Article LIMFUNC2, MML version 4.99.1005

:: LIMFUNC2:th 1
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is convergent & b1 < lim b2
   holds ex b3 being Element of NAT st
      for b4 being Element of NAT
            st b3 <= b4
         holds b1 < b2 . b4;

:: LIMFUNC2:th 2
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is convergent & lim b2 < b1
   holds ex b3 being Element of NAT st
      for b4 being Element of NAT
            st b3 <= b4
         holds b2 . b4 < b1;

:: LIMFUNC2:th 3
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
   st 0 < b1 & ].b2 - b1,b2.[ c= dom b3
for b4 being Element of REAL
      st b4 < b2
   holds ex b5 being Element of REAL st
      b4 < b5 & b5 < b2 & b5 in dom b3;

:: LIMFUNC2:th 4
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
   st 0 < b1 & ].b2,b2 + b1.[ c= dom b3
for b4 being Element of REAL
      st b2 < b4
   holds ex b5 being Element of REAL st
      b5 < b4 & b2 < b5 & b5 in dom b3;

:: LIMFUNC2:th 5
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like Relation of REAL,REAL
      st for b4 being Element of NAT holds
           b1 - (1 / (b4 + 1)) < b2 . b4 &
            b2 . b4 < b1 &
            b2 . b4 in dom b3
   holds b2 is convergent & lim b2 = b1 & rng b2 c= dom b3 & rng b2 c= (dom b3) /\ halfline b1;

:: LIMFUNC2:th 6
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like Relation of REAL,REAL
      st for b4 being Element of NAT holds
           b1 < b2 . b4 &
            b2 . b4 < b1 + (1 / (b4 + 1)) &
            b2 . b4 in dom b3
   holds b2 is convergent & lim b2 = b1 & rng b2 c= dom b3 & rng b2 c= (dom b3) /\ right_open_halfline b1;

:: LIMFUNC2:prednot 1 => LIMFUNC2:pred 1
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
  pred A1 is_left_convergent_in A2 means
    (for b1 being Element of REAL
           st b1 < a2
        holds ex b2 being Element of REAL st
           b1 < b2 & b2 < a2 & b2 in dom a1) &
     (ex b1 being Element of REAL st
        for b2 being Function-like quasi_total Relation of NAT,REAL
              st b2 is convergent & lim b2 = a2 & rng b2 c= (dom a1) /\ halfline a2
           holds a1 * b2 is convergent & lim (a1 * b2) = b1);
end;

:: LIMFUNC2:dfs 1
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
To prove
     a1 is_left_convergent_in a2
it is sufficient to prove
  thus (for b1 being Element of REAL
           st b1 < a2
        holds ex b2 being Element of REAL st
           b1 < b2 & b2 < a2 & b2 in dom a1) &
     (ex b1 being Element of REAL st
        for b2 being Function-like quasi_total Relation of NAT,REAL
              st b2 is convergent & lim b2 = a2 & rng b2 c= (dom a1) /\ halfline a2
           holds a1 * b2 is convergent & lim (a1 * b2) = b1);

:: LIMFUNC2:def 1
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
      b1 is_left_convergent_in b2
   iff
      (for b3 being Element of REAL
             st b3 < b2
          holds ex b4 being Element of REAL st
             b3 < b4 & b4 < b2 & b4 in dom b1) &
       (ex b3 being Element of REAL st
          for b4 being Function-like quasi_total Relation of NAT,REAL
                st b4 is convergent & lim b4 = b2 & rng b4 c= (dom b1) /\ halfline b2
             holds b1 * b4 is convergent & lim (b1 * b4) = b3);

:: LIMFUNC2:prednot 2 => LIMFUNC2:pred 2
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
  pred A1 is_left_divergent_to+infty_in A2 means
    (for b1 being Element of REAL
           st b1 < a2
        holds ex b2 being Element of REAL st
           b1 < b2 & b2 < a2 & b2 in dom a1) &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st b1 is convergent & lim b1 = a2 & rng b1 c= (dom a1) /\ halfline a2
        holds a1 * b1 is divergent_to+infty);
end;

:: LIMFUNC2:dfs 2
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
To prove
     a1 is_left_divergent_to+infty_in a2
it is sufficient to prove
  thus (for b1 being Element of REAL
           st b1 < a2
        holds ex b2 being Element of REAL st
           b1 < b2 & b2 < a2 & b2 in dom a1) &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st b1 is convergent & lim b1 = a2 & rng b1 c= (dom a1) /\ halfline a2
        holds a1 * b1 is divergent_to+infty);

:: LIMFUNC2:def 2
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
      b1 is_left_divergent_to+infty_in b2
   iff
      (for b3 being Element of REAL
             st b3 < b2
          holds ex b4 being Element of REAL st
             b3 < b4 & b4 < b2 & b4 in dom b1) &
       (for b3 being Function-like quasi_total Relation of NAT,REAL
             st b3 is convergent & lim b3 = b2 & rng b3 c= (dom b1) /\ halfline b2
          holds b1 * b3 is divergent_to+infty);

:: LIMFUNC2:prednot 3 => LIMFUNC2:pred 3
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
  pred A1 is_left_divergent_to-infty_in A2 means
    (for b1 being Element of REAL
           st b1 < a2
        holds ex b2 being Element of REAL st
           b1 < b2 & b2 < a2 & b2 in dom a1) &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st b1 is convergent & lim b1 = a2 & rng b1 c= (dom a1) /\ halfline a2
        holds a1 * b1 is divergent_to-infty);
end;

:: LIMFUNC2:dfs 3
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
To prove
     a1 is_left_divergent_to-infty_in a2
it is sufficient to prove
  thus (for b1 being Element of REAL
           st b1 < a2
        holds ex b2 being Element of REAL st
           b1 < b2 & b2 < a2 & b2 in dom a1) &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st b1 is convergent & lim b1 = a2 & rng b1 c= (dom a1) /\ halfline a2
        holds a1 * b1 is divergent_to-infty);

:: LIMFUNC2:def 3
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
      b1 is_left_divergent_to-infty_in b2
   iff
      (for b3 being Element of REAL
             st b3 < b2
          holds ex b4 being Element of REAL st
             b3 < b4 & b4 < b2 & b4 in dom b1) &
       (for b3 being Function-like quasi_total Relation of NAT,REAL
             st b3 is convergent & lim b3 = b2 & rng b3 c= (dom b1) /\ halfline b2
          holds b1 * b3 is divergent_to-infty);

:: LIMFUNC2:prednot 4 => LIMFUNC2:pred 4
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
  pred A1 is_right_convergent_in A2 means
    (for b1 being Element of REAL
           st a2 < b1
        holds ex b2 being Element of REAL st
           b2 < b1 & a2 < b2 & b2 in dom a1) &
     (ex b1 being Element of REAL st
        for b2 being Function-like quasi_total Relation of NAT,REAL
              st b2 is convergent & lim b2 = a2 & rng b2 c= (dom a1) /\ right_open_halfline a2
           holds a1 * b2 is convergent & lim (a1 * b2) = b1);
end;

:: LIMFUNC2:dfs 4
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
To prove
     a1 is_right_convergent_in a2
it is sufficient to prove
  thus (for b1 being Element of REAL
           st a2 < b1
        holds ex b2 being Element of REAL st
           b2 < b1 & a2 < b2 & b2 in dom a1) &
     (ex b1 being Element of REAL st
        for b2 being Function-like quasi_total Relation of NAT,REAL
              st b2 is convergent & lim b2 = a2 & rng b2 c= (dom a1) /\ right_open_halfline a2
           holds a1 * b2 is convergent & lim (a1 * b2) = b1);

:: LIMFUNC2:def 4
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
      b1 is_right_convergent_in b2
   iff
      (for b3 being Element of REAL
             st b2 < b3
          holds ex b4 being Element of REAL st
             b4 < b3 & b2 < b4 & b4 in dom b1) &
       (ex b3 being Element of REAL st
          for b4 being Function-like quasi_total Relation of NAT,REAL
                st b4 is convergent & lim b4 = b2 & rng b4 c= (dom b1) /\ right_open_halfline b2
             holds b1 * b4 is convergent & lim (b1 * b4) = b3);

:: LIMFUNC2:prednot 5 => LIMFUNC2:pred 5
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
  pred A1 is_right_divergent_to+infty_in A2 means
    (for b1 being Element of REAL
           st a2 < b1
        holds ex b2 being Element of REAL st
           b2 < b1 & a2 < b2 & b2 in dom a1) &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st b1 is convergent & lim b1 = a2 & rng b1 c= (dom a1) /\ right_open_halfline a2
        holds a1 * b1 is divergent_to+infty);
end;

:: LIMFUNC2:dfs 5
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
To prove
     a1 is_right_divergent_to+infty_in a2
it is sufficient to prove
  thus (for b1 being Element of REAL
           st a2 < b1
        holds ex b2 being Element of REAL st
           b2 < b1 & a2 < b2 & b2 in dom a1) &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st b1 is convergent & lim b1 = a2 & rng b1 c= (dom a1) /\ right_open_halfline a2
        holds a1 * b1 is divergent_to+infty);

:: LIMFUNC2:def 5
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
      b1 is_right_divergent_to+infty_in b2
   iff
      (for b3 being Element of REAL
             st b2 < b3
          holds ex b4 being Element of REAL st
             b4 < b3 & b2 < b4 & b4 in dom b1) &
       (for b3 being Function-like quasi_total Relation of NAT,REAL
             st b3 is convergent & lim b3 = b2 & rng b3 c= (dom b1) /\ right_open_halfline b2
          holds b1 * b3 is divergent_to+infty);

:: LIMFUNC2:prednot 6 => LIMFUNC2:pred 6
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
  pred A1 is_right_divergent_to-infty_in A2 means
    (for b1 being Element of REAL
           st a2 < b1
        holds ex b2 being Element of REAL st
           b2 < b1 & a2 < b2 & b2 in dom a1) &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st b1 is convergent & lim b1 = a2 & rng b1 c= (dom a1) /\ right_open_halfline a2
        holds a1 * b1 is divergent_to-infty);
end;

:: LIMFUNC2:dfs 6
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
To prove
     a1 is_right_divergent_to-infty_in a2
it is sufficient to prove
  thus (for b1 being Element of REAL
           st a2 < b1
        holds ex b2 being Element of REAL st
           b2 < b1 & a2 < b2 & b2 in dom a1) &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st b1 is convergent & lim b1 = a2 & rng b1 c= (dom a1) /\ right_open_halfline a2
        holds a1 * b1 is divergent_to-infty);

:: LIMFUNC2:def 6
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
      b1 is_right_divergent_to-infty_in b2
   iff
      (for b3 being Element of REAL
             st b2 < b3
          holds ex b4 being Element of REAL st
             b4 < b3 & b2 < b4 & b4 in dom b1) &
       (for b3 being Function-like quasi_total Relation of NAT,REAL
             st b3 is convergent & lim b3 = b2 & rng b3 c= (dom b1) /\ right_open_halfline b2
          holds b1 * b3 is divergent_to-infty);

:: LIMFUNC2:th 13
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL holds
      b2 is_left_convergent_in b1
   iff
      (for b3 being Element of REAL
             st b3 < b1
          holds ex b4 being Element of REAL st
             b3 < b4 & b4 < b1 & b4 in dom b2) &
       (ex b3 being Element of REAL st
          for b4 being Element of REAL
                st 0 < b4
             holds ex b5 being Element of REAL st
                b5 < b1 &
                 (for b6 being Element of REAL
                       st b5 < b6 & b6 < b1 & b6 in dom b2
                    holds abs ((b2 . b6) - b3) < b4));

:: LIMFUNC2:th 14
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL holds
      b2 is_left_divergent_to+infty_in b1
   iff
      (for b3 being Element of REAL
             st b3 < b1
          holds ex b4 being Element of REAL st
             b3 < b4 & b4 < b1 & b4 in dom b2) &
       (for b3 being Element of REAL holds
          ex b4 being Element of REAL st
             b4 < b1 &
              (for b5 being Element of REAL
                    st b4 < b5 & b5 < b1 & b5 in dom b2
                 holds b3 < b2 . b5));

:: LIMFUNC2:th 15
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL holds
      b2 is_left_divergent_to-infty_in b1
   iff
      (for b3 being Element of REAL
             st b3 < b1
          holds ex b4 being Element of REAL st
             b3 < b4 & b4 < b1 & b4 in dom b2) &
       (for b3 being Element of REAL holds
          ex b4 being Element of REAL st
             b4 < b1 &
              (for b5 being Element of REAL
                    st b4 < b5 & b5 < b1 & b5 in dom b2
                 holds b2 . b5 < b3));

:: LIMFUNC2:th 16
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL holds
      b2 is_right_convergent_in b1
   iff
      (for b3 being Element of REAL
             st b1 < b3
          holds ex b4 being Element of REAL st
             b4 < b3 & b1 < b4 & b4 in dom b2) &
       (ex b3 being Element of REAL st
          for b4 being Element of REAL
                st 0 < b4
             holds ex b5 being Element of REAL st
                b1 < b5 &
                 (for b6 being Element of REAL
                       st b6 < b5 & b1 < b6 & b6 in dom b2
                    holds abs ((b2 . b6) - b3) < b4));

:: LIMFUNC2:th 17
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL holds
      b2 is_right_divergent_to+infty_in b1
   iff
      (for b3 being Element of REAL
             st b1 < b3
          holds ex b4 being Element of REAL st
             b4 < b3 & b1 < b4 & b4 in dom b2) &
       (for b3 being Element of REAL holds
          ex b4 being Element of REAL st
             b1 < b4 &
              (for b5 being Element of REAL
                    st b5 < b4 & b1 < b5 & b5 in dom b2
                 holds b3 < b2 . b5));

:: LIMFUNC2:th 18
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL holds
      b2 is_right_divergent_to-infty_in b1
   iff
      (for b3 being Element of REAL
             st b1 < b3
          holds ex b4 being Element of REAL st
             b4 < b3 & b1 < b4 & b4 in dom b2) &
       (for b3 being Element of REAL holds
          ex b4 being Element of REAL st
             b1 < b4 &
              (for b5 being Element of REAL
                    st b5 < b4 & b1 < b5 & b5 in dom b2
                 holds b2 . b5 < b3));

:: LIMFUNC2:th 19
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_left_divergent_to+infty_in b1 &
         b3 is_left_divergent_to+infty_in b1 &
         (for b4 being Element of REAL
               st b4 < b1
            holds ex b5 being Element of REAL st
               b4 < b5 & b5 < b1 & b5 in (dom b2) /\ dom b3)
   holds b2 + b3 is_left_divergent_to+infty_in b1 & b2 (#) b3 is_left_divergent_to+infty_in b1;

:: LIMFUNC2:th 20
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_left_divergent_to-infty_in b1 &
         b3 is_left_divergent_to-infty_in b1 &
         (for b4 being Element of REAL
               st b4 < b1
            holds ex b5 being Element of REAL st
               b4 < b5 & b5 < b1 & b5 in (dom b2) /\ dom b3)
   holds b2 + b3 is_left_divergent_to-infty_in b1 & b2 (#) b3 is_left_divergent_to+infty_in b1;

:: LIMFUNC2:th 21
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_right_divergent_to+infty_in b1 &
         b3 is_right_divergent_to+infty_in b1 &
         (for b4 being Element of REAL
               st b1 < b4
            holds ex b5 being Element of REAL st
               b5 < b4 & b1 < b5 & b5 in (dom b2) /\ dom b3)
   holds b2 + b3 is_right_divergent_to+infty_in b1 & b2 (#) b3 is_right_divergent_to+infty_in b1;

:: LIMFUNC2:th 22
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_right_divergent_to-infty_in b1 &
         b3 is_right_divergent_to-infty_in b1 &
         (for b4 being Element of REAL
               st b1 < b4
            holds ex b5 being Element of REAL st
               b5 < b4 & b1 < b5 & b5 in (dom b2) /\ dom b3)
   holds b2 + b3 is_right_divergent_to-infty_in b1 & b2 (#) b3 is_right_divergent_to+infty_in b1;

:: LIMFUNC2:th 23
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_left_divergent_to+infty_in b1 &
         (for b4 being Element of REAL
               st b4 < b1
            holds ex b5 being Element of REAL st
               b4 < b5 & b5 < b1 & b5 in dom (b2 + b3)) &
         (ex b4 being Element of REAL st
            0 < b4 & b3 is_bounded_below_on ].b1 - b4,b1.[)
   holds b2 + b3 is_left_divergent_to+infty_in b1;

:: LIMFUNC2:th 24
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_left_divergent_to+infty_in b1 &
         (for b4 being Element of REAL
               st b4 < b1
            holds ex b5 being Element of REAL st
               b4 < b5 & b5 < b1 & b5 in dom (b2 (#) b3)) &
         (ex b4, b5 being Element of REAL st
            0 < b4 &
             0 < b5 &
             (for b6 being Element of REAL
                   st b6 in (dom b3) /\ ].b1 - b4,b1.[
                holds b5 <= b3 . b6))
   holds b2 (#) b3 is_left_divergent_to+infty_in b1;

:: LIMFUNC2:th 25
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_right_divergent_to+infty_in b1 &
         (for b4 being Element of REAL
               st b1 < b4
            holds ex b5 being Element of REAL st
               b5 < b4 & b1 < b5 & b5 in dom (b2 + b3)) &
         (ex b4 being Element of REAL st
            0 < b4 & b3 is_bounded_below_on ].b1,b1 + b4.[)
   holds b2 + b3 is_right_divergent_to+infty_in b1;

:: LIMFUNC2:th 26
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_right_divergent_to+infty_in b1 &
         (for b4 being Element of REAL
               st b1 < b4
            holds ex b5 being Element of REAL st
               b5 < b4 & b1 < b5 & b5 in dom (b2 (#) b3)) &
         (ex b4, b5 being Element of REAL st
            0 < b4 &
             0 < b5 &
             (for b6 being Element of REAL
                   st b6 in (dom b3) /\ ].b1,b1 + b4.[
                holds b5 <= b3 . b6))
   holds b2 (#) b3 is_right_divergent_to+infty_in b1;

:: LIMFUNC2:th 27
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL holds
   (b3 is_left_divergent_to+infty_in b1 & 0 < b2 implies b2 (#) b3 is_left_divergent_to+infty_in b1) &
    (b3 is_left_divergent_to+infty_in b1 & b2 < 0 implies b2 (#) b3 is_left_divergent_to-infty_in b1) &
    (b3 is_left_divergent_to-infty_in b1 & 0 < b2 implies b2 (#) b3 is_left_divergent_to-infty_in b1) &
    (b3 is_left_divergent_to-infty_in b1 & b2 < 0 implies b2 (#) b3 is_left_divergent_to+infty_in b1);

:: LIMFUNC2:th 28
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL holds
   (b3 is_right_divergent_to+infty_in b1 & 0 < b2 implies b2 (#) b3 is_right_divergent_to+infty_in b1) &
    (b3 is_right_divergent_to+infty_in b1 & b2 < 0 implies b2 (#) b3 is_right_divergent_to-infty_in b1) &
    (b3 is_right_divergent_to-infty_in b1 & 0 < b2 implies b2 (#) b3 is_right_divergent_to-infty_in b1) &
    (b3 is_right_divergent_to-infty_in b1 & b2 < 0 implies b2 (#) b3 is_right_divergent_to+infty_in b1);

:: LIMFUNC2:th 29
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st (b2 is_left_divergent_to+infty_in b1 or b2 is_left_divergent_to-infty_in b1)
   holds abs b2 is_left_divergent_to+infty_in b1;

:: LIMFUNC2:th 30
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st (b2 is_right_divergent_to+infty_in b1 or b2 is_right_divergent_to-infty_in b1)
   holds abs b2 is_right_divergent_to+infty_in b1;

:: LIMFUNC2:th 31
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st (ex b3 being Element of REAL st
            0 < b3 & b2 is_non_decreasing_on ].b1 - b3,b1.[ & not b2 is_bounded_above_on ].b1 - b3,b1.[) &
         (for b3 being Element of REAL
               st b3 < b1
            holds ex b4 being Element of REAL st
               b3 < b4 & b4 < b1 & b4 in dom b2)
   holds b2 is_left_divergent_to+infty_in b1;

:: LIMFUNC2:th 32
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st (ex b3 being Element of REAL st
            0 < b3 & b2 is_increasing_on ].b1 - b3,b1.[ & not b2 is_bounded_above_on ].b1 - b3,b1.[) &
         (for b3 being Element of REAL
               st b3 < b1
            holds ex b4 being Element of REAL st
               b3 < b4 & b4 < b1 & b4 in dom b2)
   holds b2 is_left_divergent_to+infty_in b1;

:: LIMFUNC2:th 33
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st (ex b3 being Element of REAL st
            0 < b3 & b2 is_non_increasing_on ].b1 - b3,b1.[ & not b2 is_bounded_below_on ].b1 - b3,b1.[) &
         (for b3 being Element of REAL
               st b3 < b1
            holds ex b4 being Element of REAL st
               b3 < b4 & b4 < b1 & b4 in dom b2)
   holds b2 is_left_divergent_to-infty_in b1;

:: LIMFUNC2:th 34
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st (ex b3 being Element of REAL st
            0 < b3 & b2 is_decreasing_on ].b1 - b3,b1.[ & not b2 is_bounded_below_on ].b1 - b3,b1.[) &
         (for b3 being Element of REAL
               st b3 < b1
            holds ex b4 being Element of REAL st
               b3 < b4 & b4 < b1 & b4 in dom b2)
   holds b2 is_left_divergent_to-infty_in b1;

:: LIMFUNC2:th 35
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st (ex b3 being Element of REAL st
            0 < b3 & b2 is_non_increasing_on ].b1,b1 + b3.[ & not b2 is_bounded_above_on ].b1,b1 + b3.[) &
         (for b3 being Element of REAL
               st b1 < b3
            holds ex b4 being Element of REAL st
               b4 < b3 & b1 < b4 & b4 in dom b2)
   holds b2 is_right_divergent_to+infty_in b1;

:: LIMFUNC2:th 36
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st (ex b3 being Element of REAL st
            0 < b3 & b2 is_decreasing_on ].b1,b1 + b3.[ & not b2 is_bounded_above_on ].b1,b1 + b3.[) &
         (for b3 being Element of REAL
               st b1 < b3
            holds ex b4 being Element of REAL st
               b4 < b3 & b1 < b4 & b4 in dom b2)
   holds b2 is_right_divergent_to+infty_in b1;

:: LIMFUNC2:th 37
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st (ex b3 being Element of REAL st
            0 < b3 & b2 is_non_decreasing_on ].b1,b1 + b3.[ & not b2 is_bounded_below_on ].b1,b1 + b3.[) &
         (for b3 being Element of REAL
               st b1 < b3
            holds ex b4 being Element of REAL st
               b4 < b3 & b1 < b4 & b4 in dom b2)
   holds b2 is_right_divergent_to-infty_in b1;

:: LIMFUNC2:th 38
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st (ex b3 being Element of REAL st
            0 < b3 & b2 is_increasing_on ].b1,b1 + b3.[ & not b2 is_bounded_below_on ].b1,b1 + b3.[) &
         (for b3 being Element of REAL
               st b1 < b3
            holds ex b4 being Element of REAL st
               b4 < b3 & b1 < b4 & b4 in dom b2)
   holds b2 is_right_divergent_to-infty_in b1;

:: LIMFUNC2:th 39
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_left_divergent_to+infty_in b1 &
         (for b4 being Element of REAL
               st b4 < b1
            holds ex b5 being Element of REAL st
               b4 < b5 & b5 < b1 & b5 in dom b3) &
         (ex b4 being Element of REAL st
            0 < b4 &
             (dom b3) /\ ].b1 - b4,b1.[ c= (dom b2) /\ ].b1 - b4,b1.[ &
             (for b5 being Element of REAL
                   st b5 in (dom b3) /\ ].b1 - b4,b1.[
                holds b2 . b5 <= b3 . b5))
   holds b3 is_left_divergent_to+infty_in b1;

:: LIMFUNC2:th 40
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_left_divergent_to-infty_in b1 &
         (for b4 being Element of REAL
               st b4 < b1
            holds ex b5 being Element of REAL st
               b4 < b5 & b5 < b1 & b5 in dom b3) &
         (ex b4 being Element of REAL st
            0 < b4 &
             (dom b3) /\ ].b1 - b4,b1.[ c= (dom b2) /\ ].b1 - b4,b1.[ &
             (for b5 being Element of REAL
                   st b5 in (dom b3) /\ ].b1 - b4,b1.[
                holds b3 . b5 <= b2 . b5))
   holds b3 is_left_divergent_to-infty_in b1;

:: LIMFUNC2:th 41
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_right_divergent_to+infty_in b1 &
         (for b4 being Element of REAL
               st b1 < b4
            holds ex b5 being Element of REAL st
               b5 < b4 & b1 < b5 & b5 in dom b3) &
         (ex b4 being Element of REAL st
            0 < b4 &
             (dom b3) /\ ].b1,b1 + b4.[ c= (dom b2) /\ ].b1,b1 + b4.[ &
             (for b5 being Element of REAL
                   st b5 in (dom b3) /\ ].b1,b1 + b4.[
                holds b2 . b5 <= b3 . b5))
   holds b3 is_right_divergent_to+infty_in b1;

:: LIMFUNC2:th 42
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_right_divergent_to-infty_in b1 &
         (for b4 being Element of REAL
               st b1 < b4
            holds ex b5 being Element of REAL st
               b5 < b4 & b1 < b5 & b5 in dom b3) &
         (ex b4 being Element of REAL st
            0 < b4 &
             (dom b3) /\ ].b1,b1 + b4.[ c= (dom b2) /\ ].b1,b1 + b4.[ &
             (for b5 being Element of REAL
                   st b5 in (dom b3) /\ ].b1,b1 + b4.[
                holds b3 . b5 <= b2 . b5))
   holds b3 is_right_divergent_to-infty_in b1;

:: LIMFUNC2:th 43
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_left_divergent_to+infty_in b1 &
         (ex b4 being Element of REAL st
            0 < b4 &
             ].b1 - b4,b1.[ c= (dom b3) /\ dom b2 &
             (for b5 being Element of REAL
                   st b5 in ].b1 - b4,b1.[
                holds b2 . b5 <= b3 . b5))
   holds b3 is_left_divergent_to+infty_in b1;

:: LIMFUNC2:th 44
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_left_divergent_to-infty_in b1 &
         (ex b4 being Element of REAL st
            0 < b4 &
             ].b1 - b4,b1.[ c= (dom b3) /\ dom b2 &
             (for b5 being Element of REAL
                   st b5 in ].b1 - b4,b1.[
                holds b3 . b5 <= b2 . b5))
   holds b3 is_left_divergent_to-infty_in b1;

:: LIMFUNC2:th 45
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_right_divergent_to+infty_in b1 &
         (ex b4 being Element of REAL st
            0 < b4 &
             ].b1,b1 + b4.[ c= (dom b3) /\ dom b2 &
             (for b5 being Element of REAL
                   st b5 in ].b1,b1 + b4.[
                holds b2 . b5 <= b3 . b5))
   holds b3 is_right_divergent_to+infty_in b1;

:: LIMFUNC2:th 46
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_right_divergent_to-infty_in b1 &
         (ex b4 being Element of REAL st
            0 < b4 &
             ].b1,b1 + b4.[ c= (dom b3) /\ dom b2 &
             (for b5 being Element of REAL
                   st b5 in ].b1,b1 + b4.[
                holds b3 . b5 <= b2 . b5))
   holds b3 is_right_divergent_to-infty_in b1;

:: LIMFUNC2:funcnot 1 => LIMFUNC2:func 1
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
  assume a1 is_left_convergent_in a2;
  func lim_left(A1,A2) -> Element of REAL means
    for b1 being Function-like quasi_total Relation of NAT,REAL
          st b1 is convergent & lim b1 = a2 & rng b1 c= (dom a1) /\ halfline a2
       holds a1 * b1 is convergent & lim (a1 * b1) = it;
end;

:: LIMFUNC2:def 7
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
   st b1 is_left_convergent_in b2
for b3 being Element of REAL holds
      b3 = lim_left(b1,b2)
   iff
      for b4 being Function-like quasi_total Relation of NAT,REAL
            st b4 is convergent & lim b4 = b2 & rng b4 c= (dom b1) /\ halfline b2
         holds b1 * b4 is convergent & lim (b1 * b4) = b3;

:: LIMFUNC2:funcnot 2 => LIMFUNC2:func 2
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
  assume a1 is_right_convergent_in a2;
  func lim_right(A1,A2) -> Element of REAL means
    for b1 being Function-like quasi_total Relation of NAT,REAL
          st b1 is convergent & lim b1 = a2 & rng b1 c= (dom a1) /\ right_open_halfline a2
       holds a1 * b1 is convergent & lim (a1 * b1) = it;
end;

:: LIMFUNC2:def 8
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
   st b1 is_right_convergent_in b2
for b3 being Element of REAL holds
      b3 = lim_right(b1,b2)
   iff
      for b4 being Function-like quasi_total Relation of NAT,REAL
            st b4 is convergent & lim b4 = b2 & rng b4 c= (dom b1) /\ right_open_halfline b2
         holds b1 * b4 is convergent & lim (b1 * b4) = b3;

:: LIMFUNC2:th 49
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
      st b3 is_left_convergent_in b1
   holds    lim_left(b3,b1) = b2
   iff
      for b4 being Element of REAL
            st 0 < b4
         holds ex b5 being Element of REAL st
            b5 < b1 &
             (for b6 being Element of REAL
                   st b5 < b6 & b6 < b1 & b6 in dom b3
                holds abs ((b3 . b6) - b2) < b4);

:: LIMFUNC2:th 50
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
      st b3 is_right_convergent_in b1
   holds    lim_right(b3,b1) = b2
   iff
      for b4 being Element of REAL
            st 0 < b4
         holds ex b5 being Element of REAL st
            b1 < b5 &
             (for b6 being Element of REAL
                   st b6 < b5 & b1 < b6 & b6 in dom b3
                holds abs ((b3 . b6) - b2) < b4);

:: LIMFUNC2:th 51
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
      st b3 is_left_convergent_in b1
   holds b2 (#) b3 is_left_convergent_in b1 &
    lim_left(b2 (#) b3,b1) = b2 * lim_left(b3,b1);

:: LIMFUNC2:th 52
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_left_convergent_in b1
   holds - b2 is_left_convergent_in b1 & lim_left(- b2,b1) = - lim_left(b2,b1);

:: LIMFUNC2:th 53
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_left_convergent_in b1 &
         b3 is_left_convergent_in b1 &
         (for b4 being Element of REAL
               st b4 < b1
            holds ex b5 being Element of REAL st
               b4 < b5 & b5 < b1 & b5 in dom (b2 + b3))
   holds b2 + b3 is_left_convergent_in b1 &
    lim_left(b2 + b3,b1) = (lim_left(b2,b1)) + lim_left(b3,b1);

:: LIMFUNC2:th 54
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_left_convergent_in b1 &
         b3 is_left_convergent_in b1 &
         (for b4 being Element of REAL
               st b4 < b1
            holds ex b5 being Element of REAL st
               b4 < b5 & b5 < b1 & b5 in dom (b2 - b3))
   holds b2 - b3 is_left_convergent_in b1 &
    lim_left(b2 - b3,b1) = (lim_left(b2,b1)) - lim_left(b3,b1);

:: LIMFUNC2:th 55
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_left_convergent_in b1 & b2 " {0} = {} & lim_left(b2,b1) <> 0
   holds b2 ^ is_left_convergent_in b1 & lim_left(b2 ^,b1) = (lim_left(b2,b1)) ";

:: LIMFUNC2:th 56
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_left_convergent_in b1
   holds abs b2 is_left_convergent_in b1 &
    lim_left(abs b2,b1) = abs lim_left(b2,b1);

:: LIMFUNC2:th 57
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_left_convergent_in b1 &
         lim_left(b2,b1) <> 0 &
         (for b3 being Element of REAL
               st b3 < b1
            holds ex b4 being Element of REAL st
               b3 < b4 & b4 < b1 & b4 in dom b2 & b2 . b4 <> 0)
   holds b2 ^ is_left_convergent_in b1 & lim_left(b2 ^,b1) = (lim_left(b2,b1)) ";

:: LIMFUNC2:th 58
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_left_convergent_in b1 &
         b3 is_left_convergent_in b1 &
         (for b4 being Element of REAL
               st b4 < b1
            holds ex b5 being Element of REAL st
               b4 < b5 & b5 < b1 & b5 in dom (b2 (#) b3))
   holds b2 (#) b3 is_left_convergent_in b1 &
    lim_left(b2 (#) b3,b1) = (lim_left(b2,b1)) * lim_left(b3,b1);

:: LIMFUNC2:th 59
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_left_convergent_in b1 &
         b3 is_left_convergent_in b1 &
         lim_left(b3,b1) <> 0 &
         (for b4 being Element of REAL
               st b4 < b1
            holds ex b5 being Element of REAL st
               b4 < b5 & b5 < b1 & b5 in dom (b2 / b3))
   holds b2 / b3 is_left_convergent_in b1 &
    lim_left(b2 / b3,b1) = (lim_left(b2,b1)) / lim_left(b3,b1);

:: LIMFUNC2:th 60
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
      st b3 is_right_convergent_in b1
   holds b2 (#) b3 is_right_convergent_in b1 &
    lim_right(b2 (#) b3,b1) = b2 * lim_right(b3,b1);

:: LIMFUNC2:th 61
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_right_convergent_in b1
   holds - b2 is_right_convergent_in b1 & lim_right(- b2,b1) = - lim_right(b2,b1);

:: LIMFUNC2:th 62
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_right_convergent_in b1 &
         b3 is_right_convergent_in b1 &
         (for b4 being Element of REAL
               st b1 < b4
            holds ex b5 being Element of REAL st
               b5 < b4 & b1 < b5 & b5 in dom (b2 + b3))
   holds b2 + b3 is_right_convergent_in b1 &
    lim_right(b2 + b3,b1) = (lim_right(b2,b1)) + lim_right(b3,b1);

:: LIMFUNC2:th 63
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_right_convergent_in b1 &
         b3 is_right_convergent_in b1 &
         (for b4 being Element of REAL
               st b1 < b4
            holds ex b5 being Element of REAL st
               b5 < b4 & b1 < b5 & b5 in dom (b2 - b3))
   holds b2 - b3 is_right_convergent_in b1 &
    lim_right(b2 - b3,b1) = (lim_right(b2,b1)) - lim_right(b3,b1);

:: LIMFUNC2:th 64
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_right_convergent_in b1 & b2 " {0} = {} & lim_right(b2,b1) <> 0
   holds b2 ^ is_right_convergent_in b1 & lim_right(b2 ^,b1) = (lim_right(b2,b1)) ";

:: LIMFUNC2:th 65
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_right_convergent_in b1
   holds abs b2 is_right_convergent_in b1 &
    lim_right(abs b2,b1) = abs lim_right(b2,b1);

:: LIMFUNC2:th 66
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_right_convergent_in b1 &
         lim_right(b2,b1) <> 0 &
         (for b3 being Element of REAL
               st b1 < b3
            holds ex b4 being Element of REAL st
               b4 < b3 & b1 < b4 & b4 in dom b2 & b2 . b4 <> 0)
   holds b2 ^ is_right_convergent_in b1 & lim_right(b2 ^,b1) = (lim_right(b2,b1)) ";

:: LIMFUNC2:th 67
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_right_convergent_in b1 &
         b3 is_right_convergent_in b1 &
         (for b4 being Element of REAL
               st b1 < b4
            holds ex b5 being Element of REAL st
               b5 < b4 & b1 < b5 & b5 in dom (b2 (#) b3))
   holds b2 (#) b3 is_right_convergent_in b1 &
    lim_right(b2 (#) b3,b1) = (lim_right(b2,b1)) * lim_right(b3,b1);

:: LIMFUNC2:th 68
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_right_convergent_in b1 &
         b3 is_right_convergent_in b1 &
         lim_right(b3,b1) <> 0 &
         (for b4 being Element of REAL
               st b1 < b4
            holds ex b5 being Element of REAL st
               b5 < b4 & b1 < b5 & b5 in dom (b2 / b3))
   holds b2 / b3 is_right_convergent_in b1 &
    lim_right(b2 / b3,b1) = (lim_right(b2,b1)) / lim_right(b3,b1);

:: LIMFUNC2:th 69
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_left_convergent_in b1 &
         lim_left(b2,b1) = 0 &
         (for b4 being Element of REAL
               st b4 < b1
            holds ex b5 being Element of REAL st
               b4 < b5 & b5 < b1 & b5 in dom (b2 (#) b3)) &
         (ex b4 being Element of REAL st
            0 < b4 & b3 is_bounded_on ].b1 - b4,b1.[)
   holds b2 (#) b3 is_left_convergent_in b1 & lim_left(b2 (#) b3,b1) = 0;

:: LIMFUNC2:th 70
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_right_convergent_in b1 &
         lim_right(b2,b1) = 0 &
         (for b4 being Element of REAL
               st b1 < b4
            holds ex b5 being Element of REAL st
               b5 < b4 & b1 < b5 & b5 in dom (b2 (#) b3)) &
         (ex b4 being Element of REAL st
            0 < b4 & b3 is_bounded_on ].b1,b1 + b4.[)
   holds b2 (#) b3 is_right_convergent_in b1 & lim_right(b2 (#) b3,b1) = 0;

:: LIMFUNC2:th 71
theorem
for b1 being Element of REAL
for b2, b3, b4 being Function-like Relation of REAL,REAL
      st b2 is_left_convergent_in b1 &
         b3 is_left_convergent_in b1 &
         lim_left(b2,b1) = lim_left(b3,b1) &
         (for b5 being Element of REAL
               st b5 < b1
            holds ex b6 being Element of REAL st
               b5 < b6 & b6 < b1 & b6 in dom b4) &
         (ex b5 being Element of REAL st
            0 < b5 &
             (for b6 being Element of REAL
                   st b6 in (dom b4) /\ ].b1 - b5,b1.[
                holds b2 . b6 <= b4 . b6 & b4 . b6 <= b3 . b6) &
             ((dom b2) /\ ].b1 - b5,b1.[ c= (dom b3) /\ ].b1 - b5,b1.[ &
              (dom b4) /\ ].b1 - b5,b1.[ c= (dom b2) /\ ].b1 - b5,b1.[ or (dom b3) /\ ].b1 - b5,b1.[ c= (dom b2) /\ ].b1 - b5,b1.[ &
              (dom b4) /\ ].b1 - b5,b1.[ c= (dom b3) /\ ].b1 - b5,b1.[))
   holds b4 is_left_convergent_in b1 & lim_left(b4,b1) = lim_left(b2,b1);

:: LIMFUNC2:th 72
theorem
for b1 being Element of REAL
for b2, b3, b4 being Function-like Relation of REAL,REAL
      st b2 is_left_convergent_in b1 &
         b3 is_left_convergent_in b1 &
         lim_left(b2,b1) = lim_left(b3,b1) &
         (ex b5 being Element of REAL st
            0 < b5 &
             ].b1 - b5,b1.[ c= ((dom b2) /\ dom b3) /\ dom b4 &
             (for b6 being Element of REAL
                   st b6 in ].b1 - b5,b1.[
                holds b2 . b6 <= b4 . b6 & b4 . b6 <= b3 . b6))
   holds b4 is_left_convergent_in b1 & lim_left(b4,b1) = lim_left(b2,b1);

:: LIMFUNC2:th 73
theorem
for b1 being Element of REAL
for b2, b3, b4 being Function-like Relation of REAL,REAL
      st b2 is_right_convergent_in b1 &
         b3 is_right_convergent_in b1 &
         lim_right(b2,b1) = lim_right(b3,b1) &
         (for b5 being Element of REAL
               st b1 < b5
            holds ex b6 being Element of REAL st
               b6 < b5 & b1 < b6 & b6 in dom b4) &
         (ex b5 being Element of REAL st
            0 < b5 &
             (for b6 being Element of REAL
                   st b6 in (dom b4) /\ ].b1,b1 + b5.[
                holds b2 . b6 <= b4 . b6 & b4 . b6 <= b3 . b6) &
             ((dom b2) /\ ].b1,b1 + b5.[ c= (dom b3) /\ ].b1,b1 + b5.[ &
              (dom b4) /\ ].b1,b1 + b5.[ c= (dom b2) /\ ].b1,b1 + b5.[ or (dom b3) /\ ].b1,b1 + b5.[ c= (dom b2) /\ ].b1,b1 + b5.[ &
              (dom b4) /\ ].b1,b1 + b5.[ c= (dom b3) /\ ].b1,b1 + b5.[))
   holds b4 is_right_convergent_in b1 & lim_right(b4,b1) = lim_right(b2,b1);

:: LIMFUNC2:th 74
theorem
for b1 being Element of REAL
for b2, b3, b4 being Function-like Relation of REAL,REAL
      st b2 is_right_convergent_in b1 &
         b3 is_right_convergent_in b1 &
         lim_right(b2,b1) = lim_right(b3,b1) &
         (ex b5 being Element of REAL st
            0 < b5 &
             ].b1,b1 + b5.[ c= ((dom b2) /\ dom b3) /\ dom b4 &
             (for b6 being Element of REAL
                   st b6 in ].b1,b1 + b5.[
                holds b2 . b6 <= b4 . b6 & b4 . b6 <= b3 . b6))
   holds b4 is_right_convergent_in b1 & lim_right(b4,b1) = lim_right(b2,b1);

:: LIMFUNC2:th 75
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_left_convergent_in b1 &
         b3 is_left_convergent_in b1 &
         (ex b4 being Element of REAL st
            0 < b4 &
             ((dom b2) /\ ].b1 - b4,b1.[ c= (dom b3) /\ ].b1 - b4,b1.[ &
              (for b5 being Element of REAL
                    st b5 in (dom b2) /\ ].b1 - b4,b1.[
                 holds b2 . b5 <= b3 . b5) or (dom b3) /\ ].b1 - b4,b1.[ c= (dom b2) /\ ].b1 - b4,b1.[ &
              (for b5 being Element of REAL
                    st b5 in (dom b3) /\ ].b1 - b4,b1.[
                 holds b2 . b5 <= b3 . b5)))
   holds lim_left(b2,b1) <= lim_left(b3,b1);

:: LIMFUNC2:th 76
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_right_convergent_in b1 &
         b3 is_right_convergent_in b1 &
         (ex b4 being Element of REAL st
            0 < b4 &
             ((dom b2) /\ ].b1,b1 + b4.[ c= (dom b3) /\ ].b1,b1 + b4.[ &
              (for b5 being Element of REAL
                    st b5 in (dom b2) /\ ].b1,b1 + b4.[
                 holds b2 . b5 <= b3 . b5) or (dom b3) /\ ].b1,b1 + b4.[ c= (dom b2) /\ ].b1,b1 + b4.[ &
              (for b5 being Element of REAL
                    st b5 in (dom b3) /\ ].b1,b1 + b4.[
                 holds b2 . b5 <= b3 . b5)))
   holds lim_right(b2,b1) <= lim_right(b3,b1);

:: LIMFUNC2:th 77
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st (b2 is_left_divergent_to+infty_in b1 or b2 is_left_divergent_to-infty_in b1) &
         (for b3 being Element of REAL
               st b3 < b1
            holds ex b4 being Element of REAL st
               b3 < b4 & b4 < b1 & b4 in dom b2 & b2 . b4 <> 0)
   holds b2 ^ is_left_convergent_in b1 & lim_left(b2 ^,b1) = 0;

:: LIMFUNC2:th 78
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st (b2 is_right_divergent_to+infty_in b1 or b2 is_right_divergent_to-infty_in b1) &
         (for b3 being Element of REAL
               st b1 < b3
            holds ex b4 being Element of REAL st
               b4 < b3 & b1 < b4 & b4 in dom b2 & b2 . b4 <> 0)
   holds b2 ^ is_right_convergent_in b1 & lim_right(b2 ^,b1) = 0;

:: LIMFUNC2:th 79
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_left_convergent_in b1 &
         lim_left(b2,b1) = 0 &
         (ex b3 being Element of REAL st
            0 < b3 &
             (for b4 being Element of REAL
                   st b4 in (dom b2) /\ ].b1 - b3,b1.[
                holds 0 < b2 . b4))
   holds b2 ^ is_left_divergent_to+infty_in b1;

:: LIMFUNC2:th 80
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_left_convergent_in b1 &
         lim_left(b2,b1) = 0 &
         (ex b3 being Element of REAL st
            0 < b3 &
             (for b4 being Element of REAL
                   st b4 in (dom b2) /\ ].b1 - b3,b1.[
                holds b2 . b4 < 0))
   holds b2 ^ is_left_divergent_to-infty_in b1;

:: LIMFUNC2:th 81
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_right_convergent_in b1 &
         lim_right(b2,b1) = 0 &
         (ex b3 being Element of REAL st
            0 < b3 &
             (for b4 being Element of REAL
                   st b4 in (dom b2) /\ ].b1,b1 + b3.[
                holds 0 < b2 . b4))
   holds b2 ^ is_right_divergent_to+infty_in b1;

:: LIMFUNC2:th 82
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_right_convergent_in b1 &
         lim_right(b2,b1) = 0 &
         (ex b3 being Element of REAL st
            0 < b3 &
             (for b4 being Element of REAL
                   st b4 in (dom b2) /\ ].b1,b1 + b3.[
                holds b2 . b4 < 0))
   holds b2 ^ is_right_divergent_to-infty_in b1;

:: LIMFUNC2:th 83
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_left_convergent_in b1 &
         lim_left(b2,b1) = 0 &
         (for b3 being Element of REAL
               st b3 < b1
            holds ex b4 being Element of REAL st
               b3 < b4 & b4 < b1 & b4 in dom b2 & b2 . b4 <> 0) &
         (ex b3 being Element of REAL st
            0 < b3 &
             (for b4 being Element of REAL
                   st b4 in (dom b2) /\ ].b1 - b3,b1.[
                holds 0 <= b2 . b4))
   holds b2 ^ is_left_divergent_to+infty_in b1;

:: LIMFUNC2:th 84
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_left_convergent_in b1 &
         lim_left(b2,b1) = 0 &
         (for b3 being Element of REAL
               st b3 < b1
            holds ex b4 being Element of REAL st
               b3 < b4 & b4 < b1 & b4 in dom b2 & b2 . b4 <> 0) &
         (ex b3 being Element of REAL st
            0 < b3 &
             (for b4 being Element of REAL
                   st b4 in (dom b2) /\ ].b1 - b3,b1.[
                holds b2 . b4 <= 0))
   holds b2 ^ is_left_divergent_to-infty_in b1;

:: LIMFUNC2:th 85
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_right_convergent_in b1 &
         lim_right(b2,b1) = 0 &
         (for b3 being Element of REAL
               st b1 < b3
            holds ex b4 being Element of REAL st
               b4 < b3 & b1 < b4 & b4 in dom b2 & b2 . b4 <> 0) &
         (ex b3 being Element of REAL st
            0 < b3 &
             (for b4 being Element of REAL
                   st b4 in (dom b2) /\ ].b1,b1 + b3.[
                holds 0 <= b2 . b4))
   holds b2 ^ is_right_divergent_to+infty_in b1;

:: LIMFUNC2:th 86
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_right_convergent_in b1 &
         lim_right(b2,b1) = 0 &
         (for b3 being Element of REAL
               st b1 < b3
            holds ex b4 being Element of REAL st
               b4 < b3 & b1 < b4 & b4 in dom b2 & b2 . b4 <> 0) &
         (ex b3 being Element of REAL st
            0 < b3 &
             (for b4 being Element of REAL
                   st b4 in (dom b2) /\ ].b1,b1 + b3.[
                holds b2 . b4 <= 0))
   holds b2 ^ is_right_divergent_to-infty_in b1;