reserve x,x0,x1,y,y0,y1,z,z0,z1,r,r1,s,p,p1 for Real;
reserve u,u0 for Element of REAL 3;
reserve n for Element of NAT;
reserve s1 for Real_Sequence;
reserve f,f1,f2 for PartFunc of REAL 3,REAL;
reserve R,R1 for RestFunc;
reserve L,L1 for LinearFunc;

theorem
  f1 is_hpartial_differentiable`13_in u0 &
  f2 is_hpartial_differentiable`13_in u0 implies
  pdiff1(f1,1)(#)pdiff1(f2,1) is_partial_differentiable_in u0,3
proof
  assume f1 is_hpartial_differentiable`13_in u0 &
  f2 is_hpartial_differentiable`13_in u0;
  then pdiff1(f1,1) is_partial_differentiable_in u0,3 &
  pdiff1(f2,1) is_partial_differentiable_in u0,3 by Th21;
  hence thesis by PDIFF_4:30;
end;
