Article BHSP_3, MML version 4.99.1005

:: BHSP_3:attrnot 1 => BHSP_3: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 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 dist(a2 . b3,a2 . b4) < b1;
end;

:: BHSP_3:dfs 1
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 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 dist(a2 . b3,a2 . b4) < b1;

:: BHSP_3:def 1
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 Cauchy(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 dist(b2 . b5,b2 . b6) < b3;

:: BHSP_3:prednot 1 => BHSP_3:attr 1
notation
  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;
  synonym a2 is_Cauchy_sequence for Cauchy;
end;

:: BHSP_3:th 1
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 constant
   holds b2 is Cauchy(b1);

:: BHSP_3:th 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 Cauchy(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 ||.(b2 . b5) - (b2 . b6).|| < b3;

:: BHSP_3:th 3
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 Cauchy(b1) & b3 is Cauchy(b1)
   holds b2 + b3 is Cauchy(b1);

:: BHSP_3:th 4
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 Cauchy(b1) & b3 is Cauchy(b1)
   holds b2 - b3 is Cauchy(b1);

:: BHSP_3:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b3 is Cauchy(b1)
   holds b2 * b3 is Cauchy(b1);

:: BHSP_3:th 6
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 Cauchy(b1)
   holds - b2 is Cauchy(b1);

:: BHSP_3: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 the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b3 is Cauchy(b1)
   holds b3 + b2 is Cauchy(b1);

:: BHSP_3:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b3 is Cauchy(b1)
   holds b3 - b2 is Cauchy(b1);

:: BHSP_3: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 convergent(b1)
   holds b2 is Cauchy(b1);

:: BHSP_3:prednot 2 => BHSP_3:pred 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2, a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
  pred A2 is_compared_to A3 means
    for b1 being Element of REAL
          st 0 < b1
       holds ex b2 being Element of NAT st
          for b3 being Element of NAT
                st b2 <= b3
             holds dist(a2 . b3,a3 . b3) < b1;
end;

:: BHSP_3:dfs 2
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2, a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
     a2 is_compared_to a3
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 being Element of NAT
                st b2 <= b3
             holds dist(a2 . b3,a3 . b3) < b1;

:: BHSP_3:def 2
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 holds
   b2 is_compared_to b3
iff
   for b4 being Element of REAL
         st 0 < b4
      holds ex b5 being Element of NAT st
         for b6 being Element of NAT
               st b5 <= b6
            holds dist(b2 . b6,b3 . b6) < b4;

:: BHSP_3: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 holds
   b2 is_compared_to b2;

:: BHSP_3:th 11
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_compared_to b3
   holds b3 is_compared_to b2;

:: BHSP_3:prednot 3 => BHSP_3:pred 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2, a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
  redefine pred a2 is_compared_to a3;
  symmetry;
::  for a1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
::  for a2, a3 being Function-like quasi_total Relation of NAT,the carrier of a1
::        st a2 is_compared_to a3
::     holds a3 is_compared_to a2;
  reflexivity;
::  for a1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
::  for a2 being Function-like quasi_total Relation of NAT,the carrier of a1 holds
::     a2 is_compared_to a2;
end;

:: BHSP_3:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is_compared_to b3 & b3 is_compared_to b4
   holds b2 is_compared_to b4;

:: BHSP_3:th 13
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 holds
   b2 is_compared_to b3
iff
   for b4 being Element of REAL
         st 0 < b4
      holds ex b5 being Element of NAT st
         for b6 being Element of NAT
               st b5 <= b6
            holds ||.(b2 . b6) - (b3 . b6).|| < b4;

:: BHSP_3:th 14
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 ex b4 being Element of NAT st
           for b5 being Element of NAT
                 st b4 <= b5
              holds b2 . b5 = b3 . b5
   holds b2 is_compared_to b3;

:: BHSP_3:th 15
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 Cauchy(b1) & b2 is_compared_to b3
   holds b3 is Cauchy(b1);

:: BHSP_3:th 16
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 convergent(b1) & b2 is_compared_to b3
   holds b3 is convergent(b1);

:: BHSP_3:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b3 is convergent(b1) & lim b3 = b2 & b3 is_compared_to b4
   holds b4 is convergent(b1) & lim b4 = b2;

:: BHSP_3:attrnot 2 => BHSP_3: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 bounded means
    ex b1 being Element of REAL st
       0 < b1 &
        (for b2 being Element of NAT holds
           ||.a2 . b2.|| <= b1);
end;

:: BHSP_3:dfs 3
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 bounded
it is sufficient to prove
  thus ex b1 being Element of REAL st
       0 < b1 &
        (for b2 being Element of NAT holds
           ||.a2 . b2.|| <= b1);

:: BHSP_3: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
      b2 is bounded(b1)
   iff
      ex b3 being Element of REAL st
         0 < b3 &
          (for b4 being Element of NAT holds
             ||.b2 . b4.|| <= b3);

:: BHSP_3:th 18
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 bounded(b1) & b3 is bounded(b1)
   holds b2 + b3 is bounded(b1);

:: BHSP_3:th 19
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 bounded(b1)
   holds - b2 is bounded(b1);

:: BHSP_3:th 20
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 bounded(b1) & b3 is bounded(b1)
   holds b2 - b3 is bounded(b1);

:: BHSP_3:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b3 is bounded(b1)
   holds b2 * b3 is bounded(b1);

:: BHSP_3:th 22
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 constant
   holds b2 is bounded(b1);

:: BHSP_3: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
for b3 being Element of NAT holds
   ex b4 being Element of REAL st
      0 < b4 &
       (for b5 being Element of NAT
             st b5 <= b3
          holds ||.b2 . b5.|| < b4);

:: BHSP_3:th 24
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 convergent(b1)
   holds b2 is bounded(b1);

:: BHSP_3:th 25
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 bounded(b1) & b2 is_compared_to b3
   holds b3 is bounded(b1);

:: BHSP_3:funcnot 1 => BHSP_3:func 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 natural-valued increasing Relation of NAT,REAL;
  let a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
  redefine func a3 * a2 -> Function-like quasi_total Relation of NAT,the carrier of a1;
end;

:: BHSP_3:modenot 1 => BHSP_3:mode 1
definition
  let a1 be 1-sorted;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  mode subsequence of A2 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
    ex b1 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
       it = b1 * a2;
end;

:: BHSP_3:dfs 4
definiens
  let a1 be 1-sorted;
  let a2, a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
     a3 is subsequence of a2
it is sufficient to prove
  thus ex b1 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
       a3 = b1 * a2;

:: BHSP_3:def 4
theorem
for b1 being 1-sorted
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   b3 is subsequence of b2
iff
   ex b4 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
      b3 = b4 * b2;

:: BHSP_3: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
for b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL
for b4 being Element of NAT holds
   (b2 * b3) . b4 = b2 . (b3 . b4);

:: BHSP_3:th 27
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 subsequence of b2;

:: BHSP_3:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is subsequence of b3 & b3 is subsequence of b4
   holds b2 is subsequence of b4;

:: BHSP_3:th 29
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 constant & b3 is subsequence of b2
   holds b3 is constant;

:: BHSP_3:th 30
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 constant & b3 is subsequence of b2
   holds b2 = b3;

:: BHSP_3:th 31
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 bounded(b1) & b3 is subsequence of b2
   holds b3 is bounded(b1);

:: BHSP_3:th 32
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 convergent(b1) & b3 is subsequence of b2
   holds b3 is convergent(b1);

:: BHSP_3:th 33
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 subsequence of b3 & b3 is convergent(b1)
   holds lim b2 = lim b3;

:: BHSP_3:th 34
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 Cauchy(b1) & b3 is subsequence of b2
   holds b3 is Cauchy(b1);

:: BHSP_3:funcnot 2 => BHSP_3: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;
  let a3 be Element of NAT;
  func A2 ^\ A3 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
    for b1 being Element of NAT holds
       it . b1 = a2 . (b1 + a3);
end;

:: BHSP_3: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 being Element of NAT
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b4 = b2 ^\ b3
   iff
      for b5 being Element of NAT holds
         b4 . b5 = b2 . (b5 + b3);

:: BHSP_3:th 35
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 ^\ 0 = b2;

:: BHSP_3:th 36
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
(b2 ^\ b3) ^\ b4 = (b2 ^\ b4) ^\ b3;

:: BHSP_3: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
for b3, b4 being Element of NAT holds
(b2 ^\ b3) ^\ b4 = b2 ^\ (b3 + b4);

:: BHSP_3:th 38
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
for b4 being Element of NAT holds
   (b2 + b3) ^\ b4 = (b2 ^\ b4) + (b3 ^\ b4);

:: BHSP_3: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
   (- b2) ^\ b3 = - (b2 ^\ b3);

:: BHSP_3:th 40
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
for b4 being Element of NAT holds
   (b2 - b3) ^\ b4 = (b2 ^\ b4) - (b3 ^\ b4);

:: BHSP_3:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
for b4 being Element of NAT holds
   (b2 * b3) ^\ b4 = b2 * (b3 ^\ b4);

:: BHSP_3: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 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL
for b4 being Element of NAT holds
   (b2 * b3) ^\ b4 = b2 * (b3 ^\ b4);

:: BHSP_3: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 being Element of NAT holds
   b2 ^\ b3 is subsequence of b2;

:: BHSP_3: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
for b3 being Element of NAT
      st b2 is convergent(b1)
   holds b2 ^\ b3 is convergent(b1) & lim (b2 ^\ b3) = lim b2;

:: BHSP_3:th 46
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 convergent(b1) &
         (ex b4 being Element of NAT st
            b2 = b3 ^\ b4)
   holds b3 is convergent(b1);

:: BHSP_3:th 47
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 Cauchy(b1) &
         (ex b4 being Element of NAT st
            b2 = b3 ^\ b4)
   holds b3 is Cauchy(b1);

:: BHSP_3:th 48
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
      st b2 is Cauchy(b1)
   holds b2 ^\ b3 is Cauchy(b1);

:: BHSP_3:th 49
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
for b4 being Element of NAT
      st b2 is_compared_to b3
   holds b2 ^\ b4 is_compared_to b3 ^\ b4;

:: BHSP_3:th 50
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
      st b2 is bounded(b1)
   holds b2 ^\ b3 is bounded(b1);

:: BHSP_3:th 51
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
      st b2 is constant
   holds b2 ^\ b3 is constant;

:: BHSP_3:attrnot 3 => BHSP_3:attr 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  attr a1 is complete means
    for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
          st b1 is Cauchy(a1)
       holds b1 is convergent(a1);
end;

:: BHSP_3:dfs 6
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
To prove
     a1 is complete
it is sufficient to prove
  thus for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
          st b1 is Cauchy(a1)
       holds b1 is convergent(a1);

:: BHSP_3:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
      b1 is complete
   iff
      for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
            st b2 is Cauchy(b1)
         holds b2 is convergent(b1);

:: BHSP_3:prednot 4 => BHSP_3:attr 3
notation
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  synonym a1 is_complete_space for complete;
end;

:: BHSP_3:th 53
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 complete & b2 is Cauchy(b1)
   holds b2 is bounded(b1);

:: BHSP_3:attrnot 4 => BHSP_3:attr 4
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  attr a1 is Hilbert means
    a1 is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR &
     a1 is complete;
end;

:: BHSP_3:dfs 7
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
To prove
     a1 is Hilbert
it is sufficient to prove
  thus a1 is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR &
     a1 is complete;

:: BHSP_3:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
      b1 is Hilbert
   iff
      b1 is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR &
       b1 is complete;

:: BHSP_3:prednot 5 => BHSP_3:attr 4
notation
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  synonym a1 is_Hilbert_space for Hilbert;
end;