reserve i,n,m for Nat;

theorem
for m,n be non zero Nat, f be PartFunc of REAL m,REAL n,
    Z be Subset of REAL m st
 f is_differentiable_on Z
  ex Z0 be Subset of REAL-NS m st Z=Z0 & Z0 is open
proof
   let m,n be non zero Nat;
   let f be PartFunc of REAL m,REAL n,
       Z be Subset of REAL m;
   assume A1: f is_differentiable_on Z;
   the carrier of REAL-NS m = REAL m &
   the carrier of REAL-NS n = REAL n by REAL_NS1:def 4; then
   reconsider g=f as PartFunc of REAL-NS m,REAL-NS n;
   reconsider Z0=Z as Subset of REAL-NS m by REAL_NS1:def 4;
   take Z0;
   g is_differentiable_on Z0 by A1,Th30;
   hence thesis by NDIFF_1:32;
end;
