theorem Th9: rng P c= union Subt rng P
  proof
    let x be object;
    assume A1: x in rng P;
    then reconsider x1 = x as Element of l;
A2: x in tau1.x1 & tau1.x1 c= Sub.x1 by LTLAXIO3:6, LTLAXIO3:25;
    Sub.x1 in Subt rng P by A1;
    hence thesis by A2,TARSKI:def 4;
  end;
