Spanning tree, forwarding from from root, how?

Hello!
I have very simple, and  I guess, stupid question.
I understand how switches choose forwarding ports by calculation cost to root.
But how path from root to other switches is calculated?
Let's say we have just two switches, connected by two  1G links.
One of switches is root, how it will find which por use as designated?
As I see on my hardware root uses both.
But if one of links is 1G and another 10G? How root will choose which port to use?
Thank you!

Yes, I think that link with less cost will be choosed.
But how? Is there any explanation how it works?
All manuals/examples I can find are about calculation path to root, to down from root.
Could you point me to manual with explanation how forwarding path is calculated down from root?
Thank you!

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