Article BHSP_6, MML version 4.99.1005

:: BHSP_6:funcnot 1 => BHSP_6:func 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2 be finite Element of bool the carrier of a1;
  assume the addF of a1 is commutative(the carrier of a1) & the addF of a1 is associative(the carrier of a1) & the addF of a1 is having_a_unity(the carrier of a1);
  func setsum A2 -> Element of the carrier of a1 means
    ex b1 being FinSequence of the carrier of a1 st
       b1 is one-to-one & rng b1 = a2 & it = (the addF of a1) "**" b1;
end;

:: BHSP_6:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
   st the addF of b1 is commutative(the carrier of b1) & the addF of b1 is associative(the carrier of b1) & the addF of b1 is having_a_unity(the carrier of b1)
for b2 being finite Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
      b3 = setsum b2
   iff
      ex b4 being FinSequence of the carrier of b1 st
         b4 is one-to-one & rng b4 = b2 & b3 = (the addF of b1) "**" b4;

:: BHSP_6:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
   st the addF of b1 is commutative(the carrier of b1) & the addF of b1 is associative(the carrier of b1) & the addF of b1 is having_a_unity(the carrier of b1)
for b2 being finite Element of bool the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
      st b2 c= dom b3 &
         (for b4 being set
               st b4 in dom b3
            holds b3 . b4 = b4)
   holds setsum b2 = setopfunc(b2,the carrier of b1,the carrier of b1,b3,the addF of b1);

:: BHSP_6:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
   st the addF of b1 is commutative(the carrier of b1) & the addF of b1 is associative(the carrier of b1) & the addF of b1 is having_a_unity(the carrier of b1)
for b2, b3 being finite Element of bool the carrier of b1
   st b2 misses b3
for b4 being finite Element of bool the carrier of b1
      st b4 = b2 \/ b3
   holds setsum b4 = (setsum b2) + setsum b3;

:: BHSP_6:attrnot 1 => BHSP_6:attr 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2 be Element of bool the carrier of a1;
  attr a2 is summable_set means
    ex b1 being Element of the carrier of a1 st
       for b2 being Element of REAL
             st 0 < b2
          holds ex b3 being finite Element of bool the carrier of a1 st
             b3 is not empty &
              b3 c= a2 &
              (for b4 being finite Element of bool the carrier of a1
                    st b3 c= b4 & b4 c= a2
                 holds ||.b1 - setsum b4.|| < b2);
end;

:: BHSP_6:dfs 2
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is summable_set
it is sufficient to prove
  thus ex b1 being Element of the carrier of a1 st
       for b2 being Element of REAL
             st 0 < b2
          holds ex b3 being finite Element of bool the carrier of a1 st
             b3 is not empty &
              b3 c= a2 &
              (for b4 being finite Element of bool the carrier of a1
                    st b3 c= b4 & b4 c= a2
                 holds ||.b1 - setsum b4.|| < b2);

:: BHSP_6:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1 holds
      b2 is summable_set(b1)
   iff
      ex b3 being Element of the carrier of b1 st
         for b4 being Element of REAL
               st 0 < b4
            holds ex b5 being finite Element of bool the carrier of b1 st
               b5 is not empty &
                b5 c= b2 &
                (for b6 being finite Element of bool the carrier of b1
                      st b5 c= b6 & b6 c= b2
                   holds ||.b3 - setsum b6.|| < b4);

:: BHSP_6:funcnot 2 => BHSP_6:func 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2 be Element of bool the carrier of a1;
  assume a2 is summable_set(a1);
  func sum A2 -> Element of the carrier of a1 means
    for b1 being Element of REAL
          st 0 < b1
       holds ex b2 being finite Element of bool the carrier of a1 st
          b2 is not empty &
           b2 c= a2 &
           (for b3 being finite Element of bool the carrier of a1
                 st b2 c= b3 & b3 c= a2
              holds ||.it - setsum b3.|| < b1);
end;

:: BHSP_6:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
   st b2 is summable_set(b1)
for b3 being Element of the carrier of b1 holds
      b3 = sum b2
   iff
      for b4 being Element of REAL
            st 0 < b4
         holds ex b5 being finite Element of bool the carrier of b1 st
            b5 is not empty &
             b5 c= b2 &
             (for b6 being finite Element of bool the carrier of b1
                   st b5 c= b6 & b6 c= b2
                holds ||.b3 - setsum b6.|| < b4);

:: BHSP_6:attrnot 2 => BHSP_6:attr 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2 be Function-like quasi_total additive homogeneous Relation of the carrier of a1,REAL;
  attr a2 is Bounded means
    ex b1 being Element of REAL st
       0 < b1 &
        (for b2 being Element of the carrier of a1 holds
           abs (a2 . b2) <= b1 * ||.b2.||);
end;

:: BHSP_6:dfs 4
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2 be Function-like quasi_total additive homogeneous Relation of the carrier of a1,REAL;
To prove
     a2 is Bounded
it is sufficient to prove
  thus ex b1 being Element of REAL st
       0 < b1 &
        (for b2 being Element of the carrier of a1 holds
           abs (a2 . b2) <= b1 * ||.b2.||);

:: BHSP_6:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,REAL holds
      b2 is Bounded(b1)
   iff
      ex b3 being Element of REAL st
         0 < b3 &
          (for b4 being Element of the carrier of b1 holds
             abs (b2 . b4) <= b3 * ||.b4.||);

:: BHSP_6:attrnot 3 => BHSP_6:attr 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2 be Element of bool the carrier of a1;
  attr a2 is weakly_summable_set means
    ex b1 being Element of the carrier of a1 st
       for b2 being Function-like quasi_total additive homogeneous Relation of the carrier of a1,REAL
          st b2 is Bounded(a1)
       for b3 being Element of REAL
             st 0 < b3
          holds ex b4 being finite Element of bool the carrier of a1 st
             b4 is not empty &
              b4 c= a2 &
              (for b5 being finite Element of bool the carrier of a1
                    st b4 c= b5 & b5 c= a2
                 holds abs (b2 . (b1 - setsum b5)) < b3);
end;

:: BHSP_6:dfs 5
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is weakly_summable_set
it is sufficient to prove
  thus ex b1 being Element of the carrier of a1 st
       for b2 being Function-like quasi_total additive homogeneous Relation of the carrier of a1,REAL
          st b2 is Bounded(a1)
       for b3 being Element of REAL
             st 0 < b3
          holds ex b4 being finite Element of bool the carrier of a1 st
             b4 is not empty &
              b4 c= a2 &
              (for b5 being finite Element of bool the carrier of a1
                    st b4 c= b5 & b5 c= a2
                 holds abs (b2 . (b1 - setsum b5)) < b3);

:: BHSP_6:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1 holds
      b2 is weakly_summable_set(b1)
   iff
      ex b3 being Element of the carrier of b1 st
         for b4 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,REAL
            st b4 is Bounded(b1)
         for b5 being Element of REAL
               st 0 < b5
            holds ex b6 being finite Element of bool the carrier of b1 st
               b6 is not empty &
                b6 c= b2 &
                (for b7 being finite Element of bool the carrier of b1
                      st b6 c= b7 & b7 c= b2
                   holds abs (b4 . (b3 - setsum b7)) < b5);

:: BHSP_6:prednot 1 => BHSP_6:pred 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2 be Element of bool the carrier of a1;
  let a3 be Function-like quasi_total Relation of the carrier of a1,REAL;
  pred A2 is_summable_set_by A3 means
    ex b1 being Element of REAL st
       for b2 being Element of REAL
             st 0 < b2
          holds ex b3 being finite Element of bool the carrier of a1 st
             b3 is not empty &
              b3 c= a2 &
              (for b4 being finite Element of bool the carrier of a1
                    st b3 c= b4 & b4 c= a2
                 holds abs (b1 - setopfunc(b4,the carrier of a1,REAL,a3,addreal)) < b2);
end;

:: BHSP_6:dfs 6
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2 be Element of bool the carrier of a1;
  let a3 be Function-like quasi_total Relation of the carrier of a1,REAL;
To prove
     a2 is_summable_set_by a3
it is sufficient to prove
  thus ex b1 being Element of REAL st
       for b2 being Element of REAL
             st 0 < b2
          holds ex b3 being finite Element of bool the carrier of a1 st
             b3 is not empty &
              b3 c= a2 &
              (for b4 being finite Element of bool the carrier of a1
                    st b3 c= b4 & b4 c= a2
                 holds abs (b1 - setopfunc(b4,the carrier of a1,REAL,a3,addreal)) < b2);

:: BHSP_6:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,REAL holds
      b2 is_summable_set_by b3
   iff
      ex b4 being Element of REAL st
         for b5 being Element of REAL
               st 0 < b5
            holds ex b6 being finite Element of bool the carrier of b1 st
               b6 is not empty &
                b6 c= b2 &
                (for b7 being finite Element of bool the carrier of b1
                      st b6 c= b7 & b7 c= b2
                   holds abs (b4 - setopfunc(b7,the carrier of b1,REAL,b3,addreal)) < b5);

:: BHSP_6:funcnot 3 => BHSP_6:func 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2 be Element of bool the carrier of a1;
  let a3 be Function-like quasi_total Relation of the carrier of a1,REAL;
  assume a2 is_summable_set_by a3;
  func sum_byfunc(A2,A3) -> Element of REAL means
    for b1 being Element of REAL
          st 0 < b1
       holds ex b2 being finite Element of bool the carrier of a1 st
          b2 is not empty &
           b2 c= a2 &
           (for b3 being finite Element of bool the carrier of a1
                 st b2 c= b3 & b3 c= a2
              holds abs (it - setopfunc(b3,the carrier of a1,REAL,a3,addreal)) < b1);
end;

:: BHSP_6:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,REAL
   st b2 is_summable_set_by b3
for b4 being Element of REAL holds
      b4 = sum_byfunc(b2,b3)
   iff
      for b5 being Element of REAL
            st 0 < b5
         holds ex b6 being finite Element of bool the carrier of b1 st
            b6 is not empty &
             b6 c= b2 &
             (for b7 being finite Element of bool the carrier of b1
                   st b6 c= b7 & b7 c= b2
                holds abs (b4 - setopfunc(b7,the carrier of b1,REAL,b3,addreal)) < b5);

:: BHSP_6:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
      st b2 is summable_set(b1)
   holds b2 is weakly_summable_set(b1);

:: BHSP_6:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,REAL
for b3 being FinSequence of the carrier of b1
   st 1 <= len b3
for b4 being FinSequence of REAL
      st dom b3 = dom b4 &
         (for b5 being Element of NAT
               st b5 in dom b4
            holds b4 . b5 = b2 . (b3 . b5))
   holds b2 . ((the addF of b1) "**" b3) = addreal "**" b4;

:: BHSP_6:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
   st the addF of b1 is commutative(the carrier of b1) & the addF of b1 is associative(the carrier of b1) & the addF of b1 is having_a_unity(the carrier of b1)
for b2 being finite Element of bool the carrier of b1
   st b2 is not empty
for b3 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,REAL holds
   b3 . setsum b2 = setopfunc(b2,the carrier of b1,REAL,b3,addreal);

:: BHSP_6:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
   st the addF of b1 is commutative(the carrier of b1) & the addF of b1 is associative(the carrier of b1) & the addF of b1 is having_a_unity(the carrier of b1)
for b2 being Element of bool the carrier of b1
      st b2 is weakly_summable_set(b1)
   holds ex b3 being Element of the carrier of b1 st
      for b4 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,REAL
         st b4 is Bounded(b1)
      for b5 being Element of REAL
            st 0 < b5
         holds ex b6 being finite Element of bool the carrier of b1 st
            b6 is not empty &
             b6 c= b2 &
             (for b7 being finite Element of bool the carrier of b1
                   st b6 c= b7 & b7 c= b2
                holds abs ((b4 . b3) - setopfunc(b7,the carrier of b1,REAL,b4,addreal)) < b5);

:: BHSP_6:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
   st the addF of b1 is commutative(the carrier of b1) & the addF of b1 is associative(the carrier of b1) & the addF of b1 is having_a_unity(the carrier of b1)
for b2 being Element of bool the carrier of b1
   st b2 is weakly_summable_set(b1)
for b3 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,REAL
      st b3 is Bounded(b1)
   holds b2 is_summable_set_by b3;

:: BHSP_6:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
   st the addF of b1 is commutative(the carrier of b1) & the addF of b1 is associative(the carrier of b1) & the addF of b1 is having_a_unity(the carrier of b1)
for b2 being Element of bool the carrier of b1
   st b2 is summable_set(b1)
for b3 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,REAL
      st b3 is Bounded(b1)
   holds b2 is_summable_set_by b3;

:: BHSP_6:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being finite Element of bool the carrier of b1
      st b2 is not empty
   holds b2 is summable_set(b1);

:: BHSP_6:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
   st the addF of b1 is commutative(the carrier of b1) & the addF of b1 is associative(the carrier of b1) & the addF of b1 is having_a_unity(the carrier of b1) & b1 is Hilbert
for b2 being Element of bool the carrier of b1 holds
      b2 is summable_set(b1)
   iff
      for b3 being Element of REAL
            st 0 < b3
         holds ex b4 being finite Element of bool the carrier of b1 st
            b4 is not empty &
             b4 c= b2 &
             (for b5 being finite Element of bool the carrier of b1
                   st b5 is not empty & b5 c= b2 & b4 misses b5
                holds ||.setsum b5.|| < b3);