
theorem Th28:
  for L being RelStr, x be set holds x is lower Subset of L iff x
  is upper Subset of L opp
proof
  let L be RelStr, x be set;
  hereby
    assume x is lower Subset of L;
    then reconsider X = x as lower Subset of L;
    reconsider Y = X as Subset of L opp;
    Y is upper
    proof
      let x,y be Element of L opp;
      assume that
A1:   x in Y and
A2:   x <= y;
      ~x >= ~y by A2,Th1;
      hence thesis by A1,WAYBEL_0:def 19;
    end;
    hence x is upper Subset of L opp;
  end;
  assume x is upper Subset of L opp;
  then reconsider X = x as upper Subset of L opp;
  reconsider Y = X as Subset of L;
  Y is lower
  proof
    let x,y be Element of L;
    assume that
A3: x in Y and
A4: x >= y;
    x~ <= y~ by A4,LATTICE3:9;
    hence thesis by A3,WAYBEL_0:def 20;
  end;
  hence thesis;
end;
