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Generally, when an Attack Position monster with lower ATK than the opposing Attack Position monster battles (or a Defense Position monster with low DEF is attacked by a monster with higher ATK than that DEF value), the card with the lower stat is destroyed. However, when an Attack Position monster battles an Attack Position monster that has equal ATK, both monsters are destroyed, and when your Attack Position monster battles a Defense Position monster that has DEF equal to or higher than your monster's ATK, neither monster is destroyed by battle.
Whether or not monsters are destroyed by battle is determined during damage calculation.
Many effect requirements can be fulfilled when a monster is destroyed this way, an example being "Skyscraper 2 - Hero City".
Some cards, such as "Mystic Swordsman LV2" and "Drillroid", have effects that destroy battling monsters during the Damage Step. However, cards destroyed this way are destroyed by that card effect, so are not considered to be destroyed by battle.