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
  z = <*x0,y0*> & f is_hpartial_differentiable`11_in z implies SVF1(1,
  pdiff1(f,1),z) is_differentiable_in x0
proof
  assume that
A1: z = <*x0,y0*> and
A2: f is_hpartial_differentiable`11_in z;
  consider x1,y1 such that
A3: z = <*x1,y1*> and
A4: ex N being Neighbourhood of x1 st N c= dom SVF1(1,pdiff1(f,1),z) &
ex L,R st for x st x in N holds SVF1(1,pdiff1(f,1),z).x - SVF1(1,pdiff1(f,1),z)
  .x1 = L.(x-x1) + R.(x-x1) by A2;
  x0 = x1 by A1,A3,FINSEQ_1:77;
  hence thesis by A4,FDIFF_1:def 4;
end;
