
theorem
  for m,n be non zero Nat,
        f be PartFunc of REAL m,REAL n,
        x be Element of REAL m
   st f is_differentiable_in x
  holds
    for i,j be Nat
     st 1 <= i <= m
      & 1 <= j <= n
    holds f is_partial_differentiable_in x,i,j
proof
  let m,n be non zero Nat;
  let f be PartFunc of REAL m,REAL n;
  let x be Element of REAL m;

  assume
  A1: f is_differentiable_in x;

  let i,j be Nat;
  assume
  A2: 1 <= i <= m;
  assume 1 <= j <= n; then
  consider fj be PartFunc of REAL m,REAL such that
  A4: fj = proj(j,n) * f
    & fj is_differentiable_in x by A1,Th19;

  fj is_partial_differentiable_in x,i by A2,A4,PDIFF_7:23;
  hence thesis by A4;
end;
