
theorem
  for S, T being RelStr st S, T are_isomorphic & S is lower-bounded
  holds T is lower-bounded
proof
  let S, T be RelStr;
  given f being Function of S, T such that
A1: f is isomorphic;
  per cases;
  suppose
    S is non empty & T is non empty;
    then reconsider s = S, t = T as non empty RelStr;
    given X being Element of S such that
A2: X is_<=_than the carrier of S;
    reconsider x = X as Element of s;
    reconsider g = f as Function of s, t;
    reconsider y = g.x as Element of T;
    take y;
    y is_<=_than g.:[#]S by A1,A2,YELLOW_2:13;
    then y is_<=_than rng g by RELSET_1:22;
    hence thesis by A1,WAYBEL_0:66;
  end;
  suppose
    S is empty or T is empty;
    then T is empty by A1,WAYBEL_0:def 38;
    then reconsider T as bounded RelStr;
    T is lower-bounded;
    hence thesis;
  end;
end;
