Article PROB_1, MML version 4.99.1005

:: PROB_1:th 3
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
      st ex b3 being Element of NAT st
           for b4 being Element of NAT
                 st b3 <= b4
              holds b2 . b4 = b1
   holds b2 is convergent & lim b2 = b1;

:: PROB_1:attrnot 1 => PROB_1:attr 1
definition
  let a1 be set;
  let a2 be Element of bool bool a1;
  attr a2 is compl-closed means
    for b1 being Element of bool a1
          st b1 in a2
       holds b1 ` in a2;
end;

:: PROB_1:dfs 1
definiens
  let a1 be set;
  let a2 be Element of bool bool a1;
To prove
     a2 is compl-closed
it is sufficient to prove
  thus for b1 being Element of bool a1
          st b1 in a2
       holds b1 ` in a2;

:: PROB_1:def 1
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
      b2 is compl-closed(b1)
   iff
      for b3 being Element of bool b1
            st b3 in b2
         holds b3 ` in b2;

:: PROB_1:exreg 1
registration
  let a1 be set;
  cluster non empty cap-closed compl-closed Element of bool bool a1;
end;

:: PROB_1:modenot 1
definition
  let a1 be set;
  mode Field_Subset of a1 is non empty cap-closed compl-closed Element of bool bool a1;
end;

:: PROB_1:th 4
theorem
for b1 being set
for b2, b3 being Element of bool b1 holds
{b2,b3} is Element of bool bool b1;

:: PROB_1:th 6
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1 holds
   ex b3 being Element of bool b1 st
      b3 in b2;

:: PROB_1:th 9
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1
for b3, b4 being set
      st b3 in b2 & b4 in b2
   holds b3 \/ b4 in b2;

:: PROB_1:th 10
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1 holds
   {} in b2;

:: PROB_1:th 11
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1 holds
   b1 in b2;

:: PROB_1:th 12
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1
for b3, b4 being Element of bool b1
      st b3 in b2 & b4 in b2
   holds b3 \ b4 in b2;

:: PROB_1:th 13
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1
for b3, b4 being set
      st b3 in b2 & b4 in b2
   holds (b3 \ b4) \/ b4 in b2;

:: PROB_1:th 14
theorem
for b1 being set holds
   {{},b1} is non empty cap-closed compl-closed Element of bool bool b1;

:: PROB_1:th 15
theorem
for b1 being set holds
   bool b1 is non empty cap-closed compl-closed Element of bool bool b1;

:: PROB_1:th 16
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1 holds
   {{},b1} c= b2 & b2 c= bool b1;

:: PROB_1:th 18
theorem
for b1 being set holds
   (for b2 being set
          st b2 in [:NAT,{b1}:]
       holds ex b3, b4 being set st
          [b3,b4] = b2) &
    (for b2, b3, b4 being set
          st [b2,b3] in [:NAT,{b1}:] &
             [b2,b4] in [:NAT,{b1}:]
       holds b3 = b4);

:: PROB_1:modenot 2
definition
  let a1 be set;
  mode SetSequence of a1 is Function-like quasi_total Relation of NAT,bool a1;
end;

:: PROB_1:funcnot 1 => PROB_1:func 1
definition
  let a1 be 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;

:: PROB_1:th 23
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
   union proj2 b2 is Element of bool b1;

:: PROB_1:funcnot 2 => PROB_1:func 2
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;

:: PROB_1:th 25
theorem
for b1, b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
      b2 in Union b3
   iff
      ex b4 being Element of NAT st
         b2 in b3 . b4;

:: PROB_1:th 26
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
   ex b3 being Function-like quasi_total Relation of NAT,bool b1 st
      for b4 being Element of NAT holds
         b3 . b4 = (b2 . b4) `;

:: PROB_1:funcnot 3 => PROB_1:func 3
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
  func Complement A2 -> Function-like quasi_total Relation of NAT,bool a1 means
    for b1 being Element of NAT holds
       it . b1 = (a2 . b1) `;
  involutiveness;
::  for a1 being set
::  for a2 being Function-like quasi_total Relation of NAT,bool a1 holds
::     Complement Complement a2 = a2;
end;

:: PROB_1:def 4
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
   b3 = Complement b2
iff
   for b4 being Element of NAT holds
      b3 . b4 = (b2 . b4) `;

:: PROB_1:funcnot 4 => PROB_1:func 4
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
  func Intersection A2 -> Element of bool a1 equals
    (Union Complement a2) `;
end;

:: PROB_1:def 5
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
   Intersection b2 = (Union Complement b2) `;

:: PROB_1:th 29
theorem
for b1, b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
      b2 in Intersection b3
   iff
      for b4 being Element of NAT holds
         b2 in b3 . b4;

:: PROB_1:th 30
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3, b4 being Element of bool b1
      st b2 . 0 = b3 &
         (for b5 being Element of NAT
               st b5 <> 0
            holds b2 . b5 = b4)
   holds Intersection b2 = b3 /\ b4;

:: PROB_1:attrnot 2 => PROB_1:attr 2
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
  attr a2 is non-increasing means
    for b1, b2 being Element of NAT
          st b1 <= b2
       holds a2 . b2 c= a2 . b1;
end;

:: PROB_1:dfs 4
definiens
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
To prove
     a2 is non-increasing
it is sufficient to prove
  thus for b1, b2 being Element of NAT
          st b1 <= b2
       holds a2 . b2 c= a2 . b1;

:: PROB_1:def 6
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
      b2 is non-increasing(b1)
   iff
      for b3, b4 being Element of NAT
            st b3 <= b4
         holds b2 . b4 c= b2 . b3;

:: PROB_1:attrnot 3 => PROB_1:attr 3
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
  attr a2 is non-decreasing means
    for b1, b2 being Element of NAT
          st b1 <= b2
       holds a2 . b1 c= a2 . b2;
end;

:: PROB_1:dfs 5
definiens
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
To prove
     a2 is non-decreasing
it is sufficient to prove
  thus for b1, b2 being Element of NAT
          st b1 <= b2
       holds a2 . b1 c= a2 . b2;

:: PROB_1:def 7
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
      b2 is non-decreasing(b1)
   iff
      for b3, b4 being Element of NAT
            st b3 <= b4
         holds b2 . b3 c= b2 . b4;

:: PROB_1:attrnot 4 => PROB_1:attr 4
definition
  let a1 be set;
  let a2 be Element of bool bool a1;
  attr a2 is sigma-multiplicative means
    for b1 being Function-like quasi_total Relation of NAT,bool a1
          st for b2 being Element of NAT holds
               b1 . b2 in a2
       holds Intersection b1 in a2;
end;

:: PROB_1:dfs 6
definiens
  let a1 be set;
  let a2 be Element of bool bool a1;
To prove
     a2 is sigma-multiplicative
it is sufficient to prove
  thus for b1 being Function-like quasi_total Relation of NAT,bool a1
          st for b2 being Element of NAT holds
               b1 . b2 in a2
       holds Intersection b1 in a2;

:: PROB_1:def 8
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
      b2 is sigma-multiplicative(b1)
   iff
      for b3 being Function-like quasi_total Relation of NAT,bool b1
            st for b4 being Element of NAT holds
                 b3 . b4 in b2
         holds Intersection b3 in b2;

:: PROB_1:exreg 2
registration
  let a1 be set;
  cluster non empty compl-closed sigma-multiplicative Element of bool bool a1;
end;

:: PROB_1:modenot 3
definition
  let a1 be set;
  mode SigmaField of a1 is non empty compl-closed sigma-multiplicative Element of bool bool a1;
end;

:: PROB_1:th 32
theorem
for b1 being set
for b2 being non empty set holds
      b2 is non empty compl-closed sigma-multiplicative Element of bool bool b1
   iff
      b2 c= bool b1 &
       (for b3 being Function-like quasi_total Relation of NAT,bool b1
             st for b4 being Element of NAT holds
                  b3 . b4 in b2
          holds Intersection b3 in b2) &
       (for b3 being Element of bool b1
             st b3 in b2
          holds b3 ` in b2);

:: PROB_1:th 35
theorem
for b1, b2 being set
      st b1 is non empty compl-closed sigma-multiplicative Element of bool bool b2
   holds b1 is non empty cap-closed compl-closed Element of bool bool b2;

:: PROB_1:condreg 1
registration
  let a1 be set;
  cluster non empty compl-closed sigma-multiplicative -> cap-closed (Element of bool bool a1);
end;

:: PROB_1:th 38
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1 holds
   ex b3 being Element of bool b1 st
      b3 in b2;

:: PROB_1:th 41
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Element of bool b1
      st b3 in b2 & b4 in b2
   holds b3 \/ b4 in b2;

:: PROB_1:th 42
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1 holds
   {} in b2;

:: PROB_1:th 43
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1 holds
   b1 in b2;

:: PROB_1:th 44
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Element of bool b1
      st b3 in b2 & b4 in b2
   holds b3 \ b4 in b2;

:: PROB_1:modenot 4 => PROB_1:mode 1
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  mode SetSequence of A2 -> Function-like quasi_total Relation of NAT,bool a1 means
    for b1 being Element of NAT holds
       it . b1 in a2;
end;

:: PROB_1:dfs 7
definiens
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be Function-like quasi_total Relation of NAT,bool a1;
To prove
     a3 is SetSequence of a2
it is sufficient to prove
  thus for b1 being Element of NAT holds
       a3 . b1 in a2;

:: PROB_1:def 9
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
      b3 is SetSequence of b2
   iff
      for b4 being Element of NAT holds
         b3 . b4 in b2;

:: PROB_1:th 46
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
   Union b3 in b2;

:: PROB_1:modenot 5 => PROB_1:mode 2
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  mode Event of A2 -> Element of bool a1 means
    it in a2;
end;

:: PROB_1:dfs 8
definiens
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be Element of bool a1;
To prove
     a3 is Event of a2
it is sufficient to prove
  thus a3 in a2;

:: PROB_1:def 10
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Element of bool b1 holds
      b3 is Event of b2
   iff
      b3 in b2;

:: PROB_1:th 48
theorem
for b1, b2 being set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
      st b2 in b3
   holds b2 is Event of b3;

:: PROB_1:th 49
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Event of b2 holds
b3 /\ b4 is Event of b2;

:: PROB_1:th 50
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Event of b2 holds
   b3 ` is Event of b2;

:: PROB_1:th 51
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Event of b2 holds
b3 \/ b4 is Event of b2;

:: PROB_1:th 52
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1 holds
   {} is Event of b2;

:: PROB_1:th 53
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1 holds
   b1 is Event of b2;

:: PROB_1:th 54
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Event of b2 holds
b3 \ b4 is Event of b2;

:: PROB_1:exreg 3
registration
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  cluster empty Event of a2;
end;

:: PROB_1:funcnot 5 => PROB_1:func 5
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  func [#] A2 -> Event of a2 equals
    a1;
end;

:: PROB_1:def 11
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1 holds
   [#] b2 = b1;

:: PROB_1:funcnot 6 => PROB_1:func 6
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3, a4 be Event of a2;
  redefine func a3 /\ a4 -> Event of a2;
  commutativity;
::  for a1 being set
::  for a2 being non empty compl-closed sigma-multiplicative Element of bool bool a1
::  for a3, a4 being Event of a2 holds
::  a3 /\ a4 = a4 /\ a3;
  idempotence;
::  for a1 being set
::  for a2 being non empty compl-closed sigma-multiplicative Element of bool bool a1
::  for a3 being Event of a2 holds
::     a3 /\ a3 = a3;
end;

:: PROB_1:funcnot 7 => PROB_1:func 7
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3, a4 be Event of a2;
  redefine func a3 \/ a4 -> Event of a2;
  commutativity;
::  for a1 being set
::  for a2 being non empty compl-closed sigma-multiplicative Element of bool bool a1
::  for a3, a4 being Event of a2 holds
::  a3 \/ a4 = a4 \/ a3;
  idempotence;
::  for a1 being set
::  for a2 being non empty compl-closed sigma-multiplicative Element of bool bool a1
::  for a3 being Event of a2 holds
::     a3 \/ a3 = a3;
end;

:: PROB_1:funcnot 8 => PROB_1:func 8
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3, a4 be Event of a2;
  redefine func a3 \ a4 -> Event of a2;
end;

:: PROB_1:th 57
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1 holds
      b2 is SetSequence of b3
   iff
      for b4 being Element of NAT holds
         b2 . b4 is Event of b3;

:: PROB_1:th 58
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
      st b2 is SetSequence of b3
   holds Union b2 is Event of b3;

:: PROB_1:th 59
theorem
for b1 being non empty set
for b2 being set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1 holds
   ex b4 being Relation-like Function-like set st
      proj1 b4 = b3 &
       (for b5 being Element of bool b1
             st b5 in b3
          holds (b2 in b5 implies b4 . b5 = 1) & (b2 in b5 or b4 . b5 = 0));

:: PROB_1:th 60
theorem
for b1 being non empty set
for b2 being set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1 holds
   ex b4 being Function-like quasi_total Relation of b3,REAL st
      for b5 being Element of bool b1
            st b5 in b3
         holds (b2 in b5 implies b4 . b5 = 1) & (b2 in b5 or b4 . b5 = 0);

:: PROB_1:th 62
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2
for b4 being Function-like quasi_total Relation of b2,REAL holds
   b3 * b4 is Function-like quasi_total Relation of NAT,REAL;

:: PROB_1:funcnot 9 => PROB_1:func 9
definition
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be SetSequence of a2;
  let a4 be Function-like quasi_total Relation of a2,REAL;
  redefine func a4 * a3 -> Function-like quasi_total Relation of NAT,REAL;
end;

:: PROB_1:modenot 6 => PROB_1:mode 3
definition
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  mode Probability of A2 -> Function-like quasi_total Relation of a2,REAL means
    (for b1 being Event of a2 holds
        0 <= it . b1) &
     it . a1 = 1 &
     (for b1, b2 being Event of a2
           st b1 misses b2
        holds it . (b1 \/ b2) = (it . b1) + (it . b2)) &
     (for b1 being SetSequence of a2
           st b1 is non-increasing(a1)
        holds it * b1 is convergent & lim (it * b1) = it . Intersection b1);
end;

:: PROB_1:dfs 10
definiens
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be Function-like quasi_total Relation of a2,REAL;
To prove
     a3 is Probability of a2
it is sufficient to prove
  thus (for b1 being Event of a2 holds
        0 <= a3 . b1) &
     a3 . a1 = 1 &
     (for b1, b2 being Event of a2
           st b1 misses b2
        holds a3 . (b1 \/ b2) = (a3 . b1) + (a3 . b2)) &
     (for b1 being SetSequence of a2
           st b1 is non-increasing(a1)
        holds a3 * b1 is convergent & lim (a3 * b1) = a3 . Intersection b1);

:: PROB_1:def 13
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like quasi_total Relation of b2,REAL holds
      b3 is Probability of b2
   iff
      (for b4 being Event of b2 holds
          0 <= b3 . b4) &
       b3 . b1 = 1 &
       (for b4, b5 being Event of b2
             st b4 misses b5
          holds b3 . (b4 \/ b5) = (b3 . b4) + (b3 . b5)) &
       (for b4 being SetSequence of b2
             st b4 is non-increasing(b1)
          holds b3 * b4 is convergent & lim (b3 * b4) = b3 . Intersection b4);

:: PROB_1:th 64
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2 holds
   b3 . {} = 0;

:: PROB_1:th 66
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2 holds
   b3 . [#] b2 = 1;

:: PROB_1:th 67
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Event of b2
for b4 being Probability of b2 holds
   (b4 . (([#] b2) \ b3)) + (b4 . b3) = 1;

:: PROB_1:th 68
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Event of b2
for b4 being Probability of b2 holds
   b4 . (([#] b2) \ b3) = 1 - (b4 . b3);

:: PROB_1:th 69
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Event of b2
for b5 being Probability of b2
      st b3 c= b4
   holds b5 . (b4 \ b3) = (b5 . b4) - (b5 . b3);

:: PROB_1:th 70
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Event of b2
for b5 being Probability of b2
      st b3 c= b4
   holds b5 . b3 <= b5 . b4;

:: PROB_1:th 71
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Event of b2
for b4 being Probability of b2 holds
   b4 . b3 <= 1;

:: PROB_1:th 72
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Event of b2
for b5 being Probability of b2 holds
   b5 . (b3 \/ b4) = (b5 . b3) + (b5 . (b4 \ b3));

:: PROB_1:th 73
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Event of b2
for b5 being Probability of b2 holds
   b5 . (b3 \/ b4) = (b5 . b3) + (b5 . (b4 \ (b3 /\ b4)));

:: PROB_1:th 74
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Event of b2
for b5 being Probability of b2 holds
   b5 . (b3 \/ b4) = ((b5 . b3) + (b5 . b4)) - (b5 . (b3 /\ b4));

:: PROB_1:th 75
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Event of b2
for b5 being Probability of b2 holds
   b5 . (b3 \/ b4) <= (b5 . b3) + (b5 . b4);

:: PROB_1:th 76
theorem
for b1 being non empty set holds
   bool b1 is non empty compl-closed sigma-multiplicative Element of bool bool b1;

:: PROB_1:funcnot 10 => PROB_1:func 10
definition
  let a1 be non empty set;
  let a2 be Element of bool bool a1;
  func sigma A2 -> non empty compl-closed sigma-multiplicative Element of bool bool a1 means
    a2 c= it &
     (for b1 being set
           st a2 c= b1 &
              b1 is non empty compl-closed sigma-multiplicative Element of bool bool a1
        holds it c= b1);
end;

:: PROB_1:def 14
theorem
for b1 being non empty set
for b2 being Element of bool bool b1
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1 holds
      b3 = sigma b2
   iff
      b2 c= b3 &
       (for b4 being set
             st b2 c= b4 &
                b4 is non empty compl-closed sigma-multiplicative Element of bool bool b1
          holds b3 c= b4);

:: PROB_1:funcnot 11 => PROB_1:func 11
definition
  let a1 be real set;
  func halfline A1 -> Element of bool REAL equals
    ].-infty,a1.[;
end;

:: PROB_1:def 15
theorem
for b1 being real set holds
   halfline b1 = ].-infty,b1.[;

:: PROB_1:funcnot 12 => PROB_1:func 12
definition
  func Family_of_halflines -> Element of bool bool REAL equals
    {b1 where b1 is Element of bool REAL: ex b2 being real set st
       b1 = halfline b2};
end;

:: PROB_1:def 16
theorem
Family_of_halflines = {b1 where b1 is Element of bool REAL: ex b2 being real set st
   b1 = halfline b2};

:: PROB_1:funcnot 13 => PROB_1:func 13
definition
  func Borel_Sets -> non empty compl-closed sigma-multiplicative Element of bool bool REAL equals
    sigma Family_of_halflines;
end;

:: PROB_1:def 17
theorem
Borel_Sets = sigma Family_of_halflines;