Article WAYBEL31, MML version 4.99.1005

:: WAYBEL31:sch 1
scheme WAYBEL31:sch 1
{F1 -> RelStr}:
for b1 being Element of bool bool the carrier of F1()
      st b1 = {b2 where b2 is Element of bool the carrier of F1(): P1[b2]}
   holds uparrow union b1 = union {uparrow b2 where b2 is Element of bool the carrier of F1(): P1[b2]}


:: WAYBEL31:sch 2
scheme WAYBEL31:sch 2
{F1 -> RelStr}:
for b1 being Element of bool bool the carrier of F1()
      st b1 = {b2 where b2 is Element of bool the carrier of F1(): P1[b2]}
   holds downarrow union b1 = union {downarrow b2 where b2 is Element of bool the carrier of F1(): P1[b2]}


:: WAYBEL31:funcreg 1
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema continuous RelStr;
  let a2 be with_bottom CLbasis of a1;
  cluster InclPoset Ids subrelstr a2 -> strict algebraic;
end;

:: WAYBEL31:funcnot 1 => WAYBEL31:func 1
definition
  let a1 be reflexive transitive antisymmetric with_suprema continuous RelStr;
  func CLweight A1 -> cardinal set equals
    meet {Card b1 where b1 is with_bottom CLbasis of a1: TRUE};
end;

:: WAYBEL31:def 1
theorem
for b1 being reflexive transitive antisymmetric with_suprema continuous RelStr holds
   CLweight b1 = meet {Card b2 where b2 is with_bottom CLbasis of b1: TRUE};

:: WAYBEL31:th 3
theorem
for b1 being reflexive transitive antisymmetric with_suprema continuous RelStr
for b2 being with_bottom CLbasis of b1 holds
   CLweight b1 c= Card b2;

:: WAYBEL31:th 4
theorem
for b1 being reflexive transitive antisymmetric with_suprema continuous RelStr holds
   ex b2 being with_bottom CLbasis of b1 st
      Card b2 = CLweight b1;

:: WAYBEL31:th 5
theorem
for b1 being reflexive transitive antisymmetric lower-bounded algebraic with_suprema with_infima RelStr holds
   CLweight b1 = Card the carrier of CompactSublatt b1;

:: WAYBEL31:th 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being reflexive transitive antisymmetric with_suprema continuous RelStr
   st InclPoset the topology of b1 = b2
for b3 being with_bottom CLbasis of b2 holds
   b3 is Basis of b1;

:: WAYBEL31:th 7
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr
   st InclPoset the topology of b1 = b2
for b3 being Basis of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4
   holds finsups b4 is with_bottom CLbasis of b2;

:: WAYBEL31:th 8
theorem
for b1 being non empty TopSpace-like discerning TopStruct
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema continuous RelStr
      st InclPoset the topology of b1 = b2 & b1 is infinite
   holds weight b1 = CLweight b2;

:: WAYBEL31:th 9
theorem
for b1 being non empty TopSpace-like discerning TopStruct
for b2 being reflexive transitive antisymmetric with_suprema continuous RelStr
      st InclPoset the topology of b1 = b2
   holds Card the carrier of b1 c= Card the carrier of b2;

:: WAYBEL31:th 10
theorem
for b1 being non empty TopSpace-like discerning TopStruct
      st b1 is finite
   holds weight b1 = Card the carrier of b1;

:: WAYBEL31:th 11
theorem
for b1 being TopStruct
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr
      st InclPoset the topology of b1 = b2 & b1 is finite
   holds CLweight b2 = Card the carrier of b2;

:: WAYBEL31:th 12
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema continuous RelStr
for b2 being Scott TopAugmentation of b1
for b3 being TopSpace-like Lawson TopAugmentation of b1
for b4 being Basis of b3 holds
   {uparrow b5 where b5 is Element of bool the carrier of b3: b5 in b4} is Basis of b2;

:: WAYBEL31:th 14
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 in compactbelow b2;

:: WAYBEL31:th 15
theorem
for b1 being finite reflexive transitive antisymmetric with_suprema with_infima RelStr holds
   b1 is arithmetic;

:: WAYBEL31:condreg 1
registration
  cluster finite reflexive transitive antisymmetric with_suprema with_infima -> arithmetic (RelStr);
end;

:: WAYBEL31:exreg 1
registration
  cluster non empty trivial finite strict reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr;
end;

:: WAYBEL31:th 16
theorem
for b1 being finite reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2 being with_bottom CLbasis of b1 holds
      Card b2 = CLweight b1
   iff
      b2 = the carrier of CompactSublatt b1;

:: WAYBEL31:funcnot 2 => WAYBEL31:func 2
definition
  let a1 be non empty reflexive RelStr;
  let a2 be Element of bool the carrier of a1;
  let a3 be Element of the carrier of a1;
  func Way_Up(A3,A2) -> Element of bool the carrier of a1 equals
    (wayabove a3) \ uparrow a2;
end;

:: WAYBEL31:def 2
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
   Way_Up(b3,b2) = (wayabove b3) \ uparrow b2;

:: WAYBEL31:th 17
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of the carrier of b1 holds
   Way_Up(b2,{} b1) = wayabove b2;

:: WAYBEL31:th 18
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
      st b3 in uparrow b2
   holds Way_Up(b3,b2) = {};

:: WAYBEL31:th 19
theorem
for b1 being non empty finite reflexive transitive RelStr holds
   Ids b1 is finite;

:: WAYBEL31:th 20
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema continuous RelStr
   st b1 is infinite
for b2 being with_bottom CLbasis of b1 holds
   b2 is infinite;

:: WAYBEL31:th 23
theorem
for b1 being non empty reflexive transitive antisymmetric complete RelStr
for b2 being Element of the carrier of b1
      st b2 is compact(b1)
   holds b2 = "/\"(wayabove b2,b1);

:: WAYBEL31:th 24
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema continuous RelStr
   st b1 is infinite
for b2 being with_bottom CLbasis of b1 holds
   Card {Way_Up(b3,b4) where b3 is Element of the carrier of b1, b4 is finite Element of bool the carrier of b1: b3 in b2 & b4 c= b2} c= Card b2;

:: WAYBEL31:th 25
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr
for b2 being finite Element of bool the carrier of b1 holds
   (uparrow b2) ` is open(b1) & (downarrow b2) ` is open(b1);

:: WAYBEL31:th 26
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema continuous RelStr
for b2 being TopSpace-like Lawson TopAugmentation of b1
for b3 being with_bottom CLbasis of b1 holds
   {Way_Up(b4,b5) where b4 is Element of the carrier of b1, b5 is finite Element of bool the carrier of b1: b4 in b3 & b5 c= b3} is Basis of b2;

:: WAYBEL31:th 27
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema continuous RelStr
for b2 being Scott TopAugmentation of b1
for b3 being Basis of b2 holds
   {wayabove "/\"(b4,b2) where b4 is Element of bool the carrier of b2: b4 in b3} is Basis of b2;

:: WAYBEL31:th 28
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema continuous RelStr
for b2 being Scott TopAugmentation of b1
for b3 being Basis of b2
      st b3 is infinite
   holds {"/\"(b4,b2) where b4 is Element of bool the carrier of b2: b4 in b3} is infinite;

:: WAYBEL31:th 29
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema continuous RelStr
for b2 being Scott TopAugmentation of b1 holds
   CLweight b1 = weight b2;

:: WAYBEL31:th 30
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema continuous RelStr
for b2 being TopSpace-like Lawson TopAugmentation of b1 holds
   CLweight b1 = weight b2;

:: WAYBEL31:th 31
theorem
for b1, b2 being non empty RelStr
      st b1,b2 are_isomorphic
   holds Card the carrier of b1 = Card the carrier of b2;

:: WAYBEL31:th 32
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema continuous RelStr
for b2 being with_bottom CLbasis of b1
      st Card b2 = CLweight b1
   holds CLweight b1 = CLweight InclPoset Ids subrelstr b2;

:: WAYBEL31:funcreg 2
registration
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema continuous RelStr;
  cluster InclPoset sigma a1 -> strict with_suprema continuous;
end;

:: WAYBEL31:th 33
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema continuous RelStr holds
   CLweight b1 c= CLweight InclPoset sigma b1;

:: WAYBEL31:th 34
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema continuous RelStr
      st b1 is infinite
   holds CLweight b1 = CLweight InclPoset sigma b1;