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 th267:
  {}LTLB_WFF |- A implies {}LTLB_WFF |-0 A
proof
  assume {}LTLB_WFF |- A;then
  consider f such that
A1: f.len f = A and
A2: 1<=len f and
A3: for i be Nat st 1<=i & i<=len f holds prc f,{}LTLB_WFF,i;
    defpred P[Nat] means 1<=$1 & $1<=len f implies {}LTLB_WFF |-0 'G' f/.$1;
A4: for i being Nat st for j being Nat st j<i holds P[j] holds P[i]
    proof
      let i be Nat;
      assume
A5:   for j be Nat st j<i holds P[j];
      per cases by NAT_1:14;
      suppose i=0;
        hence P[i];
      end;
      suppose not i<1;
        assume that
A6:     1<=i and
A7:     i<=len f;
        per cases by A3,A6,A7,LTLAXIO1:def 29;
        suppose f.i in LTL_axioms;then
          f/.i in LTL_axioms by Lm1,A6,A7;then
          'G' f/.i in LTL0_axioms;
          hence thesis by th10;
        end;
        suppose f.i in {}LTLB_WFF;
          hence thesis;
        end;
        suppose ex j,k be Nat st 1<=j & j<i & 1<=k & k<i &
          (f/.j,f/.k MP_rule f/.i or f/.j,f/.k IND_rule f/.i);
          then consider j,k be Nat such that
A15:      1<=j and
A16:      j<i and
A17:      1<=k and
A18:      k<i and
A19:      f/.j,f/.k MP_rule f/.i or f/.j,f/.k IND_rule f/.i;
          j<=len f by A7,A16,XXREAL_0:2;then
A20:      {}LTLB_WFF |-0 'G' f/.j by A5,A15,A16;
          k<=len f by A7,A18,XXREAL_0:2;then
A21:      {}LTLB_WFF |-0 'G' f/.k by A5,A17,A18;
          per cases by A19;
          suppose f/.j,f/.k MP_rule f/.i;
            hence thesis by A20,A21,th11;
          end;
          suppose f/.j,f/.k IND_rule f/.i;
            then consider B,C such that
A24:        f/.j = B => C and
A25:        f/.k= B => ('X' B) and
A26:        f/.i= B => ('G' C);
            thus thesis by A24,A25,A26,th13,A20,A21;
          end;
        end;
        suppose ex j be Nat st 1<=j & j<i & f/.j NEX_rule f/.i;
          then consider j be Nat such that
A15:      1<=j and
A16:      j<i and
A19:      f/.j NEX_rule f/.i;
          j<=len f by A7,A16,XXREAL_0:2;then
          {}LTLB_WFF |-0 'G' f/.j by A5,A15,A16;
          hence thesis by A19,th12;
        end;
      end;
    end;
A37:  for i be Nat holds P[i] from NAT_1:sch 4(A4);
      A = f/.len f by A1,A2,Lm1;then
      {}LTLB_WFF |-0 'G' A by A37,A2;
      hence {}LTLB_WFF |-0 A by th9;
    end;
