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 Th5:
  not sup (X /\ B) = B implies ex B1 st B1 in B & (X /\ B) c= B1
proof
  reconsider Y = (X /\ B) as Subset of B by XBOOLE_1:17;
  assume not sup (X /\ B) = B;
  then Y is bounded;
  then consider B1 such that
A1: B1 in B & Y c= B1 by Th4;
  take B1;
  thus thesis by A1;
end;
