
theorem Th23:
  for L being RelStr, X being Subset of L holds X is lower iff downarrow X c= X
proof
  let L be RelStr, X be Subset of L;
  hereby
    assume
A1: X is lower;
    thus downarrow X c= X
    proof
      let x be object;
      assume
A2:   x in downarrow X;
      then reconsider x9 = x as Element of L;
      ex y being Element of L st x9 <= y & y in X by A2,Def15;
      hence thesis by A1;
    end;
  end;
  assume
A3: downarrow X c= X;
  let x,y be Element of L;
  assume that
A4: x in X and
A5: y <= x;
  y in downarrow X by A4,A5,Def15;
  hence thesis by A3;
end;
