Article COMSEQ_2, MML version 4.99.1005

:: COMSEQ_2:th 1
theorem
for b1, b2 being Element of COMPLEX
      st b1 <> 0c & b2 <> 0c
   holds |.b1 " - (b2 ").| = |.b1 - b2.| / (|.b1.| * |.b2.|);

:: COMSEQ_2:th 2
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of NAT holds
   ex b3 being Element of REAL st
      0 < b3 &
       (for b4 being Element of NAT
             st b4 <= b2
          holds |.b1 . b4.| < b3);

:: COMSEQ_2:funcnot 1 => COMSEQ_2:func 1
definition
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,COMPLEX;
  func A2 *' -> Function-like Relation of a1,COMPLEX means
    dom it = dom a2 &
     (for b1 being Element of a1
           st b1 in dom it
        holds it . b1 = (a2 /. b1) *');
end;

:: COMSEQ_2:def 1
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX holds
   b3 = b2 *'
iff
   dom b3 = dom b2 &
    (for b4 being Element of b1
          st b4 in dom b3
       holds b3 . b4 = (b2 /. b4) *');

:: COMSEQ_2:funcnot 2 => COMSEQ_2:func 1
definition
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,COMPLEX;
  func A2 *' -> Function-like Relation of a1,COMPLEX means
    dom it = a1 &
     (for b1 being Element of a1 holds
        it . b1 = (a2 . b1) *');
end;

:: COMSEQ_2:def 2
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of b1,COMPLEX
for b3 being Function-like Relation of b1,COMPLEX holds
      b3 = b2 *'
   iff
      dom b3 = b1 &
       (for b4 being Element of b1 holds
          b3 . b4 = (b2 . b4) *');

:: COMSEQ_2:funcreg 1
registration
  let a1 be non empty set;
  let a2 be Function-like quasi_total Relation of a1,COMPLEX;
  cluster a2 *' -> Function-like total;
end;

:: COMSEQ_2:th 3
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is non-zero
   holds b1 *' is non-zero;

:: COMSEQ_2:th 4
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
   (b1 (#) b2) *' = b1 *' (#) (b2 *');

:: COMSEQ_2:th 5
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
(b1 (#) b2) *' = b1 *' (#) (b2 *');

:: COMSEQ_2:th 6
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
   b1 *' " = b1 " *';

:: COMSEQ_2:th 7
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
(b1 /" b2) *' = b1 *' /" (b2 *');

:: COMSEQ_2:attrnot 1 => COMSEQ_2:attr 1
definition
  let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
  attr a1 is bounded means
    ex b1 being Element of REAL st
       for b2 being Element of NAT holds
          |.a1 . b2.| < b1;
end;

:: COMSEQ_2:dfs 3
definiens
  let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
To prove
     a1 is bounded
it is sufficient to prove
  thus ex b1 being Element of REAL st
       for b2 being Element of NAT holds
          |.a1 . b2.| < b1;

:: COMSEQ_2:def 3
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
      b1 is bounded
   iff
      ex b2 being Element of REAL st
         for b3 being Element of NAT holds
            |.b1 . b3.| < b2;

:: COMSEQ_2:exreg 1
registration
  cluster Relation-like Function-like non empty total quasi_total complex-valued bounded Relation of NAT,COMPLEX;
end;

:: COMSEQ_2:th 8
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
      b1 is bounded
   iff
      ex b2 being Element of REAL st
         0 < b2 &
          (for b3 being Element of NAT holds
             |.b1 . b3.| < b2);

:: COMSEQ_2:attrnot 2 => COMSEQ_2:attr 2
definition
  let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
  attr a1 is convergent means
    ex b1 being Element of COMPLEX 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 |.(a1 . b4) - b1.| < b2;
end;

:: COMSEQ_2:dfs 4
definiens
  let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
To prove
     a1 is convergent
it is sufficient to prove
  thus ex b1 being Element of COMPLEX 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 |.(a1 . b4) - b1.| < b2;

:: COMSEQ_2:def 4
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
      b1 is convergent
   iff
      ex b2 being Element of COMPLEX st
         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 |.(b1 . b5) - b2.| < b3;

:: COMSEQ_2:funcnot 3 => COMSEQ_2:func 2
definition
  let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
  assume a1 is convergent;
  func lim A1 -> Element of COMPLEX 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 |.(a1 . b3) - it.| < b1;
end;

:: COMSEQ_2:def 5
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
   st b1 is convergent
for b2 being Element of COMPLEX holds
      b2 = lim 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 |.(b1 . b5) - b2.| < b3;

:: COMSEQ_2:th 9
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st ex b2 being Element of COMPLEX st
           for b3 being Element of NAT holds
              b1 . b3 = b2
   holds b1 is convergent;

:: COMSEQ_2:th 10
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of COMPLEX
      st for b3 being Element of NAT holds
           b1 . b3 = b2
   holds lim b1 = b2;

:: COMSEQ_2:exreg 2
registration
  cluster Relation-like Function-like constant non empty total quasi_total complex-valued ext-real-valued real-valued Relation of NAT,REAL;
end;

:: COMSEQ_2:exreg 3
registration
  cluster Relation-like Function-like non empty total quasi_total complex-valued convergent Relation of NAT,COMPLEX;
end;

:: COMSEQ_2:exreg 4
registration
  cluster Relation-like Function-like non empty total quasi_total complex-valued ext-real-valued real-valued convergent Relation of NAT,REAL;
end;

:: COMSEQ_2:funcnot 4 => COMSEQ_2:func 3
definition
  let a1 be Function-like quasi_total convergent Relation of NAT,COMPLEX;
  redefine func |.a1.| -> Function-like quasi_total convergent Relation of NAT,REAL;
  projectivity;
::  for a1 being Function-like quasi_total convergent Relation of NAT,COMPLEX holds
::     |.|.a1.|.| = |.a1.|;
end;

:: COMSEQ_2:th 11
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent
   holds lim |.b1.| = |.lim b1.|;

:: COMSEQ_2:funcreg 2
registration
  let a1 be Function-like quasi_total convergent Relation of NAT,COMPLEX;
  cluster a1 *' -> Function-like convergent;
end;

:: COMSEQ_2:th 12
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent
   holds lim (b1 *') = (lim b1) *';

:: COMSEQ_2:th 13
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & b2 is convergent
   holds b1 + b2 is convergent;

:: COMSEQ_2:th 14
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & b2 is convergent
   holds lim (b1 + b2) = (lim b1) + lim b2;

:: COMSEQ_2:th 15
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & b2 is convergent
   holds lim |.b1 + b2.| = |.(lim b1) + lim b2.|;

:: COMSEQ_2:th 16
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & b2 is convergent
   holds lim ((b1 + b2) *') = (lim b1) *' + ((lim b2) *');

:: COMSEQ_2:th 17
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b2 is convergent
   holds b1 (#) b2 is convergent;

:: COMSEQ_2:th 18
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b2 is convergent
   holds lim (b1 (#) b2) = b1 * lim b2;

:: COMSEQ_2:th 19
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b2 is convergent
   holds lim |.b1 (#) b2.| = |.b1.| * |.lim b2.|;

:: COMSEQ_2:th 20
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b2 is convergent
   holds lim ((b1 (#) b2) *') = b1 *' * ((lim b2) *');

:: COMSEQ_2:th 21
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent
   holds - b1 is convergent;

:: COMSEQ_2:th 22
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent
   holds lim - b1 = - lim b1;

:: COMSEQ_2:th 23
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent
   holds lim |.- b1.| = |.lim b1.|;

:: COMSEQ_2:th 24
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent
   holds lim ((- b1) *') = - ((lim b1) *');

:: COMSEQ_2:th 25
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & b2 is convergent
   holds b1 - b2 is convergent;

:: COMSEQ_2:th 26
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & b2 is convergent
   holds lim (b1 - b2) = (lim b1) - lim b2;

:: COMSEQ_2:th 27
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & b2 is convergent
   holds lim |.b1 - b2.| = |.(lim b1) - lim b2.|;

:: COMSEQ_2:th 28
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & b2 is convergent
   holds lim ((b1 - b2) *') = (lim b1) *' - ((lim b2) *');

:: COMSEQ_2:condreg 1
registration
  cluster Function-like quasi_total convergent -> bounded (Relation of NAT,COMPLEX);
end;

:: COMSEQ_2:condreg 2
registration
  cluster Function-like quasi_total non bounded -> non convergent (Relation of NAT,COMPLEX);
end;

:: COMSEQ_2:th 29
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is Function-like quasi_total convergent Relation of NAT,COMPLEX &
         b2 is Function-like quasi_total convergent Relation of NAT,COMPLEX
   holds b1 (#) b2 is convergent;

:: COMSEQ_2:th 30
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is Function-like quasi_total convergent Relation of NAT,COMPLEX &
         b2 is Function-like quasi_total convergent Relation of NAT,COMPLEX
   holds lim (b1 (#) b2) = (lim b1) * lim b2;

:: COMSEQ_2:th 31
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & b2 is convergent
   holds lim |.b1 (#) b2.| = |.lim b1.| * |.lim b2.|;

:: COMSEQ_2:th 32
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & b2 is convergent
   holds lim ((b1 (#) b2) *') = (lim b1) *' * ((lim b2) *');

:: COMSEQ_2:th 33
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & lim b1 <> 0c
   holds ex b2 being Element of NAT st
      for b3 being Element of NAT
            st b2 <= b3
         holds |.lim b1.| / 2 < |.b1 . b3.|;

:: COMSEQ_2:th 34
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & lim b1 <> 0c & b1 is non-zero
   holds b1 " is convergent;

:: COMSEQ_2:th 35
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & lim b1 <> 0c & b1 is non-zero
   holds lim (b1 ") = (lim b1) ";

:: COMSEQ_2:th 36
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & lim b1 <> 0c & b1 is non-zero
   holds lim |.b1 ".| = |.lim b1.| ";

:: COMSEQ_2:th 37
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & lim b1 <> 0c & b1 is non-zero
   holds lim (b1 " *') = (lim b1) *' ";

:: COMSEQ_2:th 38
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & b2 is convergent & lim b2 <> 0c & b2 is non-zero
   holds b1 /" b2 is convergent;

:: COMSEQ_2:th 39
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & b2 is convergent & lim b2 <> 0c & b2 is non-zero
   holds lim (b1 /" b2) = (lim b1) / lim b2;

:: COMSEQ_2:th 40
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & b2 is convergent & lim b2 <> 0c & b2 is non-zero
   holds lim |.b1 /" b2.| = |.lim b1.| / |.lim b2.|;

:: COMSEQ_2:th 41
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & b2 is convergent & lim b2 <> 0c & b2 is non-zero
   holds lim ((b1 /" b2) *') = (lim b1) *' / ((lim b2) *');

:: COMSEQ_2:th 42
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & b2 is bounded & lim b1 = 0c
   holds b1 (#) b2 is convergent;

:: COMSEQ_2:th 43
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & b2 is bounded & lim b1 = 0c
   holds lim (b1 (#) b2) = 0c;

:: COMSEQ_2:th 44
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & b2 is bounded & lim b1 = 0c
   holds lim |.b1 (#) b2.| = 0;

:: COMSEQ_2:th 45
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & b2 is bounded & lim b1 = 0c
   holds lim ((b1 (#) b2) *') = 0c;