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: Z c= dom (f ^ ) by A1, VALUED_1:def 5;
dom (f ^ ) c= dom f by RFUNCT_1:11;
then A4: Z c= dom f by A3, XBOOLE_1:1;
then A5: ( f ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((f ^ ) `| Z) . x = - (1 / ((a + x) ^2 )) ) ) by A2, Th14;

now

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_1:28
.= (- 1) * (((f ^ ) `| Z) . x) by A5, A6, FDIFF_1:def 8
.= (- 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_1:28; :: thesis: verum