theorem Th2: :: CKSPACE1:2

for m being non zero Element of NAT
for k being Element of NAT
for X being non empty Subset of (REAL m)
for f being PartFunc of (REAL m),REAL st X is open & X c= dom f holds
( f is_continuously_differentiable_up_to_order 1,X iff ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) )