How to derive the $W_1$ Wasserstein distance written with the quantile functions (i.e. with inverse cumulative distribution functions)

Question

INTRODUCTION. Either downloading the slides on Optimal Transport (OT) from the Marco Cuturi website (go to section "Teaching ENSAE --> OT --> Optimal Transport (Spring 2023) [slides]" and click on "slides"), or just watching his lectures on YouTube (Part 1, Part 2 and Part 3), Dr. Cuturi shows the following:

1. the Monge Problem (pag. 157/432),
2. the Primal form of the Kantorovich problem (pag. 168/432),
3. the Dual form of the Kantorovich problem (pag. 170/432),
4. the definition of the Wasserstein Distance (pag. 192/432),
5. the Semi-Dual form of the Wasserstein Distance (pag. 192/432),
6. the $W_1$ Wasserstein Distance written using the inverse cumulative distribution functions (pag. 243/432, or Part 3 at minute 8:25)

QUESTION. How can I derive the object "6. the $W_1$ Wasserstein Distance written using the inverse cumulative distribution functions (pag. 243/432)" from either object 1., or 2., or 3., or 4., or 5.?

In other words, where does the object "6. the $W_1$ Wasserstein Distance written using the inverse cumulative distribution functions (pag. 243/432)" come from?

Any suggestion is welcome!