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_1:23;

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;

((f ^) `| Z) . x = - (1 / ((a + x) ^2)) ) ) by A4, A5, FDIFF_2:22; :: thesis: verum

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_1:23;

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 :: thesis: for x being Real st x in Z holds

((f ^) `| Z) . x = - (1 / ((a + x) ^2))

hence
( f ^ is_differentiable_on Z & ( 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 A2, A4, FDIFF_1:9;

((f ^) `| Z) . x = diff ((f ^),x) by A6, A7, FDIFF_1:def 7

.= - ((diff (f,x)) / ((f . x) ^2)) by A8, FDIFF_2:15

.= - (((f `| Z) . x) / ((f . x) ^2)) by A4, A7, FDIFF_1:def 7

.= - (1 / ((f . x) ^2)) by A1, A3, A7, FDIFF_1:23

.= - (1 / ((a + x) ^2)) by A2, A7 ;

hence ((f ^) `| Z) . x = - (1 / ((a + x) ^2)) ; :: thesis: verum

end;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_1:9;

((f ^) `| Z) . x = diff ((f ^),x) by A6, A7, FDIFF_1:def 7

.= - ((diff (f,x)) / ((f . x) ^2)) by A8, FDIFF_2:15

.= - (((f `| Z) . x) / ((f . x) ^2)) by A4, A7, FDIFF_1:def 7

.= - (1 / ((f . x) ^2)) by A1, A3, A7, FDIFF_1:23

.= - (1 / ((a + x) ^2)) by A2, A7 ;

hence ((f ^) `| Z) . x = - (1 / ((a + x) ^2)) ; :: thesis: verum

((f ^) `| Z) . x = - (1 / ((a + x) ^2)) ) ) by A4, A5, FDIFF_2:22; :: thesis: verum