reserve A,B,p,q,r,s for Element of LTLB_WFF,
  i,j,k,n for Element of NAT,
  X for Subset of LTLB_WFF,
  f,f1 for FinSequence of LTLB_WFF,
  g for Function of LTLB_WFF,BOOLEAN;

theorem Th59: X |- ('X' (p '&&' q)) => (('X' p) '&&' ('X' q))
  proof
   set xp = 'X' p, xq = 'X' q,np = 'not' p,nq = 'not' q,xnp = 'X' 'not' p,
   xnq = 'X' 'not' q,nxp = 'not' 'X' p, nxq = 'not' 'X' q,npq = np '&&' nq;
A1: X |- (xp => xnq) => ('X' (p => nq)) by Th58;
    (xp => nxq) => (xp => nxq) is ctaut by Th24;
    then (xp => nxq) => (xp => nxq) in LTL_axioms by LTLAXIO1:def 17;
    then A2: X |- (xp => nxq) => (xp => nxq) by LTLAXIO1:42;
    nxq => xnq in LTL_axioms by LTLAXIO1:def 17;
    then X |- nxq => xnq by LTLAXIO1:42;
    then X |- (xp => nxq) => (xp => xnq) by A2,LTLAXIO1:51;
    then A3: X |- (xp => nxq) => ('X' (p => nq)) by A1,LTLAXIO1:47;
    ('X' 'not' (p => nq)) => ('not' ('X' (p => nq))) in LTL_axioms
    by LTLAXIO1:def 17;then
    X |- ('X' 'not' (p => nq)) => ('not' ('X' (p => nq))) by LTLAXIO1:42;then
A4: X |- ('not' 'not' ('X' (p => nq))) => (('not' ('X' 'not' (p => nq))))
    by LTLAXIO1:52;
    ('X' 'not' (p => nq)) => ('not' 'not' ('X' 'not' (p => nq))) is ctaut
    by Th26;then
    ('X' 'not' (p => nq)) => ('not' 'not' ('X' 'not' (p => nq))) in LTL_axioms
    by LTLAXIO1:def 17;then
A5: X |- ('X' 'not' (p => nq)) => ('not' 'not' ('X' 'not' (p => nq)))
    by LTLAXIO1:42;
    ('X' (p => nq)) => ('not' 'not' ('X' (p => nq))) is ctaut by Th26;then
    ('X' (p => nq)) => ('not' 'not' ('X' (p => nq))) in LTL_axioms
    by LTLAXIO1:def 17;then
    X |- ('X' (p => nq)) => ('not' 'not' ('X' (p => nq))) by LTLAXIO1:42;then
    X |- (xp => nxq) => ('not' 'not' ('X' (p => nq))) by A3,LTLAXIO1:47;then
    X |- (xp => nxq) => (('not' ('X' 'not' (p => nq)))) by A4,LTLAXIO1:47;then
    X |- ('not' 'not' ('X' 'not' (p => nq))) => ('not' (xp => nxq))
    by LTLAXIO1: 52;
    hence thesis by A5,LTLAXIO1:47;
  end;
