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 th10:
  A in LTL0_axioms or A in F implies F|-0 A
 proof
  defpred P1[set,set] means $2=A;
  A1: for k being Nat st k in Seg 1 holds
  ex x being Element of LTLB_WFF st P1[k,x];
  consider g such that
A2: dom g=Seg 1 & for k being Nat st k in Seg 1 holds P1[k,g.k]
    from FINSEQ_1:sch 5(A1);
A3: len g=1 by A2,FINSEQ_1:def 3;
  1 in Seg 1;
  then A4: g.1=A by A2;
  assume A5: A in LTL0_axioms or A in F;
  for j be Nat st 1<=j & j<=len g holds prc0 g,F,j
  proof
    let j be Nat;
    assume A6: 1<=j & j<=len g;
    per cases by A5;
    suppose A in LTL0_axioms;
      hence thesis by A3,A4,A6,XXREAL_0:1;
    end;
    suppose A in F;
      hence thesis by A3,A4,A6,XXREAL_0:1;
    end;
  end;
  hence F|-0A by A3,A4;
end;
