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 Th19:
  X is unbounded & limpoints X is bounded implies ex B1 st B1 in A
  & {succ B2 where B2 is Element of A : B2 in X & B1 in succ B2} is_club_in A
proof
  assume
A1: X is unbounded;
  assume limpoints X is bounded;
  then consider B1 such that
A2: B1 in A and
A3: limpoints X c= B1 by Th4;
  take B1;
  set SUCC={succ B2 where B2 is Element of A : B2 in X & B1 in succ B2};
  SUCC c= A
  proof
    let x be object;
    assume x in SUCC;
    then ex B2 being Element of A st x= succ B2 & B2 in X & B1 in succ B2;
    hence thesis by ORDINAL1:28;
  end;
  then reconsider SUCC as Subset of A;
  for B st B in A holds sup (SUCC /\ B)=B implies B in SUCC
  proof
    let B;
    assume B in A;
    then reconsider B0=B as Element of A;
    not sup (SUCC /\ B)=B
    proof
      set B2 = the Element of B;
      assume
A4:   sup (SUCC /\ B)=B;
      then consider B3 such that
A5:   B3 in (SUCC /\ B) and
      B2 c= B3 by ORDINAL2:21;
      B3 in SUCC by A5,XBOOLE_0:def 4;
      then
A6:   ex B4 being Element of A st B3= succ B4 & B4 in X & B1 in succ B4;
      sup (X /\ B)=B
      proof
        assume not sup (X /\ B) = B;
        then consider B5 such that
A7:     B5 in B and
A8:     (X /\ B) c= B5 by Th5;
        succ B5 in B by A7,ORDINAL1:28;
        then succ succ B5 in B by ORDINAL1:28;
        then consider B6 such that
A9:     B6 in (SUCC /\ B) and
A10:    succ succ B5 c= B6 by A4,ORDINAL2:21;
A11:    B6 in B by A9,XBOOLE_0:def 4;
        B6 in SUCC by A9,XBOOLE_0:def 4;
        then consider B7 being Element of A such that
A12:    B6 = succ B7 and
A13:    B7 in X and
        B1 in succ B7;
        B7 in succ B7 by ORDINAL1:6;
        then B7 in B by A12,A11,ORDINAL1:10;
        then B7 in (X /\ B) by A13,XBOOLE_0:def 4;
        then
A14:    B7 in B5 by A8;
        succ B5 in succ B7 by A10,A12,ORDINAL1:21;
        then succ B5 c= B7 by ORDINAL1:22;
        hence contradiction by A14,ORDINAL1:21;
      end;
      then
A15:  B0 in {B10 where B10 is Element of A: B10 is infinite limit_ordinal
      & sup ( X /\ B10) = B10};
      B3 in B by A5,XBOOLE_0:def 4;
      hence contradiction by A3,A15,A6,ORDINAL1:10;
    end;
    hence thesis;
  end;
  then
A16: SUCC is_closed_in A;
  for B2 st B2 in A ex C st C in SUCC & B2 c= C
  proof
    let B2 such that
A17: B2 in A;
    set B21 = (B2 \/ B1);
    B21 in A by A2,A17,ORDINAL3:12;
    then consider D such that
A18: D in X and
A19: B21 c= D by A1,Th6;
    take succ D;
    B21 in succ D by A19,ORDINAL1:22;
    then B1 in succ D by ORDINAL1:12,XBOOLE_1:7;
    hence succ D in SUCC by A18;
    B2 c= B21 by XBOOLE_1:7;
    then B2 c= D by A19;
    then B2 in succ D by ORDINAL1:22;
    hence thesis by ORDINAL1:def 2;
  end;
  then SUCC is unbounded by Th6;
  then sup SUCC = A;
  then SUCC is_unbounded_in A;
  hence thesis by A2,A16;
end;
