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 Th14:
  for X,Y being set holds (X is_stationary_in A & X c= Y & Y c= A
  implies Y is_stationary_in A)
proof
  let X,Y be set;
  assume
A1: X is_stationary_in A;
  then reconsider X1 = X as Subset of A;
  assume
A2: X c= Y;
  assume Y c= A;
  then reconsider Y1 = Y as Subset of A;
  for Z being Subset of A holds Z is closed unbounded implies X /\ Z is
  non empty by A1;
  then X1 is stationary;
  then Y1 is stationary by A2,Th13;
  then for Z being Subset of A holds Z is closed unbounded implies Y1 /\ Z is
  non empty;
  hence thesis;
end;
