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 Th62: X |- ('X' (p 'or' q)) => (('X' p) 'or' ('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;
    X |- (xnp '&&' xnq) => ('X' (np '&&' nq)) by LTLAXIO1:53;then
A1: X |- ('not' ('X' (np '&&' nq))) => ('not' (xnp '&&' xnq)) by LTLAXIO1:52;
    nxq => xnq in LTL_axioms by LTLAXIO1:def 17;
    then A2: X |- nxq => xnq by LTLAXIO1:42;
    nxp => xnp in LTL_axioms by LTLAXIO1:def 17;
    then X |- nxp => xnp by LTLAXIO1:42;
    then X |- (nxp '&&' nxq) => (xnp '&&' xnq) by A2,Th53;then
A3: X |- ('not' (xnp '&&' xnq)) => ('not' (nxp '&&' nxq)) by LTLAXIO1:52;
    ('X' (p 'or' q)) => ('not' 'X' (np '&&' nq)) in LTL_axioms
    by LTLAXIO1:def 17;then
    X |- ('X' (p 'or' q)) => ('not' 'X' (np '&&' nq)) by LTLAXIO1:42;then
    X |- ('X' (p 'or' q)) => ('not' (xnp '&&' xnq)) by A1, LTLAXIO1:47;
    hence thesis by A3,LTLAXIO1:47;
  end;
