reserve X for set;

theorem
  for Y being non empty Subset of BoolePoset X holds inf Y = meet Y
proof
  set L = BoolePoset X;
  let Y be non empty Subset of L;
  set y = the Element of Y;
A1: the carrier of L = bool X by LATTICE3:def 1;
  then y c= X;
  then reconsider Me = meet Y as Element of L by A1,SETFAM_1:7;
A2: now
    let b be Element of L;
    assume
A3: b is_<=_than Y;
    for Z being set st Z in Y holds b c= Z by Th2,A3;
    then b c= Me by SETFAM_1:5;
    hence Me >= b by Th2;
  end;
  for b being Element of L st b in Y holds Me <= b by Th2,SETFAM_1:3;
  then Me is_<=_than Y;
  hence thesis by A2,YELLOW_0:33;
end;
