let z0 be Element of REAL 2; :: thesis: for f1, f2 being PartFunc of (REAL 2),REAL st f1 is_hpartial_differentiable`22_in z0 & f2 is_hpartial_differentiable`22_in z0 holds
(pdiff1 (f1,2)) (#) (pdiff1 (f2,2)) is_partial_differentiable_in z0,2
let f1, f2 be PartFunc of (REAL 2),REAL; :: thesis: ( f1 is_hpartial_differentiable`22_in z0 & f2 is_hpartial_differentiable`22_in z0 implies (pdiff1 (f1,2)) (#) (pdiff1 (f2,2)) is_partial_differentiable_in z0,2 )
assume ( f1 is_hpartial_differentiable`22_in z0 & f2 is_hpartial_differentiable`22_in z0 ) ; :: thesis: (pdiff1 (f1,2)) (#) (pdiff1 (f2,2)) is_partial_differentiable_in z0,2
then ( pdiff1 (f1,2) is_partial_differentiable_in z0,2 & pdiff1 (f2,2) is_partial_differentiable_in z0,2 ) by Th12;
hence (pdiff1 (f1,2)) (#) (pdiff1 (f2,2)) is_partial_differentiable_in z0,2 by PDIFF_2:20; :: thesis: verum