let u0 be Element of REAL 3; :: thesis: for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`22_in u0 & f2 is_hpartial_differentiable`22_in u0 holds
( (pdiff1 f1,2) + (pdiff1 f2,2) is_partial_differentiable_in u0,2 & partdiff ((pdiff1 f1,2) + (pdiff1 f2,2)),u0,2 = (hpartdiff22 f1,u0) + (hpartdiff22 f2,u0) )
let f1, f2 be PartFunc of (REAL 3),REAL ; :: thesis: ( f1 is_hpartial_differentiable`22_in u0 & f2 is_hpartial_differentiable`22_in u0 implies ( (pdiff1 f1,2) + (pdiff1 f2,2) is_partial_differentiable_in u0,2 & partdiff ((pdiff1 f1,2) + (pdiff1 f2,2)),u0,2 = (hpartdiff22 f1,u0) + (hpartdiff22 f2,u0) ) )
assume that
A1: f1 is_hpartial_differentiable`22_in u0 and
A2: f2 is_hpartial_differentiable`22_in u0 ; :: thesis: ( (pdiff1 f1,2) + (pdiff1 f2,2) is_partial_differentiable_in u0,2 & partdiff ((pdiff1 f1,2) + (pdiff1 f2,2)),u0,2 = (hpartdiff22 f1,u0) + (hpartdiff22 f2,u0) )
A3: pdiff1 f1,2 is_partial_differentiable_in u0,2 by A1, Th12;
AA: pdiff1 f2,2 is_partial_differentiable_in u0,2 by A2, Th12;
then ( (pdiff1 f1,2) + (pdiff1 f2,2) is_partial_differentiable_in u0,2 & partdiff ((pdiff1 f1,2) + (pdiff1 f2,2)),u0,2 = (partdiff (pdiff1 f1,2),u0,2) + (partdiff (pdiff1 f2,2),u0,2) ) by A3, PDIFF_1:29;
then partdiff ((pdiff1 f1,2) + (pdiff1 f2,2)),u0,2 = (hpartdiff22 f1,u0) + (partdiff (pdiff1 f2,2),u0,2) by A1, Th20
.= (hpartdiff22 f1,u0) + (hpartdiff22 f2,u0) by A2, Th20 ;
hence ( (pdiff1 f1,2) + (pdiff1 f2,2) is_partial_differentiable_in u0,2 & partdiff ((pdiff1 f1,2) + (pdiff1 f2,2)),u0,2 = (hpartdiff22 f1,u0) + (hpartdiff22 f2,u0) ) by A3, AA, PDIFF_1:29; :: thesis: verum