
theorem
  for m,n be non zero Element of NAT,
  f be PartFunc of REAL-NS m,REAL-NS n,
  X be Subset of REAL-NS m st X is open holds
  (for i be Nat st 1 <= i <= m holds
  f is_partial_differentiable_on X,i &
  f`partial|(X,i) is_continuous_on X)
  iff
  f is_differentiable_on X & f`|X is_continuous_on X
proof
  let m,n be non zero Element of NAT, f be PartFunc of REAL-NS m,REAL-NS n,
  X be Subset of REAL-NS m;
assume A1: X is open;
hereby assume A2:for i be Nat st 1 <=i & i <= m holds
  f is_partial_differentiable_on X,i &
  f`partial|(X,i) is_continuous_on X;
now let j be Nat;
  assume A3: 1 <= j <= n;
  now let i be Nat;
    assume A4: 1 <= i <= m; then
    f is_partial_differentiable_on X,i &
    f`partial|(X,i) is_continuous_on X by A2;
    hence (Proj(j,n)*f) is_partial_differentiable_on X,i &
   (Proj(j,n)*f)`partial|(X,i) is_continuous_on X by Th20,A1,A3,A4;
 end;
 hence (Proj(j,n)*f) is_differentiable_on X
   & (Proj(j,n)*f)`|X is_continuous_on X by A1,PDIFF_7:49;
end;
hence f is_differentiable_on X & f`|X is_continuous_on X by A1,Th21;
end;
assume A5:f is_differentiable_on X & f`|X is_continuous_on X;
 let i be Nat;
  assume A6: 1 <=i & i <= m;
 now let j be Nat;
  assume 1 <=j & j <= n; then
  (Proj(j,n)*f) is_differentiable_on X &
  (Proj(j,n)*f)`|X is_continuous_on X by A1,A5,Th21;
  hence (Proj(j,n)*f) is_partial_differentiable_on X,i &
  (Proj(j,n)*f)`partial|(X,i) is_continuous_on X by A1,A6,PDIFF_7:49;
 end;
hence thesis by A6,A1,Th20;
end;
