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
  for z0 being Element of REAL 2 for N being Neighbourhood of proj(1,2).
  z0 st f is_hpartial_differentiable`21_in z0 & N c= dom SVF1(1,pdiff1(f,2),z0)
  holds for h be 0-convergent non-zero Real_Sequence,
  c be constant Real_Sequence st
rng c = {proj(1,2).z0} & rng (h+c) c= N holds h"(#)(SVF1(1,pdiff1(f,2),z0)/*(h+
c) - SVF1(1,pdiff1(f,2),z0)/*c) is convergent & hpartdiff21(f,z0) = lim (h"(#)(
  SVF1(1,pdiff1(f,2),z0)/*(h+c) - SVF1(1,pdiff1(f,2),z0)/*c))
proof
  let z0 be Element of REAL 2;
  let N be Neighbourhood of proj(1,2).z0;
  assume that
A1: f is_hpartial_differentiable`21_in z0 and
A2: N c= dom SVF1(1,pdiff1(f,2),z0);
  let h be 0-convergent non-zero Real_Sequence,
  c be constant Real_Sequence such
  that
A3: rng c = {proj(1,2).z0} & rng (h+c) c= N;
  consider x0,y0 being Element of REAL such that
A4: z0 = <*x0,y0*> by FINSEQ_2:100;
A5: pdiff1(f,2) is_partial_differentiable_in z0,1 by A1,Th11;
  then partdiff(pdiff1(f,2),z0,1) = diff(SVF1(1,pdiff1(f,2),z0),x0) by A4,
PDIFF_2:13
    .= hpartdiff21(f,z0) by A1,A4,Th7;
  hence thesis by A2,A3,A5,PDIFF_2:17;
end;
