reserve a,b,c for boolean object;
reserve p,q,r,s,A,B,C for Element of LTLB_WFF,
        F,G,X,Y for Subset of LTLB_WFF,
        i,j,k,n for Element of NAT,
        f,f1,f2,g for FinSequence of LTLB_WFF;
reserve M for LTLModel;

theorem Th42:
  p in LTL_axioms or p in X implies X|-p
 proof
  defpred P1[set,set] means $2=p;
  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 by FINSEQ_1:2,TARSKI:def 1;
  then A4: g.1=p by A2;
  assume A5: p in LTL_axioms or p in X;
  for j be Nat st 1<=j & j<=len g holds prc g,X,j
  proof
   let j be Nat;
   assume A6: 1<=j & j<=len g;
   per cases by A5;
   suppose p in LTL_axioms;
    then g.j in LTL_axioms by A3,A4,A6,XXREAL_0:1;
    hence thesis;
   end;
   suppose p in X;
    then g.j in X by A3,A4,A6,XXREAL_0:1;
    hence thesis;
   end;
  end;
  hence X|-p by A3,A4;
 end;
