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 Th1:
  X is_club_in A iff X is closed unbounded
proof
  thus X is_club_in A implies X is closed unbounded
  proof
    assume
A1: X is_club_in A;
    then X is_unbounded_in A;
    then
A2: sup X = A;
    X is_closed_in A by A1;
    then for B st B in A holds sup (X /\ B)=B implies B in X;
    hence thesis by A2;
  end;
  assume
A3: X is closed unbounded;
  then sup X = A;
  then
A4: X is_unbounded_in A;
  for B st B in A holds sup (X /\ B)=B implies B in X by A3;
  then X is_closed_in A;
  hence thesis by A4;
end;
