let c, a, b be Real; :: thesis: for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL ,REAL st Z c= dom (f1 + (c (#) f2)) & ( for x being Real st x in Z holds
f1 . x = a + (b * x) ) & f2 = #Z 2 holds
( f1 + (c (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + (c (#) f2)) `| Z) . x = b + ((2 * c) * x) ) )
let Z be open Subset of REAL ; :: thesis: for f1, f2 being PartFunc of REAL ,REAL st Z c= dom (f1 + (c (#) f2)) & ( for x being Real st x in Z holds
f1 . x = a + (b * x) ) & f2 = #Z 2 holds
( f1 + (c (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + (c (#) f2)) `| Z) . x = b + ((2 * c) * x) ) )
let f1, f2 be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (f1 + (c (#) f2)) & ( for x being Real st x in Z holds
f1 . x = a + (b * x) ) & f2 = #Z 2 implies ( f1 + (c (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + (c (#) f2)) `| Z) . x = b + ((2 * c) * x) ) ) )
assume A1: ( Z c= dom (f1 + (c (#) f2)) & ( for x being Real st x in Z holds
f1 . x = a + (b * x) ) & f2 = #Z 2 ) ; :: thesis: ( f1 + (c (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + (c (#) f2)) `| Z) . x = b + ((2 * c) * x) ) )
then Z c= (dom f1) /\ (dom (c (#) f2)) by VALUED_1:def 1;
then A2: ( Z c= dom f1 & Z c= dom (c (#) f2) ) by XBOOLE_1:18;
then A3: Z c= dom f2 by VALUED_1:def 5;
A4: for x being Real st x in Z holds
f1 . x = (b * x) + a by A1;
then A5: ( f1 is_differentiable_on Z & ( for x being Real st x in Z holds
(f1 `| Z) . x = b ) ) by A2, FDIFF_1:31;
A6: f2 is_differentiable_on Z

proof

for x being Real st x in Z holds
f2 is_differentiable_in x by A1, TAYLOR_1:2;
hence f2 is_differentiable_on Z by A3, FDIFF_1:16; :: thesis: verum

end;

then A7: c (#) f2 is_differentiable_on Z by A2, FDIFF_1:28;
A8: for x being Real st x in Z holds
(f2 `| Z) . x = 2 * x

proof

let x be Real; :: thesis: ( x in Z implies (f2 `| Z) . x = 2 * x )
assume A9: x in Z ; :: thesis: (f2 `| Z) . x = 2 * x
2 * (x #Z (2 - 1)) = 2 * x by PREPOWER:45;
then diff f2,x = 2 * x by A1, TAYLOR_1:2;
hence (f2 `| Z) . x = 2 * x by A6, A9, FDIFF_1:def 8; :: thesis: verum

end;

A10: for x being Real st x in Z holds
((c (#) f2) `| Z) . x = (2 * c) * x

proof

let x be Real; :: thesis: ( x in Z implies ((c (#) f2) `| Z) . x = (2 * c) * x )
assume A11: x in Z ; :: thesis: ((c (#) f2) `| Z) . x = (2 * c) * x
hence ((c (#) f2) `| Z) . x = c * (diff f2,x) by A2, A6, FDIFF_1:28
.= c * ((f2 `| Z) . x) by A6, A11, FDIFF_1:def 8
.= c * (2 * x) by A8, A11
.= (2 * c) * x ;
:: thesis: verum

end;

for x being Real st x in Z holds
((f1 + (c (#) f2)) `| Z) . x = b + ((2 * c) * x)

proof

let x be Real; :: thesis: ( x in Z implies ((f1 + (c (#) f2)) `| Z) . x = b + ((2 * c) * x) )
assume A12: x in Z ; :: thesis: ((f1 + (c (#) f2)) `| Z) . x = b + ((2 * c) * x)
then ((f1 + (c (#) f2)) `| Z) . x = (diff f1,x) + (diff (c (#) f2),x) by A1, A5, A7, FDIFF_1:26
.= ((f1 `| Z) . x) + (diff (c (#) f2),x) by A5, A12, FDIFF_1:def 8
.= ((f1 `| Z) . x) + (((c (#) f2) `| Z) . x) by A7, A12, FDIFF_1:def 8
.= b + (((c (#) f2) `| Z) . x) by A2, A4, A12, FDIFF_1:31
.= b + ((2 * c) * x) by A10, A12 ;
hence ((f1 + (c (#) f2)) `| Z) . x = b + ((2 * c) * x) ; :: thesis: verum

end;

hence ( f1 + (c (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + (c (#) f2)) `| Z) . x = b + ((2 * c) * x) ) ) by A1, A5, A7, FDIFF_1:26; :: thesis: verum