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 Th15: tau X = union {tau1.p where p is Element of LTLB_WFF: p in X}
  proof
    set A = {tau1.p where p is Element of l: p in X};
    hereby
      let x be object;
      assume x in tau X;
      then consider p such that
A1:   p in X and
A2:   x in tau1.p by Def5;
      tau1.p in A by A1;
      hence x in union A by TARSKI:def 4,A2;
    end;
    let x be object;
    assume x in union A;
    then consider y such that
A3: x in y and
A4: y in A by TARSKI:def 4;
A5: ex p st y = tau1.p & p in X by A4;
    then reconsider x1 = x as Element of l by A3;
    thus x in tau X by A5,A3,Def5;
  end;
