theorem Th12:
  u = <*x0,y0,z0*> & f is_hpartial_differentiable`13_in u implies
  hpartdiff13(f,u) = diff(SVF1(3,pdiff1(f,1),u),z0)
proof
    set r = hpartdiff13(f,u);
    assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`13_in u;
    consider x1,y1,z1 being Real such that
A3: u = <*x1,y1,z1*> & ex N being Neighbourhood of z1 st
    N c= dom SVF1(3,pdiff1(f,1),u) & ex L,R st for z st z in N holds
    SVF1(3,pdiff1(f,1),u).z - SVF1(3,pdiff1(f,1),u).z1 = L.(z-z1) + R.(z-z1)
    by A2;
    consider N being Neighbourhood of z1 such that
A4: N c= dom SVF1(3,pdiff1(f,1),u) & ex L,R st for z st z in N holds
    SVF1(3,pdiff1(f,1),u).z - SVF1(3,pdiff1(f,1),u).z1 = L.(z-z1) + R.(z-z1)
    by A3;
    consider L,R such that
A5: for z st z in N holds
    SVF1(3,pdiff1(f,1),u).z - SVF1(3,pdiff1(f,1),u).z1 = L.(z-z1) + R.(z-z1)
    by A4;
A6: x0 = x1 & y0 = y1 & z0 = z1 by A1,A3,FINSEQ_1:78;
A7: r = L.1 by A2,A3,A4,A5,Def12;
 SVF1(3,pdiff1(f,1),u) is_differentiable_in z0 by A4,A6,FDIFF_1:def 4;
    hence thesis by A4,A5,A6,A7,FDIFF_1:def 5;
end;
