let a, b, c be Real; 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; 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; ( 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 that
A1:
Z c= dom (f1 + (c (#) f2))
and
A2:
for x being Real st x in Z holds
f1 . x = a + (b * x)
and
A3:
f2 = #Z 2
; ( 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) ) )
A4:
for x being Real st x in Z holds
f2 is_differentiable_in x
by A3, TAYLOR_1:2;
A5:
Z c= (dom f1) /\ (dom (c (#) f2))
by A1, VALUED_1:def 1;
then A6:
Z c= dom f1
by XBOOLE_1:18;
A7:
for x being Real st x in Z holds
f1 . x = (b * x) + a
by A2;
then A8:
f1 is_differentiable_on Z
by A6, FDIFF_1:23;
A9:
Z c= dom (c (#) f2)
by A5, XBOOLE_1:18;
then
Z c= dom f2
by VALUED_1:def 5;
then A10:
f2 is_differentiable_on Z
by A4, FDIFF_1:9;
then A11:
c (#) f2 is_differentiable_on Z
by A9, FDIFF_1:20;
A12:
for x being Real st x in Z holds
(f2 `| Z) . x = 2 * x
A14:
for x being Real st x in Z holds
((c (#) f2) `| Z) . x = (2 * c) * x
for x being Real st x in Z holds
((f1 + (c (#) f2)) `| Z) . x = b + ((2 * c) * x)
proof
let x be
Real;
( x in Z implies ((f1 + (c (#) f2)) `| Z) . x = b + ((2 * c) * x) )
assume A16:
x in Z
;
((f1 + (c (#) f2)) `| Z) . x = b + ((2 * c) * x)
then ((f1 + (c (#) f2)) `| Z) . x =
(diff (f1,x)) + (diff ((c (#) f2),x))
by A1, A8, A11, FDIFF_1:18
.=
((f1 `| Z) . x) + (diff ((c (#) f2),x))
by A8, A16, FDIFF_1:def 7
.=
((f1 `| Z) . x) + (((c (#) f2) `| Z) . x)
by A11, A16, FDIFF_1:def 7
.=
b + (((c (#) f2) `| Z) . x)
by A6, A7, A16, FDIFF_1:23
.=
b + ((2 * c) * x)
by A14, A16
;
hence
((f1 + (c (#) f2)) `| Z) . x = b + ((2 * c) * x)
;
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, A8, A11, FDIFF_1:18; verum