theorem
  u = <*x0,y0,z0*> & f is_hpartial_differentiable`11_in u implies
  SVF1(1,pdiff1(f,1),u) is_differentiable_in x0
proof
    assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`11_in u;
    consider x1,y1,z1 such that
A3: u = <*x1,y1,z1*> & ex N being Neighbourhood of x1 st
    N c= dom SVF1(1,pdiff1(f,1),u) & ex L,R st for x st x in N holds
    SVF1(1,pdiff1(f,1),u).x - SVF1(1,pdiff1(f,1),u).x1 = L.(x-x1) + R.(x-x1)
    by A2;
    x0 = x1 by A1,A3,FINSEQ_1:78;
    hence thesis by A3,FDIFF_1:def 4;
end;
