Article FDIFF_4, MML version 4.99.1005

:: FDIFF_4:th 1
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 (ln * b3) &
         (for b4 being Element of REAL
               st b4 in b2
            holds b3 . b4 = b1 + b4 & 0 < b3 . b4)
   holds ln * b3 is_differentiable_on b2 &
    (for b4 being Element of REAL
          st b4 in b2
       holds ((ln * b3) `| b2) . b4 = 1 / (b1 + b4));

:: FDIFF_4:th 2
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 (ln * b3) &
         (for b4 being Element of REAL
               st b4 in b2
            holds b3 . b4 = b4 - b1 & 0 < b3 . b4)
   holds ln * b3 is_differentiable_on b2 &
    (for b4 being Element of REAL
          st b4 in b2
       holds ((ln * b3) `| b2) . b4 = 1 / (b4 - b1));

:: FDIFF_4:th 3
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 - (ln * b3) &
         (for b4 being Element of REAL
               st b4 in b2
            holds b3 . b4 = b1 - b4 & 0 < b3 . b4)
   holds - (ln * b3) is_differentiable_on b2 &
    (for b4 being Element of REAL
          st b4 in b2
       holds ((- (ln * b3)) `| b2) . b4 = 1 / (b1 - b4));

:: FDIFF_4: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 ((id b2) - (b1 (#) b3)) &
         b3 = ln * b4 &
         (for b5 being Element of REAL
               st b5 in b2
            holds b4 . b5 = b1 + b5 & 0 < b4 . b5)
   holds (id b2) - (b1 (#) b3) is_differentiable_on b2 &
    (for b5 being Element of REAL
          st b5 in b2
       holds (((id b2) - (b1 (#) b3)) `| b2) . b5 = b5 / (b1 + b5));

:: FDIFF_4:th 5
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 * b1) (#) b3) - id b2) &
         b3 = ln * b4 &
         (for b5 being Element of REAL
               st b5 in b2
            holds b4 . b5 = b1 + b5 & 0 < b4 . b5)
   holds ((2 * b1) (#) b3) - id b2 is_differentiable_on b2 &
    (for b5 being Element of REAL
          st b5 in b2
       holds ((((2 * b1) (#) b3) - id b2) `| b2) . b5 = (b1 - b5) / (b1 + b5));

:: FDIFF_4:th 6
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 ((id b2) - ((2 * b1) (#) b3)) &
         b3 = ln * b4 &
         (for b5 being Element of REAL
               st b5 in b2
            holds b4 . b5 = b5 + b1 & 0 < b4 . b5)
   holds (id b2) - ((2 * b1) (#) b3) is_differentiable_on b2 &
    (for b5 being Element of REAL
          st b5 in b2
       holds (((id b2) - ((2 * b1) (#) b3)) `| b2) . b5 = (b5 - b1) / (b5 + b1));

:: FDIFF_4:th 7
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 ((id b2) + ((2 * b1) (#) b3)) &
         b3 = ln * b4 &
         (for b5 being Element of REAL
               st b5 in b2
            holds b4 . b5 = b5 - b1 & 0 < b4 . b5)
   holds (id b2) + ((2 * b1) (#) b3) is_differentiable_on b2 &
    (for b5 being Element of REAL
          st b5 in b2
       holds (((id b2) + ((2 * b1) (#) b3)) `| b2) . b5 = (b5 + b1) / (b5 - b1));

:: FDIFF_4:th 8
theorem
for b1, b2 being Element of REAL
for b3 being open Element of bool REAL
for b4, b5 being Function-like Relation of REAL,REAL
      st b3 c= proj1 ((id b3) + ((b1 - b2) (#) b4)) &
         b4 = ln * b5 &
         (for b6 being Element of REAL
               st b6 in b3
            holds b5 . b6 = b6 + b2 & 0 < b5 . b6)
   holds (id b3) + ((b1 - b2) (#) b4) is_differentiable_on b3 &
    (for b6 being Element of REAL
          st b6 in b3
       holds (((id b3) + ((b1 - b2) (#) b4)) `| b3) . b6 = (b6 + b1) / (b6 + b2));

:: FDIFF_4:th 9
theorem
for b1, b2 being Element of REAL
for b3 being open Element of bool REAL
for b4, b5 being Function-like Relation of REAL,REAL
      st b3 c= proj1 ((id b3) + ((b1 + b2) (#) b4)) &
         b4 = ln * b5 &
         (for b6 being Element of REAL
               st b6 in b3
            holds b5 . b6 = b6 - b2 & 0 < b5 . b6)
   holds (id b3) + ((b1 + b2) (#) b4) is_differentiable_on b3 &
    (for b6 being Element of REAL
          st b6 in b3
       holds (((id b3) + ((b1 + b2) (#) b4)) `| b3) . b6 = (b6 + b1) / (b6 - b2));

:: FDIFF_4:th 10
theorem
for b1, b2 being Element of REAL
for b3 being open Element of bool REAL
for b4, b5 being Function-like Relation of REAL,REAL
      st b3 c= proj1 ((id b3) - ((b1 + b2) (#) b4)) &
         b4 = ln * b5 &
         (for b6 being Element of REAL
               st b6 in b3
            holds b5 . b6 = b6 + b2 & 0 < b5 . b6)
   holds (id b3) - ((b1 + b2) (#) b4) is_differentiable_on b3 &
    (for b6 being Element of REAL
          st b6 in b3
       holds (((id b3) - ((b1 + b2) (#) b4)) `| b3) . b6 = (b6 - b1) / (b6 + b2));

:: FDIFF_4:th 11
theorem
for b1, b2 being Element of REAL
for b3 being open Element of bool REAL
for b4, b5 being Function-like Relation of REAL,REAL
      st b3 c= proj1 ((id b3) + ((b1 - b2) (#) b4)) &
         b4 = ln * b5 &
         (for b6 being Element of REAL
               st b6 in b3
            holds b5 . b6 = b6 - b1 & 0 < b5 . b6)
   holds (id b3) + ((b1 - b2) (#) b4) is_differentiable_on b3 &
    (for b6 being Element of REAL
          st b6 in b3
       holds (((id b3) + ((b1 - b2) (#) b4)) `| b3) . b6 = (b6 - b2) / (b6 - b1));

:: FDIFF_4:th 12
theorem
for b1, b2, b3 being Element of REAL
for b4 being open Element of bool REAL
for b5, b6 being Function-like Relation of REAL,REAL
      st b4 c= proj1 (b5 + (b1 (#) b6)) &
         (for b7 being Element of REAL
               st b7 in b4
            holds b5 . b7 = b2 + (b3 * b7)) &
         b6 = #Z 2
   holds b5 + (b1 (#) b6) is_differentiable_on b4 &
    (for b7 being Element of REAL
          st b7 in b4
       holds ((b5 + (b1 (#) b6)) `| b4) . b7 = b3 + ((2 * b1) * b7));

:: FDIFF_4:th 13
theorem
for b1, b2, b3 being Element of REAL
for b4 being open Element of bool REAL
for b5, b6 being Function-like Relation of REAL,REAL
      st b4 c= proj1 (ln * (b5 + (b1 (#) b6))) &
         b6 = #Z 2 &
         (for b7 being Element of REAL
               st b7 in b4
            holds b5 . b7 = b2 + (b3 * b7) &
             0 < (b5 + (b1 (#) b6)) . b7)
   holds ln * (b5 + (b1 (#) b6)) is_differentiable_on b4 &
    (for b7 being Element of REAL
          st b7 in b4
       holds ((ln * (b5 + (b1 (#) b6))) `| b4) . b7 = (b3 + ((2 * b1) * b7)) / ((b2 + (b3 * b7)) + (b1 * (b7 |^ 2))));

:: FDIFF_4:th 14
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 b3 &
         (for b4 being Element of REAL
               st b4 in b2
            holds b3 . b4 = b1 + b4 & b3 . b4 <> 0)
   holds b3 ^ is_differentiable_on b2 &
    (for b4 being Element of REAL
          st b4 in b2
       holds (b3 ^ `| b2) . b4 = - (1 / ((b1 + b4) ^2)));

:: FDIFF_4:th 15
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) (#) (b3 ^)) &
         (for b4 being Element of REAL
               st b4 in b2
            holds b3 . b4 = b1 + b4 & b3 . b4 <> 0)
   holds (- 1) (#) (b3 ^) is_differentiable_on b2 &
    (for b4 being Element of REAL
          st b4 in b2
       holds (((- 1) (#) (b3 ^)) `| b2) . b4 = 1 / ((b1 + b4) ^2));

:: FDIFF_4: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 b3 &
         (for b4 being Element of REAL
               st b4 in b2
            holds b3 . b4 = b1 - b4 & b3 . b4 <> 0)
   holds b3 ^ is_differentiable_on b2 &
    (for b4 being Element of REAL
          st b4 in b2
       holds (b3 ^ `| b2) . b4 = 1 / ((b1 - b4) ^2));

:: FDIFF_4: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 (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_4:th 18
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 (ln * (b3 + b4)) &
         b4 = #Z 2 &
         (for b5 being Element of REAL
               st b5 in b2
            holds b3 . b5 = b1 ^2 & 0 < (b3 + b4) . b5)
   holds ln * (b3 + b4) is_differentiable_on b2 &
    (for b5 being Element of REAL
          st b5 in b2
       holds ((ln * (b3 + b4)) `| b2) . b5 = (2 * b5) / (b1 ^2 + (b5 |^ 2)));

:: FDIFF_4:th 19
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 - (ln * (b3 - b4)) &
         b4 = #Z 2 &
         (for b5 being Element of REAL
               st b5 in b2
            holds b3 . b5 = b1 ^2 & 0 < (b3 - b4) . b5)
   holds - (ln * (b3 - b4)) is_differentiable_on b2 &
    (for b5 being Element of REAL
          st b5 in b2
       holds ((- (ln * (b3 - b4))) `| b2) . b5 = (2 * b5) / (b1 ^2 - (b5 |^ 2)));

:: FDIFF_4:th 20
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) &
         b4 = #Z 3
   holds b3 + b4 is_differentiable_on b2 &
    (for b5 being Element of REAL
          st b5 in b2
       holds ((b3 + b4) `| b2) . b5 = 3 * (b5 |^ 2));

:: FDIFF_4:th 21
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 (ln * (b3 + b4)) &
         b4 = #Z 3 &
         (for b5 being Element of REAL
               st b5 in b2
            holds b3 . b5 = b1 & 0 < (b3 + b4) . b5)
   holds ln * (b3 + b4) is_differentiable_on b2 &
    (for b5 being Element of REAL
          st b5 in b2
       holds ((ln * (b3 + b4)) `| b2) . b5 = (3 * (b5 |^ 2)) / (b1 + (b5 |^ 3)));

:: FDIFF_4:th 22
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 (ln * (b3 / b4)) &
         (for b5 being Element of REAL
               st b5 in b2
            holds b3 . b5 = b1 + b5 & 0 < b3 . b5 & b4 . b5 = b1 - b5 & 0 < b4 . b5)
   holds ln * (b3 / b4) is_differentiable_on b2 &
    (for b5 being Element of REAL
          st b5 in b2
       holds ((ln * (b3 / b4)) `| b2) . b5 = (2 * b1) / (b1 ^2 - (b5 ^2)));

:: FDIFF_4:th 23
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 (ln * (b3 / b4)) &
         (for b5 being Element of REAL
               st b5 in b2
            holds b3 . b5 = b5 - b1 & 0 < b3 . b5 & b4 . b5 = b5 + b1 & 0 < b4 . b5)
   holds ln * (b3 / b4) is_differentiable_on b2 &
    (for b5 being Element of REAL
          st b5 in b2
       holds ((ln * (b3 / b4)) `| b2) . b5 = (2 * b1) / (b5 ^2 - (b1 ^2)));

:: FDIFF_4:th 24
theorem
for b1, b2 being Element of REAL
for b3 being open Element of bool REAL
for b4, b5 being Function-like Relation of REAL,REAL
      st b3 c= proj1 (ln * (b4 / b5)) &
         (for b6 being Element of REAL
               st b6 in b3
            holds b4 . b6 = b6 - b1 & 0 < b4 . b6 & b5 . b6 = b6 - b2 & 0 < b5 . b6)
   holds ln * (b4 / b5) is_differentiable_on b3 &
    (for b6 being Element of REAL
          st b6 in b3
       holds ((ln * (b4 / b5)) `| b3) . b6 = (b1 - b2) / ((b6 - b1) * (b6 - b2)));

:: FDIFF_4:th 25
theorem
for b1, b2 being Element of REAL
for b3 being open Element of bool REAL
for b4, b5, b6 being Function-like Relation of REAL,REAL
      st b3 c= proj1 ((1 / (b1 - b2)) (#) b4) &
         b4 = ln * (b5 / b6) &
         (for b7 being Element of REAL
               st b7 in b3
            holds b5 . b7 = b7 - b1 & 0 < b5 . b7 & b6 . b7 = b7 - b2 & 0 < b6 . b7 & b1 - b2 <> 0)
   holds (1 / (b1 - b2)) (#) b4 is_differentiable_on b3 &
    (for b7 being Element of REAL
          st b7 in b3
       holds (((1 / (b1 - b2)) (#) b4) `| b3) . b7 = 1 / ((b7 - b1) * (b7 - b2)));

:: FDIFF_4:th 26
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 (ln * (b3 / b4)) &
         b4 = #Z 2 &
         (for b5 being Element of REAL
               st b5 in b2
            holds b3 . b5 = b5 - b1 & 0 < b3 . b5 & 0 < b4 . b5 & b5 <> 0)
   holds ln * (b3 / b4) is_differentiable_on b2 &
    (for b5 being Element of REAL
          st b5 in b2
       holds ((ln * (b3 / b4)) `| b2) . b5 = ((2 * b1) - b5) / (b5 * (b5 - b1)));

:: FDIFF_4:th 27
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 ((#R (3 / 2)) * b3) &
         (for b4 being Element of REAL
               st b4 in b2
            holds b3 . b4 = b1 + b4 & 0 < b3 . b4)
   holds (#R (3 / 2)) * b3 is_differentiable_on b2 &
    (for b4 being Element of REAL
          st b4 in b2
       holds (((#R (3 / 2)) * b3) `| b2) . b4 = (3 / 2) * ((b1 + b4) #R (1 / 2)));

:: FDIFF_4:th 28
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 ((2 / 3) (#) ((#R (3 / 2)) * b3)) &
         (for b4 being Element of REAL
               st b4 in b2
            holds b3 . b4 = b1 + b4 & 0 < b3 . b4)
   holds (2 / 3) (#) ((#R (3 / 2)) * b3) is_differentiable_on b2 &
    (for b4 being Element of REAL
          st b4 in b2
       holds (((2 / 3) (#) ((#R (3 / 2)) * b3)) `| b2) . b4 = (b1 + b4) #R (1 / 2));

:: FDIFF_4:th 29
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 ((- (2 / 3)) (#) ((#R (3 / 2)) * b3)) &
         (for b4 being Element of REAL
               st b4 in b2
            holds b3 . b4 = b1 - b4 & 0 < b3 . b4)
   holds (- (2 / 3)) (#) ((#R (3 / 2)) * b3) is_differentiable_on b2 &
    (for b4 being Element of REAL
          st b4 in b2
       holds (((- (2 / 3)) (#) ((#R (3 / 2)) * b3)) `| b2) . b4 = (b1 - b4) #R (1 / 2));

:: FDIFF_4:th 30
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 (2 (#) ((#R (1 / 2)) * b3)) &
         (for b4 being Element of REAL
               st b4 in b2
            holds b3 . b4 = b1 + b4 & 0 < b3 . b4)
   holds 2 (#) ((#R (1 / 2)) * b3) is_differentiable_on b2 &
    (for b4 being Element of REAL
          st b4 in b2
       holds ((2 (#) ((#R (1 / 2)) * b3)) `| b2) . b4 = (b1 + b4) #R - (1 / 2));

:: FDIFF_4:th 31
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 ((- 2) (#) ((#R (1 / 2)) * b3)) &
         (for b4 being Element of REAL
               st b4 in b2
            holds b3 . b4 = b1 - b4 & 0 < b3 . b4)
   holds (- 2) (#) ((#R (1 / 2)) * b3) is_differentiable_on b2 &
    (for b4 being Element of REAL
          st b4 in b2
       holds (((- 2) (#) ((#R (1 / 2)) * b3)) `| b2) . b4 = (b1 - b4) #R - (1 / 2));

:: FDIFF_4:th 32
theorem
for b1, b2 being Element of REAL
for b3 being open Element of bool REAL
for b4 being Function-like Relation of REAL,REAL
      st b3 c= proj1 ((2 / (3 * b1)) (#) ((#R (3 / 2)) * b4)) &
         (for b5 being Element of REAL
               st b5 in b3
            holds b4 . b5 = b2 + (b1 * b5) & b1 <> 0 & 0 < b4 . b5)
   holds (2 / (3 * b1)) (#) ((#R (3 / 2)) * b4) is_differentiable_on b3 &
    (for b5 being Element of REAL
          st b5 in b3
       holds (((2 / (3 * b1)) (#) ((#R (3 / 2)) * b4)) `| b3) . b5 = (b2 + (b1 * b5)) #R (1 / 2));

:: FDIFF_4:th 33
theorem
for b1, b2 being Element of REAL
for b3 being open Element of bool REAL
for b4 being Function-like Relation of REAL,REAL
      st b3 c= proj1 ((- (2 / (3 * b1))) (#) ((#R (3 / 2)) * b4)) &
         (for b5 being Element of REAL
               st b5 in b3
            holds b4 . b5 = b2 - (b1 * b5) & b1 <> 0 & 0 < b4 . b5)
   holds (- (2 / (3 * b1))) (#) ((#R (3 / 2)) * b4) is_differentiable_on b3 &
    (for b5 being Element of REAL
          st b5 in b3
       holds (((- (2 / (3 * b1))) (#) ((#R (3 / 2)) * b4)) `| b3) . b5 = (b2 - (b1 * b5)) #R (1 / 2));

:: FDIFF_4:th 34
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 ((#R (1 / 2)) * b3) &
         b3 = b4 + b5 &
         b5 = #Z 2 &
         (for b6 being Element of REAL
               st b6 in b2
            holds b4 . b6 = b1 ^2 & 0 < b3 . b6)
   holds (#R (1 / 2)) * b3 is_differentiable_on b2 &
    (for b6 being Element of REAL
          st b6 in b2
       holds (((#R (1 / 2)) * b3) `| b2) . b6 = b6 * ((b1 ^2 + (b6 |^ 2)) #R - (1 / 2)));

:: FDIFF_4:th 35
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 - ((#R (1 / 2)) * b3) &
         b3 = b4 - b5 &
         b5 = #Z 2 &
         (for b6 being Element of REAL
               st b6 in b2
            holds b4 . b6 = b1 ^2 & 0 < b3 . b6)
   holds - ((#R (1 / 2)) * b3) is_differentiable_on b2 &
    (for b6 being Element of REAL
          st b6 in b2
       holds ((- ((#R (1 / 2)) * b3)) `| b2) . b6 = b6 * ((b1 ^2 - (b6 |^ 2)) #R - (1 / 2)));

:: FDIFF_4:th 36
theorem
for b1 being open Element of bool REAL
for b2, b3, b4 being Function-like Relation of REAL,REAL
      st b1 c= proj1 (2 (#) ((#R (1 / 2)) * b2)) &
         b2 = b3 + b4 &
         b4 = #Z 2 &
         (for b5 being Element of REAL
               st b5 in b1
            holds b3 . b5 = b5 & 0 < b2 . b5)
   holds 2 (#) ((#R (1 / 2)) * b2) is_differentiable_on b1 &
    (for b5 being Element of REAL
          st b5 in b1
       holds ((2 (#) ((#R (1 / 2)) * b2)) `| b1) . b5 = ((2 * b5) + 1) * (((b5 |^ 2) + b5) #R - (1 / 2)));

:: FDIFF_4:th 37
theorem
for b1, b2 being Element of REAL
for b3 being open Element of bool REAL
for b4 being Function-like Relation of REAL,REAL
      st b3 c= proj1 (sin * b4) &
         (for b5 being Element of REAL
               st b5 in b3
            holds b4 . b5 = (b1 * b5) + b2)
   holds sin * b4 is_differentiable_on b3 &
    (for b5 being Element of REAL
          st b5 in b3
       holds ((sin * b4) `| b3) . b5 = b1 * (cos . ((b1 * b5) + b2)));

:: FDIFF_4:th 38
theorem
for b1, b2 being Element of REAL
for b3 being open Element of bool REAL
for b4 being Function-like Relation of REAL,REAL
      st b3 c= proj1 (cos * b4) &
         (for b5 being Element of REAL
               st b5 in b3
            holds b4 . b5 = (b1 * b5) + b2)
   holds cos * b4 is_differentiable_on b3 &
    (for b5 being Element of REAL
          st b5 in b3
       holds ((cos * b4) `| b3) . b5 = - (b1 * (sin . ((b1 * b5) + b2))));

:: FDIFF_4:th 39
theorem
for b1 being open Element of bool REAL
      st for b2 being Element of REAL
              st b2 in b1
           holds cos . b2 <> 0
   holds cos ^ is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds (cos ^ `| b1) . b2 = (sin . b2) / ((cos . b2) ^2));

:: FDIFF_4:th 40
theorem
for b1 being open Element of bool REAL
      st for b2 being Element of REAL
              st b2 in b1
           holds sin . b2 <> 0
   holds sin ^ is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds (sin ^ `| b1) . b2 = - ((cos . b2) / ((sin . b2) ^2)));

:: FDIFF_4:th 41
theorem
for b1 being open Element of bool REAL
      st b1 c= proj1 (sin (#) cos)
   holds sin (#) cos is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sin (#) cos) `| b1) . b2 = cos (2 * b2));

:: FDIFF_4:th 42
theorem
for b1 being open Element of bool REAL
      st b1 c= proj1 (ln * cos) &
         (for b2 being Element of REAL
               st b2 in b1
            holds 0 < cos . b2)
   holds ln * cos is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((ln * cos) `| b1) . b2 = - tan b2);

:: FDIFF_4:th 43
theorem
for b1 being open Element of bool REAL
      st b1 c= proj1 (ln * sin) &
         (for b2 being Element of REAL
               st b2 in b1
            holds 0 < sin . b2)
   holds ln * sin is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((ln * sin) `| b1) . b2 = cot b2);

:: FDIFF_4:th 44
theorem
for b1 being open Element of bool REAL
      st b1 c= proj1 ((- id b1) (#) cos)
   holds (- id b1) (#) cos is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds (((- id b1) (#) cos) `| b1) . b2 = (- (cos . b2)) + (b2 * (sin . b2)));

:: FDIFF_4:th 45
theorem
for b1 being open Element of bool REAL
      st b1 c= proj1 ((id b1) (#) sin)
   holds (id b1) (#) sin is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds (((id b1) (#) sin) `| b1) . b2 = (sin . b2) + (b2 * (cos . b2)));

:: FDIFF_4:th 46
theorem
for b1 being open Element of bool REAL
      st b1 c= proj1 (((- id b1) (#) cos) + sin)
   holds ((- id b1) (#) cos) + sin is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((((- id b1) (#) cos) + sin) `| b1) . b2 = b2 * (sin . b2));

:: FDIFF_4:th 47
theorem
for b1 being open Element of bool REAL
      st b1 c= proj1 (((id b1) (#) sin) + cos)
   holds ((id b1) (#) sin) + cos is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((((id b1) (#) sin) + cos) `| b1) . b2 = b2 * (cos . b2));

:: FDIFF_4:th 48
theorem
for b1 being open Element of bool REAL
      st b1 c= proj1 (2 (#) ((#R (1 / 2)) * sin)) &
         (for b2 being Element of REAL
               st b2 in b1
            holds 0 < sin . b2)
   holds 2 (#) ((#R (1 / 2)) * sin) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((2 (#) ((#R (1 / 2)) * sin)) `| b1) . b2 = (cos . b2) * ((sin . b2) #R - (1 / 2)));

:: FDIFF_4:th 49
theorem
for b1 being open Element of bool REAL
      st b1 c= proj1 ((1 / 2) (#) ((#Z 2) * sin))
   holds (1 / 2) (#) ((#Z 2) * sin) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds (((1 / 2) (#) ((#Z 2) * sin)) `| b1) . b2 = (sin . b2) * (cos . b2));

:: FDIFF_4:th 50
theorem
for b1 being open Element of bool REAL
      st b1 c= proj1 (sin + ((1 / 2) (#) ((#Z 2) * sin))) &
         (for b2 being Element of REAL
               st b2 in b1
            holds 0 < sin . b2 & sin . b2 < 1)
   holds sin + ((1 / 2) (#) ((#Z 2) * sin)) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sin + ((1 / 2) (#) ((#Z 2) * sin))) `| b1) . b2 = ((cos . b2) |^ 3) / (1 - (sin . b2)));

:: FDIFF_4:th 51
theorem
for b1 being open Element of bool REAL
      st b1 c= proj1 (((1 / 2) (#) ((#Z 2) * sin)) - cos) &
         (for b2 being Element of REAL
               st b2 in b1
            holds 0 < sin . b2 & cos . b2 < 1)
   holds ((1 / 2) (#) ((#Z 2) * sin)) - cos is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((((1 / 2) (#) ((#Z 2) * sin)) - cos) `| b1) . b2 = ((sin . b2) |^ 3) / (1 - (cos . b2)));

:: FDIFF_4:th 52
theorem
for b1 being open Element of bool REAL
      st b1 c= proj1 (sin - ((1 / 2) (#) ((#Z 2) * sin))) &
         (for b2 being Element of REAL
               st b2 in b1
            holds 0 < sin . b2 & - 1 < sin . b2)
   holds sin - ((1 / 2) (#) ((#Z 2) * sin)) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sin - ((1 / 2) (#) ((#Z 2) * sin))) `| b1) . b2 = ((cos . b2) |^ 3) / (1 + (sin . b2)));

:: FDIFF_4:th 53
theorem
for b1 being open Element of bool REAL
      st b1 c= proj1 ((- cos) - ((1 / 2) (#) ((#Z 2) * sin))) &
         (for b2 being Element of REAL
               st b2 in b1
            holds 0 < sin . b2 & - 1 < cos . b2)
   holds (- cos) - ((1 / 2) (#) ((#Z 2) * sin)) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds (((- cos) - ((1 / 2) (#) ((#Z 2) * sin))) `| b1) . b2 = ((sin . b2) |^ 3) / (1 + (cos . b2)));

:: FDIFF_4:th 54
theorem
for b1 being Element of NAT
for b2 being open Element of bool REAL
      st b2 c= proj1 ((1 / b1) (#) ((#Z b1) * sin)) &
         0 < b1
   holds (1 / b1) (#) ((#Z b1) * sin) is_differentiable_on b2 &
    (for b3 being Element of REAL
          st b3 in b2
       holds (((1 / b1) (#) ((#Z b1) * sin)) `| b2) . b3 = ((sin . b3) #Z (b1 - 1)) * (cos . b3));

:: FDIFF_4:th 55
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 - 1)
   holds exp_R (#) b2 is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds ((exp_R (#) b2) `| b1) . b3 = b3 * (exp_R . b3));

:: FDIFF_4:th 56
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 / (exp_R + b2))) &
         (for b3 being Element of REAL
               st b3 in b1
            holds b2 . b3 = 1)
   holds ln * (exp_R / (exp_R + b2)) is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds ((ln * (exp_R / (exp_R + b2))) `| b1) . b3 = 1 / ((exp_R . b3) + 1));

:: FDIFF_4:th 57
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) / exp_R)) &
         (for b3 being Element of REAL
               st b3 in b1
            holds b2 . b3 = 1 & 0 < (exp_R - b2) . b3)
   holds ln * ((exp_R - b2) / exp_R) is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds ((ln * ((exp_R - b2) / exp_R)) `| b1) . b3 = 1 / ((exp_R . b3) - 1));