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;
