reserve A,B for limit_ordinal infinite Ordinal;
reserve B1,B2,B3,B5,B6,D, C for Ordinal;
reserve X for set;
reserve X for Subset of A;

theorem Th18:
  limpoints X is closed
proof
  let B;
  assume B in A;
  then reconsider Bl=B as Element of A;
  assume
A1: sup ((limpoints X) /\ B) =B;
  sup (X /\ B)=B
  proof
    assume sup (X /\ B) <> B;
    then consider B1 such that
A2: B1 in B and
A3: (X /\ B) c= B1 by Th5;
    sup ((limpoints X) /\ B) c= sup succ B1 by A3,Th16,ORDINAL2:22;
    then
A4: sup ((limpoints X) /\ B) c= succ B1 by ORDINAL2:18;
    succ B1 in B by A2,ORDINAL1:28;
    then succ B1 in succ B1 by A1,A4;
    hence contradiction;
  end;
  then sup (X /\ Bl)=Bl;
  hence thesis;
end;
