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 & partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),u0,2) = (hpartdiff12 (f1,u0)) + (hpartdiff12 (f2,u0)) )
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 & partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),u0,2) = (hpartdiff12 (f1,u0)) + (hpartdiff12 (f2,u0)) ) )
assume that
A1: f1 is_hpartial_differentiable`12_in u0 and
A2: f2 is_hpartial_differentiable`12_in u0 ; :: thesis: ( (pdiff1 (f1,1)) + (pdiff1 (f2,1)) is_partial_differentiable_in u0,2 & partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),u0,2) = (hpartdiff12 (f1,u0)) + (hpartdiff12 (f2,u0)) )
A3: pdiff1 (f1,1) is_partial_differentiable_in u0,2 by A1, Th20;
A4: pdiff1 (f2,1) is_partial_differentiable_in u0,2 by A2, Th20;
then ( (pdiff1 (f1,1)) + (pdiff1 (f2,1)) is_partial_differentiable_in u0,2 & partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),u0,2) = (partdiff ((pdiff1 (f1,1)),u0,2)) + (partdiff ((pdiff1 (f2,1)),u0,2)) ) by A3, PDIFF_1:29;
then partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),u0,2) = (hpartdiff12 (f1,u0)) + (partdiff ((pdiff1 (f2,1)),u0,2)) by A1, Th29
.= (hpartdiff12 (f1,u0)) + (hpartdiff12 (f2,u0)) by A2, Th29 ;
hence ( (pdiff1 (f1,1)) + (pdiff1 (f2,1)) is_partial_differentiable_in u0,2 & partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),u0,2) = (hpartdiff12 (f1,u0)) + (hpartdiff12 (f2,u0)) ) by A3, A4, PDIFF_1:29; :: thesis: verum