An example of two cardinals that are equivalent in the $n$-order logic and not equivalent in the $(n+1)$-order logic

It is proved that the property of two models to be equivalent in the $n$th order logic is definable in the $(n+1)$th order logic. Basing on this fact, there is given an (nonconstructive) “example” of two $n$-order equivalent cardinal numbers that are not $(n+1)$-order equivalent.