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 Th10:
  p 'U' q in tau1.(A '&&' B)implies(p 'U' q in tau1.A or p 'U' q in tau1.B)
  proof
    set a = p 'U' q,nb = 'not' B;
    assume a in tau1.(A '&&' B);
    then a in tau1.(A => nb) by Th9;
    then a in {A => nb} \/ tau1.A \/ tau1.nb by Def4;
    then A1: a in {A => nb} \/ (tau1.A \/ tau1.nb) by XBOOLE_1:4;
    a <> A => nb by HILBERT2:22;
    then not a in {A => nb} by TARSKI:def 1;
    then a in tau1.A \/ tau1.nb by A1,XBOOLE_0:def 3;
    then a in tau1.A or a in tau1.nb by XBOOLE_0:def 3;
    hence thesis by Th9;
  end;
