I don't understand the proof of correctness of Dijkstra's Algorithm

Dijkstra's Algorithm:

https://preview.redd.it/uo9weib8fn0h1.png?width=615&format=png&auto=webp&s=080d59dcd8148cfce986fa08616e27e855bc715b

https://preview.redd.it/vx824t7afn0h1.png?width=614&format=png&auto=webp&s=ffa594a70dda5595dec2110a52d59001b7f5d6b0

Proof of correctness of Dijkstra's Algorithm:

https://preview.redd.it/16gvshlffn0h1.png?width=612&format=png&auto=webp&s=660ff5e6e517d47008da2efb5ca1f73ee6386124

https://preview.redd.it/mv4s26vgfn0h1.png?width=612&format=png&auto=webp&s=de73178038ae685dbb8ef4c81a7b4d9f339367b9

https://preview.redd.it/2laqd55ifn0h1.png?width=612&format=png&auto=webp&s=aa1245fbef8e5f01313b3a3285556af10fb06fd5

Specifically, I don't understand the second part of the proof, the part after the illustration.

I'll start with this question: Why is LSP(a,v) >= LSP(a,s) + w(s,t)? Also, what is 'r' in '...then t would be added to T instead of r'?

This is what I could come up with (but it seems to me it's unrelated to the proof above):

https://preview.redd.it/lp7bfz6kfn0h1.png?width=1212&format=png&auto=webp&s=faa24f45229f1fbde31f6d0e379ca1bf3053a5a1