Article FDIFF_9, MML version 4.99.1005

:: FDIFF_9:funcnot 1 => FDIFF_9:func 1
definition
  func sec -> Function-like Relation of REAL,REAL equals
    cos ^;
end;

:: FDIFF_9:def 1
theorem
sec = cos ^;

:: FDIFF_9:funcnot 2 => FDIFF_9:func 2
definition
  func cosec -> Function-like Relation of REAL,REAL equals
    sin ^;
end;

:: FDIFF_9:def 2
theorem
cosec = sin ^;

:: FDIFF_9:th 1
theorem
for b1 being Element of REAL
      st cos . b1 <> 0
   holds sec is_differentiable_in b1 &
    diff(sec,b1) = (sin . b1) / ((cos . b1) ^2);

:: FDIFF_9:th 2
theorem
for b1 being Element of REAL
      st sin . b1 <> 0
   holds cosec is_differentiable_in b1 &
    diff(cosec,b1) = - ((cos . b1) / ((sin . b1) ^2));

:: FDIFF_9:th 3
theorem
for b1 being Element of REAL
for b2 being natural set holds
   (1 / b1) #Z b2 = 1 / (b1 #Z b2);

:: FDIFF_9:th 4
theorem
for b1 being open Element of bool REAL
      st b1 c= dom sec
   holds sec is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds (sec `| b1) . b2 = (sin . b2) / ((cos . b2) ^2));

:: FDIFF_9:th 5
theorem
for b1 being open Element of bool REAL
      st b1 c= dom cosec
   holds cosec is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds (cosec `| b1) . b2 = - ((cos . b2) / ((sin . b2) ^2)));

:: FDIFF_9:th 6
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= dom (sec * b4) &
         (for b5 being Element of REAL
               st b5 in b3
            holds b4 . b5 = (b1 * b5) + b2)
   holds sec * b4 is_differentiable_on b3 &
    (for b5 being Element of REAL
          st b5 in b3
       holds ((sec * b4) `| b3) . b5 = (b1 * (sin . ((b1 * b5) + b2))) / ((cos . ((b1 * b5) + b2)) ^2));

:: FDIFF_9:th 7
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= dom (cosec * b4) &
         (for b5 being Element of REAL
               st b5 in b3
            holds b4 . b5 = (b1 * b5) + b2)
   holds cosec * b4 is_differentiable_on b3 &
    (for b5 being Element of REAL
          st b5 in b3
       holds ((cosec * b4) `| b3) . b5 = - ((b1 * (cos . ((b1 * b5) + b2))) / ((sin . ((b1 * b5) + b2)) ^2)));

:: FDIFF_9:th 8
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= dom (sec * (b2 ^)) &
         (for b3 being Element of REAL
               st b3 in b1
            holds b2 . b3 = b3)
   holds sec * (b2 ^) is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds ((sec * (b2 ^)) `| b1) . b3 = - ((sin . (1 / b3)) / (b3 ^2 * ((cos . (1 / b3)) ^2))));

:: FDIFF_9:th 9
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= dom (cosec * (b2 ^)) &
         (for b3 being Element of REAL
               st b3 in b1
            holds b2 . b3 = b3)
   holds cosec * (b2 ^) is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds ((cosec * (b2 ^)) `| b1) . b3 = (cos . (1 / b3)) / (b3 ^2 * ((sin . (1 / b3)) ^2)));

:: FDIFF_9:th 10
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= dom (sec * (b5 + (b1 (#) b6))) &
         b6 = #Z 2 &
         (for b7 being Element of REAL
               st b7 in b4
            holds b5 . b7 = b2 + (b3 * b7))
   holds sec * (b5 + (b1 (#) b6)) is_differentiable_on b4 &
    (for b7 being Element of REAL
          st b7 in b4
       holds ((sec * (b5 + (b1 (#) b6))) `| b4) . b7 = ((b3 + ((2 * b1) * b7)) * (sin . ((b2 + (b3 * b7)) + (b1 * (b7 ^2))))) / ((cos . ((b2 + (b3 * b7)) + (b1 * (b7 ^2)))) ^2));

:: FDIFF_9:th 11
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= dom (cosec * (b5 + (b1 (#) b6))) &
         b6 = #Z 2 &
         (for b7 being Element of REAL
               st b7 in b4
            holds b5 . b7 = b2 + (b3 * b7))
   holds cosec * (b5 + (b1 (#) b6)) is_differentiable_on b4 &
    (for b7 being Element of REAL
          st b7 in b4
       holds ((cosec * (b5 + (b1 (#) b6))) `| b4) . b7 = - (((b3 + ((2 * b1) * b7)) * (cos . ((b2 + (b3 * b7)) + (b1 * (b7 ^2))))) / ((sin . ((b2 + (b3 * b7)) + (b1 * (b7 ^2)))) ^2)));

:: FDIFF_9:th 12
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (sec * exp_R)
   holds sec * exp_R is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sec * exp_R) `| b1) . b2 = ((exp_R . b2) * (sin . (exp_R . b2))) / ((cos . (exp_R . b2)) ^2));

:: FDIFF_9:th 13
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cosec * exp_R)
   holds cosec * exp_R is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cosec * exp_R) `| b1) . b2 = - (((exp_R . b2) * (cos . (exp_R . b2))) / ((sin . (exp_R . b2)) ^2)));

:: FDIFF_9:th 14
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (sec * ln)
   holds sec * ln is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sec * ln) `| b1) . b2 = (sin . (ln . b2)) / (b2 * ((cos . (ln . b2)) ^2)));

:: FDIFF_9:th 15
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cosec * ln)
   holds cosec * ln is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cosec * ln) `| b1) . b2 = - ((cos . (ln . b2)) / (b2 * ((sin . (ln . b2)) ^2))));

:: FDIFF_9:th 16
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (exp_R * sec)
   holds exp_R * sec is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((exp_R * sec) `| b1) . b2 = ((exp_R . (sec . b2)) * (sin . b2)) / ((cos . b2) ^2));

:: FDIFF_9:th 17
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (exp_R * cosec)
   holds exp_R * cosec is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((exp_R * cosec) `| b1) . b2 = - (((exp_R . (cosec . b2)) * (cos . b2)) / ((sin . b2) ^2)));

:: FDIFF_9:th 18
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (ln * sec)
   holds ln * sec is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((ln * sec) `| b1) . b2 = (sin . b2) / (cos . b2));

:: FDIFF_9:th 19
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (ln * cosec)
   holds ln * cosec is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((ln * cosec) `| b1) . b2 = - ((cos . b2) / (sin . b2)));

:: FDIFF_9:th 20
theorem
for b1 being natural set
for b2 being open Element of bool REAL
      st b2 c= dom ((#Z b1) * sec) & 1 <= b1
   holds (#Z b1) * sec is_differentiable_on b2 &
    (for b3 being Element of REAL
          st b3 in b2
       holds (((#Z b1) * sec) `| b2) . b3 = (b1 * (sin . b3)) / ((cos . b3) #Z (b1 + 1)));

:: FDIFF_9:th 21
theorem
for b1 being natural set
for b2 being open Element of bool REAL
      st b2 c= dom ((#Z b1) * cosec) & 1 <= b1
   holds (#Z b1) * cosec is_differentiable_on b2 &
    (for b3 being Element of REAL
          st b3 in b2
       holds (((#Z b1) * cosec) `| b2) . b3 = - ((b1 * (cos . b3)) / ((sin . b3) #Z (b1 + 1))));

:: FDIFF_9:th 22
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (sec - id b1)
   holds sec - id b1 is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sec - id b1) `| b1) . b2 = ((sin . b2) - ((cos . b2) ^2)) / ((cos . b2) ^2));

:: FDIFF_9:th 23
theorem
for b1 being open Element of bool REAL
      st b1 c= dom ((- cosec) - id b1)
   holds (- cosec) - id b1 is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds (((- cosec) - id b1) `| b1) . b2 = ((cos . b2) - ((sin . b2) ^2)) / ((sin . b2) ^2));

:: FDIFF_9:th 24
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (exp_R (#) sec)
   holds exp_R (#) sec is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((exp_R (#) sec) `| b1) . b2 = ((exp_R . b2) / (cos . b2)) + (((exp_R . b2) * (sin . b2)) / ((cos . b2) ^2)));

:: FDIFF_9:th 25
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (exp_R (#) cosec)
   holds exp_R (#) cosec is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((exp_R (#) cosec) `| b1) . b2 = ((exp_R . b2) / (sin . b2)) - (((exp_R . b2) * (cos . b2)) / ((sin . b2) ^2)));

:: FDIFF_9:th 26
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= dom (((1 / b1) (#) (sec * b3)) - id b2) &
         (for b4 being Element of REAL
               st b4 in b2
            holds b3 . b4 = b1 * b4 & b1 <> 0)
   holds ((1 / b1) (#) (sec * b3)) - id b2 is_differentiable_on b2 &
    (for b4 being Element of REAL
          st b4 in b2
       holds ((((1 / b1) (#) (sec * b3)) - id b2) `| b2) . b4 = ((sin . (b1 * b4)) - ((cos . (b1 * b4)) ^2)) / ((cos . (b1 * b4)) ^2));

:: FDIFF_9: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= dom (((- (1 / b1)) (#) (cosec * b3)) - id b2) &
         (for b4 being Element of REAL
               st b4 in b2
            holds b3 . b4 = b1 * b4 & b1 <> 0)
   holds ((- (1 / b1)) (#) (cosec * b3)) - id b2 is_differentiable_on b2 &
    (for b4 being Element of REAL
          st b4 in b2
       holds ((((- (1 / b1)) (#) (cosec * b3)) - id b2) `| b2) . b4 = ((cos . (b1 * b4)) - ((sin . (b1 * b4)) ^2)) / ((sin . (b1 * b4)) ^2));

:: FDIFF_9:th 28
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= dom (b4 (#) sec) &
         (for b5 being Element of REAL
               st b5 in b3
            holds b4 . b5 = (b1 * b5) + b2)
   holds b4 (#) sec is_differentiable_on b3 &
    (for b5 being Element of REAL
          st b5 in b3
       holds ((b4 (#) sec) `| b3) . b5 = (b1 / (cos . b5)) + ((((b1 * b5) + b2) * (sin . b5)) / ((cos . b5) ^2)));

:: FDIFF_9:th 29
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= dom (b4 (#) cosec) &
         (for b5 being Element of REAL
               st b5 in b3
            holds b4 . b5 = (b1 * b5) + b2)
   holds b4 (#) cosec is_differentiable_on b3 &
    (for b5 being Element of REAL
          st b5 in b3
       holds ((b4 (#) cosec) `| b3) . b5 = (b1 / (sin . b5)) - ((((b1 * b5) + b2) * (cos . b5)) / ((sin . b5) ^2)));

:: FDIFF_9:th 30
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (ln (#) sec)
   holds ln (#) sec is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((ln (#) sec) `| b1) . b2 = ((1 / (cos . b2)) / b2) + (((ln . b2) * (sin . b2)) / ((cos . b2) ^2)));

:: FDIFF_9:th 31
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (ln (#) cosec)
   holds ln (#) cosec is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((ln (#) cosec) `| b1) . b2 = ((1 / (sin . b2)) / b2) - (((ln . b2) * (cos . b2)) / ((sin . b2) ^2)));

:: FDIFF_9:th 32
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= dom (b2 ^ (#) sec) &
         (for b3 being Element of REAL
               st b3 in b1
            holds b2 . b3 = b3)
   holds b2 ^ (#) sec is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds ((b2 ^ (#) sec) `| b1) . b3 = (- ((1 / (cos . b3)) / (b3 ^2))) + (((sin . b3) / b3) / ((cos . b3) ^2)));

:: FDIFF_9:th 33
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= dom (b2 ^ (#) cosec) &
         (for b3 being Element of REAL
               st b3 in b1
            holds b2 . b3 = b3)
   holds b2 ^ (#) cosec is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds ((b2 ^ (#) cosec) `| b1) . b3 = (- ((1 / (sin . b3)) / (b3 ^2))) - (((cos . b3) / b3) / ((sin . b3) ^2)));

:: FDIFF_9:th 34
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (sec * sin)
   holds sec * sin is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sec * sin) `| b1) . b2 = ((cos . b2) * (sin . (sin . b2))) / ((cos . (sin . b2)) ^2));

:: FDIFF_9:th 35
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (sec * cos)
   holds sec * cos is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sec * cos) `| b1) . b2 = - (((sin . b2) * (sin . (cos . b2))) / ((cos . (cos . b2)) ^2)));

:: FDIFF_9:th 36
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cosec * sin)
   holds cosec * sin is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cosec * sin) `| b1) . b2 = - (((cos . b2) * (cos . (sin . b2))) / ((sin . (sin . b2)) ^2)));

:: FDIFF_9:th 37
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cosec * cos)
   holds cosec * cos is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cosec * cos) `| b1) . b2 = ((sin . b2) * (cos . (cos . b2))) / ((sin . (cos . b2)) ^2));

:: FDIFF_9:th 38
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (sec * tan)
   holds sec * tan is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sec * tan) `| b1) . b2 = ((sin . (tan . b2)) / ((cos . b2) ^2)) / ((cos . (tan . b2)) ^2));

:: FDIFF_9:th 39
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (sec * cot)
   holds sec * cot is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sec * cot) `| b1) . b2 = - (((sin . (cot . b2)) / ((sin . b2) ^2)) / ((cos . (cot . b2)) ^2)));

:: FDIFF_9:th 40
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cosec * tan)
   holds cosec * tan is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cosec * tan) `| b1) . b2 = - (((cos . (tan . b2)) / ((cos . b2) ^2)) / ((sin . (tan . b2)) ^2)));

:: FDIFF_9:th 41
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cosec * cot)
   holds cosec * cot is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cosec * cot) `| b1) . b2 = ((cos . (cot . b2)) / ((sin . b2) ^2)) / ((sin . (cot . b2)) ^2));

:: FDIFF_9:th 42
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (tan (#) sec)
   holds tan (#) sec is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((tan (#) sec) `| b1) . b2 = ((1 / ((cos . b2) ^2)) / (cos . b2)) + (((tan . b2) * (sin . b2)) / ((cos . b2) ^2)));

:: FDIFF_9:th 43
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cot (#) sec)
   holds cot (#) sec is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cot (#) sec) `| b1) . b2 = (- ((1 / ((sin . b2) ^2)) / (cos . b2))) + (((cot . b2) * (sin . b2)) / ((cos . b2) ^2)));

:: FDIFF_9:th 44
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (tan (#) cosec)
   holds tan (#) cosec is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((tan (#) cosec) `| b1) . b2 = ((1 / ((cos . b2) ^2)) / (sin . b2)) - (((tan . b2) * (cos . b2)) / ((sin . b2) ^2)));

:: FDIFF_9:th 45
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cot (#) cosec)
   holds cot (#) cosec is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cot (#) cosec) `| b1) . b2 = (- ((1 / ((sin . b2) ^2)) / (sin . b2))) - (((cot . b2) * (cos . b2)) / ((sin . b2) ^2)));