reserve A,B,C,D,p,q,r for Element of LTLB_WFF,
        F,G,X for Subset of LTLB_WFF,
        M for LTLModel,
        i,j,n for Element of NAT,
        f,f1,f2,g for FinSequence of LTLB_WFF;

theorem
  {prop i} |- 'G' prop i & not {prop i} |-0 'G' prop i
proof
  thus {prop i} |- 'G' prop i
  proof
    prop i in {prop i} by TARSKI:def 1;then
    {prop i} |- prop i by LTLAXIO1:42;
    hence thesis by LTLAXIO1:54;
  end;
  thus not {prop i} |-0 'G' prop i
  proof
    assume {prop i} |-0 'G' prop i;then
A2: {prop i} |=0 'G' prop i by th266;
    not {prop i} |=0 'X' prop i by th268,th14;then
    consider M such that
A1: M |=0 {prop i} & not M |=0 'X' prop i;
    M |=0 'G' prop i by A2,A1;then
    (SAT M).[0+1,prop i] =1 by LTLAXIO1:10;
    hence contradiction by LTLAXIO1:9,A1;
  end;
end;
