Article TOPS_1, MML version 4.99.1005

:: TOPS_1:th 21
theorem
for b1 being 1-sorted
for b2, b3 being Element of bool the carrier of b1
      st b2 ` = b3 `
   holds b2 = b3;

:: TOPS_1:th 22
theorem
for b1 being TopSpace-like TopStruct holds
   {} b1 is closed(b1);

:: TOPS_1:funcreg 1
registration
  let a1 be TopSpace-like TopStruct;
  cluster {} a1 -> closed;
end;

:: TOPS_1:th 26
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   Cl Cl b2 = Cl b2;

:: TOPS_1:th 27
theorem
for b1 being TopStruct holds
   Cl [#] b1 = [#] b1;

:: TOPS_1:funcreg 2
registration
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  cluster Cl a2 -> closed;
end;

:: TOPS_1:th 29
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is closed(b1)
   iff
      b2 ` is open(b1);

:: TOPS_1:funcreg 3
registration
  let a1 be TopSpace-like TopStruct;
  let a2 be closed Element of bool the carrier of a1;
  cluster a2 ` -> open;
end;

:: TOPS_1:th 30
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is open(b1)
   iff
      b2 ` is closed(b1);

:: TOPS_1:exreg 1
registration
  let a1 be TopSpace-like TopStruct;
  cluster open Element of bool the carrier of a1;
end;

:: TOPS_1:funcreg 4
registration
  let a1 be TopSpace-like TopStruct;
  let a2 be open Element of bool the carrier of a1;
  cluster a2 ` -> closed;
end;

:: TOPS_1:th 31
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is closed(b1) & b3 c= b2
   holds Cl b3 c= b2;

:: TOPS_1:th 32
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
(Cl b2) \ Cl b3 c= Cl (b2 \ b3);

:: TOPS_1:th 34
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is closed(b1) & b3 is closed(b1)
   holds Cl (b2 /\ b3) = (Cl b2) /\ Cl b3;

:: TOPS_1:th 35
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is closed(b1) & b3 is closed(b1)
   holds b2 /\ b3 is closed(b1);

:: TOPS_1:th 36
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is closed(b1) & b3 is closed(b1)
   holds b2 \/ b3 is closed(b1);

:: TOPS_1:th 37
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is open(b1) & b3 is open(b1)
   holds b2 \/ b3 is open(b1);

:: TOPS_1:th 38
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is open(b1) & b3 is open(b1)
   holds b2 /\ b3 is open(b1);

:: TOPS_1:th 39
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
      b3 in Cl b2
   iff
      for b4 being Element of bool the carrier of b1
            st b4 is open(b1) & b3 in b4
         holds b2 meets b4;

:: TOPS_1:th 40
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is open(b1)
   holds b2 /\ Cl b3 c= Cl (b2 /\ b3);

:: TOPS_1:th 41
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is open(b1)
   holds Cl (b2 /\ Cl b3) = Cl (b2 /\ b3);

:: TOPS_1:funcnot 1 => TOPS_1:func 1
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  func Int A2 -> Element of bool the carrier of a1 equals
    (Cl (a2 `)) `;
  projectivity;
::  for a1 being TopStruct
::  for a2 being Element of bool the carrier of a1 holds
::     Int Int a2 = Int a2;
end;

:: TOPS_1:def 1
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   Int b2 = (Cl (b2 `)) `;

:: TOPS_1:th 43
theorem
for b1 being TopSpace-like TopStruct holds
   Int [#] b1 = [#] b1;

:: TOPS_1:th 44
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   Int b2 c= b2;

:: TOPS_1:th 45
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   Int Int b2 = Int b2;

:: TOPS_1:th 46
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
(Int b2) /\ Int b3 = Int (b2 /\ b3);

:: TOPS_1:th 47
theorem
for b1 being TopStruct holds
   Int {} b1 = {} b1;

:: TOPS_1:funcreg 5
registration
  let a1 be TopStruct;
  cluster Int {} a1 -> empty;
end;

:: TOPS_1:th 48
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 c= b3
   holds Int b2 c= Int b3;

:: TOPS_1:th 49
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
(Int b2) \/ Int b3 c= Int (b2 \/ b3);

:: TOPS_1:th 50
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
Int (b2 \ b3) c= (Int b2) \ Int b3;

:: TOPS_1:th 51
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
   Int b2 is open(b1);

:: TOPS_1:funcreg 6
registration
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  cluster Int a2 -> open;
end;

:: TOPS_1:th 52
theorem
for b1 being TopSpace-like TopStruct holds
   {} b1 is open(b1);

:: TOPS_1:th 53
theorem
for b1 being TopSpace-like TopStruct holds
   [#] b1 is open(b1);

:: TOPS_1:funcreg 7
registration
  let a1 be TopSpace-like TopStruct;
  cluster {} a1 -> open;
end;

:: TOPS_1:funcreg 8
registration
  let a1 be TopSpace-like TopStruct;
  cluster [#] a1 -> open;
end;

:: TOPS_1:condreg 1
registration
  let a1 be TopSpace-like TopStruct;
  cluster empty -> open closed (Element of bool the carrier of a1);
end;

:: TOPS_1:exreg 2
registration
  let a1 be TopSpace-like TopStruct;
  cluster open closed Element of bool the carrier of a1;
end;

:: TOPS_1:exreg 3
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster non empty open closed Element of bool the carrier of a1;
end;

:: TOPS_1:th 54
theorem
for b1 being TopSpace-like TopStruct
for b2 being set
for b3 being Element of bool the carrier of b1 holds
      b2 in Int b3
   iff
      ex b4 being Element of bool the carrier of b1 st
         b4 is open(b1) & b4 c= b3 & b2 in b4;

:: TOPS_1:th 55
theorem
for b1 being TopSpace-like TopStruct
for b2 being TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2 holds
   (b4 is open(b2) implies Int b4 = b4) & (Int b3 = b3 implies b3 is open(b1));

:: TOPS_1:th 56
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is open(b1) & b2 c= b3
   holds b2 c= Int b3;

:: TOPS_1:th 57
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is open(b1)
   iff
      for b3 being set holds
            b3 in b2
         iff
            ex b4 being Element of bool the carrier of b1 st
               b4 is open(b1) & b4 c= b2 & b3 in b4;

:: TOPS_1:th 58
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   Cl Int b2 = Cl Int Cl Int b2;

:: TOPS_1:th 59
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is open(b1)
   holds Cl Int Cl b2 = Cl b2;

:: TOPS_1:funcnot 2 => TOPS_1:func 2
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  func Fr A2 -> Element of bool the carrier of a1 equals
    (Cl a2) /\ Cl (a2 `);
end;

:: TOPS_1:def 2
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   Fr b2 = (Cl b2) /\ Cl (b2 `);

:: TOPS_1:th 60
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
   Fr b2 is closed(b1);

:: TOPS_1:funcreg 9
registration
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  cluster Fr a2 -> closed;
end;

:: TOPS_1:th 61
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
      b3 in Fr b2
   iff
      for b4 being Element of bool the carrier of b1
            st b4 is open(b1) & b3 in b4
         holds b2 meets b4 & b2 ` meets b4;

:: TOPS_1:th 62
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   Fr b2 = Fr (b2 `);

:: TOPS_1:th 64
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   Fr b2 = ((Cl (b2 `)) /\ b2) \/ ((Cl b2) \ b2);

:: TOPS_1:th 65
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   Cl b2 = b2 \/ Fr b2;

:: TOPS_1:th 66
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
Fr (b2 /\ b3) c= (Fr b2) \/ Fr b3;

:: TOPS_1:th 67
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
Fr (b2 \/ b3) c= (Fr b2) \/ Fr b3;

:: TOPS_1:th 68
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   Fr Fr b2 c= Fr b2;

:: TOPS_1:th 69
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is closed(b1)
   holds Fr b2 c= b2;

:: TOPS_1:th 70
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
(Fr b2) \/ Fr b3 = ((Fr (b2 \/ b3)) \/ Fr (b2 /\ b3)) \/ ((Fr b2) /\ Fr b3);

:: TOPS_1:th 71
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   Fr Int b2 c= Fr b2;

:: TOPS_1:th 72
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   Fr Cl b2 c= Fr b2;

:: TOPS_1:th 73
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   Int b2 misses Fr b2;

:: TOPS_1:th 74
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   Int b2 = b2 \ Fr b2;

:: TOPS_1:th 75
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
   Fr Fr Fr b2 = Fr Fr b2;

:: TOPS_1:th 76
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is open(b1)
   iff
      Fr b2 = (Cl b2) \ b2;

:: TOPS_1:th 77
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is closed(b1)
   iff
      Fr b2 = b2 \ Int b2;

:: TOPS_1:attrnot 1 => TOPS_1:attr 1
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is dense means
    Cl a2 = [#] a1;
end;

:: TOPS_1:dfs 3
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is dense
it is sufficient to prove
  thus Cl a2 = [#] a1;

:: TOPS_1:def 3
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is dense(b1)
   iff
      Cl b2 = [#] b1;

:: TOPS_1:funcreg 10
registration
  let a1 be TopStruct;
  cluster [#] a1 -> dense;
end;

:: TOPS_1:th 79
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is dense(b1) & b2 c= b3
   holds b3 is dense(b1);

:: TOPS_1:th 80
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is dense(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 <> {} & b3 is open(b1)
         holds b2 meets b3;

:: TOPS_1:th 81
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
   st b2 is dense(b1)
for b3 being Element of bool the carrier of b1
      st b3 is open(b1)
   holds Cl b3 = Cl (b3 /\ b2);

:: TOPS_1:th 82
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is dense(b1) & b3 is dense(b1) & b3 is open(b1)
   holds b2 /\ b3 is dense(b1);

:: TOPS_1:attrnot 2 => TOPS_1:attr 2
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is boundary means
    a2 ` is dense(a1);
end;

:: TOPS_1:dfs 4
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is boundary
it is sufficient to prove
  thus a2 ` is dense(a1);

:: TOPS_1:def 4
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is boundary(b1)
   iff
      b2 ` is dense(b1);

:: TOPS_1:condreg 2
registration
  let a1 be TopStruct;
  cluster empty -> boundary (Element of bool the carrier of a1);
end;

:: TOPS_1:exreg 4
registration
  let a1 be TopStruct;
  cluster empty Element of bool the carrier of a1;
end;

:: TOPS_1:th 84
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is boundary(b1)
   iff
      Int b2 = {};

:: TOPS_1:funcreg 11
registration
  let a1 be TopStruct;
  let a2 be boundary Element of bool the carrier of a1;
  cluster Int a2 -> empty;
end;

:: TOPS_1:th 85
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is boundary(b1) & b3 is boundary(b1) & b3 is closed(b1)
   holds b2 \/ b3 is boundary(b1);

:: TOPS_1:th 86
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is boundary(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 c= b2 & b3 is open(b1)
         holds b3 = {};

:: TOPS_1:th 87
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is closed(b1)
   holds    b2 is boundary(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 <> {} & b3 is open(b1)
         holds ex b4 being Element of bool the carrier of b1 st
            b4 c= b3 & b4 <> {} & b4 is open(b1) & b2 misses b4;

:: TOPS_1:th 88
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is boundary(b1)
   iff
      b2 c= Fr b2;

:: TOPS_1:attrnot 3 => TOPS_1:attr 3
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is nowhere_dense means
    Cl a2 is boundary(a1);
end;

:: TOPS_1:dfs 5
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is nowhere_dense
it is sufficient to prove
  thus Cl a2 is boundary(a1);

:: TOPS_1:def 5
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is nowhere_dense(b1)
   iff
      Cl b2 is boundary(b1);

:: TOPS_1:condreg 3
registration
  let a1 be TopSpace-like TopStruct;
  cluster empty -> nowhere_dense (Element of bool the carrier of a1);
end;

:: TOPS_1:exreg 5
registration
  let a1 be TopSpace-like TopStruct;
  cluster empty Element of bool the carrier of a1;
end;

:: TOPS_1:th 90
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is nowhere_dense(b1) & b3 is nowhere_dense(b1)
   holds b2 \/ b3 is nowhere_dense(b1);

:: TOPS_1:th 91
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is nowhere_dense(b1)
   holds b2 ` is dense(b1);

:: TOPS_1:funcreg 12
registration
  let a1 be TopSpace-like TopStruct;
  let a2 be nowhere_dense Element of bool the carrier of a1;
  cluster a2 ` -> dense;
end;

:: TOPS_1:th 92
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is nowhere_dense(b1)
   holds b2 is boundary(b1);

:: TOPS_1:condreg 4
registration
  let a1 be TopSpace-like TopStruct;
  cluster nowhere_dense -> boundary (Element of bool the carrier of a1);
end;

:: TOPS_1:th 93
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is boundary(b1) & b2 is closed(b1)
   holds b2 is nowhere_dense(b1);

:: TOPS_1:condreg 5
registration
  let a1 be TopSpace-like TopStruct;
  cluster closed boundary -> nowhere_dense (Element of bool the carrier of a1);
end;

:: TOPS_1:th 94
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is closed(b1)
   holds    b2 is nowhere_dense(b1)
   iff
      b2 = Fr b2;

:: TOPS_1:th 95
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is open(b1)
   holds Fr b2 is nowhere_dense(b1);

:: TOPS_1:funcreg 13
registration
  let a1 be TopSpace-like TopStruct;
  let a2 be open Element of bool the carrier of a1;
  cluster Fr a2 -> nowhere_dense;
end;

:: TOPS_1:th 96
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is closed(b1)
   holds Fr b2 is nowhere_dense(b1);

:: TOPS_1:funcreg 14
registration
  let a1 be TopSpace-like TopStruct;
  let a2 be closed Element of bool the carrier of a1;
  cluster Fr a2 -> nowhere_dense;
end;

:: TOPS_1:th 97
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is open(b1) & b2 is nowhere_dense(b1)
   holds b2 = {};

:: TOPS_1:condreg 6
registration
  let a1 be TopSpace-like TopStruct;
  cluster open nowhere_dense -> empty (Element of bool the carrier of a1);
end;

:: TOPS_1:attrnot 4 => TOPS_1:attr 4
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is condensed means
    Int Cl a2 c= a2 & a2 c= Cl Int a2;
end;

:: TOPS_1:dfs 6
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is condensed
it is sufficient to prove
  thus Int Cl a2 c= a2 & a2 c= Cl Int a2;

:: TOPS_1:def 6
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is condensed(b1)
   iff
      Int Cl b2 c= b2 & b2 c= Cl Int b2;

:: TOPS_1:attrnot 5 => TOPS_1:attr 5
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is closed_condensed means
    a2 = Cl Int a2;
end;

:: TOPS_1:dfs 7
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is closed_condensed
it is sufficient to prove
  thus a2 = Cl Int a2;

:: TOPS_1:def 7
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is closed_condensed(b1)
   iff
      b2 = Cl Int b2;

:: TOPS_1:attrnot 6 => TOPS_1:attr 6
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is open_condensed means
    a2 = Int Cl a2;
end;

:: TOPS_1:dfs 8
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is open_condensed
it is sufficient to prove
  thus a2 = Int Cl a2;

:: TOPS_1:def 8
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is open_condensed(b1)
   iff
      b2 = Int Cl b2;

:: TOPS_1:th 101
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is open_condensed(b1)
   iff
      b2 ` is closed_condensed(b1);

:: TOPS_1:th 102
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is closed_condensed(b1)
   holds Fr Int b2 = Fr b2;

:: TOPS_1:th 103
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is closed_condensed(b1)
   holds Fr b2 c= Cl Int b2;

:: TOPS_1:th 104
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is open_condensed(b1)
   holds Fr b2 = Fr Cl b2 & Fr Cl b2 = (Cl b2) \ b2;

:: TOPS_1:th 105
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is open(b1) & b2 is closed(b1)
   holds    b2 is closed_condensed(b1)
   iff
      b2 is open_condensed(b1);

:: TOPS_1:th 106
theorem
for b1 being TopSpace-like TopStruct
for b2 being TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2 holds
   (b4 is closed(b2) & b4 is condensed(b2) implies b4 is closed_condensed(b2)) &
    (b3 is closed_condensed(b1) implies b3 is closed(b1) & b3 is condensed(b1));

:: TOPS_1:th 107
theorem
for b1 being TopSpace-like TopStruct
for b2 being TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2 holds
   (b4 is open(b2) & b4 is condensed(b2) implies b4 is open_condensed(b2)) & (b3 is open_condensed(b1) implies b3 is open(b1) & b3 is condensed(b1));

:: TOPS_1:th 108
theorem
for b1 being 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 b2 \/ b3 is closed_condensed(b1);

:: TOPS_1:th 109
theorem
for b1 being 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 b2 /\ b3 is open_condensed(b1);

:: TOPS_1:th 110
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is condensed(b1)
   holds Int Fr b2 = {};

:: TOPS_1:th 111
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is condensed(b1)
   holds Int b2 is condensed(b1) & Cl b2 is condensed(b1);