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

theorem Th54:
  for X be Subset of REAL m, f be PartFunc of REAL m,REAL st
  X c= dom f & X is open
holds
   f is_differentiable_on X
 iff
   for x be Element of REAL m st x in X holds f is_differentiable_in x
proof
   let X be Subset of REAL m, f be PartFunc of REAL m,REAL;
   set  g=<>*f;
   assume
A1: X c= dom f;
   assume X is open; then
A2:g is_differentiable_on X
    iff
   X c=dom g &
   for x be Element of REAL m st x in X holds g is_differentiable_in x
              by PDIFF_6:32;
   thus f is_differentiable_on X implies
   for x being Element of REAL m st x in X
    holds f is_differentiable_in x by A1,A2,Th53;
   assume for x be Element of REAL m st x in X holds f is_differentiable_in x;
   hence f is_differentiable_on X by A1,A2,Th3,Th53,PDIFF_7:def 1;
end;
