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 Th9:
  for f be PartFunc of REAL 2,REAL for z be Element of REAL 2 holds
  f is_partial_differentiable_in z,1 iff
  ex x0,y0 being Real st z = <*x0,y0*> &
  ex N being Neighbourhood of x0 st N c= dom SVF1(1,f,z) & ex L,R st for x st x
  in N holds SVF1(1,f,z).x - SVF1(1,f,z).x0 = L.(x-x0) + R.(x-x0)
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,1;
    thus ex x0,y0 being Real
      st z = <*x0,y0*> & ex N being Neighbourhood of x0
st N c= dom SVF1(1,f,z) & ex L,R st for x st x in N holds SVF1(1,f,z).x - SVF1(
    1,f,z).x0 = L.(x-x0) + R.(x-x0)
    proof
      consider x0,y0 such that
A2:   z = <*x0,y0*> and
A3:   SVF1(1,f,z) is_differentiable_in x0 by A1,Th5;
      ex N being Neighbourhood of x0 st N c= dom SVF1(1,f,z) & ex L,R st
for x st x in N holds SVF1(1,f,z).x - SVF1(1,f,z).x0 = L.(x-x0) + R.(x-x0) by
A3,FDIFF_1:def 4;
      hence thesis by A2;
    end;
  end;
  given x0,y0 being Real such that
A4: z = <*x0,y0*> and
A5: ex N being Neighbourhood of
  x0 st N c= dom SVF1(1,f,z) & ex L,R st for x st x in N holds SVF1(1,f,z).x -
  SVF1(1,f,z).x0 = L.(x-x0) + R.(x-x0);
::   consider x0,y0 such that
:: A4: z = <*x0,y0*> and
:: A5: ex N being Neighbourhood of x0 st N c= dom SVF1(1,f,z) & ex L,R st
::   for x st x in N holds SVF1(1,f,z).x - SVF1(1,f,z).x0 = L.(x-x0) + R.(x-x0);
  SVF1(1,f,z) is_differentiable_in x0 by A5,FDIFF_1:def 4;
  hence thesis by A4,Th5;
end;
