reserve u,v,x,x1,x2,y,y1,y2,z,p,a for object,
        A,B,X,X1,X2,X3,X4,Y,Y1,Y2,Z,N,M for set;
reserve e for object, X,X1,X2,Y1,Y2 for set;

theorem
  e in [:X1,Y1:] & e in [:X2,Y2:] implies e in [:X1 /\ X2, Y1 /\ Y2:]
proof
  assume e in [:X1,Y1:] & e in [:X2,Y2:];
  then e in [:X1,Y1:] /\ [:X2,Y2:] by XBOOLE_0:def 4;
  hence thesis by Th99;
end;
