let r be Real; :: thesis: for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( ((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) )
let Z be open Subset of REAL; :: thesis: for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( ((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) )
let f, f1, f2 be PartFunc of REAL,REAL; :: thesis: ( Z c= dom (((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) implies ( ((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) ) )
assume that
A1: Z c= dom (((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) and
A2: r <> 0 and
A3: for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) and
A4: for x being Real st x in Z holds
f1 . x = 1 and
A5: f2 = (#Z 2) * f and
A6: for x being Real st x in Z holds
f . x = x / r ; :: thesis: ( ((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) )
A7: Z c= (dom ((id Z) (#) (arccot * f))) /\ (dom ((r / 2) (#) (ln * (f1 + f2)))) by A1, VALUED_1:def 1;
then A8: Z c= dom ((r / 2) (#) (ln * (f1 + f2))) by XBOOLE_1:18;
then A9: (r / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z by A2, A4, A5, A6, Th108;
A10: Z c= dom ((id Z) (#) (arccot * f)) by A7, XBOOLE_1:18;
then A11: (id Z) (#) (arccot * f) is_differentiable_on Z by A3, Th106;
for x being Real st x in Z holds
((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r)

proof

let x be Real; :: thesis: ( x in Z implies ((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) )
assume A12: x in Z ; :: thesis: ((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r)
hence ((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = (diff (((id Z) (#) (arccot * f)),x)) + (diff (((r / 2) (#) (ln * (f1 + f2))),x)) by A1, A11, A9, FDIFF_1:18
.= ((((id Z) (#) (arccot * f)) `| Z) . x) + (diff (((r / 2) (#) (ln * (f1 + f2))),x)) by A11, A12, FDIFF_1:def 7
.= ((((id Z) (#) (arccot * f)) `| Z) . x) + ((((r / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A9, A12, FDIFF_1:def 7
.= ((arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2))))) + ((((r / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A3, A10, A12, Th106
.= ((arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2))))) + (x / (r * (1 + ((x / r) ^2)))) by A2, A4, A5, A6, A8, A12, Th108
.= arccot . (x / r) ;
:: thesis: verum

end;

hence ( ((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) ) by A1, A11, A9, FDIFF_1:18; :: thesis: verum