let z0 be Element of REAL 2; :: thesis: for f1, f2 being PartFunc of (REAL 2),REAL st f1 is_hpartial_differentiable`11_in z0 & f2 is_hpartial_differentiable`11_in z0 holds
( (pdiff1 f1,1) + (pdiff1 f2,1) is_partial_differentiable_in z0,1 & partdiff ((pdiff1 f1,1) + (pdiff1 f2,1)),z0,1 = (hpartdiff11 f1,z0) + (hpartdiff11 f2,z0) )
let f1, f2 be PartFunc of (REAL 2),REAL ; :: thesis: ( f1 is_hpartial_differentiable`11_in z0 & f2 is_hpartial_differentiable`11_in z0 implies ( (pdiff1 f1,1) + (pdiff1 f2,1) is_partial_differentiable_in z0,1 & partdiff ((pdiff1 f1,1) + (pdiff1 f2,1)),z0,1 = (hpartdiff11 f1,z0) + (hpartdiff11 f2,z0) ) )
assume that
A1: f1 is_hpartial_differentiable`11_in z0 and
A2: f2 is_hpartial_differentiable`11_in z0 ; :: thesis: ( (pdiff1 f1,1) + (pdiff1 f2,1) is_partial_differentiable_in z0,1 & partdiff ((pdiff1 f1,1) + (pdiff1 f2,1)),z0,1 = (hpartdiff11 f1,z0) + (hpartdiff11 f2,z0) )
A3: pdiff1 f1,1 is_partial_differentiable_in z0,1 by A1, Th9;
AA: pdiff1 f2,1 is_partial_differentiable_in z0,1 by A2, Th9;
then A4: ( (pdiff1 f1,1) + (pdiff1 f2,1) is_partial_differentiable_in z0,1 & partdiff ((pdiff1 f1,1) + (pdiff1 f2,1)),z0,1 = (partdiff (pdiff1 f1,1),z0,1) + (partdiff (pdiff1 f2,1),z0,1) ) by A3, PDIFF_1:29;
partdiff ((pdiff1 f1,1) + (pdiff1 f2,1)),z0,1 = (hpartdiff11 f1,z0) + (partdiff (pdiff1 f2,1),z0,1) by A1, A4, Th17
.= (hpartdiff11 f1,z0) + (hpartdiff11 f2,z0) by A2, Th17 ;
hence ( (pdiff1 f1,1) + (pdiff1 f2,1) is_partial_differentiable_in z0,1 & partdiff ((pdiff1 f1,1) + (pdiff1 f2,1)),z0,1 = (hpartdiff11 f1,z0) + (hpartdiff11 f2,z0) ) by A3, AA, PDIFF_1:29; :: thesis: verum