reserve x,x0,x1,x2,y,y0,y1,y2,r,r1,s,p,p1 for Real;
reserve z,z0 for Element of REAL 2;
reserve n,m,k for Element of NAT;
reserve Z for Subset of REAL 2;
reserve s1 for Real_Sequence;
reserve f,f1,f2 for PartFunc of REAL 2,REAL;
reserve R,R1,R2 for RestFunc;
reserve L,L1,L2 for LinearFunc;

theorem
  z = <*x0,y0*> & f is_partial_differentiable_in z,2 implies ex N being
  Neighbourhood of y0 st N c= dom SVF1(2,f,z) & ex L,R st for y st y in N holds
  SVF1(2,f,z).y - SVF1(2,f,z).y0 = L.(y-y0) + R.(y-y0)
proof
  assume that
A1: z = <*x0,y0*> and
A2: f is_partial_differentiable_in z,2;
  ex x1,y1 st z = <*x1,y1*> & SVF1(2,f,z) is_differentiable_in y1 by A2,Th6;
  then SVF1(2,f,z) is_differentiable_in y0 by A1,FINSEQ_1:77;
  hence thesis by FDIFF_1:def 4;
end;
