
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 upper filtered & Y is upper filtered holds X /\ Y is filtered
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 upper filtered and
A3: Y is upper filtered;
A4: X /\ Y is upper by A2,A3,Th29;
  per cases by A1;
  suppose
A5: L is with_infima;
    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,Th41;
      x"/\"y in Y by A3,A5,A9,A11,Th41;
      hence x"/\"y in X /\ Y by A12,XBOOLE_0:def 4;
    end;
    hence thesis by A4,A5,Th41;
  end;
  suppose
A13: L is with_suprema;
    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_0:22;
A27: z >= zy by A13,YELLOW_0:22;
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_0:22;
  end;
end;
