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No.
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Lee has higher scaling and base damage so it makes sense that he would deal more damage. If you're playing aatrox to be a damage dealer you're playing him wrong though.
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Build lifesteal and bam, problem with damaging yourself with Blood Price mostly solved. Blade of the ruined king + Hydra combined with tank items. (Randuins and Sunfire or thornmail as armor options, spirit visage and banshee's veil for MR)
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And no, Lee sin actually doesn't beat aatrox in sustain unless you also aren't building lifesteal on aatrox, using blood thirst under 50% HP to gain the tripled heal and completely not realizing that the heal on Aatroxs W is not affected by an enemies armor (which Lee Sin's lifesteal/spell vamp is affected by armor. ).
Aatrox level 5 W has a base heal of 40 with a 25% bonus AD ratio so yeah that looks a bit small, but... under 50% HP that triples to 120 + 75% bonus AD ratio that isn't affected by an opponents armor.
Aatrox sustain above 50% HP is inferior to lee's, but below 50% is a different story.
At level 18.
Lee Sin 116 AD
Aatrox 115 AD
Lee sin with his W lifesteal up (25%) lifesteal, will gain back 29 health per hit for a total of 87.
Below 50% HP aatrox's base heal is 120.
So not really that close, even further away once you remember that armor affects lee sin's lifesteal, but not aatroxs W.
Let's throw in a hydra for both characters
Lee Sin now has 191 AD. Lee Sins 32% lifesteal now heals him for 61 HP every hit against a target with zero armor. 183 HP in 3 hits.
Aatrox now has 190 AD. 120 + 56 (which is 75% of 75) = 176 HP in 3 hits unnaffected by armor. 191 X .12 (hydra lifesteal) adds an additional 23 health every hit (*3 makes it 69 additional sustain every 3 hits, though this is affected by armor). So 245 HP every 3 hits for Aatrox, of which, only 69 is affected by an enemies armor.
Oh and of course I'm forgetting the loads of attack speed in aatroxs kit. Lee sin gets a decent chunk, but nothing compared to aatrox.