reserve x,x0,x1,y,y0,y1,r,r1,s,p,p1 for Real;
reserve z,z0 for Element of REAL 2;
reserve n for Element of NAT;
reserve s1 for Real_Sequence;
reserve f,f1,f2 for PartFunc of REAL 2,REAL;
reserve R,R1 for RestFunc;
reserve L,L1 for LinearFunc;

theorem
  f1 is_hpartial_differentiable`12_in z0 & f2
  is_hpartial_differentiable`12_in z0 implies pdiff1(f1,1)+pdiff1(f2,1)
is_partial_differentiable_in z0,2 & partdiff(pdiff1(f1,1)+pdiff1(f2,1),z0,2) =
  hpartdiff12(f1,z0) + hpartdiff12(f2,z0)
proof
  assume that
A1: f1 is_hpartial_differentiable`12_in z0 and
A2: f2 is_hpartial_differentiable`12_in z0;
A3: pdiff1(f1,1) is_partial_differentiable_in z0,2 & pdiff1(f2,1)
  is_partial_differentiable_in z0,2 by A1,A2,Th10;
  then
  partdiff(pdiff1(f1,1)+pdiff1(f2,1),z0,2) = partdiff(pdiff1(f1,1),z0,2) +
  partdiff(pdiff1(f2,1),z0,2) by PDIFF_1:29;
  then
  partdiff(pdiff1(f1,1)+pdiff1(f2,1),z0,2) = hpartdiff12(f1,z0) + partdiff
  (pdiff1(f2,1),z0,2) by A1,Th14
    .= hpartdiff12(f1,z0) + hpartdiff12(f2,z0) by A2,Th14;
  hence thesis by A3,PDIFF_1:29;
end;
