Article PROB_4, MML version 4.99.1005

:: PROB_4:funcnot 1 => PROB_4:func 1
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;
  let a4 be Element of NAT;
  redefine func a3 . a4 -> Element of a2;
end;

:: PROB_4:th 1
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
   rng b3 c= b2;

:: PROB_4:th 2
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Relation-like Function-like set holds
      b3 is SetSequence of b2
   iff
      b3 is Function-like quasi_total Relation of NAT,b2;

:: PROB_4:sch 1
scheme PROB_4:sch 1
{F1 -> set,
  F2 -> non empty compl-closed sigma-multiplicative Element of bool bool F1(),
  F3 -> Element of F2()}:
ex b1 being SetSequence of F2() st
   for b2 being Element of NAT holds
      b1 . b2 = F3(b2)


:: PROB_4:exreg 1
registration
  let a1 be set;
  cluster Relation-like Function-like non empty total quasi_total disjoint_valued Relation of NAT,bool a1;
end;

:: PROB_4:exreg 2
registration
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  cluster Relation-like Function-like non empty total quasi_total disjoint_valued SetSequence of a2;
end;

:: PROB_4:th 3
theorem
for b1 being set
for b2, b3 being Element of bool b1 holds
ex b4 being Function-like quasi_total Relation of NAT,bool b1 st
   b4 . 0 = b2 &
    b4 . 1 = b3 &
    (for b5 being Element of NAT
          st 1 < b5
       holds b4 . b5 = {});

:: PROB_4:th 4
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 b3 misses b4 &
         b2 . 0 = b3 &
         b2 . 1 = b4 &
         (for b5 being Element of NAT
               st 1 < b5
            holds b2 . b5 = {})
   holds b2 is disjoint_valued & Union b2 = b3 \/ b4;

:: PROB_4:th 5
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 Union b3 in b2) &
       (for b3 being Element of bool b1
             st b3 in b2
          holds b3 ` in b2);

:: PROB_4: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 Probability of b2
for b4, b5 being Event of b2 holds
b3 . (b4 \ b5) = (b3 . (b4 \/ b5)) - (b3 . b5);

:: PROB_4: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 Probability of b2
for b4, b5 being Event of b2
      st b4 c= b5 & b3 . b5 = 0
   holds b3 . b4 = 0;

:: PROB_4:th 8
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
      for b5 being Element of NAT holds
         b4 . (b3 . b5) = 0
   iff
      b4 . Union b3 = 0;

:: PROB_4: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 Probability of b2 holds
      for b5 being set
            st b5 in rng b3
         holds b4 . b5 = 0
   iff
      b4 . union rng b3 = 0;

:: PROB_4:th 10
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like quasi_total Relation of NAT,ExtREAL
      st b1 = b2
   holds Partial_Sums b1 = Ser b2;

:: PROB_4:th 11
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like quasi_total Relation of NAT,ExtREAL
      st b1 = b2 & b1 is bounded_above
   holds sup b1 = sup rng b2;

:: PROB_4:th 12
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like quasi_total Relation of NAT,ExtREAL
      st b1 = b2 & b1 is bounded_below
   holds inf b1 = inf rng b2;

:: PROB_4:th 13
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like quasi_total Relation of NAT,ExtREAL
      st b1 = b2 & b1 is nonnegative-yielding & b1 is summable
   holds Sum b1 = SUM b2;

:: PROB_4:th 14
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 is sigma_Measure of b2;

:: PROB_4:funcnot 2 => PROB_4: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 Probability of a2;
  func P2M A3 -> sigma_Measure of a2 equals
    a3;
end;

:: PROB_4: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 Probability of b2 holds
   P2M b3 = b3;

:: PROB_4:th 15
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
      st b3 . b1 = R_EAL 1
   holds b3 is Probability of b2;

:: PROB_4:funcnot 3 => PROB_4: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 sigma_Measure of a2;
  assume a3 . a1 = R_EAL 1;
  func M2P A3 -> Probability of a2 equals
    a3;
end;

:: PROB_4: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 sigma_Measure of b2
      st b3 . b1 = R_EAL 1
   holds M2P b3 = b3;

:: PROB_4:th 16
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
      st b2 is non-decreasing(b1)
   holds Partial_Union b2 = b2;

:: PROB_4:th 17
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
      st b2 is non-decreasing(b1)
   holds (Partial_Diff_Union b2) . 0 = b2 . 0 &
    (for b3 being Element of NAT holds
       (Partial_Diff_Union b2) . (b3 + 1) = (b2 . (b3 + 1)) \ (b2 . b3));

:: PROB_4:th 18
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
   st b2 is non-decreasing(b1)
for b3 being Element of NAT holds
   b2 . (b3 + 1) = ((Partial_Diff_Union b2) . (b3 + 1)) \/ (b2 . b3);

:: PROB_4:th 19
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
   st b2 is non-decreasing(b1)
for b3 being Element of NAT holds
   (Partial_Diff_Union b2) . (b3 + 1) misses b2 . b3;

:: PROB_4:th 20
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
      st b3 is non-decreasing(b1)
   holds @Partial_Union b3 = b3;

:: PROB_4:th 21
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
      st b3 is non-decreasing(b1)
   holds (@Partial_Diff_Union b3) . 0 = b3 . 0 &
    (for b4 being Element of NAT holds
       (@Partial_Diff_Union b3) . (b4 + 1) = (b3 . (b4 + 1)) \ (b3 . b4));

:: PROB_4:th 22
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
   st b3 is non-decreasing(b1)
for b4 being Element of NAT holds
   (@Partial_Diff_Union b3) . (b4 + 1) misses b3 . b4;

:: PROB_4:prednot 1 => PROB_4: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;
  pred A3 is_complete A2 means
    for b1 being Element of bool a1
    for b2 being set
          st b2 in a2 & b1 c= b2 & a3 . b2 = 0
       holds b1 in a2;
end;

:: PROB_4:dfs 3
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;
To prove
     a3 is_complete a2
it is sufficient to prove
  thus for b1 being Element of bool a1
    for b2 being set
          st b2 in a2 & b1 c= b2 & a3 . b2 = 0
       holds b1 in a2;

:: PROB_4:def 3
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 is_complete b2
   iff
      for b4 being Element of bool b1
      for b5 being set
            st b5 in b2 & b4 c= b5 & b3 . b5 = 0
         holds b4 in b2;

:: PROB_4: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 holds
      b3 is_complete b2
   iff
      P2M b3 is_complete b2;

:: PROB_4:modenot 1 => PROB_4:mode 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;
  mode thin of A3 -> Element of bool a1 means
    ex b1 being set st
       b1 in a2 & it c= b1 & a3 . b1 = 0;
end;

:: PROB_4: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 Probability of a2;
  let a4 be Element of bool a1;
To prove
     a4 is thin of a3
it is sufficient to prove
  thus ex b1 being set st
       b1 in a2 & a4 c= b1 & a3 . b1 = 0;

:: PROB_4: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 Probability of b2
for b4 being Element of bool b1 holds
      b4 is thin of b3
   iff
      ex b5 being set st
         b5 in b2 & b4 c= b5 & b3 . b5 = 0;

:: PROB_4: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 being Element of bool b1 holds
      b4 is thin of b3
   iff
      b4 is thin of P2M b3;

:: PROB_4: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 holds
   {} is thin of b3;

:: PROB_4: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, b5 being set
   st b4 in b2 & b5 in b2
for b6, b7 being thin of b3
      st b4 \/ b6 = b5 \/ b7
   holds b3 . b4 = b3 . b5;

:: PROB_4:funcnot 4 => PROB_4: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;
  func COM(A2,A3) -> non empty Element of bool bool a1 means
    for b1 being set holds
          b1 in it
       iff
          ex b2 being set st
             b2 in a2 &
              (ex b3 being thin of a3 st
                 b1 = b2 \/ b3);
end;

:: PROB_4: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 being non empty Element of bool bool b1 holds
      b4 = COM(b2,b3)
   iff
      for b5 being set holds
            b5 in b4
         iff
            ex b6 being set st
               b6 in b2 &
                (ex b7 being thin of b3 st
                   b5 = b6 \/ b7);

:: PROB_4: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 thin of b3 holds
   b4 in COM(b2,b3);

:: PROB_4: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 holds
   COM(b2,b3) = COM(b2,P2M b3);

:: PROB_4:funcnot 5 => PROB_4:func 5
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 Element of COM(a2,a3);
  func P_COM2M_COM A4 -> Element of COM(a2,P2M a3) equals
    a4;
end;

:: PROB_4: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 being Element of COM(b2,b3) holds
   P_COM2M_COM b4 = b4;

:: PROB_4:th 29
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
   b2 c= COM(b2,b3);

:: PROB_4:funcnot 6 => PROB_4:func 6
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 Element of COM(a2,a3);
  func ProbPart A4 -> non empty Element of bool bool a1 means
    for b1 being set holds
          b1 in it
       iff
          b1 in a2 & b1 c= a4 & a4 \ b1 is thin of a3;
end;

:: PROB_4: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 Element of COM(b2,b3)
for b5 being non empty Element of bool bool b1 holds
      b5 = ProbPart b4
   iff
      for b6 being set holds
            b6 in b5
         iff
            b6 in b2 & b6 c= b4 & b4 \ b6 is thin of b3;

:: PROB_4:th 30
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 Element of COM(b2,b3) holds
   ProbPart b4 = MeasPart P_COM2M_COM b4;

:: PROB_4: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 being Element of COM(b2,b3)
for b5, b6 being set
      st b5 in ProbPart b4 & b6 in ProbPart b4
   holds b3 . b5 = b3 . b6;

:: PROB_4: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 being Function-like quasi_total Relation of NAT,COM(b2,b3) holds
   ex b5 being SetSequence of b2 st
      for b6 being Element of NAT holds
         b5 . b6 in ProbPart (b4 . b6);

:: PROB_4: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 being Function-like quasi_total Relation of NAT,COM(b2,b3)
for b5 being SetSequence of b2 holds
   ex b6 being Function-like quasi_total Relation of NAT,bool b1 st
      for b7 being Element of NAT holds
         b6 . b7 = (b4 . b7) \ (b5 . b7);

:: PROB_4: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 being Function-like quasi_total Relation of NAT,bool b1
      st for b5 being Element of NAT holds
           b4 . b5 is thin of b3
   holds ex b5 being SetSequence of b2 st
      for b6 being Element of NAT holds
         b4 . b6 c= b5 . b6 & b3 . (b5 . b6) = 0;

:: PROB_4: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 being non empty Element of bool bool b1
      st for b5 being set holds
              b5 in b4
           iff
              ex b6 being set st
                 b6 in b2 &
                  (ex b7 being thin of b3 st
                     b5 = b6 \/ b7)
   holds b4 is non empty compl-closed sigma-multiplicative Element of bool bool b1;

:: PROB_4:funcnot 7 => PROB_4:func 7
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;
  redefine func COM(a2,a3) -> non empty compl-closed sigma-multiplicative Element of bool bool a1;
end;

:: PROB_4:modenot 2 => PROB_4:mode 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;
  redefine mode thin of a3 -> Event of COM(a2,a3);
end;

:: PROB_4: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 set holds
      b4 in COM(b2,b3)
   iff
      ex b5, b6 being set st
         b5 in b2 & b6 in b2 & b5 c= b4 & b4 c= b6 & b3 . (b6 \ b5) = 0;

:: PROB_4: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 being Probability of b2
for b4 being non empty Element of bool bool b1
      st for b5 being set holds
              b5 in b4
           iff
              ex b6, b7 being set st
                 b6 in b2 & b7 in b2 & b6 c= b5 & b5 c= b7 & b3 . (b7 \ b6) = 0
   holds b4 = COM(b2,b3);

:: PROB_4:funcnot 8 => PROB_4:func 8
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;
  func COM A3 -> Probability of COM(a2,a3) means
    for b1 being set
       st b1 in a2
    for b2 being thin of a3 holds
       it . (b1 \/ b2) = a3 . b1;
end;

:: PROB_4:def 8
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 Probability of COM(b2,b3) holds
      b4 = COM b3
   iff
      for b5 being set
         st b5 in b2
      for b6 being thin of b3 holds
         b4 . (b5 \/ b6) = b3 . b5;

:: PROB_4: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 holds
   COM b3 = COM P2M b3;

:: PROB_4: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 being Probability of b2 holds
   COM b3 is_complete COM(b2,b3);

:: PROB_4: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 being Event of b2 holds
   b3 . b4 = (COM b3) . b4;

:: PROB_4:th 41
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 thin of b3 holds
   (COM b3) . b4 = 0;

:: PROB_4: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 being Element of COM(b2,b3)
for b5 being set
      st b5 in ProbPart b4
   holds b3 . b5 = (COM b3) . b4;