Article SEQFUNC, MML version 4.99.1005

:: SEQFUNC:modenot 1 => SEQFUNC:mode 1
definition
  let a1, a2 be set;
  mode Functional_Sequence of A1,A2 -> Relation-like Function-like set means
    proj1 it = NAT & proj2 it c= PFuncs(a1,a2);
end;

:: SEQFUNC:dfs 1
definiens
  let a1, a2 be set;
  let a3 be Relation-like Function-like set;
To prove
     a3 is Functional_Sequence of a1,a2
it is sufficient to prove
  thus proj1 a3 = NAT & proj2 a3 c= PFuncs(a1,a2);

:: SEQFUNC:def 1
theorem
for b1, b2 being set
for b3 being Relation-like Function-like set holds
      b3 is Functional_Sequence of b1,b2
   iff
      proj1 b3 = NAT & proj2 b3 c= PFuncs(b1,b2);

:: SEQFUNC:funcnot 1 => SEQFUNC:func 1
definition
  let a1, a2 be set;
  let a3 be Functional_Sequence of a1,a2;
  let a4 be natural set;
  redefine func a3 . a4 -> Function-like Relation of a1,a2;
end;

:: SEQFUNC:th 1
theorem
for b1, b2 being set
for b3 being Relation-like Function-like set holds
      b3 is Functional_Sequence of b1,b2
   iff
      proj1 b3 = NAT &
       (for b4 being Element of NAT holds
          b3 . b4 is Function-like Relation of b1,b2);

:: SEQFUNC:th 2
theorem
for b1, b2 being set
for b3, b4 being Functional_Sequence of b2,b1
      st for b5 being Element of NAT holds
           b3 . b5 = b4 . b5
   holds b3 = b4;

:: SEQFUNC:sch 1
scheme SEQFUNC:sch 1
{F1 -> set,
  F2 -> set,
  F3 -> Function-like Relation of F1(),F2()}:
ex b1 being Functional_Sequence of F1(),F2() st
   for b2 being natural set holds
      b1 . b2 = F3(b2)


:: SEQFUNC:funcnot 2 => SEQFUNC:func 2
definition
  let a1 be non empty set;
  let a2 be Functional_Sequence of a1,REAL;
  let a3 be real set;
  func A3 (#) A2 -> Functional_Sequence of a1,REAL means
    for b1 being natural set holds
       it . b1 = a3 (#) (a2 . b1);
end;

:: SEQFUNC:def 2
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL
for b3 being real set
for b4 being Functional_Sequence of b1,REAL holds
      b4 = b3 (#) b2
   iff
      for b5 being natural set holds
         b4 . b5 = b3 (#) (b2 . b5);

:: SEQFUNC:funcnot 3 => SEQFUNC:func 3
definition
  let a1 be non empty set;
  let a2 be Functional_Sequence of a1,REAL;
  func A2 " -> Functional_Sequence of a1,REAL means
    for b1 being natural set holds
       it . b1 = (a2 . b1) ^;
end;

:: SEQFUNC:def 3
theorem
for b1 being non empty set
for b2, b3 being Functional_Sequence of b1,REAL holds
   b3 = b2 "
iff
   for b4 being natural set holds
      b3 . b4 = (b2 . b4) ^;

:: SEQFUNC:funcnot 4 => SEQFUNC:func 4
definition
  let a1 be non empty set;
  let a2 be Functional_Sequence of a1,REAL;
  func - A2 -> Functional_Sequence of a1,REAL means
    for b1 being natural set holds
       it . b1 = - (a2 . b1);
end;

:: SEQFUNC:def 4
theorem
for b1 being non empty set
for b2, b3 being Functional_Sequence of b1,REAL holds
   b3 = - b2
iff
   for b4 being natural set holds
      b3 . b4 = - (b2 . b4);

:: SEQFUNC:funcnot 5 => SEQFUNC:func 5
definition
  let a1 be non empty set;
  let a2 be Functional_Sequence of a1,REAL;
  func abs A2 -> Functional_Sequence of a1,REAL means
    for b1 being natural set holds
       it . b1 = abs (a2 . b1);
end;

:: SEQFUNC:def 5
theorem
for b1 being non empty set
for b2, b3 being Functional_Sequence of b1,REAL holds
   b3 = abs b2
iff
   for b4 being natural set holds
      b3 . b4 = abs (b2 . b4);

:: SEQFUNC:funcnot 6 => SEQFUNC:func 6
definition
  let a1 be non empty set;
  let a2, a3 be Functional_Sequence of a1,REAL;
  func A2 + A3 -> Functional_Sequence of a1,REAL means
    for b1 being natural set holds
       it . b1 = (a2 . b1) + (a3 . b1);
end;

:: SEQFUNC:def 6
theorem
for b1 being non empty set
for b2, b3, b4 being Functional_Sequence of b1,REAL holds
   b4 = b2 + b3
iff
   for b5 being natural set holds
      b4 . b5 = (b2 . b5) + (b3 . b5);

:: SEQFUNC:funcnot 7 => SEQFUNC:func 7
definition
  let a1 be non empty set;
  let a2, a3 be Functional_Sequence of a1,REAL;
  func A2 - A3 -> Functional_Sequence of a1,REAL equals
    a2 + - a3;
end;

:: SEQFUNC:def 7
theorem
for b1 being non empty set
for b2, b3 being Functional_Sequence of b1,REAL holds
b2 - b3 = b2 + - b3;

:: SEQFUNC:funcnot 8 => SEQFUNC:func 8
definition
  let a1 be non empty set;
  let a2, a3 be Functional_Sequence of a1,REAL;
  func A2 (#) A3 -> Functional_Sequence of a1,REAL means
    for b1 being natural set holds
       it . b1 = (a2 . b1) (#) (a3 . b1);
end;

:: SEQFUNC:def 8
theorem
for b1 being non empty set
for b2, b3, b4 being Functional_Sequence of b1,REAL holds
   b4 = b2 (#) b3
iff
   for b5 being natural set holds
      b4 . b5 = (b2 . b5) (#) (b3 . b5);

:: SEQFUNC:funcnot 9 => SEQFUNC:func 9
definition
  let a1 be non empty set;
  let a2, a3 be Functional_Sequence of a1,REAL;
  func A3 / A2 -> Functional_Sequence of a1,REAL equals
    a3 (#) (a2 ");
end;

:: SEQFUNC:def 9
theorem
for b1 being non empty set
for b2, b3 being Functional_Sequence of b1,REAL holds
b3 / b2 = b3 (#) (b2 ");

:: SEQFUNC:th 3
theorem
for b1 being non empty set
for b2, b3, b4 being Functional_Sequence of b1,REAL holds
   b2 = b3 / b4
iff
   for b5 being Element of NAT holds
      b2 . b5 = (b3 . b5) / (b4 . b5);

:: SEQFUNC:th 4
theorem
for b1 being non empty set
for b2, b3, b4 being Functional_Sequence of b1,REAL holds
   b2 = b3 - b4
iff
   for b5 being Element of NAT holds
      b2 . b5 = (b3 . b5) - (b4 . b5);

:: SEQFUNC:th 5
theorem
for b1 being non empty set
for b2, b3, b4 being Functional_Sequence of b1,REAL holds
b2 + b3 = b3 + b2 &
 (b2 + b3) + b4 = b2 + (b3 + b4);

:: SEQFUNC:th 6
theorem
for b1 being non empty set
for b2, b3, b4 being Functional_Sequence of b1,REAL holds
b2 (#) b3 = b3 (#) b2 &
 (b2 (#) b3) (#) b4 = b2 (#) (b3 (#) b4);

:: SEQFUNC:th 7
theorem
for b1 being non empty set
for b2, b3, b4 being Functional_Sequence of b1,REAL holds
(b2 + b3) (#) b4 = (b2 (#) b4) + (b3 (#) b4) &
 b4 (#) (b2 + b3) = (b4 (#) b2) + (b4 (#) b3);

:: SEQFUNC:th 8
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL holds
   - b2 = (- 1) (#) b2;

:: SEQFUNC:th 9
theorem
for b1 being non empty set
for b2, b3, b4 being Functional_Sequence of b1,REAL holds
(b2 - b3) (#) b4 = (b2 (#) b4) - (b3 (#) b4) &
 (b4 (#) b2) - (b4 (#) b3) = b4 (#) (b2 - b3);

:: SEQFUNC:th 10
theorem
for b1 being non empty set
for b2 being Element of REAL
for b3, b4 being Functional_Sequence of b1,REAL holds
b2 (#) (b3 + b4) = (b2 (#) b3) + (b2 (#) b4) &
 b2 (#) (b3 - b4) = (b2 (#) b3) - (b2 (#) b4);

:: SEQFUNC:th 11
theorem
for b1 being non empty set
for b2, b3 being Element of REAL
for b4 being Functional_Sequence of b1,REAL holds
   (b2 * b3) (#) b4 = b2 (#) (b3 (#) b4);

:: SEQFUNC:th 12
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL holds
   1 (#) b2 = b2;

:: SEQFUNC:th 13
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL holds
   - - b2 = b2;

:: SEQFUNC:th 14
theorem
for b1 being non empty set
for b2, b3 being Functional_Sequence of b1,REAL holds
b2 " (#) (b3 ") = (b2 (#) b3) ";

:: SEQFUNC:th 15
theorem
for b1 being non empty set
for b2 being Element of REAL
for b3 being Functional_Sequence of b1,REAL
      st b2 <> 0
   holds (b2 (#) b3) " = b2 " (#) (b3 ");

:: SEQFUNC:th 16
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL holds
   (abs b2) " = abs (b2 ");

:: SEQFUNC:th 17
theorem
for b1 being non empty set
for b2, b3 being Functional_Sequence of b1,REAL holds
abs (b2 (#) b3) = (abs b2) (#) abs b3;

:: SEQFUNC:th 18
theorem
for b1 being non empty set
for b2, b3 being Functional_Sequence of b1,REAL holds
abs (b2 / b3) = (abs b2) / abs b3;

:: SEQFUNC:th 19
theorem
for b1 being non empty set
for b2 being Element of REAL
for b3 being Functional_Sequence of b1,REAL holds
   abs (b2 (#) b3) = (abs b2) (#) abs b3;

:: SEQFUNC:prednot 1 => SEQFUNC:pred 1
definition
  let a1, a2 be set;
  let a3 be Functional_Sequence of a1,a2;
  let a4 be set;
  pred A4 common_on_dom A3 means
    a4 <> {} &
     (for b1 being Element of NAT holds
        a4 c= dom (a3 . b1));
end;

:: SEQFUNC:dfs 10
definiens
  let a1, a2 be set;
  let a3 be Functional_Sequence of a1,a2;
  let a4 be set;
To prove
     a4 common_on_dom a3
it is sufficient to prove
  thus a4 <> {} &
     (for b1 being Element of NAT holds
        a4 c= dom (a3 . b1));

:: SEQFUNC:def 10
theorem
for b1, b2 being set
for b3 being Functional_Sequence of b1,b2
for b4 being set holds
      b4 common_on_dom b3
   iff
      b4 <> {} &
       (for b5 being Element of NAT holds
          b4 c= dom (b3 . b5));

:: SEQFUNC:funcnot 10 => SEQFUNC:func 10
definition
  let a1 be non empty set;
  let a2 be Functional_Sequence of a1,REAL;
  let a3 be Element of a1;
  func A2 # A3 -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = (a2 . b1) . a3;
end;

:: SEQFUNC:def 11
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL
for b3 being Element of b1
for b4 being Function-like quasi_total Relation of NAT,REAL holds
      b4 = b2 # b3
   iff
      for b5 being Element of NAT holds
         b4 . b5 = (b2 . b5) . b3;

:: SEQFUNC:prednot 2 => SEQFUNC:pred 2
definition
  let a1 be non empty set;
  let a2 be Functional_Sequence of a1,REAL;
  let a3 be set;
  pred A2 is_point_conv_on A3 means
    a3 common_on_dom a2 &
     (ex b1 being Function-like Relation of a1,REAL st
        a3 = dom b1 &
         (for b2 being Element of a1
            st b2 in a3
         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 abs (((a2 . b5) . b2) - (b1 . b2)) < b3));
end;

:: SEQFUNC:dfs 12
definiens
  let a1 be non empty set;
  let a2 be Functional_Sequence of a1,REAL;
  let a3 be set;
To prove
     a2 is_point_conv_on a3
it is sufficient to prove
  thus a3 common_on_dom a2 &
     (ex b1 being Function-like Relation of a1,REAL st
        a3 = dom b1 &
         (for b2 being Element of a1
            st b2 in a3
         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 abs (((a2 . b5) . b2) - (b1 . b2)) < b3));

:: SEQFUNC:def 12
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL
for b3 being set holds
      b2 is_point_conv_on b3
   iff
      b3 common_on_dom b2 &
       (ex b4 being Function-like Relation of b1,REAL st
          b3 = dom b4 &
           (for b5 being Element of b1
              st b5 in b3
           for b6 being Element of REAL
                 st 0 < b6
              holds ex b7 being Element of NAT st
                 for b8 being Element of NAT
                       st b7 <= b8
                    holds abs (((b2 . b8) . b5) - (b4 . b5)) < b6));

:: SEQFUNC:th 20
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL
for b3 being set holds
      b2 is_point_conv_on b3
   iff
      b3 common_on_dom b2 &
       (ex b4 being Function-like Relation of b1,REAL st
          b3 = dom b4 &
           (for b5 being Element of b1
                 st b5 in b3
              holds b2 # b5 is convergent & lim (b2 # b5) = b4 . b5));

:: SEQFUNC:th 21
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL
for b3 being set holds
      b2 is_point_conv_on b3
   iff
      b3 common_on_dom b2 &
       (for b4 being Element of b1
             st b4 in b3
          holds b2 # b4 is convergent);

:: SEQFUNC:prednot 3 => SEQFUNC:pred 3
definition
  let a1 be non empty set;
  let a2 be Functional_Sequence of a1,REAL;
  let a3 be set;
  pred A2 is_unif_conv_on A3 means
    a3 common_on_dom a2 &
     (ex b1 being Function-like Relation of a1,REAL st
        a3 = dom b1 &
         (for b2 being Element of REAL
               st 0 < b2
            holds ex b3 being Element of NAT st
               for b4 being Element of NAT
               for b5 being Element of a1
                     st b3 <= b4 & b5 in a3
                  holds abs (((a2 . b4) . b5) - (b1 . b5)) < b2));
end;

:: SEQFUNC:dfs 13
definiens
  let a1 be non empty set;
  let a2 be Functional_Sequence of a1,REAL;
  let a3 be set;
To prove
     a2 is_unif_conv_on a3
it is sufficient to prove
  thus a3 common_on_dom a2 &
     (ex b1 being Function-like Relation of a1,REAL st
        a3 = dom b1 &
         (for b2 being Element of REAL
               st 0 < b2
            holds ex b3 being Element of NAT st
               for b4 being Element of NAT
               for b5 being Element of a1
                     st b3 <= b4 & b5 in a3
                  holds abs (((a2 . b4) . b5) - (b1 . b5)) < b2));

:: SEQFUNC:def 13
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL
for b3 being set holds
      b2 is_unif_conv_on b3
   iff
      b3 common_on_dom b2 &
       (ex b4 being Function-like Relation of b1,REAL st
          b3 = dom b4 &
           (for b5 being Element of REAL
                 st 0 < b5
              holds ex b6 being Element of NAT st
                 for b7 being Element of NAT
                 for b8 being Element of b1
                       st b6 <= b7 & b8 in b3
                    holds abs (((b2 . b7) . b8) - (b4 . b8)) < b5));

:: SEQFUNC:funcnot 11 => SEQFUNC:func 11
definition
  let a1 be non empty set;
  let a2 be Functional_Sequence of a1,REAL;
  let a3 be set;
  assume a2 is_point_conv_on a3;
  func lim(A2,A3) -> Function-like Relation of a1,REAL means
    dom it = a3 &
     (for b1 being Element of a1
           st b1 in dom it
        holds it . b1 = lim (a2 # b1));
end;

:: SEQFUNC:def 14
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL
for b3 being set
   st b2 is_point_conv_on b3
for b4 being Function-like Relation of b1,REAL holds
      b4 = lim(b2,b3)
   iff
      dom b4 = b3 &
       (for b5 being Element of b1
             st b5 in dom b4
          holds b4 . b5 = lim (b2 # b5));

:: SEQFUNC:th 22
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL
for b3 being set
for b4 being Function-like Relation of b1,REAL
      st b2 is_point_conv_on b3
   holds    b4 = lim(b2,b3)
   iff
      dom b4 = b3 &
       (for b5 being Element of b1
          st b5 in b3
       for b6 being Element of REAL
             st 0 < b6
          holds ex b7 being Element of NAT st
             for b8 being Element of NAT
                   st b7 <= b8
                holds abs (((b2 . b8) . b5) - (b4 . b5)) < b6);

:: SEQFUNC:th 23
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL
for b3 being set
      st b2 is_unif_conv_on b3
   holds b2 is_point_conv_on b3;

:: SEQFUNC:th 24
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL
for b3, b4 being set
      st b3 c= b4 & b3 <> {} & b4 common_on_dom b2
   holds b3 common_on_dom b2;

:: SEQFUNC:th 25
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL
for b3, b4 being set
      st b3 c= b4 & b3 <> {} & b2 is_point_conv_on b4
   holds b2 is_point_conv_on b3 & (lim(b2,b4)) | b3 = lim(b2,b3);

:: SEQFUNC:th 26
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL
for b3, b4 being set
      st b3 c= b4 & b3 <> {} & b2 is_unif_conv_on b4
   holds b2 is_unif_conv_on b3;

:: SEQFUNC:th 27
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL
for b3 being set
   st b3 common_on_dom b2
for b4 being Element of b1
      st b4 in b3
   holds {b4} common_on_dom b2;

:: SEQFUNC:th 28
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL
for b3 being set
   st b2 is_point_conv_on b3
for b4 being Element of b1
      st b4 in b3
   holds {b4} common_on_dom b2;

:: SEQFUNC:th 29
theorem
for b1 being non empty set
for b2, b3 being Functional_Sequence of b1,REAL
for b4 being Element of b1
      st {b4} common_on_dom b2 & {b4} common_on_dom b3
   holds (b2 # b4) + (b3 # b4) = (b2 + b3) # b4 &
    (b2 # b4) - (b3 # b4) = (b2 - b3) # b4 &
    (b2 # b4) (#) (b3 # b4) = (b2 (#) b3) # b4;

:: SEQFUNC:th 30
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL
for b3 being Element of b1
      st {b3} common_on_dom b2
   holds (abs b2) # b3 = abs (b2 # b3) & (- b2) # b3 = - (b2 # b3);

:: SEQFUNC:th 31
theorem
for b1 being non empty set
for b2 being Element of REAL
for b3 being Functional_Sequence of b1,REAL
for b4 being Element of b1
      st {b4} common_on_dom b3
   holds (b2 (#) b3) # b4 = b2 (#) (b3 # b4);

:: SEQFUNC:th 32
theorem
for b1 being non empty set
for b2, b3 being Functional_Sequence of b1,REAL
for b4 being set
   st b4 common_on_dom b2 & b4 common_on_dom b3
for b5 being Element of b1
      st b5 in b4
   holds (b2 # b5) + (b3 # b5) = (b2 + b3) # b5 &
    (b2 # b5) - (b3 # b5) = (b2 - b3) # b5 &
    (b2 # b5) (#) (b3 # b5) = (b2 (#) b3) # b5;

:: SEQFUNC:th 33
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL
for b3 being set
   st b3 common_on_dom b2
for b4 being Element of b1
      st b4 in b3
   holds (abs b2) # b4 = abs (b2 # b4) & (- b2) # b4 = - (b2 # b4);

:: SEQFUNC:th 34
theorem
for b1 being non empty set
for b2 being Element of REAL
for b3 being Functional_Sequence of b1,REAL
for b4 being set
   st b4 common_on_dom b3
for b5 being Element of b1
      st b5 in b4
   holds (b2 (#) b3) # b5 = b2 (#) (b3 # b5);

:: SEQFUNC:th 35
theorem
for b1 being non empty set
for b2, b3 being Functional_Sequence of b1,REAL
for b4 being set
   st b2 is_point_conv_on b4 & b3 is_point_conv_on b4
for b5 being Element of b1
      st b5 in b4
   holds (b2 # b5) + (b3 # b5) = (b2 + b3) # b5 &
    (b2 # b5) - (b3 # b5) = (b2 - b3) # b5 &
    (b2 # b5) (#) (b3 # b5) = (b2 (#) b3) # b5;

:: SEQFUNC:th 36
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL
for b3 being set
   st b2 is_point_conv_on b3
for b4 being Element of b1
      st b4 in b3
   holds (abs b2) # b4 = abs (b2 # b4) & (- b2) # b4 = - (b2 # b4);

:: SEQFUNC:th 37
theorem
for b1 being non empty set
for b2 being Element of REAL
for b3 being Functional_Sequence of b1,REAL
for b4 being set
   st b3 is_point_conv_on b4
for b5 being Element of b1
      st b5 in b4
   holds (b2 (#) b3) # b5 = b2 (#) (b3 # b5);

:: SEQFUNC:th 38
theorem
for b1 being non empty set
for b2, b3 being Functional_Sequence of b1,REAL
for b4 being set
      st b4 common_on_dom b2 & b4 common_on_dom b3
   holds b4 common_on_dom b2 + b3 & b4 common_on_dom b2 - b3 & b4 common_on_dom b2 (#) b3;

:: SEQFUNC:th 39
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL
for b3 being set
      st b3 common_on_dom b2
   holds b3 common_on_dom abs b2 & b3 common_on_dom - b2;

:: SEQFUNC:th 40
theorem
for b1 being non empty set
for b2 being Element of REAL
for b3 being Functional_Sequence of b1,REAL
for b4 being set
      st b4 common_on_dom b3
   holds b4 common_on_dom b2 (#) b3;

:: SEQFUNC:th 41
theorem
for b1 being non empty set
for b2, b3 being Functional_Sequence of b1,REAL
for b4 being set
      st b2 is_point_conv_on b4 & b3 is_point_conv_on b4
   holds b2 + b3 is_point_conv_on b4 &
    lim(b2 + b3,b4) = (lim(b2,b4)) + lim(b3,b4) &
    b2 - b3 is_point_conv_on b4 &
    lim(b2 - b3,b4) = (lim(b2,b4)) - lim(b3,b4) &
    b2 (#) b3 is_point_conv_on b4 &
    lim(b2 (#) b3,b4) = (lim(b2,b4)) (#) lim(b3,b4);

:: SEQFUNC:th 42
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL
for b3 being set
      st b2 is_point_conv_on b3
   holds abs b2 is_point_conv_on b3 & lim(abs b2,b3) = abs lim(b2,b3) & - b2 is_point_conv_on b3 & lim(- b2,b3) = - lim(b2,b3);

:: SEQFUNC:th 43
theorem
for b1 being non empty set
for b2 being Element of REAL
for b3 being Functional_Sequence of b1,REAL
for b4 being set
      st b3 is_point_conv_on b4
   holds b2 (#) b3 is_point_conv_on b4 &
    lim(b2 (#) b3,b4) = b2 (#) lim(b3,b4);

:: SEQFUNC:th 44
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,REAL
for b3 being set holds
      b2 is_unif_conv_on b3
   iff
      b3 common_on_dom b2 &
       b2 is_point_conv_on b3 &
       (for b4 being Element of REAL
             st 0 < b4
          holds ex b5 being Element of NAT st
             for b6 being Element of NAT
             for b7 being Element of b1
                   st b5 <= b6 & b7 in b3
                holds abs (((b2 . b6) . b7) - ((lim(b2,b3)) . b7)) < b4);

:: SEQFUNC:th 45
theorem
for b1 being set
for b2 being Functional_Sequence of REAL,REAL
      st b2 is_unif_conv_on b1 &
         (for b3 being Element of NAT holds
            b2 . b3 is_continuous_on b1)
   holds lim(b2,b1) is_continuous_on b1;