theorem
  f is_hpartial_differentiable`12_in z0 implies r(#)pdiff1(f,1)
  is_partial_differentiable_in z0,2 & partdiff((r(#)pdiff1(f,1)),z0,2) = r*
  hpartdiff12(f,z0)
proof
  assume
A1: f is_hpartial_differentiable`12_in z0;
   reconsider r as Real;
A2: pdiff1(f,1) is_partial_differentiable_in z0,2 by Th10,A1;
  partdiff((r(#)pdiff1(f,1)),z0,2) = r*partdiff(pdiff1(f,1),z0,2) by
PDIFF_1:33,A2;
  hence thesis by A1,A2,Th14,PDIFF_1:33;
end;
