theorem
for f be PartFunc of REAL,REAL n,
    x0 be Real holds
f is_differentiable_in x0 iff
(for i be Element of NAT st 1<=i & i <=n holds
      Proj(i,n)*f is_differentiable_in x0)
proof
  let f be PartFunc of REAL,REAL n, x0 be Real;
  thus f is_differentiable_in x0 implies
   (for i be Element of NAT st 1<=i & i <=n holds
      Proj(i,n)*f is_differentiable_in x0)
  by Th25;
  assume
A1: for i be Element of NAT st 1<=i & i <=n holds
      Proj(i,n)*f is_differentiable_in x0;
    reconsider g=f as PartFunc of REAL,REAL-NS n by REAL_NS1:def 4;
    for i be Element of NAT st 1<=i & i <=n holds
                  Proj(i,n)*g is_differentiable_in x0 by A1;
    then g is_differentiable_in x0 by Th25;
    hence thesis;
end;
