let z0 be Element of REAL 2; :: thesis: for f1, f2 being PartFunc of REAL 2, REAL st f1 is_hpartial_differentiable`21_in z0 & f2 is_hpartial_differentiable`21_in z0 holds
(pdiff2 f1,z0) (#) (pdiff2 f2,z0) is_partial_differentiable`1_in z0
let f1, f2 be PartFunc of REAL 2, REAL ; :: thesis: ( f1 is_hpartial_differentiable`21_in z0 & f2 is_hpartial_differentiable`21_in z0 implies (pdiff2 f1,z0) (#) (pdiff2 f2,z0) is_partial_differentiable`1_in z0 )
assume ( f1 is_hpartial_differentiable`21_in z0 & f2 is_hpartial_differentiable`21_in z0 ) ; :: thesis: (pdiff2 f1,z0) (#) (pdiff2 f2,z0) is_partial_differentiable`1_in z0
then ( pdiff2 f1,z0 is_partial_differentiable`1_in z0 & pdiff2 f2,z0 is_partial_differentiable`1_in z0 ) by Th11;
hence (pdiff2 f1,z0) (#) (pdiff2 f2,z0) is_partial_differentiable`1_in z0 by PDIFF_2:23; :: thesis: verum