
theorem Th27:
  for L being RelStr, X,Y being Subset of L st X is lower & Y is lower
  holds X /\ Y is lower & X \/ Y is lower
proof
  let L be RelStr;
  let X,Y be Subset of L such that
A1: for x,y being Element of L st x in X & y <= x holds y in X and
A2: for x,y being Element of L st x in Y & y <= x holds y in Y;
  hereby
    let x,y be Element of L;
    assume
A3: x in X /\ Y;
    then
A4: x in X by XBOOLE_0:def 4;
A5: x in Y by A3,XBOOLE_0:def 4;
    assume
A6: y <= x;
    then
A7: y in X by A1,A4;
    y in Y by A2,A5,A6;
    hence y in X /\ Y by A7,XBOOLE_0:def 4;
  end;
  let x,y be Element of L;
  assume x in X \/ Y;
  then
A8: x in X or x in Y by XBOOLE_0:def 3;
  assume y <= x;
  then y in X or y in Y by A1,A2,A8;
  hence thesis by XBOOLE_0:def 3;
end;
