theorem
  u = <*x0,y0,z0*> & f is_partial_differentiable_in u,3 implies
  ex N being Neighbourhood of z0 st N c= dom SVF1(3,f,u) &
  ex L,R st for z st z in N holds
  SVF1(3,f,u).z - SVF1(3,f,u).z0 = L.(z-z0) + R.(z-z0)
proof
    assume that
A1: u = <*x0,y0,z0*> and
A2: f is_partial_differentiable_in u,3;
    consider x1,y1,z1 such that
A3: u = <*x1,y1,z1*> &
    SVF1(3,f,u) is_differentiable_in z1 by A2,Th9;
    SVF1(3,f,u) is_differentiable_in z0 by A1,A3,FINSEQ_1:78;
    hence thesis by FDIFF_1:def 4;
end;
