theorem
  u = <*x0,y0,z0*> & f is_hpartial_differentiable`12_in u implies
  SVF1(2,pdiff1(f,1),u) is_differentiable_in y0
proof
    assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`12_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,1),u) & ex L,R st for y st y in N holds
    SVF1(2,pdiff1(f,1),u).y - SVF1(2,pdiff1(f,1),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;
