Article LIMFUNC3, MML version 4.99.1005

:: LIMFUNC3:th 1
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 (proj2 b2 c= (proj1 b3) /\ halfline b1 or proj2 b2 c= (proj1 b3) /\ right_open_halfline b1)
   holds proj2 b2 c= (proj1 b3) \ {b1};

:: LIMFUNC3:th 2
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
           0 < abs (b1 - (b2 . b4)) &
            abs (b1 - (b2 . b4)) < 1 / (b4 + 1) &
            b2 . b4 in proj1 b3
   holds b2 is convergent &
    lim b2 = b1 &
    proj2 b2 c= proj1 b3 &
    proj2 b2 c= (proj1 b3) \ {b1};

:: LIMFUNC3:th 3
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 b2 is convergent &
      lim b2 = b1 &
      proj2 b2 c= (proj1 b3) \ {b1}
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 0 < abs (b1 - (b2 . b6)) & abs (b1 - (b2 . b6)) < b4 & b2 . b6 in proj1 b3;

:: LIMFUNC3:th 4
theorem
for b1, b2 being Element of REAL
      st 0 < b1
   holds ].b2 - b1,b2 + b1.[ \ {b2} = ].b2 - b1,b2.[ \/ ].b2,b2 + b1.[;

:: LIMFUNC3:th 5
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
   st 0 < b1 &
      ].b2 - b1,b2.[ \/ ].b2,b2 + b1.[ c= proj1 b3
for b4, b5 being Element of REAL
      st b4 < b2 & b2 < b5
   holds ex b6, b7 being Element of REAL st
      b4 < b6 & b6 < b2 & b6 in proj1 b3 & b7 < b5 & b2 < b7 & b7 in proj1 b3;

:: LIMFUNC3: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 - (1 / (b4 + 1)) < b2 . b4 &
            b2 . b4 < b1 &
            b2 . b4 in proj1 b3
   holds b2 is convergent &
    lim b2 = b1 &
    proj2 b2 c= (proj1 b3) \ {b1};

:: LIMFUNC3:th 7
theorem
for b1, b2 being Element of REAL
for b3 being Function-like quasi_total Relation of NAT,REAL
      st b3 is convergent & lim b3 = b1 & 0 < b2
   holds ex b4 being Element of NAT st
      for b5 being Element of NAT
            st b4 <= b5
         holds b1 - b2 < b3 . b5 & b3 . b5 < b1 + b2;

:: LIMFUNC3:th 8
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL holds
      for b3, b4 being Element of REAL
            st b3 < b1 & b1 < b4
         holds ex b5, b6 being Element of REAL st
            b3 < b5 & b5 < b1 & b5 in proj1 b2 & b6 < b4 & b1 < b6 & b6 in proj1 b2
   iff
      (for b3 being Element of REAL
             st b3 < b1
          holds ex b4 being Element of REAL st
             b3 < b4 & b4 < b1 & b4 in proj1 b2) &
       (for b3 being Element of REAL
             st b1 < b3
          holds ex b4 being Element of REAL st
             b4 < b3 & b1 < b4 & b4 in proj1 b2);

:: LIMFUNC3:prednot 1 => LIMFUNC3:pred 1
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
  pred A1 is_convergent_in A2 means
    (for b1, b2 being Element of REAL
           st b1 < a2 & a2 < b2
        holds ex b3, b4 being Element of REAL st
           b1 < b3 & b3 < a2 & b3 in proj1 a1 & b4 < b2 & a2 < b4 & b4 in proj1 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 &
                 proj2 b2 c= (proj1 a1) \ {a2}
           holds a1 * b2 is convergent & lim (a1 * b2) = b1);
end;

:: LIMFUNC3:dfs 1
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
To prove
     a1 is_convergent_in a2
it is sufficient to prove
  thus (for b1, b2 being Element of REAL
           st b1 < a2 & a2 < b2
        holds ex b3, b4 being Element of REAL st
           b1 < b3 & b3 < a2 & b3 in proj1 a1 & b4 < b2 & a2 < b4 & b4 in proj1 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 &
                 proj2 b2 c= (proj1 a1) \ {a2}
           holds a1 * b2 is convergent & lim (a1 * b2) = b1);

:: LIMFUNC3:def 1
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
      b1 is_convergent_in b2
   iff
      (for b3, b4 being Element of REAL
             st b3 < b2 & b2 < b4
          holds ex b5, b6 being Element of REAL st
             b3 < b5 & b5 < b2 & b5 in proj1 b1 & b6 < b4 & b2 < b6 & b6 in proj1 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 &
                   proj2 b4 c= (proj1 b1) \ {b2}
             holds b1 * b4 is convergent & lim (b1 * b4) = b3);

:: LIMFUNC3:prednot 2 => LIMFUNC3:pred 2
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
  pred A1 is_divergent_to+infty_in A2 means
    (for b1, b2 being Element of REAL
           st b1 < a2 & a2 < b2
        holds ex b3, b4 being Element of REAL st
           b1 < b3 & b3 < a2 & b3 in proj1 a1 & b4 < b2 & a2 < b4 & b4 in proj1 a1) &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st b1 is convergent &
              lim b1 = a2 &
              proj2 b1 c= (proj1 a1) \ {a2}
        holds a1 * b1 is divergent_to+infty);
end;

:: LIMFUNC3:dfs 2
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
To prove
     a1 is_divergent_to+infty_in a2
it is sufficient to prove
  thus (for b1, b2 being Element of REAL
           st b1 < a2 & a2 < b2
        holds ex b3, b4 being Element of REAL st
           b1 < b3 & b3 < a2 & b3 in proj1 a1 & b4 < b2 & a2 < b4 & b4 in proj1 a1) &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st b1 is convergent &
              lim b1 = a2 &
              proj2 b1 c= (proj1 a1) \ {a2}
        holds a1 * b1 is divergent_to+infty);

:: LIMFUNC3:def 2
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
      b1 is_divergent_to+infty_in b2
   iff
      (for b3, b4 being Element of REAL
             st b3 < b2 & b2 < b4
          holds ex b5, b6 being Element of REAL st
             b3 < b5 & b5 < b2 & b5 in proj1 b1 & b6 < b4 & b2 < b6 & b6 in proj1 b1) &
       (for b3 being Function-like quasi_total Relation of NAT,REAL
             st b3 is convergent &
                lim b3 = b2 &
                proj2 b3 c= (proj1 b1) \ {b2}
          holds b1 * b3 is divergent_to+infty);

:: LIMFUNC3:prednot 3 => LIMFUNC3:pred 3
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
  pred A1 is_divergent_to-infty_in A2 means
    (for b1, b2 being Element of REAL
           st b1 < a2 & a2 < b2
        holds ex b3, b4 being Element of REAL st
           b1 < b3 & b3 < a2 & b3 in proj1 a1 & b4 < b2 & a2 < b4 & b4 in proj1 a1) &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st b1 is convergent &
              lim b1 = a2 &
              proj2 b1 c= (proj1 a1) \ {a2}
        holds a1 * b1 is divergent_to-infty);
end;

:: LIMFUNC3:dfs 3
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
To prove
     a1 is_divergent_to-infty_in a2
it is sufficient to prove
  thus (for b1, b2 being Element of REAL
           st b1 < a2 & a2 < b2
        holds ex b3, b4 being Element of REAL st
           b1 < b3 & b3 < a2 & b3 in proj1 a1 & b4 < b2 & a2 < b4 & b4 in proj1 a1) &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st b1 is convergent &
              lim b1 = a2 &
              proj2 b1 c= (proj1 a1) \ {a2}
        holds a1 * b1 is divergent_to-infty);

:: LIMFUNC3:def 3
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
      b1 is_divergent_to-infty_in b2
   iff
      (for b3, b4 being Element of REAL
             st b3 < b2 & b2 < b4
          holds ex b5, b6 being Element of REAL st
             b3 < b5 & b5 < b2 & b5 in proj1 b1 & b6 < b4 & b2 < b6 & b6 in proj1 b1) &
       (for b3 being Function-like quasi_total Relation of NAT,REAL
             st b3 is convergent &
                lim b3 = b2 &
                proj2 b3 c= (proj1 b1) \ {b2}
          holds b1 * b3 is divergent_to-infty);

:: LIMFUNC3:th 12
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL holds
      b2 is_convergent_in b1
   iff
      (for b3, b4 being Element of REAL
             st b3 < b1 & b1 < b4
          holds ex b5, b6 being Element of REAL st
             b3 < b5 & b5 < b1 & b5 in proj1 b2 & b6 < b4 & b1 < b6 & b6 in proj1 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
                0 < b5 &
                 (for b6 being Element of REAL
                       st 0 < abs (b1 - b6) & abs (b1 - b6) < b5 & b6 in proj1 b2
                    holds abs ((b2 . b6) - b3) < b4));

:: LIMFUNC3:th 13
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL holds
      b2 is_divergent_to+infty_in b1
   iff
      (for b3, b4 being Element of REAL
             st b3 < b1 & b1 < b4
          holds ex b5, b6 being Element of REAL st
             b3 < b5 & b5 < b1 & b5 in proj1 b2 & b6 < b4 & b1 < b6 & b6 in proj1 b2) &
       (for b3 being Element of REAL holds
          ex b4 being Element of REAL st
             0 < b4 &
              (for b5 being Element of REAL
                    st 0 < abs (b1 - b5) & abs (b1 - b5) < b4 & b5 in proj1 b2
                 holds b3 < b2 . b5));

:: LIMFUNC3:th 14
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL holds
      b2 is_divergent_to-infty_in b1
   iff
      (for b3, b4 being Element of REAL
             st b3 < b1 & b1 < b4
          holds ex b5, b6 being Element of REAL st
             b3 < b5 & b5 < b1 & b5 in proj1 b2 & b6 < b4 & b1 < b6 & b6 in proj1 b2) &
       (for b3 being Element of REAL holds
          ex b4 being Element of REAL st
             0 < b4 &
              (for b5 being Element of REAL
                    st 0 < abs (b1 - b5) & abs (b1 - b5) < b4 & b5 in proj1 b2
                 holds b2 . b5 < b3));

:: LIMFUNC3:th 15
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL holds
      b2 is_divergent_to+infty_in b1
   iff
      b2 is_left_divergent_to+infty_in b1 & b2 is_right_divergent_to+infty_in b1;

:: LIMFUNC3:th 16
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL holds
      b2 is_divergent_to-infty_in b1
   iff
      b2 is_left_divergent_to-infty_in b1 & b2 is_right_divergent_to-infty_in b1;

:: LIMFUNC3:th 17
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_divergent_to+infty_in b1 &
         b3 is_divergent_to+infty_in b1 &
         (for b4, b5 being Element of REAL
               st b4 < b1 & b1 < b5
            holds ex b6, b7 being Element of REAL st
               b4 < b6 & b6 < b1 & b6 in (proj1 b2) /\ proj1 b3 & b7 < b5 & b1 < b7 & b7 in (proj1 b2) /\ proj1 b3)
   holds b2 + b3 is_divergent_to+infty_in b1 & b2 (#) b3 is_divergent_to+infty_in b1;

:: LIMFUNC3:th 18
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_divergent_to-infty_in b1 &
         b3 is_divergent_to-infty_in b1 &
         (for b4, b5 being Element of REAL
               st b4 < b1 & b1 < b5
            holds ex b6, b7 being Element of REAL st
               b4 < b6 & b6 < b1 & b6 in (proj1 b2) /\ proj1 b3 & b7 < b5 & b1 < b7 & b7 in (proj1 b2) /\ proj1 b3)
   holds b2 + b3 is_divergent_to-infty_in b1 & b2 (#) b3 is_divergent_to+infty_in b1;

:: LIMFUNC3:th 19
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_divergent_to+infty_in b1 &
         (for b4, b5 being Element of REAL
               st b4 < b1 & b1 < b5
            holds ex b6, b7 being Element of REAL st
               b4 < b6 & b6 < b1 & b6 in proj1 (b2 + b3) & b7 < b5 & b1 < b7 & b7 in proj1 (b2 + b3)) &
         (ex b4 being Element of REAL st
            0 < b4 &
             b3 is_bounded_below_on ].b1 - b4,b1.[ \/ ].b1,b1 + b4.[)
   holds b2 + b3 is_divergent_to+infty_in b1;

:: LIMFUNC3:th 20
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_divergent_to+infty_in b1 &
         (for b4, b5 being Element of REAL
               st b4 < b1 & b1 < b5
            holds ex b6, b7 being Element of REAL st
               b4 < b6 & b6 < b1 & b6 in proj1 (b2 (#) b3) & b7 < b5 & b1 < b7 & b7 in proj1 (b2 (#) b3)) &
         (ex b4, b5 being Element of REAL st
            0 < b4 &
             0 < b5 &
             (for b6 being Element of REAL
                   st b6 in (proj1 b3) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[)
                holds b5 <= b3 . b6))
   holds b2 (#) b3 is_divergent_to+infty_in b1;

:: LIMFUNC3:th 21
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL holds
   (b3 is_divergent_to+infty_in b1 & 0 < b2 implies b2 (#) b3 is_divergent_to+infty_in b1) &
    (b3 is_divergent_to+infty_in b1 & b2 < 0 implies b2 (#) b3 is_divergent_to-infty_in b1) &
    (b3 is_divergent_to-infty_in b1 & 0 < b2 implies b2 (#) b3 is_divergent_to-infty_in b1) &
    (b3 is_divergent_to-infty_in b1 & b2 < 0 implies b2 (#) b3 is_divergent_to+infty_in b1);

:: LIMFUNC3:th 22
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st (b2 is_divergent_to+infty_in b1 or b2 is_divergent_to-infty_in b1)
   holds abs b2 is_divergent_to+infty_in b1;

:: LIMFUNC3:th 23
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.[ & b2 is_non_increasing_on ].b1,b1 + b3.[ & not b2 is_bounded_above_on ].b1 - b3,b1.[ & not b2 is_bounded_above_on ].b1,b1 + b3.[) &
         (for b3, b4 being Element of REAL
               st b3 < b1 & b1 < b4
            holds ex b5, b6 being Element of REAL st
               b3 < b5 & b5 < b1 & b5 in proj1 b2 & b6 < b4 & b1 < b6 & b6 in proj1 b2)
   holds b2 is_divergent_to+infty_in b1;

:: LIMFUNC3:th 24
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.[ & b2 is_decreasing_on ].b1,b1 + b3.[ & not b2 is_bounded_above_on ].b1 - b3,b1.[ & not b2 is_bounded_above_on ].b1,b1 + b3.[) &
         (for b3, b4 being Element of REAL
               st b3 < b1 & b1 < b4
            holds ex b5, b6 being Element of REAL st
               b3 < b5 & b5 < b1 & b5 in proj1 b2 & b6 < b4 & b1 < b6 & b6 in proj1 b2)
   holds b2 is_divergent_to+infty_in b1;

:: LIMFUNC3:th 25
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.[ & b2 is_non_decreasing_on ].b1,b1 + b3.[ & not b2 is_bounded_below_on ].b1 - b3,b1.[ & not b2 is_bounded_below_on ].b1,b1 + b3.[) &
         (for b3, b4 being Element of REAL
               st b3 < b1 & b1 < b4
            holds ex b5, b6 being Element of REAL st
               b3 < b5 & b5 < b1 & b5 in proj1 b2 & b6 < b4 & b1 < b6 & b6 in proj1 b2)
   holds b2 is_divergent_to-infty_in b1;

:: LIMFUNC3:th 26
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.[ & b2 is_increasing_on ].b1,b1 + b3.[ & not b2 is_bounded_below_on ].b1 - b3,b1.[ & not b2 is_bounded_below_on ].b1,b1 + b3.[) &
         (for b3, b4 being Element of REAL
               st b3 < b1 & b1 < b4
            holds ex b5, b6 being Element of REAL st
               b3 < b5 & b5 < b1 & b5 in proj1 b2 & b6 < b4 & b1 < b6 & b6 in proj1 b2)
   holds b2 is_divergent_to-infty_in b1;

:: LIMFUNC3:th 27
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_divergent_to+infty_in b1 &
         (for b4, b5 being Element of REAL
               st b4 < b1 & b1 < b5
            holds ex b6, b7 being Element of REAL st
               b4 < b6 & b6 < b1 & b6 in proj1 b3 & b7 < b5 & b1 < b7 & b7 in proj1 b3) &
         (ex b4 being Element of REAL st
            0 < b4 &
             (proj1 b3) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[) c= (proj1 b2) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[) &
             (for b5 being Element of REAL
                   st b5 in (proj1 b3) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[)
                holds b2 . b5 <= b3 . b5))
   holds b3 is_divergent_to+infty_in b1;

:: LIMFUNC3:th 28
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_divergent_to-infty_in b1 &
         (for b4, b5 being Element of REAL
               st b4 < b1 & b1 < b5
            holds ex b6, b7 being Element of REAL st
               b4 < b6 & b6 < b1 & b6 in proj1 b3 & b7 < b5 & b1 < b7 & b7 in proj1 b3) &
         (ex b4 being Element of REAL st
            0 < b4 &
             (proj1 b3) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[) c= (proj1 b2) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[) &
             (for b5 being Element of REAL
                   st b5 in (proj1 b3) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[)
                holds b3 . b5 <= b2 . b5))
   holds b3 is_divergent_to-infty_in b1;

:: LIMFUNC3:th 29
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_divergent_to+infty_in b1 &
         (ex b4 being Element of REAL st
            0 < b4 &
             ].b1 - b4,b1.[ \/ ].b1,b1 + b4.[ c= (proj1 b3) /\ proj1 b2 &
             (for b5 being Element of REAL
                   st b5 in ].b1 - b4,b1.[ \/ ].b1,b1 + b4.[
                holds b2 . b5 <= b3 . b5))
   holds b3 is_divergent_to+infty_in b1;

:: LIMFUNC3:th 30
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_divergent_to-infty_in b1 &
         (ex b4 being Element of REAL st
            0 < b4 &
             ].b1 - b4,b1.[ \/ ].b1,b1 + b4.[ c= (proj1 b3) /\ proj1 b2 &
             (for b5 being Element of REAL
                   st b5 in ].b1 - b4,b1.[ \/ ].b1,b1 + b4.[
                holds b3 . b5 <= b2 . b5))
   holds b3 is_divergent_to-infty_in b1;

:: LIMFUNC3:funcnot 1 => LIMFUNC3:func 1
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
  assume a1 is_convergent_in a2;
  func lim(A1,A2) -> Element of REAL means
    for b1 being Function-like quasi_total Relation of NAT,REAL
          st b1 is convergent &
             lim b1 = a2 &
             proj2 b1 c= (proj1 a1) \ {a2}
       holds a1 * b1 is convergent & lim (a1 * b1) = it;
end;

:: LIMFUNC3:def 4
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
   st b1 is_convergent_in b2
for b3 being Element of REAL holds
      b3 = lim(b1,b2)
   iff
      for b4 being Function-like quasi_total Relation of NAT,REAL
            st b4 is convergent &
               lim b4 = b2 &
               proj2 b4 c= (proj1 b1) \ {b2}
         holds b1 * b4 is convergent & lim (b1 * b4) = b3;

:: LIMFUNC3:th 32
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
      st b3 is_convergent_in b1
   holds    lim(b3,b1) = b2
   iff
      for b4 being Element of REAL
            st 0 < b4
         holds ex b5 being Element of REAL st
            0 < b5 &
             (for b6 being Element of REAL
                   st 0 < abs (b1 - b6) & abs (b1 - b6) < b5 & b6 in proj1 b3
                holds abs ((b3 . b6) - b2) < b4);

:: LIMFUNC3:th 33
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_convergent_in b1
   holds b2 is_left_convergent_in b1 & b2 is_right_convergent_in b1 & lim_left(b2,b1) = lim_right(b2,b1) & lim(b2,b1) = lim_left(b2,b1) & lim(b2,b1) = lim_right(b2,b1);

:: LIMFUNC3:th 34
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_left_convergent_in b1 & b2 is_right_convergent_in b1 & lim_left(b2,b1) = lim_right(b2,b1)
   holds b2 is_convergent_in b1 & lim(b2,b1) = lim_left(b2,b1) & lim(b2,b1) = lim_right(b2,b1);

:: LIMFUNC3:th 35
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
      st b3 is_convergent_in b1
   holds b2 (#) b3 is_convergent_in b1 &
    lim(b2 (#) b3,b1) = b2 * lim(b3,b1);

:: LIMFUNC3:th 36
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_convergent_in b1
   holds - b2 is_convergent_in b1 & lim(- b2,b1) = - lim(b2,b1);

:: LIMFUNC3:th 37
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_convergent_in b1 &
         b3 is_convergent_in b1 &
         (for b4, b5 being Element of REAL
               st b4 < b1 & b1 < b5
            holds ex b6, b7 being Element of REAL st
               b4 < b6 & b6 < b1 & b6 in proj1 (b2 + b3) & b7 < b5 & b1 < b7 & b7 in proj1 (b2 + b3))
   holds b2 + b3 is_convergent_in b1 &
    lim(b2 + b3,b1) = (lim(b2,b1)) + lim(b3,b1);

:: LIMFUNC3:th 38
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_convergent_in b1 &
         b3 is_convergent_in b1 &
         (for b4, b5 being Element of REAL
               st b4 < b1 & b1 < b5
            holds ex b6, b7 being Element of REAL st
               b4 < b6 & b6 < b1 & b6 in proj1 (b2 - b3) & b7 < b5 & b1 < b7 & b7 in proj1 (b2 - b3))
   holds b2 - b3 is_convergent_in b1 &
    lim(b2 - b3,b1) = (lim(b2,b1)) - lim(b3,b1);

:: LIMFUNC3:th 39
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_convergent_in b1 & b2 " {0} = {} & lim(b2,b1) <> 0
   holds b2 ^ is_convergent_in b1 & lim(b2 ^,b1) = (lim(b2,b1)) ";

:: LIMFUNC3:th 40
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_convergent_in b1
   holds abs b2 is_convergent_in b1 & lim(abs b2,b1) = abs lim(b2,b1);

:: LIMFUNC3:th 41
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_convergent_in b1 &
         lim(b2,b1) <> 0 &
         (for b3, b4 being Element of REAL
               st b3 < b1 & b1 < b4
            holds ex b5, b6 being Element of REAL st
               b3 < b5 & b5 < b1 & b5 in proj1 b2 & b6 < b4 & b1 < b6 & b6 in proj1 b2 & b2 . b5 <> 0 & b2 . b6 <> 0)
   holds b2 ^ is_convergent_in b1 & lim(b2 ^,b1) = (lim(b2,b1)) ";

:: LIMFUNC3:th 42
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_convergent_in b1 &
         b3 is_convergent_in b1 &
         (for b4, b5 being Element of REAL
               st b4 < b1 & b1 < b5
            holds ex b6, b7 being Element of REAL st
               b4 < b6 & b6 < b1 & b6 in proj1 (b2 (#) b3) & b7 < b5 & b1 < b7 & b7 in proj1 (b2 (#) b3))
   holds b2 (#) b3 is_convergent_in b1 &
    lim(b2 (#) b3,b1) = (lim(b2,b1)) * lim(b3,b1);

:: LIMFUNC3:th 43
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_convergent_in b1 &
         b3 is_convergent_in b1 &
         lim(b3,b1) <> 0 &
         (for b4, b5 being Element of REAL
               st b4 < b1 & b1 < b5
            holds ex b6, b7 being Element of REAL st
               b4 < b6 & b6 < b1 & b6 in proj1 (b2 / b3) & b7 < b5 & b1 < b7 & b7 in proj1 (b2 / b3))
   holds b2 / b3 is_convergent_in b1 &
    lim(b2 / b3,b1) = (lim(b2,b1)) / lim(b3,b1);

:: LIMFUNC3:th 44
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_convergent_in b1 &
         lim(b2,b1) = 0 &
         (for b4, b5 being Element of REAL
               st b4 < b1 & b1 < b5
            holds ex b6, b7 being Element of REAL st
               b4 < b6 & b6 < b1 & b6 in proj1 (b2 (#) b3) & b7 < b5 & b1 < b7 & b7 in proj1 (b2 (#) b3)) &
         (ex b4 being Element of REAL st
            0 < b4 &
             b3 is_bounded_on ].b1 - b4,b1.[ \/ ].b1,b1 + b4.[)
   holds b2 (#) b3 is_convergent_in b1 & lim(b2 (#) b3,b1) = 0;

:: LIMFUNC3:th 45
theorem
for b1 being Element of REAL
for b2, b3, b4 being Function-like Relation of REAL,REAL
      st b2 is_convergent_in b1 &
         b3 is_convergent_in b1 &
         lim(b2,b1) = lim(b3,b1) &
         (for b5, b6 being Element of REAL
               st b5 < b1 & b1 < b6
            holds ex b7, b8 being Element of REAL st
               b5 < b7 & b7 < b1 & b7 in proj1 b4 & b8 < b6 & b1 < b8 & b8 in proj1 b4) &
         (ex b5 being Element of REAL st
            0 < b5 &
             (for b6 being Element of REAL
                   st b6 in (proj1 b4) /\ (].b1 - b5,b1.[ \/ ].b1,b1 + b5.[)
                holds b2 . b6 <= b4 . b6 & b4 . b6 <= b3 . b6) &
             ((proj1 b2) /\ (].b1 - b5,b1.[ \/ ].b1,b1 + b5.[) c= (proj1 b3) /\ (].b1 - b5,b1.[ \/ ].b1,b1 + b5.[) &
              (proj1 b4) /\ (].b1 - b5,b1.[ \/ ].b1,b1 + b5.[) c= (proj1 b2) /\ (].b1 - b5,b1.[ \/ ].b1,b1 + b5.[) or (proj1 b3) /\ (].b1 - b5,b1.[ \/ ].b1,b1 + b5.[) c= (proj1 b2) /\ (].b1 - b5,b1.[ \/ ].b1,b1 + b5.[) &
              (proj1 b4) /\ (].b1 - b5,b1.[ \/ ].b1,b1 + b5.[) c= (proj1 b3) /\ (].b1 - b5,b1.[ \/ ].b1,b1 + b5.[)))
   holds b4 is_convergent_in b1 & lim(b4,b1) = lim(b2,b1);

:: LIMFUNC3:th 46
theorem
for b1 being Element of REAL
for b2, b3, b4 being Function-like Relation of REAL,REAL
      st b2 is_convergent_in b1 &
         b3 is_convergent_in b1 &
         lim(b2,b1) = lim(b3,b1) &
         (ex b5 being Element of REAL st
            0 < b5 &
             ].b1 - b5,b1.[ \/ ].b1,b1 + b5.[ c= ((proj1 b2) /\ proj1 b3) /\ proj1 b4 &
             (for b6 being Element of REAL
                   st b6 in ].b1 - b5,b1.[ \/ ].b1,b1 + b5.[
                holds b2 . b6 <= b4 . b6 & b4 . b6 <= b3 . b6))
   holds b4 is_convergent_in b1 & lim(b4,b1) = lim(b2,b1);

:: LIMFUNC3:th 47
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_convergent_in b1 &
         b3 is_convergent_in b1 &
         (ex b4 being Element of REAL st
            0 < b4 &
             ((proj1 b2) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[) c= (proj1 b3) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[) &
              (for b5 being Element of REAL
                    st b5 in (proj1 b2) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[)
                 holds b2 . b5 <= b3 . b5) or (proj1 b3) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[) c= (proj1 b2) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[) &
              (for b5 being Element of REAL
                    st b5 in (proj1 b3) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[)
                 holds b2 . b5 <= b3 . b5)))
   holds lim(b2,b1) <= lim(b3,b1);

:: LIMFUNC3:th 48
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st (b2 is_divergent_to+infty_in b1 or b2 is_divergent_to-infty_in b1) &
         (for b3, b4 being Element of REAL
               st b3 < b1 & b1 < b4
            holds ex b5, b6 being Element of REAL st
               b3 < b5 & b5 < b1 & b5 in proj1 b2 & b6 < b4 & b1 < b6 & b6 in proj1 b2 & b2 . b5 <> 0 & b2 . b6 <> 0)
   holds b2 ^ is_convergent_in b1 & lim(b2 ^,b1) = 0;

:: LIMFUNC3:th 49
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_convergent_in b1 &
         lim(b2,b1) = 0 &
         (for b3, b4 being Element of REAL
               st b3 < b1 & b1 < b4
            holds ex b5, b6 being Element of REAL st
               b3 < b5 & b5 < b1 & b5 in proj1 b2 & b6 < b4 & b1 < b6 & b6 in proj1 b2 & b2 . b5 <> 0 & b2 . b6 <> 0) &
         (ex b3 being Element of REAL st
            0 < b3 &
             (for b4 being Element of REAL
                   st b4 in (proj1 b2) /\ (].b1 - b3,b1.[ \/ ].b1,b1 + b3.[)
                holds 0 <= b2 . b4))
   holds b2 ^ is_divergent_to+infty_in b1;

:: LIMFUNC3:th 50
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_convergent_in b1 &
         lim(b2,b1) = 0 &
         (for b3, b4 being Element of REAL
               st b3 < b1 & b1 < b4
            holds ex b5, b6 being Element of REAL st
               b3 < b5 & b5 < b1 & b5 in proj1 b2 & b6 < b4 & b1 < b6 & b6 in proj1 b2 & b2 . b5 <> 0 & b2 . b6 <> 0) &
         (ex b3 being Element of REAL st
            0 < b3 &
             (for b4 being Element of REAL
                   st b4 in (proj1 b2) /\ (].b1 - b3,b1.[ \/ ].b1,b1 + b3.[)
                holds b2 . b4 <= 0))
   holds b2 ^ is_divergent_to-infty_in b1;

:: LIMFUNC3:th 51
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_convergent_in b1 &
         lim(b2,b1) = 0 &
         (ex b3 being Element of REAL st
            0 < b3 &
             (for b4 being Element of REAL
                   st b4 in (proj1 b2) /\ (].b1 - b3,b1.[ \/ ].b1,b1 + b3.[)
                holds 0 < b2 . b4))
   holds b2 ^ is_divergent_to+infty_in b1;

:: LIMFUNC3:th 52
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_convergent_in b1 &
         lim(b2,b1) = 0 &
         (ex b3 being Element of REAL st
            0 < b3 &
             (for b4 being Element of REAL
                   st b4 in (proj1 b2) /\ (].b1 - b3,b1.[ \/ ].b1,b1 + b3.[)
                holds b2 . b4 < 0))
   holds b2 ^ is_divergent_to-infty_in b1;