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
  f is_hpartial_differentiable`13_in u0 implies
  r(#)pdiff1(f,1) is_partial_differentiable_in u0,3 &
  partdiff((r(#)pdiff1(f,1)),u0,3) = r*hpartdiff13(f,u0)
proof
    assume
A1: f is_hpartial_differentiable`13_in u0;
    then pdiff1(f,1) is_partial_differentiable_in u0,3 by Th21;then
    r(#)pdiff1(f,1) is_partial_differentiable_in u0,3 &
    partdiff((r(#)pdiff1(f,1)),u0,3) = r*partdiff(pdiff1(f,1),u0,3)
    by PDIFF_1:33;
    hence thesis by A1,Th30;
end;
