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 Th11:
  B1 in A & X is closed unbounded implies X \ B1 is closed unbounded
proof
  assume
A1: B1 in A;
  assume
A2: X is closed unbounded;
  thus (X \ B1) is closed
  proof
    let B such that
A3: B in A;
    assume
A4: sup ((X \ B1) /\ B)=B;
    sup (X /\ B) c= sup B by ORDINAL2:22,XBOOLE_1:17;
    then
A5: sup (X /\ B) c= B by ORDINAL2:18;
    ((X \ B1) /\ B) c= (X /\ B) by XBOOLE_1:26,36;
    then B c= sup (X /\ B) by A4,ORDINAL2:22;
    then sup (X /\ B)=B by A5,XBOOLE_0:def 10;
    then
A6: B in X by A2,A3;
    assume not B in (X \ B1);
    then B in B1 by A6,XBOOLE_0:def 5;
    then
A7: B c= B1 by ORDINAL1:def 2;
    (X \ B) misses B by XBOOLE_1:79;
    then (X \ B1) misses B by A7,XBOOLE_1:34,63;
    then (X \ B1) /\ B = {} by XBOOLE_0:def 7;
    hence contradiction by A4,ORDINAL2:18;
  end;
  for B2 st B2 in A ex C st C in (X \ B1) & B2 c= C
  proof
    let B2 such that
A8: B2 in A;
    per cases by ORDINAL1:16;
    suppose
A9:   B1 in B2;
      consider D such that
A10:  D in X and
A11:  B2 c= D by A2,A8,Th6;
      take D;
      B1 in D by A9,A11;
      hence D in (X \ B1) by A10,XBOOLE_0:def 5;
      thus thesis by A11;
    end;
    suppose
A12:  B2 c= B1;
      consider D such that
A13:  D in X and
A14:  B1 c= D by A1,A2,Th6;
      take D;
      not D in B1 by A14,ORDINAL1:5;
      hence D in (X \ B1) by A13,XBOOLE_0:def 5;
      thus thesis by A12,A14;
    end;
  end;
  hence thesis by Th6;
end;
