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 ;
A5: for x being Real st x in Z holds
f . x <> 0 by A2;
then A6: f ^ is_differentiable_on Z by ;
now :: thesis: for x being Real st x in Z holds
((f ^) `| Z) . x = - (1 / ((a + x) ^2))
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 ;
((f ^) `| Z) . x = diff ((f ^),x) by
.= - ((diff (f,x)) / ((f . x) ^2)) by
.= - (((f `| Z) . x) / ((f . x) ^2)) by
.= - (1 / ((f . x) ^2)) by
.= - (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 ; :: thesis: verum