theorem :: PDIFF_9:58

for m being non zero Element of NAT
for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),REAL st X c= dom f & X c= dom g & f is_differentiable_on X & g is_differentiable_on X holds
( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) ) )