let f be PartFunc of COMPLEX ,COMPLEX ; :: thesis: for Z being open Subset of COMPLEX st Z c= dom f & f | Z is constant holds
( f is_differentiable_on Z & ( for x being Complex st x in Z holds
(f `| Z) /. x = 0c ) )
let Z be open Subset of COMPLEX ; :: thesis: ( Z c= dom f & f | Z is constant implies ( f is_differentiable_on Z & ( for x being Complex st x in Z holds
(f `| Z) /. x = 0c ) ) )
assume A1: ( Z c= dom f & f | Z is constant ) ; :: thesis: ( f is_differentiable_on Z & ( for x being Complex st x in Z holds
(f `| Z) /. x = 0c ) )
then consider a1 being Complex such that
A2: for x being Complex st x in Z /\ (dom f) holds
f /. x = a1 by PARTFUN2:54;
set L = cf;
for x being Complex holds cf /. x = 0c * x by FUNCOP_1:13;
then reconsider L = cf as C_LINEAR by Def4;
set R = cf;
A3: dom cf = COMPLEX by FUNCOP_1:19;

now

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

A4: now

let n be Nat; :: thesis: ((h " ) (#) (cf /* h)) . n = 0c
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 VALUED_1:5
.= ((h " ) . n) * (cf /. (h . n)) by A5, FUNCT_2:186, X
.= ((h " ) . n) * 0c by FUNCOP_1:13
.= 0c ; :: thesis: verum

end;

then A6: (h " ) (#) (cf /* h) is constant by VALUED_0:def 18;
((h " ) (#) (cf /* h)) . 0 = 0c by A4;
hence ( (h " ) (#) (cf /* h) is convergent & lim ((h " ) (#) (cf /* h)) = 0c ) by A6, CFCONT_1:48, CFCONT_1:49; :: thesis: verum

end;

then reconsider R = cf as C_REST by Def3;

A7: now

let x0 be Complex; :: 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, Th12;
A11: N c= dom f by A1, A10, XBOOLE_1:1;
for x being Complex st x in N holds
(f /. x) - (f /. x0) = (L /. (x - x0)) + (R /. (x - x0))

proof

let x be Complex; :: 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) = a1 - (f /. x0) by A2
.= a1 - a1 by A2, A9
.= (L /. (x - x0)) + 0c 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, Def6; :: thesis: verum

end;

hence A14: f is_differentiable_on Z by A1, Th18; :: thesis: for x being Complex st x in Z holds
(f `| Z) /. x = 0c
let x0 be Complex; :: thesis: ( x0 in Z implies (f `| Z) /. x0 = 0c )
assume A15: x0 in Z ; :: thesis: (f `| Z) /. x0 = 0c
then A16: x0 in Z /\ (dom f) by A1, XBOOLE_0:def 4;
A17: f is_differentiable_in x0 by A7, A15;
consider N being Neighbourhood of x0 such that
A18: N c= Z by A15, Th12;
A19: N c= dom f by A1, A18, XBOOLE_1:1;
A20: for x being Complex st x in N holds
(f /. x) - (f /. x0) = (L /. (x - x0)) + (R /. (x - x0))

proof

let x be Complex; :: 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 A18, A19, XBOOLE_0:def 4;
hence (f /. x) - (f /. x0) = a1 - (f /. x0) by A2
.= a1 - a1 by A2, A16
.= (L /. (x - x0)) + 0c 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 A14, A15, Def12
.= L /. 1r by A17, A19, A20, Def7
.= 0c by FUNCOP_1:13 ; :: thesis: verum