theorem
  f is_hpartial_differentiable`12_in u0 implies
  r(#)pdiff1(f,1) is_partial_differentiable_in u0,2 &
  partdiff((r(#)pdiff1(f,1)),u0,2) = r*hpartdiff12(f,u0)
proof
    assume
A1: f is_hpartial_differentiable`12_in u0;
    then pdiff1(f,1) is_partial_differentiable_in u0,2 by Th20;then
    r(#)pdiff1(f,1) is_partial_differentiable_in u0,2 &
    partdiff((r(#)pdiff1(f,1)),u0,2) = r*partdiff(pdiff1(f,1),u0,2)
    by PDIFF_1:33;
    hence thesis by A1,Th29;
end;
