One addendum to that - It seems like a LOT of the cavalry shown so far has been very well armed with a variety of weapons. Typically mounted soldiers on about 2/3 of the continent seem to carry a horsebow, a lance and a sword. Depending on the situation they drop the lance or leave the bow cased.
Exceptions to this appear to be the easten nations. Illian, Tear, Cairhein, Andor and Mayene appear to come more from the school of thought about the glory of the charge. Murandy, Altara, Gheldean, Amadicia appear to be in between... some of them are strictly lancers... some lancers/horsebows. Tarabon, Domani, Borderlanders and Seanchan are almost all equipped with both lance and bow. That's my general feel at least...
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