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 ) ) )
set R = cf;
defpred S₁[ set ] means $1 in COMPLEX ;
A1: 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 )

now

let n be Nat; :: thesis: ((h " ) (#) (cf /* h)) . n = 0c
A2: ( n in NAT & rng h c= dom cf ) by A1, ORDINAL1:def 13;
thus ((h " ) (#) (cf /* h)) . n = ((h " ) . n) * ((cf /* h) . n) by VALUED_1:5
.= ((h " ) . n) * (cf /. (h . n)) by A2, FUNCT_2:186
.= ((h " ) . n) * 0c by FUNCOP_1:13
.= 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_1:48, CFCONT_1:49; :: thesis: verum

end;

then reconsider R = cf as C_REST by Def3;
assume that
A3: Z c= dom f and
A4: 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 ) )
deffunc H₁( Complex) -> Element of COMPLEX = a * $1;
consider L being Function of COMPLEX ,COMPLEX such that
A5: 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 A5;
then reconsider L = L as C_LINEAR by Def4;

A6: now

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: 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 A4, A8
.= ((a * x) + b) - ((a * x0) + b) by A4, A7
.= (a * (x - x0)) + 0c
.= (L /. (x - x0)) + 0c by A5
.= (L /. (x - x0)) + (R /. (x - x0)) by FUNCOP_1:13 ;
:: thesis: verum

end;

N c= dom f by A3, A8, XBOOLE_1:1;
hence f is_differentiable_in x0 by A9, Def6; :: thesis: verum

end;

hence A10: f is_differentiable_on Z by A3, Th15; :: 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 A11: x0 in Z ; :: thesis: (f `| Z) /. x0 = a
then consider N being Neighbourhood of x0 such that
A12: N c= Z by Th9;
A13: 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 A4, A12
.= ((a * x) + b) - ((a * x0) + b) by A4, A11
.= (a * (x - x0)) + 0c
.= (L /. (x - x0)) + 0c by A5
.= (L /. (x - x0)) + (R /. (x - x0)) by FUNCOP_1:13 ;
:: thesis: verum

end;

A14: N c= dom f by A3, A12, XBOOLE_1:1;
A15: f is_differentiable_in x0 by A6, A11;
thus (f `| Z) /. x0 = diff f,x0 by A10, A11, Def12
.= L /. 1r by A15, A14, A13, Def7
.= a * 1r by A5
.= a ; :: thesis: verum