theorem
  f1 is_hpartial_differentiable`12_in u0 &
  f2 is_hpartial_differentiable`12_in u0 implies
  pdiff1(f1,1)+pdiff1(f2,1) is_partial_differentiable_in u0,2 &
  partdiff(pdiff1(f1,1)+pdiff1(f2,1),u0,2)
  = hpartdiff12(f1,u0) + hpartdiff12(f2,u0)
proof
    assume that
A1: f1 is_hpartial_differentiable`12_in u0 and
A2: f2 is_hpartial_differentiable`12_in u0;
A3: pdiff1(f1,1) is_partial_differentiable_in u0,2 by A1,Th20;
A4: pdiff1(f2,1) is_partial_differentiable_in u0,2 by A2,Th20;then
 pdiff1(f1,1)+pdiff1(f2,1) is_partial_differentiable_in u0,2 &
    partdiff(pdiff1(f1,1)+pdiff1(f2,1),u0,2)
    = partdiff(pdiff1(f1,1),u0,2) + partdiff(pdiff1(f2,1),u0,2)
    by A3,PDIFF_1:29;
    then partdiff(pdiff1(f1,1)+pdiff1(f2,1),u0,2)
    = hpartdiff12(f1,u0) + partdiff(pdiff1(f2,1),u0,2) by A1,Th29
   .= hpartdiff12(f1,u0) + hpartdiff12(f2,u0) by A2,Th29;
    hence thesis by A3,A4,PDIFF_1:29;
end;
