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 Th16:
  X /\ B3 c= B1 implies B3 /\ limpoints X c= succ B1
proof
  assume
A1: X /\ B3 c= B1;
  defpred P[set] means $1is infinite limit_ordinal & sup (X /\ $1) = $1;
  assume not B3 /\ limpoints X c= succ B1;
  then consider x being object such that
A2: x in B3 /\ limpoints X and
A3: not x in succ B1;
  reconsider x1=x as Element of B3 by A2,XBOOLE_0:def 4;
  succ B1 c= x1 by A3,ORDINAL1:16;
  then
A4: B1 in x1 by ORDINAL1:21;
A5: x1 in {B2 where B2 is Element of A: P[B2]} by A2,XBOOLE_0:def 4;
A6: P[x1] from CARD_FIL:sch 1(A5);
  then reconsider x2=x1 as infinite limit_ordinal Ordinal;
  reconsider Y = (X /\ x2) as Subset of x2 by XBOOLE_1:17;
  Y is unbounded by A6;
  then consider C such that
A7: C in Y and
A8: B1 c= C by A4,Th6;
A9: C in X by A7,XBOOLE_0:def 4;
  x in B3 by A2,XBOOLE_0:def 4;
  then C in B3 by A7,ORDINAL1:10;
  then C in X /\ B3 by A9,XBOOLE_0:def 4;
  then C in B1 by A1;
  then C in C by A8;
  hence contradiction;
end;
