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 ) ) )
reconsider cf = COMPLEX --> 0c as Function of COMPLEX,COMPLEX ;
set R = cf;

now :: thesis: for h being non-zero 0 -convergent Complex_Sequence holds
( (h ") (#) (cf /* h) is convergent & lim ((h ") (#) (cf /* h)) = 0c )

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

now :: thesis: for n being Nat holds ((h ") (#) (cf /* h)) . n = 0c

let n be Nat; :: thesis: ((h ") (#) (cf /* h)) . n = 0c
A2: ( n in NAT & rng h c= dom cf ) by ORDINAL1:def 12;
thus ((h ") (#) (cf /* h)) . n = ((h ") . n) * ((cf /* h) . n) by VALUED_1:5
.= ((h ") . n) * (cf /. (h . n)) by A2, FUNCT_2:109
.= ((h ") . n) * 0c
.= 0c ; :: thesis: verum

end;

then ( (h ") (#) (cf /* h) is constant & ((h ") (#) (cf /* h)) . 0 = 0c ) by VALUED_0:def 18;
hence ( (h ") (#) (cf /* h) is convergent & lim ((h ") (#) (cf /* h)) = 0c ) by CFCONT_1:26, CFCONT_1:27; :: thesis: verum

end;

then reconsider R = cf as C_RestFunc by Def3;
set L = cf;
for x being Complex holds cf /. x = 0c * x by XCMPLX_0:def 2, FUNCOP_1:7;
then reconsider L = cf as C_LinearFunc by Def4;
assume that
A3: Z c= dom f and
A4: f | Z is constant ; :: thesis: ( f is_differentiable_on Z & ( for x being Complex st x in Z holds
(f `| Z) /. x = 0c ) )
consider a1 being Element of COMPLEX such that
A5: for x being Element of COMPLEX st x in Z /\ (dom f) holds
f /. x = a1 by A4, PARTFUN2:35;

A6: now :: thesis: for x0 being Complex st x0 in Z holds
f is_differentiable_in x0

let x0 be Complex; :: thesis: ( x0 in Z implies f is_differentiable_in x0 )
assume A7: x0 in Z ; :: thesis: f is_differentiable_in x0
then consider N being Neighbourhood of x0 such that
A8: N c= Z by Th9;
A9: N c= dom f by A3, A8;
A10: x0 in Z /\ (dom f) by A3, A7, XBOOLE_0:def 4;
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)) )
A11: x - x0 in COMPLEX by XCMPLX_0:def 2;
assume x in N ; :: thesis: (f /. x) - (f /. x0) = (L /. (x - x0)) + (R /. (x - x0))
then x in Z /\ (dom f) by A8, A9, XBOOLE_0:def 4;
hence (f /. x) - (f /. x0) = a1 - (f /. x0) by A5
.= a1 - a1 by A5, A10
.= (L /. (x - x0)) + 0c by FUNCOP_1:7, A11
.= (L /. (x - x0)) + (R /. (x - x0)) by FUNCOP_1:7, A11 ;
:: thesis: verum

end;

hence f is_differentiable_in x0 by A9; :: thesis: verum

end;

hence A12: f is_differentiable_on Z by A3, Th15; :: 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 A13: x0 in Z ; :: thesis: (f `| Z) /. x0 = 0c
then consider N being Neighbourhood of x0 such that
A14: N c= Z by Th9;
A15: N c= dom f by A3, A14;
A16: x0 in Z /\ (dom f) by A3, A13, XBOOLE_0:def 4;
A17: 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)) )
A18: x - x0 in COMPLEX by XCMPLX_0:def 2;
assume x in N ; :: thesis: (f /. x) - (f /. x0) = (L /. (x - x0)) + (R /. (x - x0))
then x in Z /\ (dom f) by A14, A15, XBOOLE_0:def 4;
hence (f /. x) - (f /. x0) = a1 - (f /. x0) by A5
.= a1 - a1 by A5, A16
.= (L /. (x - x0)) + 0c by FUNCOP_1:7, A18
.= (L /. (x - x0)) + (R /. (x - x0)) by FUNCOP_1:7, A18 ;
:: thesis: verum

end;

A19: f is_differentiable_in x0 by A6, A13;
thus (f `| Z) /. x0 = diff (f,x0) by A12, A13, Def12
.= L /. 1r by A19, A15, A17, Def7
.= 0c ; :: thesis: verum