Article CFCONT_1, MML version 4.99.1005

:: CFCONT_1:funcnot 1 => CFCONT_1:func 1
definition
  let a1 be Function-like Relation of COMPLEX,COMPLEX;
  let a2 be Function-like quasi_total Relation of NAT,COMPLEX;
  assume rng a2 c= dom a1;
  func A1 * A2 -> Function-like quasi_total Relation of NAT,COMPLEX equals
    a2 * a1;
end;

:: CFCONT_1:def 1
theorem
for b1 being Function-like Relation of COMPLEX,COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st rng b2 c= dom b1
   holds b1 * b2 = b2 * b1;

:: CFCONT_1:prednot 1 => CFCONT_1:pred 1
definition
  let a1 be Function-like Relation of COMPLEX,COMPLEX;
  let a2 be Element of COMPLEX;
  pred A1 is_continuous_in A2 means
    a2 in dom a1 &
     (for b1 being Function-like quasi_total Relation of NAT,COMPLEX
           st rng b1 c= dom a1 & b1 is convergent & lim b1 = a2
        holds a1 * b1 is convergent & a1 /. a2 = lim (a1 * b1));
end;

:: CFCONT_1:dfs 2
definiens
  let a1 be Function-like Relation of COMPLEX,COMPLEX;
  let a2 be Element of COMPLEX;
To prove
     a1 is_continuous_in a2
it is sufficient to prove
  thus a2 in dom a1 &
     (for b1 being Function-like quasi_total Relation of NAT,COMPLEX
           st rng b1 c= dom a1 & b1 is convergent & lim b1 = a2
        holds a1 * b1 is convergent & a1 /. a2 = lim (a1 * b1));

:: CFCONT_1:def 2
theorem
for b1 being Function-like Relation of COMPLEX,COMPLEX
for b2 being Element of COMPLEX holds
      b1 is_continuous_in b2
   iff
      b2 in dom b1 &
       (for b3 being Function-like quasi_total Relation of NAT,COMPLEX
             st rng b3 c= dom b1 & b3 is convergent & lim b3 = b2
          holds b1 * b3 is convergent & b1 /. b2 = lim (b1 * b3));

:: CFCONT_1:th 2
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
   b1 = b2 - b3
iff
   for b4 being Element of NAT holds
      b1 . b4 = (b2 . b4) - (b3 . b4);

:: CFCONT_1:th 3
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
   rng (b2 ^\ b1) c= rng b2;

:: CFCONT_1:th 4
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
      st rng b2 c= dom b3
   holds b2 . b1 in dom b3;

:: CFCONT_1:th 5
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
      b1 in rng b2
   iff
      ex b3 being Element of NAT st
         b1 = b2 . b3;

:: CFCONT_1:th 6
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
   b2 . b1 in rng b2;

:: CFCONT_1:th 7
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is subsequence of b2
   holds rng b1 c= rng b2;

:: CFCONT_1:th 8
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is subsequence of b2 & b2 is non-zero
   holds b1 is non-zero;

:: CFCONT_1:th 9
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL holds
   (b1 + b2) * b3 = (b1 * b3) + (b2 * b3) &
    (b1 - b2) * b3 = (b1 * b3) - (b2 * b3) &
    (b1 (#) b2) * b3 = (b1 * b3) (#) (b2 * b3);

:: CFCONT_1:th 10
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL holds
   (b1 (#) b2) * b3 = b1 (#) (b2 * b3);

:: CFCONT_1:th 11
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL holds
   (- b1) * b2 = - (b1 * b2) &
    |.b1.| * b2 = |.b1 * b2.|;

:: CFCONT_1:th 12
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL holds
   (b1 * b2) " = b1 " * b2;

:: CFCONT_1:th 13
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL holds
   (b1 /" b2) * b3 = (b1 * b3) /" (b2 * b3);

:: CFCONT_1:th 14
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of bool COMPLEX
      st for b3 being Element of NAT holds
           b1 . b3 in b2
   holds rng b1 c= b2;

:: CFCONT_1:th 16
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
      st rng b2 c= dom b3
   holds (b3 * b2) . b1 = b3 /. (b2 . b1);

:: CFCONT_1:th 17
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
      st rng b2 c= dom b3
   holds (b3 * b2) ^\ b1 = b3 * (b2 ^\ b1);

:: CFCONT_1:th 18
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2, b3 being Function-like Relation of COMPLEX,COMPLEX
      st rng b1 c= (dom b2) /\ dom b3
   holds (b2 + b3) * b1 = (b2 * b1) + (b3 * b1) &
    (b2 - b3) * b1 = (b2 * b1) - (b3 * b1) &
    (b2 (#) b3) * b1 = (b2 * b1) (#) (b3 * b1);

:: CFCONT_1:th 19
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
      st rng b2 c= dom b3
   holds (b1 (#) b3) * b2 = b1 (#) (b3 * b2);

:: CFCONT_1:th 20
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX
      st rng b1 c= dom b2
   holds - (b2 * b1) = (- b2) * b1;

:: CFCONT_1:th 21
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX
      st rng b1 c= dom (b2 ^)
   holds b2 * b1 is non-zero;

:: CFCONT_1:th 22
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX
      st rng b1 c= dom (b2 ^)
   holds b2 ^ * b1 = (b2 * b1) ";

:: CFCONT_1:th 23
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX
for b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL
      st rng b1 c= dom b2
   holds Re ((b2 * b1) * b3) = Re (b2 * (b1 * b3));

:: CFCONT_1:th 24
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX
for b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL
      st rng b1 c= dom b2
   holds Im ((b2 * b1) * b3) = Im (b2 * (b1 * b3));

:: CFCONT_1:th 25
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX
for b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL
      st rng b1 c= dom b2
   holds (b2 * b1) * b3 = b2 * (b1 * b3);

:: CFCONT_1:th 26
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
      st rng b1 c= dom b3 & b2 is subsequence of b1
   holds b3 * b2 is subsequence of b3 * b1;

:: CFCONT_1:th 27
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
      st b3 is total(COMPLEX, COMPLEX)
   holds (b3 * b2) . b1 = b3 /. (b2 . b1);

:: CFCONT_1:th 28
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
      st b3 is total(COMPLEX, COMPLEX)
   holds b3 * (b2 ^\ b1) = (b3 * b2) ^\ b1;

:: CFCONT_1:th 29
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2, b3 being Function-like Relation of COMPLEX,COMPLEX
      st b2 is total(COMPLEX, COMPLEX) & b3 is total(COMPLEX, COMPLEX)
   holds (b2 + b3) * b1 = (b2 * b1) + (b3 * b1) &
    (b2 - b3) * b1 = (b2 * b1) - (b3 * b1) &
    (b2 (#) b3) * b1 = (b2 * b1) (#) (b3 * b1);

:: CFCONT_1:th 30
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
      st b3 is total(COMPLEX, COMPLEX)
   holds (b1 (#) b3) * b2 = b1 (#) (b3 * b2);

:: CFCONT_1:th 31
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
      st rng b2 c= dom (b3 | b1)
   holds (b3 | b1) * b2 = b3 * b2;

:: CFCONT_1:th 32
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Element of bool COMPLEX
for b4 being Function-like Relation of COMPLEX,COMPLEX
      st rng b2 c= dom (b4 | b1) &
         (rng b2 c= dom (b4 | b3) or b1 c= b3)
   holds (b4 | b1) * b2 = (b4 | b3) * b2;

:: CFCONT_1:th 33
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
      st rng b2 c= dom (b3 | b1) & b3 " {0} = {}
   holds (b3 ^ | b1) * b2 = ((b3 | b1) * b2) ";

:: CFCONT_1:attrnot 1 => FUNCT_1:attr 3
definition
  let a1 be Relation-like Function-like set;
  attr a1 is constant means
    ex b1 being Element of COMPLEX st
       for b2 being Element of NAT holds
          a1 . b2 = b1;
end;

:: CFCONT_1:dfs 3
definiens
  let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
To prove
     a1 is constant
it is sufficient to prove
  thus ex b1 being Element of COMPLEX st
       for b2 being Element of NAT holds
          a1 . b2 = b1;

:: CFCONT_1:def 4
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
      b1 is constant
   iff
      ex b2 being Element of COMPLEX st
         for b3 being Element of NAT holds
            b1 . b3 = b2;

:: CFCONT_1:th 34
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
      b1 is constant
   iff
      ex b2 being Element of COMPLEX st
         rng b1 = {b2};

:: CFCONT_1:th 35
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
      b1 is constant
   iff
      for b2 being Element of NAT holds
         b1 . b2 = b1 . (b2 + 1);

:: CFCONT_1:th 36
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
      b1 is constant
   iff
      for b2, b3 being Element of NAT holds
      b1 . b2 = b1 . (b2 + b3);

:: CFCONT_1:th 37
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
      b1 is constant
   iff
      for b2, b3 being Element of NAT holds
      b1 . b2 = b1 . b3;

:: CFCONT_1:th 38
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
   b2 ^\ b1 is subsequence of b2;

:: CFCONT_1:th 39
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is subsequence of b2 & b2 is convergent
   holds b1 is convergent;

:: CFCONT_1:th 40
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is subsequence of b2 & b2 is convergent
   holds lim b1 = lim b2;

:: CFCONT_1:th 41
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent &
         (ex b3 being Element of NAT st
            for b4 being Element of NAT
                  st b3 <= b4
               holds b2 . b4 = b1 . b4)
   holds b2 is convergent;

:: CFCONT_1:th 42
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent &
         (ex b3 being Element of NAT st
            for b4 being Element of NAT
                  st b3 <= b4
               holds b2 . b4 = b1 . b4)
   holds lim b1 = lim b2;

:: CFCONT_1:th 43
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b2 is convergent
   holds b2 ^\ b1 is convergent & lim (b2 ^\ b1) = lim b2;

:: CFCONT_1:th 44
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent &
         (ex b3 being Element of NAT st
            b1 = b2 ^\ b3)
   holds b2 is convergent;

:: CFCONT_1:th 45
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent &
         (ex b3 being Element of NAT st
            b1 = b2 ^\ b3)
   holds lim b2 = lim b1;

:: CFCONT_1:th 46
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & lim b1 <> 0
   holds ex b2 being Element of NAT st
      b1 ^\ b2 is non-zero;

:: CFCONT_1:th 47
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is convergent & lim b1 <> 0
   holds ex b2 being Function-like quasi_total Relation of NAT,COMPLEX st
      b2 is subsequence of b1 & b2 is non-zero;

:: CFCONT_1:th 48
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is constant
   holds b1 is convergent;

:: CFCONT_1:th 49
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st (b2 is constant & b1 in rng b2 or b2 is constant &
         (ex b3 being Element of NAT st
            b2 . b3 = b1))
   holds lim b2 = b1;

:: CFCONT_1:th 50
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
   st b1 is constant
for b2 being Element of NAT holds
   lim b1 = b1 . b2;

:: CFCONT_1:th 51
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
   st b1 is convergent & lim b1 <> 0
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b2 is subsequence of b1 & b2 is non-zero
   holds lim (b2 ") = (lim b1) ";

:: CFCONT_1:th 52
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is constant & b2 is convergent
   holds lim (b1 + b2) = (b1 . 0) + lim b2 &
    lim (b1 - b2) = (b1 . 0) - lim b2 &
    lim (b2 - b1) = (lim b2) - (b1 . 0) &
    lim (b1 (#) b2) = (b1 . 0) * lim b2;

:: CFCONT_1:sch 1
scheme CFCONT_1:sch 1
ex b1 being Function-like quasi_total Relation of NAT,COMPLEX st
   for b2 being Element of NAT holds
      P1[b2, b1 . b2]
provided
   for b1 being Element of NAT holds
      ex b2 being Element of COMPLEX st
         P1[b1, b2];


:: CFCONT_1:th 53
theorem
for b1 being Element of COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX holds
      b2 is_continuous_in b1
   iff
      b1 in dom b2 &
       (for b3 being Function-like quasi_total Relation of NAT,COMPLEX
             st rng b3 c= dom b2 &
                b3 is convergent &
                lim b3 = b1 &
                (for b4 being Element of NAT holds
                   b3 . b4 <> b1)
          holds b2 * b3 is convergent & b2 /. b1 = lim (b2 * b3));

:: CFCONT_1:th 54
theorem
for b1 being Element of COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX holds
      b2 is_continuous_in b1
   iff
      b1 in dom b2 &
       (for b3 being Element of REAL
             st 0 < b3
          holds ex b4 being Element of REAL st
             0 < b4 &
              (for b5 being Element of COMPLEX
                    st b5 in dom b2 & |.b5 - b1.| < b4
                 holds |.(b2 /. b5) - (b2 /. b1).| < b3));

:: CFCONT_1:th 55
theorem
for b1 being Element of COMPLEX
for b2, b3 being Function-like Relation of COMPLEX,COMPLEX
      st b2 is_continuous_in b1 & b3 is_continuous_in b1
   holds b2 + b3 is_continuous_in b1 & b2 - b3 is_continuous_in b1 & b2 (#) b3 is_continuous_in b1;

:: CFCONT_1:th 56
theorem
for b1, b2 being Element of COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
      st b3 is_continuous_in b1
   holds b2 (#) b3 is_continuous_in b1;

:: CFCONT_1:th 57
theorem
for b1 being Element of COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX
      st b2 is_continuous_in b1
   holds - b2 is_continuous_in b1;

:: CFCONT_1:th 58
theorem
for b1 being Element of COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX
      st b2 is_continuous_in b1 & b2 /. b1 <> 0
   holds b2 ^ is_continuous_in b1;

:: CFCONT_1:th 59
theorem
for b1 being Element of COMPLEX
for b2, b3 being Function-like Relation of COMPLEX,COMPLEX
      st b2 is_continuous_in b1 & b2 /. b1 <> 0 & b3 is_continuous_in b1
   holds b3 / b2 is_continuous_in b1;

:: CFCONT_1:prednot 2 => CFCONT_1:pred 2
definition
  let a1 be Function-like Relation of COMPLEX,COMPLEX;
  let a2 be set;
  pred A1 is_continuous_on A2 means
    a2 c= dom a1 &
     (for b1 being Element of COMPLEX
           st b1 in a2
        holds a1 | a2 is_continuous_in b1);
end;

:: CFCONT_1:dfs 4
definiens
  let a1 be Function-like Relation of COMPLEX,COMPLEX;
  let a2 be set;
To prove
     a1 is_continuous_on a2
it is sufficient to prove
  thus a2 c= dom a1 &
     (for b1 being Element of COMPLEX
           st b1 in a2
        holds a1 | a2 is_continuous_in b1);

:: CFCONT_1:def 5
theorem
for b1 being Function-like Relation of COMPLEX,COMPLEX
for b2 being set holds
      b1 is_continuous_on b2
   iff
      b2 c= dom b1 &
       (for b3 being Element of COMPLEX
             st b3 in b2
          holds b1 | b2 is_continuous_in b3);

:: CFCONT_1:th 60
theorem
for b1 being set
for b2 being Function-like Relation of COMPLEX,COMPLEX holds
      b2 is_continuous_on b1
   iff
      b1 c= dom b2 &
       (for b3 being Function-like quasi_total Relation of NAT,COMPLEX
             st rng b3 c= b1 & b3 is convergent & lim b3 in b1
          holds b2 * b3 is convergent & b2 /. lim b3 = lim (b2 * b3));

:: CFCONT_1:th 61
theorem
for b1 being set
for b2 being Function-like Relation of COMPLEX,COMPLEX holds
      b2 is_continuous_on b1
   iff
      b1 c= dom b2 &
       (for b3 being Element of COMPLEX
       for b4 being Element of REAL
             st b3 in b1 & 0 < b4
          holds ex b5 being Element of REAL st
             0 < b5 &
              (for b6 being Element of COMPLEX
                    st b6 in b1 & |.b6 - b3.| < b5
                 holds |.(b2 /. b6) - (b2 /. b3).| < b4));

:: CFCONT_1:th 62
theorem
for b1 being set
for b2 being Function-like Relation of COMPLEX,COMPLEX holds
      b2 is_continuous_on b1
   iff
      b2 | b1 is_continuous_on b1;

:: CFCONT_1:th 63
theorem
for b1, b2 being set
for b3 being Function-like Relation of COMPLEX,COMPLEX
      st b3 is_continuous_on b1 & b2 c= b1
   holds b3 is_continuous_on b2;

:: CFCONT_1:th 64
theorem
for b1 being Element of COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX
      st b1 in dom b2
   holds b2 is_continuous_on {b1};

:: CFCONT_1:th 65
theorem
for b1 being set
for b2, b3 being Function-like Relation of COMPLEX,COMPLEX
      st b2 is_continuous_on b1 & b3 is_continuous_on b1
   holds b2 + b3 is_continuous_on b1 & b2 - b3 is_continuous_on b1 & b2 (#) b3 is_continuous_on b1;

:: CFCONT_1:th 66
theorem
for b1, b2 being set
for b3, b4 being Function-like Relation of COMPLEX,COMPLEX
      st b3 is_continuous_on b1 & b4 is_continuous_on b2
   holds b3 + b4 is_continuous_on b1 /\ b2 & b3 - b4 is_continuous_on b1 /\ b2 & b3 (#) b4 is_continuous_on b1 /\ b2;

:: CFCONT_1:th 67
theorem
for b1 being Element of COMPLEX
for b2 being set
for b3 being Function-like Relation of COMPLEX,COMPLEX
      st b3 is_continuous_on b2
   holds b1 (#) b3 is_continuous_on b2;

:: CFCONT_1:th 68
theorem
for b1 being set
for b2 being Function-like Relation of COMPLEX,COMPLEX
      st b2 is_continuous_on b1
   holds - b2 is_continuous_on b1;

:: CFCONT_1:th 69
theorem
for b1 being set
for b2 being Function-like Relation of COMPLEX,COMPLEX
      st b2 is_continuous_on b1 & b2 " {0} = {}
   holds b2 ^ is_continuous_on b1;

:: CFCONT_1:th 70
theorem
for b1 being set
for b2 being Function-like Relation of COMPLEX,COMPLEX
      st b2 is_continuous_on b1 &
         (b2 | b1) " {0} = {}
   holds b2 ^ is_continuous_on b1;

:: CFCONT_1:th 71
theorem
for b1 being set
for b2, b3 being Function-like Relation of COMPLEX,COMPLEX
      st b2 is_continuous_on b1 & b2 " {0} = {} & b3 is_continuous_on b1
   holds b3 / b2 is_continuous_on b1;

:: CFCONT_1:th 72
theorem
for b1 being Function-like Relation of COMPLEX,COMPLEX
      st b1 is total(COMPLEX, COMPLEX) &
         (for b2, b3 being Element of COMPLEX holds
         b1 /. (b2 + b3) = (b1 /. b2) + (b1 /. b3)) &
         (ex b2 being Element of COMPLEX st
            b1 is_continuous_in b2)
   holds b1 is_continuous_on COMPLEX;

:: CFCONT_1:attrnot 2 => CFCONT_1:attr 1
definition
  let a1 be set;
  attr a1 is compact means
    for b1 being Function-like quasi_total Relation of NAT,COMPLEX
          st rng b1 c= a1
       holds ex b2 being Function-like quasi_total Relation of NAT,COMPLEX st
          b2 is subsequence of b1 & b2 is convergent & lim b2 in a1;
end;

:: CFCONT_1:dfs 5
definiens
  let a1 be set;
To prove
     a1 is compact
it is sufficient to prove
  thus for b1 being Function-like quasi_total Relation of NAT,COMPLEX
          st rng b1 c= a1
       holds ex b2 being Function-like quasi_total Relation of NAT,COMPLEX st
          b2 is subsequence of b1 & b2 is convergent & lim b2 in a1;

:: CFCONT_1:def 6
theorem
for b1 being set holds
      b1 is compact
   iff
      for b2 being Function-like quasi_total Relation of NAT,COMPLEX
            st rng b2 c= b1
         holds ex b3 being Function-like quasi_total Relation of NAT,COMPLEX st
            b3 is subsequence of b2 & b3 is convergent & lim b3 in b1;

:: CFCONT_1:th 73
theorem
for b1 being Function-like Relation of COMPLEX,COMPLEX
      st dom b1 is compact & b1 is_continuous_on dom b1
   holds rng b1 is compact;

:: CFCONT_1:th 74
theorem
for b1 being Element of bool COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX
      st b1 c= dom b2 & b1 is compact & b2 is_continuous_on b1
   holds b2 .: b1 is compact;