
theorem
  for m,n be non zero Element of NAT,
        X be Subset of REAL-NS m,
        f be PartFunc of REAL-NS m,REAL-NS n
    st X is open & X c= dom f
  holds
    f is_differentiable_on X & f `| X is_continuous_on X
      iff
    for i,j be Nat st 1 <= i <= m & 1 <= j <= n
    holds
      Proj(j,n) * f is_partial_differentiable_on X,i
    & (Proj(j,n) * f) `partial| (X,i) is_continuous_on X
proof
  let m,n be non zero Element of NAT,
        X be Subset of REAL-NS m,
        f be PartFunc of REAL-NS m,REAL-NS n;
  assume
  A1: X is open & X c= dom f;

  hereby
    assume
    A2: f is_differentiable_on X & f `| X is_continuous_on X;
    let i,j be Nat;
    assume
    A3: 1 <= i <= m;
    assume
    A4: 1 <= j <= n;

    f is_partial_differentiable_on X,i
    & f `partial| (X,i) is_continuous_on X by A1,A2,A3,PDIFF_8:22;

    hence
      Proj(j,n) * f is_partial_differentiable_on X,i
    & (Proj(j,n) * f) `partial| (X,i) is_continuous_on X
      by A1,A3,A4,PDIFF_8:20;
  end;

  assume
  A5: for i,j be Nat st 1 <= i <= m & 1 <= j <= n
      holds
        Proj(j,n) * f is_partial_differentiable_on X,i
      & (Proj(j,n) * f) `partial| (X,i) is_continuous_on X;

  for i be Nat st 1 <= i <= m
  holds
    f is_partial_differentiable_on X,i
  & f `partial| (X,i) is_continuous_on X
  proof
    let i be Nat;
    assume
    A6: 1 <= i <= m; then

    for j be Nat st 1 <= j <= n
    holds
      Proj(j,n) * f is_partial_differentiable_on X,i
    & (Proj(j,n) * f) `partial| (X,i) is_continuous_on X by A5;
    hence
      f is_partial_differentiable_on X,i
    & f `partial| (X,i) is_continuous_on X by A1,A6,PDIFF_8:20;
  end;
  hence f is_differentiable_on X & f `| X is_continuous_on X
    by A1,PDIFF_8:22;
end;
