reserve m,n for non zero Element of NAT;
reserve i,j,k for Element of NAT;
reserve Z for set;

theorem Th55:
for X be Subset of REAL m, f be PartFunc of REAL m,REAL st
 X c= dom f & f is_differentiable_on X holds X is open
proof
   let X be Subset of REAL m, f be PartFunc of REAL m,REAL;
   reconsider g=<>*f as PartFunc of REAL m,REAL 1;
   assume X c= dom f & f is_differentiable_on X; then
   g is_differentiable_on X by Th53;
   hence thesis by PDIFF_6:33;
end;
