Article KURATO_2, MML version 4.99.1005

:: KURATO_2:th 2
theorem
for b1 being Relation-like Function-like set
for b2 being set
      st b2 in proj1 b1
   holds meet b1 c= b1 . b2;

:: KURATO_2:th 3
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
   b2 = b3
iff
   for b4 being Element of NAT holds
      b2 . b4 = b3 . b4;

:: KURATO_2:th 4
theorem
for b1, b2, b3, b4 being set
      st b1 meets b2 & b3 meets b4
   holds [:b1,b3:] meets [:b2,b4:];

:: KURATO_2:condreg 1
registration
  let a1 be set;
  cluster Function-like quasi_total -> non empty (Relation of NAT,bool a1);
end;

:: KURATO_2:exreg 1
registration
  let a1 be non empty set;
  cluster non empty Relation-like non-empty Function-like quasi_total total Relation of NAT,bool a1;
end;

:: KURATO_2:modenot 1
definition
  let a1 be 1-sorted;
  mode SetSequence of a1 is Function-like quasi_total Relation of NAT,bool the carrier of a1;
end;

:: KURATO_2:sch 1
scheme KURATO_2:sch 1
{F1 -> set,
  F2 -> Element of bool F1()}:
ex b1 being Function-like quasi_total Relation of NAT,bool F1() st
   for b2 being Element of NAT holds
      b1 . b2 = F2(b2)


:: KURATO_2:funcnot 1 => KURATO_2:func 1
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
  redefine func Union a2 -> Element of bool a1;
end;

:: KURATO_2:funcnot 2 => KURATO_2:func 2
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
  redefine func meet a2 -> Element of bool a1;
end;

:: KURATO_2:funcnot 3 => KURATO_2:func 3
definition
  let a1 be non empty set;
  let a2 be Function-like quasi_total Relation of NAT,a1;
  let a3 be Element of NAT;
  func A2 ^\ A3 -> Function-like quasi_total Relation of NAT,a1 means
    for b1 being Element of NAT holds
       it . b1 = a2 . (b1 + a3);
end;

:: KURATO_2:def 2
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of NAT,b1
for b3 being Element of NAT
for b4 being Function-like quasi_total Relation of NAT,b1 holds
      b4 = b2 ^\ b3
   iff
      for b5 being Element of NAT holds
         b4 . b5 = b2 . (b5 + b3);

:: KURATO_2:funcnot 4 => KURATO_2:func 4
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
  func lim_inf A2 -> Element of bool a1 means
    ex b1 being Function-like quasi_total Relation of NAT,bool a1 st
       it = Union b1 &
        (for b2 being Element of NAT holds
           b1 . b2 = meet (a2 ^\ b2));
end;

:: KURATO_2:def 3
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being Element of bool b1 holds
      b3 = lim_inf b2
   iff
      ex b4 being Function-like quasi_total Relation of NAT,bool b1 st
         b3 = Union b4 &
          (for b5 being Element of NAT holds
             b4 . b5 = meet (b2 ^\ b5));

:: KURATO_2:funcnot 5 => KURATO_2:func 5
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
  func lim_sup A2 -> Element of bool a1 means
    ex b1 being Function-like quasi_total Relation of NAT,bool a1 st
       it = meet b1 &
        (for b2 being Element of NAT holds
           b1 . b2 = Union (a2 ^\ b2));
end;

:: KURATO_2:def 4
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being Element of bool b1 holds
      b3 = lim_sup b2
   iff
      ex b4 being Function-like quasi_total Relation of NAT,bool b1 st
         b3 = meet b4 &
          (for b5 being Element of NAT holds
             b4 . b5 = Union (b2 ^\ b5));

:: KURATO_2:th 6
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being set holds
      b3 in meet b2
   iff
      for b4 being Element of NAT holds
         b3 in b2 . b4;

:: KURATO_2:th 7
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being set holds
      b3 in lim_inf b2
   iff
      ex b4 being Element of NAT st
         for b5 being Element of NAT holds
            b3 in b2 . (b4 + b5);

:: KURATO_2:th 8
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being set holds
      b3 in lim_sup b2
   iff
      for b4 being Element of NAT holds
         ex b5 being Element of NAT st
            b3 in b2 . (b4 + b5);

:: KURATO_2:th 9
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
   lim_inf b2 c= lim_sup b2;

:: KURATO_2:th 10
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
   meet b2 c= lim_inf b2;

:: KURATO_2:th 11
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
   lim_sup b2 c= Union b2;

:: KURATO_2:th 12
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
   lim_inf b2 = (lim_sup Complement b2) `;

:: KURATO_2:th 13
theorem
for b1 being set
for b2, b3, b4 being Function-like quasi_total Relation of NAT,bool b1
      st for b5 being Element of NAT holds
           b4 . b5 = (b2 . b5) /\ (b3 . b5)
   holds lim_inf b4 = (lim_inf b2) /\ lim_inf b3;

:: KURATO_2:th 14
theorem
for b1 being set
for b2, b3, b4 being Function-like quasi_total Relation of NAT,bool b1
      st for b5 being Element of NAT holds
           b4 . b5 = (b2 . b5) \/ (b3 . b5)
   holds lim_sup b4 = (lim_sup b2) \/ lim_sup b3;

:: KURATO_2:th 15
theorem
for b1 being set
for b2, b3, b4 being Function-like quasi_total Relation of NAT,bool b1
      st for b5 being Element of NAT holds
           b4 . b5 = (b2 . b5) \/ (b3 . b5)
   holds (lim_inf b2) \/ lim_inf b3 c= lim_inf b4;

:: KURATO_2:th 16
theorem
for b1 being set
for b2, b3, b4 being Function-like quasi_total Relation of NAT,bool b1
      st for b5 being Element of NAT holds
           b4 . b5 = (b2 . b5) /\ (b3 . b5)
   holds lim_sup b4 c= (lim_sup b2) /\ lim_sup b3;

:: KURATO_2:th 17
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being Element of bool b1
      st for b4 being Element of NAT holds
           b2 . b4 = b3
   holds lim_sup b2 = b3;

:: KURATO_2:th 18
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being Element of bool b1
      st for b4 being Element of NAT holds
           b2 . b4 = b3
   holds lim_inf b2 = b3;

:: KURATO_2:th 19
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1
for b4 being Element of bool b1
      st for b5 being Element of NAT holds
           b3 . b5 = b4 \+\ (b2 . b5)
   holds b4 \+\ lim_inf b2 c= lim_sup b3;

:: KURATO_2:th 20
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1
for b4 being Element of bool b1
      st for b5 being Element of NAT holds
           b3 . b5 = b4 \+\ (b2 . b5)
   holds b4 \+\ lim_sup b2 c= lim_sup b3;

:: KURATO_2:attrnot 1 => KURATO_2:attr 1
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
  attr a2 is descending means
    for b1 being Element of NAT holds
       a2 . (b1 + 1) c= a2 . b1;
end;

:: KURATO_2:dfs 4
definiens
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
To prove
     a2 is descending
it is sufficient to prove
  thus for b1 being Element of NAT holds
       a2 . (b1 + 1) c= a2 . b1;

:: KURATO_2:def 5
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
      b2 is descending(b1)
   iff
      for b3 being Element of NAT holds
         b2 . (b3 + 1) c= b2 . b3;

:: KURATO_2:attrnot 2 => KURATO_2:attr 2
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
  attr a2 is ascending means
    for b1 being Element of NAT holds
       a2 . b1 c= a2 . (b1 + 1);
end;

:: KURATO_2:dfs 5
definiens
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
To prove
     a2 is ascending
it is sufficient to prove
  thus for b1 being Element of NAT holds
       a2 . b1 c= a2 . (b1 + 1);

:: KURATO_2:def 6
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
      b2 is ascending(b1)
   iff
      for b3 being Element of NAT holds
         b2 . b3 c= b2 . (b3 + 1);

:: KURATO_2:th 21
theorem
for b1 being Relation-like Function-like set
   st for b2 being Element of NAT holds
        b1 . (b2 + 1) c= b1 . b2
for b2, b3 being Element of NAT
      st b2 <= b3
   holds b1 . b3 c= b1 . b2;

:: KURATO_2:th 22
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
   st b2 is descending(b1)
for b3, b4 being Element of NAT
      st b4 <= b3
   holds b2 . b3 c= b2 . b4;

:: KURATO_2:th 23
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
   st b2 is ascending(b1)
for b3, b4 being Element of NAT
      st b4 <= b3
   holds b2 . b4 c= b2 . b3;

:: KURATO_2:th 24
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being set
      st b2 is descending(b1) &
         (ex b4 being Element of NAT st
            for b5 being Element of NAT
                  st b4 < b5
               holds b3 in b2 . b5)
   holds b3 in meet b2;

:: KURATO_2:th 25
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
      st b2 is descending(b1)
   holds lim_inf b2 = meet b2;

:: KURATO_2:th 26
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
      st b2 is ascending(b1)
   holds lim_sup b2 = Union b2;

:: KURATO_2:attrnot 3 => KURATO_2:attr 3
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
  attr a2 is convergent means
    lim_sup a2 = lim_inf a2;
end;

:: KURATO_2:dfs 6
definiens
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
To prove
     a2 is convergent
it is sufficient to prove
  thus lim_sup a2 = lim_inf a2;

:: KURATO_2:def 7
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
      b2 is convergent(b1)
   iff
      lim_sup b2 = lim_inf b2;

:: KURATO_2:th 27
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
      st b2 is constant
   holds the_value_of b2 is Element of bool b1;

:: KURATO_2:attrnot 4 => KURATO_2:attr 4
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
  redefine attr a2 is constant means
    ex b1 being Element of bool a1 st
       for b2 being Element of NAT holds
          a2 . b2 = b1;
end;

:: KURATO_2:dfs 7
definiens
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
To prove
     a1 is constant
it is sufficient to prove
  thus ex b1 being Element of bool a1 st
       for b2 being Element of NAT holds
          a2 . b2 = b1;

:: KURATO_2:def 8
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
      b2 is constant
   iff
      ex b3 being Element of bool b1 st
         for b4 being Element of NAT holds
            b2 . b4 = b3;

:: KURATO_2:condreg 2
registration
  let a1 be set;
  cluster Function-like constant quasi_total -> descending ascending convergent (Relation of NAT,bool a1);
end;

:: KURATO_2:exreg 2
registration
  let a1 be set;
  cluster non empty Relation-like Function-like constant quasi_total total Relation of NAT,bool a1;
end;

:: KURATO_2:funcnot 6 => KURATO_2:func 6
definition
  let a1 be set;
  let a2 be Function-like quasi_total convergent Relation of NAT,bool a1;
  func Lim_K A2 -> Element of bool a1 means
    it = lim_sup a2 & it = lim_inf a2;
end;

:: KURATO_2:def 9
theorem
for b1 being set
for b2 being Function-like quasi_total convergent Relation of NAT,bool b1
for b3 being Element of bool b1 holds
      b3 = Lim_K b2
   iff
      b3 = lim_sup b2 & b3 = lim_inf b2;

:: KURATO_2:th 28
theorem
for b1 being set
for b2 being Function-like quasi_total convergent Relation of NAT,bool b1
for b3 being set holds
      b3 in Lim_K b2
   iff
      ex b4 being Element of NAT st
         for b5 being Element of NAT holds
            b3 in b2 . (b4 + b5);

:: KURATO_2:funcreg 1
registration
  let a1 be FinSequence of the carrier of TOP-REAL 2;
  cluster L~ a1 -> closed;
end;

:: KURATO_2:th 30
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of Euclid b1
for b3 being real set holds
   Ball(b2,b3) is open Element of bool the carrier of TOP-REAL b1;

:: KURATO_2:th 32
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of Euclid b1
for b3, b4 being Element of the carrier of TOP-REAL b1
for b5 being real set
      st b2 = b4 & b3 in Ball(b2,b5)
   holds |.b3 - b4.| < b5;

:: KURATO_2:th 33
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of Euclid b1
for b3, b4 being Element of the carrier of TOP-REAL b1
for b5 being real set
      st b2 = b4 & |.b3 - b4.| < b5
   holds b3 in Ball(b2,b5);

:: KURATO_2:th 34
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
for b3 being Element of bool the carrier of TOP-REAL b1
      st b2 in Cl b3
   holds ex b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 st
      rng b4 c= b3 & b4 is convergent(b1) & lim b4 = b2;

:: KURATO_2:funcreg 2
registration
  let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
  cluster TopSpaceMetr a1 -> first-countable;
end;

:: KURATO_2:funcreg 3
registration
  let a1 be Element of NAT;
  cluster TOP-REAL a1 -> strict TopSpace-like first-countable;
end;

:: KURATO_2:th 36
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of the carrier of Euclid b1
      st b3 = b4
   holds    b3 in Cl b2
   iff
      for b5 being real set
            st 0 < b5
         holds Ball(b4,b5) meets b2;

:: KURATO_2:th 37
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of the carrier of Euclid b1
      st b4 = b2 & b2 <> b3
   holds ex b5 being Element of REAL st
      not b3 in Ball(b4,b5);

:: KURATO_2:th 38
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1 holds
      b2 is not Bounded(b1)
   iff
      for b3 being Element of REAL
            st 0 < b3
         holds ex b4, b5 being Element of the carrier of Euclid b1 st
            b4 in b2 & b5 in b2 & b3 < dist(b4,b5);

:: KURATO_2:th 39
theorem
for b1 being Element of NAT
for b2, b3 being real set
for b4, b5 being Element of the carrier of Euclid b1
      st Ball(b4,b2) meets Ball(b5,b3)
   holds dist(b4,b5) < b2 + b3;

:: KURATO_2:th 40
theorem
for b1 being Element of NAT
for b2, b3, b4 being real set
for b5, b6, b7 being Element of the carrier of Euclid b1
      st Ball(b5,b2) meets Ball(b7,b4) & Ball(b7,b4) meets Ball(b6,b3)
   holds dist(b5,b6) < (b2 + b3) + (2 * b4);

:: KURATO_2:th 41
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
for b5 being Element of bool the carrier of [:b1,b2:] holds
      b5 is a_neighborhood of [:{b3},{b4}:]
   iff
      b5 is a_neighborhood of [b3,b4];

:: KURATO_2:funcnot 7 => KURATO_2:func 7
definition
  let a1 be non empty set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
  let a3 be Element of NAT;
  redefine func a2 . a3 -> Element of bool a1;
end;

:: KURATO_2:th 42
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being Function-like quasi_total natural-valued Relation of NAT,REAL holds
   b2 * b3 is Function-like quasi_total Relation of NAT,bool the carrier of b1;

:: KURATO_2:th 43
theorem
id NAT is Function-like quasi_total natural-valued increasing Relation of NAT,REAL;

:: KURATO_2:modenot 2 => KURATO_2:mode 1
definition
  let a1 be non empty 1-sorted;
  let a2 be Function-like quasi_total Relation of NAT,bool the carrier of a1;
  mode subsequence of A2 -> Function-like quasi_total Relation of NAT,bool the carrier of a1 means
    ex b1 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
       it = a2 * b1;
end;

:: KURATO_2:dfs 9
definiens
  let a1 be non empty 1-sorted;
  let a2, a3 be Function-like quasi_total Relation of NAT,bool the carrier of a1;
To prove
     a3 is subsequence of a2
it is sufficient to prove
  thus ex b1 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
       a3 = a2 * b1;

:: KURATO_2:def 10
theorem
for b1 being non empty 1-sorted
for b2, b3 being Function-like quasi_total Relation of NAT,bool the carrier of b1 holds
   b3 is subsequence of b2
iff
   ex b4 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
      b3 = b2 * b4;

:: KURATO_2:th 44
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1 holds
   b2 is subsequence of b2;

:: KURATO_2:th 45
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being subsequence of b2 holds
   rng b3 c= rng b2;

:: KURATO_2:th 46
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being subsequence of b2
for b4 being subsequence of b3 holds
   b4 is subsequence of b2;

:: KURATO_2:th 47
theorem
for b1 being non empty 1-sorted
for b2, b3 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b4 being Element of bool the carrier of b1
      st b3 is subsequence of b2 &
         (for b5 being Element of NAT holds
            b2 . b5 = b4)
   holds b3 = b2;

:: KURATO_2:th 48
theorem
for b1 being non empty 1-sorted
for b2 being Function-like constant quasi_total Relation of NAT,bool the carrier of b1
for b3 being subsequence of b2 holds
   b2 = b3;

:: KURATO_2:th 49
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being subsequence of b2
for b4 being Element of NAT holds
   ex b5 being Element of NAT st
      b4 <= b5 & b3 . b4 = b2 . b5;

:: KURATO_2:condreg 3
registration
  let a1 be non empty 1-sorted;
  let a2 be Function-like constant quasi_total Relation of NAT,bool the carrier of a1;
  cluster -> constant (subsequence of a2);
end;

:: KURATO_2:sch 2
scheme KURATO_2:sch 2
{F1 -> non empty TopSpace-like TopStruct,
  F2 -> Function-like quasi_total Relation of NAT,bool the carrier of F1()}:
ex b1 being subsequence of F2() st
   for b2 being Element of NAT holds
      P1[b1 . b2]
provided
   for b1 being Element of NAT holds
      ex b2 being Element of NAT st
         b1 <= b2 & P1[F2() . b2];


:: KURATO_2:funcnot 8 => KURATO_2:func 8
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Function-like quasi_total Relation of NAT,bool the carrier of a1;
  func Lim_inf A2 -> Element of bool the carrier of a1 means
    for b1 being Element of the carrier of a1 holds
          b1 in it
       iff
          for b2 being a_neighborhood of b1 holds
             ex b3 being Element of NAT st
                for b4 being Element of NAT
                      st b3 < b4
                   holds a2 . b4 meets b2;
end;

:: KURATO_2:def 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being Element of bool the carrier of b1 holds
      b3 = Lim_inf b2
   iff
      for b4 being Element of the carrier of b1 holds
            b4 in b3
         iff
            for b5 being a_neighborhood of b4 holds
               ex b6 being Element of NAT st
                  for b7 being Element of NAT
                        st b6 < b7
                     holds b2 . b7 meets b5;

:: KURATO_2:th 50
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL b1
for b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of the carrier of Euclid b1
      st b3 = b4
   holds    b3 in Lim_inf b2
   iff
      for b5 being real set
            st 0 < b5
         holds ex b6 being Element of NAT st
            for b7 being Element of NAT
                  st b6 < b7
               holds b2 . b7 meets Ball(b4,b5);

:: KURATO_2:th 51
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1 holds
   Cl Lim_inf b2 = Lim_inf b2;

:: KURATO_2:th 52
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1 holds
   Lim_inf b2 is closed(b1);

:: KURATO_2:th 53
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Function-like quasi_total Relation of NAT,bool the carrier of b1
      st b2 is subsequence of b3
   holds Lim_inf b3 c= Lim_inf b2;

:: KURATO_2:th 54
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Function-like quasi_total Relation of NAT,bool the carrier of b1
      st for b4 being Element of NAT holds
           b2 . b4 c= b3 . b4
   holds Lim_inf b2 c= Lim_inf b3;

:: KURATO_2:th 55
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Function-like quasi_total Relation of NAT,bool the carrier of b1
      st for b5 being Element of NAT holds
           b4 . b5 = (b2 . b5) \/ (b3 . b5)
   holds (Lim_inf b2) \/ Lim_inf b3 c= Lim_inf b4;

:: KURATO_2:th 56
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Function-like quasi_total Relation of NAT,bool the carrier of b1
      st for b5 being Element of NAT holds
           b4 . b5 = (b2 . b5) /\ (b3 . b5)
   holds Lim_inf b4 c= (Lim_inf b2) /\ Lim_inf b3;

:: KURATO_2:th 57
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Function-like quasi_total Relation of NAT,bool the carrier of b1
      st for b4 being Element of NAT holds
           b3 . b4 = Cl (b2 . b4)
   holds Lim_inf b3 = Lim_inf b2;

:: KURATO_2:th 58
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL b1
for b3 being Element of the carrier of TOP-REAL b1
      st ex b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 st
           b4 is convergent(b1) &
            (for b5 being Element of NAT holds
               b4 . b5 in b2 . b5) &
            b3 = lim b4
   holds b3 in Lim_inf b2;

:: KURATO_2:th 59
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,bool the carrier of b1
      st for b4 being Element of NAT holds
           b3 . b4 c= b2
   holds Lim_inf b3 c= Cl b2;

:: KURATO_2:th 60
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being Element of bool the carrier of b1
      st for b4 being Element of NAT holds
           b2 . b4 = b3
   holds Lim_inf b2 = Cl b3;

:: KURATO_2:th 61
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being closed Element of bool the carrier of b1
      st for b4 being Element of NAT holds
           b2 . b4 = b3
   holds Lim_inf b2 = b3;

:: KURATO_2:th 62
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL b1
for b3 being Element of bool the carrier of TOP-REAL b1
      st b3 is Bounded(b1) &
         (for b4 being Element of NAT holds
            b2 . b4 c= b3)
   holds Lim_inf b2 is Bounded(b1);

:: KURATO_2:th 63
theorem
for b1 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 is Bounded(2) &
         (for b3 being Element of NAT holds
            b1 . b3 c= b2)
   holds Lim_inf b1 is compact(TOP-REAL 2);

:: KURATO_2:th 64
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL b1
for b4 being Function-like quasi_total Relation of NAT,bool the carrier of [:TOP-REAL b1,TOP-REAL b1:]
      st for b5 being Element of NAT holds
           b4 . b5 = [:b2 . b5,b3 . b5:]
   holds [:Lim_inf b2,Lim_inf b3:] = Lim_inf b4;

:: KURATO_2:th 65
theorem
for b1 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2 holds
   lim_inf b1 c= Lim_inf b1;

:: KURATO_2:th 66
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   Fr ((UBD L~ Cage(b1,b2)) `) = L~ Cage(b1,b2);

:: KURATO_2:funcnot 9 => KURATO_2:func 9
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Function-like quasi_total Relation of NAT,bool the carrier of a1;
  func Lim_sup A2 -> Element of bool the carrier of a1 means
    for b1 being set holds
          b1 in it
       iff
          ex b2 being subsequence of a2 st
             b1 in Lim_inf b2;
end;

:: KURATO_2:def 12
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being Element of bool the carrier of b1 holds
      b3 = Lim_sup b2
   iff
      for b4 being set holds
            b4 in b3
         iff
            ex b5 being subsequence of b2 st
               b4 in Lim_inf b5;

:: KURATO_2:th 67
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
for b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of the carrier of Euclid b1
      st b3 = b4
   holds    b3 is_a_cluster_point_of b2
   iff
      for b5 being real set
      for b6 being Element of NAT
            st 0 < b5
         holds ex b7 being Element of NAT st
            b6 <= b7 & b2 . b7 in Ball(b4,b5);

:: KURATO_2:th 68
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1 holds
   Lim_inf b2 c= Lim_sup b2;

:: KURATO_2:th 69
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2
      st (for b4 being Element of NAT holds
            b1 . b4 c= b2 . b4) &
         b3 is subsequence of b1
   holds ex b4 being subsequence of b2 st
      for b5 being Element of NAT holds
         b3 . b5 c= b4 . b5;

:: KURATO_2:th 70
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2
      st (for b4 being Element of NAT holds
            b1 . b4 c= b2 . b4) &
         b3 is subsequence of b2
   holds ex b4 being subsequence of b1 st
      for b5 being Element of NAT holds
         b4 . b5 c= b3 . b5;

:: KURATO_2:th 71
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2
      st for b3 being Element of NAT holds
           b1 . b3 c= b2 . b3
   holds Lim_sup b1 c= Lim_sup b2;

:: KURATO_2:th 72
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2
      st for b4 being Element of NAT holds
           b3 . b4 = (b1 . b4) \/ (b2 . b4)
   holds (Lim_sup b1) \/ Lim_sup b2 c= Lim_sup b3;

:: KURATO_2:th 73
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2
      st for b4 being Element of NAT holds
           b3 . b4 = (b1 . b4) /\ (b2 . b4)
   holds Lim_sup b3 c= (Lim_sup b1) /\ Lim_sup b2;

:: KURATO_2:th 74
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2
for b3, b4 being Function-like quasi_total Relation of NAT,bool the carrier of [:TOP-REAL 2,TOP-REAL 2:]
      st (for b5 being Element of NAT holds
            b3 . b5 = [:b1 . b5,b2 . b5:]) &
         b4 is subsequence of b3
   holds ex b5, b6 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2 st
      b5 is subsequence of b1 &
       b6 is subsequence of b2 &
       (for b7 being Element of NAT holds
          b4 . b7 = [:b5 . b7,b6 . b7:]);

:: KURATO_2:th 75
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of NAT,bool the carrier of [:TOP-REAL 2,TOP-REAL 2:]
      st for b4 being Element of NAT holds
           b3 . b4 = [:b1 . b4,b2 . b4:]
   holds Lim_sup b3 c= [:Lim_sup b1,Lim_sup b2:];

:: KURATO_2:th 76
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being Element of bool the carrier of b1
      st for b4 being Element of NAT holds
           b2 . b4 = b3
   holds Lim_inf b2 = Lim_sup b2;

:: KURATO_2:th 77
theorem
for b1 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
      st for b3 being Element of NAT holds
           b1 . b3 = b2
   holds Lim_sup b1 = Cl b2;

:: KURATO_2:th 78
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2
      st for b3 being Element of NAT holds
           b2 . b3 = Cl (b1 . b3)
   holds Lim_sup b2 = Lim_sup b1;