Article COMSEQ_3, MML version 4.99.1005

:: COMSEQ_3:th 1
theorem
for b1 being Element of NAT holds
   b1 + 1 <> 0c & (b1 + 1) * <i> <> 0c;

:: COMSEQ_3:th 2
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = 0
for b2 being Element of NAT holds
   (Partial_Sums abs b1) . b2 = 0;

:: COMSEQ_3:th 3
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st for b2 being Element of NAT holds
           b1 . b2 = 0
   holds b1 is absolutely_summable;

:: COMSEQ_3:condreg 1
registration
  cluster Function-like quasi_total summable -> convergent (Relation of NAT,REAL);
end;

:: COMSEQ_3:condreg 2
registration
  cluster Function-like quasi_total absolutely_summable -> summable (Relation of NAT,REAL);
end;

:: COMSEQ_3:exreg 1
registration
  cluster Relation-like Function-like non empty total quasi_total complex-valued ext-real-valued real-valued absolutely_summable Relation of NAT,REAL;
end;

:: COMSEQ_3:th 4
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st b1 is convergent
for b2 being Element of REAL
      st 0 < b2
   holds ex b3 being Element of NAT st
      for b4, b5 being Element of NAT
            st b3 <= b4 & b3 <= b5
         holds abs ((b1 . b4) - (b1 . b5)) < b2;

:: COMSEQ_3:th 5
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of REAL
   st for b3 being Element of NAT holds
        b1 . b3 <= b2
for b3, b4 being Element of NAT holds
((Partial_Sums b1) . (b3 + b4)) - ((Partial_Sums b1) . b3) <= b2 * b4;

:: COMSEQ_3:th 6
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of REAL
   st for b3 being Element of NAT holds
        b1 . b3 <= b2
for b3 being Element of NAT holds
   (Partial_Sums b1) . b3 <= b2 * (b3 + 1);

:: COMSEQ_3:th 7
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Element of NAT
for b4 being Element of REAL
      st for b5 being Element of NAT
              st b5 <= b3
           holds b1 . b5 <= b4 * (b2 . b5)
   holds (Partial_Sums b1) . b3 <= b4 * ((Partial_Sums b2) . b3);

:: COMSEQ_3:th 8
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Element of NAT
for b4 being Element of REAL
   st for b5 being Element of NAT
           st b5 <= b3
        holds b1 . b5 <= b4 * (b2 . b5)
for b5 being Element of NAT
   st b5 <= b3
for b6 being Element of NAT
      st b5 + b6 <= b3
   holds ((Partial_Sums b1) . (b5 + b6)) - ((Partial_Sums b1) . b5) <= b4 * (((Partial_Sums b2) . (b5 + b6)) - ((Partial_Sums b2) . b5));

:: COMSEQ_3:th 9
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st for b2 being Element of NAT holds
           0 <= b1 . b2
   holds (for b2, b3 being Element of NAT
          st b2 <= b3
       holds abs (((Partial_Sums b1) . b3) - ((Partial_Sums b1) . b2)) = ((Partial_Sums b1) . b3) - ((Partial_Sums b1) . b2)) &
    (for b2 being Element of NAT holds
       abs ((Partial_Sums b1) . b2) = (Partial_Sums b1) . b2);

:: COMSEQ_3:th 10
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & b2 is convergent & lim (b1 - b2) = 0c
   holds lim b1 = lim b2;

:: COMSEQ_3:funcnot 1 => COMSEQ_3:func 1
definition
  let a1 be complex set;
  func A1 GeoSeq -> Function-like quasi_total Relation of NAT,COMPLEX means
    it . 0 = 1r &
     (for b1 being Element of NAT holds
        it . (b1 + 1) = (it . b1) * a1);
end;

:: COMSEQ_3:def 1
theorem
for b1 being complex set
for b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
      b2 = b1 GeoSeq
   iff
      b2 . 0 = 1r &
       (for b3 being Element of NAT holds
          b2 . (b3 + 1) = (b2 . b3) * b1);

:: COMSEQ_3:funcnot 2 => NEWTON:func 2
notation
  let a1 be complex set;
  let a2 be natural set;
  synonym a1 #N a2 for a1 |^ a2;
end;

:: COMSEQ_3:funcnot 3 => COMSEQ_3:func 2
definition
  let a1 be complex set;
  let a2 be natural set;
  redefine func A1 #N A2 -> Element of COMPLEX equals
    a1 GeoSeq . a2;
end;

:: COMSEQ_3:def 2
theorem
for b1 being complex set
for b2 being natural set holds
   b1 #N b2 = b1 GeoSeq . b2;

:: COMSEQ_3:th 11
theorem
for b1 being Element of COMPLEX holds
   b1 #N 0 = 1r;

:: COMSEQ_3:funcnot 4 => COMSEQ_3:func 3
definition
  let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
  func Re A1 -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = Re (a1 . b1);
end;

:: COMSEQ_3:def 3
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Function-like quasi_total Relation of NAT,REAL holds
      b2 = Re b1
   iff
      for b3 being Element of NAT holds
         b2 . b3 = Re (b1 . b3);

:: COMSEQ_3:funcnot 5 => COMSEQ_3:func 4
definition
  let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
  func Im A1 -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = Im (a1 . b1);
end;

:: COMSEQ_3:def 4
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Function-like quasi_total Relation of NAT,REAL holds
      b2 = Im b1
   iff
      for b3 being Element of NAT holds
         b2 . b3 = Im (b1 . b3);

:: COMSEQ_3:th 12
theorem
for b1 being Element of COMPLEX holds
   |.b1.| <= (abs Re b1) + abs Im b1;

:: COMSEQ_3:th 13
theorem
for b1 being Element of COMPLEX holds
   abs Re b1 <= |.b1.| & abs Im b1 <= |.b1.|;

:: COMSEQ_3:th 14
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st Re b1 = Re b2 & Im b1 = Im b2
   holds b1 = b2;

:: COMSEQ_3:th 15
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
(Re b1) + Re b2 = Re (b1 + b2) &
 (Im b1) + Im b2 = Im (b1 + b2);

:: COMSEQ_3:th 16
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
   - Re b1 = Re - b1 & - Im b1 = Im - b1;

:: COMSEQ_3:th 17
theorem
for b1 being Element of REAL
for b2 being Element of COMPLEX holds
   b1 * Re b2 = Re (b1 * b2) & b1 * Im b2 = Im (b1 * b2);

:: COMSEQ_3:th 18
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
(Re b1) - Re b2 = Re (b1 - b2) &
 (Im b1) - Im b2 = Im (b1 - b2);

:: COMSEQ_3:th 19
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of REAL holds
   b2 (#) Re b1 = Re (b2 (#) b1) & b2 (#) Im b1 = Im (b2 (#) b1);

:: COMSEQ_3:th 20
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of COMPLEX holds
   Re (b2 (#) b1) = ((Re b2) (#) Re b1) - ((Im b2) (#) Im b1) &
    Im (b2 (#) b1) = ((Re b2) (#) Im b1) + ((Im b2) (#) Re b1);

:: COMSEQ_3:th 21
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
Re (b1 (#) b2) = ((Re b1) (#) Re b2) - ((Im b1) (#) Im b2) &
 Im (b1 (#) b2) = ((Re b1) (#) Im b2) + ((Im b1) (#) Re b2);

:: COMSEQ_3:funcnot 6 => COMSEQ_3:func 5
definition
  let a1 be Function-like quasi_total natural-valued increasing Relation of NAT,REAL;
  let a2 be Function-like quasi_total Relation of NAT,COMPLEX;
  redefine func a2 * a1 -> Function-like quasi_total Relation of NAT,COMPLEX;
end;

:: COMSEQ_3:th 22
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL holds
   Re (b1 * b2) = (Re b1) * b2 & Im (b1 * b2) = (Im b1) * b2;

:: COMSEQ_3:funcnot 7 => COMSEQ_3:func 6
definition
  let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
  let a2 be Element of NAT;
  func A1 ^\ A2 -> Function-like quasi_total Relation of NAT,COMPLEX means
    for b1 being Element of NAT holds
       it . b1 = a1 . (b1 + a2);
end;

:: COMSEQ_3:def 6
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of NAT
for b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
      b3 = b1 ^\ b2
   iff
      for b4 being Element of NAT holds
         b3 . b4 = b1 . (b4 + b2);

:: COMSEQ_3:th 23
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of NAT holds
   (Re b1) ^\ b2 = Re (b1 ^\ b2) & (Im b1) ^\ b2 = Im (b1 ^\ b2);

:: COMSEQ_3:funcnot 8 => COMSEQ_3:func 7
definition
  let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
  func Partial_Sums A1 -> Function-like quasi_total Relation of NAT,COMPLEX means
    it . 0 = a1 . 0 &
     (for b1 being Element of NAT holds
        it . (b1 + 1) = (it . b1) + (a1 . (b1 + 1)));
end;

:: COMSEQ_3:def 7
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
   b2 = Partial_Sums b1
iff
   b2 . 0 = b1 . 0 &
    (for b3 being Element of NAT holds
       b2 . (b3 + 1) = (b2 . b3) + (b1 . (b3 + 1)));

:: COMSEQ_3:funcnot 9 => COMSEQ_3:func 8
definition
  let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
  func Sum A1 -> Element of COMPLEX equals
    lim Partial_Sums a1;
end;

:: COMSEQ_3:def 8
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
   Sum b1 = lim Partial_Sums b1;

:: COMSEQ_3:th 24
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
   st for b2 being Element of NAT holds
        b1 . b2 = 0c
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = 0c;

:: COMSEQ_3:th 25
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
   st for b2 being Element of NAT holds
        b1 . b2 = 0c
for b2 being Element of NAT holds
   (Partial_Sums |.b1.|) . b2 = 0;

:: COMSEQ_3:th 26
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
   Partial_Sums Re b1 = Re Partial_Sums b1 & Partial_Sums Im b1 = Im Partial_Sums b1;

:: COMSEQ_3:th 27
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
(Partial_Sums b1) + Partial_Sums b2 = Partial_Sums (b1 + b2);

:: COMSEQ_3:th 28
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
(Partial_Sums b1) - Partial_Sums b2 = Partial_Sums (b1 - b2);

:: COMSEQ_3:th 29
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of COMPLEX holds
   Partial_Sums (b2 (#) b1) = b2 (#) Partial_Sums b1;

:: COMSEQ_3:th 30
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of NAT holds
   |.(Partial_Sums b1) . b2.| <= (Partial_Sums |.b1.|) . b2;

:: COMSEQ_3:th 31
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2, b3 being Element of NAT holds
|.((Partial_Sums b1) . b2) - ((Partial_Sums b1) . b3).| <= abs (((Partial_Sums |.b1.|) . b2) - ((Partial_Sums |.b1.|) . b3));

:: COMSEQ_3:th 32
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of NAT holds
   (Partial_Sums Re b1) ^\ b2 = Re ((Partial_Sums b1) ^\ b2) &
    (Partial_Sums Im b1) ^\ b2 = Im ((Partial_Sums b1) ^\ b2);

:: COMSEQ_3:th 33
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st for b3 being Element of NAT holds
           b1 . b3 = b2 . 0
   holds Partial_Sums (b2 ^\ 1) = ((Partial_Sums b2) ^\ 1) - b1;

:: COMSEQ_3:th 34
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
   Partial_Sums |.b1.| is non-decreasing;

:: COMSEQ_3:funcreg 1
registration
  let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
  cluster Partial_Sums |.a1.| -> Function-like quasi_total non-decreasing;
end;

:: COMSEQ_3:th 35
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Element of NAT
      st for b4 being Element of NAT
              st b4 <= b3
           holds b1 . b4 = b2 . b4
   holds (Partial_Sums b1) . b3 = (Partial_Sums b2) . b3;

:: COMSEQ_3:th 36
theorem
for b1 being Element of COMPLEX
   st 1r <> b1
for b2 being Element of NAT holds
   (Partial_Sums (b1 GeoSeq)) . b2 = (1r - (b1 #N (b2 + 1))) / (1r - b1);

:: COMSEQ_3:th 37
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of COMPLEX
   st b2 <> 1r &
      (for b3 being Element of NAT holds
         b1 . (b3 + 1) = b2 * (b1 . b3))
for b3 being Element of NAT holds
   (Partial_Sums b1) . b3 = (b1 . 0) * ((1r - (b2 #N (b3 + 1))) / (1r - b2));

:: COMSEQ_3:th 38
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like quasi_total Relation of NAT,COMPLEX
      st for b4 being Element of NAT holds
           Re (b3 . b4) = b1 . b4 & Im (b3 . b4) = b2 . b4
   holds    b1 is convergent & b2 is convergent
   iff
      b3 is convergent;

:: COMSEQ_3:th 39
theorem
for b1, b2 being Function-like quasi_total convergent Relation of NAT,REAL
for b3 being Function-like quasi_total Relation of NAT,COMPLEX
      st for b4 being Element of NAT holds
           Re (b3 . b4) = b1 . b4 & Im (b3 . b4) = b2 . b4
   holds b3 is convergent &
    lim b3 = (lim b1) + ((lim b2) * <i>);

:: COMSEQ_3:th 40
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like quasi_total convergent Relation of NAT,COMPLEX
      st for b4 being Element of NAT holds
           Re (b3 . b4) = b1 . b4 & Im (b3 . b4) = b2 . b4
   holds b1 is convergent & b2 is convergent & lim b1 = Re lim b3 & lim b2 = Im lim b3;

:: COMSEQ_3:th 41
theorem
for b1 being Function-like quasi_total convergent Relation of NAT,COMPLEX holds
   Re b1 is convergent & Im b1 is convergent & lim Re b1 = Re lim b1 & lim Im b1 = Im lim b1;

:: COMSEQ_3:funcreg 2
registration
  let a1 be Function-like quasi_total convergent Relation of NAT,COMPLEX;
  cluster Re a1 -> Function-like quasi_total convergent;
end;

:: COMSEQ_3:funcreg 3
registration
  let a1 be Function-like quasi_total convergent Relation of NAT,COMPLEX;
  cluster Im a1 -> Function-like quasi_total convergent;
end;

:: COMSEQ_3:th 42
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st Re b1 is convergent & Im b1 is convergent
   holds b1 is convergent & Re lim b1 = lim Re b1 & Im lim b1 = lim Im b1;

:: COMSEQ_3:th 43
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of COMPLEX
      st 0 < |.b2.| &
         |.b2.| < 1 &
         b1 . 0 = b2 &
         (for b3 being Element of NAT holds
            b1 . (b3 + 1) = (b1 . b3) * b2)
   holds b1 is convergent & lim b1 = 0c;

:: COMSEQ_3:th 44
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of COMPLEX
      st |.b2.| < 1 &
         (for b3 being Element of NAT holds
            b1 . b3 = b2 #N (b3 + 1))
   holds b1 is convergent & lim b1 = 0c;

:: COMSEQ_3:th 45
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of REAL
      st 0 < b2 &
         (ex b3 being Element of NAT st
            for b4 being Element of NAT
                  st b3 <= b4
               holds b2 <= |.b1 . b4.|) &
         |.b1.| is convergent
   holds lim |.b1.| <> 0;

:: COMSEQ_3:th 46
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
      b1 is convergent
   iff
      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 |.(b1 . b4) - (b1 . b3).| < b2;

:: COMSEQ_3:th 47
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
   st b1 is convergent
for b2 being Element of REAL
      st 0 < b2
   holds ex b3 being Element of NAT st
      for b4, b5 being Element of NAT
            st b3 <= b4 & b3 <= b5
         holds |.(b1 . b4) - (b1 . b5).| < b2;

:: COMSEQ_3:th 48
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st (for b3 being Element of NAT holds
            |.b2 . b3.| <= b1 . b3) &
         b1 is convergent &
         lim b1 = 0
   holds b2 is convergent & lim b2 = 0c;

:: COMSEQ_3:modenot 1 => COMSEQ_3:mode 1
definition
  let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
  mode subsequence of A1 -> Function-like quasi_total Relation of NAT,COMPLEX means
    ex b1 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
       it = a1 * b1;
end;

:: COMSEQ_3:dfs 8
definiens
  let a1, a2 be Function-like quasi_total Relation of NAT,COMPLEX;
To prove
     a2 is subsequence of a1
it is sufficient to prove
  thus ex b1 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
       a2 = a1 * b1;

:: COMSEQ_3:def 9
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
   b2 is subsequence of b1
iff
   ex b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
      b2 = b1 * b3;

:: COMSEQ_3:th 49
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is subsequence of b2
   holds Re b1 is subsequence of Re b2 & Im b1 is subsequence of Im b2;

:: COMSEQ_3:th 50
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is subsequence of b2 & b2 is subsequence of b3
   holds b1 is subsequence of b3;

:: COMSEQ_3:th 51
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is bounded
   holds ex b2 being Function-like quasi_total Relation of NAT,COMPLEX st
      b2 is subsequence of b1 & b2 is convergent;

:: COMSEQ_3:attrnot 1 => COMSEQ_3:attr 1
definition
  let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
  attr a1 is summable means
    Partial_Sums a1 is convergent;
end;

:: COMSEQ_3:dfs 9
definiens
  let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
To prove
     a1 is summable
it is sufficient to prove
  thus Partial_Sums a1 is convergent;

:: COMSEQ_3:def 10
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
      b1 is summable
   iff
      Partial_Sums b1 is convergent;

:: COMSEQ_3:exreg 2
registration
  cluster Relation-like Function-like non empty total quasi_total complex-valued summable Relation of NAT,COMPLEX;
end;

:: COMSEQ_3:funcreg 4
registration
  let a1 be Function-like quasi_total summable Relation of NAT,COMPLEX;
  cluster Partial_Sums a1 -> Function-like quasi_total convergent;
end;

:: COMSEQ_3:attrnot 2 => COMSEQ_3:attr 2
definition
  let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
  attr a1 is absolutely_summable means
    |.a1.| is summable;
end;

:: COMSEQ_3:dfs 10
definiens
  let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
To prove
     a1 is absolutely_summable
it is sufficient to prove
  thus |.a1.| is summable;

:: COMSEQ_3:def 11
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
      b1 is absolutely_summable
   iff
      |.b1.| is summable;

:: COMSEQ_3:th 52
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st for b2 being Element of NAT holds
           b1 . b2 = 0c
   holds b1 is absolutely_summable;

:: COMSEQ_3:exreg 3
registration
  cluster Relation-like Function-like non empty total quasi_total complex-valued absolutely_summable Relation of NAT,COMPLEX;
end;

:: COMSEQ_3:funcreg 5
registration
  let a1 be Function-like quasi_total absolutely_summable Relation of NAT,COMPLEX;
  cluster |.a1.| -> Function-like quasi_total summable;
end;

:: COMSEQ_3:funcnot 10 => COMSEQ_3:func 9
definition
  let a1 be Function-like quasi_total absolutely_summable Relation of NAT,COMPLEX;
  redefine func |.a1.| -> Function-like quasi_total summable Relation of NAT,REAL;
  projectivity;
::  for a1 being Function-like quasi_total absolutely_summable Relation of NAT,COMPLEX holds
::     |.|.a1.|.| = |.a1.|;
end;

:: COMSEQ_3:th 53
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is summable
   holds b1 is convergent & lim b1 = 0c;

:: COMSEQ_3:condreg 3
registration
  cluster Function-like quasi_total summable -> convergent (Relation of NAT,COMPLEX);
end;

:: COMSEQ_3:th 54
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is summable
   holds Re b1 is summable &
    Im b1 is summable &
    Sum b1 = (Sum Re b1) + ((Sum Im b1) * <i>);

:: COMSEQ_3:funcreg 6
registration
  let a1 be Function-like quasi_total summable Relation of NAT,COMPLEX;
  cluster Re a1 -> Function-like quasi_total summable;
end;

:: COMSEQ_3:funcreg 7
registration
  let a1 be Function-like quasi_total summable Relation of NAT,COMPLEX;
  cluster Im a1 -> Function-like quasi_total summable;
end;

:: COMSEQ_3:th 55
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is summable & b2 is summable
   holds b1 + b2 is summable &
    Sum (b1 + b2) = (Sum b1) + Sum b2;

:: COMSEQ_3:th 56
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is summable & b2 is summable
   holds b1 - b2 is summable &
    Sum (b1 - b2) = (Sum b1) - Sum b2;

:: COMSEQ_3:funcreg 8
registration
  let a1, a2 be Function-like quasi_total summable Relation of NAT,COMPLEX;
  cluster a1 + a2 -> Function-like quasi_total summable;
end;

:: COMSEQ_3:funcreg 9
registration
  let a1, a2 be Function-like quasi_total summable Relation of NAT,COMPLEX;
  cluster a1 - a2 -> Function-like quasi_total summable;
end;

:: COMSEQ_3:th 57
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of COMPLEX
      st b1 is summable
   holds b2 (#) b1 is summable & Sum (b2 (#) b1) = b2 * Sum b1;

:: COMSEQ_3:funcreg 10
registration
  let a1 be Element of COMPLEX;
  let a2 be Function-like quasi_total summable Relation of NAT,COMPLEX;
  cluster a1 (#) a2 -> Function-like quasi_total summable;
end;

:: COMSEQ_3:th 58
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st Re b1 is summable & Im b1 is summable
   holds b1 is summable &
    Sum b1 = (Sum Re b1) + ((Sum Im b1) * <i>);

:: COMSEQ_3:th 59
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
   st b1 is summable
for b2 being Element of NAT holds
   b1 ^\ b2 is summable;

:: COMSEQ_3:funcreg 11
registration
  let a1 be Function-like quasi_total summable Relation of NAT,COMPLEX;
  let a2 be Element of NAT;
  cluster a1 ^\ a2 -> Function-like quasi_total summable;
end;

:: COMSEQ_3:th 60
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st ex b2 being Element of NAT st
           b1 ^\ b2 is summable
   holds b1 is summable;

:: COMSEQ_3:th 61
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
   st b1 is summable
for b2 being Element of NAT holds
   Sum b1 = ((Partial_Sums b1) . b2) + Sum (b1 ^\ (b2 + 1));

:: COMSEQ_3:th 62
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
      Partial_Sums |.b1.| is bounded_above
   iff
      b1 is absolutely_summable;

:: COMSEQ_3:funcreg 12
registration
  let a1 be Function-like quasi_total absolutely_summable Relation of NAT,COMPLEX;
  cluster Partial_Sums |.a1.| -> Function-like quasi_total bounded_above;
end;

:: COMSEQ_3:th 63
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
      b1 is summable
   iff
      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 |.((Partial_Sums b1) . b4) - ((Partial_Sums b1) . b3).| < b2;

:: COMSEQ_3:th 64
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is absolutely_summable
   holds b1 is summable;

:: COMSEQ_3:condreg 4
registration
  cluster Function-like quasi_total absolutely_summable -> summable (Relation of NAT,COMPLEX);
end;

:: COMSEQ_3:exreg 4
registration
  cluster Relation-like Function-like non empty total quasi_total complex-valued absolutely_summable Relation of NAT,COMPLEX;
end;

:: COMSEQ_3:th 65
theorem
for b1 being Element of COMPLEX
      st |.b1.| < 1
   holds b1 GeoSeq is summable &
    Sum (b1 GeoSeq) = 1r / (1r - b1);

:: COMSEQ_3:th 66
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of COMPLEX
      st |.b2.| < 1 &
         (for b3 being Element of NAT holds
            b1 . (b3 + 1) = b2 * (b1 . b3))
   holds b1 is summable &
    Sum b1 = (b1 . 0) / (1r - b2);

:: COMSEQ_3:th 67
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is summable &
         (ex b3 being Element of NAT st
            for b4 being Element of NAT
                  st b3 <= b4
               holds |.b2 . b4.| <= b1 . b4)
   holds b2 is absolutely_summable;

:: COMSEQ_3:th 68
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st (for b3 being Element of NAT holds
            0 <= |.b1.| . b3 &
             |.b1.| . b3 <= |.b2.| . b3) &
         b2 is absolutely_summable
   holds b1 is absolutely_summable &
    Sum |.b1.| <= Sum |.b2.|;

:: COMSEQ_3:th 69
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st (for b2 being Element of NAT holds
            0 < |.b1.| . b2) &
         (ex b2 being Element of NAT st
            for b3 being Element of NAT
                  st b2 <= b3
               holds 1 <= (|.b1.| . (b3 + 1)) / (|.b1.| . b3))
   holds b1 is not absolutely_summable;

:: COMSEQ_3:th 70
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st (for b3 being Element of NAT holds
            b1 . b3 = b3 -root (|.b2.| . b3)) &
         b1 is convergent &
         lim b1 < 1
   holds b2 is absolutely_summable;

:: COMSEQ_3:th 71
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st (for b3 being Element of NAT holds
            b1 . b3 = b3 -root (|.b2.| . b3)) &
         (ex b3 being Element of NAT st
            for b4 being Element of NAT
                  st b3 <= b4
               holds 1 <= b1 . b4)
   holds |.b2.| is not summable;

:: COMSEQ_3:th 72
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st (for b3 being Element of NAT holds
            b1 . b3 = b3 -root (|.b2.| . b3)) &
         b1 is convergent &
         1 < lim b1
   holds b2 is not absolutely_summable;

:: COMSEQ_3:th 73
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st |.b2.| is non-increasing &
         (for b3 being Element of NAT holds
            b1 . b3 = (2 to_power b3) * (|.b2.| . (2 to_power b3)))
   holds    b2 is absolutely_summable
   iff
      b1 is summable;

:: COMSEQ_3:th 74
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of REAL
      st 1 < b2 &
         (for b3 being Element of NAT
               st 1 <= b3
            holds |.b1.| . b3 = 1 / (b3 to_power b2))
   holds b1 is absolutely_summable;

:: COMSEQ_3:th 75
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of REAL
      st b2 <= 1 &
         (for b3 being Element of NAT
               st 1 <= b3
            holds |.b1.| . b3 = 1 / (b3 to_power b2))
   holds b1 is not absolutely_summable;

:: COMSEQ_3:th 76
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st (for b3 being Element of NAT holds
            b2 . b3 <> 0c &
             b1 . b3 = (|.b2.| . (b3 + 1)) / (|.b2.| . b3)) &
         b1 is convergent &
         lim b1 < 1
   holds b2 is absolutely_summable;

:: COMSEQ_3:th 77
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st (for b2 being Element of NAT holds
            b1 . b2 <> 0c) &
         (ex b2 being Element of NAT st
            for b3 being Element of NAT
                  st b2 <= b3
               holds 1 <= (|.b1.| . (b3 + 1)) / (|.b1.| . b3))
   holds b1 is not absolutely_summable;