Math card trick 2

It is the explanation of the trick shown below in the embedded video.

Figure 1: showing  how the cards are arranged to start with. It shows the card (red) which is the randomly chosen card by the observer from the set of randomly chosen 9 cards. It is always kept in the 44th position from the top of the deck.

1. Now there are 4 piles created starting from the top of the deck. ( counting 10 downwards and stopping when I get a card match).
2. Let x be the number for the card match in the first pile, such that 0<= x <=10.
3. It means that the number of cards making a pile can range from 1 to 11 .
4. So if you get a card match at x, you stop there, so the number of cards in Pile 1 = 11 - x.
5. Similarly let y be the card match for Pile 2, number of cards in Pile 2 = 11 - y
6. Let z be the card match for Pile 3 , number of cards in Pile 3 = 11 - z
7. Let a be the card match for Pile 4, number of cards in Pile 4 = 11 - a
8. Now lets link x,y,z and a.
9. We know that in the worst case each pile should have gotten 11 cards each from the deck.
10. So now if Pile 1 card match is x then, it leaves out x  cards, that should have  otherwise  belonged to Pile 1 , in the deck. That leaves a total of (41 +x) cards in the deck.
11. Now again we shell out (11 - y) cards out (since in the second pile, card match was y). That will leave a total of (30 + x + y) cards in the deck.
12. Similarly we give away (11 -z) and (11 - a) number of cards for Piles 3 and 4 respectively. Finally that will leave us with (8 + x+ y+ z+ a) number of cards in the deck.
13. Now lets find out what will the position from the top of the deck of our random card.
14. Our random card is in the 9 th position from the bottom of this new deck too. ( since we will have disturbed it all along).
15. In a pack of  (8 + x+ y+ z+ a) cards , the number of cards above the 9th card is (  x+ y+ z+ a - 1)
16. So (x + y + z + a) th  card from the top will our random card, which was purposefully kept in the top of the 9 card pile previously.