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A-T were often crossed with warmbloods, but they themselves are not warmbloods.
Along with the Caspian Horse and Persian Horse, the A-T is believed to be one of the oldest pure breeds of horse still existing today and is even more ancient than the Arabian, based on DNA studies.
A warmblood, by the way, is not a draft-thoroughbred cross, or any other cross - such cross breds are called 'Half Breds' or 'Cross Breds'. Warmbloods originate from a number of European regional registries with the purpose of first producing fast cavalry horses, and then producing FEI riding/driving sport horses.