theorem
  f is_hpartial_differentiable`11_in u0 implies
  r(#)pdiff1(f,1) is_partial_differentiable_in u0,1 &
  partdiff((r(#)pdiff1(f,1)),u0,1) = r*hpartdiff11(f,u0)
proof
    assume
A1: f is_hpartial_differentiable`11_in u0;
    then pdiff1(f,1) is_partial_differentiable_in u0,1 by Th19;then
    r(#)pdiff1(f,1) is_partial_differentiable_in u0,1 &
    partdiff((r(#)pdiff1(f,1)),u0,1) = r*partdiff(pdiff1(f,1),u0,1)
    by PDIFF_1:33;
    hence thesis by A1,Th28;
end;
