theorem Th21:
  f is_differentiable_on X & Z c= X implies f is_differentiable_on Z
proof
  assume A1: f is_differentiable_on X & Z c= X;
  then
A2: X c= dom f & for x st x in X holds f|X is_differentiable_in x;
A3: Z c= dom f by A1,A2;
  reconsider g=f as PartFunc of REAL,REAL-NS n by REAL_NS1:def 4;
  now
    let x;
    assume x in X;
    then f|X is_differentiable_in x by A1;
    hence g|X is_differentiable_in x;
  end;
  then g is_differentiable_on X by A2,NDIFF_3:def 5;
  then
A4: g is_differentiable_on Z by A1,NDIFF_3:24;
  now
    let x;
    assume x in Z;
    then g|Z is_differentiable_in x by A4,NDIFF_3:def 5;
    hence f|Z is_differentiable_in x;
  end;
  hence thesis by A3;
end;
