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 ) & 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 ) & 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 ) & 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 A1: ( Z c= dom (f1 - f2) & ( for x being Real st x in Z holds
f1 . x = a ) & 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) ) )
then Z c= (dom f1) /\ (dom f2) by VALUED_1:12;
then A2: ( Z c= dom f1 & Z c= dom f2 ) by XBOOLE_1:18;
A3: for x being Real st x in Z holds
f1 . x = (0 * x) + a by A1;
then A4: ( f1 is_differentiable_on Z & ( for x being Real st x in Z holds
(f1 `| Z) . x = 0 ) ) by A2, FDIFF_1:31;
for x being Real st x in Z holds
f2 is_differentiable_in x by A1, TAYLOR_1:2;
then A5: f2 is_differentiable_on Z by A2, FDIFF_1:16;
A6: for x being Real st x in Z holds
(f2 `| Z) . x = 2 * (x #Z (2 - 1))

proof

let x be Real; :: thesis: ( x in Z implies (f2 `| Z) . x = 2 * (x #Z (2 - 1)) )
assume A7: x in Z ; :: thesis: (f2 `| Z) . x = 2 * (x #Z (2 - 1))
diff f2,x = 2 * (x #Z (2 - 1)) by A1, TAYLOR_1:2;
hence (f2 `| Z) . x = 2 * (x #Z (2 - 1)) by A5, A7, FDIFF_1:def 8; :: thesis: verum

end;

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 A8: x in Z ; :: thesis: ((f1 - f2) `| Z) . x = - (2 * x)
then ((f1 - f2) `| Z) . x = (diff f1,x) - (diff f2,x) by A1, A4, A5, FDIFF_1:27;
hence ((f1 - f2) `| Z) . x = ((f1 `| Z) . x) - (diff f2,x) by A4, A8, FDIFF_1:def 8
.= ((f1 `| Z) . x) - ((f2 `| Z) . x) by A5, A8, FDIFF_1:def 8
.= 0 - ((f2 `| Z) . x) by A2, A3, A8, FDIFF_1:31
.= 0 - (2 * (x #Z (2 - 1))) by A6, A8
.= 0 - (2 * (x |^ 1)) by PREPOWER:46
.= - (2 * (x |^ 1))
.= - (2 * x) by NEWTON:10 ;
:: 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, A4, A5, FDIFF_1:27; :: thesis: verum