reserve k,n,m for Nat,
  a,x,X,Y for set,
  D,D1,D2,S for non empty set,
  p,q for FinSequence of NAT;
reserve F,F1,G,G1,H,H1,H2 for LTL-formula;
reserve sq,sq9 for FinSequence;
reserve L,L9 for FinSequence;
reserve j for Nat;
reserve j1 for Element of NAT;
reserve V for LTLModel;
reserve Kai for Function of atomic_LTL,the BasicAssign of V;
reserve f,f1,f2 for Function of LTL_WFF,the carrier of V;
reserve BASSIGN for non empty Subset of ModelSP(Inf_seq(S));
reserve t for Element of Inf_seq(S);
reserve f,g for Assign of Inf_seqModel(S,BASSIGN);
reserve r for Element of Inf_seq(AtomicFamily);

theorem
  H is atomic implies (r |= H iff H in (CastSeq(r,AtomicFamily)).0)
proof
  assume
A1: H is atomic;
  then
A2: H in atomic_LTL;
A3: r |= H iff r|= Evaluate(H,AtomicKai);
  ex f be Function of LTL_WFF, the carrier of Inf_seqModel( AtomicFamily,
AtomicBasicAsgn) st f is-Evaluation-for AtomicKai & Evaluate(H, AtomicKai) = f.
  H by Def35;
  then Evaluate(H,AtomicKai) = AtomicKai.H by A1
    .= AtomicAsgn(H) by A2,Def62;
  then r |= H iff (Fid(AtomicAsgn(H),Inf_seq(AtomicFamily))).r=TRUE by A3;
  then r |= H iff AtomicFunc(H,r) = TRUE by Def60;
  hence thesis by Def59;
end;
