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 Th50:
  X|-(p '&&' q)=>r & X|-p=>s implies X|-(p '&&' q)=>(s '&&' r)
 proof
  assume that
   A1: X|-(p '&&' q)=>r and
   A2: X|-p=>s;
  set A=((p '&&' q)=>r)=>((p=>s)=>((p '&&' q)=>(s '&&' r)));
  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).q=0 or(VAL f).q=1 by XBOOLEAN:def 3;
   A5: (VAL f).s=0 or(VAL f).s=1 by XBOOLEAN:def 3;
   A6: (VAL f).r=0 or(VAL f).r=1 by XBOOLEAN:def 3;
   thus(VAL f).A=(VAL f).((p '&&' q)=>r)=>(VAL f).((p=>s)=>((p '&&' q)=>(s '&&'
r))) by Def15
    .=((VAL f).(p '&&' q)=>(VAL f).r)=>(VAL f).((p=>s)=>((p '&&' q)=>(s '&&' r)
)) by Def15
    .=(((VAL f).p '&'(VAL f).q)=>(VAL f).r)=>(VAL f).((p=>s)=>((p '&&' q)=>(s
'&&' r))) by Th31
    .=(((VAL f).p '&'(VAL f).q)=>(VAL f).r)=>((VAL f).(p=>s)=>(VAL f).((p '&&'
q)=>(s '&&' r))) by Def15
    .=(((VAL f).p '&'(VAL f).q)=>(VAL f).r)=>(((VAL f).p=>(VAL f).s)=>(VAL f).(
(p '&&' q)=>(s '&&' r))) by Def15
    .=(((VAL f).p '&'(VAL f).q)=>(VAL f).r)=>(((VAL f).p=>(VAL f).s)=>((VAL f).
(p '&&' q)=>(VAL f).(s '&&' r))) by Def15
    .=(((VAL f).p '&'(VAL f).q)=>(VAL f).r)=>(((VAL f).p=>(VAL f).s)=>(((VAL f)
.p '&'(VAL f).q)=>(VAL f).(s '&&' r))) by Th31
    .=1 by A3,A4,A6,A5,Th31;
  end;
  then A is LTL_TAUT_OF_PL;
  then A in LTL_axioms by Def17;
  then X|-A by Th42;
  then X|-((p=>s)=>((p '&&' q)=>(s '&&' r))) by A1,Th43;
  hence thesis by A2,Th43;
 end;
