let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL ,REAL st Z c= dom f & ex r being Real st rng f = {r} 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 & ex r being Real st rng f = {r} 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 ; :: thesis: ( for r being Real holds not rng f = {r} or ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = 0 ) ) )
given r being Real such that A2: rng f = {r} ; :: thesis: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = 0 ) )

A3: now

let x0 be Real; :: thesis: ( x0 in dom f implies f . x0 = r )
assume x0 in dom f ; :: thesis: f . x0 = r
then f . x0 in {r} by A2, FUNCT_1:def 5;
hence f . x0 = r by TARSKI:def 1; :: thesis: verum

end;

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;
A4: 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 )

A5: now

let n be Nat; :: thesis: ((h " ) (#) (cf /* h)) . n = 0
X: n in NAT by ORDINAL1:def 13;
A6: rng h c= dom cf by A4;
thus ((h " ) (#) (cf /* h)) . n = ((h " ) . n) * ((cf /* h) . n) by X, SEQ_1:12
.= ((h " ) . n) * (cf . (h . n)) by A6, X, FUNCT_2:185
.= ((h " ) . n) * 0 by FUNCOP_1:13
.= 0 ; :: thesis: verum

end;

then A7: (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 A5;
hence lim ((h " ) (#) (cf /* h)) = 0 by A7, SEQ_4:40; :: thesis: verum

end;

then reconsider R = cf as REST by Def3;

A8: now

let x0 be Real; :: thesis: ( x0 in Z implies f is_differentiable_in x0 )
assume A9: x0 in Z ; :: thesis: f is_differentiable_in x0
then consider N being Neighbourhood of x0 such that
A10: N c= Z by 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))
hence (f . x) - (f . x0) = r - (f . x0) by A3, A11
.= r - r by A1, A3, 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: f is_differentiable_in x0 by A8;
then ex N being Neighbourhood of x0 st
( N c= dom f & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) ) by Def5;
then consider N being Neighbourhood of x0 such that
A15: N c= dom f ;
A16: 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))
hence (f . x) - (f . x0) = r - (f . x0) by A3, A15
.= r - r by A1, A3, A13
.= (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 A14, A15, A16, Def6
.= 0 by FUNCOP_1:13 ; :: thesis: verum