let a be Real; :: thesis: for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 - f2) & ( for x being Real st x in Z holds
f1 . x = a ^2 ) & f2 = #Z 2 holds
( f1 - f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 - f2) `| Z) . x = - (2 * x) ) )
let Z be open Subset of REAL; :: thesis: for f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 - f2) & ( for x being Real st x in Z holds
f1 . x = a ^2 ) & f2 = #Z 2 holds
( f1 - f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 - f2) `| Z) . x = - (2 * x) ) )
let f1, f2 be PartFunc of REAL,REAL; :: thesis: ( Z c= dom (f1 - f2) & ( for x being Real st x in Z holds
f1 . x = a ^2 ) & f2 = #Z 2 implies ( f1 - f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 - f2) `| Z) . x = - (2 * x) ) ) )
assume that
A1: Z c= dom (f1 - f2) and
A2: for x being Real st x in Z holds
f1 . x = a ^2 and
A3: f2 = #Z 2 ; :: thesis: ( f1 - f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 - f2) `| Z) . x = - (2 * x) ) )
A4: f1 + ((- 1) (#) f2) = f1 - f2 ;
A5: for x being Real st x in Z holds
f1 . x = (a ^2) + (0 * x) by A2;
for x being Real st x in Z holds
((f1 - f2) `| Z) . x = - (2 * x)

proof

let x be Real; :: thesis: ( x in Z implies ((f1 - f2) `| Z) . x = - (2 * x) )
assume x in Z ; :: thesis: ((f1 - f2) `| Z) . x = - (2 * x)
hence ((f1 - f2) `| Z) . x = 0 + ((2 * (- 1)) * x) by A1, A3, A5, FDIFF_4:12
.= - (2 * x) ;
:: thesis: verum

end;

hence ( f1 - f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 - f2) `| Z) . x = - (2 * x) ) ) by A1, A3, A4, A5, FDIFF_4:12; :: thesis: verum