
theorem Th31:
  for L being RelStr holds L opp is lower-bounded iff L is upper-bounded
proof
  let L be RelStr;
  thus L opp is lower-bounded implies L is upper-bounded
  proof
    given x being Element of L opp such that
A1: x is_<=_than the carrier of L opp;
    take ~x;
    thus thesis by A1,Th9;
  end;
  given x being Element of L such that
A2: x is_>=_than the carrier of L;
  take x~;
  thus thesis by A2,Th8;
end;
