Article TOPRNS_1, MML version 4.99.1005

:: TOPRNS_1:modenot 1
definition
  let a1 be Element of NAT;
  mode Real_Sequence of a1 is Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
end;

:: TOPRNS_1:th 2
theorem
for b1 being Element of NAT
for b2 being Relation-like Function-like set holds
      b2 is Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
   iff
      proj1 b2 = NAT &
       (for b3 being Element of NAT holds
          b2 . b3 is Element of the carrier of TOP-REAL b1);

:: TOPRNS_1:attrnot 1 => TOPRNS_1:attr 1
definition
  let a1 be Element of NAT;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
  attr a2 is non-zero means
    proj2 a2 c= (the carrier of TOP-REAL a1) \ {0.REAL a1};
end;

:: TOPRNS_1:dfs 1
definiens
  let a1 be Element of NAT;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
To prove
     a2 is non-zero
it is sufficient to prove
  thus proj2 a2 c= (the carrier of TOP-REAL a1) \ {0.REAL a1};

:: TOPRNS_1:def 1
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
      b2 is non-zero(b1)
   iff
      proj2 b2 c= (the carrier of TOP-REAL b1) \ {0.REAL b1};

:: TOPRNS_1:th 3
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
      b2 is non-zero(b1)
   iff
      for b3 being set
            st b3 in NAT
         holds b2 . b3 <> 0.REAL b1;

:: TOPRNS_1:th 4
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
      b2 is non-zero(b1)
   iff
      for b3 being Element of NAT holds
         b2 . b3 <> 0.REAL b1;

:: TOPRNS_1:th 5
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
      st for b4 being set
              st b4 in NAT
           holds b2 . b4 = b3 . b4
   holds b2 = b3;

:: TOPRNS_1:th 6
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
      st for b4 being Element of NAT holds
           b2 . b4 = b3 . b4
   holds b2 = b3;

:: TOPRNS_1:sch 1
scheme TOPRNS_1:sch 1
{F1 -> Element of NAT,
  F2 -> Element of the carrier of TOP-REAL F1()}:
ex b1 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL F1() st
   for b2 being Element of NAT holds
      b1 . b2 = F2(b2)


:: TOPRNS_1:funcnot 1 => TOPRNS_1:func 1
definition
  let a1 be Element of NAT;
  let a2, a3 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
  func A2 + A3 -> Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1 means
    for b1 being Element of NAT holds
       it . b1 = (a2 . b1) + (a3 . b1);
end;

:: TOPRNS_1:def 2
theorem
for b1 being Element of NAT
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
   b4 = b2 + b3
iff
   for b5 being Element of NAT holds
      b4 . b5 = (b2 . b5) + (b3 . b5);

:: TOPRNS_1:funcnot 2 => TOPRNS_1:func 2
definition
  let a1 be Element of REAL;
  let a2 be Element of NAT;
  let a3 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a2;
  func A1 * A3 -> Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a2 means
    for b1 being Element of NAT holds
       it . b1 = a1 * (a3 . b1);
end;

:: TOPRNS_1:def 3
theorem
for b1 being Element of REAL
for b2 being Element of NAT
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b2 holds
   b4 = b1 * b3
iff
   for b5 being Element of NAT holds
      b4 . b5 = b1 * (b3 . b5);

:: TOPRNS_1:funcnot 3 => TOPRNS_1:func 3
definition
  let a1 be Element of NAT;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
  func - A2 -> Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1 means
    for b1 being Element of NAT holds
       it . b1 = - (a2 . b1);
end;

:: TOPRNS_1:def 4
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
   b3 = - b2
iff
   for b4 being Element of NAT holds
      b3 . b4 = - (b2 . b4);

:: TOPRNS_1:funcnot 4 => TOPRNS_1:func 4
definition
  let a1 be Element of NAT;
  let a2, a3 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
  func A2 - A3 -> Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1 equals
    a2 + - a3;
end;

:: TOPRNS_1:def 5
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 - b3 = b2 + - b3;

:: TOPRNS_1:funcnot 5 => TOPRNS_1:func 5
definition
  let a1 be Element of NAT;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
  func |.A2.| -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = |.a2 . b1.|;
end;

:: TOPRNS_1:def 7
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
for b3 being Function-like quasi_total Relation of NAT,REAL holds
      b3 = |.b2.|
   iff
      for b4 being Element of NAT holds
         b3 . b4 = |.b2 . b4.|;

:: TOPRNS_1:th 8
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3 being Element of the carrier of TOP-REAL b1 holds
   (abs b2) * |.b3.| = |.b2 * b3.|;

:: TOPRNS_1:th 9
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
   |.b2 * b3.| = (abs b2) (#) |.b3.|;

:: TOPRNS_1:th 10
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 + b3 = b3 + b2;

:: TOPRNS_1:th 11
theorem
for b1 being Element of NAT
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
(b2 + b3) + b4 = b2 + (b3 + b4);

:: TOPRNS_1:th 12
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
   - b2 = (- 1) * b2;

:: TOPRNS_1:th 13
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 * (b3 + b4) = (b2 * b3) + (b2 * b4);

:: TOPRNS_1:th 14
theorem
for b1 being Element of NAT
for b2, b3 being Element of REAL
for b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
   (b2 * b3) * b4 = b2 * (b3 * b4);

:: TOPRNS_1:th 15
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 * (b3 - b4) = (b2 * b3) - (b2 * b4);

:: TOPRNS_1:th 16
theorem
for b1 being Element of NAT
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 - (b3 + b4) = (b2 - b3) - b4;

:: TOPRNS_1:th 17
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
   1 * b2 = b2;

:: TOPRNS_1:th 18
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
   - - b2 = b2;

:: TOPRNS_1:th 19
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 - - b3 = b2 + b3;

:: TOPRNS_1:th 20
theorem
for b1 being Element of NAT
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 - (b3 - b4) = (b2 - b3) + b4;

:: TOPRNS_1:th 21
theorem
for b1 being Element of NAT
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 + (b3 - b4) = (b2 + b3) - b4;

:: TOPRNS_1:th 22
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
      st b2 <> 0 & b3 is non-zero(b1)
   holds b2 * b3 is non-zero(b1);

:: TOPRNS_1:th 23
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
      st b2 is non-zero(b1)
   holds - b2 is non-zero(b1);

:: TOPRNS_1:th 24
theorem
for b1 being Element of NAT holds
   |.0.REAL b1.| = 0;

:: TOPRNS_1:th 25
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
      st |.b2.| = 0
   holds b2 = 0.REAL b1;

:: TOPRNS_1:th 26
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
   0 <= |.b2.|;

:: TOPRNS_1:th 27
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
   |.- b2.| = |.b2.|;

:: TOPRNS_1:th 28
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|.b2 - b3.| = |.b3 - b2.|;

:: TOPRNS_1:th 29
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
   |.b2 - b3.| = 0
iff
   b2 = b3;

:: TOPRNS_1:th 30
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|.b2 + b3.| <= |.b2.| + |.b3.|;

:: TOPRNS_1:th 31
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|.b2 - b3.| <= |.b2.| + |.b3.|;

:: TOPRNS_1:th 32
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|.b2.| - |.b3.| <= |.b2 + b3.|;

:: TOPRNS_1:th 33
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|.b2.| - |.b3.| <= |.b2 - b3.|;

:: TOPRNS_1:th 34
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
      st b2 <> b3
   holds 0 < |.b2 - b3.|;

:: TOPRNS_1:th 35
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1 holds
|.b2 - b3.| <= |.b2 - b4.| + |.b4 - b3.|;

:: TOPRNS_1:th 36
theorem
for b1 being Element of NAT
for b2, b3 being Element of REAL
for b4, b5 being Element of the carrier of TOP-REAL b1
      st 0 <= b2 & |.b4.| < |.b5.| & b2 < b3
   holds |.b4.| * b2 < |.b5.| * b3;

:: TOPRNS_1:th 38
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3 being Element of the carrier of TOP-REAL b1 holds
      - |.b3.| < b2 & b2 < |.b3.|
   iff
      abs b2 < |.b3.|;

:: TOPRNS_1:attrnot 2 => TOPRNS_1:attr 2
definition
  let a1 be Element of NAT;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
  attr a2 is bounded means
    ex b1 being Element of REAL st
       for b2 being Element of NAT holds
          |.a2 . b2.| < b1;
end;

:: TOPRNS_1:dfs 7
definiens
  let a1 be Element of NAT;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
To prove
     a2 is bounded
it is sufficient to prove
  thus ex b1 being Element of REAL st
       for b2 being Element of NAT holds
          |.a2 . b2.| < b1;

:: TOPRNS_1:def 8
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
      b2 is bounded(b1)
   iff
      ex b3 being Element of REAL st
         for b4 being Element of NAT holds
            |.b2 . b4.| < b3;

:: TOPRNS_1:th 39
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
for b3 being Element of NAT holds
   ex b4 being Element of REAL st
      0 < b4 &
       (for b5 being Element of NAT
             st b5 <= b3
          holds |.b2 . b5.| < b4);

:: TOPRNS_1:attrnot 3 => TOPRNS_1:attr 3
definition
  let a1 be Element of NAT;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
  attr a2 is convergent means
    ex b1 being Element of the carrier of TOP-REAL a1 st
       for b2 being Element of REAL
             st 0 < b2
          holds ex b3 being Element of NAT st
             for b4 being Element of NAT
                   st b3 <= b4
                holds |.(a2 . b4) - b1.| < b2;
end;

:: TOPRNS_1:dfs 8
definiens
  let a1 be Element of NAT;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
To prove
     a2 is convergent
it is sufficient to prove
  thus ex b1 being Element of the carrier of TOP-REAL a1 st
       for b2 being Element of REAL
             st 0 < b2
          holds ex b3 being Element of NAT st
             for b4 being Element of NAT
                   st b3 <= b4
                holds |.(a2 . b4) - b1.| < b2;

:: TOPRNS_1:def 9
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
      b2 is convergent(b1)
   iff
      ex b3 being Element of the carrier of TOP-REAL b1 st
         for b4 being Element of REAL
               st 0 < b4
            holds ex b5 being Element of NAT st
               for b6 being Element of NAT
                     st b5 <= b6
                  holds |.(b2 . b6) - b3.| < b4;

:: TOPRNS_1:funcnot 6 => TOPRNS_1:func 6
definition
  let a1 be Element of NAT;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
  assume a2 is convergent(a1);
  func lim A2 -> Element of the carrier of TOP-REAL a1 means
    for b1 being Element of REAL
          st 0 < b1
       holds ex b2 being Element of NAT st
          for b3 being Element of NAT
                st b2 <= b3
             holds |.(a2 . b3) - it.| < b1;
end;

:: TOPRNS_1:def 10
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
   st b2 is convergent(b1)
for b3 being Element of the carrier of TOP-REAL b1 holds
      b3 = lim b2
   iff
      for b4 being Element of REAL
            st 0 < b4
         holds ex b5 being Element of NAT st
            for b6 being Element of NAT
                  st b5 <= b6
               holds |.(b2 . b6) - b3.| < b4;

:: TOPRNS_1:th 41
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
      st b2 is convergent(b1) & b3 is convergent(b1)
   holds b2 + b3 is convergent(b1);

:: TOPRNS_1:th 42
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
      st b2 is convergent(b1) & b3 is convergent(b1)
   holds lim (b2 + b3) = (lim b2) + lim b3;

:: TOPRNS_1:th 43
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
      st b3 is convergent(b1)
   holds b2 * b3 is convergent(b1);

:: TOPRNS_1:th 44
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
      st b3 is convergent(b1)
   holds lim (b2 * b3) = b2 * lim b3;

:: TOPRNS_1:th 45
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
      st b2 is convergent(b1)
   holds - b2 is convergent(b1);

:: TOPRNS_1:th 46
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
      st b2 is convergent(b1)
   holds lim - b2 = - lim b2;

:: TOPRNS_1:th 47
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
      st b2 is convergent(b1) & b3 is convergent(b1)
   holds b2 - b3 is convergent(b1);

:: TOPRNS_1:th 48
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
      st b2 is convergent(b1) & b3 is convergent(b1)
   holds lim (b2 - b3) = (lim b2) - lim b3;

:: TOPRNS_1:th 50
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
      st b2 is convergent(b1)
   holds b2 is bounded(b1);

:: TOPRNS_1:th 51
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
      st b2 is convergent(b1) & lim b2 <> 0.REAL b1
   holds ex b3 being Element of NAT st
      for b4 being Element of NAT
            st b3 <= b4
         holds |.lim b2.| / 2 < |.b2 . b4.|;