let X be set ; :: thesis: for f being PartFunc of COMPLEX,COMPLEX
for Z being open Subset of COMPLEX st f is_differentiable_on X & Z c= X holds
f is_differentiable_on Z
let f be PartFunc of COMPLEX,COMPLEX; :: thesis: for Z being open Subset of COMPLEX st f is_differentiable_on X & Z c= X holds
f is_differentiable_on Z
let Z be open Subset of COMPLEX; :: thesis: ( f is_differentiable_on X & Z c= X implies f is_differentiable_on Z )
assume that
A1: f is_differentiable_on X and
A2: Z c= X ; :: thesis: f is_differentiable_on Z
A3: f | X is differentiable by A1;
X c= dom f by A1;
hence Z c= dom f by A2; :: according to CFDIFF_1:def 9 :: thesis: f | Z is differentiable
let x0 be Complex; :: according to CFDIFF_1:def 8 :: thesis: ( x0 in dom (f | Z) implies f | Z is_differentiable_in x0 )
assume A4: x0 in dom (f | Z) ; :: thesis: f | Z is_differentiable_in x0
then A5: x0 in dom f by RELAT_1:57;
A6: x0 in Z by A4, RELAT_1:57;
then consider N1 being Neighbourhood of x0 such that
A7: N1 c= Z by Th9;
x0 in dom (f | X) by A2, A5, A6, RELAT_1:57;
then f | X is_differentiable_in x0 by A3;
then consider N being Neighbourhood of x0 such that
A8: N c= dom (f | X) and
A9: ex L being C_LinearFunc ex R being C_RestFunc st
for x being Complex st x in N holds
((f | X) /. x) - ((f | X) /. x0) = (L /. (x - x0)) + (R /. (x - x0)) ;
consider N2 being Neighbourhood of x0 such that
A10: N2 c= N and
A11: N2 c= N1 by Lm1;
A12: N2 c= Z by A7, A11;
take N2 ; :: according to CFDIFF_1:def 6 :: thesis: ( N2 c= dom (f | Z) & ex L being C_LinearFunc ex R being C_RestFunc st
for z being Complex st z in N2 holds
((f | Z) /. z) - ((f | Z) /. x0) = (L /. (z - x0)) + (R /. (z - x0)) )
dom (f | X) = (dom f) /\ X by RELAT_1:61;
then dom (f | X) c= dom f by XBOOLE_1:17;
then N c= dom f by A8;
then N2 c= dom f by A10;
then N2 c= (dom f) /\ Z by A12, XBOOLE_1:19;
hence A13: N2 c= dom (f | Z) by RELAT_1:61; :: thesis: ex L being C_LinearFunc ex R being C_RestFunc st
for z being Complex st z in N2 holds
((f | Z) /. z) - ((f | Z) /. x0) = (L /. (z - x0)) + (R /. (z - x0))
consider L being C_LinearFunc, R being C_RestFunc such that
A14: for x being Complex st x in N holds
((f | X) /. x) - ((f | X) /. x0) = (L /. (x - x0)) + (R /. (x - x0)) by A9;
take L ; :: thesis: ex R being C_RestFunc st
for z being Complex st z in N2 holds
((f | Z) /. z) - ((f | Z) /. x0) = (L /. (z - x0)) + (R /. (z - x0))
take R ; :: thesis: for z being Complex st z in N2 holds
((f | Z) /. z) - ((f | Z) /. x0) = (L /. (z - x0)) + (R /. (z - x0))
let x be Complex; :: thesis: ( x in N2 implies ((f | Z) /. x) - ((f | Z) /. x0) = (L /. (x - x0)) + (R /. (x - x0)) )
assume A15: x in N2 ; :: thesis: ((f | Z) /. x) - ((f | Z) /. x0) = (L /. (x - x0)) + (R /. (x - x0))
then ((f | X) /. x) - ((f | X) /. x0) = (L /. (x - x0)) + (R /. (x - x0)) by A10, A14;
then A16: ((f | X) /. x) - (f /. x0) = (L /. (x - x0)) + (R /. (x - x0)) by A2, A5, A6, PARTFUN2:17;
x in N by A10, A15;
then (f /. x) - (f /. x0) = (L /. (x - x0)) + (R /. (x - x0)) by A8, A16, PARTFUN2:15;
then (f /. x) - ((f | Z) /. x0) = (L /. (x - x0)) + (R /. (x - x0)) by A5, A6, PARTFUN2:17;
hence ((f | Z) /. x) - ((f | Z) /. x0) = (L /. (x - x0)) + (R /. (x - x0)) by A13, A15, PARTFUN2:15; :: thesis: verum