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 X |- (('X' p) 'or' ('X' q)) => 'X' (p 'or' 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;
    ('not' 'X' ('not' npq)) => ('X' 'not' 'not' npq) in LTL_axioms
    by LTLAXIO1:def 17;then
A1: X |- ('not' 'X' ('not' npq)) => ('X' 'not' 'not' npq) by LTLAXIO1:42;
    ('not' 'not' npq) => npq is ctaut by Th25;
    then ('not' 'not' npq) => npq in LTL_axioms by LTLAXIO1:def 17;
    then X |- ('not' 'not' npq) => npq by LTLAXIO1:42;
    then A2: X |- 'X' (('not' 'not' npq) => npq) by LTLAXIO1:44;
    ('X' (('not' 'not' npq) => npq)) =>
  (('X' ('not' 'not' npq)) => ('X' npq)) in LTL_axioms by LTLAXIO1:def 17;then
    X |- ('X' (('not' 'not' npq) => npq)) =>
    (('X' ('not' 'not' npq)) => ('X' npq)) by LTLAXIO1:42;
    then X |- (('X' ('not' 'not' npq)) => ('X' npq)) by LTLAXIO1:43, A2;
    then A3: X |- ('not' 'X' ('not' npq)) => ('X' npq) by LTLAXIO1:47,A1;
    X |- ('X' npq) => (xnp '&&' xnq) by Th59;then
A4: X |- ('not' 'X' ('not' npq)) => (xnp '&&' xnq) by LTLAXIO1:47, A3;
    xnq => nxq in LTL_axioms by LTLAXIO1:def 17;
    then A5: X |- xnq => nxq by LTLAXIO1:42;
    ('not' 'not' 'X' ('not' npq)) => ('X' ('not' npq)) is ctaut by Th25;then
    ('not' 'not' 'X' ('not' npq)) => ('X' ('not' npq)) in LTL_axioms
    by LTLAXIO1:def 17;then
A6: X |- ('not' 'not' 'X' ('not' npq)) => ('X' ('not' npq)) by LTLAXIO1:42;
    xnp => nxp in LTL_axioms by LTLAXIO1:def 17;
    then X |- xnp => nxp by LTLAXIO1:42;
    then X |- (xnp '&&' xnq) => (nxp '&&' nxq) by A5,Th53;then
    X |- ('not' 'X' ('not' npq)) => (nxp '&&' nxq) by LTLAXIO1:47, A4;then
    X |- ('not' (nxp '&&' nxq)) => ('not' 'not' 'X' ('not' npq))
    by LTLAXIO1: 52;
    hence thesis by A6,LTLAXIO1:47;
  end;
