Article LOPBAN_3, MML version 4.99.1005

:: LOPBAN_3:th 1
theorem
for b1 being non empty right_complementable add-associative right_zeroed NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
   st for b3 being Element of NAT holds
        b2 . b3 = 0. b1
for b3 being Element of NAT holds
   (Partial_Sums b2) . b3 = 0. b1;

:: LOPBAN_3:attrnot 1 => LOPBAN_3:attr 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  attr a2 is summable means
    Partial_Sums a2 is convergent(a1);
end;

:: LOPBAN_3:dfs 1
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
     a2 is summable
it is sufficient to prove
  thus Partial_Sums a2 is convergent(a1);

:: LOPBAN_3:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is summable(b1)
   iff
      Partial_Sums b2 is convergent(b1);

:: LOPBAN_3:exreg 1
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster Relation-like Function-like non empty total quasi_total summable Relation of NAT,the carrier of a1;
end;

:: LOPBAN_3:funcnot 1 => LOPBAN_3:func 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  func Sum A2 -> Element of the carrier of a1 equals
    lim Partial_Sums a2;
end;

:: LOPBAN_3:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   Sum b2 = lim Partial_Sums b2;

:: LOPBAN_3:attrnot 2 => LOPBAN_3:attr 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  attr a2 is norm_summable means
    ||.a2.|| is summable;
end;

:: LOPBAN_3:dfs 3
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
     a2 is norm_summable
it is sufficient to prove
  thus ||.a2.|| is summable;

:: LOPBAN_3:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is norm_summable(b1)
   iff
      ||.b2.|| is summable;

:: LOPBAN_3:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of NAT holds
   0 <= ||.b2.|| . b3;

:: LOPBAN_3:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3, b4 being Element of the carrier of b1 holds
||.b2 - b3.|| = ||.(b2 - b4) + (b4 - b3).||;

:: LOPBAN_3:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
   st b2 is convergent(b1)
for b3 being Element of REAL
      st 0 < b3
   holds ex b4 being Element of NAT st
      for b5 being Element of NAT
            st b4 <= b5
         holds ||.(b2 . b5) - (b2 . b4).|| < b3;

:: LOPBAN_3:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is CCauchy(b1)
   iff
      for b3 being Element of REAL
            st 0 < b3
         holds ex b4 being Element of NAT st
            for b5 being Element of NAT
                  st b4 <= b5
               holds ||.(b2 . b5) - (b2 . b4).|| < b3;

:: LOPBAN_3:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
   st for b3 being Element of NAT holds
        b2 . b3 = 0. b1
for b3 being Element of NAT holds
   (Partial_Sums ||.b2.||) . b3 = 0;

:: LOPBAN_3:attrnot 3 => LOPBAN_3:attr 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  redefine attr a2 is constant means
    ex b1 being Element of the carrier of a1 st
       for b2 being Element of NAT holds
          a2 . b2 = b1;
end;

:: LOPBAN_3:dfs 4
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
     a1 is constant
it is sufficient to prove
  thus ex b1 being Element of the carrier of a1 st
       for b2 being Element of NAT holds
          a2 . b2 = b1;

:: LOPBAN_3:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is constant
   iff
      ex b3 being Element of the carrier of b1 st
         for b4 being Element of NAT holds
            b2 . b4 = b3;

:: LOPBAN_3:funcnot 2 => LOPBAN_3:func 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  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;

:: LOPBAN_3:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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);

:: LOPBAN_3:funcnot 3 => LOPBAN_3:func 3
definition
  let a1 be non empty 1-sorted;
  let a2 be Function-like natural-valued quasi_total 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;

:: LOPBAN_3:funcnot 4 => LOPBAN_3:func 4
definition
  let a1 be non empty 1-sorted;
  let a2 be Function-like natural-valued quasi_total 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;

:: LOPBAN_3:th 7
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like natural-valued quasi_total increasing Relation of NAT,REAL
for b4 being Element of NAT holds
   (b2 * b3) . b4 = b2 . (b3 . b4);

:: LOPBAN_3:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   b2 ^\ 0 = b2;

:: LOPBAN_3:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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;

:: LOPBAN_3:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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);

:: LOPBAN_3:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b3 is subsequence of b2 & b2 is convergent(b1)
   holds b3 is convergent(b1);

:: LOPBAN_3:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b3 is subsequence of b2 & b2 is convergent(b1)
   holds lim b3 = lim b2;

:: LOPBAN_3:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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;

:: LOPBAN_3:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
for b4 being Element of NAT
      st b2 is convergent(b1)
   holds b2 ^\ b4 is convergent(b1) & lim (b2 ^\ b4) = lim b2;

:: LOPBAN_3:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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);

:: LOPBAN_3:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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 lim b3 = lim b2;

:: LOPBAN_3:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is constant
   holds b2 is convergent(b1);

:: LOPBAN_3:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st for b3 being Element of NAT holds
           b2 . b3 = 0. b1
   holds b2 is norm_summable(b1);

:: LOPBAN_3:exreg 2
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster Relation-like Function-like non empty total quasi_total norm_summable Relation of NAT,the carrier of a1;
end;

:: LOPBAN_3:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is summable(b1)
   holds b2 is convergent(b1) & lim b2 = 0. b1;

:: LOPBAN_3:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
(Partial_Sums b2) + Partial_Sums b3 = Partial_Sums (b2 + b3);

:: LOPBAN_3:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
(Partial_Sums b2) - Partial_Sums b3 = Partial_Sums (b2 - b3);

:: LOPBAN_3:funcreg 1
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like quasi_total norm_summable Relation of NAT,the carrier of a1;
  cluster ||.a2.|| -> Function-like quasi_total summable;
end;

:: LOPBAN_3:condreg 1
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster Function-like quasi_total summable -> convergent (Relation of NAT,the carrier of a1);
end;

:: LOPBAN_3:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is summable(b1) & b3 is summable(b1)
   holds b2 + b3 is summable(b1) &
    Sum (b2 + b3) = (Sum b2) + Sum b3;

:: LOPBAN_3:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is summable(b1) & b3 is summable(b1)
   holds b2 - b3 is summable(b1) &
    Sum (b2 - b3) = (Sum b2) - Sum b3;

:: LOPBAN_3:funcreg 2
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2, a3 be Function-like quasi_total summable Relation of NAT,the carrier of a1;
  cluster a2 + a3 -> Function-like quasi_total summable;
end;

:: LOPBAN_3:funcreg 3
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2, a3 be Function-like quasi_total summable Relation of NAT,the carrier of a1;
  cluster a2 - a3 -> Function-like quasi_total summable;
end;

:: LOPBAN_3:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of REAL holds
   Partial_Sums (b3 * b2) = b3 * Partial_Sums b2;

:: LOPBAN_3:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total summable Relation of NAT,the carrier of b1
for b3 being Element of REAL holds
   b3 * b2 is summable(b1) & Sum (b3 * b2) = b3 * Sum b2;

:: LOPBAN_3:funcreg 4
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Element of REAL;
  let a3 be Function-like quasi_total summable Relation of NAT,the carrier of a1;
  cluster a2 * a3 -> Function-like quasi_total summable;
end;

:: LOPBAN_3:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st for b4 being Element of NAT holds
           b3 . b4 = b2 . 0
   holds Partial_Sums (b2 ^\ 1) = ((Partial_Sums b2) ^\ 1) - b3;

:: LOPBAN_3:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
   st b2 is summable(b1)
for b3 being Element of NAT holds
   b2 ^\ b3 is summable(b1);

:: LOPBAN_3:funcreg 5
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like quasi_total summable Relation of NAT,the carrier of a1;
  let a3 be Element of NAT;
  cluster a2 ^\ a3 -> Function-like quasi_total summable;
end;

:: LOPBAN_3:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      Partial_Sums ||.b2.|| is bounded_above
   iff
      b2 is norm_summable(b1);

:: LOPBAN_3:funcreg 6
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like quasi_total norm_summable Relation of NAT,the carrier of a1;
  cluster Partial_Sums ||.a2.|| -> Function-like quasi_total bounded_above;
end;

:: LOPBAN_3:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is summable(b1)
   iff
      for b3 being Element of REAL
            st 0 < b3
         holds ex b4 being Element of NAT st
            for b5 being Element of NAT
                  st b4 <= b5
               holds ||.((Partial_Sums b2) . b5) - ((Partial_Sums b2) . b4).|| < b3;

:: LOPBAN_3:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3, b4 being Element of NAT
      st b3 <= b4
   holds ||.((Partial_Sums b2) . b4) - ((Partial_Sums b2) . b3).|| <= abs (((Partial_Sums ||.b2.||) . b4) - ((Partial_Sums ||.b2.||) . b3));

:: LOPBAN_3:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is norm_summable(b1)
   holds b2 is summable(b1);

:: LOPBAN_3:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is summable &
         (ex b4 being Element of NAT st
            for b5 being Element of NAT
                  st b4 <= b5
               holds ||.b3 . b5.|| <= b2 . b5)
   holds b3 is norm_summable(b1);

:: LOPBAN_3:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st (for b4 being Element of NAT holds
            0 <= ||.b2.|| . b4 &
             ||.b2.|| . b4 <= ||.b3.|| . b4) &
         b3 is norm_summable(b1)
   holds b2 is norm_summable(b1) &
    Sum ||.b2.|| <= Sum ||.b3.||;

:: LOPBAN_3:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st (for b3 being Element of NAT holds
            0 < ||.b2.|| . b3) &
         (ex b3 being Element of NAT st
            for b4 being Element of NAT
                  st b3 <= b4
               holds 1 <= (||.b2.|| . (b4 + 1)) / (||.b2.|| . b4))
   holds b2 is not norm_summable(b1);

:: LOPBAN_3:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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
      st (for b4 being Element of NAT holds
            b3 . b4 = b4 -root (||.b2.|| . b4)) &
         b3 is convergent &
         lim b3 < 1
   holds b2 is norm_summable(b1);

:: LOPBAN_3:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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
      st (for b4 being Element of NAT holds
            b3 . b4 = b4 -root (||.b2.|| . b4)) &
         (ex b4 being Element of NAT st
            for b5 being Element of NAT
                  st b4 <= b5
               holds 1 <= b3 . b5)
   holds ||.b2.|| is not summable;

:: LOPBAN_3:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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
      st (for b4 being Element of NAT holds
            b3 . b4 = b4 -root (||.b2.|| . b4)) &
         b3 is convergent &
         1 < lim b3
   holds b2 is not norm_summable(b1);

:: LOPBAN_3:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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
      st ||.b2.|| is non-increasing &
         (for b4 being Element of NAT holds
            b3 . b4 = (2 to_power b4) * (||.b2.|| . (2 to_power b4)))
   holds    b2 is norm_summable(b1)
   iff
      b3 is summable;

:: LOPBAN_3:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of REAL
      st 1 < b3 &
         (for b4 being Element of NAT
               st 1 <= b4
            holds ||.b2.|| . b4 = 1 / (b4 to_power b3))
   holds b2 is norm_summable(b1);

:: LOPBAN_3:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of REAL
      st b3 <= 1 &
         (for b4 being Element of NAT
               st 1 <= b4
            holds ||.b2.|| . b4 = 1 / (b4 to_power b3))
   holds b2 is not norm_summable(b1);

:: LOPBAN_3:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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
      st (for b4 being Element of NAT holds
            b2 . b4 <> 0. b1 &
             b3 . b4 = (||.b2.|| . (b4 + 1)) / (||.b2.|| . b4)) &
         b3 is convergent &
         lim b3 < 1
   holds b2 is norm_summable(b1);

:: LOPBAN_3:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st (for b3 being Element of NAT holds
            b2 . b3 <> 0. b1) &
         (ex b3 being Element of NAT st
            for b4 being Element of NAT
                  st b3 <= b4
               holds 1 <= (||.b2.|| . (b4 + 1)) / (||.b2.|| . b4))
   holds b2 is not norm_summable(b1);

:: LOPBAN_3:condreg 2
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR;
  cluster Function-like quasi_total norm_summable -> summable (Relation of NAT,the carrier of a1);
end;

:: LOPBAN_3:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like Algebra-like associative right-distributive right_unital Banach_Algebra-like Normed_AlgebraStr
for b2, b3, b4 being Element of the carrier of b1
for b5, b6 being Element of REAL holds
b2 + b3 = b3 + b2 &
 (b2 + b3) + b4 = b2 + (b3 + b4) &
 b2 + 0. b1 = b2 &
 (ex b7 being Element of the carrier of b1 st
    b2 + b7 = 0. b1) &
 (b2 * b3) * b4 = b2 * (b3 * b4) &
 1 * b2 = b2 &
 0 * b2 = 0. b1 &
 b5 * 0. b1 = 0. b1 &
 (- 1) * b2 = - b2 &
 b2 * 1. b1 = b2 &
 (1. b1) * b2 = b2 &
 b2 * (b3 + b4) = (b2 * b3) + (b2 * b4) &
 (b3 + b4) * b2 = (b3 * b2) + (b4 * b2) &
 b5 * (b2 * b3) = (b5 * b2) * b3 &
 b5 * (b2 + b3) = (b5 * b2) + (b5 * b3) &
 (b5 + b6) * b2 = (b5 * b2) + (b6 * b2) &
 (b5 * b6) * b2 = b5 * (b6 * b2) &
 (b5 * b6) * (b2 * b3) = (b5 * b2) * (b6 * b3) &
 b5 * (b2 * b3) = b2 * (b5 * b3) &
 (0. b1) * b2 = 0. b1 &
 b2 * 0. b1 = 0. b1 &
 b2 * (b3 - b4) = (b2 * b3) - (b2 * b4) &
 (b3 - b4) * b2 = (b3 * b2) - (b4 * b2) &
 (b2 + b3) - b4 = b2 + (b3 - b4) &
 (b2 - b3) + b4 = b2 - (b3 - b4) &
 (b2 - b3) - b4 = b2 - (b3 + b4) &
 b2 + b3 = (b2 - b4) + (b4 + b3) &
 b2 - b3 = (b2 - b4) + (b4 - b3) &
 b2 = (b2 - b3) + b3 &
 b2 = b3 - (b3 - b2) &
 (||.b2.|| = 0 implies b2 = 0. b1) &
 (b2 = 0. b1 implies ||.b2.|| = 0) &
 ||.b5 * b2.|| = (abs b5) * ||.b2.|| &
 ||.b2 + b3.|| <= ||.b2.|| + ||.b3.|| &
 ||.b2 * b3.|| <= ||.b2.|| * ||.b3.|| &
 ||.1. b1.|| = 1 &
 b1 is complete;

:: LOPBAN_3:condreg 3
registration
  cluster non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like Algebra-like associative right-distributive right_unital Banach_Algebra-like -> well-unital (Normed_AlgebraStr);
end;

:: LOPBAN_3:attrnot 4 => ALGSTR_0:attr 25
definition
  let a1 be multLoopStr;
  let a2 be Element of the carrier of a1;
  attr a2 is invertible means
    ex b1 being Element of the carrier of a1 st
       a2 * b1 = 1. a1 & b1 * a2 = 1. a1;
end;

:: LOPBAN_3:dfs 6
definiens
  let a1 be non empty associative well-unital multLoopStr;
  let a2 be Element of the carrier of a1;
To prove
     a2 is invertible
it is sufficient to prove
  thus ex b1 being Element of the carrier of a1 st
       a2 * b1 = 1. a1 & b1 * a2 = 1. a1;

:: LOPBAN_3:def 8
theorem
for b1 being non empty associative well-unital multLoopStr
for b2 being Element of the carrier of b1 holds
      b2 is invertible(b1)
   iff
      ex b3 being Element of the carrier of b1 st
         b2 * b3 = 1. b1 & b3 * b2 = 1. b1;

:: LOPBAN_3:funcnot 5 => LOPBAN_3:func 5
definition
  let a1 be non empty Normed_AlgebraStr;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  let a3 be Element of the carrier of a1;
  func A3 * A2 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
    for b1 being Element of NAT holds
       it . b1 = a3 * (a2 . b1);
end;

:: LOPBAN_3:def 9
theorem
for b1 being non empty Normed_AlgebraStr
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,the carrier of b1 holds
      b4 = b3 * b2
   iff
      for b5 being Element of NAT holds
         b4 . b5 = b3 * (b2 . b5);

:: LOPBAN_3:funcnot 6 => LOPBAN_3:func 6
definition
  let a1 be non empty Normed_AlgebraStr;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  let a3 be Element of the carrier of a1;
  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;

:: LOPBAN_3:def 10
theorem
for b1 being non empty Normed_AlgebraStr
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,the carrier of b1 holds
      b4 = b2 * b3
   iff
      for b5 being Element of NAT holds
         b4 . b5 = (b2 . b5) * b3;

:: LOPBAN_3:funcnot 7 => LOPBAN_3:func 7
definition
  let a1 be non empty Normed_AlgebraStr;
  let a2, a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
  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 . b1);
end;

:: LOPBAN_3:def 11
theorem
for b1 being non empty Normed_AlgebraStr
for b2, b3, 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 . b5);

:: LOPBAN_3:funcnot 8 => ALGSTR_0:func 9
notation
  let a1 be non empty associative well-unital multLoopStr;
  let a2 be Element of the carrier of a1;
  synonym a2 " for / a2;
end;

:: LOPBAN_3:funcnot 9 => ALGSTR_0:func 9
definition
  let a1 be multLoopStr;
  let a2 be Element of the carrier of a1;
  assume a2 is invertible(a1);
  func A2 " -> Element of the carrier of a1 means
    a2 * it = 1. a1 & it * a2 = 1. a1;
end;

:: LOPBAN_3:def 12
theorem
for b1 being non empty associative well-unital multLoopStr
for b2 being Element of the carrier of b1
   st b2 is invertible(b1)
for b3 being Element of the carrier of b1 holds
      b3 = / b2
   iff
      b2 * b3 = 1. b1 & b3 * b2 = 1. b1;

:: LOPBAN_3:funcnot 10 => LOPBAN_3:func 8
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like Algebra-like associative right-distributive right_unital Banach_Algebra-like Normed_AlgebraStr;
  let a2 be Element of the carrier of a1;
  func A2 GeoSeq -> Function-like quasi_total Relation of NAT,the carrier of a1 means
    it . 0 = 1. a1 &
     (for b1 being Element of NAT holds
        it . (b1 + 1) = (it . b1) * a2);
end;

:: LOPBAN_3:def 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like Algebra-like associative right-distributive right_unital Banach_Algebra-like Normed_AlgebraStr
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b3 = b2 GeoSeq
   iff
      b3 . 0 = 1. b1 &
       (for b4 being Element of NAT holds
          b3 . (b4 + 1) = (b3 . b4) * b2);

:: LOPBAN_3:funcnot 11 => LOPBAN_3:func 9
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like Algebra-like associative right-distributive right_unital Banach_Algebra-like Normed_AlgebraStr;
  let a2 be Element of the carrier of a1;
  let a3 be Element of NAT;
  func A2 #N A3 -> Element of the carrier of a1 equals
    a2 GeoSeq . a3;
end;

:: LOPBAN_3:def 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like Algebra-like associative right-distributive right_unital Banach_Algebra-like Normed_AlgebraStr
for b2 being Element of the carrier of b1
for b3 being Element of NAT holds
   b2 #N b3 = b2 GeoSeq . b3;

:: LOPBAN_3:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like Algebra-like associative right-distributive right_unital Banach_Algebra-like Normed_AlgebraStr
for b2 being Element of the carrier of b1 holds
   b2 #N 0 = 1. b1;

:: LOPBAN_3:th 45
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like Algebra-like associative right-distributive right_unital Banach_Algebra-like Normed_AlgebraStr
for b2 being Element of the carrier of b1
      st ||.b2.|| < 1
   holds b2 GeoSeq is summable(b1) & b2 GeoSeq is norm_summable(b1);

:: LOPBAN_3:th 46
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like Algebra-like associative right-distributive right_unital Banach_Algebra-like Normed_AlgebraStr
for b2 being Element of the carrier of b1
      st ||.(1. b1) - b2.|| < 1
   holds ((1. b1) - b2) GeoSeq is summable(b1) & ((1. b1) - b2) GeoSeq is norm_summable(b1);

:: LOPBAN_3:th 47
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like Algebra-like associative right-distributive right_unital Banach_Algebra-like Normed_AlgebraStr
for b2 being Element of the carrier of b1
      st ||.(1. b1) - b2.|| < 1
   holds b2 is invertible(b1) &
    / b2 = Sum (((1. b1) - b2) GeoSeq);