Article BHSP_4, MML version 4.99.1005

:: BHSP_4:funcnot 1 => BHSP_4:func 1
definition
  let a1 be non empty addLoopStr;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  func Partial_Sums A2 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
    it . 0 = a2 . 0 &
     (for b1 being Element of NAT holds
        it . (b1 + 1) = (it . b1) + (a2 . (b1 + 1)));
end;

:: BHSP_4:def 1
theorem
for b1 being non empty addLoopStr
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   b3 = Partial_Sums b2
iff
   b3 . 0 = b2 . 0 &
    (for b4 being Element of NAT holds
       b3 . (b4 + 1) = (b3 . b4) + (b2 . (b4 + 1)));

:: BHSP_4:th 1
theorem
for b1 being non empty Abelian add-associative addLoopStr
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
(Partial_Sums b2) + Partial_Sums b3 = Partial_Sums (b2 + b3);

:: BHSP_4:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
(Partial_Sums b2) - Partial_Sums b3 = Partial_Sums (b2 - b3);

:: BHSP_4:th 3
theorem
for b1 being Element of REAL
for b2 being non empty RealLinearSpace-like RLSStruct
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
   Partial_Sums (b1 * b3) = b1 * Partial_Sums b3;

:: BHSP_4: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 Relation of NAT,the carrier of b1 holds
   Partial_Sums - b2 = - Partial_Sums b2;

:: BHSP_4:th 5
theorem
for b1, b2 being Element of REAL
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b4, b5 being Function-like quasi_total Relation of NAT,the carrier of b3 holds
(b1 * Partial_Sums b4) + (b2 * Partial_Sums b5) = Partial_Sums ((b1 * b4) + (b2 * b5));

:: BHSP_4:attrnot 1 => BHSP_4:attr 1
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 Relation of NAT,the carrier of a1;
  attr a2 is summable means
    Partial_Sums a2 is convergent(a1);
end;

:: BHSP_4:dfs 2
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 Relation of NAT,the carrier of a1;
To prove
     a2 is summable
it is sufficient to prove
  thus Partial_Sums a2 is convergent(a1);

:: BHSP_4:def 2
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 Relation of NAT,the carrier of b1 holds
      b2 is summable(b1)
   iff
      Partial_Sums b2 is convergent(b1);

:: BHSP_4:funcnot 2 => BHSP_4:func 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 Relation of NAT,the carrier of a1;
  func Sum A2 -> Element of the carrier of a1 equals
    lim Partial_Sums a2;
end;

:: BHSP_4:def 3
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 Relation of NAT,the carrier of b1 holds
   Sum b2 = lim Partial_Sums b2;

:: BHSP_4:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is summable(b1) & b3 is summable(b1)
   holds b2 + b3 is summable(b1) &
    Sum (b2 + b3) = (Sum b2) + Sum b3;

:: BHSP_4:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is summable(b1) & b3 is summable(b1)
   holds b2 - b3 is summable(b1) &
    Sum (b2 - b3) = (Sum b2) - Sum b3;

:: BHSP_4:th 8
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
      st b3 is summable(b2)
   holds b1 * b3 is summable(b2) & Sum (b1 * b3) = b1 * Sum b3;

:: BHSP_4:th 9
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 Relation of NAT,the carrier of b1
      st b2 is summable(b1)
   holds b2 is convergent(b1) & lim b2 = 0. b1;

:: BHSP_4:th 10
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 Relation of NAT,the carrier of b1
      st b1 is Hilbert
   holds    b2 is summable(b1)
   iff
      for b3 being Element of REAL
            st 0 < b3
         holds ex b4 being Element of NAT st
            for b5, b6 being Element of NAT
                  st b4 <= b5 & b4 <= b6
               holds ||.((Partial_Sums b2) . b5) - ((Partial_Sums b2) . b6).|| < b3;

:: BHSP_4:th 11
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 Relation of NAT,the carrier of b1
      st b2 is summable(b1)
   holds Partial_Sums b2 is bounded(b1);

:: BHSP_4:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st for b4 being Element of NAT holds
           b3 . b4 = b2 . 0
   holds Partial_Sums (b2 ^\ 1) = ((Partial_Sums b2) ^\ 1) - b3;

:: BHSP_4:th 13
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 Relation of NAT,the carrier of b1
   st b2 is summable(b1)
for b3 being Element of NAT holds
   b2 ^\ b3 is summable(b1);

:: BHSP_4:th 14
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 Relation of NAT,the carrier of b1
      st ex b3 being Element of NAT st
           b2 ^\ b3 is summable(b1)
   holds b2 is summable(b1);

:: BHSP_4:funcnot 3 => BHSP_4:func 3
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 Relation of NAT,the carrier of a1;
  let a3 be Element of NAT;
  func Sum(A2,A3) -> Element of the carrier of a1 equals
    (Partial_Sums a2) . a3;
end;

:: BHSP_4: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 Relation of NAT,the carrier of b1
for b3 being Element of NAT holds
   Sum(b2,b3) = (Partial_Sums b2) . b3;

:: BHSP_4:th 16
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 Relation of NAT,the carrier of b1 holds
   Sum(b2,0) = b2 . 0;

:: BHSP_4:th 17
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 Relation of NAT,the carrier of b1 holds
   Sum(b2,1) = (Sum(b2,0)) + (b2 . 1);

:: BHSP_4:th 18
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 Relation of NAT,the carrier of b1 holds
   Sum(b2,1) = (b2 . 0) + (b2 . 1);

:: BHSP_4:th 19
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
   Sum(b3,b1 + 1) = (Sum(b3,b1)) + (b3 . (b1 + 1));

:: BHSP_4:th 20
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
   b3 . (b1 + 1) = (Sum(b3,b1 + 1)) - Sum(b3,b1);

:: BHSP_4:th 21
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 Relation of NAT,the carrier of b1 holds
   b2 . 1 = (Sum(b2,1)) - Sum(b2,0);

:: BHSP_4:funcnot 4 => BHSP_4:func 4
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 Relation of NAT,the carrier of a1;
  let a3, a4 be Element of NAT;
  func Sum(A2,A3,A4) -> Element of the carrier of a1 equals
    (Sum(a2,a3)) - Sum(a2,a4);
end;

:: BHSP_4:def 5
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 Relation of NAT,the carrier of b1
for b3, b4 being Element of NAT holds
Sum(b2,b3,b4) = (Sum(b2,b3)) - Sum(b2,b4);

:: BHSP_4:th 23
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 Relation of NAT,the carrier of b1 holds
   Sum(b2,1,0) = b2 . 1;

:: BHSP_4:th 24
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
   Sum(b3,b1 + 1,b1) = b3 . (b1 + 1);

:: BHSP_4:th 25
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 Relation of NAT,the carrier of b1
      st b1 is Hilbert
   holds    b2 is summable(b1)
   iff
      for b3 being Element of REAL
            st 0 < b3
         holds ex b4 being Element of NAT st
            for b5, b6 being Element of NAT
                  st b4 <= b5 & b4 <= b6
               holds ||.(Sum(b2,b5)) - Sum(b2,b6).|| < b3;

:: BHSP_4:th 26
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 Relation of NAT,the carrier of b1
      st b1 is Hilbert
   holds    b2 is summable(b1)
   iff
      for b3 being Element of REAL
            st 0 < b3
         holds ex b4 being Element of NAT st
            for b5, b6 being Element of NAT
                  st b4 <= b5 & b4 <= b6
               holds ||.Sum(b2,b5,b6).|| < b3;

:: BHSP_4:attrnot 2 => BHSP_4: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 Relation of NAT,the carrier of a1;
  attr a2 is absolutely_summable means
    ||.a2.|| is summable;
end;

:: BHSP_4:dfs 6
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 Relation of NAT,the carrier of a1;
To prove
     a2 is absolutely_summable
it is sufficient to prove
  thus ||.a2.|| is summable;

:: BHSP_4:def 8
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 Relation of NAT,the carrier of b1 holds
      b2 is absolutely_summable(b1)
   iff
      ||.b2.|| is summable;

:: BHSP_4:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is absolutely_summable(b1) & b3 is absolutely_summable(b1)
   holds b2 + b3 is absolutely_summable(b1);

:: BHSP_4:th 28
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
      st b3 is absolutely_summable(b2)
   holds b1 * b3 is absolutely_summable(b2);

:: BHSP_4:th 29
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
      st (for b4 being Element of NAT holds
            ||.b3.|| . b4 <= b1 . b4) &
         b1 is summable
   holds b3 is absolutely_summable(b2);

:: BHSP_4:th 30
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
      st (for b4 being Element of NAT holds
            b3 . b4 <> 0. b2 &
             b1 . b4 = ||.b3 . (b4 + 1).|| / ||.b3 . b4.||) &
         b1 is convergent &
         lim b1 < 1
   holds b3 is absolutely_summable(b2);

:: BHSP_4:th 31
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
      st 0 < b1 &
         (ex b4 being Element of NAT st
            for b5 being Element of NAT
                  st b4 <= b5
               holds b1 <= ||.b3 . b5.||) &
         b3 is convergent(b2)
   holds lim b3 <> 0. b2;

:: BHSP_4:th 32
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 Relation of NAT,the carrier of b1
      st (for b3 being Element of NAT holds
            b2 . b3 <> 0. b1) &
         (ex b3 being Element of NAT st
            for b4 being Element of NAT
                  st b3 <= b4
               holds 1 <= ||.b2 . (b4 + 1).|| / ||.b2 . b4.||)
   holds b2 is not summable(b1);

:: BHSP_4:th 33
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
      st (for b4 being Element of NAT holds
            b3 . b4 <> 0. b2) &
         (for b4 being Element of NAT holds
            b1 . b4 = ||.b3 . (b4 + 1).|| / ||.b3 . b4.||) &
         b1 is convergent &
         1 < lim b1
   holds b3 is not summable(b2);

:: BHSP_4:th 34
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
      st (for b4 being Element of NAT holds
            b1 . b4 = b4 -root ||.b3 . b4.||) &
         b1 is convergent &
         lim b1 < 1
   holds b3 is absolutely_summable(b2);

:: BHSP_4:th 35
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
      st (for b4 being Element of NAT holds
            b1 . b4 = b4 -root (||.b3.|| . b4)) &
         (ex b4 being Element of NAT st
            for b5 being Element of NAT
                  st b4 <= b5
               holds 1 <= b1 . b5)
   holds b3 is not summable(b2);

:: BHSP_4:th 36
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
      st (for b4 being Element of NAT holds
            b1 . b4 = b4 -root (||.b3.|| . b4)) &
         b1 is convergent &
         1 < lim b1
   holds b3 is not summable(b2);

:: BHSP_4:th 37
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 Relation of NAT,the carrier of b1 holds
   Partial_Sums ||.b2.|| is non-decreasing;

:: BHSP_4:th 38
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 Relation of NAT,the carrier of b1
for b3 being Element of NAT holds
   0 <= (Partial_Sums ||.b2.||) . b3;

:: BHSP_4:th 39
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 Relation of NAT,the carrier of b1
for b3 being Element of NAT holds
   ||.(Partial_Sums b2) . b3.|| <= (Partial_Sums ||.b2.||) . b3;

:: BHSP_4:th 40
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 Relation of NAT,the carrier of b1
for b3 being Element of NAT holds
   ||.Sum(b2,b3).|| <= Sum(||.b2.||,b3);

:: BHSP_4:th 41
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 Relation of NAT,the carrier of b1
for b3, b4 being Element of NAT holds
||.((Partial_Sums b2) . b4) - ((Partial_Sums b2) . b3).|| <= abs (((Partial_Sums ||.b2.||) . b4) - ((Partial_Sums ||.b2.||) . b3));

:: BHSP_4:th 42
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 Relation of NAT,the carrier of b1
for b3, b4 being Element of NAT holds
||.(Sum(b2,b4)) - Sum(b2,b3).|| <= abs ((Sum(||.b2.||,b4)) - Sum(||.b2.||,b3));

:: BHSP_4:th 43
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 Relation of NAT,the carrier of b1
for b3, b4 being Element of NAT holds
||.Sum(b2,b4,b3).|| <= abs Sum(||.b2.||,b4,b3);

:: BHSP_4:th 44
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 Relation of NAT,the carrier of b1
      st b1 is Hilbert & b2 is absolutely_summable(b1)
   holds b2 is summable(b1);

:: BHSP_4:funcnot 5 => BHSP_4:func 5
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 Relation of NAT,the carrier of a1;
  let a3 be Function-like quasi_total Relation of NAT,REAL;
  func A3 * A2 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
    for b1 being Element of NAT holds
       it . b1 = (a3 . b1) * (a2 . b1);
end;

:: BHSP_4:def 9
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 Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,REAL
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b4 = b3 * b2
   iff
      for b5 being Element of NAT holds
         b4 . b5 = (b3 . b5) * (b2 . b5);

:: BHSP_4:th 45
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
b1 * (b3 + b4) = (b1 * b3) + (b1 * b4);

:: BHSP_4:th 46
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b4 being Function-like quasi_total Relation of NAT,the carrier of b3 holds
   (b1 + b2) * b4 = (b1 * b4) + (b2 * b4);

:: BHSP_4:th 47
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b4 being Function-like quasi_total Relation of NAT,the carrier of b3 holds
   (b1 (#) b2) * b4 = b1 * (b2 * b4);

:: BHSP_4:th 48
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b4 being Function-like quasi_total Relation of NAT,the carrier of b3 holds
   (b1 (#) b2) * b4 = b1 * (b2 * b4);

:: BHSP_4:th 49
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
   b1 * - b3 = (- b1) * b3;

:: BHSP_4:th 50
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
      st b1 is convergent & b3 is convergent(b2)
   holds b1 * b3 is convergent(b2);

:: BHSP_4:th 51
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
      st b1 is bounded & b3 is bounded(b2)
   holds b1 * b3 is bounded(b2);

:: BHSP_4:th 52
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
      st b1 is convergent & b3 is convergent(b2)
   holds b1 * b3 is convergent(b2) &
    lim (b1 * b3) = (lim b1) * lim b3;

:: BHSP_4:attrnot 3 => BHSP_4:attr 3
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  attr a1 is Cauchy means
    for b1 being Element of REAL
          st 0 < b1
       holds ex b2 being Element of NAT st
          for b3, b4 being Element of NAT
                st b2 <= b3 & b2 <= b4
             holds abs ((a1 . b3) - (a1 . b4)) < b1;
end;

:: BHSP_4:dfs 8
definiens
  let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
     a1 is Cauchy
it is sufficient to prove
  thus for b1 being Element of REAL
          st 0 < b1
       holds ex b2 being Element of NAT st
          for b3, b4 being Element of NAT
                st b2 <= b3 & b2 <= b4
             holds abs ((a1 . b3) - (a1 . b4)) < b1;

:: BHSP_4:def 10
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is Cauchy
   iff
      for b2 being Element of REAL
            st 0 < b2
         holds ex b3 being Element of NAT st
            for b4, b5 being Element of NAT
                  st b3 <= b4 & b3 <= b5
               holds abs ((b1 . b4) - (b1 . b5)) < b2;

:: BHSP_4:prednot 1 => BHSP_4:attr 3
notation
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  synonym a1 is_Cauchy_sequence for Cauchy;
end;

:: BHSP_4:th 53
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
      st b2 is Hilbert & b3 is Cauchy(b2) & b1 is Cauchy
   holds b1 * b3 is Cauchy(b2);

:: BHSP_4:th 54
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
for b4 being Element of NAT holds
   (Partial_Sums ((b1 - (b1 ^\ 1)) * Partial_Sums b3)) . b4 = ((Partial_Sums (b1 * b3)) . (b4 + 1)) - ((b1 * Partial_Sums b3) . (b4 + 1));

:: BHSP_4:th 55
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
for b4 being Element of NAT holds
   (Partial_Sums (b1 * b3)) . (b4 + 1) = ((b1 * Partial_Sums b3) . (b4 + 1)) - ((Partial_Sums (((b1 ^\ 1) - b1) * Partial_Sums b3)) . b4);

:: BHSP_4:th 56
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
for b4 being Element of NAT holds
   Sum(b1 * b3,b4 + 1) = ((b1 * Partial_Sums b3) . (b4 + 1)) - Sum(((b1 ^\ 1) - b1) * Partial_Sums b3,b4);