Article TDLAT_2, MML version 4.99.1005

:: TDLAT_2:th 1
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
   Int Cl Int b2 c= Int Cl b2 & Int Cl Int b2 c= Cl Int b2;

:: TDLAT_2:th 2
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
   Cl Int b2 c= Cl Int Cl b2 & Int Cl b2 c= Cl Int Cl b2;

:: TDLAT_2:th 3
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b3 is closed(b1) & Cl Int (b2 /\ b3) = b2
   holds b2 c= b3;

:: TDLAT_2:th 4
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is open(b1) & Int Cl (b2 \/ b3) = b3
   holds b2 c= b3;

:: TDLAT_2:th 5
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 c= Cl Int b2
   holds b2 \/ Int Cl b2 c= Cl Int (b2 \/ Int Cl b2);

:: TDLAT_2:th 6
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st Int Cl b2 c= b2
   holds Int Cl (b2 /\ Cl Int b2) c= b2 /\ Cl Int b2;

:: TDLAT_2:funcnot 1 => PCOMPS_1:func 2
notation
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool bool the carrier of a1;
  synonym Cl a2 for clf a2;
end;

:: TDLAT_2:th 7
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   clf b2 = {b3 where b3 is Element of bool the carrier of b1: ex b4 being Element of bool the carrier of b1 st
      b3 = Cl b4 & b4 in b2};

:: TDLAT_2:th 8
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   clf b2 = clf clf b2;

:: TDLAT_2:th 9
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 = {}
   iff
      clf b2 = {};

:: TDLAT_2:th 10
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1 holds
clf (b2 /\ b3) c= (clf b2) /\ clf b3;

:: TDLAT_2:th 11
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1 holds
(clf b2) \ clf b3 c= clf (b2 \ b3);

:: TDLAT_2:th 12
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
for b3 being Element of bool the carrier of b1
      st b3 in b2
   holds meet clf b2 c= Cl b3 & Cl b3 c= union clf b2;

:: TDLAT_2:th 13
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   meet b2 c= meet clf b2;

:: TDLAT_2:th 14
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   Cl meet b2 c= meet clf b2;

:: TDLAT_2:th 15
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   union clf b2 c= Cl union b2;

:: TDLAT_2:funcnot 2 => TDLAT_2:func 1
definition
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool bool the carrier of a1;
  func Int A2 -> Element of bool bool the carrier of a1 means
    for b1 being Element of bool the carrier of a1 holds
          b1 in it
       iff
          ex b2 being Element of bool the carrier of a1 st
             b1 = Int b2 & b2 in a2;
end;

:: TDLAT_2:def 1
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1 holds
   b3 = Int b2
iff
   for b4 being Element of bool the carrier of b1 holds
         b4 in b3
      iff
         ex b5 being Element of bool the carrier of b1 st
            b4 = Int b5 & b5 in b2;

:: TDLAT_2:th 16
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   Int b2 = {b3 where b3 is Element of bool the carrier of b1: ex b4 being Element of bool the carrier of b1 st
      b3 = Int b4 & b4 in b2};

:: TDLAT_2:th 17
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   Int b2 = Int Int b2;

:: TDLAT_2:th 18
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   Int b2 is open(b1);

:: TDLAT_2:th 19
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 = {}
   iff
      Int b2 = {};

:: TDLAT_2:th 20
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1
      st b3 = {b2}
   holds Int b3 = {Int b2};

:: TDLAT_2:th 21
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1
      st b2 c= b3
   holds Int b2 c= Int b3;

:: TDLAT_2:th 22
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1 holds
Int (b2 \/ b3) = (Int b2) \/ Int b3;

:: TDLAT_2:th 23
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1 holds
Int (b2 /\ b3) c= (Int b2) /\ Int b3;

:: TDLAT_2:th 24
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1 holds
(Int b2) \ Int b3 c= Int (b2 \ b3);

:: TDLAT_2:th 25
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
for b3 being Element of bool the carrier of b1
      st b3 in b2
   holds Int b3 c= union Int b2 & meet Int b2 c= Int b3;

:: TDLAT_2:th 26
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   union Int b2 c= union b2;

:: TDLAT_2:th 27
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   meet Int b2 c= meet b2;

:: TDLAT_2:th 28
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   union Int b2 c= Int union b2;

:: TDLAT_2:th 29
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   Int meet b2 c= meet Int b2;

:: TDLAT_2:th 30
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is finite
   holds Int meet b2 = meet Int b2;

:: TDLAT_2:th 31
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   clf Int b2 = {b3 where b3 is Element of bool the carrier of b1: ex b4 being Element of bool the carrier of b1 st
      b3 = Cl Int b4 & b4 in b2};

:: TDLAT_2:th 32
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   Int clf b2 = {b3 where b3 is Element of bool the carrier of b1: ex b4 being Element of bool the carrier of b1 st
      b3 = Int Cl b4 & b4 in b2};

:: TDLAT_2:th 33
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   clf Int clf b2 = {b3 where b3 is Element of bool the carrier of b1: ex b4 being Element of bool the carrier of b1 st
      b3 = Cl Int Cl b4 & b4 in b2};

:: TDLAT_2:th 34
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   Int clf Int b2 = {b3 where b3 is Element of bool the carrier of b1: ex b4 being Element of bool the carrier of b1 st
      b3 = Int Cl Int b4 & b4 in b2};

:: TDLAT_2:th 35
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   clf Int clf Int b2 = clf Int b2;

:: TDLAT_2:th 36
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   Int clf Int clf b2 = Int clf b2;

:: TDLAT_2:th 37
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   union Int clf b2 c= union clf Int clf b2;

:: TDLAT_2:th 38
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   meet Int clf b2 c= meet clf Int clf b2;

:: TDLAT_2:th 39
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   union clf Int b2 c= union clf Int clf b2;

:: TDLAT_2:th 40
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   meet clf Int b2 c= meet clf Int clf b2;

:: TDLAT_2:th 41
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   union Int clf Int b2 c= union Int clf b2;

:: TDLAT_2:th 42
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   meet Int clf Int b2 c= meet Int clf b2;

:: TDLAT_2:th 43
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   union Int clf Int b2 c= union clf Int b2;

:: TDLAT_2:th 44
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   meet Int clf Int b2 c= meet clf Int b2;

:: TDLAT_2:th 45
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   union clf Int clf b2 c= union clf b2;

:: TDLAT_2:th 46
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   meet clf Int clf b2 c= meet clf b2;

:: TDLAT_2:th 47
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   union Int b2 c= union Int clf Int b2;

:: TDLAT_2:th 48
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   meet Int b2 c= meet Int clf Int b2;

:: TDLAT_2:th 49
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   union clf Int b2 c= Cl Int union b2;

:: TDLAT_2:th 50
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   Cl Int meet b2 c= meet clf Int b2;

:: TDLAT_2:th 51
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   union Int clf b2 c= Int Cl union b2;

:: TDLAT_2:th 52
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   Int Cl meet b2 c= meet Int clf b2;

:: TDLAT_2:th 53
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   union clf Int clf b2 c= Cl Int Cl union b2;

:: TDLAT_2:th 54
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   Cl Int Cl meet b2 c= meet clf Int clf b2;

:: TDLAT_2:th 55
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   union Int clf Int b2 c= Int Cl Int union b2;

:: TDLAT_2:th 56
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   Int Cl Int meet b2 c= meet Int clf Int b2;

:: TDLAT_2:th 57
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st for b3 being Element of bool the carrier of b1
              st b3 in b2
           holds b3 c= Cl Int b3
   holds union b2 c= Cl Int union b2 &
    Cl union b2 = Cl Int Cl union b2;

:: TDLAT_2:th 58
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st for b3 being Element of bool the carrier of b1
              st b3 in b2
           holds Int Cl b3 c= b3
   holds Int Cl meet b2 c= meet b2 &
    Int Cl Int meet b2 = Int meet b2;

:: TDLAT_2:th 59
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b3 is condensed(b1)
   holds    (Int Cl (b2 \/ b3)) \/ (b2 \/ b3) = b3
   iff
      b2 c= b3;

:: TDLAT_2:th 60
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is condensed(b1)
   holds    (Cl Int (b2 /\ b3)) /\ (b2 /\ b3) = b2
   iff
      b2 c= b3;

:: TDLAT_2:th 61
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is closed_condensed(b1) & b3 is closed_condensed(b1)
   holds    Int b2 c= Int b3
   iff
      b2 c= b3;

:: TDLAT_2:th 62
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is open_condensed(b1) & b3 is open_condensed(b1)
   holds    Cl b2 c= Cl b3
   iff
      b2 c= b3;

:: TDLAT_2:th 63
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is closed_condensed(b1) & b2 c= b3
   holds Cl Int (b2 /\ b3) = b2;

:: TDLAT_2:th 64
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b3 is open_condensed(b1) & b2 c= b3
   holds Int Cl (b2 \/ b3) = b3;

:: TDLAT_2:attrnot 1 => TDLAT_2:attr 1
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of bool bool the carrier of a1;
  attr a2 is domains-family means
    for b1 being Element of bool the carrier of a1
          st b1 in a2
       holds b1 is condensed(a1);
end;

:: TDLAT_2:dfs 2
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of bool bool the carrier of a1;
To prove
     a2 is domains-family
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st b1 in a2
       holds b1 is condensed(a1);

:: TDLAT_2:def 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 is domains-family(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 in b2
         holds b3 is condensed(b1);

:: TDLAT_2:th 65
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 c= Domains_of b1
   iff
      b2 is domains-family(b1);

:: TDLAT_2:th 66
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is domains-family(b1)
   holds union b2 c= Cl Int union b2 &
    Cl union b2 = Cl Int Cl union b2;

:: TDLAT_2:th 67
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is domains-family(b1)
   holds Int Cl meet b2 c= meet b2 &
    Int Cl Int meet b2 = Int meet b2;

:: TDLAT_2:th 68
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is domains-family(b1)
   holds (union b2) \/ Int Cl union b2 is condensed(b1);

:: TDLAT_2:th 69
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   (for b3 being Element of bool the carrier of b1
          st b3 in b2
       holds b3 c= (union b2) \/ Int Cl union b2) &
    (for b3 being Element of bool the carrier of b1
          st b3 is condensed(b1) &
             (for b4 being Element of bool the carrier of b1
                   st b4 in b2
                holds b4 c= b3)
       holds (union b2) \/ Int Cl union b2 c= b3);

:: TDLAT_2:th 70
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is domains-family(b1)
   holds (meet b2) /\ Cl Int meet b2 is condensed(b1);

:: TDLAT_2:th 71
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   (for b3 being Element of bool the carrier of b1
          st b3 in b2
       holds (meet b2) /\ Cl Int meet b2 c= b3) &
    (b2 = {} or for b3 being Element of bool the carrier of b1
          st b3 is condensed(b1) &
             (for b4 being Element of bool the carrier of b1
                   st b4 in b2
                holds b3 c= b4)
       holds b3 c= (meet b2) /\ Cl Int meet b2);

:: TDLAT_2:attrnot 2 => TDLAT_2:attr 2
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of bool bool the carrier of a1;
  attr a2 is closed-domains-family means
    for b1 being Element of bool the carrier of a1
          st b1 in a2
       holds b1 is closed_condensed(a1);
end;

:: TDLAT_2:dfs 3
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of bool bool the carrier of a1;
To prove
     a2 is closed-domains-family
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st b1 in a2
       holds b1 is closed_condensed(a1);

:: TDLAT_2:def 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 is closed-domains-family(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 in b2
         holds b3 is closed_condensed(b1);

:: TDLAT_2:th 72
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 c= Closed_Domains_of b1
   iff
      b2 is closed-domains-family(b1);

:: TDLAT_2:th 73
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is closed-domains-family(b1)
   holds b2 is domains-family(b1);

:: TDLAT_2:th 74
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is closed-domains-family(b1)
   holds b2 is closed(b1);

:: TDLAT_2:th 75
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is domains-family(b1)
   holds clf b2 is closed-domains-family(b1);

:: TDLAT_2:th 76
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is closed-domains-family(b1)
   holds Cl union b2 is closed_condensed(b1) & Cl Int meet b2 is closed_condensed(b1);

:: TDLAT_2:th 77
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   (for b3 being Element of bool the carrier of b1
          st b3 in b2
       holds b3 c= Cl union b2) &
    (for b3 being Element of bool the carrier of b1
          st b3 is closed_condensed(b1) &
             (for b4 being Element of bool the carrier of b1
                   st b4 in b2
                holds b4 c= b3)
       holds Cl union b2 c= b3);

:: TDLAT_2:th 78
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   (b2 is closed(b1) implies for b3 being Element of bool the carrier of b1
          st b3 in b2
       holds Cl Int meet b2 c= b3) &
    (b2 = {} or for b3 being Element of bool the carrier of b1
          st b3 is closed_condensed(b1) &
             (for b4 being Element of bool the carrier of b1
                   st b4 in b2
                holds b3 c= b4)
       holds b3 c= Cl Int meet b2);

:: TDLAT_2:attrnot 3 => TDLAT_2:attr 3
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of bool bool the carrier of a1;
  attr a2 is open-domains-family means
    for b1 being Element of bool the carrier of a1
          st b1 in a2
       holds b1 is open_condensed(a1);
end;

:: TDLAT_2:dfs 4
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of bool bool the carrier of a1;
To prove
     a2 is open-domains-family
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st b1 in a2
       holds b1 is open_condensed(a1);

:: TDLAT_2:def 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 is open-domains-family(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 in b2
         holds b3 is open_condensed(b1);

:: TDLAT_2:th 79
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 c= Open_Domains_of b1
   iff
      b2 is open-domains-family(b1);

:: TDLAT_2:th 80
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is open-domains-family(b1)
   holds b2 is domains-family(b1);

:: TDLAT_2:th 81
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is open-domains-family(b1)
   holds b2 is open(b1);

:: TDLAT_2:th 82
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is domains-family(b1)
   holds Int b2 is open-domains-family(b1);

:: TDLAT_2:th 83
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is open-domains-family(b1)
   holds Int meet b2 is open_condensed(b1) & Int Cl union b2 is open_condensed(b1);

:: TDLAT_2:th 84
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   (b2 is open(b1) implies for b3 being Element of bool the carrier of b1
          st b3 in b2
       holds b3 c= Int Cl union b2) &
    (for b3 being Element of bool the carrier of b1
          st b3 is open_condensed(b1) &
             (for b4 being Element of bool the carrier of b1
                   st b4 in b2
                holds b4 c= b3)
       holds Int Cl union b2 c= b3);

:: TDLAT_2:th 85
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   (for b3 being Element of bool the carrier of b1
          st b3 in b2
       holds Int meet b2 c= b3) &
    (b2 = {} or for b3 being Element of bool the carrier of b1
          st b3 is open_condensed(b1) &
             (for b4 being Element of bool the carrier of b1
                   st b4 in b2
                holds b3 c= b4)
       holds b3 c= Int meet b2);

:: TDLAT_2:th 86
theorem
for b1 being non empty TopSpace-like TopStruct holds
   the carrier of Domains_Lattice b1 = Domains_of b1;

:: TDLAT_2:th 87
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of Domains_Lattice b1
for b4, b5 being Element of Domains_of b1
      st b2 = b4 & b3 = b5
   holds b2 "\/" b3 = (Int Cl (b4 \/ b5)) \/ (b4 \/ b5) &
    b2 "/\" b3 = (Cl Int (b4 /\ b5)) /\ (b4 /\ b5);

:: TDLAT_2:th 88
theorem
for b1 being non empty TopSpace-like TopStruct holds
   Bottom Domains_Lattice b1 = {} b1 & Top Domains_Lattice b1 = [#] b1;

:: TDLAT_2:th 89
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of Domains_Lattice b1
for b4, b5 being Element of Domains_of b1
      st b2 = b4 & b3 = b5
   holds    b2 [= b3
   iff
      b4 c= b5;

:: TDLAT_2:th 90
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of Domains_Lattice b1 holds
   ex b3 being Element of the carrier of Domains_Lattice b1 st
      b2 is_less_than b3 &
       (for b4 being Element of the carrier of Domains_Lattice b1
             st b2 is_less_than b4
          holds b3 [= b4);

:: TDLAT_2:th 91
theorem
for b1 being non empty TopSpace-like TopStruct holds
   Domains_Lattice b1 is complete;

:: TDLAT_2:th 92
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
   st b2 is domains-family(b1)
for b3 being Element of bool the carrier of Domains_Lattice b1
      st b3 = b2
   holds "\/"(b3,Domains_Lattice b1) = (union b2) \/ Int Cl union b2;

:: TDLAT_2:th 93
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
   st b2 is domains-family(b1)
for b3 being Element of bool the carrier of Domains_Lattice b1
      st b3 = b2
   holds (b3 = {} or "/\"(b3,Domains_Lattice b1) = (meet b2) /\ Cl Int meet b2) &
    (b3 = {} implies "/\"(b3,Domains_Lattice b1) = [#] b1);

:: TDLAT_2:th 94
theorem
for b1 being non empty TopSpace-like TopStruct holds
   the carrier of Closed_Domains_Lattice b1 = Closed_Domains_of b1;

:: TDLAT_2:th 95
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of Closed_Domains_Lattice b1
for b4, b5 being Element of Closed_Domains_of b1
      st b2 = b4 & b3 = b5
   holds b2 "\/" b3 = b4 \/ b5 & b2 "/\" b3 = Cl Int (b4 /\ b5);

:: TDLAT_2:th 96
theorem
for b1 being non empty TopSpace-like TopStruct holds
   Bottom Closed_Domains_Lattice b1 = {} b1 & Top Closed_Domains_Lattice b1 = [#] b1;

:: TDLAT_2:th 97
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of Closed_Domains_Lattice b1
for b4, b5 being Element of Closed_Domains_of b1
      st b2 = b4 & b3 = b5
   holds    b2 [= b3
   iff
      b4 c= b5;

:: TDLAT_2:th 98
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of Closed_Domains_Lattice b1 holds
   ex b3 being Element of the carrier of Closed_Domains_Lattice b1 st
      b2 is_less_than b3 &
       (for b4 being Element of the carrier of Closed_Domains_Lattice b1
             st b2 is_less_than b4
          holds b3 [= b4);

:: TDLAT_2:th 99
theorem
for b1 being non empty TopSpace-like TopStruct holds
   Closed_Domains_Lattice b1 is complete;

:: TDLAT_2:th 100
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
   st b2 is closed-domains-family(b1)
for b3 being Element of bool the carrier of Closed_Domains_Lattice b1
      st b3 = b2
   holds "\/"(b3,Closed_Domains_Lattice b1) = Cl union b2;

:: TDLAT_2:th 101
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
   st b2 is closed-domains-family(b1)
for b3 being Element of bool the carrier of Closed_Domains_Lattice b1
      st b3 = b2
   holds (b3 = {} or "/\"(b3,Closed_Domains_Lattice b1) = Cl Int meet b2) &
    (b3 = {} implies "/\"(b3,Closed_Domains_Lattice b1) = [#] b1);

:: TDLAT_2:th 102
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
   st b2 is closed-domains-family(b1)
for b3 being Element of bool the carrier of Domains_Lattice b1
      st b3 = b2
   holds (b3 = {} or "/\"(b3,Domains_Lattice b1) = Cl Int meet b2) &
    (b3 = {} implies "/\"(b3,Domains_Lattice b1) = [#] b1);

:: TDLAT_2:th 103
theorem
for b1 being non empty TopSpace-like TopStruct holds
   the carrier of Open_Domains_Lattice b1 = Open_Domains_of b1;

:: TDLAT_2:th 104
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of Open_Domains_Lattice b1
for b4, b5 being Element of Open_Domains_of b1
      st b2 = b4 & b3 = b5
   holds b2 "\/" b3 = Int Cl (b4 \/ b5) & b2 "/\" b3 = b4 /\ b5;

:: TDLAT_2:th 105
theorem
for b1 being non empty TopSpace-like TopStruct holds
   Bottom Open_Domains_Lattice b1 = {} b1 & Top Open_Domains_Lattice b1 = [#] b1;

:: TDLAT_2:th 106
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of Open_Domains_Lattice b1
for b4, b5 being Element of Open_Domains_of b1
      st b2 = b4 & b3 = b5
   holds    b2 [= b3
   iff
      b4 c= b5;

:: TDLAT_2:th 107
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of Open_Domains_Lattice b1 holds
   ex b3 being Element of the carrier of Open_Domains_Lattice b1 st
      b2 is_less_than b3 &
       (for b4 being Element of the carrier of Open_Domains_Lattice b1
             st b2 is_less_than b4
          holds b3 [= b4);

:: TDLAT_2:th 108
theorem
for b1 being non empty TopSpace-like TopStruct holds
   Open_Domains_Lattice b1 is complete;

:: TDLAT_2:th 109
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
   st b2 is open-domains-family(b1)
for b3 being Element of bool the carrier of Open_Domains_Lattice b1
      st b3 = b2
   holds "\/"(b3,Open_Domains_Lattice b1) = Int Cl union b2;

:: TDLAT_2:th 110
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
   st b2 is open-domains-family(b1)
for b3 being Element of bool the carrier of Open_Domains_Lattice b1
      st b3 = b2
   holds (b3 = {} or "/\"(b3,Open_Domains_Lattice b1) = Int meet b2) &
    (b3 = {} implies "/\"(b3,Open_Domains_Lattice b1) = [#] b1);

:: TDLAT_2:th 111
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
   st b2 is open-domains-family(b1)
for b3 being Element of bool the carrier of Domains_Lattice b1
      st b3 = b2
   holds "\/"(b3,Domains_Lattice b1) = Int Cl union b2;