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
  F \/ {A} |-0 B implies F |-0 A => B
proof
  assume F \/ {A} |-0 B;then
  consider f such that
A1: f.len f = B and
A2: 1<=len f and
A3: for i be Nat st 1<=i & i<=len f holds prc0 f,F \/ {A},i;
    defpred P[Nat] means
    1<=$1 & $1<=len f implies F |-0 A => 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,Def29;
        suppose A8: f.i in LTL0_axioms;
A9:       F |-0 f/.i => (A=>f/.i) by th15,LTLAXIO1:34;
          f/.i in LTL0_axioms by A6,A7,A8,Lm1;
          then F |-0 f/.i by th10;
          hence thesis by A9,th11a;
        end;
        suppose A10: f.i in F \/ {A};
          per cases by A10,XBOOLE_0:def 3;
          suppose A11: f.i in F;
            A12: F |-0 f/.i=>(A=>f/.i) by th15,LTLAXIO1:34;
            f/.i in F by A6,A7,A11,Lm1;
            then F |-0 f/.i by th10;
            hence thesis by A12,th11a;
          end;
          suppose f.i in {A};then
            f.i=A by TARSKI:def 1;then
B1:         f/.i=A by A6,A7,Lm1;
            A => A is LTL_TAUT_OF_PL by LTLAXIO2:24;then
            A => A in LTL_axioms by LTLAXIO1:def 17;
            hence thesis by B1,th15;
          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
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 MP0_rule f/.i
          or f/.j,f/.k IND0_rule f/.i;
          j<=len f by A7,A16,XXREAL_0:2;then
A20:      F|-0 A => f/.j by A5,A15,A16;
          k<=len f by A7,A18,XXREAL_0:2;then
A21:      F|-0 A => f/.k by A5,A17,A18;
          per cases by A19;
          suppose A22: f/.j,f/.k MP_rule f/.i;
A23:        F |-0 (A=>(f/.j=>f/.i))=>((A=>f/.j)=>(A=>f/.i))
            by th15,LTLAXIO1:35;
            F|-0 (A=>f/.j)=>(A=>f/.i) by A23,th11a,A21,A22;
            hence F |-0 A => f/.i by A20,th11a;
          end;
          suppose f/.j,f/.k MP0_rule f/.i;
            then consider C,D such that
A24:        f/.j= 'G' C and
A25:        f/.k= 'G' (C => D) and
A26:        f/.i= 'G' D;
B1:         {}LTLB_WFF c= F;
            {}LTLB_WFF |-0 f/.k => (f/.j => f/.i)
            by th267,LTLAXIO1:60,A24,A25,A26;then
B2:         F |-0 f/.k => (f/.j => f/.i) by mon,B1;
            (A => f/.k) => ((f/.k => (f/.j => f/.i))
            => (A => (f/.j => f/.i))) is LTL_TAUT_OF_PL by th16;then
            (A => f/.k) => ((f/.k => (f/.j => f/.i))
            => (A => (f/.j => f/.i))) in LTL_axioms by LTLAXIO1:def 17;then
            F |-0 (A => f/.k) => ((f/.k =>
            (f/.j => f/.i)) => (A => (f/.j => f/.i))) by th15;then
            F |-0 ((f/.k => (f/.j => f/.i))
            => (A => (f/.j => f/.i))) by th11a,A21;then
B3:         F |-0 A => (f/.j => f/.i) by B2,th11a;
            (A => (f/.j => f/.i)) => ((A => f/.j) => (A => f/.i))
            is LTL_TAUT_OF_PL by th17;then
            (A => (f/.j => f/.i)) => ((A => f/.j) => (A => f/.i))
            in LTL_axioms by LTLAXIO1:def 17;then
            F |-0 (A => (f/.j => f/.i))
            => ((A => f/.j) => (A => f/.i)) by th15;then
            F |-0 (A => f/.j) => (A => f/.i) by th11a,B3;
            hence F|-0 A => f/.i by th11a,A20;
          end;
          suppose f/.j,f/.k IND0_rule f/.i;
            then consider C,D such that
A24:        f/.j = 'G' (C=>D) and
A25:        f/.k= 'G' (C => ('X' C)) and
A26:        f/.i= 'G' (C => ('G' D));
            A => f/.j => (A => f/.k => (A => (f/.j '&&' f/.k)))
            is LTL_TAUT_OF_PL by LTLAXIO2:40;then
            A => f/.j => (A => f/.k => (A => (f/.j '&&' f/.k)))
            in LTL_axioms by LTLAXIO1:def 17;then
            F |-0 A => f/.j => (A => f/.k => (A => (f/.j '&&' f/.k)))
            by th15;then
            F |-0 A => f/.k => (A => (f/.j '&&' f/.k))
            by th11a,A20;then
B10:        F |-0 A => (f/.j '&&' f/.k) by th11a,A21;
B12:        {}LTLB_WFF c= F;
            {}LTLB_WFF |- f/.j => (f/.k => f/.i) by th20,A24,A25,A26;then
            {}LTLB_WFF |- (f/.j '&&' f/.k) => f/.i by LTLAXIO1:48;then
B11:        F |-0 (f/.j '&&' f/.k) => f/.i by mon,B12,th267;
            (A => (f/.j '&&' f/.k)) => (((f/.j '&&' f/.k) => f/.i)
            => (A => f/.i)) is LTL_TAUT_OF_PL by th16;then
            (A => (f/.j '&&' f/.k)) => (((f/.j '&&' f/.k) => f/.i)
            => (A => f/.i)) in LTL_axioms by LTLAXIO1:def 17;then
            F |-0 (A => (f/.j '&&' f/.k)) => (((f/.j '&&' f/.k) => f/.i)
            => (A => f/.i)) by th15;then
            F |-0 (((f/.j '&&' f/.k) => f/.i) => (A => f/.i)) by th11a,B10;
            hence thesis by th11a,B11;
          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,q,r such that
A32:      1<=j and
A33:      j<i and
A34:      f/.j NEX0_rule f/.i or f/.j REFL0_rule f/.i;
B4:       j<=len f by A7,A33,XXREAL_0:2;then
B4a:       F |-0 A => f/.j by A5,A32,A33;
          per cases by A34;
          suppose f/.j NEX0_rule f/.i;
            then consider C such that
A24:        f/.j= 'G' C and
A26:        f/.i= 'G' 'X' C;
B8:         {}LTLB_WFF c= F;
            {}LTLB_WFF |-0 f/.j => f/.i by th19,th267,A24,A26;then
B9:         F |-0 f/.j => f/.i by mon,B8;
            (A => f/.j) => ((f/.j => f/.i) => (A => f/.i))
            is LTL_TAUT_OF_PL by th16;then
            (A => f/.j) => ((f/.j => f/.i) => (A => f/.i))
            in LTL_axioms by LTLAXIO1:def 17;then
            F |-0 (A => f/.j) => ((f/.j => f/.i) => (A => f/.i))
            by th15;then
            F |-0 ((f/.j => f/.i) => (A => f/.i)) by th11a,B4a;
            hence thesis by B9,th11a;
          end;
          suppose f/.j REFL0_rule f/.i;then
B6:         F |-0 A => 'G' f/.i by B4,A5,A32,A33;
B5:         {}LTLB_WFF c= F;
            {}LTLB_WFF |-0 ('G' f/.i) => f/.i by th267,th18;then
B7:         F |-0 ('G' f/.i) => f/.i by mon,B5;
            (A => ('G' f/.i)) => ((('G' f/.i) => f/.i) => (A => f/.i)) is
            LTL_TAUT_OF_PL by th16;then
            (A => ('G' f/.i)) => ((('G' f/.i) => f/.i) => (A => f/.i)) in
            LTL_axioms by LTLAXIO1:def 17;then
            F |-0 (A => ('G' f/.i)) => ((('G' f/.i) => f/.i) => (A => f/.i))
            by th15;then
            F |-0 ((('G' f/.i) => f/.i) => (A => f/.i)) by B6,th11a;
            hence thesis by th11a,B7;
          end;
        end;
      end;
    end;
A37: for i be Nat holds P[i] from NAT_1:sch 4(A4);
     B = f/.len f by A1,A2,Lm1;
     hence F |-0 A => B by A2,A37;
   end;
