theorem
  f is_hpartial_differentiable`11_in z0 implies r(#)pdiff1(f,1)
  is_partial_differentiable_in z0,1 & partdiff((r(#)pdiff1(f,1)),z0,1) = r*
  hpartdiff11(f,z0)
proof
  assume
A1: f is_hpartial_differentiable`11_in z0;
   reconsider r as Real;
A2: pdiff1(f,1) is_partial_differentiable_in z0,1 by Th9,A1;
  then partdiff((r(#)pdiff1(f,1)),z0,1) = r*partdiff(pdiff1(f,1),z0,1) by
PDIFF_1:33;
  hence thesis by A1,A2,Th13,PDIFF_1:33;
end;
