let a, b be Complex; :: thesis: for f being PartFunc of COMPLEX ,COMPLEX
for Z being open Subset of COMPLEX st Z c= dom f & ( for x being Complex st x in Z holds
f /. x = (a * x) + b ) holds
( f is_differentiable_on Z & ( for x being Complex st x in Z holds
(f `| Z) /. x = a ) )
let f be PartFunc of COMPLEX ,COMPLEX ; :: thesis: for Z being open Subset of COMPLEX st Z c= dom f & ( for x being Complex st x in Z holds
f /. x = (a * x) + b ) holds
( f is_differentiable_on Z & ( for x being Complex st x in Z holds
(f `| Z) /. x = a ) )
let Z be open Subset of COMPLEX ; :: thesis: ( Z c= dom f & ( for x being Complex st x in Z holds
f /. x = (a * x) + b ) implies ( f is_differentiable_on Z & ( for x being Complex st x in Z holds
(f `| Z) /. x = a ) ) )
assume A1: ( Z c= dom f & ( for x being Complex st x in Z holds
f /. x = (a * x) + b ) ) ; :: thesis: ( f is_differentiable_on Z & ( for x being Complex st x in Z holds
(f `| Z) /. x = a ) )
defpred S₁[ set ] means $1 in COMPLEX ;
deffunc H₁( Complex) -> Element of COMPLEX = a * $1;
consider L being Function of COMPLEX ,COMPLEX such that
A2: for x being Element of COMPLEX holds L . x = H₁(x) from FUNCT_2:sch 4();
for z being Complex holds L /. z = a * z by A2;
then reconsider L = L as C_LINEAR by Def4;
set R = cf;
A6: 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 )

A7: now

let n be Nat; :: thesis: ((h " ) (#) (cf /* h)) . n = 0c
X: n in NAT by ORDINAL1:def 13;
A8: rng h c= dom cf by A6;
thus ((h " ) (#) (cf /* h)) . n = ((h " ) . n) * ((cf /* h) . n) by VALUED_1:5
.= ((h " ) . n) * (cf /. (h . n)) by A8, FUNCT_2:186, X
.= ((h " ) . n) * 0c by FUNCOP_1:13
.= 0c ; :: thesis: verum

end;

then A9: (h " ) (#) (cf /* h) is constant by VALUED_0:def 18;
((h " ) (#) (cf /* h)) . 0 = 0c by A7;
hence ( (h " ) (#) (cf /* h) is convergent & lim ((h " ) (#) (cf /* h)) = 0c ) by A9, CFCONT_1:48, CFCONT_1:49; :: thesis: verum

end;

then reconsider R = cf as C_REST by Def3;

A10: now

let x0 be Complex; :: thesis: ( x0 in Z implies f is_differentiable_in x0 )
assume A11: x0 in Z ; :: thesis: f is_differentiable_in x0
then consider N being Neighbourhood of x0 such that
A12: N c= Z by Th12;
A13: N c= dom f by A1, A12, 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))
hence (f /. x) - (f /. x0) = ((a * x) + b) - (f /. x0) by A1, A12
.= ((a * x) + b) - ((a * x0) + b) by A1, A11
.= (a * (x - x0)) + 0c
.= (L /. (x - x0)) + 0c by A2
.= (L /. (x - x0)) + (R /. (x - x0)) by FUNCOP_1:13 ;
:: thesis: verum

end;

hence f is_differentiable_in x0 by A13, Def6; :: thesis: verum

end;

hence A15: f is_differentiable_on Z by A1, Th18; :: thesis: for x being Complex st x in Z holds
(f `| Z) /. x = a
let x0 be Complex; :: thesis: ( x0 in Z implies (f `| Z) /. x0 = a )
assume A16: x0 in Z ; :: thesis: (f `| Z) /. x0 = a
then A17: f is_differentiable_in x0 by A10;
consider N being Neighbourhood of x0 such that
A18: N c= Z by A16, 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))
hence (f /. x) - (f /. x0) = ((a * x) + b) - (f /. x0) by A1, A18
.= ((a * x) + b) - ((a * x0) + b) by A1, A16
.= (a * (x - x0)) + 0c
.= (L /. (x - x0)) + 0c by A2
.= (L /. (x - x0)) + (R /. (x - x0)) by FUNCOP_1:13 ;
:: thesis: verum

end;

thus (f `| Z) /. x0 = diff f,x0 by A15, A16, Def12
.= L /. 1r by A17, A19, A20, Def7
.= a * 1r by A2
.= a ; :: thesis: verum