theorem
  f1 is_hpartial_differentiable`21_in z0 & f2
  is_hpartial_differentiable`21_in z0 implies pdiff1(f1,2)-pdiff1(f2,2)
is_partial_differentiable_in z0,1 & partdiff(pdiff1(f1,2)-pdiff1(f2,2),z0,1) =
  hpartdiff21(f1,z0) - hpartdiff21(f2,z0)
proof
  assume that
A1: f1 is_hpartial_differentiable`21_in z0 and
A2: f2 is_hpartial_differentiable`21_in z0;
A3: pdiff1(f1,2) is_partial_differentiable_in z0,1 & pdiff1(f2,2)
  is_partial_differentiable_in z0,1 by A1,A2,Th11;
  then
  partdiff(pdiff1(f1,2)-pdiff1(f2,2),z0,1) = partdiff(pdiff1(f1,2),z0,1) -
  partdiff(pdiff1(f2,2),z0,1) by PDIFF_1:31;
  then
  partdiff(pdiff1(f1,2)-pdiff1(f2,2),z0,1) = hpartdiff21(f1,z0) - partdiff
  (pdiff1(f2,2),z0,1) by A1,Th15
    .= hpartdiff21(f1,z0) - hpartdiff21(f2,z0) by A2,Th15;
  hence thesis by A3,PDIFF_1:31;
end;
