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
  {z} = X \/ Y & X <> Y implies X = {} or Y = {}
proof
  assume {z} = X \/ Y;
  then X={z} & Y={z} or X={} & Y={z} or X={z} & Y={} by Th36;
  hence thesis;
end;
