Article PROB_2, MML version 4.99.1005

:: PROB_2:th 4
theorem
for b1, b2, b3, b4 being Element of REAL
      st b1 <> 0 & b2 <> 0
   holds    b4 / b2 = b3 / b1
   iff
      b4 * b1 = b3 * b2;

:: PROB_2:th 5
theorem
for b1 being Element of REAL
for b2, b3 being Function-like quasi_total Relation of NAT,REAL
      st b2 is convergent &
         (for b4 being Element of NAT holds
            b3 . b4 = b1 - (b2 . b4))
   holds b3 is convergent & lim b3 = b1 - lim b2;

:: PROB_2:funcnot 1 => PROB_2:func 1
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 Element of NAT;
  redefine func a3 . a4 -> Event of a2;
end;

:: PROB_2:funcnot 2 => PROB_2:func 2
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;
  func @Intersection A3 -> Event of a2 equals
    Intersection a3;
end;

:: PROB_2:def 1
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 holds
   @Intersection b3 = Intersection b3;

:: PROB_2:th 9
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 Event of b2 holds
   ex b5 being SetSequence of b2 st
      for b6 being Element of NAT holds
         b5 . b6 = (b3 . b6) /\ b4;

:: PROB_2:th 10
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 SetSequence of b2
for b5 being Event of b2
      st b3 is non-increasing(b1) &
         (for b6 being Element of NAT holds
            b4 . b6 = (b3 . b6) /\ b5)
   holds b4 is non-increasing(b1);

:: PROB_2:th 11
theorem
for b1 being non empty set
for b2 being Element of NAT
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4 being SetSequence of b3
for b5 being Function-like quasi_total Relation of b3,REAL holds
   (b5 * b4) . b2 = b5 . (b4 . b2);

:: PROB_2:th 12
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 SetSequence of b2
for b5 being Event of b2
      st for b6 being Element of NAT holds
           b3 . b6 = (b4 . b6) /\ b5
   holds (Intersection b4) /\ b5 = Intersection b3;

:: PROB_2:th 13
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 Probability of b2
      st for b5 being Event of b2 holds
           b3 . b5 = b4 . b5
   holds b3 = b4;

:: PROB_2:th 14
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 being Element of NAT holds
         b2 . (b3 + 1) c= b2 . b3;

:: PROB_2:th 15
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 being Element of NAT holds
         b2 . b3 c= b2 . (b3 + 1);

:: PROB_2:th 16
theorem
for b1 being non empty set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1
      st for b4 being Element of NAT holds
           b2 . b4 = b3 . b4
   holds b2 = b3;

:: PROB_2:th 17
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
      b2 is non-increasing(b1)
   iff
      Complement b2 is non-decreasing(b1);

:: PROB_2:funcnot 3 => PROB_2:func 3
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;
  func @Complement A3 -> SetSequence of a2 equals
    Complement a3;
end;

:: PROB_2:def 2
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 holds
   @Complement b3 = Complement b3;

:: PROB_2:attrnot 1 => PROB_2:attr 1
definition
  let a1 be Relation-like Function-like set;
  attr a1 is disjoint_valued means
    for b1, b2 being set
          st b1 <> b2
       holds a1 . b1 misses a1 . b2;
end;

:: PROB_2:dfs 3
definiens
  let a1 be Relation-like Function-like set;
To prove
     a1 is disjoint_valued
it is sufficient to prove
  thus for b1, b2 being set
          st b1 <> b2
       holds a1 . b1 misses a1 . b2;

:: PROB_2:def 3
theorem
for b1 being Relation-like Function-like set holds
      b1 is disjoint_valued
   iff
      for b2, b3 being set
            st b2 <> b3
         holds b1 . b2 misses b1 . b3;

:: PROB_2:attrnot 2 => PROB_2:attr 2
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;
  redefine attr a3 is disjoint_valued means
    for b1, b2 being Element of NAT
          st b1 <> b2
       holds a3 . b1 misses a3 . b2;
end;

:: PROB_2:dfs 4
definiens
  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;
To prove
     a1 is disjoint_valued
it is sufficient to prove
  thus for b1, b2 being Element of NAT
          st b1 <> b2
       holds a3 . b1 misses a3 . b2;

:: PROB_2:def 4
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 holds
      b3 is disjoint_valued
   iff
      for b4, b5 being Element of NAT
            st b4 <> b5
         holds b3 . b4 misses b3 . b5;

:: PROB_2:th 20
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-decreasing(b1)
          holds b3 * b4 is convergent & lim (b3 * b4) = b3 . Union b4);

:: PROB_2:th 21
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, b5, b6 being Event of b2 holds
b3 . ((b4 \/ b5) \/ b6) = ((((b3 . b4) + (b3 . b5)) + (b3 . b6)) - (((b3 . (b4 /\ b5)) + (b3 . (b5 /\ b6))) + (b3 . (b4 /\ b6)))) + (b3 . ((b4 /\ b5) /\ b6));

:: PROB_2:th 22
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, b5 being Event of b2 holds
b3 . (b4 \ (b4 /\ b5)) = (b3 . b4) - (b3 . (b4 /\ b5));

:: PROB_2:th 23
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, b5 being Event of b2 holds
b3 . (b4 /\ b5) <= b3 . b5 & b3 . (b4 /\ b5) <= b3 . b4;

:: PROB_2:th 24
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, b5, b6 being Event of b2
      st b4 = b5 `
   holds b3 . b6 = (b3 . (b6 /\ b5)) + (b3 . (b6 /\ b4));

:: PROB_2:th 25
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, b5 being Event of b2 holds
((b3 . b4) + (b3 . b5)) - 1 <= b3 . (b4 /\ b5);

:: PROB_2:th 26
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 Event of b2 holds
   b3 . b4 = 1 - (b3 . (([#] b2) \ b4));

:: PROB_2:th 27
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 Event of b2 holds
      b3 . b4 < 1
   iff
      0 < b3 . (([#] b2) \ b4);

:: PROB_2:th 28
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 Event of b2 holds
      b3 . (([#] b2) \ b4) < 1
   iff
      0 < b3 . b4;

:: PROB_2:prednot 1 => PROB_2:pred 1
definition
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be Probability of a2;
  let a4, a5 be Event of a2;
  pred A4,A5 are_independent_respect_to A3 means
    a3 . (a4 /\ a5) = (a3 . a4) * (a3 . a5);
end;

:: PROB_2:dfs 5
definiens
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be Probability of a2;
  let a4, a5 be Event of a2;
To prove
     a4,a5 are_independent_respect_to a3
it is sufficient to prove
  thus a3 . (a4 /\ a5) = (a3 . a4) * (a3 . a5);

:: PROB_2:def 5
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, b5 being Event of b2 holds
   b4,b5 are_independent_respect_to b3
iff
   b3 . (b4 /\ b5) = (b3 . b4) * (b3 . b5);

:: PROB_2:prednot 2 => PROB_2:pred 2
definition
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be Probability of a2;
  let a4, a5, a6 be Event of a2;
  pred A4,A5,A6 are_independent_respect_to A3 means
    a3 . ((a4 /\ a5) /\ a6) = ((a3 . a4) * (a3 . a5)) * (a3 . a6) &
     a3 . (a4 /\ a5) = (a3 . a4) * (a3 . a5) &
     a3 . (a4 /\ a6) = (a3 . a4) * (a3 . a6) &
     a3 . (a5 /\ a6) = (a3 . a5) * (a3 . a6);
end;

:: PROB_2:dfs 6
definiens
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be Probability of a2;
  let a4, a5, a6 be Event of a2;
To prove
     a4,a5,a6 are_independent_respect_to a3
it is sufficient to prove
  thus a3 . ((a4 /\ a5) /\ a6) = ((a3 . a4) * (a3 . a5)) * (a3 . a6) &
     a3 . (a4 /\ a5) = (a3 . a4) * (a3 . a5) &
     a3 . (a4 /\ a6) = (a3 . a4) * (a3 . a6) &
     a3 . (a5 /\ a6) = (a3 . a5) * (a3 . a6);

:: PROB_2:def 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 Probability of b2
for b4, b5, b6 being Event of b2 holds
   b4,b5,b6 are_independent_respect_to b3
iff
   b3 . ((b4 /\ b5) /\ b6) = ((b3 . b4) * (b3 . b5)) * (b3 . b6) &
    b3 . (b4 /\ b5) = (b3 . b4) * (b3 . b5) &
    b3 . (b4 /\ b6) = (b3 . b4) * (b3 . b6) &
    b3 . (b5 /\ b6) = (b3 . b5) * (b3 . b6);

:: PROB_2:th 31
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, b5 being Event of b2
      st b4,b5 are_independent_respect_to b3
   holds b5,b4 are_independent_respect_to b3;

:: PROB_2:th 32
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, b5, b6 being Event of b2 holds
   b4,b5,b6 are_independent_respect_to b3
iff
   b3 . ((b4 /\ b5) /\ b6) = ((b3 . b4) * (b3 . b5)) * (b3 . b6) &
    b4,b5 are_independent_respect_to b3 &
    b5,b6 are_independent_respect_to b3 &
    b4,b6 are_independent_respect_to b3;

:: PROB_2:th 33
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, b5, b6 being Event of b2
      st b4,b5,b6 are_independent_respect_to b3
   holds b5,b4,b6 are_independent_respect_to b3;

:: PROB_2:th 34
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, b5, b6 being Event of b2
      st b4,b5,b6 are_independent_respect_to b3
   holds b4,b6,b5 are_independent_respect_to b3;

:: PROB_2:th 35
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, b5 being Event of b2
      st b5 = {}
   holds b4,b5 are_independent_respect_to b3;

:: PROB_2:th 36
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 Event of b2 holds
   b4,[#] b2 are_independent_respect_to b3;

:: PROB_2:th 37
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,b4 are_independent_respect_to b5
   holds b3,([#] b2) \ b4 are_independent_respect_to b5;

:: PROB_2:th 38
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, b5 being Event of b2
      st b4,b5 are_independent_respect_to b3
   holds ([#] b2) \ b4,([#] b2) \ b5 are_independent_respect_to b3;

:: PROB_2:th 39
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4, b5 being Event of b2
for b6 being Probability of b2
      st b3,b4 are_independent_respect_to b6 & b3,b5 are_independent_respect_to b6 & b4 misses b5
   holds b3,b4 \/ b5 are_independent_respect_to b6;

:: PROB_2:th 40
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, b5 being Event of b2
      st b4,b5 are_independent_respect_to b3 & b3 . b4 < 1 & b3 . b5 < 1
   holds b3 . (b4 \/ b5) < 1;

:: PROB_2:funcnot 4 => PROB_2:func 4
definition
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be Probability of a2;
  let a4 be Event of a2;
  assume 0 < a3 . a4;
  func A3 .|. A4 -> Probability of a2 means
    for b1 being Event of a2 holds
       it . b1 = (a3 . (b1 /\ a4)) / (a3 . a4);
end;

:: PROB_2:def 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 Probability of b2
for b4 being Event of b2
   st 0 < b3 . b4
for b5 being Probability of b2 holds
      b5 = b3 .|. b4
   iff
      for b6 being Event of b2 holds
         b5 . b6 = (b3 . (b6 /\ b4)) / (b3 . b4);

:: PROB_2: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 Probability of b2
for b4, b5 being Event of b2
      st 0 < b3 . b4
   holds b3 . (b5 /\ b4) = ((b3 .|. b4) . b5) * (b3 . b4);

:: PROB_2: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 Probability of b2
for b4, b5, b6 being Event of b2
      st 0 < b3 . (b4 /\ b5)
   holds b3 . ((b4 /\ b5) /\ b6) = ((b3 . b4) * ((b3 .|. b4) . b5)) * ((b3 .|. (b4 /\ b5)) . b6);

:: PROB_2: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 Probability of b2
for b4, b5, b6 being Event of b2
      st b6 = b5 ` & 0 < b3 . b5 & 0 < b3 . b6
   holds b3 . b4 = (((b3 .|. b5) . b4) * (b3 . b5)) + (((b3 .|. b6) . b4) * (b3 . b6));

:: PROB_2: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 Probability of b2
for b4, b5, b6, b7 being Event of b2
      st b5 misses b6 & b7 = (b5 \/ b6) ` & 0 < b3 . b5 & 0 < b3 . b6 & 0 < b3 . b7
   holds b3 . b4 = ((((b3 .|. b5) . b4) * (b3 . b5)) + (((b3 .|. b6) . b4) * (b3 . b6))) + (((b3 .|. b7) . b4) * (b3 . b7));

:: PROB_2: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 Probability of b2
for b4, b5 being Event of b2
      st 0 < b3 . b5
   holds    (b3 .|. b5) . b4 = b3 . b4
   iff
      b4,b5 are_independent_respect_to b3;

:: PROB_2: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 Probability of b2
for b4, b5 being Event of b2
      st 0 < b3 . b5 &
         b3 . b5 < 1 &
         (b3 .|. b5) . b4 = (b3 .|. (([#] b2) \ b5)) . b4
   holds b4,b5 are_independent_respect_to b3;

:: PROB_2:th 48
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, b5 being Event of b2
      st 0 < b3 . b5
   holds (((b3 . b4) + (b3 . b5)) - 1) / (b3 . b5) <= (b3 .|. b5) . b4;

:: PROB_2: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, b4 being Event of b2
for b5 being Probability of b2
      st 0 < b5 . b3 & 0 < b5 . b4
   holds (b5 .|. b4) . b3 = (((b5 .|. b3) . b4) * (b5 . b3)) / (b5 . b4);

:: PROB_2: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, b4, b5 being Event of b2
for b6 being Probability of b2
      st 0 < b6 . b3 & b5 = b4 ` & 0 < b6 . b4 & 0 < b6 . b5
   holds (b6 .|. b3) . b4 = (((b6 .|. b4) . b3) * (b6 . b4)) / ((((b6 .|. b4) . b3) * (b6 . b4)) + (((b6 .|. b5) . b3) * (b6 . b5))) &
    (b6 .|. b3) . b5 = (((b6 .|. b5) . b3) * (b6 . b5)) / ((((b6 .|. b4) . b3) * (b6 . b4)) + (((b6 .|. b5) . b3) * (b6 . b5)));

:: PROB_2: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, b4, b5, b6 being Event of b2
for b7 being Probability of b2
      st 0 < b7 . b3 & 0 < b7 . b4 & 0 < b7 . b5 & 0 < b7 . b6 & b4 misses b5 & b6 = (b4 \/ b5) `
   holds (b7 .|. b3) . b4 = (((b7 .|. b4) . b3) * (b7 . b4)) / (((((b7 .|. b4) . b3) * (b7 . b4)) + (((b7 .|. b5) . b3) * (b7 . b5))) + (((b7 .|. b6) . b3) * (b7 . b6))) &
    (b7 .|. b3) . b5 = (((b7 .|. b5) . b3) * (b7 . b5)) / (((((b7 .|. b4) . b3) * (b7 . b4)) + (((b7 .|. b5) . b3) * (b7 . b5))) + (((b7 .|. b6) . b3) * (b7 . b6))) &
    (b7 .|. b3) . b6 = (((b7 .|. b6) . b3) * (b7 . b6)) / (((((b7 .|. b4) . b3) * (b7 . b4)) + (((b7 .|. b5) . b3) * (b7 . b5))) + (((b7 .|. b6) . b3) * (b7 . b6)));

:: PROB_2:th 52
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 0 < b5 . b4
   holds 1 - ((b5 . (([#] b2) \ b3)) / (b5 . b4)) <= (b5 .|. b4) . b3;