theorem Th31:
  f is_hpartial_differentiable`21_in u implies
  hpartdiff21(f,u) = partdiff(pdiff1(f,2),u,1)
proof
    assume
A1: f is_hpartial_differentiable`21_in u;
    consider x0,y0,z0 being Element of REAL such that
A2: u = <*x0,y0,z0*> by FINSEQ_2:103;
    hpartdiff21(f,u) = diff(SVF1(1,pdiff1(f,2),u),x0) by A1,A2,Th13
    .= partdiff(pdiff1(f,2),u,1) by A2,PDIFF_4:19;
    hence thesis;
end;
