let X be set ; :: thesis: for S, T being 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 S, T be 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 that
A2: f is_differentiable_on X and
A3: Z c= X ; :: thesis:
X c= dom f by A2;
hence A4: Z c= dom f by A3; :: 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 A5: x0 in Z ; :: thesis:
then consider N1 being Neighbourhood of x0 such that
A6: N1 c= Z by ;
f | X is_differentiable_in x0 by A2, A3, A5;
then consider N being Neighbourhood of x0 such that
A7: N c= dom (f | X) and
A8: ex L being Point of ex R being RestFunc 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)) ;
consider N2 being Neighbourhood of x0 such that
A9: N2 c= N and
A10: N2 c= N1 by Th1;
A11: N2 c= Z by ;
take N2 ; :: according to NDIFF_1:def 6 :: thesis: ( N2 c= dom (f | Z) & ex L being Point of ex R being RestFunc 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_1:61;
then dom (f | X) c= dom f by XBOOLE_1:17;
then N c= dom f by A7;
then N2 c= dom f by A9;
then N2 c= (dom f) /\ Z by ;
hence A12: N2 c= dom (f | Z) by RELAT_1:61; :: thesis: ex L being Point of ex R being RestFunc 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 being RestFunc of S,T such that
A13: for x being Point of S st x in N holds
((f | X) /. x) - ((f | X) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by A8;
take L ; :: thesis: ex R being RestFunc 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 A14: 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 ;
then A15: ((f | X) /. x) - (f /. x0) = (L . (x - x0)) + (R /. (x - x0)) by ;
x in N by ;
then (f /. x) - (f /. x0) = (L . (x - x0)) + (R /. (x - x0)) by ;
then (f /. x) - ((f | Z) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by ;
hence ((f | Z) /. x) - ((f | Z) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by ; :: thesis: verum