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 Th6:
  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(2,f,z) is_differentiable_in y0)
  iff f is_partial_differentiable_in z,2
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(2,f,z) is_differentiable_in y0) implies
  f is_partial_differentiable_in z,2 by Th2;
  assume
A2: f is_partial_differentiable_in z,2;
  proj(2,2).z = y0 by A1,Th2;
  then SVF1(2,f,z) is_differentiable_in y0 by A2;
  hence thesis by A1;
end;
