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 If you have three Lightning channels, and two of them have plenty of spending capacity, but you want to send a transaction that is larger than the balance in either individual channel… does your node “do the work” for you to make a combined payment from both channels? Would it likewise split out a larger incoming payment to multiple channels, and “fill the first cup before it starts filling the next” 

How does it know to do this, if each channel is opened to a different peer? #asknostr 
 You’ll have to use AMP. You can do this with @ZEUS. 

https://docs.lightning.engineering/lightning-network-tools/lnd/amp 
 Prob easier to just have a fat channel with lots of liquidity in both directions though 
 Renting fat channels is a good option. Phoenix offers this in their app. 
 Thanks my dude. Lots of learning to do 🫡 
 If you wanna go down the rabbit hole, read at least the first few chapters of Mastering Lightning

https://github.com/lnbook/lnbook 
 Another great resource I have learnt a lot from is the voltage blog. Many well written and easy to understand articles the Lightning topics I have had questions for. 

This post discusses multipart payments which your question is about. 
https://www.voltage.cloud/blog/multipart-payments-on-lightning-network-break-down-amp-and-mmp

And this may be a good place to start learning about running your own personal node in general.

https://www.voltage.cloud/blog/lightning-node-type-series-1-the-consumer-node-or-how-to-live-on-lightning

Not sure how far along the journey you are but I have found this my goto site for learning something about lightning.