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 Th51:
  X|-p=>(q=>r) & X|-r=>s implies X|-p=>(q=>s)
 proof
  assume that
   A1: X|-p=>(q=>r) and
   A2: X|-r=>s;
  set A=(p=>(q=>r))=>((r=>s)=>(p=>(q=>s)));
  now let f be Function of LTLB_WFF,BOOLEAN;
   A3: (VAL f).p=0 or(VAL f).p=1 by XBOOLEAN:def 3;
   A4: (VAL f).r=0 or(VAL f).r=1 by XBOOLEAN:def 3;
   set B=(VAL f).(p=>(q=>r)),C=(VAL f).(r=>s),D=(VAL f).(p=>(q=>s));
   A5: (VAL f).q=0 or(VAL f).q=1 by XBOOLEAN:def 3;
   A6: (VAL f).s=0 or(VAL f).s=1 by XBOOLEAN:def 3;
   A7: (VAL f).(p=>(q=>s))=(VAL f).p=>(VAL f).(q=>s) by Def15
    .=(VAL f).p=>((VAL f).q=>(VAL f).s) by Def15;
   A8: (VAL f).(p=>(q=>r))=(VAL f).p=>(VAL f).(q=>r) by Def15
    .=(VAL f).p=>((VAL f).q=>(VAL f).r) by Def15;
   thus(VAL f).A=B=>(VAL f).((r=>s)=>(p=>(q=>s))) by Def15
    .=B=>(C=>D) by Def15
    .=1 by A3,A5,A4,A6,A8,A7,Def15;
  end;
  then A is LTL_TAUT_OF_PL;
  then A in LTL_axioms by Def17;
  then X|-A by Th42;
  then X|-((r=>s)=>(p=>(q=>s))) by A1,Th43;
  hence X|-p=>(q=>s) by A2,Th43;
 end;
