
theorem Th26:
  for L being RelStr, x be set holds x is directed Subset of L iff
  x is filtered Subset of L opp
proof
  let L be RelStr, x be set;
  hereby
    assume x is directed Subset of L;
    then reconsider X = x as directed Subset of L;
    reconsider Y = X as Subset of L opp;
    Y is filtered
    proof
      let x,y be Element of L opp;
      assume x in Y & y in Y;
      then consider z being Element of L such that
A1:   z in X & z >= ~x & z >= ~y by WAYBEL_0:def 1;
      take z~;
      ~(z~) = z~;
      hence thesis by A1,Th1;
    end;
    hence x is filtered Subset of L opp;
  end;
  assume x is filtered Subset of L opp;
  then reconsider X = x as filtered Subset of L opp;
  reconsider Y = X as Subset of L;
  Y is directed
  proof
    let x,y be Element of L;
    assume x in Y & y in Y;
    then consider z being Element of L opp such that
A2: z in X & z <= x~ & z <= y~ by WAYBEL_0:def 2;
    take ~z;
    (~z)~ = ~z;
    hence thesis by A2,LATTICE3:9;
  end;
  hence thesis;
end;
