Article BHSP_2, MML version 4.99.1005

:: BHSP_2:attrnot 1 => BHSP_2: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 convergent means
    ex b1 being Element of the carrier of a1 st
       for b2 being Element of REAL
             st 0 < b2
          holds ex b3 being Element of NAT st
             for b4 being Element of NAT
                   st b3 <= b4
                holds dist(a2 . b4,b1) < b2;
end;

:: BHSP_2: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 convergent
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 Element of NAT st
             for b4 being Element of NAT
                   st b3 <= b4
                holds dist(a2 . b4,b1) < b2;

:: BHSP_2: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 convergent(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 Element of NAT st
               for b6 being Element of NAT
                     st b5 <= b6
                  holds dist(b2 . b6,b3) < b4;

:: BHSP_2: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 convergent(b1);

:: BHSP_2:th 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
      st b2 is convergent(b1) &
         (ex b4 being Element of NAT st
            for b5 being Element of NAT
                  st b4 <= b5
               holds b3 . b5 = b2 . b5)
   holds b3 is convergent(b1);

:: BHSP_2: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 convergent(b1) & b3 is convergent(b1)
   holds b2 + b3 is convergent(b1);

:: BHSP_2: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 convergent(b1) & b3 is convergent(b1)
   holds b2 - b3 is convergent(b1);

:: BHSP_2: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 convergent(b1)
   holds b2 * b3 is convergent(b1);

:: BHSP_2: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 convergent(b1)
   holds - b2 is convergent(b1);

:: BHSP_2: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 convergent(b1)
   holds b3 + b2 is convergent(b1);

:: BHSP_2: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 convergent(b1)
   holds b3 - b2 is convergent(b1);

:: BHSP_2: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 holds
      b2 is convergent(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 Element of NAT st
               for b6 being Element of NAT
                     st b5 <= b6
                  holds ||.(b2 . b6) - b3.|| < b4;

:: BHSP_2:funcnot 1 => BHSP_2: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 Relation of NAT,the carrier of a1;
  assume a2 is convergent(a1);
  func lim A2 -> Element of the carrier of a1 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,it) < b1;
end;

:: BHSP_2: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
   st b2 is convergent(b1)
for b3 being Element of the carrier of b1 holds
      b3 = lim b2
   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) < b4;

:: BHSP_2:th 10
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 constant & b2 in proj2 b3
   holds lim b3 = b2;

:: BHSP_2:th 11
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 constant &
         (ex b4 being Element of NAT st
            b3 . b4 = b2)
   holds lim b3 = b2;

:: BHSP_2: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 b2 is convergent(b1) &
         (ex b4 being Element of NAT st
            for b5 being Element of NAT
                  st b4 <= b5
               holds b3 . b5 = b2 . b5)
   holds lim b2 = lim b3;

:: BHSP_2: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
      st b2 is convergent(b1) & b3 is convergent(b1)
   holds lim (b2 + b3) = (lim b2) + lim b3;

:: BHSP_2: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 b2 is convergent(b1) & b3 is convergent(b1)
   holds lim (b2 - b3) = (lim b2) - lim b3;

:: BHSP_2:th 15
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 convergent(b1)
   holds lim (b2 * b3) = b2 * lim b3;

:: BHSP_2: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
      st b2 is convergent(b1)
   holds lim - b2 = - lim b2;

:: BHSP_2: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 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b3 is convergent(b1)
   holds lim (b3 + b2) = (lim b3) + b2;

:: BHSP_2:th 18
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 convergent(b1)
   holds lim (b3 - b2) = (lim b3) - b2;

:: BHSP_2:th 19
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 convergent(b1)
   holds    lim b3 = b2
   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 ||.(b3 . b6) - b2.|| < b4;

:: BHSP_2:funcnot 2 => BHSP_2: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 ||.A2.|| -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = ||.a2 . b1.||;
end;

:: BHSP_2: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
for b3 being Function-like quasi_total Relation of NAT,REAL holds
      b3 = ||.b2.||
   iff
      for b4 being Element of NAT holds
         b3 . b4 = ||.b2 . b4.||;

:: BHSP_2:th 20
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 convergent;

:: BHSP_2: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 the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b3 is convergent(b1) & lim b3 = b2
   holds ||.b3.|| is convergent & lim ||.b3.|| = ||.b2.||;

:: BHSP_2:th 22
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 convergent(b1) & lim b3 = b2
   holds ||.b3 - b2.|| is convergent & lim ||.b3 - b2.|| = 0;

:: BHSP_2:funcnot 3 => BHSP_2: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 the carrier of a1;
  func dist(A2,A3) -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = dist(a2 . b1,a3);
end;

:: BHSP_2: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 the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,REAL holds
      b4 = dist(b2,b3)
   iff
      for b5 being Element of NAT holds
         b4 . b5 = dist(b2 . b5,b3);

:: BHSP_2:th 23
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 convergent(b1) & lim b3 = b2
   holds dist(b3,b2) is convergent;

:: BHSP_2:th 24
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 convergent(b1) & lim b3 = b2
   holds dist(b3,b2) is convergent & lim dist(b3,b2) = 0;

:: BHSP_2:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4, b5 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b4 is convergent(b1) & lim b4 = b2 & b5 is convergent(b1) & lim b5 = b3
   holds ||.b4 + b5.|| is convergent &
    lim ||.b4 + b5.|| = ||.b2 + b3.||;

:: BHSP_2:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4, b5 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b4 is convergent(b1) & lim b4 = b2 & b5 is convergent(b1) & lim b5 = b3
   holds ||.(b4 + b5) - (b2 + b3).|| is convergent &
    lim ||.(b4 + b5) - (b2 + b3).|| = 0;

:: BHSP_2:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4, b5 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b4 is convergent(b1) & lim b4 = b2 & b5 is convergent(b1) & lim b5 = b3
   holds ||.b4 - b5.|| is convergent &
    lim ||.b4 - b5.|| = ||.b2 - b3.||;

:: BHSP_2:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4, b5 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b4 is convergent(b1) & lim b4 = b2 & b5 is convergent(b1) & lim b5 = b3
   holds ||.(b4 - b5) - (b2 - b3).|| is convergent &
    lim ||.(b4 - b5) - (b2 - b3).|| = 0;

:: BHSP_2:th 29
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 Element of REAL
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b4 is convergent(b1) & lim b4 = b2
   holds ||.b3 * b4.|| is convergent &
    lim ||.b3 * b4.|| = ||.b3 * b2.||;

:: BHSP_2:th 30
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 Element of REAL
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b4 is convergent(b1) & lim b4 = b2
   holds ||.(b3 * b4) - (b3 * b2).|| is convergent &
    lim ||.(b3 * b4) - (b3 * b2).|| = 0;

:: BHSP_2:th 31
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 convergent(b1) & lim b3 = b2
   holds ||.- b3.|| is convergent &
    lim ||.- b3.|| = ||.- b2.||;

:: BHSP_2:th 32
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 convergent(b1) & lim b3 = b2
   holds ||.(- b3) - - b2.|| is convergent &
    lim ||.(- b3) - - b2.|| = 0;

:: BHSP_2:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b4 is convergent(b1) & lim b4 = b2
   holds ||.(b4 + b3) - (b2 + b3).|| is convergent &
    lim ||.(b4 + b3) - (b2 + b3).|| = 0;

:: BHSP_2:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b4 is convergent(b1) & lim b4 = b2
   holds ||.b4 - b3.|| is convergent &
    lim ||.b4 - b3.|| = ||.b2 - b3.||;

:: BHSP_2:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b4 is convergent(b1) & lim b4 = b2
   holds ||.(b4 - b3) - (b2 - b3).|| is convergent &
    lim ||.(b4 - b3) - (b2 - b3).|| = 0;

:: BHSP_2:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4, b5 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b4 is convergent(b1) & lim b4 = b2 & b5 is convergent(b1) & lim b5 = b3
   holds dist(b4 + b5,b2 + b3) is convergent &
    lim dist(b4 + b5,b2 + b3) = 0;

:: BHSP_2:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4, b5 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b4 is convergent(b1) & lim b4 = b2 & b5 is convergent(b1) & lim b5 = b3
   holds dist(b4 - b5,b2 - b3) is convergent &
    lim dist(b4 - b5,b2 - b3) = 0;

:: BHSP_2:th 38
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 Element of REAL
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b4 is convergent(b1) & lim b4 = b2
   holds dist(b3 * b4,b3 * b2) is convergent &
    lim dist(b3 * b4,b3 * b2) = 0;

:: BHSP_2:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b4 is convergent(b1) & lim b4 = b2
   holds dist(b4 + b3,b2 + b3) is convergent &
    lim dist(b4 + b3,b2 + b3) = 0;

:: BHSP_2:funcnot 4 => BHSP_2:func 4
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2 be Element of the carrier of a1;
  let a3 be Element of REAL;
  func Ball(A2,A3) -> Element of bool the carrier of a1 equals
    {b1 where b1 is Element of the carrier of a1: ||.a2 - b1.|| < a3};
end;

:: BHSP_2: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 the carrier of b1
for b3 being Element of REAL holds
   Ball(b2,b3) = {b4 where b4 is Element of the carrier of b1: ||.b2 - b4.|| < b3};

:: BHSP_2:funcnot 5 => BHSP_2:func 5
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2 be Element of the carrier of a1;
  let a3 be Element of REAL;
  func cl_Ball(A2,A3) -> Element of bool the carrier of a1 equals
    {b1 where b1 is Element of the carrier of a1: ||.a2 - b1.|| <= a3};
end;

:: BHSP_2: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 the carrier of b1
for b3 being Element of REAL holds
   cl_Ball(b2,b3) = {b4 where b4 is Element of the carrier of b1: ||.b2 - b4.|| <= b3};

:: BHSP_2:funcnot 6 => BHSP_2:func 6
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2 be Element of the carrier of a1;
  let a3 be Element of REAL;
  func Sphere(A2,A3) -> Element of bool the carrier of a1 equals
    {b1 where b1 is Element of the carrier of a1: ||.a2 - b1.|| = a3};
end;

:: BHSP_2: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 the carrier of b1
for b3 being Element of REAL holds
   Sphere(b2,b3) = {b4 where b4 is Element of the carrier of b1: ||.b2 - b4.|| = b3};

:: BHSP_2:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL holds
      b2 in Ball(b3,b4)
   iff
      ||.b3 - b2.|| < b4;

:: BHSP_2:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL holds
      b2 in Ball(b3,b4)
   iff
      dist(b3,b2) < b4;

:: BHSP_2:th 42
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 Element of REAL
      st 0 < b3
   holds b2 in Ball(b2,b3);

:: BHSP_2:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Element of the carrier of b1
for b5 being Element of REAL
      st b2 in Ball(b3,b5) & b4 in Ball(b3,b5)
   holds dist(b2,b4) < 2 * b5;

:: BHSP_2:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Element of the carrier of b1
for b5 being Element of REAL
      st b2 in Ball(b3,b5)
   holds b2 - b4 in Ball(b3 - b4,b5);

:: BHSP_2:th 45
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL
      st b2 in Ball(b3,b4)
   holds b2 - b3 in Ball(0. b1,b4);

:: BHSP_2:th 46
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of REAL
      st b2 in Ball(b3,b4) & b4 <= b5
   holds b2 in Ball(b3,b5);

:: BHSP_2:th 47
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL holds
      b2 in cl_Ball(b3,b4)
   iff
      ||.b3 - b2.|| <= b4;

:: BHSP_2:th 48
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL holds
      b2 in cl_Ball(b3,b4)
   iff
      dist(b3,b2) <= b4;

:: BHSP_2:th 49
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 Element of REAL
      st 0 <= b3
   holds b2 in cl_Ball(b2,b3);

:: BHSP_2:th 50
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL
      st b2 in Ball(b3,b4)
   holds b2 in cl_Ball(b3,b4);

:: BHSP_2:th 51
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL holds
      b2 in Sphere(b3,b4)
   iff
      ||.b3 - b2.|| = b4;

:: BHSP_2:th 52
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL holds
      b2 in Sphere(b3,b4)
   iff
      dist(b3,b2) = b4;

:: BHSP_2:th 53
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL
      st b2 in Sphere(b3,b4)
   holds b2 in cl_Ball(b3,b4);

:: BHSP_2:th 54
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 Element of REAL holds
   Ball(b2,b3) c= cl_Ball(b2,b3);

:: BHSP_2:th 55
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 Element of REAL holds
   Sphere(b2,b3) c= cl_Ball(b2,b3);

:: BHSP_2:th 56
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 Element of REAL holds
   (Ball(b2,b3)) \/ Sphere(b2,b3) = cl_Ball(b2,b3);