Article REAL_3, MML version 4.99.1005

:: REAL_3:funcreg 1
registration
  let a1 be natural set;
  cluster a1 div 0 -> empty integer;
end;

:: REAL_3:funcreg 2
registration
  let a1 be natural set;
  cluster a1 mod 0 -> empty integer;
end;

:: REAL_3:funcreg 3
registration
  let a1 be natural set;
  cluster 0 div a1 -> empty integer;
end;

:: REAL_3:funcreg 4
registration
  let a1 be natural set;
  cluster 0 mod a1 -> empty integer;
end;

:: REAL_3:funcreg 5
registration
  let a1 be complex set;
  cluster a1 - a1 -> empty;
end;

:: REAL_3:funcreg 6
registration
  let a1 be complex set;
  cluster a1 / 0 -> empty;
end;

:: REAL_3:funcreg 7
registration
  cluster [\0/] -> empty integer;
end;

:: REAL_3:th 1
theorem
for b1 being real set
      st 0 < b1 & b1 < 1
   holds 1 < 1 / b1;

:: REAL_3:th 2
theorem
for b1 being integer set
for b2 being real set
      st b1 <= b2 & b2 < b1 + 1
   holds [\b2/] = b1;

:: REAL_3:th 3
theorem
for b1, b2 being natural set holds
[\b1 / b2/] = b1 div b2;

:: REAL_3:th 4
theorem
for b1, b2 being natural set
      st b1 mod b2 = 0
   holds b1 / b2 = b1 div b2;

:: REAL_3:th 5
theorem
for b1, b2 being natural set
      st b1 / b2 = b1 div b2
   holds b1 mod b2 = 0;

:: REAL_3:th 6
theorem
for b1, b2 being natural set holds
frac (b1 / b2) = (b1 mod b2) / b2;

:: REAL_3:th 7
theorem
for b1 being rational set
      st 0 <= b1
   holds ex b2, b3 being natural set st
      b3 <> 0 & b1 = b2 / b3;

:: REAL_3:attrnot 1 => VALUED_0:attr 5
notation
  let a1 be Relation-like set;
  synonym integer-yielding for integer-valued;
end;

:: REAL_3:exreg 1
registration
  cluster Relation-like Function-like non empty quasi_total complex-valued ext-real-valued real-valued integer-valued total Relation of NAT,REAL;
end;

:: REAL_3:modenot 1
definition
  mode Integer_Sequence is Function-like quasi_total integer-valued Relation of NAT,REAL;
end;

:: REAL_3:th 8
theorem
for b1 being Relation-like Function-like set holds
      b1 is Function-like quasi_total integer-valued Relation of NAT,REAL
   iff
      proj1 b1 = NAT &
       (for b2 being set
             st b2 in NAT
          holds b1 . b2 is integer);

:: REAL_3:th 9
theorem
for b1 being Relation-like Function-like set holds
      b1 is Function-like quasi_total Relation of NAT,INT
   iff
      b1 is Function-like quasi_total integer-valued Relation of NAT,REAL;

:: REAL_3:th 10
theorem
for b1 being Relation-like Function-like set holds
      b1 is Function-like quasi_total natural-valued Relation of NAT,REAL
   iff
      proj1 b1 = NAT &
       (for b2 being set
             st b2 in NAT
          holds b1 . b2 is natural);

:: REAL_3:th 11
theorem
for b1 being Relation-like Function-like set holds
      b1 is Function-like quasi_total Relation of NAT,NAT
   iff
      b1 is Function-like quasi_total natural-valued Relation of NAT,REAL;

:: REAL_3:funcnot 1 => REAL_3:func 1
definition
  let a1, a2 be natural set;
  func modSeq(A1,A2) -> Function-like quasi_total natural-valued Relation of NAT,REAL means
    it . 0 = a1 mod a2 &
     it . 1 = a2 mod (a1 mod a2) &
     (for b1 being natural set holds
        it . (b1 + 2) = (it . b1) mod (it . (b1 + 1)));
end;

:: REAL_3:def 2
theorem
for b1, b2 being natural set
for b3 being Function-like quasi_total natural-valued Relation of NAT,REAL holds
      b3 = modSeq(b1,b2)
   iff
      b3 . 0 = b1 mod b2 &
       b3 . 1 = b2 mod (b1 mod b2) &
       (for b4 being natural set holds
          b3 . (b4 + 2) = (b3 . b4) mod (b3 . (b4 + 1)));

:: REAL_3:funcnot 2 => REAL_3:func 2
definition
  let a1, a2 be natural set;
  func divSeq(A1,A2) -> Function-like quasi_total natural-valued Relation of NAT,REAL means
    it . 0 = a1 div a2 &
     it . 1 = a2 div (a1 mod a2) &
     (for b1 being natural set holds
        it . (b1 + 2) = ((modSeq(a1,a2)) . b1) div ((modSeq(a1,a2)) . (b1 + 1)));
end;

:: REAL_3:def 3
theorem
for b1, b2 being natural set
for b3 being Function-like quasi_total natural-valued Relation of NAT,REAL holds
      b3 = divSeq(b1,b2)
   iff
      b3 . 0 = b1 div b2 &
       b3 . 1 = b2 div (b1 mod b2) &
       (for b4 being natural set holds
          b3 . (b4 + 2) = ((modSeq(b1,b2)) . b4) div ((modSeq(b1,b2)) . (b4 + 1)));

:: REAL_3:th 12
theorem
for b1, b2 being natural set holds
(divSeq(b1,b2)) . 1 = b2 div ((modSeq(b1,b2)) . 0);

:: REAL_3:th 13
theorem
for b1, b2 being natural set holds
(modSeq(b1,b2)) . 1 = b2 mod ((modSeq(b1,b2)) . 0);

:: REAL_3:th 14
theorem
for b1, b2, b3, b4 being natural set
      st b1 <= b2 & (modSeq(b3,b4)) . b1 = 0
   holds (modSeq(b3,b4)) . b2 = 0;

:: REAL_3:th 15
theorem
for b1, b2, b3, b4 being natural set
      st b1 < b2 &
         (modSeq(b3,b4)) . b1 <= (modSeq(b3,b4)) . b2
   holds (modSeq(b3,b4)) . b1 = 0;

:: REAL_3:th 16
theorem
for b1, b2, b3 being natural set
      st (divSeq(b1,b2)) . (b3 + 1) = 0
   holds (modSeq(b1,b2)) . b3 = 0;

:: REAL_3:th 17
theorem
for b1, b2, b3, b4 being natural set
      st b1 <> 0 & b1 <= b2 & (divSeq(b3,b4)) . b1 = 0
   holds (divSeq(b3,b4)) . b2 = 0;

:: REAL_3:th 18
theorem
for b1, b2, b3, b4 being natural set
      st b1 < b2 & (modSeq(b3,b4)) . b1 = 0
   holds (divSeq(b3,b4)) . b2 = 0;

:: REAL_3:th 19
theorem
for b1, b2 being natural set
      st b1 <> 0
   holds b2 = (((divSeq(b2,b1)) . 0) * b1) + ((modSeq(b2,b1)) . 0);

:: REAL_3:th 20
theorem
for b1, b2 being natural set
      st b1 <> 0
   holds b2 / b1 = ((divSeq(b2,b1)) . 0) + (1 / (b1 / ((modSeq(b2,b1)) . 0)));

:: REAL_3:th 21
theorem
for b1 being natural set holds
   divSeq(b1,0) = NAT --> 0;

:: REAL_3:th 22
theorem
for b1 being natural set holds
   modSeq(b1,0) = NAT --> 0;

:: REAL_3:th 23
theorem
for b1 being natural set holds
   divSeq(0,b1) = NAT --> 0;

:: REAL_3:th 24
theorem
for b1 being natural set holds
   modSeq(0,b1) = NAT --> 0;

:: REAL_3:th 25
theorem
for b1, b2 being natural set holds
ex b3 being natural set st
   (divSeq(b1,b2)) . b3 = 0 & (modSeq(b1,b2)) . b3 = 0;

:: REAL_3:funcnot 3 => REAL_3:func 3
definition
  let a1 be real set;
  func remainders_for_scf A1 -> Function-like quasi_total Relation of NAT,REAL means
    it . 0 = a1 &
     (for b1 being natural set holds
        it . (b1 + 1) = 1 / frac (it . b1));
end;

:: REAL_3:def 4
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
      b2 = remainders_for_scf b1
   iff
      b2 . 0 = b1 &
       (for b3 being natural set holds
          b2 . (b3 + 1) = 1 / frac (b2 . b3));

:: REAL_3:funcnot 4 => REAL_3:func 3
notation
  let a1 be real set;
  synonym rfs a1 for remainders_for_scf a1;
end;

:: REAL_3:funcnot 5 => REAL_3:func 4
definition
  let a1 be real set;
  func SimpleContinuedFraction A1 -> Function-like quasi_total integer-valued Relation of NAT,REAL means
    for b1 being natural set holds
       it . b1 = [\(remainders_for_scf a1) . b1/];
end;

:: REAL_3:def 5
theorem
for b1 being real set
for b2 being Function-like quasi_total integer-valued Relation of NAT,REAL holds
      b2 = SimpleContinuedFraction b1
   iff
      for b3 being natural set holds
         b2 . b3 = [\(remainders_for_scf b1) . b3/];

:: REAL_3:funcnot 6 => REAL_3:func 4
notation
  let a1 be real set;
  synonym scf a1 for SimpleContinuedFraction a1;
end;

:: REAL_3:th 26
theorem
for b1 being natural set
for b2 being real set holds
   (remainders_for_scf b2) . (b1 + 1) = 1 / (((remainders_for_scf b2) . b1) - ((SimpleContinuedFraction b2) . b1));

:: REAL_3:th 27
theorem
for b1, b2 being natural set
for b3 being real set
      st (remainders_for_scf b3) . b1 = 0 & b1 <= b2
   holds (remainders_for_scf b3) . b2 = 0;

:: REAL_3:th 28
theorem
for b1, b2 being natural set
for b3 being real set
      st (remainders_for_scf b3) . b1 = 0 & b1 <= b2
   holds (SimpleContinuedFraction b3) . b2 = 0;

:: REAL_3:th 29
theorem
for b1 being natural set
for b2 being integer set holds
   (remainders_for_scf b2) . (b1 + 1) = 0;

:: REAL_3:th 30
theorem
for b1 being natural set
for b2 being integer set holds
   (SimpleContinuedFraction b2) . 0 = b2 & (SimpleContinuedFraction b2) . (b1 + 1) = 0;

:: REAL_3:th 31
theorem
for b1 being natural set
for b2 being integer set
      st 1 < b2
   holds (remainders_for_scf (1 / b2)) . 1 = b2 &
    (remainders_for_scf (1 / b2)) . (b1 + 2) = 0;

:: REAL_3:th 32
theorem
for b1 being natural set
for b2 being integer set
      st 1 < b2
   holds (SimpleContinuedFraction (1 / b2)) . 0 = 0 &
    (SimpleContinuedFraction (1 / b2)) . 1 = b2 &
    (SimpleContinuedFraction (1 / b2)) . (b1 + 2) = 0;

:: REAL_3:th 33
theorem
for b1 being natural set
for b2 being real set
      st for b3 being natural set holds
           (SimpleContinuedFraction b2) . b3 = 0
   holds (remainders_for_scf b2) . b1 = 0;

:: REAL_3:th 34
theorem
for b1 being real set
      st for b2 being natural set holds
           (SimpleContinuedFraction b1) . b2 = 0
   holds b1 = 0;

:: REAL_3:th 35
theorem
for b1 being real set holds
   frac b1 = b1 - ((SimpleContinuedFraction b1) . 0);

:: REAL_3:th 36
theorem
for b1 being natural set
for b2 being real set holds
   (remainders_for_scf b2) . (b1 + 1) = (remainders_for_scf (1 / frac b2)) . b1;

:: REAL_3:th 37
theorem
for b1 being natural set
for b2 being real set holds
   (SimpleContinuedFraction b2) . (b1 + 1) = (SimpleContinuedFraction (1 / frac b2)) . b1;

:: REAL_3:th 38
theorem
for b1 being natural set
for b2 being real set
      st 1 <= b1
   holds 0 <= (SimpleContinuedFraction b2) . b1;

:: REAL_3:th 39
theorem
for b1 being natural set
for b2 being real set
      st 1 <= b1
   holds (SimpleContinuedFraction b2) . b1 in NAT;

:: REAL_3:th 40
theorem
for b1 being natural set
for b2 being real set
      st 1 <= b1 & (SimpleContinuedFraction b2) . b1 <> 0
   holds 1 <= (SimpleContinuedFraction b2) . b1;

:: REAL_3:th 41
theorem
for b1, b2, b3 being natural set holds
(SimpleContinuedFraction (b1 / b2)) . b3 = (divSeq(b1,b2)) . b3 &
 (remainders_for_scf (b1 / b2)) . 1 = b2 / ((modSeq(b1,b2)) . 0) &
 (remainders_for_scf (b1 / b2)) . (b3 + 2) = ((modSeq(b1,b2)) . b3) / ((modSeq(b1,b2)) . (b3 + 1));

:: REAL_3:th 42
theorem
for b1 being real set holds
      b1 is rational
   iff
      ex b2 being natural set st
         for b3 being natural set
               st b2 <= b3
            holds (SimpleContinuedFraction b1) . b3 = 0;

:: REAL_3:th 43
theorem
for b1 being real set
      st for b2 being natural set holds
           (SimpleContinuedFraction b1) . b2 <> 0
   holds b1 is not rational;

:: REAL_3:funcnot 7 => REAL_3:func 5
definition
  let a1 be real set;
  func convergent_numerators A1 -> Function-like quasi_total Relation of NAT,REAL means
    it . 0 = (SimpleContinuedFraction a1) . 0 &
     it . 1 = (((SimpleContinuedFraction a1) . 1) * ((SimpleContinuedFraction a1) . 0)) + 1 &
     (for b1 being natural set holds
        it . (b1 + 2) = (((SimpleContinuedFraction a1) . (b1 + 2)) * (it . (b1 + 1))) + (it . b1));
end;

:: REAL_3:def 6
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
      b2 = convergent_numerators b1
   iff
      b2 . 0 = (SimpleContinuedFraction b1) . 0 &
       b2 . 1 = (((SimpleContinuedFraction b1) . 1) * ((SimpleContinuedFraction b1) . 0)) + 1 &
       (for b3 being natural set holds
          b2 . (b3 + 2) = (((SimpleContinuedFraction b1) . (b3 + 2)) * (b2 . (b3 + 1))) + (b2 . b3));

:: REAL_3:funcnot 8 => REAL_3:func 6
definition
  let a1 be real set;
  func convergent_denominators A1 -> Function-like quasi_total Relation of NAT,REAL means
    it . 0 = 1 &
     it . 1 = (SimpleContinuedFraction a1) . 1 &
     (for b1 being natural set holds
        it . (b1 + 2) = (((SimpleContinuedFraction a1) . (b1 + 2)) * (it . (b1 + 1))) + (it . b1));
end;

:: REAL_3:def 7
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
      b2 = convergent_denominators b1
   iff
      b2 . 0 = 1 &
       b2 . 1 = (SimpleContinuedFraction b1) . 1 &
       (for b3 being natural set holds
          b2 . (b3 + 2) = (((SimpleContinuedFraction b1) . (b3 + 2)) * (b2 . (b3 + 1))) + (b2 . b3));

:: REAL_3:funcnot 9 => REAL_3:func 5
notation
  let a1 be real set;
  synonym c_n a1 for convergent_numerators a1;
end;

:: REAL_3:funcnot 10 => REAL_3:func 6
notation
  let a1 be real set;
  synonym c_d a1 for convergent_denominators a1;
end;

:: REAL_3:th 44
theorem
for b1 being real set
   st 0 < (SimpleContinuedFraction b1) . 0
for b2 being natural set holds
   (convergent_numerators b1) . b2 in NAT;

:: REAL_3:th 45
theorem
for b1 being real set
   st 0 < (SimpleContinuedFraction b1) . 0
for b2 being natural set holds
   0 < (convergent_numerators b1) . b2;

:: REAL_3:th 46
theorem
for b1 being real set
   st 0 < (SimpleContinuedFraction b1) . 0
for b2 being natural set holds
   ((SimpleContinuedFraction b1) . (b2 + 2)) * ((convergent_numerators b1) . (b2 + 1)) < (convergent_numerators b1) . (b2 + 2);

:: REAL_3:th 47
theorem
for b1 being real set
for b2, b3 being natural set
   st 0 < (SimpleContinuedFraction b1) . 0
for b4 being natural set
      st b2 = (convergent_numerators b1) . (b4 + 1) & b3 = (convergent_numerators b1) . b4
   holds b2 hcf b3 = 1;

:: REAL_3:th 48
theorem
for b1 being real set
   st 0 < (SimpleContinuedFraction b1) . 0 &
      (for b2 being natural set holds
         (SimpleContinuedFraction b1) . b2 <> 0)
for b2 being natural set holds
   tau |^ b2 <= (convergent_numerators b1) . b2;

:: REAL_3:th 49
theorem
for b1 being natural set
for b2 being real set
   st 0 < (SimpleContinuedFraction b2) . 0 &
      (for b3 being natural set holds
         (SimpleContinuedFraction b2) . b3 <= b1)
for b3 being natural set holds
   (convergent_numerators b2) . b3 <= ((b1 + sqrt (b1 ^2 + 4)) / 2) |^ (b3 + 1);

:: REAL_3:th 50
theorem
for b1 being natural set
for b2 being real set holds
   (convergent_denominators b2) . b1 in NAT;

:: REAL_3:th 51
theorem
for b1 being natural set
for b2 being real set holds
   0 <= (convergent_denominators b2) . b1;

:: REAL_3:th 52
theorem
for b1 being real set
   st 0 < (SimpleContinuedFraction b1) . 1
for b2 being natural set holds
   0 < (convergent_denominators b1) . b2;

:: REAL_3:th 53
theorem
for b1 being natural set
for b2 being real set holds
   ((SimpleContinuedFraction b2) . (b1 + 2)) * ((convergent_denominators b2) . (b1 + 1)) <= (convergent_denominators b2) . (b1 + 2);

:: REAL_3:th 54
theorem
for b1 being real set
   st 0 < (SimpleContinuedFraction b1) . 1
for b2 being natural set holds
   ((SimpleContinuedFraction b1) . (b2 + 2)) * ((convergent_denominators b1) . (b2 + 1)) < (convergent_denominators b1) . (b2 + 2);

:: REAL_3:th 55
theorem
for b1 being real set
   st for b2 being natural set holds
        0 < (SimpleContinuedFraction b1) . b2
for b2 being natural set
      st 1 <= b2
   holds 1 / (((convergent_denominators b1) . b2) * ((convergent_denominators b1) . (b2 + 1))) < 1 / (((SimpleContinuedFraction b1) . (b2 + 1)) * (((convergent_denominators b1) . b2) ^2));

:: REAL_3:th 56
theorem
for b1 being natural set
for b2 being real set
   st for b3 being natural set holds
        (SimpleContinuedFraction b2) . b3 <= b1
for b3 being natural set holds
   (convergent_denominators b2) . (b3 + 1) <= ((b1 + sqrt (b1 ^2 + 4)) / 2) |^ (b3 + 1);

:: REAL_3:th 57
theorem
for b1 being natural set
for b2 being real set
for b3, b4 being natural set
      st b3 = (convergent_denominators b2) . (b1 + 1) & b4 = (convergent_denominators b2) . b1
   holds b3 hcf b4 = 1;

:: REAL_3:th 58
theorem
for b1 being real set
   st for b2 being natural set holds
        0 < (SimpleContinuedFraction b1) . b2
for b2 being natural set holds
   1 / ((SimpleContinuedFraction b1) . (b2 + 2)) <= ((convergent_denominators b1) . (b2 + 1)) / ((convergent_denominators b1) . b2);

:: REAL_3:th 59
theorem
for b1 being real set
   st for b2 being natural set holds
        0 < (SimpleContinuedFraction b1) . b2
for b2 being natural set holds
   (convergent_denominators b1) . (b2 + 2) <= (2 * ((SimpleContinuedFraction b1) . (b2 + 2))) * ((convergent_denominators b1) . (b2 + 1));

:: REAL_3:th 60
theorem
for b1 being real set
   st for b2 being natural set holds
        (SimpleContinuedFraction b1) . b2 <> 0
for b2 being natural set holds
   1 / (((SimpleContinuedFraction b1) . (b2 + 1)) * (((convergent_denominators b1) . b2) ^2)) <= 1 / (((convergent_denominators b1) . b2) ^2);

:: REAL_3:th 61
theorem
for b1 being real set
   st for b2 being natural set holds
        (SimpleContinuedFraction b1) . b2 <> 0
for b2 being natural set holds
   tau |^ b2 <= (convergent_denominators b1) . (b2 + 1);

:: REAL_3:th 62
theorem
for b1 being natural set
for b2 being real set
   st 0 < b1 &
      (for b3 being natural set holds
         b1 <= (SimpleContinuedFraction b2) . b3)
for b3 being natural set holds
   ((b1 + sqrt (b1 ^2 + 4)) / 2) |^ b3 <= (convergent_denominators b2) . (b3 + 1);

:: REAL_3:th 63
theorem
for b1 being natural set
for b2 being real set holds
   ((convergent_numerators b2) . (b1 + 2)) / ((convergent_denominators b2) . (b1 + 2)) = ((((SimpleContinuedFraction b2) . (b1 + 2)) * ((convergent_numerators b2) . (b1 + 1))) + ((convergent_numerators b2) . b1)) / ((((SimpleContinuedFraction b2) . (b1 + 2)) * ((convergent_denominators b2) . (b1 + 1))) + ((convergent_denominators b2) . b1));

:: REAL_3:th 64
theorem
for b1 being natural set
for b2 being real set holds
   (((convergent_numerators b2) . (b1 + 1)) * ((convergent_denominators b2) . b1)) - (((convergent_numerators b2) . b1) * ((convergent_denominators b2) . (b1 + 1))) = (- 1) |^ b1;

:: REAL_3:th 65
theorem
for b1 being natural set
for b2 being real set
      st for b3 being natural set holds
           (convergent_denominators b2) . b3 <> 0
   holds (((convergent_numerators b2) . (b1 + 1)) / ((convergent_denominators b2) . (b1 + 1))) - (((convergent_numerators b2) . b1) / ((convergent_denominators b2) . b1)) = ((- 1) |^ b1) / (((convergent_denominators b2) . (b1 + 1)) * ((convergent_denominators b2) . b1));

:: REAL_3:th 66
theorem
for b1 being natural set
for b2 being real set holds
   (((convergent_numerators b2) . (b1 + 2)) * ((convergent_denominators b2) . b1)) - (((convergent_numerators b2) . b1) * ((convergent_denominators b2) . (b1 + 2))) = ((- 1) |^ b1) * ((SimpleContinuedFraction b2) . (b1 + 2));

:: REAL_3:th 67
theorem
for b1 being natural set
for b2 being real set
      st for b3 being natural set holds
           (convergent_denominators b2) . b3 <> 0
   holds (((convergent_numerators b2) . (b1 + 2)) / ((convergent_denominators b2) . (b1 + 2))) - (((convergent_numerators b2) . b1) / ((convergent_denominators b2) . b1)) = (((- 1) |^ b1) * ((SimpleContinuedFraction b2) . (b1 + 2))) / (((convergent_denominators b2) . (b1 + 2)) * ((convergent_denominators b2) . b1));

:: REAL_3:th 68
theorem
for b1 being real set
   st for b2 being natural set holds
        (SimpleContinuedFraction b1) . b2 <> 0
for b2 being natural set
      st 1 <= b2
   holds ((convergent_numerators b1) . b2) / ((convergent_denominators b1) . b2) = (((convergent_numerators b1) . (b2 + 1)) - ((convergent_numerators b1) . (b2 - 1))) / (((convergent_denominators b1) . (b2 + 1)) - ((convergent_denominators b1) . (b2 - 1)));

:: REAL_3:th 69
theorem
for b1 being real set
   st for b2 being natural set holds
        (convergent_denominators b1) . b2 <> 0
for b2 being natural set holds
   abs ((((convergent_numerators b1) . (b2 + 1)) / ((convergent_denominators b1) . (b2 + 1))) - (((convergent_numerators b1) . b2) / ((convergent_denominators b1) . b2))) = 1 / abs (((convergent_denominators b1) . (b2 + 1)) * ((convergent_denominators b1) . b2));

:: REAL_3:th 70
theorem
for b1 being real set
   st 0 < (SimpleContinuedFraction b1) . 1
for b2 being natural set holds
   ((convergent_numerators b1) . (2 * b2)) / ((convergent_denominators b1) . (2 * b2)) < ((convergent_numerators b1) . ((2 * b2) + 1)) / ((convergent_denominators b1) . ((2 * b2) + 1));

:: REAL_3:funcnot 11 => REAL_3:func 7
definition
  let a1 be real set;
  func convergents_of_continued_fractions A1 -> Function-like quasi_total Relation of NAT,REAL equals
    (convergent_numerators a1) /" convergent_denominators a1;
end;

:: REAL_3:def 8
theorem
for b1 being real set holds
   convergents_of_continued_fractions b1 = (convergent_numerators b1) /" convergent_denominators b1;

:: REAL_3:funcnot 12 => REAL_3:func 7
notation
  let a1 be real set;
  synonym cocf a1 for convergents_of_continued_fractions a1;
end;

:: REAL_3:th 71
theorem
for b1 being real set holds
   (convergents_of_continued_fractions b1) . 0 = (SimpleContinuedFraction b1) . 0;

:: REAL_3:th 72
theorem
for b1 being real set
      st (SimpleContinuedFraction b1) . 1 <> 0
   holds (convergents_of_continued_fractions b1) . 1 = ((SimpleContinuedFraction b1) . 0) + (1 / ((SimpleContinuedFraction b1) . 1));

:: REAL_3:th 73
theorem
for b1 being real set
      st for b2 being natural set holds
           0 < (SimpleContinuedFraction b1) . b2
   holds (convergents_of_continued_fractions b1) . 2 = ((SimpleContinuedFraction b1) . 0) + (1 / (((SimpleContinuedFraction b1) . 1) + (1 / ((SimpleContinuedFraction b1) . 2))));

:: REAL_3:th 74
theorem
for b1 being real set
      st for b2 being natural set holds
           0 < (SimpleContinuedFraction b1) . b2
   holds (convergents_of_continued_fractions b1) . 3 = ((SimpleContinuedFraction b1) . 0) + (1 / (((SimpleContinuedFraction b1) . 1) + (1 / (((SimpleContinuedFraction b1) . 2) + (1 / ((SimpleContinuedFraction b1) . 3))))));

:: REAL_3:th 75
theorem
for b1 being real set
   st for b2 being natural set holds
        0 < (SimpleContinuedFraction b1) . b2
for b2 being natural set
      st 1 <= b2
   holds ((convergent_numerators b1) . ((2 * b2) + 1)) / ((convergent_denominators b1) . ((2 * b2) + 1)) < ((convergent_numerators b1) . ((2 * b2) - 1)) / ((convergent_denominators b1) . ((2 * b2) - 1));

:: REAL_3:th 76
theorem
for b1 being real set
   st for b2 being natural set holds
        0 < (SimpleContinuedFraction b1) . b2
for b2 being natural set
      st 1 <= b2
   holds ((convergent_numerators b1) . ((2 * b2) - 2)) / ((convergent_denominators b1) . ((2 * b2) - 2)) < ((convergent_numerators b1) . (2 * b2)) / ((convergent_denominators b1) . (2 * b2));

:: REAL_3:th 77
theorem
for b1 being real set
   st for b2 being natural set holds
        0 < (SimpleContinuedFraction b1) . b2
for b2 being natural set
      st 1 <= b2
   holds ((convergent_numerators b1) . (2 * b2)) / ((convergent_denominators b1) . (2 * b2)) < ((convergent_numerators b1) . ((2 * b2) - 1)) / ((convergent_denominators b1) . ((2 * b2) - 1));

:: REAL_3:funcnot 13 => REAL_3:func 8
definition
  let a1 be real set;
  func backContinued_fraction A1 -> Function-like quasi_total Relation of NAT,REAL means
    it . 0 = (SimpleContinuedFraction a1) . 0 &
     (for b1 being natural set holds
        it . (b1 + 1) = (1 / (it . b1)) + ((SimpleContinuedFraction a1) . (b1 + 1)));
end;

:: REAL_3:def 9
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
      b2 = backContinued_fraction b1
   iff
      b2 . 0 = (SimpleContinuedFraction b1) . 0 &
       (for b3 being natural set holds
          b2 . (b3 + 1) = (1 / (b2 . b3)) + ((SimpleContinuedFraction b1) . (b3 + 1)));

:: REAL_3:funcnot 14 => REAL_3:func 8
notation
  let a1 be real set;
  synonym bcf a1 for backContinued_fraction a1;
end;

:: REAL_3:th 78
theorem
for b1 being real set
   st 0 < (SimpleContinuedFraction b1) . 0
for b2 being natural set holds
   (backContinued_fraction b1) . (b2 + 1) = ((convergent_numerators b1) . (b2 + 1)) / ((convergent_numerators b1) . b2);