Article LIMFUNC1, MML version 4.99.1005

:: LIMFUNC1:funcnot 1 => LIMFUNC1:func 1
definition
  let a1, a2 be Element of NAT;
  redefine func max(a1,a2) -> Element of NAT;
  commutativity;
::  for a1, a2 being Element of NAT holds
::  max(a1,a2) = max(a2,a1);
  idempotence;
::  for a1 being Element of NAT holds
::     max(a1,a1) = a1;
end;

:: LIMFUNC1:funcnot 2 => PROB_1:func 11
notation
  let a1 be real set;
  synonym left_open_halfline a1 for halfline a1;
end;

:: LIMFUNC1:funcnot 3 => LIMFUNC1:func 2
definition
  let a1 be real set;
  func left_closed_halfline A1 -> Element of bool REAL equals
    ].-infty,a1.];
end;

:: LIMFUNC1:def 1
theorem
for b1 being real set holds
   left_closed_halfline b1 = ].-infty,b1.];

:: LIMFUNC1:funcnot 4 => LIMFUNC1:func 3
definition
  let a1 be real set;
  func right_closed_halfline A1 -> Element of bool REAL equals
    [.a1,+infty.[;
end;

:: LIMFUNC1:def 2
theorem
for b1 being real set holds
   right_closed_halfline b1 = [.b1,+infty.[;

:: LIMFUNC1:funcnot 5 => LIMFUNC1:func 4
definition
  let a1 be real set;
  func right_open_halfline A1 -> Element of bool REAL equals
    ].a1,+infty.[;
end;

:: LIMFUNC1:def 3
theorem
for b1 being real set holds
   right_open_halfline b1 = ].b1,+infty.[;

:: LIMFUNC1:th 8
theorem
for b1, b2 being real set
      st b1 <= b2
   holds right_open_halfline b2 c= right_open_halfline b1;

:: LIMFUNC1:th 9
theorem
for b1, b2 being real set
      st b1 <= b2
   holds right_closed_halfline b2 c= right_closed_halfline b1;

:: LIMFUNC1:th 10
theorem
for b1 being real set holds
   right_open_halfline b1 c= right_closed_halfline b1;

:: LIMFUNC1:th 11
theorem
for b1, b2 being real set holds
].b1,b2.[ c= right_open_halfline b1;

:: LIMFUNC1:th 12
theorem
for b1, b2 being real set holds
[.b1,b2.] c= right_closed_halfline b1;

:: LIMFUNC1:th 13
theorem
for b1, b2 being real set
      st b1 <= b2
   holds halfline b1 c= halfline b2;

:: LIMFUNC1:th 14
theorem
for b1, b2 being real set
      st b1 <= b2
   holds left_closed_halfline b1 c= left_closed_halfline b2;

:: LIMFUNC1:th 15
theorem
for b1 being real set holds
   halfline b1 c= left_closed_halfline b1;

:: LIMFUNC1:th 16
theorem
for b1, b2 being real set holds
].b1,b2.[ c= halfline b2;

:: LIMFUNC1:th 17
theorem
for b1, b2 being real set holds
[.b1,b2.] c= left_closed_halfline b2;

:: LIMFUNC1:th 18
theorem
for b1, b2 being real set holds
(halfline b1) /\ right_open_halfline b2 = ].b2,b1.[;

:: LIMFUNC1:th 19
theorem
for b1, b2 being real set holds
(left_closed_halfline b1) /\ right_closed_halfline b2 = [.b2,b1.];

:: LIMFUNC1:th 20
theorem
for b1, b2, b3 being real set
      st b1 <= b2
   holds ].b2,b3.[ c= right_open_halfline b1 & [.b2,b3.] c= right_closed_halfline b1;

:: LIMFUNC1:th 21
theorem
for b1, b2, b3 being real set
      st b1 < b2
   holds [.b2,b3.] c= right_open_halfline b1;

:: LIMFUNC1:th 22
theorem
for b1, b2, b3 being real set
      st b1 <= b2
   holds ].b3,b1.[ c= halfline b2 & [.b3,b1.] c= left_closed_halfline b2;

:: LIMFUNC1:th 23
theorem
for b1, b2, b3 being real set
      st b1 < b2
   holds [.b3,b1.] c= halfline b2;

:: LIMFUNC1:th 24
theorem
for b1 being real set holds
   REAL \ right_open_halfline b1 = left_closed_halfline b1 & REAL \ right_closed_halfline b1 = halfline b1 & REAL \ halfline b1 = right_closed_halfline b1 & REAL \ left_closed_halfline b1 = right_open_halfline b1;

:: LIMFUNC1:th 25
theorem
for b1, b2 being real set holds
REAL \ ].b1,b2.[ = (left_closed_halfline b1) \/ right_closed_halfline b2 &
 REAL \ [.b1,b2.] = (halfline b1) \/ right_open_halfline b2;

:: LIMFUNC1:th 26
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
   (b1 is non-decreasing implies b1 is bounded_below) & (b1 is non-increasing implies b1 is bounded_above);

:: LIMFUNC1:th 27
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st b1 is non-empty & b1 is convergent & lim b1 = 0 & b1 is non-decreasing
for b2 being Element of NAT holds
   b1 . b2 < 0;

:: LIMFUNC1:th 28
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st b1 is non-empty & b1 is convergent & lim b1 = 0 & b1 is non-increasing
for b2 being Element of NAT holds
   0 < b1 . b2;

:: LIMFUNC1:th 29
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is convergent & 0 < lim b1
   holds ex b2 being Element of NAT st
      for b3 being Element of NAT
            st b2 <= b3
         holds 0 < b1 . b3;

:: LIMFUNC1:th 30
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is convergent & 0 < lim b1
   holds ex b2 being Element of NAT st
      for b3 being Element of NAT
            st b2 <= b3
         holds (lim b1) / 2 < b1 . b3;

:: LIMFUNC1:attrnot 1 => LIMFUNC1:attr 1
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  attr a1 is divergent_to+infty means
    for b1 being Element of REAL holds
       ex b2 being Element of NAT st
          for b3 being Element of NAT
                st b2 <= b3
             holds b1 < a1 . b3;
end;

:: LIMFUNC1:dfs 4
definiens
  let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
     a1 is divergent_to+infty
it is sufficient to prove
  thus for b1 being Element of REAL holds
       ex b2 being Element of NAT st
          for b3 being Element of NAT
                st b2 <= b3
             holds b1 < a1 . b3;

:: LIMFUNC1:def 4
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is divergent_to+infty
   iff
      for b2 being Element of REAL holds
         ex b3 being Element of NAT st
            for b4 being Element of NAT
                  st b3 <= b4
               holds b2 < b1 . b4;

:: LIMFUNC1:attrnot 2 => LIMFUNC1:attr 2
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  attr a1 is divergent_to-infty means
    for b1 being Element of REAL holds
       ex b2 being Element of NAT st
          for b3 being Element of NAT
                st b2 <= b3
             holds a1 . b3 < b1;
end;

:: LIMFUNC1:dfs 5
definiens
  let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
     a1 is divergent_to-infty
it is sufficient to prove
  thus for b1 being Element of REAL holds
       ex b2 being Element of NAT st
          for b3 being Element of NAT
                st b2 <= b3
             holds a1 . b3 < b1;

:: LIMFUNC1:def 5
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is divergent_to-infty
   iff
      for b2 being Element of REAL holds
         ex b3 being Element of NAT st
            for b4 being Element of NAT
                  st b3 <= b4
               holds b1 . b4 < b2;

:: LIMFUNC1:th 33
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st (b1 is divergent_to+infty or b1 is divergent_to-infty)
   holds ex b2 being Element of NAT st
      for b3 being Element of NAT
            st b2 <= b3
         holds b1 ^\ b3 is non-empty;

:: LIMFUNC1:th 34
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL holds
   (b2 ^\ b1 is divergent_to+infty implies b2 is divergent_to+infty) & (b2 ^\ b1 is divergent_to-infty implies b2 is divergent_to-infty);

:: LIMFUNC1:th 35
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is divergent_to+infty & b2 is divergent_to+infty
   holds b1 + b2 is divergent_to+infty;

:: LIMFUNC1:th 36
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is divergent_to+infty & b2 is bounded_below
   holds b1 + b2 is divergent_to+infty;

:: LIMFUNC1:th 37
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is divergent_to+infty & b2 is divergent_to+infty
   holds b1 (#) b2 is divergent_to+infty;

:: LIMFUNC1:th 38
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is divergent_to-infty & b2 is divergent_to-infty
   holds b1 + b2 is divergent_to-infty;

:: LIMFUNC1:th 39
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is divergent_to-infty & b2 is bounded_above
   holds b1 + b2 is divergent_to-infty;

:: LIMFUNC1:th 40
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of REAL holds
   (b1 is divergent_to+infty & 0 < b2 implies b2 (#) b1 is divergent_to+infty) &
    (b1 is divergent_to+infty & b2 < 0 implies b2 (#) b1 is divergent_to-infty) &
    (b1 is divergent_to+infty & b2 = 0 implies rng (b2 (#) b1) = {0} & b2 (#) b1 is constant);

:: LIMFUNC1:th 41
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of REAL holds
   (b1 is divergent_to-infty & 0 < b2 implies b2 (#) b1 is divergent_to-infty) &
    (b1 is divergent_to-infty & b2 < 0 implies b2 (#) b1 is divergent_to+infty) &
    (b1 is divergent_to-infty & b2 = 0 implies rng (b2 (#) b1) = {0} & b2 (#) b1 is constant);

:: LIMFUNC1:th 42
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
   (b1 is divergent_to+infty implies - b1 is divergent_to-infty) & (b1 is divergent_to-infty implies - b1 is divergent_to+infty);

:: LIMFUNC1:th 43
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded_below & b2 is divergent_to-infty
   holds b1 - b2 is divergent_to+infty;

:: LIMFUNC1:th 44
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded_above & b2 is divergent_to+infty
   holds b1 - b2 is divergent_to-infty;

:: LIMFUNC1:th 45
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is divergent_to+infty & b2 is convergent
   holds b1 + b2 is divergent_to+infty;

:: LIMFUNC1:th 46
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is divergent_to-infty & b2 is convergent
   holds b1 + b2 is divergent_to-infty;

:: LIMFUNC1:th 47
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st for b2 being Element of NAT holds
           b1 . b2 = b2
   holds b1 is divergent_to+infty;

:: LIMFUNC1:th 48
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st for b2 being Element of NAT holds
           b1 . b2 = - b2
   holds b1 is divergent_to-infty;

:: LIMFUNC1:th 49
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is divergent_to+infty &
         (ex b3 being Element of REAL st
            0 < b3 &
             (for b4 being Element of NAT holds
                b3 <= b2 . b4))
   holds b1 (#) b2 is divergent_to+infty;

:: LIMFUNC1:th 50
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is divergent_to-infty &
         (ex b3 being Element of REAL st
            0 < b3 &
             (for b4 being Element of NAT holds
                b3 <= b2 . b4))
   holds b1 (#) b2 is divergent_to-infty;

:: LIMFUNC1:th 51
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is divergent_to-infty & b2 is divergent_to-infty
   holds b1 (#) b2 is divergent_to+infty;

:: LIMFUNC1:th 52
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st (b1 is divergent_to+infty or b1 is divergent_to-infty)
   holds abs b1 is divergent_to+infty;

:: LIMFUNC1:th 53
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is divergent_to+infty & b2 is subsequence of b1
   holds b2 is divergent_to+infty;

:: LIMFUNC1:th 54
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is divergent_to-infty & b2 is subsequence of b1
   holds b2 is divergent_to-infty;

:: LIMFUNC1:th 55
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is divergent_to+infty & b2 is convergent & 0 < lim b2
   holds b1 (#) b2 is divergent_to+infty;

:: LIMFUNC1:th 56
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-decreasing & b1 is not bounded_above
   holds b1 is divergent_to+infty;

:: LIMFUNC1:th 57
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-increasing & b1 is not bounded_below
   holds b1 is divergent_to-infty;

:: LIMFUNC1:th 58
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is increasing & b1 is not bounded_above
   holds b1 is divergent_to+infty;

:: LIMFUNC1:th 59
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is decreasing & b1 is not bounded_below
   holds b1 is divergent_to-infty;

:: LIMFUNC1:th 60
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is monotone & b1 is not convergent & b1 is not divergent_to+infty
   holds b1 is divergent_to-infty;

:: LIMFUNC1:th 61
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st (b1 is divergent_to+infty or b1 is divergent_to-infty)
   holds b1 " is convergent & lim (b1 ") = 0;

:: LIMFUNC1:th 62
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-empty &
         b1 is convergent &
         lim b1 = 0 &
         (ex b2 being Element of NAT st
            for b3 being Element of NAT
                  st b2 <= b3
               holds 0 < b1 . b3)
   holds b1 " is divergent_to+infty;

:: LIMFUNC1:th 63
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-empty &
         b1 is convergent &
         lim b1 = 0 &
         (ex b2 being Element of NAT st
            for b3 being Element of NAT
                  st b2 <= b3
               holds b1 . b3 < 0)
   holds b1 " is divergent_to-infty;

:: LIMFUNC1:th 64
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-empty & b1 is convergent & lim b1 = 0 & b1 is non-decreasing
   holds b1 " is divergent_to-infty;

:: LIMFUNC1:th 65
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-empty & b1 is convergent & lim b1 = 0 & b1 is non-increasing
   holds b1 " is divergent_to+infty;

:: LIMFUNC1:th 66
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-empty & b1 is convergent & lim b1 = 0 & b1 is increasing
   holds b1 " is divergent_to-infty;

:: LIMFUNC1:th 67
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-empty & b1 is convergent & lim b1 = 0 & b1 is decreasing
   holds b1 " is divergent_to+infty;

:: LIMFUNC1:th 68
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded & (b2 is divergent_to+infty or b2 is divergent_to-infty) & b2 is non-empty
   holds b1 /" b2 is convergent & lim (b1 /" b2) = 0;

:: LIMFUNC1:th 69
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is divergent_to+infty &
         (for b3 being Element of NAT holds
            b1 . b3 <= b2 . b3)
   holds b2 is divergent_to+infty;

:: LIMFUNC1:th 70
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is divergent_to-infty &
         (for b3 being Element of NAT holds
            b2 . b3 <= b1 . b3)
   holds b2 is divergent_to-infty;

:: LIMFUNC1:attrnot 3 => LIMFUNC1:attr 3
definition
  let a1 be Function-like Relation of REAL,REAL;
  attr a1 is convergent_in+infty means
    (for b1 being Element of REAL holds
        ex b2 being Element of REAL st
           b1 < 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 divergent_to+infty & rng b2 c= dom a1
           holds a1 * b2 is convergent & lim (a1 * b2) = b1);
end;

:: LIMFUNC1:dfs 6
definiens
  let a1 be Function-like Relation of REAL,REAL;
To prove
     a1 is convergent_in+infty
it is sufficient to prove
  thus (for b1 being Element of REAL holds
        ex b2 being Element of REAL st
           b1 < 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 divergent_to+infty & rng b2 c= dom a1
           holds a1 * b2 is convergent & lim (a1 * b2) = b1);

:: LIMFUNC1:def 6
theorem
for b1 being Function-like Relation of REAL,REAL holds
      b1 is convergent_in+infty
   iff
      (for b2 being Element of REAL holds
          ex b3 being Element of REAL st
             b2 < b3 & b3 in dom b1) &
       (ex b2 being Element of REAL st
          for b3 being Function-like quasi_total Relation of NAT,REAL
                st b3 is divergent_to+infty & rng b3 c= dom b1
             holds b1 * b3 is convergent & lim (b1 * b3) = b2);

:: LIMFUNC1:attrnot 4 => LIMFUNC1:attr 4
definition
  let a1 be Function-like Relation of REAL,REAL;
  attr a1 is divergent_in+infty_to+infty means
    (for b1 being Element of REAL holds
        ex b2 being Element of REAL st
           b1 < b2 & b2 in dom a1) &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st b1 is divergent_to+infty & rng b1 c= dom a1
        holds a1 * b1 is divergent_to+infty);
end;

:: LIMFUNC1:dfs 7
definiens
  let a1 be Function-like Relation of REAL,REAL;
To prove
     a1 is divergent_in+infty_to+infty
it is sufficient to prove
  thus (for b1 being Element of REAL holds
        ex b2 being Element of REAL st
           b1 < b2 & b2 in dom a1) &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st b1 is divergent_to+infty & rng b1 c= dom a1
        holds a1 * b1 is divergent_to+infty);

:: LIMFUNC1:def 7
theorem
for b1 being Function-like Relation of REAL,REAL holds
      b1 is divergent_in+infty_to+infty
   iff
      (for b2 being Element of REAL holds
          ex b3 being Element of REAL st
             b2 < b3 & b3 in dom b1) &
       (for b2 being Function-like quasi_total Relation of NAT,REAL
             st b2 is divergent_to+infty & rng b2 c= dom b1
          holds b1 * b2 is divergent_to+infty);

:: LIMFUNC1:attrnot 5 => LIMFUNC1:attr 5
definition
  let a1 be Function-like Relation of REAL,REAL;
  attr a1 is divergent_in+infty_to-infty means
    (for b1 being Element of REAL holds
        ex b2 being Element of REAL st
           b1 < b2 & b2 in dom a1) &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st b1 is divergent_to+infty & rng b1 c= dom a1
        holds a1 * b1 is divergent_to-infty);
end;

:: LIMFUNC1:dfs 8
definiens
  let a1 be Function-like Relation of REAL,REAL;
To prove
     a1 is divergent_in+infty_to-infty
it is sufficient to prove
  thus (for b1 being Element of REAL holds
        ex b2 being Element of REAL st
           b1 < b2 & b2 in dom a1) &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st b1 is divergent_to+infty & rng b1 c= dom a1
        holds a1 * b1 is divergent_to-infty);

:: LIMFUNC1:def 8
theorem
for b1 being Function-like Relation of REAL,REAL holds
      b1 is divergent_in+infty_to-infty
   iff
      (for b2 being Element of REAL holds
          ex b3 being Element of REAL st
             b2 < b3 & b3 in dom b1) &
       (for b2 being Function-like quasi_total Relation of NAT,REAL
             st b2 is divergent_to+infty & rng b2 c= dom b1
          holds b1 * b2 is divergent_to-infty);

:: LIMFUNC1:attrnot 6 => LIMFUNC1:attr 6
definition
  let a1 be Function-like Relation of REAL,REAL;
  attr a1 is convergent_in-infty means
    (for b1 being Element of REAL holds
        ex b2 being Element of REAL st
           b2 < b1 & 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 divergent_to-infty & rng b2 c= dom a1
           holds a1 * b2 is convergent & lim (a1 * b2) = b1);
end;

:: LIMFUNC1:dfs 9
definiens
  let a1 be Function-like Relation of REAL,REAL;
To prove
     a1 is convergent_in-infty
it is sufficient to prove
  thus (for b1 being Element of REAL holds
        ex b2 being Element of REAL st
           b2 < b1 & 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 divergent_to-infty & rng b2 c= dom a1
           holds a1 * b2 is convergent & lim (a1 * b2) = b1);

:: LIMFUNC1:def 9
theorem
for b1 being Function-like Relation of REAL,REAL holds
      b1 is convergent_in-infty
   iff
      (for b2 being Element of REAL holds
          ex b3 being Element of REAL st
             b3 < b2 & b3 in dom b1) &
       (ex b2 being Element of REAL st
          for b3 being Function-like quasi_total Relation of NAT,REAL
                st b3 is divergent_to-infty & rng b3 c= dom b1
             holds b1 * b3 is convergent & lim (b1 * b3) = b2);

:: LIMFUNC1:attrnot 7 => LIMFUNC1:attr 7
definition
  let a1 be Function-like Relation of REAL,REAL;
  attr a1 is divergent_in-infty_to+infty means
    (for b1 being Element of REAL holds
        ex b2 being Element of REAL st
           b2 < b1 & b2 in dom a1) &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st b1 is divergent_to-infty & rng b1 c= dom a1
        holds a1 * b1 is divergent_to+infty);
end;

:: LIMFUNC1:dfs 10
definiens
  let a1 be Function-like Relation of REAL,REAL;
To prove
     a1 is divergent_in-infty_to+infty
it is sufficient to prove
  thus (for b1 being Element of REAL holds
        ex b2 being Element of REAL st
           b2 < b1 & b2 in dom a1) &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st b1 is divergent_to-infty & rng b1 c= dom a1
        holds a1 * b1 is divergent_to+infty);

:: LIMFUNC1:def 10
theorem
for b1 being Function-like Relation of REAL,REAL holds
      b1 is divergent_in-infty_to+infty
   iff
      (for b2 being Element of REAL holds
          ex b3 being Element of REAL st
             b3 < b2 & b3 in dom b1) &
       (for b2 being Function-like quasi_total Relation of NAT,REAL
             st b2 is divergent_to-infty & rng b2 c= dom b1
          holds b1 * b2 is divergent_to+infty);

:: LIMFUNC1:attrnot 8 => LIMFUNC1:attr 8
definition
  let a1 be Function-like Relation of REAL,REAL;
  attr a1 is divergent_in-infty_to-infty means
    (for b1 being Element of REAL holds
        ex b2 being Element of REAL st
           b2 < b1 & b2 in dom a1) &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st b1 is divergent_to-infty & rng b1 c= dom a1
        holds a1 * b1 is divergent_to-infty);
end;

:: LIMFUNC1:dfs 11
definiens
  let a1 be Function-like Relation of REAL,REAL;
To prove
     a1 is divergent_in-infty_to-infty
it is sufficient to prove
  thus (for b1 being Element of REAL holds
        ex b2 being Element of REAL st
           b2 < b1 & b2 in dom a1) &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st b1 is divergent_to-infty & rng b1 c= dom a1
        holds a1 * b1 is divergent_to-infty);

:: LIMFUNC1:def 11
theorem
for b1 being Function-like Relation of REAL,REAL holds
      b1 is divergent_in-infty_to-infty
   iff
      (for b2 being Element of REAL holds
          ex b3 being Element of REAL st
             b3 < b2 & b3 in dom b1) &
       (for b2 being Function-like quasi_total Relation of NAT,REAL
             st b2 is divergent_to-infty & rng b2 c= dom b1
          holds b1 * b2 is divergent_to-infty);

:: LIMFUNC1:th 77
theorem
for b1 being Function-like Relation of REAL,REAL holds
      b1 is convergent_in+infty
   iff
      (for b2 being Element of REAL holds
          ex b3 being Element of REAL st
             b2 < b3 & b3 in dom b1) &
       (ex b2 being Element of REAL st
          for b3 being Element of REAL
                st 0 < b3
             holds ex b4 being Element of REAL st
                for b5 being Element of REAL
                      st b4 < b5 & b5 in dom b1
                   holds abs ((b1 . b5) - b2) < b3);

:: LIMFUNC1:th 78
theorem
for b1 being Function-like Relation of REAL,REAL holds
      b1 is convergent_in-infty
   iff
      (for b2 being Element of REAL holds
          ex b3 being Element of REAL st
             b3 < b2 & b3 in dom b1) &
       (ex b2 being Element of REAL st
          for b3 being Element of REAL
                st 0 < b3
             holds ex b4 being Element of REAL st
                for b5 being Element of REAL
                      st b5 < b4 & b5 in dom b1
                   holds abs ((b1 . b5) - b2) < b3);

:: LIMFUNC1:th 79
theorem
for b1 being Function-like Relation of REAL,REAL holds
      b1 is divergent_in+infty_to+infty
   iff
      (for b2 being Element of REAL holds
          ex b3 being Element of REAL st
             b2 < b3 & b3 in dom b1) &
       (for b2 being Element of REAL holds
          ex b3 being Element of REAL st
             for b4 being Element of REAL
                   st b3 < b4 & b4 in dom b1
                holds b2 < b1 . b4);

:: LIMFUNC1:th 80
theorem
for b1 being Function-like Relation of REAL,REAL holds
      b1 is divergent_in+infty_to-infty
   iff
      (for b2 being Element of REAL holds
          ex b3 being Element of REAL st
             b2 < b3 & b3 in dom b1) &
       (for b2 being Element of REAL holds
          ex b3 being Element of REAL st
             for b4 being Element of REAL
                   st b3 < b4 & b4 in dom b1
                holds b1 . b4 < b2);

:: LIMFUNC1:th 81
theorem
for b1 being Function-like Relation of REAL,REAL holds
      b1 is divergent_in-infty_to+infty
   iff
      (for b2 being Element of REAL holds
          ex b3 being Element of REAL st
             b3 < b2 & b3 in dom b1) &
       (for b2 being Element of REAL holds
          ex b3 being Element of REAL st
             for b4 being Element of REAL
                   st b4 < b3 & b4 in dom b1
                holds b2 < b1 . b4);

:: LIMFUNC1:th 82
theorem
for b1 being Function-like Relation of REAL,REAL holds
      b1 is divergent_in-infty_to-infty
   iff
      (for b2 being Element of REAL holds
          ex b3 being Element of REAL st
             b3 < b2 & b3 in dom b1) &
       (for b2 being Element of REAL holds
          ex b3 being Element of REAL st
             for b4 being Element of REAL
                   st b4 < b3 & b4 in dom b1
                holds b1 . b4 < b2);

:: LIMFUNC1:th 83
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is divergent_in+infty_to+infty &
         b2 is divergent_in+infty_to+infty &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b3 < b4 & b4 in (dom b1) /\ dom b2)
   holds b1 + b2 is divergent_in+infty_to+infty & b1 (#) b2 is divergent_in+infty_to+infty;

:: LIMFUNC1:th 84
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is divergent_in+infty_to-infty &
         b2 is divergent_in+infty_to-infty &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b3 < b4 & b4 in (dom b1) /\ dom b2)
   holds b1 + b2 is divergent_in+infty_to-infty & b1 (#) b2 is divergent_in+infty_to+infty;

:: LIMFUNC1:th 85
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is divergent_in-infty_to+infty &
         b2 is divergent_in-infty_to+infty &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b4 < b3 & b4 in (dom b1) /\ dom b2)
   holds b1 + b2 is divergent_in-infty_to+infty & b1 (#) b2 is divergent_in-infty_to+infty;

:: LIMFUNC1:th 86
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is divergent_in-infty_to-infty &
         b2 is divergent_in-infty_to-infty &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b4 < b3 & b4 in (dom b1) /\ dom b2)
   holds b1 + b2 is divergent_in-infty_to-infty & b1 (#) b2 is divergent_in-infty_to+infty;

:: LIMFUNC1:th 87
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is divergent_in+infty_to+infty &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b3 < b4 & b4 in dom (b1 + b2)) &
         (ex b3 being Element of REAL st
            b2 is_bounded_below_on right_open_halfline b3)
   holds b1 + b2 is divergent_in+infty_to+infty;

:: LIMFUNC1:th 88
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is divergent_in+infty_to+infty &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b3 < b4 & b4 in dom (b1 (#) b2)) &
         (ex b3, b4 being Element of REAL st
            0 < b3 &
             (for b5 being Element of REAL
                   st b5 in (dom b2) /\ right_open_halfline b4
                holds b3 <= b2 . b5))
   holds b1 (#) b2 is divergent_in+infty_to+infty;

:: LIMFUNC1:th 89
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is divergent_in-infty_to+infty &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b4 < b3 & b4 in dom (b1 + b2)) &
         (ex b3 being Element of REAL st
            b2 is_bounded_below_on halfline b3)
   holds b1 + b2 is divergent_in-infty_to+infty;

:: LIMFUNC1:th 90
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is divergent_in-infty_to+infty &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b4 < b3 & b4 in dom (b1 (#) b2)) &
         (ex b3, b4 being Element of REAL st
            0 < b3 &
             (for b5 being Element of REAL
                   st b5 in (dom b2) /\ halfline b4
                holds b3 <= b2 . b5))
   holds b1 (#) b2 is divergent_in-infty_to+infty;

:: LIMFUNC1:th 91
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
   (b1 is divergent_in+infty_to+infty & 0 < b2 implies b2 (#) b1 is divergent_in+infty_to+infty) &
    (b1 is divergent_in+infty_to+infty & b2 < 0 implies b2 (#) b1 is divergent_in+infty_to-infty) &
    (b1 is divergent_in+infty_to-infty & 0 < b2 implies b2 (#) b1 is divergent_in+infty_to-infty) &
    (b1 is divergent_in+infty_to-infty & b2 < 0 implies b2 (#) b1 is divergent_in+infty_to+infty);

:: LIMFUNC1:th 92
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
   (b1 is divergent_in-infty_to+infty & 0 < b2 implies b2 (#) b1 is divergent_in-infty_to+infty) &
    (b1 is divergent_in-infty_to+infty & b2 < 0 implies b2 (#) b1 is divergent_in-infty_to-infty) &
    (b1 is divergent_in-infty_to-infty & 0 < b2 implies b2 (#) b1 is divergent_in-infty_to-infty) &
    (b1 is divergent_in-infty_to-infty & b2 < 0 implies b2 (#) b1 is divergent_in-infty_to+infty);

:: LIMFUNC1:th 93
theorem
for b1 being Function-like Relation of REAL,REAL
      st (b1 is divergent_in+infty_to+infty or b1 is divergent_in+infty_to-infty)
   holds abs b1 is divergent_in+infty_to+infty;

:: LIMFUNC1:th 94
theorem
for b1 being Function-like Relation of REAL,REAL
      st (b1 is divergent_in-infty_to+infty or b1 is divergent_in-infty_to-infty)
   holds abs b1 is divergent_in-infty_to+infty;

:: LIMFUNC1:th 95
theorem
for b1 being Function-like Relation of REAL,REAL
      st (ex b2 being Element of REAL st
            b1 is_non_decreasing_on right_open_halfline b2 & not b1 is_bounded_above_on right_open_halfline b2) &
         (for b2 being Element of REAL holds
            ex b3 being Element of REAL st
               b2 < b3 & b3 in dom b1)
   holds b1 is divergent_in+infty_to+infty;

:: LIMFUNC1:th 96
theorem
for b1 being Function-like Relation of REAL,REAL
      st (ex b2 being Element of REAL st
            b1 is_increasing_on right_open_halfline b2 & not b1 is_bounded_above_on right_open_halfline b2) &
         (for b2 being Element of REAL holds
            ex b3 being Element of REAL st
               b2 < b3 & b3 in dom b1)
   holds b1 is divergent_in+infty_to+infty;

:: LIMFUNC1:th 97
theorem
for b1 being Function-like Relation of REAL,REAL
      st (ex b2 being Element of REAL st
            b1 is_non_increasing_on right_open_halfline b2 & not b1 is_bounded_below_on right_open_halfline b2) &
         (for b2 being Element of REAL holds
            ex b3 being Element of REAL st
               b2 < b3 & b3 in dom b1)
   holds b1 is divergent_in+infty_to-infty;

:: LIMFUNC1:th 98
theorem
for b1 being Function-like Relation of REAL,REAL
      st (ex b2 being Element of REAL st
            b1 is_decreasing_on right_open_halfline b2 & not b1 is_bounded_below_on right_open_halfline b2) &
         (for b2 being Element of REAL holds
            ex b3 being Element of REAL st
               b2 < b3 & b3 in dom b1)
   holds b1 is divergent_in+infty_to-infty;

:: LIMFUNC1:th 99
theorem
for b1 being Function-like Relation of REAL,REAL
      st (ex b2 being Element of REAL st
            b1 is_non_increasing_on halfline b2 & not b1 is_bounded_above_on halfline b2) &
         (for b2 being Element of REAL holds
            ex b3 being Element of REAL st
               b3 < b2 & b3 in dom b1)
   holds b1 is divergent_in-infty_to+infty;

:: LIMFUNC1:th 100
theorem
for b1 being Function-like Relation of REAL,REAL
      st (ex b2 being Element of REAL st
            b1 is_decreasing_on halfline b2 & not b1 is_bounded_above_on halfline b2) &
         (for b2 being Element of REAL holds
            ex b3 being Element of REAL st
               b3 < b2 & b3 in dom b1)
   holds b1 is divergent_in-infty_to+infty;

:: LIMFUNC1:th 101
theorem
for b1 being Function-like Relation of REAL,REAL
      st (ex b2 being Element of REAL st
            b1 is_non_decreasing_on halfline b2 & not b1 is_bounded_below_on halfline b2) &
         (for b2 being Element of REAL holds
            ex b3 being Element of REAL st
               b3 < b2 & b3 in dom b1)
   holds b1 is divergent_in-infty_to-infty;

:: LIMFUNC1:th 102
theorem
for b1 being Function-like Relation of REAL,REAL
      st (ex b2 being Element of REAL st
            b1 is_increasing_on halfline b2 & not b1 is_bounded_below_on halfline b2) &
         (for b2 being Element of REAL holds
            ex b3 being Element of REAL st
               b3 < b2 & b3 in dom b1)
   holds b1 is divergent_in-infty_to-infty;

:: LIMFUNC1:th 103
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is divergent_in+infty_to+infty &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b3 < b4 & b4 in dom b2) &
         (ex b3 being Element of REAL st
            (dom b2) /\ right_open_halfline b3 c= (dom b1) /\ right_open_halfline b3 &
             (for b4 being Element of REAL
                   st b4 in (dom b2) /\ right_open_halfline b3
                holds b1 . b4 <= b2 . b4))
   holds b2 is divergent_in+infty_to+infty;

:: LIMFUNC1:th 104
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is divergent_in+infty_to-infty &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b3 < b4 & b4 in dom b2) &
         (ex b3 being Element of REAL st
            (dom b2) /\ right_open_halfline b3 c= (dom b1) /\ right_open_halfline b3 &
             (for b4 being Element of REAL
                   st b4 in (dom b2) /\ right_open_halfline b3
                holds b2 . b4 <= b1 . b4))
   holds b2 is divergent_in+infty_to-infty;

:: LIMFUNC1:th 105
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is divergent_in-infty_to+infty &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b4 < b3 & b4 in dom b2) &
         (ex b3 being Element of REAL st
            (dom b2) /\ halfline b3 c= (dom b1) /\ halfline b3 &
             (for b4 being Element of REAL
                   st b4 in (dom b2) /\ halfline b3
                holds b1 . b4 <= b2 . b4))
   holds b2 is divergent_in-infty_to+infty;

:: LIMFUNC1:th 106
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is divergent_in-infty_to-infty &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b4 < b3 & b4 in dom b2) &
         (ex b3 being Element of REAL st
            (dom b2) /\ halfline b3 c= (dom b1) /\ halfline b3 &
             (for b4 being Element of REAL
                   st b4 in (dom b2) /\ halfline b3
                holds b2 . b4 <= b1 . b4))
   holds b2 is divergent_in-infty_to-infty;

:: LIMFUNC1:th 107
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is divergent_in+infty_to+infty &
         (ex b3 being Element of REAL st
            right_open_halfline b3 c= (dom b2) /\ dom b1 &
             (for b4 being Element of REAL
                   st b4 in right_open_halfline b3
                holds b1 . b4 <= b2 . b4))
   holds b2 is divergent_in+infty_to+infty;

:: LIMFUNC1:th 108
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is divergent_in+infty_to-infty &
         (ex b3 being Element of REAL st
            right_open_halfline b3 c= (dom b2) /\ dom b1 &
             (for b4 being Element of REAL
                   st b4 in right_open_halfline b3
                holds b2 . b4 <= b1 . b4))
   holds b2 is divergent_in+infty_to-infty;

:: LIMFUNC1:th 109
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is divergent_in-infty_to+infty &
         (ex b3 being Element of REAL st
            halfline b3 c= (dom b2) /\ dom b1 &
             (for b4 being Element of REAL
                   st b4 in halfline b3
                holds b1 . b4 <= b2 . b4))
   holds b2 is divergent_in-infty_to+infty;

:: LIMFUNC1:th 110
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is divergent_in-infty_to-infty &
         (ex b3 being Element of REAL st
            halfline b3 c= (dom b2) /\ dom b1 &
             (for b4 being Element of REAL
                   st b4 in halfline b3
                holds b2 . b4 <= b1 . b4))
   holds b2 is divergent_in-infty_to-infty;

:: LIMFUNC1:funcnot 6 => LIMFUNC1:func 5
definition
  let a1 be Function-like Relation of REAL,REAL;
  assume a1 is convergent_in+infty;
  func lim_in+infty A1 -> Element of REAL means
    for b1 being Function-like quasi_total Relation of NAT,REAL
          st b1 is divergent_to+infty & rng b1 c= dom a1
       holds a1 * b1 is convergent & lim (a1 * b1) = it;
end;

:: LIMFUNC1:def 12
theorem
for b1 being Function-like Relation of REAL,REAL
   st b1 is convergent_in+infty
for b2 being Element of REAL holds
      b2 = lim_in+infty b1
   iff
      for b3 being Function-like quasi_total Relation of NAT,REAL
            st b3 is divergent_to+infty & rng b3 c= dom b1
         holds b1 * b3 is convergent & lim (b1 * b3) = b2;

:: LIMFUNC1:funcnot 7 => LIMFUNC1:func 6
definition
  let a1 be Function-like Relation of REAL,REAL;
  assume a1 is convergent_in-infty;
  func lim_in-infty A1 -> Element of REAL means
    for b1 being Function-like quasi_total Relation of NAT,REAL
          st b1 is divergent_to-infty & rng b1 c= dom a1
       holds a1 * b1 is convergent & lim (a1 * b1) = it;
end;

:: LIMFUNC1:def 13
theorem
for b1 being Function-like Relation of REAL,REAL
   st b1 is convergent_in-infty
for b2 being Element of REAL holds
      b2 = lim_in-infty b1
   iff
      for b3 being Function-like quasi_total Relation of NAT,REAL
            st b3 is divergent_to-infty & rng b3 c= dom b1
         holds b1 * b3 is convergent & lim (b1 * b3) = b2;

:: LIMFUNC1:th 113
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
      st b1 is convergent_in-infty
   holds    lim_in-infty b1 = b2
   iff
      for b3 being Element of REAL
            st 0 < b3
         holds ex b4 being Element of REAL st
            for b5 being Element of REAL
                  st b5 < b4 & b5 in dom b1
               holds abs ((b1 . b5) - b2) < b3;

:: LIMFUNC1:th 114
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
      st b1 is convergent_in+infty
   holds    lim_in+infty b1 = b2
   iff
      for b3 being Element of REAL
            st 0 < b3
         holds ex b4 being Element of REAL st
            for b5 being Element of REAL
                  st b4 < b5 & b5 in dom b1
               holds abs ((b1 . b5) - b2) < b3;

:: LIMFUNC1:th 115
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
      st b1 is convergent_in+infty
   holds b2 (#) b1 is convergent_in+infty &
    lim_in+infty (b2 (#) b1) = b2 * lim_in+infty b1;

:: LIMFUNC1:th 116
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is convergent_in+infty
   holds - b1 is convergent_in+infty & lim_in+infty - b1 = - lim_in+infty b1;

:: LIMFUNC1:th 117
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is convergent_in+infty &
         b2 is convergent_in+infty &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b3 < b4 & b4 in dom (b1 + b2))
   holds b1 + b2 is convergent_in+infty &
    lim_in+infty (b1 + b2) = (lim_in+infty b1) + lim_in+infty b2;

:: LIMFUNC1:th 118
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is convergent_in+infty &
         b2 is convergent_in+infty &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b3 < b4 & b4 in dom (b1 - b2))
   holds b1 - b2 is convergent_in+infty &
    lim_in+infty (b1 - b2) = (lim_in+infty b1) - lim_in+infty b2;

:: LIMFUNC1:th 119
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is convergent_in+infty & b1 " {0} = {} & lim_in+infty b1 <> 0
   holds b1 ^ is convergent_in+infty & lim_in+infty (b1 ^) = (lim_in+infty b1) ";

:: LIMFUNC1:th 120
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is convergent_in+infty
   holds abs b1 is convergent_in+infty & lim_in+infty abs b1 = abs lim_in+infty b1;

:: LIMFUNC1:th 121
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is convergent_in+infty &
         lim_in+infty b1 <> 0 &
         (for b2 being Element of REAL holds
            ex b3 being Element of REAL st
               b2 < b3 & b3 in dom b1 & b1 . b3 <> 0)
   holds b1 ^ is convergent_in+infty & lim_in+infty (b1 ^) = (lim_in+infty b1) ";

:: LIMFUNC1:th 122
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is convergent_in+infty &
         b2 is convergent_in+infty &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b3 < b4 & b4 in dom (b1 (#) b2))
   holds b1 (#) b2 is convergent_in+infty &
    lim_in+infty (b1 (#) b2) = (lim_in+infty b1) * lim_in+infty b2;

:: LIMFUNC1:th 123
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is convergent_in+infty &
         b2 is convergent_in+infty &
         lim_in+infty b2 <> 0 &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b3 < b4 & b4 in dom (b1 / b2))
   holds b1 / b2 is convergent_in+infty &
    lim_in+infty (b1 / b2) = (lim_in+infty b1) / lim_in+infty b2;

:: LIMFUNC1:th 124
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
      st b1 is convergent_in-infty
   holds b2 (#) b1 is convergent_in-infty &
    lim_in-infty (b2 (#) b1) = b2 * lim_in-infty b1;

:: LIMFUNC1:th 125
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is convergent_in-infty
   holds - b1 is convergent_in-infty & lim_in-infty - b1 = - lim_in-infty b1;

:: LIMFUNC1:th 126
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is convergent_in-infty &
         b2 is convergent_in-infty &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b4 < b3 & b4 in dom (b1 + b2))
   holds b1 + b2 is convergent_in-infty &
    lim_in-infty (b1 + b2) = (lim_in-infty b1) + lim_in-infty b2;

:: LIMFUNC1:th 127
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is convergent_in-infty &
         b2 is convergent_in-infty &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b4 < b3 & b4 in dom (b1 - b2))
   holds b1 - b2 is convergent_in-infty &
    lim_in-infty (b1 - b2) = (lim_in-infty b1) - lim_in-infty b2;

:: LIMFUNC1:th 128
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is convergent_in-infty & b1 " {0} = {} & lim_in-infty b1 <> 0
   holds b1 ^ is convergent_in-infty & lim_in-infty (b1 ^) = (lim_in-infty b1) ";

:: LIMFUNC1:th 129
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is convergent_in-infty
   holds abs b1 is convergent_in-infty & lim_in-infty abs b1 = abs lim_in-infty b1;

:: LIMFUNC1:th 130
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is convergent_in-infty &
         lim_in-infty b1 <> 0 &
         (for b2 being Element of REAL holds
            ex b3 being Element of REAL st
               b3 < b2 & b3 in dom b1 & b1 . b3 <> 0)
   holds b1 ^ is convergent_in-infty & lim_in-infty (b1 ^) = (lim_in-infty b1) ";

:: LIMFUNC1:th 131
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is convergent_in-infty &
         b2 is convergent_in-infty &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b4 < b3 & b4 in dom (b1 (#) b2))
   holds b1 (#) b2 is convergent_in-infty &
    lim_in-infty (b1 (#) b2) = (lim_in-infty b1) * lim_in-infty b2;

:: LIMFUNC1:th 132
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is convergent_in-infty &
         b2 is convergent_in-infty &
         lim_in-infty b2 <> 0 &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b4 < b3 & b4 in dom (b1 / b2))
   holds b1 / b2 is convergent_in-infty &
    lim_in-infty (b1 / b2) = (lim_in-infty b1) / lim_in-infty b2;

:: LIMFUNC1:th 133
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is convergent_in+infty &
         lim_in+infty b1 = 0 &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b3 < b4 & b4 in dom (b1 (#) b2)) &
         (ex b3 being Element of REAL st
            b2 is_bounded_on right_open_halfline b3)
   holds b1 (#) b2 is convergent_in+infty & lim_in+infty (b1 (#) b2) = 0;

:: LIMFUNC1:th 134
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is convergent_in-infty &
         lim_in-infty b1 = 0 &
         (for b3 being Element of REAL holds
            ex b4 being Element of REAL st
               b4 < b3 & b4 in dom (b1 (#) b2)) &
         (ex b3 being Element of REAL st
            b2 is_bounded_on halfline b3)
   holds b1 (#) b2 is convergent_in-infty & lim_in-infty (b1 (#) b2) = 0;

:: LIMFUNC1:th 135
theorem
for b1, b2, b3 being Function-like Relation of REAL,REAL
      st b1 is convergent_in+infty &
         b2 is convergent_in+infty &
         lim_in+infty b1 = lim_in+infty b2 &
         (for b4 being Element of REAL holds
            ex b5 being Element of REAL st
               b4 < b5 & b5 in dom b3) &
         (ex b4 being Element of REAL st
            ((dom b1) /\ right_open_halfline b4 c= (dom b2) /\ right_open_halfline b4 &
              (dom b3) /\ right_open_halfline b4 c= (dom b1) /\ right_open_halfline b4 or (dom b2) /\ right_open_halfline b4 c= (dom b1) /\ right_open_halfline b4 &
              (dom b3) /\ right_open_halfline b4 c= (dom b2) /\ right_open_halfline b4) &
             (for b5 being Element of REAL
                   st b5 in (dom b3) /\ right_open_halfline b4
                holds b1 . b5 <= b3 . b5 & b3 . b5 <= b2 . b5))
   holds b3 is convergent_in+infty & lim_in+infty b3 = lim_in+infty b1;

:: LIMFUNC1:th 136
theorem
for b1, b2, b3 being Function-like Relation of REAL,REAL
      st b1 is convergent_in+infty &
         b2 is convergent_in+infty &
         lim_in+infty b1 = lim_in+infty b2 &
         (ex b4 being Element of REAL st
            right_open_halfline b4 c= ((dom b1) /\ dom b2) /\ dom b3 &
             (for b5 being Element of REAL
                   st b5 in right_open_halfline b4
                holds b1 . b5 <= b3 . b5 & b3 . b5 <= b2 . b5))
   holds b3 is convergent_in+infty & lim_in+infty b3 = lim_in+infty b1;

:: LIMFUNC1:th 137
theorem
for b1, b2, b3 being Function-like Relation of REAL,REAL
      st b1 is convergent_in-infty &
         b2 is convergent_in-infty &
         lim_in-infty b1 = lim_in-infty b2 &
         (for b4 being Element of REAL holds
            ex b5 being Element of REAL st
               b5 < b4 & b5 in dom b3) &
         (ex b4 being Element of REAL st
            ((dom b1) /\ halfline b4 c= (dom b2) /\ halfline b4 &
              (dom b3) /\ halfline b4 c= (dom b1) /\ halfline b4 or (dom b2) /\ halfline b4 c= (dom b1) /\ halfline b4 &
              (dom b3) /\ halfline b4 c= (dom b2) /\ halfline b4) &
             (for b5 being Element of REAL
                   st b5 in (dom b3) /\ halfline b4
                holds b1 . b5 <= b3 . b5 & b3 . b5 <= b2 . b5))
   holds b3 is convergent_in-infty & lim_in-infty b3 = lim_in-infty b1;

:: LIMFUNC1:th 138
theorem
for b1, b2, b3 being Function-like Relation of REAL,REAL
      st b1 is convergent_in-infty &
         b2 is convergent_in-infty &
         lim_in-infty b1 = lim_in-infty b2 &
         (ex b4 being Element of REAL st
            halfline b4 c= ((dom b1) /\ dom b2) /\ dom b3 &
             (for b5 being Element of REAL
                   st b5 in halfline b4
                holds b1 . b5 <= b3 . b5 & b3 . b5 <= b2 . b5))
   holds b3 is convergent_in-infty & lim_in-infty b3 = lim_in-infty b1;

:: LIMFUNC1:th 139
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is convergent_in+infty &
         b2 is convergent_in+infty &
         (ex b3 being Element of REAL st
            ((dom b1) /\ right_open_halfline b3 c= (dom b2) /\ right_open_halfline b3 &
             (for b4 being Element of REAL
                   st b4 in (dom b1) /\ right_open_halfline b3
                holds b1 . b4 <= b2 . b4) or (dom b2) /\ right_open_halfline b3 c= (dom b1) /\ right_open_halfline b3 &
             (for b4 being Element of REAL
                   st b4 in (dom b2) /\ right_open_halfline b3
                holds b1 . b4 <= b2 . b4)))
   holds lim_in+infty b1 <= lim_in+infty b2;

:: LIMFUNC1:th 140
theorem
for b1, b2 being Function-like Relation of REAL,REAL
      st b1 is convergent_in-infty &
         b2 is convergent_in-infty &
         (ex b3 being Element of REAL st
            ((dom b1) /\ halfline b3 c= (dom b2) /\ halfline b3 &
             (for b4 being Element of REAL
                   st b4 in (dom b1) /\ halfline b3
                holds b1 . b4 <= b2 . b4) or (dom b2) /\ halfline b3 c= (dom b1) /\ halfline b3 &
             (for b4 being Element of REAL
                   st b4 in (dom b2) /\ halfline b3
                holds b1 . b4 <= b2 . b4)))
   holds lim_in-infty b1 <= lim_in-infty b2;

:: LIMFUNC1:th 141
theorem
for b1 being Function-like Relation of REAL,REAL
      st (b1 is divergent_in+infty_to+infty or b1 is divergent_in+infty_to-infty) &
         (for b2 being Element of REAL holds
            ex b3 being Element of REAL st
               b2 < b3 & b3 in dom b1 & b1 . b3 <> 0)
   holds b1 ^ is convergent_in+infty & lim_in+infty (b1 ^) = 0;

:: LIMFUNC1:th 142
theorem
for b1 being Function-like Relation of REAL,REAL
      st (b1 is divergent_in-infty_to+infty or b1 is divergent_in-infty_to-infty) &
         (for b2 being Element of REAL holds
            ex b3 being Element of REAL st
               b3 < b2 & b3 in dom b1 & b1 . b3 <> 0)
   holds b1 ^ is convergent_in-infty & lim_in-infty (b1 ^) = 0;

:: LIMFUNC1:th 143
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is convergent_in+infty &
         lim_in+infty b1 = 0 &
         (for b2 being Element of REAL holds
            ex b3 being Element of REAL st
               b2 < b3 & b3 in dom b1 & b1 . b3 <> 0) &
         (ex b2 being Element of REAL st
            for b3 being Element of REAL
                  st b3 in (dom b1) /\ right_open_halfline b2
               holds 0 <= b1 . b3)
   holds b1 ^ is divergent_in+infty_to+infty;

:: LIMFUNC1:th 144
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is convergent_in+infty &
         lim_in+infty b1 = 0 &
         (for b2 being Element of REAL holds
            ex b3 being Element of REAL st
               b2 < b3 & b3 in dom b1 & b1 . b3 <> 0) &
         (ex b2 being Element of REAL st
            for b3 being Element of REAL
                  st b3 in (dom b1) /\ right_open_halfline b2
               holds b1 . b3 <= 0)
   holds b1 ^ is divergent_in+infty_to-infty;

:: LIMFUNC1:th 145
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is convergent_in-infty &
         lim_in-infty b1 = 0 &
         (for b2 being Element of REAL holds
            ex b3 being Element of REAL st
               b3 < b2 & b3 in dom b1 & b1 . b3 <> 0) &
         (ex b2 being Element of REAL st
            for b3 being Element of REAL
                  st b3 in (dom b1) /\ halfline b2
               holds 0 <= b1 . b3)
   holds b1 ^ is divergent_in-infty_to+infty;

:: LIMFUNC1:th 146
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is convergent_in-infty &
         lim_in-infty b1 = 0 &
         (for b2 being Element of REAL holds
            ex b3 being Element of REAL st
               b3 < b2 & b3 in dom b1 & b1 . b3 <> 0) &
         (ex b2 being Element of REAL st
            for b3 being Element of REAL
                  st b3 in (dom b1) /\ halfline b2
               holds b1 . b3 <= 0)
   holds b1 ^ is divergent_in-infty_to-infty;

:: LIMFUNC1:th 147
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is convergent_in+infty &
         lim_in+infty b1 = 0 &
         (ex b2 being Element of REAL st
            for b3 being Element of REAL
                  st b3 in (dom b1) /\ right_open_halfline b2
               holds 0 < b1 . b3)
   holds b1 ^ is divergent_in+infty_to+infty;

:: LIMFUNC1:th 148
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is convergent_in+infty &
         lim_in+infty b1 = 0 &
         (ex b2 being Element of REAL st
            for b3 being Element of REAL
                  st b3 in (dom b1) /\ right_open_halfline b2
               holds b1 . b3 < 0)
   holds b1 ^ is divergent_in+infty_to-infty;

:: LIMFUNC1:th 149
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is convergent_in-infty &
         lim_in-infty b1 = 0 &
         (ex b2 being Element of REAL st
            for b3 being Element of REAL
                  st b3 in (dom b1) /\ halfline b2
               holds 0 < b1 . b3)
   holds b1 ^ is divergent_in-infty_to+infty;

:: LIMFUNC1:th 150
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is convergent_in-infty &
         lim_in-infty b1 = 0 &
         (ex b2 being Element of REAL st
            for b3 being Element of REAL
                  st b3 in (dom b1) /\ halfline b2
               holds b1 . b3 < 0)
   holds b1 ^ is divergent_in-infty_to-infty;

:: LIMFUNC1:th 151
theorem
for b1 being real set holds
   ].-infty,b1.[ = {b2 where b2 is Element of REAL: b2 < b1};

:: LIMFUNC1:th 152
theorem
for b1 being real set holds
   ].b1,+infty.[ = {b2 where b2 is Element of REAL: b1 < b2};

:: LIMFUNC1:th 153
theorem
for b1 being real set holds
   ].-infty,b1.] = {b2 where b2 is Element of REAL: b2 <= b1};

:: LIMFUNC1:th 154
theorem
for b1 being real set holds
   [.b1,+infty.[ = {b2 where b2 is Element of REAL: b1 <= b2};

:: LIMFUNC1:th 155
theorem
REAL = ].-infty,+infty.[;