Article YELLOW_9, MML version 4.99.1005

:: YELLOW_9:sch 1
scheme YELLOW_9:sch 1
{F1 -> non empty set,
  F2 -> Element of bool F1(),
  F3 -> Element of bool F1(),
  F4 -> Element of F1()}:
F2() = {F4(b1) where b1 is Element of F1(): b1 in F3()}
provided
   F3() = {F4(b1) where b1 is Element of F1(): b1 in F2()}
and
   for b1 being Element of F1() holds
      F4(F4(b1)) = b1;


:: YELLOW_9:sch 2
scheme YELLOW_9:sch 2
{F1 -> non empty RelStr,
  F2 -> Element of bool bool the carrier of F1(),
  F3 -> set,
  F4 -> Element of bool the carrier of F1()}:
COMPLEMENT F2() = {F4(b1) ` where b1 is Element of the carrier of F1(): b1 in F3()}
provided
   F2() = {F4(b1) where b1 is Element of the carrier of F1(): b1 in F3()};


:: YELLOW_9:sch 3
scheme YELLOW_9:sch 3
{F1 -> non empty RelStr,
  F2 -> Element of bool bool the carrier of F1(),
  F3 -> set,
  F4 -> Element of bool the carrier of F1()}:
COMPLEMENT F2() = {F4(b1) where b1 is Element of the carrier of F1(): b1 in F3()}
provided
   F2() = {F4(b1) ` where b1 is Element of the carrier of F1(): b1 in F3()};


:: YELLOW_9:th 1
theorem
for b1 being non empty RelStr
for b2, b3 being Element of the carrier of b1 holds
   b3 in (uparrow b2) `
iff
   not b2 <= b3;

:: YELLOW_9:sch 4
scheme YELLOW_9:sch 4
{F1 -> TopSpace-like TopStruct,
  F2 -> set,
  F3 -> Relation-like Function-like set}:
F3() " union {F2(b1) where b1 is Element of bool the carrier of F1(): P1[b1]} = union {F3() " F2(b1) where b1 is Element of bool the carrier of F1(): P1[b1]}


:: YELLOW_9:th 2
theorem
for b1 being 1-sorted
for b2 being non empty 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b2 holds
   (b3 " b4) ` = b3 " (b4 `);

:: YELLOW_9:th 3
theorem
for b1 being 1-sorted
for b2 being Element of bool bool the carrier of b1 holds
   COMPLEMENT b2 = {b3 ` where b3 is Element of bool the carrier of b1: b3 in b2};

:: YELLOW_9:th 4
theorem
for b1 being non empty RelStr
for b2 being Element of bool the carrier of b1 holds
   uparrow b2 = union {uparrow b3 where b3 is Element of the carrier of b1: b3 in b2} &
    downarrow b2 = union {downarrow b3 where b3 is Element of the carrier of b1: b3 in b2};

:: YELLOW_9:th 5
theorem
for b1 being non empty RelStr
for b2 being Element of bool bool the carrier of b1
for b3 being Element of bool the carrier of b1
      st b2 = {(uparrow b4) ` where b4 is Element of the carrier of b1: b4 in b3}
   holds Intersect b2 = (uparrow b3) `;

:: YELLOW_9:exreg 1
registration
  cluster non empty trivial finite reflexive discrete strict TopRelStr;
end;

:: YELLOW_9:exreg 2
registration
  cluster non empty trivial TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete strict TopRelStr;
end;

:: YELLOW_9:exreg 3
registration
  let a1 be non empty RelStr;
  let a2 be non empty reflexive antisymmetric upper-bounded RelStr;
  cluster Relation-like Function-like quasi_total infs-preserving Relation of the carrier of a1,the carrier of a2;
end;

:: YELLOW_9:exreg 4
registration
  let a1 be non empty RelStr;
  let a2 be non empty reflexive antisymmetric lower-bounded RelStr;
  cluster Relation-like Function-like quasi_total sups-preserving Relation of the carrier of a1,the carrier of a2;
end;

:: YELLOW_9:funcnot 1 => YELLOW_9:func 1
definition
  let a1, a2 be 1-sorted;
  assume the carrier of a2 c= the carrier of a1;
  func incl(A2,A1) -> Function-like quasi_total Relation of the carrier of a2,the carrier of a1 equals
    id the carrier of a2;
end;

:: YELLOW_9:def 1
theorem
for b1, b2 being 1-sorted
      st the carrier of b2 c= the carrier of b1
   holds incl(b2,b1) = id the carrier of b2;

:: YELLOW_9:funcreg 1
registration
  let a1 be non empty RelStr;
  let a2 be non empty SubRelStr of a1;
  cluster incl(a2,a1) -> Function-like quasi_total monotone;
end;

:: YELLOW_9:funcnot 2 => YELLOW_9:func 2
definition
  let a1 be non empty RelStr;
  let a2 be non empty Element of bool the carrier of a1;
  func A2 +id -> non empty strict NetStr over a1 equals
    (incl(subrelstr a2,a1)) * ((subrelstr a2) +id);
end;

:: YELLOW_9:def 2
theorem
for b1 being non empty RelStr
for b2 being non empty Element of bool the carrier of b1 holds
   b2 +id = (incl(subrelstr b2,b1)) * ((subrelstr b2) +id);

:: YELLOW_9:funcnot 3 => YELLOW_9:func 3
definition
  let a1 be non empty RelStr;
  let a2 be non empty Element of bool the carrier of a1;
  func A2 opp+id -> non empty strict NetStr over a1 equals
    (incl(subrelstr a2,a1)) * ((subrelstr a2) opp+id);
end;

:: YELLOW_9:def 3
theorem
for b1 being non empty RelStr
for b2 being non empty Element of bool the carrier of b1 holds
   b2 opp+id = (incl(subrelstr b2,b1)) * ((subrelstr b2) opp+id);

:: YELLOW_9:th 6
theorem
for b1 being non empty RelStr
for b2 being non empty Element of bool the carrier of b1 holds
   the carrier of b2 +id = b2 &
    b2 +id is full SubRelStr of b1 &
    (for b3 being Element of the carrier of b2 +id holds
       b2 +id . b3 = b3);

:: YELLOW_9:th 7
theorem
for b1 being non empty RelStr
for b2 being non empty Element of bool the carrier of b1 holds
   the carrier of b2 opp+id = b2 &
    b2 opp+id is full SubRelStr of b1 ~ &
    (for b3 being Element of the carrier of b2 opp+id holds
       b2 opp+id . b3 = b3);

:: YELLOW_9:funcreg 2
registration
  let a1 be non empty reflexive RelStr;
  let a2 be non empty Element of bool the carrier of a1;
  cluster a2 +id -> non empty reflexive strict;
end;

:: YELLOW_9:funcreg 3
registration
  let a1 be non empty reflexive RelStr;
  let a2 be non empty Element of bool the carrier of a1;
  cluster a2 opp+id -> non empty reflexive strict;
end;

:: YELLOW_9:funcreg 4
registration
  let a1 be non empty transitive RelStr;
  let a2 be non empty Element of bool the carrier of a1;
  cluster a2 +id -> non empty transitive strict;
end;

:: YELLOW_9:funcreg 5
registration
  let a1 be non empty transitive RelStr;
  let a2 be non empty Element of bool the carrier of a1;
  cluster a2 opp+id -> non empty transitive strict;
end;

:: YELLOW_9:funcreg 6
registration
  let a1 be non empty reflexive RelStr;
  let a2 be directed Element of bool the carrier of a1;
  cluster subrelstr a2 -> strict full directed;
end;

:: YELLOW_9:funcreg 7
registration
  let a1 be non empty reflexive RelStr;
  let a2 be non empty directed Element of bool the carrier of a1;
  cluster a2 +id -> non empty strict directed;
end;

:: YELLOW_9:funcreg 8
registration
  let a1 be non empty reflexive RelStr;
  let a2 be non empty filtered Element of bool the carrier of a1;
  cluster (subrelstr a2) opp+id -> strict directed;
end;

:: YELLOW_9:funcreg 9
registration
  let a1 be non empty reflexive RelStr;
  let a2 be non empty filtered Element of bool the carrier of a1;
  cluster a2 opp+id -> non empty strict directed;
end;

:: YELLOW_9:th 8
theorem
for b1 being TopSpace-like TopStruct
      st b1 is empty
   holds the topology of b1 = {{}};

:: YELLOW_9:th 9
theorem
for b1 being non empty trivial TopSpace-like TopStruct holds
   the topology of b1 = bool the carrier of b1 &
    (for b2 being Element of the carrier of b1 holds
       the carrier of b1 = {b2} &
        the topology of b1 = {{},{b2}});

:: YELLOW_9:th 10
theorem
for b1 being non empty trivial TopSpace-like TopStruct holds
   {the carrier of b1} is Basis of b1 & {} is prebasis of b1 & {{}} is prebasis of b1;

:: YELLOW_9:th 11
theorem
for b1, b2 being set
for b3 being Element of bool bool b1
      st b3 = {b2}
   holds FinMeetCl b3 = {b2,b1} & UniCl b3 = {b2,{}};

:: YELLOW_9:th 12
theorem
for b1 being set
for b2, b3 being Element of bool bool b1
      st (b2 = b3 \/ {b1} or b3 = b2 \ {b1})
   holds Intersect b2 = Intersect b3;

:: YELLOW_9:th 13
theorem
for b1 being set
for b2, b3 being Element of bool bool b1
      st (b2 = b3 \/ {b1} or b3 = b2 \ {b1})
   holds FinMeetCl b2 = FinMeetCl b3;

:: YELLOW_9:th 14
theorem
for b1 being set
for b2 being Element of bool bool b1
   st b1 in b2
for b3 being set holds
      b3 in FinMeetCl b2
   iff
      ex b4 being non empty finite Element of bool bool b1 st
         b4 c= b2 & b3 = Intersect b4;

:: YELLOW_9:th 15
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
   UniCl UniCl b2 = UniCl b2;

:: YELLOW_9:th 16
theorem
for b1 being set
for b2 being empty Element of bool bool b1 holds
   UniCl b2 = {{}};

:: YELLOW_9:th 17
theorem
for b1 being set
for b2 being empty Element of bool bool b1 holds
   FinMeetCl b2 = {b1};

:: YELLOW_9:th 18
theorem
for b1 being set
for b2 being Element of bool bool b1
      st b2 = {{},b1}
   holds UniCl b2 = b2 & FinMeetCl b2 = b2;

:: YELLOW_9:th 19
theorem
for b1, b2 being set
for b3 being Element of bool bool b1
for b4 being Element of bool bool b2 holds
   (b3 c= b4 implies UniCl b3 c= UniCl b4) & (b3 = b4 implies UniCl b3 = UniCl b4);

:: YELLOW_9:th 20
theorem
for b1, b2 being set
for b3 being Element of bool bool b1
for b4 being Element of bool bool b2
      st b3 = b4 & b1 in b3 & b1 c= b2
   holds FinMeetCl b4 = {b2} \/ FinMeetCl b3;

:: YELLOW_9:th 21
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
   UniCl FinMeetCl UniCl b2 = UniCl FinMeetCl b2;

:: YELLOW_9:th 22
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      the topology of b1 = UniCl b2
   iff
      b2 is Basis of b1;

:: YELLOW_9:th 23
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 is prebasis of b1
   iff
      FinMeetCl b2 is Basis of b1;

:: YELLOW_9:th 24
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st UniCl b2 is prebasis of b1
   holds b2 is prebasis of b1;

:: YELLOW_9:th 25
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Basis of b1
      st the carrier of b1 = the carrier of b2 & b3 is Basis of b2
   holds the topology of b1 = the topology of b2;

:: YELLOW_9:th 26
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being prebasis of b1
      st the carrier of b1 = the carrier of b2 & b3 is prebasis of b2
   holds the topology of b1 = the topology of b2;

:: YELLOW_9:th 27
theorem
for b1 being TopSpace-like TopStruct
for b2 being Basis of b1 holds
   b2 is open(b1) & b2 is prebasis of b1;

:: YELLOW_9:th 28
theorem
for b1 being TopSpace-like TopStruct
for b2 being prebasis of b1 holds
   b2 is open(b1);

:: YELLOW_9:th 29
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being prebasis of b1 holds
   b2 \/ {the carrier of b1} is prebasis of b1;

:: YELLOW_9:th 30
theorem
for b1 being TopSpace-like TopStruct
for b2 being Basis of b1 holds
   b2 \/ {the carrier of b1} is Basis of b1;

:: YELLOW_9:th 31
theorem
for b1 being TopSpace-like TopStruct
for b2 being Basis of b1
for b3 being Element of bool the carrier of b1 holds
      b3 is open(b1)
   iff
      for b4 being Element of the carrier of b1
            st b4 in b3
         holds ex b5 being Element of bool the carrier of b1 st
            b5 in b2 & b4 in b5 & b5 c= b3;

:: YELLOW_9:th 32
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 c= the topology of b1 &
         (for b3 being Element of bool the carrier of b1
            st b3 is open(b1)
         for b4 being Element of the carrier of b1
               st b4 in b3
            holds ex b5 being Element of bool the carrier of b1 st
               b5 in b2 & b4 in b5 & b5 c= b3)
   holds b2 is Basis of b1;

:: YELLOW_9:th 33
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct
for b3 being Basis of b2
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b4 is continuous(b1, b2)
   iff
      for b5 being Element of bool the carrier of b2
            st b5 in b3
         holds b4 " (b5 `) is closed(b1);

:: YELLOW_9:th 34
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct
for b3 being Basis of b2
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b4 is continuous(b1, b2)
   iff
      for b5 being Element of bool the carrier of b2
            st b5 in b3
         holds b4 " b5 is open(b1);

:: YELLOW_9:th 35
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct
for b3 being prebasis of b2
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b4 is continuous(b1, b2)
   iff
      for b5 being Element of bool the carrier of b2
            st b5 in b3
         holds b4 " (b5 `) is closed(b1);

:: YELLOW_9:th 36
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct
for b3 being prebasis of b2
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b4 is continuous(b1, b2)
   iff
      for b5 being Element of bool the carrier of b2
            st b5 in b3
         holds b4 " b5 is open(b1);

:: YELLOW_9:th 37
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
for b4 being Basis of b1
      st for b5 being Element of bool the carrier of b1
              st b5 in b4 & b2 in b5
           holds b5 meets b3
   holds b2 in Cl b3;

:: YELLOW_9:th 38
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
for b4 being prebasis of b1
      st for b5 being finite Element of bool bool the carrier of b1
              st b5 c= b4 & b2 in Intersect b5
           holds Intersect b5 meets b3
   holds b2 in Cl b3;

:: YELLOW_9:th 39
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being prebasis of b1
for b3 being Element of the carrier of b1
for b4 being non empty transitive directed NetStr over b1
   st for b5 being Element of bool the carrier of b1
           st b5 in b2 & b3 in b5
        holds b4 is_eventually_in b5
for b5 being Element of bool the carrier of b1
      st rng netmap(b4,b1) c= b5
   holds b3 in Cl b5;

:: YELLOW_9:th 40
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Basis of b1
for b4 being Basis of b2 holds
   {[:b5,b6:] where b5 is Element of bool the carrier of b1, b6 is Element of bool the carrier of b2: b5 in b3 & b6 in b4} is Basis of [:b1,b2:];

:: YELLOW_9:th 41
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being prebasis of b1
for b4 being prebasis of b2 holds
   {[:the carrier of b1,b5:] where b5 is Element of bool the carrier of b2: b5 in b4} \/ {[:b5,the carrier of b2:] where b5 is Element of bool the carrier of b1: b5 in b3} is prebasis of [:b1,b2:];

:: YELLOW_9:th 42
theorem
for b1, b2 being set
for b3 being Element of bool bool [:b1,b2:]
for b4 being non empty Element of bool bool b1
for b5 being non empty Element of bool bool b2
      st b3 = {[:b6,b7:] where b6 is Element of bool b1, b7 is Element of bool b2: b6 in b4 & b7 in b5}
   holds Intersect b3 = [:Intersect b4,Intersect b5:];

:: YELLOW_9:th 43
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being prebasis of b1
for b4 being prebasis of b2
      st union b3 = the carrier of b1 & union b4 = the carrier of b2
   holds {[:b5,b6:] where b5 is Element of bool the carrier of b1, b6 is Element of bool the carrier of b2: b5 in b3 & b6 in b4} is prebasis of [:b1,b2:];

:: YELLOW_9:modenot 1 => YELLOW_9:mode 1
definition
  let a1 be RelStr;
  mode TopAugmentation of A1 -> TopRelStr means
    RelStr(#the carrier of it,the InternalRel of it#) = RelStr(#the carrier of a1,the InternalRel of a1#);
end;

:: YELLOW_9:dfs 4
definiens
  let a1 be RelStr;
  let a2 be TopRelStr;
To prove
     a2 is TopAugmentation of a1
it is sufficient to prove
  thus RelStr(#the carrier of a2,the InternalRel of a2#) = RelStr(#the carrier of a1,the InternalRel of a1#);

:: YELLOW_9:def 4
theorem
for b1 being RelStr
for b2 being TopRelStr holds
      b2 is TopAugmentation of b1
   iff
      RelStr(#the carrier of b2,the InternalRel of b2#) = RelStr(#the carrier of b1,the InternalRel of b1#);

:: YELLOW_9:attrnot 1 => PRE_TOPC:attr 2
notation
  let a1 be RelStr;
  let a2 be TopAugmentation of a1;
  synonym correct for TopSpace-like;
end;

:: YELLOW_9:exreg 5
registration
  let a1 be RelStr;
  cluster TopSpace-like discrete strict TopAugmentation of a1;
end;

:: YELLOW_9:th 44
theorem
for b1 being TopRelStr holds
   b1 is TopAugmentation of b1;

:: YELLOW_9:th 45
theorem
for b1 being TopRelStr
for b2 being TopAugmentation of b1 holds
   b1 is TopAugmentation of b2;

:: YELLOW_9:th 46
theorem
for b1 being RelStr
for b2 being TopAugmentation of b1
for b3 being TopAugmentation of b2 holds
   b3 is TopAugmentation of b1;

:: YELLOW_9:condreg 1
registration
  let a1 be non empty RelStr;
  cluster -> non empty (TopAugmentation of a1);
end;

:: YELLOW_9:condreg 2
registration
  let a1 be reflexive RelStr;
  cluster -> reflexive (TopAugmentation of a1);
end;

:: YELLOW_9:condreg 3
registration
  let a1 be transitive RelStr;
  cluster -> transitive (TopAugmentation of a1);
end;

:: YELLOW_9:condreg 4
registration
  let a1 be antisymmetric RelStr;
  cluster -> antisymmetric (TopAugmentation of a1);
end;

:: YELLOW_9:condreg 5
registration
  let a1 be non empty complete RelStr;
  cluster -> complete (TopAugmentation of a1);
end;

:: YELLOW_9:th 47
theorem
for b1 being non empty reflexive antisymmetric up-complete RelStr
for b2 being non empty reflexive RelStr
   st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel 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 & b3 is inaccessible_by_directed_joins(b1)
   holds b4 is inaccessible_by_directed_joins(b2);

:: YELLOW_9:th 48
theorem
for b1 being non empty reflexive RelStr
for b2 being TopAugmentation of b1
      st the topology of b2 = sigma b1
   holds b2 is TopSpace-like;

:: YELLOW_9:th 49
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being TopAugmentation of b1
      st the topology of b2 = sigma b1
   holds b2 is Scott;

:: YELLOW_9:exreg 6
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  cluster non empty TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete lower-bounded upper-bounded bounded up-complete /\-complete strict Scott TopAugmentation of a1;
end;

:: YELLOW_9:th 50
theorem
for b1, b2 being non empty reflexive transitive antisymmetric complete Scott TopRelStr
   st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel 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 & b3 is open(b1)
   holds b4 is open(b2);

:: YELLOW_9:th 51
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Scott TopAugmentation of b1 holds
   the topology of b2 = sigma b1;

:: YELLOW_9:th 52
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
      st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#)
   holds sigma b1 = sigma b2;

:: YELLOW_9:condreg 6
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  cluster Scott -> TopSpace-like (TopAugmentation of a1);
end;

:: YELLOW_9:modenot 2 => YELLOW_9:mode 2
definition
  let a1 be TopStruct;
  mode TopExtension of A1 -> TopSpace-like TopStruct means
    the carrier of a1 = the carrier of it & the topology of a1 c= the topology of it;
end;

:: YELLOW_9:dfs 5
definiens
  let a1 be TopStruct;
  let a2 be TopSpace-like TopStruct;
To prove
     a2 is TopExtension of a1
it is sufficient to prove
  thus the carrier of a1 = the carrier of a2 & the topology of a1 c= the topology of a2;

:: YELLOW_9:def 5
theorem
for b1 being TopStruct
for b2 being TopSpace-like TopStruct holds
      b2 is TopExtension of b1
   iff
      the carrier of b1 = the carrier of b2 & the topology of b1 c= the topology of b2;

:: YELLOW_9:th 53
theorem
for b1 being TopStruct holds
   ex b2 being TopExtension of b1 st
      b2 is strict & the topology of b1 is prebasis of b2;

:: YELLOW_9:exreg 7
registration
  let a1 be TopStruct;
  cluster strict TopSpace-like discrete TopExtension of a1;
end;

:: YELLOW_9:modenot 3 => YELLOW_9:mode 3
definition
  let a1, a2 be TopStruct;
  mode Refinement of A1,A2 -> TopSpace-like TopStruct means
    the carrier of it = (the carrier of a1) \/ the carrier of a2 &
     (the topology of a1) \/ the topology of a2 is prebasis of it;
end;

:: YELLOW_9:dfs 6
definiens
  let a1, a2 be TopStruct;
  let a3 be TopSpace-like TopStruct;
To prove
     a3 is Refinement of a1,a2
it is sufficient to prove
  thus the carrier of a3 = (the carrier of a1) \/ the carrier of a2 &
     (the topology of a1) \/ the topology of a2 is prebasis of a3;

:: YELLOW_9:def 6
theorem
for b1, b2 being TopStruct
for b3 being TopSpace-like TopStruct holds
      b3 is Refinement of b1,b2
   iff
      the carrier of b3 = (the carrier of b1) \/ the carrier of b2 &
       (the topology of b1) \/ the topology of b2 is prebasis of b3;

:: YELLOW_9:condreg 7
registration
  let a1 be non empty TopStruct;
  let a2 be TopStruct;
  cluster -> non empty (Refinement of a1,a2);
end;

:: YELLOW_9:condreg 8
registration
  let a1 be non empty TopStruct;
  let a2 be TopStruct;
  cluster -> non empty (Refinement of a2,a1);
end;

:: YELLOW_9:th 54
theorem
for b1, b2 being TopStruct
for b3, b4 being Refinement of b1,b2 holds
TopStruct(#the carrier of b3,the topology of b3#) = TopStruct(#the carrier of b4,the topology of b4#);

:: YELLOW_9:th 55
theorem
for b1, b2 being TopStruct
for b3 being Refinement of b1,b2 holds
   b3 is Refinement of b2,b1;

:: YELLOW_9:th 56
theorem
for b1, b2 being TopStruct
for b3 being Refinement of b1,b2
for b4 being Element of bool bool the carrier of b3
      st b4 = (the topology of b1) \/ the topology of b2
   holds the topology of b3 = UniCl FinMeetCl b4;

:: YELLOW_9:th 57
theorem
for b1, b2 being TopStruct
   st the carrier of b1 = the carrier of b2
for b3 being Refinement of b1,b2 holds
   b3 is TopExtension of b1 & b3 is TopExtension of b2;

:: YELLOW_9:th 58
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Refinement of b1,b2
for b4 being prebasis of b1
for b5 being prebasis of b2 holds
   (b4 \/ b5) \/ {the carrier of b1,the carrier of b2} is prebasis of b3;

:: YELLOW_9:th 59
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Basis of b1
for b4 being Basis of b2
for b5 being Refinement of b1,b2 holds
   (b3 \/ b4) \/ INTERSECTION(b3,b4) is Basis of b5;

:: YELLOW_9:th 60
theorem
for b1, b2 being non empty TopSpace-like TopStruct
   st the carrier of b1 = the carrier of b2
for b3 being Refinement of b1,b2 holds
   INTERSECTION(the topology of b1,the topology of b2) is Basis of b3;

:: YELLOW_9:th 61
theorem
for b1 being non empty RelStr
for b2, b3 being TopSpace-like TopAugmentation of b1
for b4 being Refinement of b2,b3 holds
   INTERSECTION(the topology of b2,the topology of b3) is Basis of b4;