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;
reserve M for non countable Aleph;
reserve X for Subset of M;
reserve N,N1 for cardinal infinite Element of M;

theorem
  omega in cf M & X is unbounded implies limpoints X is unbounded
proof
  assume
A1: omega in cf M;
  assume
A2: X is unbounded;
  for B1 st B1 in M ex C st C in limpoints X & B1 c= C
  proof
    let B1;
    assume B1 in M;
    then consider B such that
    B in M and
A3: B1 in B & B in limpoints X by A1,A2,Th24;
    take B;
    thus thesis by A3,ORDINAL1:def 2;
  end;
  hence thesis by Th6;
end;
