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 Th9: p 'U' q in tau1.('not' A) implies p 'U' q in tau1.A
  proof
    set a = p 'U' q,na = 'not' A,f = TFALSUM;
A1: a <> A => f by HILBERT2:22;
    assume a in tau1.na;
    then a in {A => f} \/ tau1.A \/ tau1.f by Def4;
    then A2: a in {A => f} \/ tau1.A or a in tau1.f by XBOOLE_0:def 3;
    a <> f by HILBERT2:23;
    then not a in {f} by TARSKI:def 1;
    then a in {A => f} or a in tau1.A by A2,Def4,XBOOLE_0:def 3;
    hence thesis by A1,TARSKI:def 1;
  end;
