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 Th53:
  X|-(('X' p)'&&'('X' q))=>('X'(p '&&' q))
 proof
  set Xp='X' p,nq='not' q,nXq='not' 'X' q,Xnq='X' 'not' q;
  Xnq=>nXq in LTL_axioms by Def17;
  then A1: X|-Xnq=>nXq by Th42;
  ('not'('X'(p=>nq)))=>('X'('not'(p=>nq))) in LTL_axioms by Def17;
  then A2: X|-('not'('X'(p=>nq)))=>('X'('not'(p=>nq))) by Th42;
  ('X'(p=>nq))=>(Xp=>Xnq) in LTL_axioms by Def17;
  then X|-('X'(p=>nq))=>(Xp=>Xnq) by Th42;
  then X|-('X'(p=>nq))=>(Xp=>nXq) by A1,Th51;
  then X|-('not'(Xp=>nXq))=>('not'('X'(p=>nq))) by Th52;
  hence thesis by A2,Th47;
 end;
