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

theorem Th8:
  z = <*x0,y0*> & f is_hpartial_differentiable`22_in z implies
  hpartdiff22(f,z) = diff(SVF1(2,pdiff1(f,2),z),y0)
proof
  set r = hpartdiff22(f,z);
  assume that
A1: z = <*x0,y0*> and
A2: f is_hpartial_differentiable`22_in z;
  consider x1,y1 such that
A3: z = <*x1,y1*> and
A4: ex N being Neighbourhood of y1 st N c= dom SVF1(2,pdiff1(f,2),z) &
ex L,R st for y st y in N holds SVF1(2,pdiff1(f,2),z).y - SVF1(2,pdiff1(f,2),z)
  .y1 = L.(y-y1) + R.(y-y1) by A2;
  consider N being Neighbourhood of y1 such that
A5: N c= dom SVF1(2,pdiff1(f,2),z) and
A6: ex L,R st for y st y in N holds SVF1(2,pdiff1(f,2),z).y - SVF1(2,
  pdiff1(f,2),z).y1 = L.(y-y1) + R.(y-y1) by A4;
A7: y0 = y1 by A1,A3,FINSEQ_1:77;
  then
A8: SVF1(2,pdiff1(f,2),z) is_differentiable_in y0 by A5,A6,FDIFF_1:def 4;
  consider L,R such that
A9: for y st y in N holds SVF1(2,pdiff1(f,2),z).y - SVF1(2,pdiff1(f,2),z
  ).y1 = L.(y-y1) + R.(y-y1) by A6;
  r = L.1 by A2,A3,A5,A9,Def9;
  hence thesis by A5,A9,A7,A8,FDIFF_1:def 5;
end;
