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 th266:
  F |-0 A implies F |=0 A
  proof
    assume F |-0 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 prc0 f,F,i;
    defpred P[Nat] means
   1<=$1 & $1<=len f implies F |=0 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 being 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,Def29;
        suppose f.i in LTL0_axioms;then
          f/.i in LTL0_axioms by Lm1,A6,A7;
          hence F |=0 f/.i by Th2;
        end;
        suppose f.i in F;
          then A9: f/.i in F by A6,A7,Lm1;
          thus F |=0 f/.i
          proof
            let M;
            assume M |=0 F;
            hence M |=0 f/.i by A9;
          end;
        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 MP0_rule f/.i
          or f/.j,f/.k IND0_rule f/.i);
          then consider j,k be Nat such that
A10:      1<=j and
A11:      j<i and
A12:      1<=k and
A13:      k<i and
A14:      f/.j,f/.k MP_rule f/.i or f/.j,f/.k MP0_rule f/.i
          or f/.j,f/.k IND0_rule f/.i;
A15:      k<=len f by A7,A13,XXREAL_0:2;then
A15a:     F |=0 f/.k by A5,A12,A13;
A16:      j<=len f by A7,A11,XXREAL_0:2;
B5:       F |=0 f/.j by A5,A10,A11,A16;
          per cases by A14;
          suppose f/.j,f/.k MP_rule f/.i;then
            F |=0 f/.j=>f/.i by A15,A5,A12,A13;
            hence F |=0 f/.i by A5,A10,A11,A16,Th3;
          end;
          suppose f/.j,f/.k MP0_rule f/.i;then
            consider A,B such that
B7:         f/.j = 'G' A and
B8:         f/.k= 'G' (A=>B) and
B9:         f/.i = 'G' B;
            thus F |=0 f/.i by Th4,B7,B8,B9,A15a,B5;
          end;
          suppose f/.j,f/.k IND0_rule f/.i;
            then consider A,B such that
A17:        f/.j= 'G' (A=>B) and
A18:        f/.k= 'G' (A=>('X' A)) & f/.i = 'G' (A=>('G' B));
            F |=0 'G' (A=>B) by A5,A10,A11,A16,A17;
            hence F |=0 f/.i by A15a,A18,Th7;
          end;
        end;
        suppose ex j be Nat st 1<=j & j<i &
          (f/.j NEX0_rule f/.i or f/.j REFL0_rule f/.i);
          then consider j be Nat such that
A19:      1<=j and
A20:      j<i and
A21:      f/.j NEX0_rule f/.i or f/.j REFL0_rule f/.i;
B11:      j<=len f by A7,A20,XXREAL_0:2;
          per cases by A21;
          suppose f/.j NEX0_rule f/.i;then
            consider A,B such that
B7:         f/.j = 'G' A and
B9:         f/.i = 'G' 'X' A;
            thus F |=0 f/.i by Th5,B7,B9,B11,A5,A20,A19;
          end;
          suppose f/.j REFL0_rule f/.i;
            hence F |=0 f/.i by Th6,B11,A5,A20,A19;
          end;
        end;
      end;
    end;
A22: for i be Nat holds P[i] from NAT_1:sch 4(A4);
     f/.len f=A by A1,A2,Lm1;
     hence F |=0 A by A2,A22;
   end;
