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 Th8:
  X is unbounded & B1 in A implies ex B3 being Element of A st B3
  in { B2 where B2 is Element of A: B2 in X & B1 in B2}
proof
  assume
A1: X is unbounded;
  assume B1 in A;
  then succ B1 in A by ORDINAL1:28;
  then consider B3 such that
A2: B3 in X and
A3: succ B1 c= B3 by A1,Th6;
  reconsider B4 = B3 as Element of A by A2;
  take B4;
  B1 in B3 by A3,ORDINAL1:21;
  hence thesis by A2;
end;
