
theorem
  for L being non empty antisymmetric RelStr
  st L is with_suprema or L is with_infima for X,Y being Subset of L
  st X is lower directed & Y is lower directed holds X /\ Y is directed
proof
  let L be non empty antisymmetric RelStr such that
A1: L is with_suprema or L is with_infima;
  let X,Y be Subset of L such that
A2: X is lower directed and
A3: Y is lower directed;
A4: X /\ Y is lower by A2,A3,Th27;
  per cases by A1;
  suppose
A5: L is with_suprema;
    now
      let x,y be Element of L;
      assume that
A6:   x in X /\ Y and
A7:   y in X /\ Y;
A8:   x in X by A6,XBOOLE_0:def 4;
A9:   x in Y by A6,XBOOLE_0:def 4;
A10:  y in X by A7,XBOOLE_0:def 4;
A11:  y in Y by A7,XBOOLE_0:def 4;
A12:  x"\/"y in X by A2,A5,A8,A10,Th40;
      x"\/"y in Y by A3,A5,A9,A11,Th40;
      hence x"\/"y in X /\ Y by A12,XBOOLE_0:def 4;
    end;
    hence thesis by A4,A5,Th40;
  end;
  suppose
A13: L is with_infima;
    let x,y be Element of L;
    assume that
A14: x in X /\ Y and
A15: y in X /\ Y;
A16: x in X by A14,XBOOLE_0:def 4;
A17: x in Y by A14,XBOOLE_0:def 4;
A18: y in X by A15,XBOOLE_0:def 4;
A19: y in Y by A15,XBOOLE_0:def 4;
    consider zx being Element of L such that
A20: zx in X and
A21: x <= zx and
A22: y <= zx by A2,A16,A18;
    consider zy being Element of L such that
A23: zy in Y and
A24: x <= zy and
A25: y <= zy by A3,A17,A19;
    take z = zx"/\"zy;
A26: z <= zx by A13,YELLOW_0:23;
A27: z <= zy by A13,YELLOW_0:23;
A28: z in X by A2,A20,A26;
    z in Y by A3,A23,A27;
    hence z in X /\ Y by A28,XBOOLE_0:def 4;
    thus thesis by A13,A21,A22,A24,A25,YELLOW_0:23;
  end;
end;
