
theorem Th29:
  for m,n be non zero Element of NAT,
        x be Point of REAL-NS m,
        f be PartFunc of REAL-NS m,REAL-NS n
    st f is_differentiable_in x
  holds diff(f,x) = Mx2Tran(Jacobian(f,x))
proof
  let m,n be non zero Element of NAT,
        x be Point of REAL-NS m,
        f be PartFunc of REAL-NS m,REAL-NS n;

  assume
  A1: f is_differentiable_in x;

  consider g be PartFunc of REAL m,REAL n,
            y be Element of REAL m such that
  A2: g = f & y = x
    & Jacobian(f,x) = Jacobian(g,y) by Def2;

  A3: g is_differentiable_in y by A1,A2;
  thus diff(f,x)
    = diff(g,y) by A1,A2,PDIFF_1:21
  .= Mx2Tran(Jacobian(f,x)) by A2,A3,Th28;
end;
