Article BHSP_7, MML version 4.99.1005

:: BHSP_7:th 1
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
      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 b6 is not empty & b6 c= b2 & b5 misses b6
                holds abs setopfunc(b6,the carrier of b1,REAL,b3,addreal) < b4);

:: BHSP_7: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 being finite OrthogonalFamily of b1
   st b2 is not empty
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
   st b2 c= proj1 b3 &
      (for b4 being Element of the carrier of b1
            st b4 in b2
         holds b3 . b4 = b4)
for b4 being Function-like quasi_total Relation of the carrier of b1,REAL
      st b2 c= proj1 b4 &
         (for b5 being Element of the carrier of b1
               st b5 in b2
            holds b4 . b5 = b5 .|. b5)
   holds (setopfunc(b2,the carrier of b1,the carrier of b1,b3,the addF of b1)) .|. setopfunc(b2,the carrier of b1,the carrier of b1,b3,the addF of b1) = setopfunc(b2,the carrier of b1,REAL,b4,addreal);

:: BHSP_7:th 3
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 OrthogonalFamily of b1
   st b2 is not empty
for b3 being Function-like quasi_total Relation of the carrier of b1,REAL
      st b2 c= proj1 b3 &
         (for b4 being Element of the carrier of b1
               st b4 in b2
            holds b3 . b4 = b4 .|. b4)
   holds (setsum b2) .|. setsum b2 = setopfunc(b2,the carrier of b1,REAL,b3,addreal);

:: BHSP_7:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being OrthogonalFamily of b1
for b3 being Element of bool the carrier of b1
      st b3 is Element of bool b2
   holds b3 is OrthogonalFamily of b1;

:: BHSP_7:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being OrthonormalFamily of b1
for b3 being Element of bool the carrier of b1
      st b3 is Element of bool b2
   holds b3 is OrthonormalFamily of b1;

:: BHSP_7: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) & b1 is Hilbert
for b2 being OrthonormalFamily of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,REAL
      st b2 c= proj1 b3 &
         (for b4 being Element of the carrier of b1
               st b4 in b2
            holds b3 . b4 = b4 .|. b4)
   holds    b2 is summable_set(b1)
   iff
      b2 is_summable_set_by b3;

:: BHSP_7:th 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
   st b2 is not empty & b2 is summable_set(b1)
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 b4 c= b5 & b5 c= b2
          holds abs (((sum b2) .|. sum b2) - ((setsum b5) .|. setsum b5)) < b3);

:: BHSP_7: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) & b1 is Hilbert
for b2 being OrthonormalFamily of b1
   st b2 is not empty
for b3 being Function-like quasi_total Relation of the carrier of b1,REAL
      st b2 c= proj1 b3 &
         (for b4 being Element of the carrier of b1
               st b4 in b2
            holds b3 . b4 = b4 .|. b4) &
         b2 is summable_set(b1)
   holds (sum b2) .|. sum b2 = sum_byfunc(b2,b3);