let u0 be Element of REAL 3; :: thesis: for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`12_in u0 & f2 is_hpartial_differentiable`12_in u0 holds
(pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in u0,2
let f1, f2 be PartFunc of (REAL 3),REAL; :: thesis: ( f1 is_hpartial_differentiable`12_in u0 & f2 is_hpartial_differentiable`12_in u0 implies (pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in u0,2 )
assume ( f1 is_hpartial_differentiable`12_in u0 & f2 is_hpartial_differentiable`12_in u0 ) ; :: thesis: (pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in u0,2
then ( pdiff1 (f1,1) is_partial_differentiable_in u0,2 & pdiff1 (f2,1) is_partial_differentiable_in u0,2 ) by Th20;
hence (pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in u0,2 by PDIFF_4:29; :: thesis: verum