Article ISOMICHI, MML version 4.99.1005

:: ISOMICHI:funcreg 1
registration
  let a1 be non trivial set;
  cluster ADTS a1 -> non trivial;
end;

:: ISOMICHI:exreg 1
registration
  cluster non empty non trivial strict TopSpace-like anti-discrete TopStruct;
end;

:: ISOMICHI:th 1
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
(Int Cl Int b2) /\ Int Cl Int b3 = Int Cl Int (b2 /\ b3);

:: ISOMICHI:th 2
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
Cl Int Cl (b2 \/ b3) = (Cl Int Cl b2) \/ Cl Int Cl b3;

:: ISOMICHI:attrnot 1 => ISOMICHI:attr 1
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is supercondensed means
    Int Cl a2 = Int a2;
end;

:: ISOMICHI:dfs 1
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is supercondensed
it is sufficient to prove
  thus Int Cl a2 = Int a2;

:: ISOMICHI:def 1
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is supercondensed(b1)
   iff
      Int Cl b2 = Int b2;

:: ISOMICHI:attrnot 2 => ISOMICHI:attr 2
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is subcondensed means
    Cl Int a2 = Cl a2;
end;

:: ISOMICHI:dfs 2
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is subcondensed
it is sufficient to prove
  thus Cl Int a2 = Cl a2;

:: ISOMICHI:def 2
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is subcondensed(b1)
   iff
      Cl Int b2 = Cl b2;

:: ISOMICHI:th 3
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 supercondensed(b1);

:: ISOMICHI:th 4
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is open(b1)
   holds b2 is subcondensed(b1);

:: ISOMICHI:attrnot 3 => TOPS_1:attr 4
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is condensed means
    Cl Int a2 = Cl a2 & Int Cl a2 = Int a2;
end;

:: ISOMICHI:dfs 3
definiens
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is condensed
it is sufficient to prove
  thus Cl Int a2 = Cl a2 & Int Cl a2 = Int a2;

:: ISOMICHI:def 3
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is condensed(b1)
   iff
      Cl Int b2 = Cl b2 & Int Cl b2 = Int b2;

:: ISOMICHI:th 5
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is condensed(b1)
   iff
      b2 is subcondensed(b1) & b2 is supercondensed(b1);

:: ISOMICHI:condreg 1
registration
  let a1 be TopSpace-like TopStruct;
  cluster condensed -> supercondensed subcondensed (Element of bool the carrier of a1);
end;

:: ISOMICHI:condreg 2
registration
  let a1 be TopSpace-like TopStruct;
  cluster supercondensed subcondensed -> condensed (Element of bool the carrier of a1);
end;

:: ISOMICHI:exreg 2
registration
  let a1 be TopSpace-like TopStruct;
  cluster condensed supercondensed subcondensed Element of bool the carrier of a1;
end;

:: ISOMICHI:th 6
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is supercondensed(b1)
   holds b2 ` is subcondensed(b1);

:: ISOMICHI:th 7
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is subcondensed(b1)
   holds b2 ` is supercondensed(b1);

:: ISOMICHI:th 8
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is supercondensed(b1)
   iff
      Int Cl b2 c= b2;

:: ISOMICHI:th 9
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is subcondensed(b1)
   iff
      b2 c= Cl Int b2;

:: ISOMICHI:condreg 3
registration
  let a1 be TopSpace-like TopStruct;
  cluster subcondensed -> semi-open (Element of bool the carrier of a1);
end;

:: ISOMICHI:condreg 4
registration
  let a1 be TopSpace-like TopStruct;
  cluster semi-open -> subcondensed (Element of bool the carrier of a1);
end;

:: ISOMICHI:th 10
theorem
for b1 being TopSpace-like 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;

:: ISOMICHI:attrnot 4 => TOPS_1:attr 6
notation
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  synonym regular_open for open_condensed;
end;

:: ISOMICHI:attrnot 5 => TOPS_1:attr 5
notation
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  synonym regular_closed for closed_condensed;
end;

:: ISOMICHI:th 11
theorem
for b1 being TopSpace-like TopStruct holds
   [#] b1 is open_condensed(b1) & [#] b1 is closed_condensed(b1);

:: ISOMICHI:funcreg 2
registration
  let a1 be TopSpace-like TopStruct;
  cluster [#] a1 -> closed_condensed open_condensed;
end;

:: ISOMICHI:th 12
theorem
for b1 being TopSpace-like TopStruct holds
   {} b1 is open_condensed(b1) & {} b1 is closed_condensed(b1);

:: ISOMICHI:funcreg 3
registration
  let a1 be TopSpace-like TopStruct;
  cluster {} a1 -> closed_condensed open_condensed;
end;

:: ISOMICHI:th 14
theorem
for b1 being TopSpace-like TopStruct holds
   Int Cl {} b1 = {} b1;

:: ISOMICHI:th 15
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is open_condensed(b1)
   holds b2 ` is closed_condensed(b1);

:: ISOMICHI:exreg 3
registration
  let a1 be TopSpace-like TopStruct;
  cluster closed_condensed open_condensed Element of bool the carrier of a1;
end;

:: ISOMICHI:funcreg 4
registration
  let a1 be TopSpace-like TopStruct;
  let a2 be open_condensed Element of bool the carrier of a1;
  cluster a2 ` -> closed_condensed;
end;

:: ISOMICHI:th 16
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is closed_condensed(b1)
   holds b2 ` is open_condensed(b1);

:: ISOMICHI:funcreg 5
registration
  let a1 be TopSpace-like TopStruct;
  let a2 be closed_condensed Element of bool the carrier of a1;
  cluster a2 ` -> open_condensed;
end;

:: ISOMICHI:condreg 5
registration
  let a1 be TopSpace-like TopStruct;
  cluster open_condensed -> open (Element of bool the carrier of a1);
end;

:: ISOMICHI:condreg 6
registration
  let a1 be TopSpace-like TopStruct;
  cluster closed_condensed -> closed (Element of bool the carrier of a1);
end;

:: ISOMICHI:th 17
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
   Int Cl b2 is open_condensed(b1) & Cl Int b2 is closed_condensed(b1);

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

:: ISOMICHI:funcreg 7
registration
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  cluster Cl Int a2 -> closed_condensed;
end;

:: ISOMICHI:th 18
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is open_condensed(b1)
   iff
      b2 is supercondensed(b1) & b2 is open(b1);

:: ISOMICHI:th 19
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is closed_condensed(b1)
   iff
      b2 is subcondensed(b1) & b2 is closed(b1);

:: ISOMICHI:condreg 7
registration
  let a1 be TopSpace-like TopStruct;
  cluster open_condensed -> open condensed (Element of bool the carrier of a1);
end;

:: ISOMICHI:condreg 8
registration
  let a1 be TopSpace-like TopStruct;
  cluster open condensed -> open_condensed (Element of bool the carrier of a1);
end;

:: ISOMICHI:condreg 9
registration
  let a1 be TopSpace-like TopStruct;
  cluster closed_condensed -> closed condensed (Element of bool the carrier of a1);
end;

:: ISOMICHI:condreg 10
registration
  let a1 be TopSpace-like TopStruct;
  cluster closed condensed -> closed_condensed (Element of bool the carrier of a1);
end;

:: ISOMICHI:th 20
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is condensed(b1)
   iff
      ex b3 being Element of bool the carrier of b1 st
         b3 is open_condensed(b1) & b3 c= b2 & b2 c= Cl b3;

:: ISOMICHI:th 21
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is condensed(b1)
   iff
      ex b3 being Element of bool the carrier of b1 st
         b3 is closed_condensed(b1) & Int b3 c= b2 & b2 c= b3;

:: ISOMICHI:funcnot 1 => TOPS_1:func 2
notation
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  synonym Bound a2 for Fr a2;
end;

:: ISOMICHI: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) \ Int a2;
end;

:: ISOMICHI:def 4
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   Fr b2 = (Cl b2) \ Int b2;

:: ISOMICHI:th 22
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
   Fr b2 is closed(b1);

:: ISOMICHI:th 23
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is condensed(b1)
   iff
      Fr b2 = (Cl Int b2) \ Int Cl b2 &
       Fr b2 = (Cl Int b2) /\ Cl Int (b2 `);

:: ISOMICHI:funcnot 3 => ISOMICHI:func 1
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  func Border A2 -> Element of bool the carrier of a1 equals
    Int Fr a2;
end;

:: ISOMICHI:def 5
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   Border b2 = Int Fr b2;

:: ISOMICHI:th 24
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
   Border b2 is open_condensed(b1) &
    Border b2 = (Int Cl b2) \ Cl Int b2 &
    Border b2 = (Int Cl b2) /\ Int Cl (b2 `);

:: ISOMICHI:funcreg 8
registration
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  cluster Border a2 -> open_condensed;
end;

:: ISOMICHI:th 25
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is supercondensed(b1)
   iff
      Int b2 is open_condensed(b1) & Border b2 is empty;

:: ISOMICHI:th 26
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is subcondensed(b1)
   iff
      Cl b2 is closed_condensed(b1) & Border b2 is empty;

:: ISOMICHI:funcreg 9
registration
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  cluster Border Border a2 -> empty;
end;

:: ISOMICHI:th 27
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is condensed(b1)
   iff
      Int b2 is open_condensed(b1) & Cl b2 is closed_condensed(b1) & Border b2 is empty;

:: ISOMICHI:th 28
theorem
for b1 being Element of bool the carrier of R^1
for b2 being real set
      st b1 = ].-infty,b2.]
   holds Int b1 = ].-infty,b2.[;

:: ISOMICHI:th 29
theorem
for b1 being Element of bool the carrier of R^1
for b2 being real set
      st b1 = [.b2,+infty.[
   holds Int b1 = ].b2,+infty.[;

:: ISOMICHI:th 30
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3 being real set
      st b1 = (].-infty,b2.] \/ IRRAT(b2,b3)) \/ [.b3,+infty.[
   holds Cl b1 = the carrier of R^1;

:: ISOMICHI:th 31
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3 being real set
      st b1 = RAT(b2,b3)
   holds Int b1 = {};

:: ISOMICHI:th 32
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3 being real set
      st b1 = IRRAT(b2,b3)
   holds Int b1 = {};

:: ISOMICHI:th 33
theorem
for b1, b2 being real set holds
].-infty,b1.] \ ].-infty,b2.[ = [.b2,b1.];

:: ISOMICHI:th 34
theorem
for b1, b2 being real set
      st b1 < b2
   holds [.b2,+infty.[ misses ].-infty,b1.[;

:: ISOMICHI:th 35
theorem
for b1, b2 being real set
      st b2 <= b1
   holds IRRAT(b1,b2) = {};

:: ISOMICHI:th 36
theorem
for b1, b2 being real set holds
IRRAT(b1,b2) c= [.b1,+infty.[;

:: ISOMICHI:th 37
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3, b4 being real set
      st b1 = ].-infty,b2.[ \/ RAT(b3,b4) &
         b2 < b3 &
         b3 < b4
   holds Int b1 = ].-infty,b2.[;

:: ISOMICHI:th 38
theorem
for b1, b2 being real set holds
[.b1,b2.] misses ].b2,+infty.[;

:: ISOMICHI:th 39
theorem
for b1 being real set holds
   [.b1,+infty.[ \ ].b1,+infty.[ = {b1};

:: ISOMICHI:th 40
theorem
for b1, b2 being real set holds
[.b1,b2.] = [.b1,+infty.[ \ ].b2,+infty.[;

:: ISOMICHI:th 41
theorem
for b1, b2 being real set
      st b1 < b2
   holds REAL = (].-infty,b1.[ \/ [.b1,b2.]) \/ ].b2,+infty.[;

:: ISOMICHI:th 42
theorem
for b1, b2 being real set holds
].b1,b2.[ = ].b1,+infty.[ \ [.b2,+infty.[;

:: ISOMICHI:th 43
theorem
for b1, b2, b3 being real set
      st b2 < b3 & b3 < b1
   holds ].-infty,b1.[ \ [.b2,b3.] = ].-infty,b2.[ \/ ].b3,b1.[;

:: ISOMICHI:th 44
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3, b4 being real set
      st b1 = ].-infty,b2.] \/ [.b3,b4.] &
         b2 < b3 &
         b3 < b4
   holds Int b1 = ].-infty,b2.[ \/ ].b3,b4.[;

:: ISOMICHI:prednot 1 => not XBOOLE_0:pred 3
notation
  let a1, a2 be set;
  antonym a1,a2 are_c=-incomparable for a1,a2 are_c=-comparable;
end;

:: ISOMICHI:th 45
theorem
for b1, b2 being set
      st b1,b2 are_c=-comparable & not b1 c= b2
   holds b2 c< b1;

:: ISOMICHI:attrnot 6 => ISOMICHI:attr 3
definition
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is 1st_class means
    Int Cl a2 c= Cl Int a2;
end;

:: ISOMICHI:dfs 6
definiens
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is 1st_class
it is sufficient to prove
  thus Int Cl a2 c= Cl Int a2;

:: ISOMICHI:def 6
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is 1st_class(b1)
   iff
      Int Cl b2 c= Cl Int b2;

:: ISOMICHI:attrnot 7 => ISOMICHI:attr 4
definition
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is 2nd_class means
    Cl Int a2 c< Int Cl a2;
end;

:: ISOMICHI:dfs 7
definiens
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is 2nd_class
it is sufficient to prove
  thus Cl Int a2 c< Int Cl a2;

:: ISOMICHI:def 7
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is 2nd_class(b1)
   iff
      Cl Int b2 c< Int Cl b2;

:: ISOMICHI:attrnot 8 => ISOMICHI:attr 5
definition
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is 3rd_class means
    not Cl Int a2,Int Cl a2 are_c=-comparable;
end;

:: ISOMICHI:dfs 8
definiens
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is 3rd_class
it is sufficient to prove
  thus not Cl Int a2,Int Cl a2 are_c=-comparable;

:: ISOMICHI:def 8
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is 3rd_class(b1)
   iff
      not Cl Int b2,Int Cl b2 are_c=-comparable;

:: ISOMICHI:th 46
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is not 1st_class(b1) & b2 is not 2nd_class(b1)
   holds b2 is 3rd_class(b1);

:: ISOMICHI:condreg 11
registration
  let a1 be TopSpace-like TopStruct;
  cluster 1st_class -> non 2nd_class non 3rd_class (Element of bool the carrier of a1);
end;

:: ISOMICHI:condreg 12
registration
  let a1 be TopSpace-like TopStruct;
  cluster 2nd_class -> non 1st_class non 3rd_class (Element of bool the carrier of a1);
end;

:: ISOMICHI:condreg 13
registration
  let a1 be TopSpace-like TopStruct;
  cluster 3rd_class -> non 1st_class non 2nd_class (Element of bool the carrier of a1);
end;

:: ISOMICHI:th 47
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is 1st_class(b1)
   iff
      Border b2 is empty;

:: ISOMICHI:condreg 14
registration
  let a1 be TopSpace-like TopStruct;
  cluster supercondensed -> 1st_class (Element of bool the carrier of a1);
end;

:: ISOMICHI:condreg 15
registration
  let a1 be TopSpace-like TopStruct;
  cluster subcondensed -> 1st_class (Element of bool the carrier of a1);
end;

:: ISOMICHI:attrnot 9 => ISOMICHI:attr 6
definition
  let a1 be TopSpace-like TopStruct;
  attr a1 is with_1st_class_subsets means
    ex b1 being Element of bool the carrier of a1 st
       b1 is 1st_class(a1);
end;

:: ISOMICHI:dfs 9
definiens
  let a1 be TopSpace-like TopStruct;
To prove
     a1 is with_1st_class_subsets
it is sufficient to prove
  thus ex b1 being Element of bool the carrier of a1 st
       b1 is 1st_class(a1);

:: ISOMICHI:def 9
theorem
for b1 being TopSpace-like TopStruct holds
      b1 is with_1st_class_subsets
   iff
      ex b2 being Element of bool the carrier of b1 st
         b2 is 1st_class(b1);

:: ISOMICHI:attrnot 10 => ISOMICHI:attr 7
definition
  let a1 be TopSpace-like TopStruct;
  attr a1 is with_2nd_class_subsets means
    ex b1 being Element of bool the carrier of a1 st
       b1 is 2nd_class(a1);
end;

:: ISOMICHI:dfs 10
definiens
  let a1 be TopSpace-like TopStruct;
To prove
     a1 is with_2nd_class_subsets
it is sufficient to prove
  thus ex b1 being Element of bool the carrier of a1 st
       b1 is 2nd_class(a1);

:: ISOMICHI:def 10
theorem
for b1 being TopSpace-like TopStruct holds
      b1 is with_2nd_class_subsets
   iff
      ex b2 being Element of bool the carrier of b1 st
         b2 is 2nd_class(b1);

:: ISOMICHI:attrnot 11 => ISOMICHI:attr 8
definition
  let a1 be TopSpace-like TopStruct;
  attr a1 is with_3rd_class_subsets means
    ex b1 being Element of bool the carrier of a1 st
       b1 is 3rd_class(a1);
end;

:: ISOMICHI:dfs 11
definiens
  let a1 be TopSpace-like TopStruct;
To prove
     a1 is with_3rd_class_subsets
it is sufficient to prove
  thus ex b1 being Element of bool the carrier of a1 st
       b1 is 3rd_class(a1);

:: ISOMICHI:def 11
theorem
for b1 being TopSpace-like TopStruct holds
      b1 is with_3rd_class_subsets
   iff
      ex b2 being Element of bool the carrier of b1 st
         b2 is 3rd_class(b1);

:: ISOMICHI:condreg 16
registration
  let a1 be non empty TopSpace-like anti-discrete TopStruct;
  cluster non empty proper -> 2nd_class (Element of bool the carrier of a1);
end;

:: ISOMICHI:exreg 4
registration
  let a1 be non empty non trivial strict TopSpace-like anti-discrete TopStruct;
  cluster 2nd_class Element of bool the carrier of a1;
end;

:: ISOMICHI:exreg 5
registration
  cluster non empty non trivial strict TopSpace-like with_1st_class_subsets with_2nd_class_subsets TopStruct;
end;

:: ISOMICHI:exreg 6
registration
  cluster non empty strict TopSpace-like with_3rd_class_subsets TopStruct;
end;

:: ISOMICHI:exreg 7
registration
  let a1 be TopSpace-like TopStruct;
  cluster 1st_class Element of bool the carrier of a1;
end;

:: ISOMICHI:exreg 8
registration
  let a1 be TopSpace-like with_2nd_class_subsets TopStruct;
  cluster 2nd_class Element of bool the carrier of a1;
end;

:: ISOMICHI:exreg 9
registration
  let a1 be TopSpace-like with_3rd_class_subsets TopStruct;
  cluster 3rd_class Element of bool the carrier of a1;
end;

:: ISOMICHI:th 48
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is 1st_class(b1)
   iff
      b2 ` is 1st_class(b1);

:: ISOMICHI:th 49
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is 2nd_class(b1)
   iff
      b2 ` is 2nd_class(b1);

:: ISOMICHI:th 50
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is 3rd_class(b1)
   iff
      b2 ` is 3rd_class(b1);

:: ISOMICHI:funcreg 10
registration
  let a1 be TopSpace-like TopStruct;
  let a2 be 1st_class Element of bool the carrier of a1;
  cluster a2 ` -> 1st_class;
end;

:: ISOMICHI:funcreg 11
registration
  let a1 be TopSpace-like with_2nd_class_subsets TopStruct;
  let a2 be 2nd_class Element of bool the carrier of a1;
  cluster a2 ` -> 2nd_class;
end;

:: ISOMICHI:funcreg 12
registration
  let a1 be TopSpace-like with_3rd_class_subsets TopStruct;
  let a2 be 3rd_class Element of bool the carrier of a1;
  cluster a2 ` -> 3rd_class;
end;

:: ISOMICHI:th 51
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is 1st_class(b1)
   holds Int Cl b2 = Int Cl Int b2 & Cl Int b2 = Cl Int Cl b2;

:: ISOMICHI:th 52
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st (Int Cl b2 = Int Cl Int b2 or Cl Int b2 = Cl Int Cl b2)
   holds b2 is 1st_class(b1);

:: ISOMICHI:th 53
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is 1st_class(b1) & b3 is 1st_class(b1)
   holds (Int Cl b2) /\ Int Cl b3 = Int Cl (b2 /\ b3) &
    (Cl Int b2) \/ Cl Int b3 = Cl Int (b2 \/ b3);

:: ISOMICHI:th 54
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is 1st_class(b1) & b3 is 1st_class(b1)
   holds b2 \/ b3 is 1st_class(b1) & b2 /\ b3 is 1st_class(b1);