1 vs 1 a lancer wont beat any other type of cav on eu1 and it usually ends up with both lancers dead if they fight each other due to recent changes (ok maybe jav cav is weakest but im discounting that 
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(we can agree to disagree i really dont care its my opinion that HX is too strong when compared to other cav)
Lancer will beat 1h/2h cavalry every time, all they can do is try get behind you but there are many ways to make sure they can't stay there/get there in first place.
Horse Crossbow I don't think is very good at all, it does barely any damage long reload time and the builds that use it most effectively require you to have very little STR and IF. Of course a lancer won't "beat" them 1v1 though unless the HX fucks up bad, but thats just what HX and HA are, you can't expect to kill them 1v1.
Regarding HX vs infantry, the only HX i've ever seen do better than even average lance/1h/2h cavalry is ROHYPNOL and that is just because he is amazingly good at the class.
The only way you could say HX is "better" than any other cavalry is that they have builds purely designed to use on horseback unlike i'd say 99% of cav who hybrid in some way. But any type of cavalry can do this and infact one of my favourite lancer builds is 12/30 which is completely dedicated cavalry and very good at it.
