theorem Th65: :: PDIFF_9:65

for m being non zero Element of NAT
for i being Element of NAT
for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds
( f + g is_partial_differentiable_on X,i & (f + g) `partial| (X,i) = (f `partial| (X,i)) + (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) )