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 Th31:
  for f be Function of LTLB_WFF,BOOLEAN,p,q holds
  (VAL f).(p '&&' q)=(VAL f).p'&'(VAL f).q
 proof
  let f be Function of LTLB_WFF,BOOLEAN,p,q;
  A1: (VAL f).p=0 or(VAL f).p=1 by XBOOLEAN:def 3;
  A2: (VAL f).q=0 or(VAL f).q=1 by XBOOLEAN:def 3;
  thus(VAL f).(p '&&' q)=(VAL f).(p=>(q=>TFALSUM))=>(VAL f).(TFALSUM) by Def15
   .=((VAL f).p=>(VAL f).(q=>TFALSUM))=>(VAL f).(TFALSUM) by Def15
   .=((VAL f).p=>((VAL f).q=>(VAL f).(TFALSUM)))=>(VAL f).(TFALSUM) by Def15
   .=((VAL f)/.p=>((VAL f).q=>FALSE))=>(VAL f).(TFALSUM) by Def15
   .=(VAL f).p '&'(VAL f).q by A1,A2,Def15;
 end;
