let z0 be Element of REAL 2; :: thesis: for f1, f2 being PartFunc of REAL 2, REAL st f1 is_partial_differentiable`1_in z0 & f2 is_partial_differentiable`1_in z0 holds
( f1 + f2 is_partial_differentiable`1_in z0 & partdiff1 (f1 + f2),z0 = (partdiff1 f1,z0) + (partdiff1 f2,z0) )
let f1, f2 be PartFunc of REAL 2, REAL ; :: thesis: ( f1 is_partial_differentiable`1_in z0 & f2 is_partial_differentiable`1_in z0 implies ( f1 + f2 is_partial_differentiable`1_in z0 & partdiff1 (f1 + f2),z0 = (partdiff1 f1,z0) + (partdiff1 f2,z0) ) )
assume that
A1: f1 is_partial_differentiable`1_in z0 and
A2: f2 is_partial_differentiable`1_in z0 ; :: thesis: ( f1 + f2 is_partial_differentiable`1_in z0 & partdiff1 (f1 + f2),z0 = (partdiff1 f1,z0) + (partdiff1 f2,z0) )
( f1 is_partial_differentiable_in z0,1 & f2 is_partial_differentiable_in z0,1 ) by Lem3, A1, A2;
then ( f1 + f2 is_partial_differentiable_in z0,1 & partdiff (f1 + f2),z0,1 = (partdiff f1,z0,1) + (partdiff f2,z0,1) ) by PDIFF_1:29;
hence ( f1 + f2 is_partial_differentiable`1_in z0 & partdiff1 (f1 + f2),z0 = (partdiff1 f1,z0) + (partdiff1 f2,z0) ) by Lem3; :: thesis: verum