Article CLVECT_2, MML version 4.99.1005

:: CLVECT_2:attrnot 1 => CLVECT_2:attr 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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;

:: CLVECT_2:dfs 1
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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;

:: CLVECT_2:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is constant
   holds b2 is convergent(b1);

:: CLVECT_2:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of COMPLEX
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);

:: CLVECT_2:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is convergent(b1)
   holds - b2 is convergent(b1);

:: CLVECT_2:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:funcnot 1 => CLVECT_2:func 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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;

:: CLVECT_2:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of COMPLEX
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;

:: CLVECT_2:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is convergent(b1)
   holds lim - b2 = - lim b2;

:: CLVECT_2:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:funcnot 2 => CLVECT_2:func 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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;

:: CLVECT_2:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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.||;

:: CLVECT_2:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is convergent(b1)
   holds ||.b2.|| is convergent;

:: CLVECT_2:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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.||;

:: CLVECT_2:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:funcnot 3 => CLVECT_2:func 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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;

:: CLVECT_2:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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.||;

:: CLVECT_2:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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.||;

:: CLVECT_2:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX
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.||;

:: CLVECT_2:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX
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;

:: CLVECT_2:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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.||;

:: CLVECT_2:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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.||;

:: CLVECT_2:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX
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;

:: CLVECT_2:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:funcnot 4 => CLVECT_2:func 4
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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;

:: CLVECT_2:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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};

:: CLVECT_2:funcnot 5 => CLVECT_2:func 5
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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;

:: CLVECT_2:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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};

:: CLVECT_2:funcnot 6 => CLVECT_2:func 6
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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;

:: CLVECT_2:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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};

:: CLVECT_2:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1
for b3 being Element of REAL
      st 0 < b3
   holds b2 in Ball(b2,b3);

:: CLVECT_2:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 45
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 46
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 47
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 48
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 49
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 50
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 51
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 52
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 53
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 54
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1
for b3 being Element of REAL holds
   Ball(b2,b3) c= cl_Ball(b2,b3);

:: CLVECT_2:th 55
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1
for b3 being Element of REAL holds
   Sphere(b2,b3) c= cl_Ball(b2,b3);

:: CLVECT_2:th 56
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:attrnot 2 => CLVECT_2:attr 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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;

:: CLVECT_2:dfs 8
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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;

:: CLVECT_2:def 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 57
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is constant
   holds b2 is Cauchy(b1);

:: CLVECT_2:th 58
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 59
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 60
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 61
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of COMPLEX
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);

:: CLVECT_2:th 62
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is Cauchy(b1)
   holds - b2 is Cauchy(b1);

:: CLVECT_2:th 63
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 64
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 65
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is convergent(b1)
   holds b2 is Cauchy(b1);

:: CLVECT_2:prednot 1 => CLVECT_2:pred 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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;

:: CLVECT_2:dfs 9
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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;

:: CLVECT_2:def 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 66
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   b2 is_compared_to b2;

:: CLVECT_2:th 67
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:prednot 2 => CLVECT_2:pred 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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 ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
::  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 ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
::  for a2 being Function-like quasi_total Relation of NAT,the carrier of a1 holds
::     a2 is_compared_to a2;
end;

:: CLVECT_2:th 68
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 69
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 70
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 71
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 72
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 73
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:attrnot 3 => CLVECT_2:attr 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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;

:: CLVECT_2:dfs 10
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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);

:: CLVECT_2:def 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 74
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 75
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is bounded(b1)
   holds - b2 is bounded(b1);

:: CLVECT_2:th 76
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 77
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of COMPLEX
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);

:: CLVECT_2:th 78
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is constant
   holds b2 is bounded(b1);

:: CLVECT_2:th 79
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 80
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is convergent(b1)
   holds b2 is bounded(b1);

:: CLVECT_2:th 81
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:funcnot 7 => CLVECT_2:func 7
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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;

:: CLVECT_2:th 82
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 83
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   b2 is subsequence of b2;

:: CLVECT_2:th 84
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 85
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 86
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 87
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 88
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 89
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 90
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:funcnot 8 => CLVECT_2:func 8
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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;

:: CLVECT_2:def 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 91
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   b2 ^\ 0 = b2;

:: CLVECT_2:th 92
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 93
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 94
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 95
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 96
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 97
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of COMPLEX
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);

:: CLVECT_2:th 98
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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 natural-valued increasing Relation of NAT,REAL holds
   (b2 * b4) ^\ b3 = b2 * (b4 ^\ b3);

:: CLVECT_2:th 99
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 100
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 101
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 102
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 103
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 104
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:th 105
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:th 106
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CLVECT_2:attrnot 4 => CLVECT_2:attr 4
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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;

:: CLVECT_2:dfs 12
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
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);

:: CLVECT_2:def 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR 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);

:: CLVECT_2:th 107
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CLVECT_2:attrnot 5 => CLVECT_2:attr 5
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  attr a1 is Hilbert means
    a1 is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR &
     a1 is complete;
end;

:: CLVECT_2:dfs 13
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
To prove
     a1 is Hilbert
it is sufficient to prove
  thus a1 is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR &
     a1 is complete;

:: CLVECT_2:def 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR holds
      b1 is Hilbert
   iff
      b1 is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR &
       b1 is complete;