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 Th10:
  for f be PartFunc of REAL 2,REAL for z be Element of REAL 2
holds f is_partial_differentiable_in z,2 iff
 ex x0,y0 being Real st z = <*x0,y0
  *> & 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
  let f be PartFunc of REAL 2,REAL;
  let z be Element of REAL 2;
  hereby
    assume
A1: f is_partial_differentiable_in z,2;
    thus ex x0,y0 being Real
   st z = <*x0,y0*> & 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
      consider x0,y0 such that
A2:   z = <*x0,y0*> and
A3:   SVF1(2,f,z) is_differentiable_in y0 by A1,Th6;
      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) by
A3,FDIFF_1:def 4;
      hence thesis by A2;
    end;
  end;
::   assume ex x0,y0 being Real st z = <*x0,y0*> & 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);
::   then
  given x0,y0 such that
A4: z = <*x0,y0*> and
A5: 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);
  SVF1(2,f,z) is_differentiable_in y0 by A5,FDIFF_1:def 4;
  hence thesis by A4,Th6;
end;
