let u0 be Element of REAL 3; :: thesis: for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`13_in u0 & f2 is_hpartial_differentiable`13_in u0 holds
( (pdiff1 f1,1) + (pdiff1 f2,1) is_partial_differentiable_in u0,3 & partdiff ((pdiff1 f1,1) + (pdiff1 f2,1)),u0,3 = (hpartdiff13 f1,u0) + (hpartdiff13 f2,u0) )
let f1, f2 be PartFunc of (REAL 3),REAL ; :: thesis: ( f1 is_hpartial_differentiable`13_in u0 & f2 is_hpartial_differentiable`13_in u0 implies ( (pdiff1 f1,1) + (pdiff1 f2,1) is_partial_differentiable_in u0,3 & partdiff ((pdiff1 f1,1) + (pdiff1 f2,1)),u0,3 = (hpartdiff13 f1,u0) + (hpartdiff13 f2,u0) ) )
assume that
A1: f1 is_hpartial_differentiable`13_in u0 and
A2: f2 is_hpartial_differentiable`13_in u0 ; :: thesis: ( (pdiff1 f1,1) + (pdiff1 f2,1) is_partial_differentiable_in u0,3 & partdiff ((pdiff1 f1,1) + (pdiff1 f2,1)),u0,3 = (hpartdiff13 f1,u0) + (hpartdiff13 f2,u0) )
A3: pdiff1 f1,1 is_partial_differentiable_in u0,3 by A1, Th10ForZ;
AA: pdiff1 f2,1 is_partial_differentiable_in u0,3 by A2, Th10ForZ;
then ( (pdiff1 f1,1) + (pdiff1 f2,1) is_partial_differentiable_in u0,3 & partdiff ((pdiff1 f1,1) + (pdiff1 f2,1)),u0,3 = (partdiff (pdiff1 f1,1),u0,3) + (partdiff (pdiff1 f2,1),u0,3) ) by A3, PDIFF_1:29;
then partdiff ((pdiff1 f1,1) + (pdiff1 f2,1)),u0,3 = (hpartdiff13 f1,u0) + (partdiff (pdiff1 f2,1),u0,3) by A1, Th18ForZ
.= (hpartdiff13 f1,u0) + (hpartdiff13 f2,u0) by A2, Th18ForZ ;
hence ( (pdiff1 f1,1) + (pdiff1 f2,1) is_partial_differentiable_in u0,3 & partdiff ((pdiff1 f1,1) + (pdiff1 f2,1)),u0,3 = (hpartdiff13 f1,u0) + (hpartdiff13 f2,u0) ) by A3, AA, PDIFF_1:29; :: thesis: verum