let a be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL ,REAL st Z c= dom f & ( for x being Real st x in Z holds
( f . x = a + x & f . x <> 0 ) ) holds
( f ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((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 f & ( for x being Real st x in Z holds
( f . x = a + x & f . x <> 0 ) ) holds
( f ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((f ^ ) `| Z) . x = - (1 / ((a + x) ^2 )) ) )
let f be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom f & ( for x being Real st x in Z holds
( f . x = a + x & f . x <> 0 ) ) implies ( f ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((f ^ ) `| Z) . x = - (1 / ((a + x) ^2 )) ) ) )
assume that
A1: Z c= dom f and
A2: for x being Real st x in Z holds
( f . x = a + x & f . x <> 0 ) ; :: thesis: ( f ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((f ^ ) `| Z) . x = - (1 / ((a + x) ^2 )) ) )
A3: for x being Real st x in Z holds
f . x = (1 * x) + a by A2;
then A4: f is_differentiable_on Z by A1, FDIFF_1:31;
A5: for x being Real st x in Z holds
f . x <> 0 by A2;
then A6: f ^ is_differentiable_on Z by A4, FDIFF_2:22;

now

let x be Real; :: thesis: ( x in Z implies ((f ^ ) `| Z) . x = - (1 / ((a + x) ^2 )) )
assume A7: x in Z ; :: thesis: ((f ^ ) `| Z) . x = - (1 / ((a + x) ^2 ))
then A8: ( f . x <> 0 & f is_differentiable_in x ) by A2, A4, FDIFF_1:16;
((f ^ ) `| Z) . x = diff (f ^ ),x by A6, A7, FDIFF_1:def 8
.= - ((diff f,x) / ((f . x) ^2 )) by A8, FDIFF_2:15
.= - (((f `| Z) . x) / ((f . x) ^2 )) by A4, A7, FDIFF_1:def 8
.= - (1 / ((f . x) ^2 )) by A1, A3, A7, FDIFF_1:31
.= - (1 / ((a + x) ^2 )) by A2, A7 ;
hence ((f ^ ) `| Z) . x = - (1 / ((a + x) ^2 )) ; :: thesis: verum

end;

hence ( f ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((f ^ ) `| Z) . x = - (1 / ((a + x) ^2 )) ) ) by A4, A5, FDIFF_2:22; :: thesis: verum