theorem Th24: p 'U' q in Sub.(p 'U' q)
  proof
    set puq = p 'U' q;
A1: tau1.puq c= tau1.(p '&&' puq) by Th13;
    puq in {puq} by TARSKI:def 1;
    then A2: puq in tau1.puq by Def4;
    tau1.untn(p,q) \/ Sub.p c= tau1.untn(p,q) \/ Sub.p \/ Sub.q by XBOOLE_1:7;
    then tau1.untn(p,q) c= tau1.untn(p,q) \/ Sub.p &
    tau1.untn(p,q) \/ Sub.p c= Sub.( puq) by XBOOLE_1:7,Def6;
    then A3: tau1.untn(p,q) c= Sub.(puq);
    tau1.(p '&&' puq) c= tau1.untn(p,q) by Th14;
    then tau1.(p '&&' puq) c= Sub.(puq) by A3;
    then tau1.puq c= Sub.(puq) by A1;
    hence thesis by A2;
  end;
