reserve x,x0,x1,y,y0,y1,r,r1,s,p,p1 for Real;
reserve z,z0 for Element of REAL 2;
reserve n for Element of NAT;
reserve s1 for Real_Sequence;
reserve f,f1,f2 for PartFunc of REAL 2,REAL;
reserve R,R1 for RestFunc;
reserve L,L1 for LinearFunc;

theorem Th13:
  f is_hpartial_differentiable`11_in z implies hpartdiff11(f,z) =
  partdiff(pdiff1(f,1),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`11_in z;
  then
A3: pdiff1(f,1) is_partial_differentiable_in z,1 by Th9;
  hpartdiff11(f,z) = diff(SVF1(1,pdiff1(f,1),z),x0) by A2,A1,Th5
    .= partdiff(pdiff1(f,1),z,1) by A1,A3,PDIFF_2:13;
  hence thesis;
end;
