Article FDIFF_6, MML version 4.99.1005

:: FDIFF_6:th 1
theorem
for b1, b2 being Element of REAL
      st 0 < b1
   holds exp_R . (b2 * log(number_e,b1)) = b1 #R b2;

:: FDIFF_6:th 2
theorem
for b1, b2 being Element of REAL
      st 0 < b1
   holds exp_R . - (b2 * log(number_e,b1)) = b1 #R - b2;

:: FDIFF_6:th 3
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3, b4 being Function-like Relation of REAL,REAL
      st b2 c= proj1 (b3 - b4) &
         (for b5 being Element of REAL
               st b5 in b2
            holds b3 . b5 = b1 ^2) &
         b4 = #Z 2
   holds b3 - b4 is_differentiable_on b2 &
    (for b5 being Element of REAL
          st b5 in b2
       holds ((b3 - b4) `| b2) . b5 = - (2 * b5));

:: FDIFF_6:th 4
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3, b4 being Function-like Relation of REAL,REAL
      st b2 c= proj1 ((b3 + b4) / (b3 - b4)) &
         b4 = #Z 2 &
         (for b5 being Element of REAL
               st b5 in b2
            holds b3 . b5 = b1 ^2 & (b3 - b4) . b5 <> 0)
   holds (b3 + b4) / (b3 - b4) is_differentiable_on b2 &
    (for b5 being Element of REAL
          st b5 in b2
       holds (((b3 + b4) / (b3 - b4)) `| b2) . b5 = ((4 * (b1 ^2)) * b5) / ((b1 ^2 - (b5 |^ 2)) ^2));

:: FDIFF_6:th 5
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3, b4, b5 being Function-like Relation of REAL,REAL
      st b2 c= proj1 b3 &
         b3 = ln * ((b4 + b5) / (b4 - b5)) &
         b5 = #Z 2 &
         (for b6 being Element of REAL
               st b6 in b2
            holds b4 . b6 = b1 ^2 & 0 < (b4 - b5) . b6 & b1 <> 0)
   holds b3 is_differentiable_on b2 &
    (for b6 being Element of REAL
          st b6 in b2
       holds (b3 `| b2) . b6 = ((4 * (b1 ^2)) * b6) / ((b1 |^ 4) - (b6 |^ 4)));

:: FDIFF_6:th 6
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3, b4, b5 being Function-like Relation of REAL,REAL
      st b2 c= proj1 ((1 / (4 * (b1 ^2))) (#) b3) &
         b3 = ln * ((b4 + b5) / (b4 - b5)) &
         b5 = #Z 2 &
         (for b6 being Element of REAL
               st b6 in b2
            holds b4 . b6 = b1 ^2 & 0 < (b4 - b5) . b6 & b1 <> 0)
   holds (1 / (4 * (b1 ^2))) (#) b3 is_differentiable_on b2 &
    (for b6 being Element of REAL
          st b6 in b2
       holds (((1 / (4 * (b1 ^2))) (#) b3) `| b2) . b6 = b6 / ((b1 |^ 4) - (b6 |^ 4)));

:: FDIFF_6:th 7
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b1 c= proj1 (b2 / (b3 + b2)) &
         b2 = #Z 2 &
         (for b4 being Element of REAL
               st b4 in b1
            holds b3 . b4 = 1 & b4 <> 0)
   holds b2 / (b3 + b2) is_differentiable_on b1 &
    (for b4 being Element of REAL
          st b4 in b1
       holds ((b2 / (b3 + b2)) `| b1) . b4 = (2 * b4) / ((1 + (b4 ^2)) ^2));

:: FDIFF_6:th 8
theorem
for b1 being open Element of bool REAL
for b2, b3, b4 being Function-like Relation of REAL,REAL
      st b1 c= proj1 ((1 / 2) (#) b2) &
         b2 = ln * (b3 / (b4 + b3)) &
         b3 = #Z 2 &
         (for b5 being Element of REAL
               st b5 in b1
            holds b4 . b5 = 1 & b5 <> 0)
   holds (1 / 2) (#) b2 is_differentiable_on b1 &
    (for b5 being Element of REAL
          st b5 in b1
       holds (((1 / 2) (#) b2) `| b1) . b5 = 1 / (b5 * (1 + (b5 ^2))));

:: FDIFF_6:th 9
theorem
for b1 being Element of NAT
for b2 being open Element of bool REAL
      st b2 c= proj1 (ln * #Z b1) &
         (for b3 being Element of REAL
               st b3 in b2
            holds 0 < b3)
   holds ln * #Z b1 is_differentiable_on b2 &
    (for b3 being Element of REAL
          st b3 in b2
       holds ((ln * #Z b1) `| b2) . b3 = b1 / b3);

:: FDIFF_6:th 10
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b1 c= proj1 (b2 ^ + (ln * (b3 / b2))) &
         (for b4 being Element of REAL
               st b4 in b1
            holds b2 . b4 = b4 & 0 < b2 . b4 & b3 . b4 = b4 - 1 & 0 < b3 . b4)
   holds b2 ^ + (ln * (b3 / b2)) is_differentiable_on b1 &
    (for b4 being Element of REAL
          st b4 in b1
       holds ((b2 ^ + (ln * (b3 / b2))) `| b1) . b4 = 1 / (b4 ^2 * (b4 - 1)));

:: FDIFF_6:th 11
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3 being Function-like Relation of REAL,REAL
      st b2 c= proj1 (exp_R * b3) &
         (for b4 being Element of REAL
               st b4 in b2
            holds b3 . b4 = b4 * log(number_e,b1)) &
         0 < b1
   holds exp_R * b3 is_differentiable_on b2 &
    (for b4 being Element of REAL
          st b4 in b2
       holds ((exp_R * b3) `| b2) . b4 = (b1 #R b4) * log(number_e,b1));

:: FDIFF_6:th 12
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3, b4 being Function-like Relation of REAL,REAL
      st b2 c= proj1 ((1 / log(number_e,b1)) (#) ((exp_R * b3) (#) b4)) &
         (for b5 being Element of REAL
               st b5 in b2
            holds b3 . b5 = b5 * log(number_e,b1) &
             b4 . b5 = b5 - (1 / log(number_e,b1))) &
         0 < b1 &
         b1 <> 1
   holds (1 / log(number_e,b1)) (#) ((exp_R * b3) (#) b4) is_differentiable_on b2 &
    (for b5 being Element of REAL
          st b5 in b2
       holds (((1 / log(number_e,b1)) (#) ((exp_R * b3) (#) b4)) `| b2) . b5 = b5 * (b1 #R b5));

:: FDIFF_6:th 13
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3 being Function-like Relation of REAL,REAL
      st b2 c= proj1 ((1 / (1 + log(number_e,b1))) (#) ((exp_R * b3) (#) exp_R)) &
         (for b4 being Element of REAL
               st b4 in b2
            holds b3 . b4 = b4 * log(number_e,b1)) &
         0 < b1 &
         b1 <> 1 / number_e
   holds (1 / (1 + log(number_e,b1))) (#) ((exp_R * b3) (#) exp_R) is_differentiable_on b2 &
    (for b4 being Element of REAL
          st b4 in b2
       holds (((1 / (1 + log(number_e,b1))) (#) ((exp_R * b3) (#) exp_R)) `| b2) . b4 = (b1 #R b4) * (exp_R . b4));

:: FDIFF_6:th 14
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= proj1 (exp_R * b2) &
         (for b3 being Element of REAL
               st b3 in b1
            holds b2 . b3 = - b3)
   holds exp_R * b2 is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds ((exp_R * b2) `| b1) . b3 = - exp_R - b3);

:: FDIFF_6:th 15
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b1 c= proj1 (b2 (#) (exp_R * b3)) &
         (for b4 being Element of REAL
               st b4 in b1
            holds b2 . b4 = (- b4) - 1 & b3 . b4 = - b4)
   holds b2 (#) (exp_R * b3) is_differentiable_on b1 &
    (for b4 being Element of REAL
          st b4 in b1
       holds ((b2 (#) (exp_R * b3)) `| b1) . b4 = b4 / exp_R b4);

:: FDIFF_6:th 16
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3 being Function-like Relation of REAL,REAL
      st b2 c= proj1 - (exp_R * b3) &
         (for b4 being Element of REAL
               st b4 in b2
            holds b3 . b4 = - (b4 * log(number_e,b1))) &
         0 < b1
   holds - (exp_R * b3) is_differentiable_on b2 &
    (for b4 being Element of REAL
          st b4 in b2
       holds ((- (exp_R * b3)) `| b2) . b4 = (b1 #R - b4) * log(number_e,b1));

:: FDIFF_6:th 17
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3, b4 being Function-like Relation of REAL,REAL
      st b2 c= proj1 ((1 / log(number_e,b1)) (#) ((- (exp_R * b3)) (#) b4)) &
         (for b5 being Element of REAL
               st b5 in b2
            holds b3 . b5 = - (b5 * log(number_e,b1)) &
             b4 . b5 = b5 + (1 / log(number_e,b1))) &
         0 < b1 &
         b1 <> 1
   holds (1 / log(number_e,b1)) (#) ((- (exp_R * b3)) (#) b4) is_differentiable_on b2 &
    (for b5 being Element of REAL
          st b5 in b2
       holds (((1 / log(number_e,b1)) (#) ((- (exp_R * b3)) (#) b4)) `| b2) . b5 = b5 / (b1 #R b5));

:: FDIFF_6:th 18
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3 being Function-like Relation of REAL,REAL
      st b2 c= proj1 ((1 / ((log(number_e,b1)) - 1)) (#) ((exp_R * b3) / exp_R)) &
         (for b4 being Element of REAL
               st b4 in b2
            holds b3 . b4 = b4 * log(number_e,b1)) &
         0 < b1 &
         b1 <> number_e
   holds (1 / ((log(number_e,b1)) - 1)) (#) ((exp_R * b3) / exp_R) is_differentiable_on b2 &
    (for b4 being Element of REAL
          st b4 in b2
       holds (((1 / ((log(number_e,b1)) - 1)) (#) ((exp_R * b3) / exp_R)) `| b2) . b4 = (b1 #R b4) / (exp_R . b4));

:: FDIFF_6:th 19
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3 being Function-like Relation of REAL,REAL
      st b2 c= proj1 ((1 / (1 - log(number_e,b1))) (#) (exp_R / (exp_R * b3))) &
         (for b4 being Element of REAL
               st b4 in b2
            holds b3 . b4 = b4 * log(number_e,b1)) &
         0 < b1 &
         b1 <> number_e
   holds (1 / (1 - log(number_e,b1))) (#) (exp_R / (exp_R * b3)) is_differentiable_on b2 &
    (for b4 being Element of REAL
          st b4 in b2
       holds (((1 / (1 - log(number_e,b1))) (#) (exp_R / (exp_R * b3))) `| b2) . b4 = (exp_R . b4) / (b1 #R b4));

:: FDIFF_6:th 20
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= proj1 (ln * (exp_R + b2)) &
         (for b3 being Element of REAL
               st b3 in b1
            holds b2 . b3 = 1)
   holds ln * (exp_R + b2) is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds ((ln * (exp_R + b2)) `| b1) . b3 = (exp_R . b3) / ((exp_R . b3) + 1));

:: FDIFF_6:th 21
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= proj1 (ln * (exp_R - b2)) &
         (for b3 being Element of REAL
               st b3 in b1
            holds b2 . b3 = 1 & 0 < (exp_R - b2) . b3)
   holds ln * (exp_R - b2) is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds ((ln * (exp_R - b2)) `| b1) . b3 = (exp_R . b3) / ((exp_R . b3) - 1));

:: FDIFF_6:th 22
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= proj1 - (ln * (b2 - exp_R)) &
         (for b3 being Element of REAL
               st b3 in b1
            holds b2 . b3 = 1 & 0 < (b2 - exp_R) . b3)
   holds - (ln * (b2 - exp_R)) is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds ((- (ln * (b2 - exp_R))) `| b1) . b3 = (exp_R . b3) / (1 - (exp_R . b3)));

:: FDIFF_6:th 23
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= proj1 (((#Z 2) * exp_R) + b2) &
         (for b3 being Element of REAL
               st b3 in b1
            holds b2 . b3 = 1)
   holds ((#Z 2) * exp_R) + b2 is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds ((((#Z 2) * exp_R) + b2) `| b1) . b3 = 2 * exp_R (2 * b3));

:: FDIFF_6:th 24
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b1 c= proj1 ((1 / 2) (#) (ln * b2)) &
         b2 = ((#Z 2) * exp_R) + b3 &
         (for b4 being Element of REAL
               st b4 in b1
            holds b3 . b4 = 1)
   holds (1 / 2) (#) (ln * b2) is_differentiable_on b1 &
    (for b4 being Element of REAL
          st b4 in b1
       holds (((1 / 2) (#) (ln * b2)) `| b1) . b4 = (exp_R b4) / ((exp_R b4) + exp_R - b4));

:: FDIFF_6:th 25
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= proj1 (((#Z 2) * exp_R) - b2) &
         (for b3 being Element of REAL
               st b3 in b1
            holds b2 . b3 = 1)
   holds ((#Z 2) * exp_R) - b2 is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds ((((#Z 2) * exp_R) - b2) `| b1) . b3 = 2 * exp_R (2 * b3));

:: FDIFF_6:th 26
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b1 c= proj1 ((1 / 2) (#) (ln * b2)) &
         b2 = ((#Z 2) * exp_R) - b3 &
         (for b4 being Element of REAL
               st b4 in b1
            holds b3 . b4 = 1 & 0 < b2 . b4)
   holds (1 / 2) (#) (ln * b2) is_differentiable_on b1 &
    (for b4 being Element of REAL
          st b4 in b1
       holds (((1 / 2) (#) (ln * b2)) `| b1) . b4 = (exp_R b4) / ((exp_R b4) - exp_R - b4));

:: FDIFF_6:th 27
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= proj1 ((#Z 2) * (exp_R - b2)) &
         (for b3 being Element of REAL
               st b3 in b1
            holds b2 . b3 = 1)
   holds (#Z 2) * (exp_R - b2) is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds (((#Z 2) * (exp_R - b2)) `| b1) . b3 = (2 * (exp_R . b3)) * ((exp_R . b3) - 1));

:: FDIFF_6:th 28
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b1 c= proj1 b2 &
         b2 = ln * (((#Z 2) * (exp_R - b3)) / exp_R) &
         (for b4 being Element of REAL
               st b4 in b1
            holds b3 . b4 = 1 & 0 < (exp_R - b3) . b4)
   holds b2 is_differentiable_on b1 &
    (for b4 being Element of REAL
          st b4 in b1
       holds (b2 `| b1) . b4 = ((exp_R . b4) + 1) / ((exp_R . b4) - 1));

:: FDIFF_6:th 29
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= proj1 ((#Z 2) * (exp_R + b2)) &
         (for b3 being Element of REAL
               st b3 in b1
            holds b2 . b3 = 1)
   holds (#Z 2) * (exp_R + b2) is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds (((#Z 2) * (exp_R + b2)) `| b1) . b3 = (2 * (exp_R . b3)) * ((exp_R . b3) + 1));

:: FDIFF_6:th 30
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b1 c= proj1 b2 &
         b2 = ln * (((#Z 2) * (exp_R + b3)) / exp_R) &
         (for b4 being Element of REAL
               st b4 in b1
            holds b3 . b4 = 1)
   holds b2 is_differentiable_on b1 &
    (for b4 being Element of REAL
          st b4 in b1
       holds (b2 `| b1) . b4 = ((exp_R . b4) - 1) / ((exp_R . b4) + 1));

:: FDIFF_6:th 31
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= proj1 ((#Z 2) * (b2 - exp_R)) &
         (for b3 being Element of REAL
               st b3 in b1
            holds b2 . b3 = 1)
   holds (#Z 2) * (b2 - exp_R) is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds (((#Z 2) * (b2 - exp_R)) `| b1) . b3 = - ((2 * (exp_R . b3)) * (1 - (exp_R . b3))));

:: FDIFF_6:th 32
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b1 c= proj1 b2 &
         b2 = ln * (exp_R / ((#Z 2) * (b3 - exp_R))) &
         (for b4 being Element of REAL
               st b4 in b1
            holds b3 . b4 = 1 & 0 < (b3 - exp_R) . b4)
   holds b2 is_differentiable_on b1 &
    (for b4 being Element of REAL
          st b4 in b1
       holds (b2 `| b1) . b4 = (1 + (exp_R . b4)) / (1 - (exp_R . b4)));

:: FDIFF_6:th 33
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b1 c= proj1 b2 &
         b2 = ln * (exp_R / ((#Z 2) * (b3 + exp_R))) &
         (for b4 being Element of REAL
               st b4 in b1
            holds b3 . b4 = 1)
   holds b2 is_differentiable_on b1 &
    (for b4 being Element of REAL
          st b4 in b1
       holds (b2 `| b1) . b4 = (1 - (exp_R . b4)) / (1 + (exp_R . b4)));

:: FDIFF_6:th 34
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b1 c= proj1 (ln * b2) &
         b2 = exp_R + (exp_R * b3) &
         (for b4 being Element of REAL
               st b4 in b1
            holds b3 . b4 = - b4)
   holds ln * b2 is_differentiable_on b1 &
    (for b4 being Element of REAL
          st b4 in b1
       holds ((ln * b2) `| b1) . b4 = ((exp_R b4) - exp_R - b4) / ((exp_R b4) + exp_R - b4));

:: FDIFF_6:th 35
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b1 c= proj1 (ln * b2) &
         b2 = exp_R - (exp_R * b3) &
         (for b4 being Element of REAL
               st b4 in b1
            holds b3 . b4 = - b4 & 0 < b2 . b4)
   holds ln * b2 is_differentiable_on b1 &
    (for b4 being Element of REAL
          st b4 in b1
       holds ((ln * b2) `| b1) . b4 = ((exp_R b4) + exp_R - b4) / ((exp_R b4) - exp_R - b4));

:: FDIFF_6:th 36
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= proj1 ((2 / 3) (#) ((#R (3 / 2)) * (b2 + exp_R))) &
         (for b3 being Element of REAL
               st b3 in b1
            holds b2 . b3 = 1)
   holds (2 / 3) (#) ((#R (3 / 2)) * (b2 + exp_R)) is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds (((2 / 3) (#) ((#R (3 / 2)) * (b2 + exp_R))) `| b1) . b3 = (exp_R . b3) * ((1 + (exp_R . b3)) #R (1 / 2)));

:: FDIFF_6:th 37
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3, b4 being Function-like Relation of REAL,REAL
      st b2 c= proj1 ((2 / (3 * log(number_e,b1))) (#) ((#R (3 / 2)) * (b3 + (exp_R * b4)))) &
         (for b5 being Element of REAL
               st b5 in b2
            holds b3 . b5 = 1 & b4 . b5 = b5 * log(number_e,b1)) &
         0 < b1 &
         b1 <> 1
   holds (2 / (3 * log(number_e,b1))) (#) ((#R (3 / 2)) * (b3 + (exp_R * b4))) is_differentiable_on b2 &
    (for b5 being Element of REAL
          st b5 in b2
       holds (((2 / (3 * log(number_e,b1))) (#) ((#R (3 / 2)) * (b3 + (exp_R * b4)))) `| b2) . b5 = (b1 #R b5) * ((1 + (b1 #R b5)) #R (1 / 2)));

:: FDIFF_6:th 38
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= proj1 ((- (1 / 2)) (#) (cos * b2)) &
         (for b3 being Element of REAL
               st b3 in b1
            holds b2 . b3 = 2 * b3)
   holds (- (1 / 2)) (#) (cos * b2) is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds (((- (1 / 2)) (#) (cos * b2)) `| b1) . b3 = sin (2 * b3));

:: FDIFF_6:th 39
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= proj1 (2 (#) ((#R (1 / 2)) * (b2 - cos))) &
         (for b3 being Element of REAL
               st b3 in b1
            holds b2 . b3 = 1 & 0 < sin . b3 & cos . b3 < 1 & - 1 < cos . b3)
   holds 2 (#) ((#R (1 / 2)) * (b2 - cos)) is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds ((2 (#) ((#R (1 / 2)) * (b2 - cos))) `| b1) . b3 = (1 + (cos . b3)) #R (1 / 2));

:: FDIFF_6:th 40
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= proj1 ((- 2) (#) ((#R (1 / 2)) * (b2 + cos))) &
         (for b3 being Element of REAL
               st b3 in b1
            holds b2 . b3 = 1 & 0 < sin . b3 & cos . b3 < 1 & - 1 < cos . b3)
   holds (- 2) (#) ((#R (1 / 2)) * (b2 + cos)) is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds (((- 2) (#) ((#R (1 / 2)) * (b2 + cos))) `| b1) . b3 = (1 - (cos . b3)) #R (1 / 2));

:: FDIFF_6:th 41
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b1 c= proj1 ((1 / 2) (#) (ln * b2)) &
         b2 = b3 + (2 (#) sin) &
         (for b4 being Element of REAL
               st b4 in b1
            holds b3 . b4 = 1 & 0 < b2 . b4)
   holds (1 / 2) (#) (ln * b2) is_differentiable_on b1 &
    (for b4 being Element of REAL
          st b4 in b1
       holds (((1 / 2) (#) (ln * b2)) `| b1) . b4 = (cos . b4) / (1 + (2 * (sin . b4))));

:: FDIFF_6:th 42
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b1 c= proj1 ((- (1 / 2)) (#) (ln * b2)) &
         b2 = b3 + (2 (#) cos) &
         (for b4 being Element of REAL
               st b4 in b1
            holds b3 . b4 = 1 & 0 < b2 . b4)
   holds (- (1 / 2)) (#) (ln * b2) is_differentiable_on b1 &
    (for b4 being Element of REAL
          st b4 in b1
       holds (((- (1 / 2)) (#) (ln * b2)) `| b1) . b4 = (sin . b4) / (1 + (2 * (cos . b4))));

:: FDIFF_6:th 43
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3 being Function-like Relation of REAL,REAL
      st b2 c= proj1 ((1 / (4 * b1)) (#) (sin * b3)) &
         (for b4 being Element of REAL
               st b4 in b2
            holds b3 . b4 = (2 * b1) * b4) &
         b1 <> 0
   holds (1 / (4 * b1)) (#) (sin * b3) is_differentiable_on b2 &
    (for b4 being Element of REAL
          st b4 in b2
       holds (((1 / (4 * b1)) (#) (sin * b3)) `| b2) . b4 = (1 / 2) * cos ((2 * b1) * b4));

:: FDIFF_6:th 44
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3, b4 being Function-like Relation of REAL,REAL
      st b2 c= proj1 (b3 - ((1 / (4 * b1)) (#) (sin * b4))) &
         (for b5 being Element of REAL
               st b5 in b2
            holds b3 . b5 = b5 / 2 & b4 . b5 = (2 * b1) * b5) &
         b1 <> 0
   holds b3 - ((1 / (4 * b1)) (#) (sin * b4)) is_differentiable_on b2 &
    (for b5 being Element of REAL
          st b5 in b2
       holds ((b3 - ((1 / (4 * b1)) (#) (sin * b4))) `| b2) . b5 = (sin (b1 * b5)) ^2);

:: FDIFF_6:th 45
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3, b4 being Function-like Relation of REAL,REAL
      st b2 c= proj1 (b3 + ((1 / (4 * b1)) (#) (sin * b4))) &
         (for b5 being Element of REAL
               st b5 in b2
            holds b3 . b5 = b5 / 2 & b4 . b5 = (2 * b1) * b5) &
         b1 <> 0
   holds b3 + ((1 / (4 * b1)) (#) (sin * b4)) is_differentiable_on b2 &
    (for b5 being Element of REAL
          st b5 in b2
       holds ((b3 + ((1 / (4 * b1)) (#) (sin * b4))) `| b2) . b5 = (cos (b1 * b5)) ^2);

:: FDIFF_6:th 46
theorem
for b1 being Element of NAT
for b2 being open Element of bool REAL
      st b2 c= proj1 ((1 / b1) (#) ((#Z b1) * cos)) &
         0 < b1
   holds (1 / b1) (#) ((#Z b1) * cos) is_differentiable_on b2 &
    (for b3 being Element of REAL
          st b3 in b2
       holds (((1 / b1) (#) ((#Z b1) * cos)) `| b2) . b3 = - (((cos . b3) #Z (b1 - 1)) * (sin . b3)));

:: FDIFF_6:th 47
theorem
for b1 being Element of NAT
for b2 being open Element of bool REAL
      st b2 c= proj1 (((1 / 3) (#) ((#Z 3) * cos)) - cos) &
         0 < b1
   holds ((1 / 3) (#) ((#Z 3) * cos)) - cos is_differentiable_on b2 &
    (for b3 being Element of REAL
          st b3 in b2
       holds ((((1 / 3) (#) ((#Z 3) * cos)) - cos) `| b2) . b3 = (sin . b3) |^ 3);

:: FDIFF_6:th 48
theorem
for b1 being Element of NAT
for b2 being open Element of bool REAL
      st b2 c= proj1 (sin - ((1 / 3) (#) ((#Z 3) * sin))) &
         0 < b1
   holds sin - ((1 / 3) (#) ((#Z 3) * sin)) is_differentiable_on b2 &
    (for b3 being Element of REAL
          st b3 in b2
       holds ((sin - ((1 / 3) (#) ((#Z 3) * sin))) `| b2) . b3 = (cos . b3) |^ 3);

:: FDIFF_6:th 49
theorem
for b1 being open Element of bool REAL
      st b1 c= proj1 (sin * ln)
   holds sin * ln is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sin * ln) `| b1) . b2 = (cos . log(number_e,b2)) / b2);

:: FDIFF_6:th 50
theorem
for b1 being open Element of bool REAL
      st b1 c= proj1 - (cos * ln)
   holds - (cos * ln) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((- (cos * ln)) `| b1) . b2 = (sin . log(number_e,b2)) / b2);