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 Th48:
  X|-p=>(q=>r) implies X|-(p '&&' q)=>r
 proof
  set A=(p=>(q=>r))=>((p '&&' q)=>r);
  now let f be Function of LTLB_WFF,BOOLEAN;
   thus(VAL f).A=(VAL f).(p=>(q=>r))=>(VAL f).((p '&&' q)=>r) by Def15
    .=((VAL f).p=>(VAL f).(q=>r))=>(VAL f).((p '&&' q)=>r) by Def15
    .=(VAL f).p=>((VAL f).q=>(VAL f).r)=>(VAL f).((p '&&' q)=>r) by Def15
    .=(VAL f).p=>((VAL f).q=>(VAL f).r)=>((VAL f).(p '&&' q)=>(VAL f).r) by
Def15
    .=(VAL f).p=>((VAL f).q=>(VAL f).r)=>((VAL f).p '&'(VAL f).q=>(VAL f).r)
by Th31
    .=1 by Th2;
  end;
  then A is LTL_TAUT_OF_PL;
  then A in LTL_axioms by Def17;
  then A1: X|-A by Th42;
  assume X|-p=>(q=>r);
  hence X|-(p '&&' q)=>r by A1,Th43;
 end;
