let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL ,REAL st Z c= dom f & f | Z is V8() holds
( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = 0 ) )
let f be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom f & f | Z is V8() implies ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = 0 ) ) )
assume A1: ( Z c= dom f & f | Z is V8() ) ; :: thesis: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = 0 ) )
then consider r being Real such that
A2: for x being Real st x in Z /\ (dom f) holds
f . x = r by PARTFUN2:76;
set L = cf;
for p being Real holds cf . p = 0 * p by FUNCOP_1:13;
then reconsider L = cf as LINEAR by Def4;
set R = cf;
A3: dom cf = REAL by FUNCOP_1:19;

now

let h be convergent_to_0 Real_Sequence; :: thesis: ( (h " ) (#) (cf /* h) is convergent & lim ((h " ) (#) (cf /* h)) = 0 )

A4: now

let n be Nat; :: thesis: ((h " ) (#) (cf /* h)) . n = 0
X: n in NAT by ORDINAL1:def 13;
A5: rng h c= dom cf by A3;
thus ((h " ) (#) (cf /* h)) . n = ((h " ) . n) * ((cf /* h) . n) by X, SEQ_1:12
.= ((h " ) . n) * (cf . (h . n)) by A5, X, FUNCT_2:185
.= ((h " ) . n) * 0 by FUNCOP_1:13
.= 0 ; :: thesis: verum

end;

then A6: (h " ) (#) (cf /* h) is V8() by VALUED_0:def 18;
hence (h " ) (#) (cf /* h) is convergent ; :: thesis: lim ((h " ) (#) (cf /* h)) = 0
((h " ) (#) (cf /* h)) . 0 = 0 by A4;
hence lim ((h " ) (#) (cf /* h)) = 0 by A6, SEQ_4:40; :: thesis: verum

end;

then reconsider R = cf as REST by Def3;

A7: now

let x0 be Real; :: thesis: ( x0 in Z implies f is_differentiable_in x0 )
assume A8: x0 in Z ; :: thesis: f is_differentiable_in x0
then A9: x0 in Z /\ (dom f) by A1, XBOOLE_0:def 4;
consider N being Neighbourhood of x0 such that
A10: N c= Z by A8, RCOMP_1:39;
A11: N c= dom f by A1, A10, XBOOLE_1:1;
for x being Real st x in N holds
(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))

proof

let x be Real; :: thesis: ( x in N implies (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) )
assume x in N ; :: thesis: (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))
then x in Z /\ (dom f) by A10, A11, XBOOLE_0:def 4;
hence (f . x) - (f . x0) = r - (f . x0) by A2
.= r - r by A2, A9
.= (L . (x - x0)) + 0 by FUNCOP_1:13
.= (L . (x - x0)) + (R . (x - x0)) by FUNCOP_1:13 ;
:: thesis: verum

end;

hence f is_differentiable_in x0 by A11, Def5; :: thesis: verum

end;

hence A12: f is_differentiable_on Z by A1, Th16; :: thesis: for x being Real st x in Z holds
(f `| Z) . x = 0
let x0 be Real; :: thesis: ( x0 in Z implies (f `| Z) . x0 = 0 )
assume A13: x0 in Z ; :: thesis: (f `| Z) . x0 = 0
then A14: x0 in Z /\ (dom f) by A1, XBOOLE_0:def 4;
A15: f is_differentiable_in x0 by A7, A13;
consider N being Neighbourhood of x0 such that
A16: N c= Z by A13, RCOMP_1:39;
A17: N c= dom f by A1, A16, XBOOLE_1:1;
A18: for x being Real st x in N holds
(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))

proof

let x be Real; :: thesis: ( x in N implies (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) )
assume x in N ; :: thesis: (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))
then x in Z /\ (dom f) by A16, A17, XBOOLE_0:def 4;
hence (f . x) - (f . x0) = r - (f . x0) by A2
.= r - r by A2, A14
.= (L . (x - x0)) + 0 by FUNCOP_1:13
.= (L . (x - x0)) + (R . (x - x0)) by FUNCOP_1:13 ;
:: thesis: verum

end;

thus (f `| Z) . x0 = diff f,x0 by A12, A13, Def8
.= L . 1 by A15, A17, A18, Def6
.= 0 by FUNCOP_1:13 ; :: thesis: verum