Article PROB_3, MML version 4.99.1005

:: PROB_3:th 1
theorem
for b1 being Relation-like Function-like FinSequence-like set holds
   not 0 in proj1 b1;

:: PROB_3:th 2
theorem
for b1 being natural set
for b2 being Relation-like Function-like FinSequence-like set holds
      b1 in proj1 b2
   iff
      b1 <> 0 & b1 <= len b2;

:: PROB_3:th 3
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
      st ex b3 being natural set st
           for b4 being natural set
                 st b3 <= b4
              holds b2 . b4 = b1
   holds b2 is convergent & lim b2 = b1;

:: PROB_3:th 4
theorem
for b1 being natural set
for b2 being non empty set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b4 being SetSequence of b3
for b5 being Probability of b3 holds
   0 <= (b5 * b4) . b1;

:: PROB_3:th 5
theorem
for b1 being natural set
for b2 being non empty set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b4, b5 being SetSequence of b3
for b6 being Probability of b3
      st b4 . b1 c= b5 . b1
   holds (b6 * b4) . b1 <= (b6 * b5) . b1;

:: PROB_3:th 6
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 Probability of b2
      st b3 is non-decreasing(b1)
   holds b4 * b3 is non-decreasing;

:: PROB_3:th 7
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 Probability of b2
      st b3 is non-increasing(b1)
   holds b4 * b3 is non-increasing;

:: PROB_3:th 8
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
      b3 . 0 = b2 . 0 &
       (for b4 being natural set holds
          b3 . (b4 + 1) = (b3 . b4) /\ (b2 . (b4 + 1)));

:: PROB_3:th 9
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
      b3 . 0 = b2 . 0 &
       (for b4 being natural set holds
          b3 . (b4 + 1) = (b3 . b4) \/ (b2 . (b4 + 1)));

:: PROB_3:funcnot 1 => PROB_3:func 1
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
  func Partial_Intersection A2 -> Function-like quasi_total Relation of NAT,bool a1 means
    it . 0 = a2 . 0 &
     (for b1 being natural set holds
        it . (b1 + 1) = (it . b1) /\ (a2 . (b1 + 1)));
end;

:: PROB_3:def 1
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
   b3 = Partial_Intersection b2
iff
   b3 . 0 = b2 . 0 &
    (for b4 being natural set holds
       b3 . (b4 + 1) = (b3 . b4) /\ (b2 . (b4 + 1)));

:: PROB_3:funcnot 2 => PROB_3:func 2
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
  func Partial_Union A2 -> Function-like quasi_total Relation of NAT,bool a1 means
    it . 0 = a2 . 0 &
     (for b1 being natural set holds
        it . (b1 + 1) = (it . b1) \/ (a2 . (b1 + 1)));
end;

:: PROB_3:def 2
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
   b3 = Partial_Union b2
iff
   b3 . 0 = b2 . 0 &
    (for b4 being natural set holds
       b3 . (b4 + 1) = (b3 . b4) \/ (b2 . (b4 + 1)));

:: PROB_3:th 10
theorem
for b1 being natural set
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2 holds
   (Partial_Intersection b3) . b1 c= b3 . b1;

:: PROB_3:th 11
theorem
for b1 being natural set
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2 holds
   b3 . b1 c= (Partial_Union b3) . b1;

:: PROB_3:th 12
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
   Partial_Intersection b2 is non-increasing(b1);

:: PROB_3:th 13
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
   Partial_Union b2 is non-decreasing(b1);

:: PROB_3:th 14
theorem
for b1 being natural set
for b2, b3 being set
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
      b3 in (Partial_Intersection b4) . b1
   iff
      for b5 being natural set
            st b5 <= b1
         holds b3 in b4 . b5;

:: PROB_3:th 15
theorem
for b1 being natural set
for b2, b3 being set
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
      b3 in (Partial_Union b4) . b1
   iff
      ex b5 being natural set st
         b5 <= b1 & b3 in b4 . b5;

:: PROB_3:th 16
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
   Intersection Partial_Intersection b2 = Intersection b2;

:: PROB_3:th 17
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
   Union Partial_Union b2 = Union b2;

:: PROB_3:th 18
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
      b3 . 0 = b2 . 0 &
       (for b4 being natural set holds
          b3 . (b4 + 1) = (b2 . (b4 + 1)) \ ((Partial_Union b2) . b4));

:: PROB_3:funcnot 3 => PROB_3:func 3
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
  func Partial_Diff_Union A2 -> Function-like quasi_total Relation of NAT,bool a1 means
    it . 0 = a2 . 0 &
     (for b1 being natural set holds
        it . (b1 + 1) = (a2 . (b1 + 1)) \ ((Partial_Union a2) . b1));
end;

:: PROB_3:def 3
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
   b3 = Partial_Diff_Union b2
iff
   b3 . 0 = b2 . 0 &
    (for b4 being natural set holds
       b3 . (b4 + 1) = (b2 . (b4 + 1)) \ ((Partial_Union b2) . b4));

:: PROB_3:th 19
theorem
for b1 being natural set
for b2, b3 being set
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
      b3 in (Partial_Diff_Union b4) . b1
   iff
      b3 in b4 . b1 &
       (for b5 being natural set
             st b5 < b1
          holds not b3 in b4 . b5);

:: PROB_3:th 20
theorem
for b1 being natural set
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2 holds
   (Partial_Diff_Union b3) . b1 c= b3 . b1;

:: PROB_3:th 21
theorem
for b1 being natural set
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2 holds
   (Partial_Diff_Union b3) . b1 c= (Partial_Union b3) . b1;

:: PROB_3:th 22
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
   Partial_Union Partial_Diff_Union b2 = Partial_Union b2;

:: PROB_3:th 23
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
   Union Partial_Diff_Union b2 = Union b2;

:: PROB_3:attrnot 1 => PROB_3:attr 1
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
  redefine attr a2 is disjoint_valued means
    for b1, b2 being natural set
          st b1 <> b2
       holds a2 . b1 misses a2 . b2;
end;

:: PROB_3:dfs 4
definiens
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,bool a1;
To prove
     a1 is disjoint_valued
it is sufficient to prove
  thus for b1, b2 being natural set
          st b1 <> b2
       holds a2 . b1 misses a2 . b2;

:: PROB_3:def 4
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
      b2 is disjoint_valued
   iff
      for b3, b4 being natural set
            st b3 <> b4
         holds b2 . b3 misses b2 . b4;

:: PROB_3:th 24
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
   Partial_Diff_Union b2 is disjoint_valued;

:: PROB_3:funcnot 4 => PROB_3:func 4
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be SetSequence of a2;
  func @Partial_Intersection A3 -> SetSequence of a2 equals
    Partial_Intersection a3;
end;

:: PROB_3:def 5
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
   @Partial_Intersection b3 = Partial_Intersection b3;

:: PROB_3:funcnot 5 => PROB_3:func 5
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be SetSequence of a2;
  func @Partial_Union A3 -> SetSequence of a2 equals
    Partial_Union a3;
end;

:: PROB_3:def 6
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
   @Partial_Union b3 = Partial_Union b3;

:: PROB_3:funcnot 6 => PROB_3:func 6
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be SetSequence of a2;
  func @Partial_Diff_Union A3 -> SetSequence of a2 equals
    Partial_Diff_Union a3;
end;

:: PROB_3:def 7
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
   @Partial_Diff_Union b3 = Partial_Diff_Union b3;

:: PROB_3:th 25
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being SetSequence of b2
      st b3 = @Partial_Intersection b4
   holds b3 . 0 = b4 . 0 &
    (for b5 being natural set holds
       b3 . (b5 + 1) = (b3 . b5) /\ (b4 . (b5 + 1)));

:: PROB_3:th 26
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being SetSequence of b2
      st b3 = @Partial_Union b4
   holds b3 . 0 = b4 . 0 &
    (for b5 being natural set holds
       b3 . (b5 + 1) = (b3 . b5) \/ (b4 . (b5 + 1)));

:: PROB_3:th 27
theorem
for b1 being natural set
for b2 being set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b4 being SetSequence of b3 holds
   (@Partial_Intersection b4) . b1 c= b4 . b1;

:: PROB_3:th 28
theorem
for b1 being natural set
for b2 being set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b4 being SetSequence of b3 holds
   b4 . b1 c= (@Partial_Union b4) . b1;

:: PROB_3:th 29
theorem
for b1 being natural set
for b2, b3 being set
for b4 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b5 being SetSequence of b4 holds
      b3 in (@Partial_Intersection b5) . b1
   iff
      for b6 being natural set
            st b6 <= b1
         holds b3 in b5 . b6;

:: PROB_3:th 30
theorem
for b1 being natural set
for b2, b3 being set
for b4 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b5 being SetSequence of b4 holds
      b3 in (@Partial_Union b5) . b1
   iff
      ex b6 being natural set st
         b6 <= b1 & b3 in b5 . b6;

:: PROB_3:th 31
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
   @Partial_Intersection b3 is non-increasing(b1);

:: PROB_3:th 32
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
   @Partial_Union b3 is non-decreasing(b1);

:: PROB_3:th 33
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
   Intersection @Partial_Intersection b3 = Intersection b3;

:: PROB_3:th 34
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 @Partial_Union b3 = Union b3;

:: PROB_3:th 35
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being SetSequence of b2
      st b3 = @Partial_Diff_Union b4
   holds b3 . 0 = b4 . 0 &
    (for b5 being natural set holds
       b3 . (b5 + 1) = (b4 . (b5 + 1)) \ ((@Partial_Union b4) . b5));

:: PROB_3:th 36
theorem
for b1 being natural set
for b2, b3 being set
for b4 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b5 being SetSequence of b4 holds
      b3 in (@Partial_Diff_Union b5) . b1
   iff
      b3 in b5 . b1 &
       (for b6 being natural set
             st b6 < b1
          holds not b3 in b5 . b6);

:: PROB_3:th 37
theorem
for b1 being natural set
for b2 being set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b4 being SetSequence of b3 holds
   (@Partial_Diff_Union b4) . b1 c= b4 . b1;

:: PROB_3:th 38
theorem
for b1 being natural set
for b2 being set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b4 being SetSequence of b3 holds
   (@Partial_Diff_Union b4) . b1 c= (@Partial_Union b4) . b1;

:: PROB_3:th 39
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
   @Partial_Union @Partial_Diff_Union b3 = @Partial_Union b3;

:: PROB_3:th 40
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 @Partial_Diff_Union b3 = Union b3;

:: PROB_3:th 41
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
   @Partial_Diff_Union b3 is disjoint_valued;

:: PROB_3:th 42
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 Probability of b2 holds
   b4 * @Partial_Union b3 is non-decreasing;

:: PROB_3:th 43
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 Probability of b2 holds
   b4 * @Partial_Intersection b3 is non-increasing;

:: PROB_3:th 44
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 Probability of b2 holds
   Partial_Sums (b4 * b3) is non-decreasing;

:: PROB_3:th 45
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 Probability of b2 holds
   (b4 * @Partial_Union b3) . 0 = (Partial_Sums (b4 * b3)) . 0;

:: PROB_3:th 46
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 Probability of b2 holds
   b4 * @Partial_Union b3 is convergent &
    lim (b4 * @Partial_Union b3) = sup (b4 * @Partial_Union b3) &
    lim (b4 * @Partial_Union b3) = b4 . Union b3;

:: PROB_3:th 47
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
   st b3 is disjoint_valued
for b4, b5 being natural set
      st b4 < b5
   holds (@Partial_Union b3) . b4 misses b3 . b5;

:: PROB_3:th 48
theorem
for b1 being natural set
for b2 being non empty set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b4 being SetSequence of b3
for b5 being Probability of b3
      st b4 is disjoint_valued
   holds (b5 * @Partial_Union b4) . b1 = (Partial_Sums (b5 * b4)) . b1;

:: PROB_3:th 49
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 Probability of b2
      st b3 is disjoint_valued
   holds b4 * @Partial_Union b3 = Partial_Sums (b4 * b3);

:: PROB_3:th 50
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 Probability of b2
      st b3 is disjoint_valued
   holds Partial_Sums (b4 * b3) is convergent &
    lim Partial_Sums (b4 * b3) = sup Partial_Sums (b4 * b3) &
    lim Partial_Sums (b4 * b3) = b4 . Union b3;

:: PROB_3:th 51
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 Probability of b2
      st b3 is disjoint_valued
   holds b4 . Union b3 = Sum (b4 * b3);

:: PROB_3:funcnot 7 => PROB_3:func 7
definition
  let a1 be set;
  let a2 be FinSequence of bool a1;
  let a3 be natural set;
  redefine func a2 . a3 -> Element of bool a1;
end;

:: PROB_3:th 52
theorem
for b1 being set holds
   ex b2 being FinSequence of bool b1 st
      for b3 being natural set
            st b3 in dom b2
         holds b2 . b3 = b1;

:: PROB_3:th 53
theorem
for b1 being set
for b2 being FinSequence of bool b1 holds
   union rng b2 is Element of bool b1;

:: PROB_3:funcnot 8 => PROB_3:func 8
definition
  let a1 be set;
  let a2 be FinSequence of bool a1;
  redefine func Union a2 -> Element of bool a1;
end;

:: PROB_3:th 54
theorem
for b1, b2 being set
for b3 being FinSequence of bool b1 holds
      b2 in Union b3
   iff
      ex b4 being natural set st
         b4 in dom b3 & b2 in b3 . b4;

:: PROB_3:funcnot 9 => PROB_3:func 9
definition
  let a1 be set;
  let a2 be FinSequence of bool a1;
  func Complement A2 -> FinSequence of bool a1 means
    len it = len a2 &
     (for b1 being natural set
           st b1 in dom it
        holds it . b1 = (a2 . b1) `);
end;

:: PROB_3:def 8
theorem
for b1 being set
for b2, b3 being FinSequence of bool b1 holds
   b3 = Complement b2
iff
   len b3 = len b2 &
    (for b4 being natural set
          st b4 in dom b3
       holds b3 . b4 = (b2 . b4) `);

:: PROB_3:funcnot 10 => PROB_3:func 10
definition
  let a1 be set;
  let a2 be FinSequence of bool a1;
  func Intersection A2 -> Element of bool a1 equals
    (Union Complement a2) `
    if a2 <> {}
    otherwise {};
end;

:: PROB_3:def 9
theorem
for b1 being set
for b2 being FinSequence of bool b1 holds
   (b2 = {} or Intersection b2 = (Union Complement b2) `) &
    (b2 = {} implies Intersection b2 = {});

:: PROB_3:th 55
theorem
for b1 being set
for b2 being FinSequence of bool b1 holds
   dom Complement b2 = dom b2;

:: PROB_3:th 56
theorem
for b1, b2 being set
for b3 being FinSequence of bool b1
      st b3 <> {}
   holds    b2 in Intersection b3
   iff
      for b4 being natural set
            st b4 in dom b3
         holds b2 in b3 . b4;

:: PROB_3:th 57
theorem
for b1, b2 being set
for b3 being FinSequence of bool b1
      st b3 <> {}
   holds    b2 in meet rng b3
   iff
      for b4 being natural set
            st b4 in dom b3
         holds b2 in b3 . b4;

:: PROB_3:th 58
theorem
for b1 being set
for b2 being FinSequence of bool b1 holds
   Intersection b2 = meet rng b2;

:: PROB_3:th 59
theorem
for b1 being set
for b2 being FinSequence of bool b1 holds
   ex b3 being Function-like quasi_total Relation of NAT,bool b1 st
      (for b4 being natural set
             st b4 in dom b2
          holds b3 . b4 = b2 . b4) &
       (for b4 being natural set
             st not b4 in dom b2
          holds b3 . b4 = {});

:: PROB_3:th 60
theorem
for b1 being set
for b2 being FinSequence of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1
      st (for b4 being natural set
               st b4 in dom b2
            holds b3 . b4 = b2 . b4) &
         (for b4 being natural set
               st not b4 in dom b2
            holds b3 . b4 = {})
   holds b3 . 0 = {} & Union b3 = Union b2;

:: PROB_3:modenot 1 => PROB_3:mode 1
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  mode FinSequence of A2 -> FinSequence of bool a1 means
    for b1 being natural set
          st b1 in dom it
       holds it . b1 in a2;
end;

:: PROB_3:dfs 10
definiens
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be FinSequence of bool a1;
To prove
     a3 is FinSequence of a2
it is sufficient to prove
  thus for b1 being natural set
          st b1 in dom a3
       holds a3 . b1 in a2;

:: PROB_3:def 10
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being FinSequence of bool b1 holds
      b3 is FinSequence of b2
   iff
      for b4 being natural set
            st b4 in dom b3
         holds b3 . b4 in b2;

:: PROB_3:funcnot 11 => PROB_3:func 11
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be FinSequence of a2;
  let a4 be natural set;
  redefine func a3 . a4 -> Event of a2;
end;

:: PROB_3:th 61
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being FinSequence of b2 holds
   ex b4 being SetSequence of b2 st
      (for b5 being natural set
             st b5 in dom b3
          holds b4 . b5 = b3 . b5) &
       (for b5 being natural set
             st not b5 in dom b3
          holds b4 . b5 = {});

:: PROB_3:th 62
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being FinSequence of b2 holds
   Union b3 in b2;

:: PROB_3:funcnot 12 => PROB_3:func 12
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be FinSequence of a2;
  func @Complement A3 -> FinSequence of a2 equals
    Complement a3;
end;

:: PROB_3:def 11
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being FinSequence of b2 holds
   @Complement b3 = Complement b3;

:: PROB_3:th 63
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being FinSequence of b2 holds
   Intersection b3 in b2;

:: PROB_3: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
for b4 being FinSequence of b2 holds
   dom (b3 * b4) = dom b4;

:: PROB_3:th 65
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
for b4 being FinSequence of b2 holds
   b3 * b4 is FinSequence of REAL;

:: PROB_3:funcnot 13 => PROB_3:func 13
definition
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be FinSequence of a2;
  let a4 be Probability of a2;
  redefine func a4 * a3 -> FinSequence of REAL;
end;

:: PROB_3: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
for b4 being FinSequence of b2 holds
   len (b3 * b4) = len b4;

:: PROB_3:th 67
theorem
for b1 being FinSequence of REAL
      st len b1 = 0
   holds Sum b1 = 0;

:: PROB_3:th 68
theorem
for b1 being FinSequence of REAL
      st 1 <= len b1
   holds ex b2 being Function-like quasi_total Relation of NAT,REAL st
      b2 . 1 = b1 . 1 &
       (for b3 being natural set
             st 0 <> b3 & b3 < len b1
          holds b2 . (b3 + 1) = (b2 . b3) + (b1 . (b3 + 1))) &
       Sum b1 = b2 . len b1;

:: PROB_3: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 being Probability of b2
for b4 being FinSequence of b2
for b5 being SetSequence of b2
      st (for b6 being natural set
               st b6 in dom b4
            holds b5 . b6 = b4 . b6) &
         (for b6 being natural set
               st not b6 in dom b4
            holds b5 . b6 = {})
   holds Partial_Sums (b3 * b5) is convergent &
    Sum (b3 * b5) = (Partial_Sums (b3 * b5)) . len b4 &
    b3 . Union b5 <= Sum (b3 * b5) &
    Sum (b3 * b4) = Sum (b3 * b5);

:: PROB_3: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 being Probability of b2
for b4 being FinSequence of b2 holds
   b3 . Union b4 <= Sum (b3 * b4) &
    (b4 is disjoint_valued implies b3 . Union b4 = Sum (b3 * b4));

:: PROB_3:attrnot 2 => PROB_3:attr 2
definition
  let a1 be set;
  let a2 be Element of bool bool a1;
  attr a2 is non-decreasing-closed means
    for b1 being Function-like quasi_total Relation of NAT,bool a1
          st b1 is non-decreasing(a1) &
             (for b2 being natural set holds
                b1 . b2 in a2)
       holds Union b1 in a2;
end;

:: PROB_3:dfs 12
definiens
  let a1 be set;
  let a2 be Element of bool bool a1;
To prove
     a2 is non-decreasing-closed
it is sufficient to prove
  thus for b1 being Function-like quasi_total Relation of NAT,bool a1
          st b1 is non-decreasing(a1) &
             (for b2 being natural set holds
                b1 . b2 in a2)
       holds Union b1 in a2;

:: PROB_3:def 12
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
      b2 is non-decreasing-closed(b1)
   iff
      for b3 being Function-like quasi_total Relation of NAT,bool b1
            st b3 is non-decreasing(b1) &
               (for b4 being natural set holds
                  b3 . b4 in b2)
         holds Union b3 in b2;

:: PROB_3:attrnot 3 => PROB_3:attr 3
definition
  let a1 be set;
  let a2 be Element of bool bool a1;
  attr a2 is non-increasing-closed means
    for b1 being Function-like quasi_total Relation of NAT,bool a1
          st b1 is non-increasing(a1) &
             (for b2 being natural set holds
                b1 . b2 in a2)
       holds Intersection b1 in a2;
end;

:: PROB_3:dfs 13
definiens
  let a1 be set;
  let a2 be Element of bool bool a1;
To prove
     a2 is non-increasing-closed
it is sufficient to prove
  thus for b1 being Function-like quasi_total Relation of NAT,bool a1
          st b1 is non-increasing(a1) &
             (for b2 being natural set holds
                b1 . b2 in a2)
       holds Intersection b1 in a2;

:: PROB_3:def 13
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
      b2 is non-increasing-closed(b1)
   iff
      for b3 being Function-like quasi_total Relation of NAT,bool b1
            st b3 is non-increasing(b1) &
               (for b4 being natural set holds
                  b3 . b4 in b2)
         holds Intersection b3 in b2;

:: PROB_3:th 71
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
      b2 is non-decreasing-closed(b1)
   iff
      for b3 being Function-like quasi_total Relation of NAT,bool b1
            st b3 is non-decreasing(b1) &
               (for b4 being natural set holds
                  b3 . b4 in b2)
         holds lim b3 in b2;

:: PROB_3:th 72
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
      b2 is non-increasing-closed(b1)
   iff
      for b3 being Function-like quasi_total Relation of NAT,bool b1
            st b3 is non-increasing(b1) &
               (for b4 being natural set holds
                  b3 . b4 in b2)
         holds lim b3 in b2;

:: PROB_3:th 73
theorem
for b1 being set holds
   bool b1 is non-decreasing-closed(b1) & bool b1 is non-increasing-closed(b1);

:: PROB_3:modenot 2 => PROB_3:mode 2
definition
  let a1 be set;
  mode MonotoneClass of A1 -> Element of bool bool a1 means
    it is non-decreasing-closed(a1) & it is non-increasing-closed(a1);
end;

:: PROB_3:dfs 14
definiens
  let a1 be set;
  let a2 be Element of bool bool a1;
To prove
     a2 is MonotoneClass of a1
it is sufficient to prove
  thus a2 is non-decreasing-closed(a1) & a2 is non-increasing-closed(a1);

:: PROB_3:def 14
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
      b2 is MonotoneClass of b1
   iff
      b2 is non-decreasing-closed(b1) & b2 is non-increasing-closed(b1);

:: PROB_3:th 74
theorem
for b1, b2 being set holds
   b1 is MonotoneClass of b2
iff
   b1 c= bool b2 &
    (for b3 being Function-like quasi_total Relation of NAT,bool b2
          st b3 is monotone(b2) &
             (for b4 being natural set holds
                b3 . b4 in b1)
       holds lim b3 in b1);

:: PROB_3:th 75
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1 holds
      b2 is non empty compl-closed sigma-multiplicative Element of bool bool b1
   iff
      b2 is MonotoneClass of b1;

:: PROB_3:th 76
theorem
for b1 being non empty set holds
   bool b1 is MonotoneClass of b1;

:: PROB_3:th 77
theorem
for b1 being non empty set
for b2 being Element of bool bool b1 holds
   ex b3 being MonotoneClass of b1 st
      b2 c= b3 &
       (for b4 being set
             st b2 c= b4 & b4 is MonotoneClass of b1
          holds b3 c= b4);

:: PROB_3:funcnot 14 => PROB_3:func 14
definition
  let a1 be non empty set;
  let a2 be Element of bool bool a1;
  func monotoneclass A2 -> MonotoneClass of a1 means
    a2 c= it &
     (for b1 being set
           st a2 c= b1 & b1 is MonotoneClass of a1
        holds it c= b1);
end;

:: PROB_3:def 15
theorem
for b1 being non empty set
for b2 being Element of bool bool b1
for b3 being MonotoneClass of b1 holds
      b3 = monotoneclass b2
   iff
      b2 c= b3 &
       (for b4 being set
             st b2 c= b4 & b4 is MonotoneClass of b1
          holds b3 c= b4);

:: PROB_3:th 78
theorem
for b1 being non empty set
for b2 being non empty cap-closed compl-closed Element of bool bool b1 holds
   monotoneclass b2 is non empty cap-closed compl-closed Element of bool bool b1;

:: PROB_3:th 79
theorem
for b1 being non empty set
for b2 being non empty cap-closed compl-closed Element of bool bool b1 holds
   sigma b2 = monotoneclass b2;