Article CLOPBAN4, MML version 4.99.1005

:: CLOPBAN4:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is convergent(b1) & b3 is convergent(b1) & lim (b2 - b3) = 0. b1
   holds lim b2 = lim b3;

:: CLOPBAN4:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1
      st for b4 being Element of NAT holds
           b2 . b4 = b3
   holds lim b2 = b3;

:: CLOPBAN4:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
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);

:: CLOPBAN4:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
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 b2 * b3 is convergent(b1);

:: CLOPBAN4:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
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);

:: CLOPBAN4:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
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 (b2 * b3) = b2 * lim b3;

:: CLOPBAN4:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
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;

:: CLOPBAN4:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
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;

:: CLOPBAN4:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_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
   Partial_Sums (b2 * b3) = b2 * Partial_Sums b3 &
    Partial_Sums (b3 * b2) = (Partial_Sums b3) * b2;

:: CLOPBAN4:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of NAT
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   ||.(Partial_Sums b3) . b2.|| <= (Partial_Sums ||.b3.||) . b2;

:: CLOPBAN4:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of NAT
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1
      st for b5 being Element of NAT
              st b5 <= b2
           holds b3 . b5 = b4 . b5
   holds (Partial_Sums b3) . b2 = (Partial_Sums b4) . b2;

:: CLOPBAN4:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
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.|| <= b3 . b4) &
         b3 is convergent &
         lim b3 = 0
   holds b2 is convergent(b1) & lim b2 = 0. b1;

:: CLOPBAN4:funcnot 1 => CLOPBAN4:func 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr;
  let a2 be Element of the carrier of a1;
  func A2 ExpSeq -> Function-like quasi_total Relation of NAT,the carrier of a1 means
    for b1 being Element of NAT holds
       it . b1 = (1r / (b1 !c)) * (a2 #N b1);
end;

:: CLOPBAN4:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_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 ExpSeq
   iff
      for b4 being Element of NAT holds
         b3 . b4 = (1r / (b4 !c)) * (b2 #N b4);

:: CLOPBAN4:sch 1
scheme CLOPBAN4:sch 1
{F1 -> non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr,
  F2 -> Element of the carrier of F1()}:
for b1 being Element of NAT holds
   ex b2 being Function-like quasi_total Relation of NAT,the carrier of F1() st
      for b3 being Element of NAT holds
         (b3 <= b1 implies b2 . b3 = F2(b1, b3)) & (b3 <= b1 or b2 . b3 = 0. F1())


:: CLOPBAN4:funcnot 2 => CLOPBAN4:func 2
definition
  let a1 be Element of NAT;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr;
  let a3, a4 be Element of the carrier of a2;
  func Expan(A1,A3,A4) -> Function-like quasi_total Relation of NAT,the carrier of a2 means
    for b1 being Element of NAT holds
       (b1 <= a1 implies it . b1 = (((Coef a1) . b1) * (a3 #N b1)) * (a4 #N (a1 -' b1))) &
        (b1 <= a1 or it . b1 = 0. a2);
end;

:: CLOPBAN4:def 4
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b3, b4 being Element of the carrier of b2
for b5 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
      b5 = Expan(b1,b3,b4)
   iff
      for b6 being Element of NAT holds
         (b6 <= b1 implies b5 . b6 = (((Coef b1) . b6) * (b3 #N b6)) * (b4 #N (b1 -' b6))) &
          (b6 <= b1 or b5 . b6 = 0. b2);

:: CLOPBAN4:funcnot 3 => CLOPBAN4:func 3
definition
  let a1 be Element of NAT;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr;
  let a3, a4 be Element of the carrier of a2;
  func Expan_e(A1,A3,A4) -> Function-like quasi_total Relation of NAT,the carrier of a2 means
    for b1 being Element of NAT holds
       (b1 <= a1 implies it . b1 = (((Coef_e a1) . b1) * (a3 #N b1)) * (a4 #N (a1 -' b1))) &
        (b1 <= a1 or it . b1 = 0. a2);
end;

:: CLOPBAN4:def 5
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b3, b4 being Element of the carrier of b2
for b5 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
      b5 = Expan_e(b1,b3,b4)
   iff
      for b6 being Element of NAT holds
         (b6 <= b1 implies b5 . b6 = (((Coef_e b1) . b6) * (b3 #N b6)) * (b4 #N (b1 -' b6))) &
          (b6 <= b1 or b5 . b6 = 0. b2);

:: CLOPBAN4:funcnot 4 => CLOPBAN4:func 4
definition
  let a1 be Element of NAT;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr;
  let a3, a4 be Element of the carrier of a2;
  func Alfa(A1,A3,A4) -> Function-like quasi_total Relation of NAT,the carrier of a2 means
    for b1 being Element of NAT holds
       (b1 <= a1 implies it . b1 = (a3 ExpSeq . b1) * ((Partial_Sums (a4 ExpSeq)) . (a1 -' b1))) &
        (b1 <= a1 or it . b1 = 0. a2);
end;

:: CLOPBAN4:def 6
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b3, b4 being Element of the carrier of b2
for b5 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
      b5 = Alfa(b1,b3,b4)
   iff
      for b6 being Element of NAT holds
         (b6 <= b1 implies b5 . b6 = (b3 ExpSeq . b6) * ((Partial_Sums (b4 ExpSeq)) . (b1 -' b6))) &
          (b6 <= b1 or b5 . b6 = 0. b2);

:: CLOPBAN4:funcnot 5 => CLOPBAN4:func 5
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr;
  let a2, a3 be Element of the carrier of a1;
  let a4 be Element of NAT;
  func Conj(A4,A2,A3) -> Function-like quasi_total Relation of NAT,the carrier of a1 means
    for b1 being Element of NAT holds
       (b1 <= a4 implies it . b1 = (a2 ExpSeq . b1) * (((Partial_Sums (a3 ExpSeq)) . a4) - ((Partial_Sums (a3 ExpSeq)) . (a4 -' b1)))) &
        (b1 <= a4 or it . b1 = 0. a1);
end;

:: CLOPBAN4:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2, b3 being Element of the carrier of b1
for b4 being Element of NAT
for b5 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b5 = Conj(b4,b2,b3)
   iff
      for b6 being Element of NAT holds
         (b6 <= b4 implies b5 . b6 = (b2 ExpSeq . b6) * (((Partial_Sums (b3 ExpSeq)) . b4) - ((Partial_Sums (b3 ExpSeq)) . (b4 -' b6)))) &
          (b6 <= b4 or b5 . b6 = 0. b1);

:: CLOPBAN4:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of the carrier of b1
for b3 being Element of NAT holds
   b2 ExpSeq . (b3 + 1) = ((1r / [*b3 + 1,0*]) * b2) * (b2 ExpSeq . b3) &
    b2 ExpSeq . 0 = 1. b1 &
    ||.b2 ExpSeq . b3.|| <= ||.b2.|| ExpSeq . b3;

:: CLOPBAN4:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of NAT
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st 0 < b2
   holds (Shift b3) . b2 = b3 . (b2 -' 1);

:: CLOPBAN4:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of NAT
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   (Partial_Sums b3) . b2 = ((Partial_Sums Shift b3) . b2) + (b3 . b2);

:: CLOPBAN4:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of NAT
for b3, b4 being Element of the carrier of b1
      st b3,b4 are_commutative
   holds (b3 + b4) #N b2 = (Partial_Sums Expan(b2,b3,b4)) . b2;

:: CLOPBAN4:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2, b3 being Element of the carrier of b1
for b4 being Element of NAT holds
   Expan_e(b4,b2,b3) = (1r / (b4 !c)) * Expan(b4,b2,b3);

:: CLOPBAN4:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of NAT
for b3, b4 being Element of the carrier of b1
      st b3,b4 are_commutative
   holds (1r / (b2 !c)) * ((b3 + b4) #N b2) = (Partial_Sums Expan_e(b2,b3,b4)) . b2;

:: CLOPBAN4:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr holds
   (0. b1) ExpSeq is norm_summable(b1) & Sum ((0. b1) ExpSeq) = 1. b1;

:: CLOPBAN4:funcreg 1
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr;
  let a2 be Element of the carrier of a1;
  cluster a2 ExpSeq -> Function-like quasi_total norm_summable;
end;

:: CLOPBAN4:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2, b3 being Element of the carrier of b1 holds
b2 ExpSeq . 0 = 1. b1 &
 (Expan(0,b2,b3)) . 0 = 1. b1;

:: CLOPBAN4:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of NAT
      st b4 <= b5
   holds (Alfa(b5 + 1,b2,b3)) . b4 = ((Alfa(b5,b2,b3)) . b4) + ((Expan_e(b5 + 1,b2,b3)) . b4);

:: CLOPBAN4:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2, b3 being Element of the carrier of b1
for b4 being Element of NAT holds
   (Partial_Sums Alfa(b4 + 1,b2,b3)) . b4 = ((Partial_Sums Alfa(b4,b2,b3)) . b4) + ((Partial_Sums Expan_e(b4 + 1,b2,b3)) . b4);

:: CLOPBAN4:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2, b3 being Element of the carrier of b1
for b4 being Element of NAT holds
   b2 ExpSeq . b4 = (Expan_e(b4,b2,b3)) . b4;

:: CLOPBAN4:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of NAT
for b3, b4 being Element of the carrier of b1
      st b3,b4 are_commutative
   holds (Partial_Sums ((b3 + b4) ExpSeq)) . b2 = (Partial_Sums Alfa(b2,b3,b4)) . b2;

:: CLOPBAN4:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of NAT
for b3, b4 being Element of the carrier of b1
      st b3,b4 are_commutative
   holds (((Partial_Sums (b3 ExpSeq)) . b2) * ((Partial_Sums (b4 ExpSeq)) . b2)) - ((Partial_Sums ((b3 + b4) ExpSeq)) . b2) = (Partial_Sums Conj(b2,b3,b4)) . b2;

:: CLOPBAN4:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of the carrier of b1
for b3 being Element of NAT holds
   0 <= ||.b2.|| ExpSeq . b3;

:: CLOPBAN4:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of the carrier of b1
for b3 being Element of NAT holds
   ||.(Partial_Sums (b2 ExpSeq)) . b3.|| <= (Partial_Sums (||.b2.|| ExpSeq)) . b3 &
    (Partial_Sums (||.b2.|| ExpSeq)) . b3 <= Sum (||.b2.|| ExpSeq) &
    ||.(Partial_Sums (b2 ExpSeq)) . b3.|| <= Sum (||.b2.|| ExpSeq);

:: CLOPBAN4:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of the carrier of b1 holds
   1 <= Sum (||.b2.|| ExpSeq);

:: CLOPBAN4:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of the carrier of b1
for b3, b4 being Element of NAT holds
abs ((Partial_Sums (||.b2.|| ExpSeq)) . b3) = (Partial_Sums (||.b2.|| ExpSeq)) . b3 &
 (b3 <= b4 implies abs (((Partial_Sums (||.b2.|| ExpSeq)) . b4) - ((Partial_Sums (||.b2.|| ExpSeq)) . b3)) = ((Partial_Sums (||.b2.|| ExpSeq)) . b4) - ((Partial_Sums (||.b2.|| ExpSeq)) . b3));

:: CLOPBAN4:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of NAT holds
abs ((Partial_Sums ||.Conj(b4,b2,b3).||) . b5) = (Partial_Sums ||.Conj(b4,b2,b3).||) . b5;

:: CLOPBAN4:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2, b3 being Element of the carrier of b1
for b4 being real set
      st 0 < b4
   holds ex b5 being Element of NAT st
      for b6 being Element of NAT
            st b5 <= b6
         holds abs ((Partial_Sums ||.Conj(b6,b2,b3).||) . b6) < b4;

:: CLOPBAN4:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
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 for b5 being Element of NAT holds
           b4 . b5 = (Partial_Sums Conj(b5,b2,b3)) . b5
   holds b4 is convergent(b1) & lim b4 = 0. b1;

:: CLOPBAN4:funcnot 6 => CLOPBAN4:func 6
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr;
  func exp_ A1 -> Function-like quasi_total Relation of the carrier of a1,the carrier of a1 means
    for b1 being Element of the carrier of a1 holds
       it . b1 = Sum (b1 ExpSeq);
end;

:: CLOPBAN4:def 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
      b2 = exp_ b1
   iff
      for b3 being Element of the carrier of b1 holds
         b2 . b3 = Sum (b3 ExpSeq);

:: CLOPBAN4:funcnot 7 => CLOPBAN4:func 7
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr;
  let a2 be Element of the carrier of a1;
  func exp A2 -> Element of the carrier of a1 equals
    (exp_ a1) . a2;
end;

:: CLOPBAN4:def 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of the carrier of b1 holds
   exp b2 = (exp_ b1) . b2;

:: CLOPBAN4:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of the carrier of b1 holds
   exp b2 = Sum (b2 ExpSeq);

:: CLOPBAN4:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2, b3 being Element of the carrier of b1
      st b2,b3 are_commutative
   holds exp (b2 + b3) = (exp b2) * exp b3 &
    exp (b3 + b2) = (exp b3) * exp b2 &
    exp (b2 + b3) = exp (b3 + b2) &
    exp b2,exp b3 are_commutative;

:: CLOPBAN4:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2, b3 being Element of the carrier of b1
      st b2,b3 are_commutative
   holds b2 * exp b3 = (exp b3) * b2;

:: CLOPBAN4:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr holds
   exp 0. b1 = 1. b1;

:: CLOPBAN4:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of the carrier of b1 holds
   (exp b2) * exp - b2 = 1. b1 &
    (exp - b2) * exp b2 = 1. b1;

:: CLOPBAN4:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of the carrier of b1 holds
   exp b2 is invertible(b1) & / exp b2 = exp - b2 & exp - b2 is invertible(b1) & / exp - b2 = exp b2;

:: CLOPBAN4:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of the carrier of b1
for b3, b4 being Element of COMPLEX holds
b3 * b2,b4 * b2 are_commutative;

:: CLOPBAN4:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of the carrier of b1
for b3, b4 being Element of COMPLEX holds
(exp (b3 * b2)) * exp (b4 * b2) = exp ((b3 + b4) * b2) &
 (exp (b4 * b2)) * exp (b3 * b2) = exp ((b4 + b3) * b2) &
 exp ((b3 + b4) * b2) = exp ((b4 + b3) * b2) &
 exp (b3 * b2),exp (b4 * b2) are_commutative;