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Corresponding Square concept in endgame

Can someone please explain to me the Corresponding Square concept in the endgame?
The king standing on a KEY SQUARE wins regardless of which side is to move. The CORRESPONDING SQUARE is the square from which the opposing king prevents the king from occupying the key square.

For example, in this position...

lichess.org/analysis/standard/4k3/8/8/8/8/1P6/8/2K5_w_-_-_0_0

... the squares a5, b5, c5 are key squares. If the white king is on one of these squares, white wins.

If the white king is on c4, the black king must move to c6 - the corresponding square for c4, where black can prevent that white enters b5 or c5. The same applies to b4 and b6, as well as a4 and a6.

That's why White only wins here with 1.Kb2! Kd7 2.Ka3! Kc6 3.Ka4!. The black king cannot now enter a6, the corresponding square for a4. After 3...Kb6 4.Kb4! it is Black's turn, and he cannot defend all three key squares. It would have to be White's turn to do so.

Another example. In the following study by Grigoriev ...

lichess.org/analysis/fromPosition/8/1p6/1P6/8/7K/8/8/1k6_w_-_-_0_1

... White cannot prevent Black from winning the b6-pawn. To hold a draw, if Black captures on b6, white must be able to move to b4, after which he occupies the corresponding square for b6. This only works with 1.Kg3! Kc2 Kf2! Kd3 Ke1! Kc4 Kd2! Kb5 Kc3! Kxb6 Kb4! and draw.
See also Chapter 10 of Averbakh & Maizelis "Comprehensive Chess Endings - Book 4 - Pawn Endings", which can be found as PDF on the internet.
An extreme example from that book:

i.ibb.co/9ZLQ4fP/corresponding-squares.png

The key squares are f4 and e5.

The squares with numbers are corresponding squares. If white enters one of the numbers, black must be able to enter the corresponding number, otherwise he loses. I leave it as an exercise for you to figure out the win for white ;-)
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