let a be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((- 1) (#) (f ^)) & ( for x being Real st x in Z holds
( f . x = a + x & f . x <> 0 ) ) holds
( (- 1) (#) (f ^) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- 1) (#) (f ^)) `| Z) . x = 1 / ((a + x) ^2) ) )
let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom ((- 1) (#) (f ^)) & ( for x being Real st x in Z holds
( f . x = a + x & f . x <> 0 ) ) holds
( (- 1) (#) (f ^) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- 1) (#) (f ^)) `| Z) . x = 1 / ((a + x) ^2) ) )
let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom ((- 1) (#) (f ^)) & ( for x being Real st x in Z holds
( f . x = a + x & f . x <> 0 ) ) implies ( (- 1) (#) (f ^) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- 1) (#) (f ^)) `| Z) . x = 1 / ((a + x) ^2) ) ) )
assume that
A1: Z c= dom ((- 1) (#) (f ^)) and
A2: for x being Real st x in Z holds
( f . x = a + x & f . x <> 0 ) ; :: thesis: ( (- 1) (#) (f ^) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- 1) (#) (f ^)) `| Z) . x = 1 / ((a + x) ^2) ) )
A3: dom (f ^) c= dom f by RFUNCT_1:1;
Z c= dom (f ^) by A1, VALUED_1:def 5;
then A4: Z c= dom f by A3, XBOOLE_1:1;
then A5: f ^ is_differentiable_on Z by A2, Th14;

now :: thesis: for x being Real st x in Z holds
(((- 1) (#) (f ^)) `| Z) . x = 1 / ((a + x) ^2)

let x be Real; :: thesis: ( x in Z implies (((- 1) (#) (f ^)) `| Z) . x = 1 / ((a + x) ^2) )
assume A6: x in Z ; :: thesis: (((- 1) (#) (f ^)) `| Z) . x = 1 / ((a + x) ^2)
hence (((- 1) (#) (f ^)) `| Z) . x = (- 1) * (diff ((f ^),x)) by A1, A5, FDIFF_1:20
.= (- 1) * (((f ^) `| Z) . x) by A5, A6, FDIFF_1:def 7
.= (- 1) * (- (1 / ((a + x) ^2))) by A2, A4, A6, Th14
.= 1 / ((a + x) ^2) ;
:: thesis: verum

end;

hence ( (- 1) (#) (f ^) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- 1) (#) (f ^)) `| Z) . x = 1 / ((a + x) ^2) ) ) by A1, A5, FDIFF_1:20; :: thesis: verum