reserve A,B,p,q,r,s for Element of LTLB_WFF,
  n for Element of NAT,
  X for Subset of LTLB_WFF,
  g for Function of LTLB_WFF,BOOLEAN,
  x,y for set;

theorem p in tau X implies tau1.p c= tau X
  proof
    assume p in tau X;
    then consider B such that
A1: B in X and
A2: p in tau1.B by Def5;
A3: tau1.B c= tau X
    by Def5,A1;
    tau1.p c= tau1.B by A2,Th8;
    hence thesis by A3;
  end;
