
theorem
  for m,n be non zero Nat,
        f be PartFunc of REAL-NS m,REAL-NS n,
        x be Element of REAL-NS 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-NS m,REAL-NS n;
  let x be Element of REAL-NS m;

  assume
  A1: f is_differentiable_in x;

  let i,j be Nat;

  assume
  A2: 1 <= i <= m;
  assume
  A3: 1 <= j <= n;

  reconsider fj = Proj(j,n) * f as PartFunc of REAL-NS m,REAL-NS 1;

  fj is_differentiable_in x by A1,A3,PDIFF_6:29;
  then fj is_partial_differentiable_in x,i by A2,PDIFF_7:21;
  hence thesis;
end;
