Article SERIES_3, MML version 4.99.1005

:: SERIES_3:funcreg 1
registration
  let a1 be real set;
  cluster |.a1.| -> complex non negative;
end;

:: SERIES_3:th 1
theorem
for b1, b2, b3 being real set
      st b2 < b1 & 0 <= b2 & 0 <= b3
   holds b2 / b1 <= (b2 + b3) / (b1 + b3);

:: SERIES_3:th 2
theorem
for b1, b2 being real positive set holds
sqrt (b1 * b2) <= (b1 + b2) / 2;

:: SERIES_3:th 3
theorem
for b1, b2 being real positive set holds
2 <= (b1 / b2) + (b2 / b1);

:: SERIES_3:th 4
theorem
for b1, b2 being real set holds
b1 * b2 <= ((b1 + b2) / 2) ^2;

:: SERIES_3:th 5
theorem
for b1, b2 being real set holds
((b1 + b2) / 2) ^2 <= (b1 ^2 + (b2 ^2)) / 2;

:: SERIES_3:th 6
theorem
for b1, b2 being real set holds
(2 * b1) * b2 <= b1 ^2 + (b2 ^2);

:: SERIES_3:th 7
theorem
for b1, b2 being real set holds
b1 * b2 <= (b1 ^2 + (b2 ^2)) / 2;

:: SERIES_3:th 8
theorem
for b1, b2 being real set holds
(2 * abs b1) * abs b2 <= b1 ^2 + (b2 ^2);

:: SERIES_3:th 9
theorem
for b1, b2 being real set holds
(4 * b1) * b2 <= (b1 + b2) ^2;

:: SERIES_3:th 10
theorem
for b1, b2, b3 being real set holds
((b1 * b2) + (b2 * b3)) + (b1 * b3) <= (b1 ^2 + (b2 ^2)) + (b3 ^2);

:: SERIES_3:th 11
theorem
for b1, b2, b3 being real set holds
3 * (((b1 * b2) + (b2 * b3)) + (b1 * b3)) <= ((b1 + b2) + b3) ^2;

:: SERIES_3:th 12
theorem
for b1, b2, b3 being real positive set holds
((3 * b1) * b2) * b3 <= ((b1 |^ 3) + (b2 |^ 3)) + (b3 |^ 3);

:: SERIES_3:th 13
theorem
for b1, b2, b3 being real positive set holds
(b1 * b2) * b3 <= (((b1 |^ 3) + (b2 |^ 3)) + (b3 |^ 3)) / 3;

:: SERIES_3:th 14
theorem
for b1, b2, b3 being real positive set holds
((b2 / b1) + (b3 / b2)) + (b1 / b3) <= (((b1 / b2) |^ 3) + ((b2 / b3) |^ 3)) + ((b3 / b1) |^ 3);

:: SERIES_3:th 15
theorem
for b1, b2, b3 being real positive set holds
3 * (3 -root ((b1 * b2) * b3)) <= (b1 + b2) + b3;

:: SERIES_3:th 16
theorem
for b1, b2, b3 being real positive set holds
3 -root ((b1 * b2) * b3) <= ((b1 + b2) + b3) / 3;

:: SERIES_3:th 17
theorem
for b1, b2, b3 being real set
      st (b1 + b2) + b3 = 1
   holds ((b1 * b2) + (b2 * b3)) + (b1 * b3) <= 1 / 3;

:: SERIES_3:th 18
theorem
for b1, b2 being real set
      st b1 + b2 = 1
   holds b1 * b2 <= 1 / 4;

:: SERIES_3:th 19
theorem
for b1, b2 being real set
      st b1 + b2 = 1
   holds 1 / 2 <= b1 ^2 + (b2 ^2);

:: SERIES_3:th 20
theorem
for b1, b2 being real positive set
      st b1 + b2 = 1
   holds 9 <= (1 + (1 / b1)) * (1 + (1 / b2));

:: SERIES_3:th 21
theorem
for b1, b2 being real set
      st b1 + b2 = 1
   holds 1 / 4 <= (b1 |^ 3) + (b2 |^ 3);

:: SERIES_3:th 22
theorem
for b1, b2 being real positive set
      st b1 + b2 = 1
   holds (b1 |^ 3) + (b2 |^ 3) < 1;

:: SERIES_3:th 23
theorem
for b1, b2 being real positive set
      st b1 + b2 = 1
   holds 25 / 4 <= (b1 + (1 / b1)) * (b2 + (1 / b2));

:: SERIES_3:th 24
theorem
for b1 being real positive set
for b2 being real set
      st abs b2 <= b1
   holds b2 ^2 <= b1 ^2;

:: SERIES_3:th 25
theorem
for b1 being real positive set
for b2 being real set
      st b1 <= abs b2
   holds b1 ^2 <= b2 ^2;

:: SERIES_3:th 26
theorem
for b1, b2 being real set holds
abs ((abs b1) - abs b2) <= (abs b1) + abs b2;

:: SERIES_3:th 27
theorem
for b1, b2, b3 being real positive set
      st (b1 * b2) * b3 = 1
   holds ((sqrt b1) + sqrt b2) + sqrt b3 <= ((1 / b1) + (1 / b2)) + (1 / b3);

:: SERIES_3:th 28
theorem
for b1, b2, b3 being real set
      st 0 < b1 & 0 < b2 & b3 < 0 & (b1 + b2) + b3 = 0
   holds 6 * ((((b1 |^ 3) + (b2 |^ 3)) + (b3 |^ 3)) ^2) <= (((b1 |^ 2) + (b2 |^ 2)) + (b3 |^ 2)) |^ 3;

:: SERIES_3:th 29
theorem
for b1, b2, b3 being real positive set
      st 1 <= b1
   holds 2 * (b1 to_power sqrt (b2 * b3)) <= (b1 to_power b2) + (b1 to_power b3);

:: SERIES_3:th 30
theorem
for b1, b2, b3 being real positive set
      st b2 <= b1 & b3 <= b2
   holds ((b1 * b2) * b3) to_power (((b1 + b2) + b3) / 3) <= ((b1 to_power b1) * (b2 to_power b2)) * (b3 to_power b3);

:: SERIES_3:th 31
theorem
for b1 being Element of NAT
for b2, b3 being real non negative set holds
(b2 |^ (b1 + 2)) + (((b1 + 2) * (b2 |^ (b1 + 1))) * b3) <= (b2 + b3) |^ (b1 + 2);

:: SERIES_3:th 32
theorem
for b1, b2 being real positive set
for b3 being Element of NAT holds
   ((b1 + b2) / 2) |^ b3 <= ((b1 |^ b3) + (b2 |^ b3)) / 2;

:: SERIES_3:th 33
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        0 < b1 . b2
for b2 being Element of NAT holds
   0 < (Partial_Sums b1) . b2;

:: SERIES_3:th 34
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        0 <= b1 . b2
for b2 being Element of NAT holds
   0 <= (Partial_Sums b1) . b2;

:: SERIES_3:th 35
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st for b3 being Element of NAT holds
           b2 . b3 < 0
   holds (Partial_Sums b2) . b1 < 0;

:: SERIES_3:th 36
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
   st b1 = b2 (#) b2
for b3 being Element of NAT holds
   0 <= (Partial_Sums b1) . b3;

:: SERIES_3:th 37
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st for b3 being Element of NAT holds
           0 < b2 . b3 & b2 . (b3 - 1) < b2 . b3
   holds (Partial_Sums b2) . b1 < (b1 + 1) * (b2 . (b1 + 1));

:: SERIES_3:th 38
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st for b3 being Element of NAT holds
           0 < b2 . b3 & b2 . (b3 - 1) <= b2 . b3
   holds (Partial_Sums b2) . b1 <= (b1 + 1) * (b2 . (b1 + 1));

:: SERIES_3:th 39
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL
   st b1 = b2 (#) b3 &
      (for b4 being Element of NAT holds
         0 <= b2 . b4 & 0 <= b3 . b4)
for b4 being Element of NAT holds
   (Partial_Sums b1) . b4 <= ((Partial_Sums b2) . b4) * ((Partial_Sums b3) . b4);

:: SERIES_3:th 40
theorem
for b1 being Element of NAT
for b2, b3, b4 being Function-like quasi_total Relation of NAT,REAL
      st b2 = b3 (#) b4 &
         (for b5 being Element of NAT holds
            b3 . b5 < 0 & b4 . b5 < 0)
   holds (Partial_Sums b2) . b1 <= ((Partial_Sums b3) . b1) * ((Partial_Sums b4) . b1);

:: SERIES_3:th 41
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of NAT holds
   abs ((Partial_Sums b1) . b2) <= (Partial_Sums abs b1) . b2;

:: SERIES_3:th 42
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL holds
   (Partial_Sums b2) . b1 <= (Partial_Sums abs b2) . b1;

:: SERIES_3:funcnot 1 => SERIES_3:func 1
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  func Partial_Product A1 -> Function-like quasi_total Relation of NAT,REAL means
    it . 0 = a1 . 0 &
     (for b1 being Element of NAT holds
        it . (b1 + 1) = (it . b1) * (a1 . (b1 + 1)));
end;

:: SERIES_3:def 1
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
   b2 = Partial_Product b1
iff
   b2 . 0 = b1 . 0 &
    (for b3 being Element of NAT holds
       b2 . (b3 + 1) = (b2 . b3) * (b1 . (b3 + 1)));

:: SERIES_3:th 43
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st for b3 being Element of NAT holds
           0 < b2 . b3
   holds 0 < (Partial_Product b2) . b1;

:: SERIES_3:th 44
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st for b3 being Element of NAT holds
           0 <= b2 . b3
   holds 0 <= (Partial_Product b2) . b1;

:: SERIES_3:th 45
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        0 < b1 . b2 & b1 . b2 < 1
for b2 being Element of NAT holds
   0 < (Partial_Product b1) . b2 & (Partial_Product b1) . b2 < 1;

:: SERIES_3:th 46
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        1 <= b1 . b2
for b2 being Element of NAT holds
   1 <= (Partial_Product b1) . b2;

:: SERIES_3:th 47
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
   st for b3 being Element of NAT holds
        0 <= b1 . b3 & 0 <= b2 . b3
for b3 being Element of NAT holds
   ((Partial_Product b1) . b3) + ((Partial_Product b2) . b3) <= (Partial_Product (b1 + b2)) . b3;

:: SERIES_3:th 48
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st for b3 being Element of NAT holds
           b2 . b3 = ((2 * b3) + 1) / ((2 * b3) + 2)
   holds (Partial_Product b2) . b1 <= 1 / sqrt ((3 * b1) + 4);

:: SERIES_3:th 49
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
   st for b3 being Element of NAT holds
        b1 . b3 = 1 + (b2 . b3) & - 1 < b2 . b3 & b2 . b3 < 0
for b3 being Element of NAT holds
   1 + ((Partial_Sums b2) . b3) <= (Partial_Product b1) . b3;

:: SERIES_3:th 50
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
   st for b3 being Element of NAT holds
        b1 . b3 = 1 + (b2 . b3) & 0 <= b2 . b3
for b3 being Element of NAT holds
   1 + ((Partial_Sums b2) . b3) <= (Partial_Product b1) . b3;

:: SERIES_3:th 51
theorem
for b1, b2, b3, b4, b5 being Function-like quasi_total Relation of NAT,REAL
   st b1 = b2 (#) b3 & b4 = b2 (#) b2 & b5 = b3 (#) b3
for b6 being Element of NAT holds
   ((Partial_Sums b1) . b6) ^2 <= ((Partial_Sums b4) . b6) * ((Partial_Sums b5) . b6);

:: SERIES_3:th 52
theorem
for b1, b2, b3, b4, b5 being Function-like quasi_total Relation of NAT,REAL
   st b1 = b2 (#) b2 &
      b3 = b4 (#) b4 &
      (for b6 being Element of NAT holds
         0 <= b2 . b6 &
          0 <= b4 . b6 &
          b5 . b6 = ((b2 . b6) + (b4 . b6)) ^2)
for b6 being Element of NAT holds
   sqrt ((Partial_Sums b5) . b6) <= (sqrt ((Partial_Sums b1) . b6)) + sqrt ((Partial_Sums b3) . b6);

:: SERIES_3:th 53
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st for b3 being Element of NAT holds
           0 < b2 . b3 & b2 . (b3 - 1) <= b2 . b3
   holds (b1 + 1) * ((b1 + 1) -root ((Partial_Product b2) . b1)) <= (Partial_Sums b2) . b1;