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;

theorem
  X <> {} implies [:{x},X:] <> {} & [:X,{x}:] <> {} by Th89;
