theorem Th32:
  f is_partial_differentiable_in x,i implies r(#)f
  is_partial_differentiable_in x,i & partdiff((r(#)f),x,i) = r*partdiff(f,x,i)
proof
  assume f is_partial_differentiable_in x,i;
  then
A1: f*reproj(i,x) is_differentiable_in Proj(i,m).x;
  r(#)(f*reproj(i,x)) = (r(#)f)*reproj(i,x) by Th27;
  then (r(#)f)*reproj(i,x) is_differentiable_in Proj(i,m).x by A1,NDIFF_1:37;
  hence r(#)f is_partial_differentiable_in x,i;
  diff(r(#)(f*reproj(i,x)),Proj(i,m).x) = partdiff(r(#)f,x,i) by Th27;
  hence thesis by A1,NDIFF_1:37;
end;
