reserve x,x0,x1,y,y0,y1,z,z0,z1,r,r1,s,p,p1 for Real;
reserve u,u0 for Element of REAL 3;
reserve n for Element of NAT;
reserve s1 for Real_Sequence;
reserve f,f1,f2 for PartFunc of REAL 3,REAL;
reserve R,R1 for RestFunc;
reserve L,L1 for LinearFunc;

theorem Th34:
  f is_hpartial_differentiable`31_in u implies
  hpartdiff31(f,u) = partdiff(pdiff1(f,3),u,1)
proof
    assume
A1: f is_hpartial_differentiable`31_in u;
    consider x0,y0,z0 being Element of REAL such that
A2: u = <*x0,y0,z0*> by FINSEQ_2:103;
    hpartdiff31(f,u) = diff(SVF1(1,pdiff1(f,3),u),x0)
    by A1,A2,Th16
    .= partdiff(pdiff1(f,3),u,1) by A2,PDIFF_4:19;
    hence thesis;
end;
