reserve x,x0,x1,y,y0,y1,z,z0,z1,r,r1,s,p,p1 for Real;
reserve u,u0 for Element of REAL 3;
reserve n for Element of NAT;
reserve s1 for Real_Sequence;
reserve f,f1,f2 for PartFunc of REAL 3,REAL;
reserve R,R1 for RestFunc;
reserve L,L1 for LinearFunc;

theorem
  u = <*x0,y0,z0*> & f is_hpartial_differentiable`22_in u implies
  SVF1(2,pdiff1(f,2),u) is_differentiable_in y0
proof
    assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`22_in u;
    consider x1,y1,z1 such that
A3: u = <*x1,y1,z1*> & ex N being Neighbourhood of y1 st
    N c= dom SVF1(2,pdiff1(f,2),u) & ex L,R st for y st y in N holds
    SVF1(2,pdiff1(f,2),u).y - SVF1(2,pdiff1(f,2),u).y1 = L.(y-y1) + R.(y-y1)
    by A2;
    y0 = y1 by A1,A3,FINSEQ_1:78;
    hence thesis by A3,FDIFF_1:def 4;
end;
