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 Th47:
  X|-p=>q & X|-q=>r implies X|-p=>r
 proof
  assume that
   A1: X|-p=>q and
   A2: X|-q=>r;
  set A=(p=>q)=>((q=>r)=>(p=>r));
  now let f be Function of LTLB_WFF,BOOLEAN;
   thus(VAL f).A=(VAL f).(p=>q)=>(VAL f).((q=>r)=>(p=>r)) by Def15
    .=((VAL f).p=>(VAL f).q)=>(VAL f).((q=>r)=>(p=>r)) by Def15
    .=((VAL f).p=>(VAL f).q)=>((VAL f).(q=>r)=>(VAL f).(p=>r)) by Def15
    .=((VAL f).p=>(VAL f).q)=>((VAL f).q=>(VAL f).r=>(VAL f).(p=>r)) by Def15
    .=((VAL f).p=>(VAL f).q)=>((VAL f).q=>(VAL f).r=>((VAL f).p=>(VAL f).r))
by Def15
    .=1 by XBOOLEAN:106;
  end;
  then (p=>q)=>((q=>r)=>(p=>r)) is LTL_TAUT_OF_PL;
  then (p=>q)=>((q=>r)=>(p=>r)) in LTL_axioms by Def17;
  then X|-(p=>q)=>((q=>r)=>(p=>r)) by Th42;
  then X|-(q=>r)=>(p=>r) by A1,Th43;
  hence X|-p=>r by A2,Th43;
 end;
