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
  f is_hpartial_differentiable`22_in z0 implies r(#)pdiff1(f,2)
  is_partial_differentiable_in z0,2 & partdiff((r(#)pdiff1(f,2)),z0,2) = r*
  hpartdiff22(f,z0)
proof
  assume
A1: f is_hpartial_differentiable`22_in z0;
  then
A2: pdiff1(f,2) is_partial_differentiable_in z0,2 by Th12;
   reconsider r as Real;
  partdiff((r(#)pdiff1(f,2)),z0,2) = r*partdiff(pdiff1(f,2),z0,2) by
PDIFF_1:33,A2;
  hence thesis by A1,A2,Th16,PDIFF_1:33;
end;
