let u0 be Element of REAL 3; :: thesis: for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`11_in u0 & f2 is_hpartial_differentiable`11_in u0 holds
( (pdiff1 f1,1) + (pdiff1 f2,1) is_partial_differentiable_in u0,1 & partdiff ((pdiff1 f1,1) + (pdiff1 f2,1)),u0,1 = (hpartdiff11 f1,u0) + (hpartdiff11 f2,u0) )
let f1, f2 be PartFunc of (REAL 3),REAL ; :: thesis: ( f1 is_hpartial_differentiable`11_in u0 & f2 is_hpartial_differentiable`11_in u0 implies ( (pdiff1 f1,1) + (pdiff1 f2,1) is_partial_differentiable_in u0,1 & partdiff ((pdiff1 f1,1) + (pdiff1 f2,1)),u0,1 = (hpartdiff11 f1,u0) + (hpartdiff11 f2,u0) ) )
assume that
A1: f1 is_hpartial_differentiable`11_in u0 and
A2: f2 is_hpartial_differentiable`11_in u0 ; :: thesis: ( (pdiff1 f1,1) + (pdiff1 f2,1) is_partial_differentiable_in u0,1 & partdiff ((pdiff1 f1,1) + (pdiff1 f2,1)),u0,1 = (hpartdiff11 f1,u0) + (hpartdiff11 f2,u0) )
A3: pdiff1 f1,1 is_partial_differentiable_in u0,1 by A1, Th9;
AA: pdiff1 f2,1 is_partial_differentiable_in u0,1 by A2, Th9;
then ( (pdiff1 f1,1) + (pdiff1 f2,1) is_partial_differentiable_in u0,1 & partdiff ((pdiff1 f1,1) + (pdiff1 f2,1)),u0,1 = (partdiff (pdiff1 f1,1),u0,1) + (partdiff (pdiff1 f2,1),u0,1) ) by A3, PDIFF_1:29;
then partdiff ((pdiff1 f1,1) + (pdiff1 f2,1)),u0,1 = (hpartdiff11 f1,u0) + (partdiff (pdiff1 f2,1),u0,1) by A1, Th17
.= (hpartdiff11 f1,u0) + (hpartdiff11 f2,u0) by A2, Th17 ;
hence ( (pdiff1 f1,1) + (pdiff1 f2,1) is_partial_differentiable_in u0,1 & partdiff ((pdiff1 f1,1) + (pdiff1 f2,1)),u0,1 = (hpartdiff11 f1,u0) + (hpartdiff11 f2,u0) ) by A3, AA, PDIFF_1:29; :: thesis: verum