theorem LM300: :: NDIFF_7:40

for X, Y, W being RealNormSpace
for Z being Subset of [:X,Y:]
for f being PartFunc of [:X,Y:],W holds
( ( f is_partial_differentiable_on`1 Z implies f * ((IsoCPNrSP (X,Y)) ") is_partial_differentiable_on ((IsoCPNrSP (X,Y)) ") " Z,1 ) & ( f * ((IsoCPNrSP (X,Y)) ") is_partial_differentiable_on ((IsoCPNrSP (X,Y)) ") " Z,1 implies f is_partial_differentiable_on`1 Z ) & ( f is_partial_differentiable_on`2 Z implies f * ((IsoCPNrSP (X,Y)) ") is_partial_differentiable_on ((IsoCPNrSP (X,Y)) ") " Z,2 ) & ( f * ((IsoCPNrSP (X,Y)) ") is_partial_differentiable_on ((IsoCPNrSP (X,Y)) ") " Z,2 implies f is_partial_differentiable_on`2 Z ) )