reserve X,Y,x for set;
reserve A for non empty preBoolean set;

theorem
  for A,B being non empty preBoolean set holds A /\ B is non empty preBoolean
proof
  let A,B be non empty preBoolean set;
  {} in A & {} in B by Th7;
  then reconsider C = A /\ B as non empty set by XBOOLE_0:def 4;
  now
    let X,Y be set;
    assume
A1: X in C & Y in C;
    then
A2: X in B & Y in B by XBOOLE_0:def 4;
    then
A3: X \/ Y in B by Th1;
A4: X \ Y in B by A2,Th1;
A5: X in A & Y in A by A1,XBOOLE_0:def 4;
    then X \/ Y in A by Th1;
    hence X \/ Y in C by A3,XBOOLE_0:def 4;
    X \ Y in A by A5,Th1;
    hence X \ Y in C by A4,XBOOLE_0:def 4;
  end;
  hence thesis by Th1;
end;
