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 th262ac1:
  {prop n} |= 'X' prop n & not {prop n} |=0 'X' prop n
proof
  thus {prop n} |= 'X' prop n
  proof
    let M;
    assume M |= {prop n};then
A2: M |= prop n by LTLAXIO1:23;
    let i;
    (SAT M).[i+1,prop n] = 1 by A2;
    hence (SAT M).[i,'X' prop n]=1 by LTLAXIO1:9;
  end;
  thus not {prop n} |=0 'X' prop n
  proof
    defpred P[Element of NAT,Element of bool props] means
    ($1 = 0 implies $2 = {prop n}) &
    (not $1 = 0 implies $2 = {}LTLB_WFF);
A3: for x being Element of NAT ex y being Element of bool props st P[x,y]
    proof
      let x being Element of NAT;
      per cases;
      suppose S1: x = 0;
        prop n in props by LTLAXIO1:def 10;then
        reconsider p0 = {prop n} as Element of bool props by ZFMISC_1:31;
        P[x,p0] by S1;
        hence ex y being Element of bool props st P[x,y];
      end;
      suppose S2:not x = 0;
        reconsider e = {}LTLB_WFF as Element of bool props by XBOOLE_1:2;
        P[x,e] by S2;
        hence ex y being Element of bool props st P[x,y];
      end;
    end;
    consider M being Function of NAT,bool props such that
A4: for x being Element of NAT holds P[x,M.x] from FUNCT_2:sch 3(A3);
    reconsider M as LTLModel;
A5: M |=0 {prop n}
    proof
      let A;
      assume A in {prop n};then
A6:   A = prop n by TARSKI:def 1;
      M.0 = {prop n} by A4;then
      prop n in M.0 by TARSKI:def 1;
      hence M |=0 A by LTLAXIO1:def 11,A6;
    end;
    not M |=0 'X' prop n
    proof
      assume M |=0 'X' prop n;then
      (SAT M).[0+1,prop n]=1 by LTLAXIO1:9;then
      prop n in M.1 by LTLAXIO1:def 11;
      hence contradiction by A4;
    end;
    hence thesis by A5;
  end;
end;
