:: deftheorem defines is_differentiable_in NDIFF_1:def 6 :
for S, T being RealNormSpace
for f being PartFunc of S,T
for x0 being Point of S holds
( f is_differentiable_in x0 iff ex N being Neighbourhood of x0 st
( N c= dom f & ex L being Point of (R_NormSpace_of_BoundedLinearOperators (S,T)) ex R being RestFunc of S,T st
for x being Point of S st x in N holds
(f /. x) - (f /. x0) = (L . (x - x0)) + (R /. (x - x0)) ) );