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 Th8: p in tau1.q implies tau1.p c= tau1.q
  proof
    defpred P1[Element of l] means $1 in tau1.q implies tau1.$1 c= tau1.q;
A1: for n holds P1[ prop n]
    proof
      let n;
      set pr = prop n;
      assume
A2:   pr in tau1.q;
      let x be object;
      assume x in tau1.pr;
      then x in {pr} by Def4;
      hence x in tau1.q by TARSKI:def 1,A2;
    end;
A3: for r, s st P1[r] & P1[s] holds P1[r 'U' s] & P1[r => s]
    proof
      let r,s;
      assume that A4: P1[r] and A5: P1[s];
      thus P1[r 'U' s]
      proof
        set f = r 'U' s;
        assume
A6:     f in tau1.q;
        let x be object;
        assume x in tau1.f;
        then x in {f} by Def4;
        hence x in tau1.q by TARSKI:def 1,A6;
      end;
      thus P1[r => s]
      proof
        set f = r => s;
        assume
A7:     f in tau1.q;
        then {f} c= tau1.q by ZFMISC_1:31;
        then {f} \/ tau1.r c= tau1.q by XBOOLE_1:8, A7,Th7,A4;then
A8:     {f} \/ tau1.r \/ tau1.s c= tau1.q by XBOOLE_1:8, A7,Th7, A5;
        let x be object;
        assume x in tau1.f;
        then x in {f} \/ tau1.r \/ tau1.s by Def4;
        hence x in tau1.q by A8;
      end;
    end;
A9: P1[TFALSUM]
    proof
      set f = TFALSUM;
      assume
A10:  f in tau1.q;
      let x be object;
      assume x in tau1.f;
      then x in {f} by Def4;
      hence x in tau1.q by TARSKI:def 1,A10;
    end;
    for p holds P1[p] from HILBERT2:sch 2(A9,A1,A3);
    hence thesis;
  end;
