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 |- ('not' (A 'U' B)) => ('X' 'not' untn(A,B))
  proof
    set p = A 'U' B, q = 'X' B, r = 'X' (A '&&' (A 'U' B));
    (q 'or' r) => p in LTL_axioms by LTLAXIO1:def 17;
    then A1: X |- (q 'or' r) => p by LTLAXIO1:42;
    X |- ('X' untn(A,B)) => (q 'or' r) by Th62;
    then X |- ('X' untn(A,B)) => p by LTLAXIO1:47,A1;
    then A2: X |- ('not' p) => ('not' ('X' untn(A,B))) by LTLAXIO1:52;
    ('not' ('X' untn(A,B))) => ('X' ('not' untn(A,B))) in LTL_axioms
    by LTLAXIO1:def 17;then
    X |- ('not' ('X' untn(A,B))) => ('X' ('not' untn(A,B))) by LTLAXIO1:42;
    hence thesis by LTLAXIO1:47,A2;
  end;
