let r be Real; :: thesis: for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) )
let Z be open Subset of REAL; :: thesis: for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) )
let f, f1, f2 be PartFunc of REAL,REAL; :: thesis: ( Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) implies ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) ) )
assume that
A1: Z c= dom (f1 + f2) and
A2: for x being Real st x in Z holds
f1 . x = 1 and
A3: f2 = (#Z 2) * f and
A4: for x being Real st x in Z holds
f . x = x / r ; :: thesis: ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) )
A5: for x being Real st x in Z holds
f1 . x = (0 * x) + 1 by A2;
A6: Z c= (dom f1) /\ (dom f2) by A1, VALUED_1:def 1;
then A7: Z c= dom f1 by XBOOLE_1:18;
then A8: f1 is_differentiable_on Z by A5, FDIFF_1:23;
A9: for x being Real st x in Z holds
f . x = ((1 / r) * x) + 0
proof
let x be Real; :: thesis: ( x in Z implies f . x = ((1 / r) * x) + 0 )
assume x in Z ; :: thesis: f . x = ((1 / r) * x) + 0
hence f . x = x / r by A4
.= ((1 / r) * x) + 0 ;
:: thesis: verum
end;
A10: for x being Real st x in Z holds
f2 is_differentiable_in x
proof
let x be Real; :: thesis: ( x in Z implies f2 is_differentiable_in x )
Z c= dom ((#Z 2) * f) by A3, A6, XBOOLE_1:18;
then for y being object st y in Z holds
y in dom f by FUNCT_1:11;
then Z c= dom f ;
then A11: f is_differentiable_on Z by A9, FDIFF_1:23;
assume x in Z ; :: thesis: f2 is_differentiable_in x
then f is_differentiable_in x by A11, FDIFF_1:9;
hence f2 is_differentiable_in x by A3, TAYLOR_1:3; :: thesis: verum
end;
Z c= dom f2 by A6, XBOOLE_1:18;
then A12: f2 is_differentiable_on Z by A10, FDIFF_1:9;
A13: for x being Real st x in Z holds
(f2 `| Z) . x = (2 * x) / (r ^2)
proof
let x be Real; :: thesis: ( x in Z implies (f2 `| Z) . x = (2 * x) / (r ^2) )
assume A14: x in Z ; :: thesis: (f2 `| Z) . x = (2 * x) / (r ^2)
Z c= dom ((#Z 2) * f) by A3, A6, XBOOLE_1:18;
then for y being object st y in Z holds
y in dom f by FUNCT_1:11;
then A15: Z c= dom f ;
then A16: f is_differentiable_on Z by A9, FDIFF_1:23;
then A17: f is_differentiable_in x by A14, FDIFF_1:9;
(f2 `| Z) . x = diff (((#Z 2) * f),x) by A3, A12, A14, FDIFF_1:def 7
.= (2 * ((f . x) #Z (2 - 1))) * (diff (f,x)) by A17, TAYLOR_1:3
.= (2 * (f . x)) * (diff (f,x)) by PREPOWER:35
.= (2 * (x / r)) * (diff (f,x)) by A4, A14
.= (2 * (x / r)) * ((f `| Z) . x) by A14, A16, FDIFF_1:def 7
.= (2 * (x / r)) * (1 / r) by A9, A14, A15, FDIFF_1:23
.= 2 * ((x / r) * (1 / r))
.= 2 * ((x * 1) / (r * r)) by XCMPLX_1:76
.= (2 * x) / (r ^2) ;
hence (f2 `| Z) . x = (2 * x) / (r ^2) ; :: thesis: verum
end;
for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2)
proof
let x be Real; :: thesis: ( x in Z implies ((f1 + f2) `| Z) . x = (2 * x) / (r ^2) )
assume A18: x in Z ; :: thesis: ((f1 + f2) `| Z) . x = (2 * x) / (r ^2)
then ((f1 + f2) `| Z) . x = (diff (f1,x)) + (diff (f2,x)) by A1, A8, A12, FDIFF_1:18
.= ((f1 `| Z) . x) + (diff (f2,x)) by A8, A18, FDIFF_1:def 7
.= ((f1 `| Z) . x) + ((f2 `| Z) . x) by A12, A18, FDIFF_1:def 7
.= 0 + ((f2 `| Z) . x) by A7, A5, A18, FDIFF_1:23
.= (2 * x) / (r ^2) by A13, A18 ;
hence ((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ; :: 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) / (r ^2) ) ) by A1, A8, A12, FDIFF_1:18; :: thesis: verum