theorem Th15:
  f is_hpartial_differentiable`21_in z implies hpartdiff21(f,z) =
  partdiff(pdiff1(f,2),z,1)
proof
  consider x0,y0 being Element of REAL such that
A1: z = <*x0,y0*> by FINSEQ_2:100;
  assume
A2: f is_hpartial_differentiable`21_in z;
  then
A3: pdiff1(f,2) is_partial_differentiable_in z,1 by Th11;
  hpartdiff21(f,z) = diff(SVF1(1,pdiff1(f,2),z),x0) by A2,A1,Th7
    .= partdiff(pdiff1(f,2),z,1) by A1,A3,PDIFF_2:13;
  hence thesis;
end;
