => WAYBEL23:attr 7
definition
  let a1 be non empty RelStr;
  let a2 be Element of bool the carrier of a1;
  attr a2 is with_top means
    Top a1 in a2;
end;

:: WAYBEL23:dfs 9
definiens
  let a1 be non empty RelStr;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is with_top
it is sufficient to prove
  thus Top a1 in a2;

:: WAYBEL23:def 9
theorem
for b1 being non empty RelStr
for b2 being Element of bool the carrier of b1 holds
      b2 is with_top(b1)
   iff
      Top b1 in b2;

:: WAYBEL23:condreg 5
registration
  let a1 be non empty RelStr;
  cluster with_bottom -> non empty (Element of bool the carrier of a1);
end;

:: WAYBEL23:condreg 6
registration
  let a1 be non empty RelStr;
  cluster with_top -> non empty (Element of bool the carrier of a1);
end;

:: WAYBEL23:exreg 2
registration
  let a1 be non empty RelStr;
  cluster with_bottom Element of bool the carrier of a1;
end;

:: WAYBEL23:exreg 3
registration
  let a1 be non empty RelStr;
  cluster with_top Element of bool the carrier of a1;
end;

:: WAYBEL23:exreg 4
registration
  let a1 be reflexive transitive antisymmetric with_suprema continuous RelStr;
  cluster with_bottom CLbasis of a1;
end;

:: WAYBEL23:exreg 5
registration
  let a1 be reflexive transitive antisymmetric with_suprema continuous RelStr;
  cluster with_top CLbasis of a1;
end;

:: WAYBEL23:th 42
theorem
for b1 being non empty antisymmetric lower-bounded RelStr
for b2 being with_bottom Element of bool the carrier of b1 holds
   subrelstr b2 is lower-bounded;

:: WAYBEL23:funcreg 10
registration
  let a1 be non empty antisymmetric lower-bounded RelStr;
  let a2 be with_bottom Element of bool the carrier of a1;
  cluster subrelstr a2 -> strict lower-bounded full;
end;

:: WAYBEL23:condreg 7
registration
  let a1 be reflexive transitive antisymmetric with_suprema continuous RelStr;
  cluster -> join-closed (CLbasis of a1);
end;

:: WAYBEL23:exreg 6
registration
  cluster non empty non trivial reflexive transitive antisymmetric with_suprema with_infima upper-bounded bounded up-complete satisfying_axiom_of_approximation continuous RelStr;
end;

:: WAYBEL23:condreg 8
registration
  let a1 be non trivial reflexive transitive antisymmetric with_suprema lower-bounded continuous RelStr;
  cluster -> non empty (CLbasis of a1);
end;

:: WAYBEL23:th 43
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr holds
   the carrier of CompactSublatt b1 is join-closed Element of bool the carrier of b1;

:: WAYBEL23:th 44
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded algebraic RelStr holds
   the carrier of CompactSublatt b1 is with_bottom CLbasis of b1;

:: WAYBEL23:th 45
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded continuous RelStr
      st the carrier of CompactSublatt b1 is CLbasis of b1
   holds b1 is algebraic;

:: WAYBEL23:th 46
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded continuous RelStr
for b2 being join-closed Element of bool the carrier of b1 holds
      b2 is CLbasis of b1
   iff
      for b3, b4 being Element of the carrier of b1
            st not b4 <= b3
         holds ex b5 being Element of the carrier of b1 st
            b5 in b2 & not b5 <= b3 & b5 is_way_below b4;

:: WAYBEL23:th 47
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded continuous RelStr
for b2 being join-closed Element of bool the carrier of b1
      st Bottom b1 in b2
   holds    b2 is CLbasis of b1
   iff
      for b3, b4 being Element of the carrier of b1
            st b3 is_way_below b4
         holds ex b5 being Element of the carrier of b1 st
            b5 in b2 & b3 <= b5 & b5 is_way_below b4;

:: WAYBEL23:th 48
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded continuous RelStr
for b2 being join-closed Element of bool the carrier of b1
      st Bottom b1 in b2
   holds    b2 is CLbasis of b1
   iff
      the carrier of CompactSublatt b1 c= b2 &
       (for b3, b4 being Element of the carrier of b1
             st not b4 <= b3
          holds ex b5 being Element of the carrier of b1 st
             b5 in b2 & not b5 <= b3 & b5 <= b4);

:: WAYBEL23:th 49
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded continuous RelStr
for b2 being join-closed Element of bool the carrier of b1
      st Bottom b1 in b2
   holds    b2 is CLbasis of b1
   iff
      for b3, b4 being Element of the carrier of b1
            st not b4 <= b3
         holds ex b5 being Element of the carrier of b1 st
            b5 in b2 & not b5 <= b3 & b5 <= b4;

:: WAYBEL23:th 50
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded RelStr
for b2 being non empty full SubRelStr of b1
   st Bottom b1 in the carrier of b2 &
      the carrier of b2 is join-closed Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
   (waybelow b3) /\ the carrier of b2 is non empty directed lower Element of bool the carrier of b2;

:: WAYBEL23:funcnot 3 => WAYBEL23:func 3
definition
  let a1 be non empty reflexive transitive RelStr;
  let a2 be non empty full SubRelStr of a1;
  func supMap A2 -> Function-like quasi_total Relation of the carrier of InclPoset Ids a2,the carrier of a1 means
    for b1 being non empty directed lower Element of bool the carrier of a2 holds
       it . b1 = "\/"(b1,a1);
end;

:: WAYBEL23:def 10
theorem
for b1 being non empty reflexive transitive RelStr
for b2 being non empty full SubRelStr of b1
for b3 being Function-like quasi_total Relation of the carrier of InclPoset Ids b2,the carrier of b1 holds
      b3 = supMap b2
   iff
      for b4 being non empty directed lower Element of bool the carrier of b2 holds
         b3 . b4 = "\/"(b4,b1);

:: WAYBEL23:funcnot 4 => WAYBEL23:func 4
definition
  let a1 be non empty reflexive transitive RelStr;
  let a2 be non empty full SubRelStr of a1;
  func idsMap A2 -> Function-like quasi_total Relation of the carrier of InclPoset Ids a2,the carrier of InclPoset Ids a1 means
    for b1 being non empty directed lower Element of bool the carrier of a2 holds
       ex b2 being Element of bool the carrier of a1 st
          b1 = b2 & it . b1 = downarrow b2;
end;

:: WAYBEL23:def 11
theorem
for b1 being non empty reflexive transitive RelStr
for b2 being non empty full SubRelStr of b1
for b3 being Function-like quasi_total Relation of the carrier of InclPoset Ids b2,the carrier of InclPoset Ids b1 holds
      b3 = idsMap b2
   iff
      for b4 being non empty directed lower Element of bool the carrier of b2 holds
         ex b5 being Element of bool the carrier of b1 st
            b4 = b5 & b3 . b4 = downarrow b5;

:: WAYBEL23:funcreg 11
registration
  let a1 be reflexive RelStr;
  let a2 be Element of bool the carrier of a1;
  cluster subrelstr a2 -> strict reflexive full;
end;

:: WAYBEL23:funcreg 12
registration
  let a1 be transitive RelStr;
  let a2 be Element of bool the carrier of a1;
  cluster subrelstr a2 -> strict transitive full;
end;

:: WAYBEL23:funcreg 13
registration
  let a1 be antisymmetric RelStr;
  let a2 be Element of bool the carrier of a1;
  cluster subrelstr a2 -> strict antisymmetric full;
end;

:: WAYBEL23:funcnot 5 => WAYBEL23:func 5
definition
  let a1 be reflexive transitive antisymmetric with_suprema lower-bounded continuous RelStr;
  let a2 be with_bottom CLbasis of a1;
  func baseMap A2 -> Function-like quasi_total Relation of the carrier of a1,the carrier of InclPoset Ids subrelstr a2 means
    for b1 being Element of the carrier of a1 holds
       it . b1 = (waybelow b1) /\ a2;
end;

:: WAYBEL23:def 12
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded continuous RelStr
for b2 being with_bottom CLbasis of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of InclPoset Ids subrelstr b2 holds
      b3 = baseMap b2
   iff
      for b4 being Element of the carrier of b1 holds
         b3 . b4 = (waybelow b4) /\ b2;

:: WAYBEL23:th 51
theorem
for b1 being non empty reflexive transitive RelStr
for b2 being non empty full SubRelStr of b1 holds
   dom supMap b2 = Ids b2 & rng supMap b2 is Element of bool the carrier of b1;

:: WAYBEL23:th 52
theorem
for b1 being non empty reflexive transitive RelStr
for b2 being non empty full SubRelStr of b1
for b3 being set holds
      b3 in dom supMap b2
   iff
      b3 is non empty directed lower Element of bool the carrier of b2;

:: WAYBEL23:th 53
theorem
for b1 being non empty reflexive transitive RelStr
for b2 being non empty full SubRelStr of b1 holds
   dom idsMap b2 = Ids b2 & rng idsMap b2 is Element of bool Ids b1;

:: WAYBEL23:th 54
theorem
for b1 being non empty reflexive transitive RelStr
for b2 being non empty full SubRelStr of b1
for b3 being set holds
      b3 in dom idsMap b2
   iff
      b3 is non empty directed lower Element of bool the carrier of b2;

:: WAYBEL23:th 55
theorem
for b1 being non empty reflexive transitive RelStr
for b2 being non empty full SubRelStr of b1
for b3 being set
      st b3 in rng idsMap b2
   holds b3 is non empty directed lower Element of bool the carrier of b1;

:: WAYBEL23:th 56
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded continuous RelStr
for b2 being with_bottom CLbasis of b1 holds
   dom baseMap b2 = the carrier of b1 &
    rng baseMap b2 is Element of bool Ids subrelstr b2;

:: WAYBEL23:th 57
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded continuous RelStr
for b2 being with_bottom CLbasis of b1
for b3 being set
      st b3 in rng baseMap b2
   holds b3 is non empty directed lower Element of bool the carrier of subrelstr b2;

:: WAYBEL23:th 58
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr
for b2 being non empty full SubRelStr of b1 holds
   supMap b2 is monotone(InclPoset Ids b2, b1);

:: WAYBEL23:th 59
theorem
for b1 being non empty reflexive transitive RelStr
for b2 being non empty full SubRelStr of b1 holds
   idsMap b2 is monotone(InclPoset Ids b2, InclPoset Ids b1);

:: WAYBEL23:th 60
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded continuous RelStr
for b2 being with_bottom CLbasis of b1 holds
   baseMap b2 is monotone(b1, InclPoset Ids subrelstr b2);

:: WAYBEL23:funcreg 14
registration
  let a1 be non empty reflexive transitive antisymmetric up-complete RelStr;
  let a2 be non empty full SubRelStr of a1;
  cluster supMap a2 -> Function-like quasi_total monotone;
end;

:: WAYBEL23:funcreg 15
registration
  let a1 be non empty reflexive transitive RelStr;
  let a2 be non empty full SubRelStr of a1;
  cluster idsMap a2 -> Function-like quasi_total monotone;
end;

:: WAYBEL23:funcreg 16
registration
  let a1 be reflexive transitive antisymmetric with_suprema lower-bounded continuous RelStr;
  let a2 be with_bottom CLbasis of a1;
  cluster baseMap a2 -> Function-like quasi_total monotone;
end;

:: WAYBEL23:th 61
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded continuous RelStr
for b2 being with_bottom CLbasis of b1 holds
   idsMap subrelstr b2 is sups-preserving(InclPoset Ids subrelstr b2, InclPoset Ids b1);

:: WAYBEL23:th 62
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr
for b2 being non empty full SubRelStr of b1 holds
   supMap b2 = (SupMap b1) * idsMap b2;

:: WAYBEL23:th 63
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded continuous RelStr
for b2 being with_bottom CLbasis of b1 holds
   [supMap subrelstr b2,baseMap b2] is Galois(InclPoset Ids subrelstr b2, b1);

:: WAYBEL23:th 64
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded continuous RelStr
for b2 being with_bottom CLbasis of b1 holds
   supMap subrelstr b2 is upper_adjoint(InclPoset Ids subrelstr b2, b1) &
    baseMap b2 is lower_adjoint(InclPoset Ids subrelstr b2, b1);

:: WAYBEL23:th 65
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded continuous RelStr
for b2 being with_bottom CLbasis of b1 holds
   rng supMap subrelstr b2 = the carrier of b1;

:: WAYBEL23:th 66
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded continuous RelStr
for b2 being with_bottom CLbasis of b1 holds
   supMap subrelstr b2 is infs-preserving(InclPoset Ids subrelstr b2, b1) &
    supMap subrelstr b2 is sups-preserving(InclPoset Ids subrelstr b2, b1);

:: WAYBEL23:th 67
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded continuous RelStr
for b2 being with_bottom CLbasis of b1 holds
   baseMap b2 is sups-preserving(b1, InclPoset Ids subrelstr b2);

:: WAYBEL23:funcreg 17
registration
  let a1 be reflexive transitive antisymmetric with_suprema lower-bounded continuous RelStr;
  let a2 be with_bottom CLbasis of a1;
  cluster supMap subrelstr a2 -> Function-like quasi_total infs-preserving sups-preserving;
end;

:: WAYBEL23:funcreg 18
registration
  let a1 be reflexive transitive antisymmetric with_suprema lower-bounded continuous RelStr;
  let a2 be with_bottom CLbasis of a1;
  cluster baseMap a2 -> Function-like quasi_total sups-preserving;
end;

:: WAYBEL23:th 69
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded continuous RelStr
for b2 being with_bottom CLbasis of b1 holds
   the carrier of CompactSublatt InclPoset Ids subrelstr b2 = {downarrow b3 where b3 is Element of the carrier of subrelstr b2: TRUE};

:: WAYBEL23:th 70
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded continuous RelStr
for b2 being with_bottom CLbasis of b1 holds
   CompactSublatt InclPoset Ids subrelstr b2,subrelstr b2 are_isomorphic;

:: WAYBEL23:th 71
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded continuous RelStr
for b2 being with_bottom CLbasis of b1
   st for b3 being with_bottom CLbasis of b1 holds
        b2 c= b3
for b3 being Element of the carrier of InclPoset Ids subrelstr b2 holds
   b3 = (waybelow "\/"(b3,b1)) /\ b2;

:: WAYBEL23:th 72
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded continuous RelStr holds
      b1 is algebraic
   iff
      the carrier of CompactSublatt b1 is with_bottom CLbasis of b1 &
       (for b2 being with_bottom CLbasis of b1 holds
          the carrier of CompactSublatt b1 c= b2);

:: WAYBEL23:th 73
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded continuous RelStr holds
      b1 is algebraic
   iff
      ex b2 being with_bottom CLbasis of b1 st
         for b3 being with_bottom CLbasis of b1 holds
            b2 c= b3;

:: WAYBEL23:th 74
theorem
for b1 being TopStruct
for b2 being Basis of b1 holds
   weight b1 c= Card b2;

:: WAYBEL23:th 75
theorem
for b1 being TopStruct holds
   ex b2 being Basis of b1 st
      Card b2 = weight b1;