reserve A,B,p,q,r for Element of LTLB_WFF,
  M for LTLModel,
  j,k,n for Element of NAT,
  i for Nat,
  X for Subset of LTLB_WFF,
  F for finite Subset of LTLB_WFF,
  f for FinSequence of LTLB_WFF,
  g for Function of LTLB_WFF,BOOLEAN,
  x,y,z for set,
  P,Q,R for PNPair;

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;
