theorem
  f is_hpartial_differentiable`31_in u0 implies
  r(#)pdiff1(f,3) is_partial_differentiable_in u0,1 &
  partdiff((r(#)pdiff1(f,3)),u0,1) = r*hpartdiff31(f,u0)
proof
    assume
A1: f is_hpartial_differentiable`31_in u0;
    then pdiff1(f,3) is_partial_differentiable_in u0,1
    by Th25;then
    r(#)pdiff1(f,3) is_partial_differentiable_in u0,1 &
    partdiff((r(#)pdiff1(f,3)),u0,1) = r*partdiff(pdiff1(f,3),u0,1)
    by PDIFF_1:33;
    hence thesis by A1,Th34;
end;
