reserve D for set;
reserve x,x0,x1,x2,y,y0,y1,y2,z,z0,z1,z2,r,s,t for Real;
reserve p,a,u,u0 for Element of REAL 3;
reserve n,m,k for Element of NAT;
reserve f,f1,f2,f3,g for PartFunc of REAL 3,REAL;
reserve R,R1,R2 for RestFunc;
reserve L,L1,L2 for LinearFunc;

theorem
  u = <*x0,y0,z0*> & f is_partial_differentiable_in u,2 implies
  ex N being Neighbourhood of y0 st N c= dom SVF1(2,f,u) &
  ex L,R st for y st y in N holds
  SVF1(2,f,u).y - SVF1(2,f,u).y0 = L.(y-y0) + R.(y-y0)
proof
    assume that
A1: u = <*x0,y0,z0*> and
A2: f is_partial_differentiable_in u,2;
    consider x1,y1,z1 such that
A3: u = <*x1,y1,z1*> & SVF1(2,f,u) is_differentiable_in y1 by A2,Th8;
    SVF1(2,f,u) is_differentiable_in y0 by A1,A3,FINSEQ_1:78;
    hence thesis by FDIFF_1:def 4;
end;
