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
proof
let x be Real; :: thesis: ( x in Z implies f . x = ((- 1) * x) + a )
assume x in Z ; :: thesis: f . x = ((- 1) * x) + a
then f . x = a - x by A2;
hence f . x = ((- 1) * x) + a ; :: thesis: verum
end;
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 / ((f . x) ^2))) by XCMPLX_1:187
.= 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