reserve a,b,c for boolean object;
reserve p,q,r,s,A,B,C for Element of LTLB_WFF,
        F,G,X,Y for Subset of LTLB_WFF,
        i,j,k,n for Element of NAT,
        f,f1,f2,g for FinSequence of LTLB_WFF;
reserve M for LTLModel;

theorem Th46:
  X|-r=>(p '&&' q) implies X|-r=>p & X|-r=>q
 proof
  assume A1: X|-r=>(p '&&' q);
  set A=(r=>(p '&&' q))=>(r=>p);
  A is LTL_TAUT_OF_PL
  proof
   let f be Function of LTLB_WFF,BOOLEAN;
   thus(VAL f).A=(VAL f).(r=>(p '&&' q))=>(VAL f).(r=>p) by Def15
    .=(VAL f).r=>(VAL f).(p '&&' q)=>(VAL f).(r=>p) by Def15
    .=(VAL f).r=>((VAL f).p '&'(VAL f).q)=>(VAL f).(r=>p) by Th31
    .=(VAL f).r=>((VAL f).p '&'(VAL f).q)=>((VAL f).r=>(VAL f).p) by Def15
    .=1 by Th1;
  end;
  then A in LTL_axioms by Def17;
  then X|-A by Th42;
  hence X|-r=>p by A1,Th43;
  set A=(r=>(p '&&' q))=>(r=>q);
  A is LTL_TAUT_OF_PL
  proof
   let f be Function of LTLB_WFF,BOOLEAN;
   thus(VAL f).A=(VAL f).(r=>(p '&&' q))=>(VAL f).(r=>q) by Def15
    .=(VAL f).r=>(VAL f).(p '&&' q)=>(VAL f).(r=>q) by Def15
    .=(VAL f).r=>((VAL f).p '&'(VAL f).q)=>(VAL f).(r=>q) by Th31
    .=(VAL f).r=>((VAL f).p '&'(VAL f).q)=>((VAL f).r=>(VAL f).q) by Def15
    .=1 by Th1;
  end;
  then A in LTL_axioms by Def17;
  then X|-A by Th42;
  hence X|-r=>q by A1,Th43;
 end;
