theorem Th5:
  for f be PartFunc of REAL 2,REAL for z be Element of REAL 2 holds
  (ex x0,y0 being Real
    st z = <*x0,y0*> & SVF1(1,f,z) is_differentiable_in x0)
  iff f is_partial_differentiable_in z,1
proof
  let f be PartFunc of REAL 2,REAL;
  let z be Element of REAL 2;
  consider x0,y0 being Element of REAL such that
A1: z = <*x0,y0*> by FINSEQ_2:100;
  thus (ex x0,y0 being Real st z = <*x0,y0*> &
  SVF1(1,f,z) is_differentiable_in x0) implies
  f is_partial_differentiable_in z,1 by Th1;
  assume
A2: f is_partial_differentiable_in z,1;
  proj(1,2).z = x0 by A1,Th1;
  then SVF1(1,f,z) is_differentiable_in x0 by A2;
  hence thesis by A1;
end;
