let X be set ; :: thesis: for T, S being non trivial RealNormSpace
for f being PartFunc of S,T
for Z being Subset of S st Z is open & f is_differentiable_on X & Z c= X holds
f is_differentiable_on Z
let T, S be non trivial RealNormSpace; :: thesis: for f being PartFunc of S,T
for Z being Subset of S st Z is open & f is_differentiable_on X & Z c= X holds
f is_differentiable_on Z
let f be PartFunc of S,T; :: thesis: for Z being Subset of S st Z is open & f is_differentiable_on X & Z c= X holds
f is_differentiable_on Z
let Z be Subset of S; :: thesis: ( Z is open & f is_differentiable_on X & Z c= X implies f is_differentiable_on Z )
assume A1: Z is open ; :: thesis: ( not f is_differentiable_on X or not Z c= X or f is_differentiable_on Z )
assume A2: ( f is_differentiable_on X & Z c= X ) ; :: thesis: f is_differentiable_on Z
then X c= dom f by Def8;
hence A3: Z c= dom f by A2, XBOOLE_1:1; :: according to NDIFF_1:def 8 :: thesis: for x being Point of S st x in Z holds
f | Z is_differentiable_in x
let x0 be Point of S; :: thesis: ( x0 in Z implies f | Z is_differentiable_in x0 )
assume A4: x0 in Z ; :: thesis: f | Z is_differentiable_in x0
then f | X is_differentiable_in x0 by A2, Def8;
then consider N being Neighbourhood of x0 such that
A5: ( N c= dom (f | X) & ex L being Point of (R_NormSpace_of_BoundedLinearOperators S,T) ex R being REST of S,T st
for x being Point of S st x in N holds
((f | X) /. x) - ((f | X) /. x0) = (L . (x - x0)) + (R /. (x - x0)) ) by Def6;
consider N1 being Neighbourhood of x0 such that
A6: N1 c= Z by A1, A4, Th2;
consider N2 being Neighbourhood of x0 such that
A7: ( N2 c= N & N2 c= N1 ) by Th1;
take N2 ; :: according to NDIFF_1:def 6 :: thesis: ( N2 c= dom (f | Z) & ex L being Point of (R_NormSpace_of_BoundedLinearOperators S,T) ex R being REST of S,T st
for x being Point of S st x in N2 holds
((f | Z) /. x) - ((f | Z) /. x0) = (L . (x - x0)) + (R /. (x - x0)) )
dom (f | X) = (dom f) /\ X by RELAT_1:90;
then dom (f | X) c= dom f by XBOOLE_1:17;
then N c= dom f by A5, XBOOLE_1:1;
then A8: N2 c= dom f by A7, XBOOLE_1:1;
N2 c= Z by A6, A7, XBOOLE_1:1;
then N2 c= (dom f) /\ Z by A8, XBOOLE_1:19;
hence A9: N2 c= dom (f | Z) by RELAT_1:90; :: thesis: ex L being Point of (R_NormSpace_of_BoundedLinearOperators S,T) ex R being REST of S,T st
for x being Point of S st x in N2 holds
((f | Z) /. x) - ((f | Z) /. x0) = (L . (x - x0)) + (R /. (x - x0))
consider L being Point of (R_NormSpace_of_BoundedLinearOperators S,T), R being REST of S,T such that
A10: for x being Point of S st x in N holds
((f | X) /. x) - ((f | X) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by A5;
take L ; :: thesis: ex R being REST of S,T st
for x being Point of S st x in N2 holds
((f | Z) /. x) - ((f | Z) /. x0) = (L . (x - x0)) + (R /. (x - x0))
take R ; :: thesis: for x being Point of S st x in N2 holds
((f | Z) /. x) - ((f | Z) /. x0) = (L . (x - x0)) + (R /. (x - x0))
let x be Point of S; :: thesis: ( x in N2 implies ((f | Z) /. x) - ((f | Z) /. x0) = (L . (x - x0)) + (R /. (x - x0)) )
assume A11: x in N2 ; :: thesis: ((f | Z) /. x) - ((f | Z) /. x0) = (L . (x - x0)) + (R /. (x - x0))
then A12: x in N by A7;
((f | X) /. x) - ((f | X) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by A7, A10, A11;
then ((f | X) /. x) - (f /. x0) = (L . (x - x0)) + (R /. (x - x0)) by A2, A3, A4, PARTFUN2:35;
then (f /. x) - (f /. x0) = (L . (x - x0)) + (R /. (x - x0)) by A5, A12, PARTFUN2:32;
then (f /. x) - ((f | Z) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by A3, A4, PARTFUN2:35;
hence ((f | Z) /. x) - ((f | Z) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by A9, A11, PARTFUN2:32; :: thesis: verum