:: FDIFF_7 semantic presentation

theorem Th1: :: FDIFF_7:1
for b1 being Real holds b1 #Z 2 = b1 ^2
proof end;

theorem Th2: :: FDIFF_7:2
for b1 being Real st b1 > 0 holds
b1 #R (1 / 2) = sqrt b1
proof end;

theorem Th3: :: FDIFF_7:3
for b1 being Real st b1 > 0 holds
b1 #R (- (1 / 2)) = 1 / (sqrt b1)
proof end;

Lemma4: for b1 being Real holds 2 * ((cos . (b1 / 2)) ^2 ) = 1 + (cos . b1)
proof end;

Lemma5: for b1 being Real holds 2 * ((sin . (b1 / 2)) ^2 ) = 1 - (cos . b1)
proof end;

theorem Th4: :: FDIFF_7:4
for b1 being Real
for b2 being open Subset of REAL st b2 c= ].(- 1),1.[ & b2 c= dom (b1 (#) arcsin ) holds
( b1 (#) arcsin is_differentiable_on b2 & ( for b3 being Real st b3 in b2 holds
((b1 (#) arcsin ) `| b2) . b3 = b1 / (sqrt (1 - (b3 ^2 ))) ) )
proof end;

theorem Th5: :: FDIFF_7:5
for b1 being Real
for b2 being open Subset of REAL st b2 c= ].(- 1),1.[ & b2 c= dom (b1 (#) arccos ) holds
( b1 (#) arccos is_differentiable_on b2 & ( for b3 being Real st b3 in b2 holds
((b1 (#) arccos ) `| b2) . b3 = - (b1 / (sqrt (1 - (b3 ^2 )))) ) )
proof end;

theorem Th6: :: FDIFF_7:6
for b1 being Real
for b2 being PartFunc of REAL , REAL st b2 is_differentiable_in b1 & b2 . b1 > - 1 & b2 . b1 < 1 holds
( arcsin * b2 is_differentiable_in b1 & diff (arcsin * b2),b1 = (diff b2,b1) / (sqrt (1 - ((b2 . b1) ^2 ))) )
proof end;

theorem Th7: :: FDIFF_7:7
for b1 being Real
for b2 being PartFunc of REAL , REAL st b2 is_differentiable_in b1 & b2 . b1 > - 1 & b2 . b1 < 1 holds
( arccos * b2 is_differentiable_in b1 & diff (arccos * b2),b1 = - ((diff b2,b1) / (sqrt (1 - ((b2 . b1) ^2 )))) )
proof end;

theorem Th8: :: FDIFF_7:8
for b1 being open Subset of REAL st b1 c= dom ((log_ number_e ) * arcsin ) & b1 c= ].(- 1),1.[ & ( for b2 being Real st b2 in b1 holds
arcsin . b2 > 0 ) holds
( (log_ number_e ) * arcsin is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
(((log_ number_e ) * arcsin ) `| b1) . b2 = 1 / ((sqrt (1 - (b2 ^2 ))) * (arcsin . b2)) ) )
proof end;

theorem Th9: :: FDIFF_7:9
for b1 being open Subset of REAL st b1 c= dom ((log_ number_e ) * arccos ) & b1 c= ].(- 1),1.[ & ( for b2 being Real st b2 in b1 holds
arccos . b2 > 0 ) holds
( (log_ number_e ) * arccos is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
(((log_ number_e ) * arccos ) `| b1) . b2 = - (1 / ((sqrt (1 - (b2 ^2 ))) * (arccos . b2))) ) )
proof end;

theorem Th10: :: FDIFF_7:10
for b1 being Nat
for b2 being open Subset of REAL st b2 c= dom ((#Z b1) * arcsin ) & b2 c= ].(- 1),1.[ holds
( (#Z b1) * arcsin is_differentiable_on b2 & ( for b3 being Real st b3 in b2 holds
(((#Z b1) * arcsin ) `| b2) . b3 = (b1 * ((arcsin . b3) #Z (b1 - 1))) / (sqrt (1 - (b3 ^2 ))) ) )
proof end;

theorem Th11: :: FDIFF_7:11
for b1 being Nat
for b2 being open Subset of REAL st b2 c= dom ((#Z b1) * arccos ) & b2 c= ].(- 1),1.[ holds
( (#Z b1) * arccos is_differentiable_on b2 & ( for b3 being Real st b3 in b2 holds
(((#Z b1) * arccos ) `| b2) . b3 = - ((b1 * ((arccos . b3) #Z (b1 - 1))) / (sqrt (1 - (b3 ^2 )))) ) )
proof end;

theorem Th12: :: FDIFF_7:12
for b1 being open Subset of REAL st b1 c= dom ((1 / 2) (#) ((#Z 2) * arcsin )) & b1 c= ].(- 1),1.[ holds
( (1 / 2) (#) ((#Z 2) * arcsin ) is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
(((1 / 2) (#) ((#Z 2) * arcsin )) `| b1) . b2 = (arcsin . b2) / (sqrt (1 - (b2 ^2 ))) ) )
proof end;

theorem Th13: :: FDIFF_7:13
for b1 being open Subset of REAL st b1 c= dom ((1 / 2) (#) ((#Z 2) * arccos )) & b1 c= ].(- 1),1.[ holds
( (1 / 2) (#) ((#Z 2) * arccos ) is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
(((1 / 2) (#) ((#Z 2) * arccos )) `| b1) . b2 = - ((arccos . b2) / (sqrt (1 - (b2 ^2 )))) ) )
proof end;

theorem Th14: :: FDIFF_7:14
for b1, b2 being Real
for b3 being open Subset of REAL
for b4 being PartFunc of REAL , REAL st b3 c= dom (arcsin * b4) & ( for b5 being Real st b5 in b3 holds
( b4 . b5 = (b1 * b5) + b2 & b4 . b5 > - 1 & b4 . b5 < 1 ) ) holds
( arcsin * b4 is_differentiable_on b3 & ( for b5 being Real st b5 in b3 holds
((arcsin * b4) `| b3) . b5 = b1 / (sqrt (1 - (((b1 * b5) + b2) ^2 ))) ) )
proof end;

theorem Th15: :: FDIFF_7:15
for b1, b2 being Real
for b3 being open Subset of REAL
for b4 being PartFunc of REAL , REAL st b3 c= dom (arccos * b4) & ( for b5 being Real st b5 in b3 holds
( b4 . b5 = (b1 * b5) + b2 & b4 . b5 > - 1 & b4 . b5 < 1 ) ) holds
( arccos * b4 is_differentiable_on b3 & ( for b5 being Real st b5 in b3 holds
((arccos * b4) `| b3) . b5 = - (b1 / (sqrt (1 - (((b1 * b5) + b2) ^2 )))) ) )
proof end;

theorem Th16: :: FDIFF_7:16
for b1 being open Subset of REAL st b1 c= dom ((id b1) (#) arcsin ) & b1 c= ].(- 1),1.[ holds
( (id b1) (#) arcsin is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
(((id b1) (#) arcsin ) `| b1) . b2 = (arcsin . b2) + (b2 / (sqrt (1 - (b2 ^2 )))) ) )
proof end;

theorem Th17: :: FDIFF_7:17
for b1 being open Subset of REAL st b1 c= dom ((id b1) (#) arccos ) & b1 c= ].(- 1),1.[ holds
( (id b1) (#) arccos is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
(((id b1) (#) arccos ) `| b1) . b2 = (arccos . b2) - (b2 / (sqrt (1 - (b2 ^2 )))) ) )
proof end;

theorem Th18: :: FDIFF_7:18
for b1, b2 being Real
for b3 being open Subset of REAL
for b4 being PartFunc of REAL , REAL st b3 c= dom (b4 (#) arcsin ) & b3 c= ].(- 1),1.[ & ( for b5 being Real st b5 in b3 holds
b4 . b5 = (b1 * b5) + b2 ) holds
( b4 (#) arcsin is_differentiable_on b3 & ( for b5 being Real st b5 in b3 holds
((b4 (#) arcsin ) `| b3) . b5 = (b1 * (arcsin . b5)) + (((b1 * b5) + b2) / (sqrt (1 - (b5 ^2 )))) ) )
proof end;

theorem Th19: :: FDIFF_7:19
for b1, b2 being Real
for b3 being open Subset of REAL
for b4 being PartFunc of REAL , REAL st b3 c= dom (b4 (#) arccos ) & b3 c= ].(- 1),1.[ & ( for b5 being Real st b5 in b3 holds
b4 . b5 = (b1 * b5) + b2 ) holds
( b4 (#) arccos is_differentiable_on b3 & ( for b5 being Real st b5 in b3 holds
((b4 (#) arccos ) `| b3) . b5 = (b1 * (arccos . b5)) - (((b1 * b5) + b2) / (sqrt (1 - (b5 ^2 )))) ) )
proof end;

theorem Th20: :: FDIFF_7:20
for b1 being open Subset of REAL
for b2 being PartFunc of REAL , REAL st b1 c= dom ((1 / 2) (#) (arcsin * b2)) & ( for b3 being Real st b3 in b1 holds
( b2 . b3 = 2 * b3 & b2 . b3 > - 1 & b2 . b3 < 1 ) ) holds
( (1 / 2) (#) (arcsin * b2) is_differentiable_on b1 & ( for b3 being Real st b3 in b1 holds
(((1 / 2) (#) (arcsin * b2)) `| b1) . b3 = 1 / (sqrt (1 - ((2 * b3) ^2 ))) ) )
proof end;

theorem Th21: :: FDIFF_7:21
for b1 being open Subset of REAL
for b2 being PartFunc of REAL , REAL st b1 c= dom ((1 / 2) (#) (arccos * b2)) & ( for b3 being Real st b3 in b1 holds
( b2 . b3 = 2 * b3 & b2 . b3 > - 1 & b2 . b3 < 1 ) ) holds
( (1 / 2) (#) (arccos * b2) is_differentiable_on b1 & ( for b3 being Real st b3 in b1 holds
(((1 / 2) (#) (arccos * b2)) `| b1) . b3 = - (1 / (sqrt (1 - ((2 * b3) ^2 )))) ) )
proof end;

theorem Th22: :: FDIFF_7:22
for b1 being open Subset of REAL
for b2, b3, b4 being PartFunc of REAL , REAL st b1 c= dom ((#R (1 / 2)) * b2) & b2 = b3 - b4 & b4 = #Z 2 & ( for b5 being Real st b5 in b1 holds
( b3 . b5 = 1 & b2 . b5 > 0 ) ) holds
( (#R (1 / 2)) * b2 is_differentiable_on b1 & ( for b5 being Real st b5 in b1 holds
(((#R (1 / 2)) * b2) `| b1) . b5 = - (b5 * ((1 - (b5 #Z 2)) #R (- (1 / 2)))) ) )
proof end;

theorem Th23: :: FDIFF_7:23
for b1 being open Subset of REAL
for b2, b3, b4 being PartFunc of REAL , REAL st b1 c= dom (((id b1) (#) arcsin ) + ((#R (1 / 2)) * b2)) & b1 c= ].(- 1),1.[ & b2 = b3 - b4 & b4 = #Z 2 & ( for b5 being Real st b5 in b1 holds
( b3 . b5 = 1 & b2 . b5 > 0 & b5 <> 0 ) ) holds
( ((id b1) (#) arcsin ) + ((#R (1 / 2)) * b2) is_differentiable_on b1 & ( for b5 being Real st b5 in b1 holds
((((id b1) (#) arcsin ) + ((#R (1 / 2)) * b2)) `| b1) . b5 = arcsin . b5 ) )
proof end;

theorem Th24: :: FDIFF_7:24
for b1 being open Subset of REAL
for b2, b3, b4 being PartFunc of REAL , REAL st b1 c= dom (((id b1) (#) arccos ) - ((#R (1 / 2)) * b2)) & b1 c= ].(- 1),1.[ & b2 = b3 - b4 & b4 = #Z 2 & ( for b5 being Real st b5 in b1 holds
( b3 . b5 = 1 & b2 . b5 > 0 & b5 <> 0 ) ) holds
( ((id b1) (#) arccos ) - ((#R (1 / 2)) * b2) is_differentiable_on b1 & ( for b5 being Real st b5 in b1 holds
((((id b1) (#) arccos ) - ((#R (1 / 2)) * b2)) `| b1) . b5 = arccos . b5 ) )
proof end;

theorem Th25: :: FDIFF_7:25
for b1 being Real
for b2 being open Subset of REAL
for b3 being PartFunc of REAL , REAL st b2 c= dom ((id b2) (#) (arcsin * b3)) & ( for b4 being Real st b4 in b2 holds
( b3 . b4 = b4 / b1 & b3 . b4 > - 1 & b3 . b4 < 1 ) ) holds
( (id b2) (#) (arcsin * b3) is_differentiable_on b2 & ( for b4 being Real st b4 in b2 holds
(((id b2) (#) (arcsin * b3)) `| b2) . b4 = (arcsin . (b4 / b1)) + (b4 / (b1 * (sqrt (1 - ((b4 / b1) ^2 ))))) ) )
proof end;

theorem Th26: :: FDIFF_7:26
for b1 being Real
for b2 being open Subset of REAL
for b3 being PartFunc of REAL , REAL st b2 c= dom ((id b2) (#) (arccos * b3)) & ( for b4 being Real st b4 in b2 holds
( b3 . b4 = b4 / b1 & b3 . b4 > - 1 & b3 . b4 < 1 ) ) holds
( (id b2) (#) (arccos * b3) is_differentiable_on b2 & ( for b4 being Real st b4 in b2 holds
(((id b2) (#) (arccos * b3)) `| b2) . b4 = (arccos . (b4 / b1)) - (b4 / (b1 * (sqrt (1 - ((b4 / b1) ^2 ))))) ) )
proof end;

theorem Th27: :: FDIFF_7:27
for b1 being Real
for b2 being open Subset of REAL
for b3, b4, b5 being PartFunc of REAL , REAL st b2 c= dom ((#R (1 / 2)) * b3) & b3 = b4 - b5 & b5 = #Z 2 & ( for b6 being Real st b6 in b2 holds
( b4 . b6 = b1 ^2 & b3 . b6 > 0 ) ) holds
( (#R (1 / 2)) * b3 is_differentiable_on b2 & ( for b6 being Real st b6 in b2 holds
(((#R (1 / 2)) * b3) `| b2) . b6 = - (b6 * (((b1 ^2 ) - (b6 #Z 2)) #R (- (1 / 2)))) ) )
proof end;

theorem Th28: :: FDIFF_7:28
for b1 being Real
for b2 being open Subset of REAL
for b3, b4, b5, b6 being PartFunc of REAL , REAL st b2 c= dom (((id b2) (#) (arcsin * b3)) + ((#R (1 / 2)) * b4)) & b2 c= ].(- 1),1.[ & b4 = b5 - b6 & b6 = #Z 2 & ( for b7 being Real st b7 in b2 holds
( b5 . b7 = b1 ^2 & b4 . b7 > 0 & b3 . b7 = b7 / b1 & b3 . b7 > - 1 & b3 . b7 < 1 & b7 <> 0 & b1 > 0 ) ) holds
( ((id b2) (#) (arcsin * b3)) + ((#R (1 / 2)) * b4) is_differentiable_on b2 & ( for b7 being Real st b7 in b2 holds
((((id b2) (#) (arcsin * b3)) + ((#R (1 / 2)) * b4)) `| b2) . b7 = arcsin . (b7 / b1) ) )
proof end;

theorem Th29: :: FDIFF_7:29
for b1 being Real
for b2 being open Subset of REAL
for b3, b4, b5, b6 being PartFunc of REAL , REAL st b2 c= dom (((id b2) (#) (arccos * b3)) - ((#R (1 / 2)) * b4)) & b2 c= ].(- 1),1.[ & b4 = b5 - b6 & b6 = #Z 2 & ( for b7 being Real st b7 in b2 holds
( b5 . b7 = b1 ^2 & b4 . b7 > 0 & b3 . b7 = b7 / b1 & b3 . b7 > - 1 & b3 . b7 < 1 & b7 <> 0 & b1 > 0 ) ) holds
( ((id b2) (#) (arccos * b3)) - ((#R (1 / 2)) * b4) is_differentiable_on b2 & ( for b7 being Real st b7 in b2 holds
((((id b2) (#) (arccos * b3)) - ((#R (1 / 2)) * b4)) `| b2) . b7 = arccos . (b7 / b1) ) )
proof end;

theorem Th30: :: FDIFF_7:30
for b1 being Nat
for b2 being open Subset of REAL st b2 c= dom ((- (1 / b1)) (#) ((#Z b1) * (sin ^ ))) & b1 > 0 & ( for b3 being Real st b3 in b2 holds
sin . b3 <> 0 ) holds
( (- (1 / b1)) (#) ((#Z b1) * (sin ^ )) is_differentiable_on b2 & ( for b3 being Real st b3 in b2 holds
(((- (1 / b1)) (#) ((#Z b1) * (sin ^ ))) `| b2) . b3 = (cos . b3) / ((sin . b3) #Z (b1 + 1)) ) )
proof end;

theorem Th31: :: FDIFF_7:31
for b1 being Nat
for b2 being open Subset of REAL st b2 c= dom ((1 / b1) (#) ((#Z b1) * (cos ^ ))) & b1 > 0 & ( for b3 being Real st b3 in b2 holds
cos . b3 <> 0 ) holds
( (1 / b1) (#) ((#Z b1) * (cos ^ )) is_differentiable_on b2 & ( for b3 being Real st b3 in b2 holds
(((1 / b1) (#) ((#Z b1) * (cos ^ ))) `| b2) . b3 = (sin . b3) / ((cos . b3) #Z (b1 + 1)) ) )
proof end;

theorem Th32: :: FDIFF_7:32
for b1 being open Subset of REAL st b1 c= dom (sin * (log_ number_e )) & ( for b2 being Real st b2 in b1 holds
b2 > 0 ) holds
( sin * (log_ number_e ) is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
((sin * (log_ number_e )) `| b1) . b2 = (cos . ((log_ number_e ) . b2)) / b2 ) )
proof end;

theorem Th33: :: FDIFF_7:33
for b1 being open Subset of REAL st b1 c= dom (cos * (log_ number_e )) & ( for b2 being Real st b2 in b1 holds
b2 > 0 ) holds
( cos * (log_ number_e ) is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
((cos * (log_ number_e )) `| b1) . b2 = - ((sin . ((log_ number_e ) . b2)) / b2) ) )
proof end;

theorem Th34: :: FDIFF_7:34
for b1 being open Subset of REAL st b1 c= dom (sin * exp_R ) holds
( sin * exp_R is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
((sin * exp_R ) `| b1) . b2 = (exp_R . b2) * (cos . (exp_R . b2)) ) )
proof end;

theorem Th35: :: FDIFF_7:35
for b1 being open Subset of REAL st b1 c= dom (cos * exp_R ) holds
( cos * exp_R is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
((cos * exp_R ) `| b1) . b2 = - ((exp_R . b2) * (sin . (exp_R . b2))) ) )
proof end;

theorem Th36: :: FDIFF_7:36
for b1 being open Subset of REAL st b1 c= dom (exp_R * cos ) holds
( exp_R * cos is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
((exp_R * cos ) `| b1) . b2 = - ((exp_R . (cos . b2)) * (sin . b2)) ) )
proof end;

theorem Th37: :: FDIFF_7:37
for b1 being open Subset of REAL st b1 c= dom (exp_R * sin ) holds
( exp_R * sin is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
((exp_R * sin ) `| b1) . b2 = (exp_R . (sin . b2)) * (cos . b2) ) )
proof end;

theorem Th38: :: FDIFF_7:38
for b1 being open Subset of REAL st b1 c= dom (sin + cos ) holds
( sin + cos is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
((sin + cos ) `| b1) . b2 = (cos . b2) - (sin . b2) ) )
proof end;

theorem Th39: :: FDIFF_7:39
for b1 being open Subset of REAL st b1 c= dom (sin - cos ) holds
( sin - cos is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
((sin - cos ) `| b1) . b2 = (cos . b2) + (sin . b2) ) )
proof end;

theorem Th40: :: FDIFF_7:40
for b1 being open Subset of REAL st b1 c= dom (exp_R (#) (sin - cos )) holds
( exp_R (#) (sin - cos ) is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
((exp_R (#) (sin - cos )) `| b1) . b2 = (2 * (exp_R . b2)) * (sin . b2) ) )
proof end;

theorem Th41: :: FDIFF_7:41
for b1 being open Subset of REAL st b1 c= dom (exp_R (#) (sin + cos )) holds
( exp_R (#) (sin + cos ) is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
((exp_R (#) (sin + cos )) `| b1) . b2 = (2 * (exp_R . b2)) * (cos . b2) ) )
proof end;

theorem Th42: :: FDIFF_7:42
for b1 being open Subset of REAL st b1 c= dom ((sin + cos ) / exp_R ) holds
( (sin + cos ) / exp_R is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
(((sin + cos ) / exp_R ) `| b1) . b2 = - ((2 * (sin . b2)) / (exp_R . b2)) ) )
proof end;

theorem Th43: :: FDIFF_7:43
for b1 being open Subset of REAL st b1 c= dom ((sin - cos ) / exp_R ) holds
( (sin - cos ) / exp_R is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
(((sin - cos ) / exp_R ) `| b1) . b2 = (2 * (cos . b2)) / (exp_R . b2) ) )
proof end;

theorem Th44: :: FDIFF_7:44
for b1 being open Subset of REAL st b1 c= dom (exp_R (#) sin ) holds
( exp_R (#) sin is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
((exp_R (#) sin ) `| b1) . b2 = (exp_R . b2) * ((sin . b2) + (cos . b2)) ) )
proof end;

theorem Th45: :: FDIFF_7:45
for b1 being open Subset of REAL st b1 c= dom (exp_R (#) cos ) holds
( exp_R (#) cos is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
((exp_R (#) cos ) `| b1) . b2 = (exp_R . b2) * ((cos . b2) - (sin . b2)) ) )
proof end;

theorem Th46: :: FDIFF_7:46
for b1 being Real st cos . b1 <> 0 holds
( sin / cos is_differentiable_in b1 & diff (sin / cos ),b1 = 1 / ((cos . b1) ^2 ) )
proof end;

theorem Th47: :: FDIFF_7:47
for b1 being Real st sin . b1 <> 0 holds
( cos / sin is_differentiable_in b1 & diff (cos / sin ),b1 = - (1 / ((sin . b1) ^2 )) )
proof end;

theorem Th48: :: FDIFF_7:48
for b1 being open Subset of REAL st b1 c= dom ((#Z 2) * (sin / cos )) & ( for b2 being Real st b2 in b1 holds
cos . b2 <> 0 ) holds
( (#Z 2) * (sin / cos ) is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
(((#Z 2) * (sin / cos )) `| b1) . b2 = (2 * (sin . b2)) / ((cos . b2) #Z 3) ) )
proof end;

theorem Th49: :: FDIFF_7:49
for b1 being open Subset of REAL st b1 c= dom ((#Z 2) * (cos / sin )) & ( for b2 being Real st b2 in b1 holds
sin . b2 <> 0 ) holds
( (#Z 2) * (cos / sin ) is_differentiable_on b1 & ( for b2 being Real st b2 in b1 holds
(((#Z 2) * (cos / sin )) `| b1) . b2 = - ((2 * (cos . b2)) / ((sin . b2) #Z 3)) ) )
proof end;

theorem Th50: :: FDIFF_7:50
for b1 being open Subset of REAL
for b2 being PartFunc of REAL , REAL st b1 c= dom ((sin / cos ) * b2) & ( for b3 being Real st b3 in b1 holds
( b2 . b3 = b3 / 2 & cos . (b2 . b3) <> 0 ) ) holds
( (sin / cos ) * b2 is_differentiable_on b1 & ( for b3 being Real st b3 in b1 holds
(((sin / cos ) * b2) `| b1) . b3 = 1 / (1 + (cos . b3)) ) )
proof end;

theorem Th51: :: FDIFF_7:51
for b1 being open Subset of REAL
for b2 being PartFunc of REAL , REAL st b1 c= dom ((cos / sin ) * b2) & ( for b3 being Real st b3 in b1 holds
( b2 . b3 = b3 / 2 & sin . (b2 . b3) <> 0 ) ) holds
( (cos / sin ) * b2 is_differentiable_on b1 & ( for b3 being Real st b3 in b1 holds
(((cos / sin ) * b2) `| b1) . b3 = - (1 / (1 - (cos . b3))) ) )
proof end;