reserve E for non empty set;
reserve a for Element of E;
reserve A, B for Subset of E;
reserve Y for set;
reserve p for FinSequence;
reserve e, e1, e2 for Singleton of E;

theorem
  for E being non empty set, e being Singleton of E, A being Event of E st e
  c= A \/ A` holds e c= A or e c= A`
proof
  let E be non empty set, e be Singleton of E, A be Event of E;
  ex a being Element of E st a in E & e = {a} by Th6;
  then consider a being Element of E such that
A1: e = {a};
  assume e c= A \/ A`;
  then a in A \/ A` by A1,ZFMISC_1:31;
  then a in A or a in A` by XBOOLE_0:def 3;
  hence thesis by A1,ZFMISC_1:31;
end;
