Well, let's look at each position locally.
$$c Diagram 1
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$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . O X X X . . . |
$$ | . . . . . . . . B . . . 1 O O O . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
- Click Here To Show Diagram Code
[go]$$c Diagram 1
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$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . O X X X . . . |
$$ | . . . . . . . . B . . . 1 O O O . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |[/go]
Just looking at the 8 stones to the right, the formation is equal (ignoring the edges of the board). However, all the stones are tactically weak, with at most 4 dame. That means that the first player has a greater than usual advantage. Yes, the

stone offers some support, but the cut is still questionable.
$$Wc Diagram 2
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . 1 X . . . . . . . . . . . . . . |
$$ | . . . X O . . . . , . . . . . . . . . |
$$ | . . . 2 . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------
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[go]$$Wc Diagram 2
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . 1 X . . . . . . . . . . . . . . |
$$ | . . . X O . . . . , . . . . . . . . . |
$$ | . . . 2 . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]
Again, ignoring the edges of the board, the position after the cut is equal, but all the stones are weak.

is good, as the proverb states.
White would in general do better to play at 2 instead. That also produces a locally equal position, but the stones are not as weak.
$$c Diagram 3
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . X . . . |
$$ | . . . . . . . . . . . . . O 1 . . . . |
$$ | . . . . . . . . . . . . . 3 2 O . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . B . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------
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[go]$$c Diagram 3
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . X . . . |
$$ | . . . . . . . . . . . . . O 1 . . . . |
$$ | . . . . . . . . . . . . . 3 2 O . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . B . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

-

gives White a good shape (extending from a crosscut). Not recommended, as a rule.
$$c Diagram 4
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . X . . . |
$$ | . . . . . . . . . . . . . O 2 3 . . . |
$$ | . . . . . . . . . . . . . 6 1 O 5 . . |
$$ | . . . . . . . . . . . . . . 4 . 7 . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . B . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------
- Click Here To Show Diagram Code
[go]$$c Diagram 4
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . X . . . |
$$ | . . . . . . . . . . . . . O 2 3 . . . |
$$ | . . . . . . . . . . . . . 6 1 O 5 . . |
$$ | . . . . . . . . . . . . . . 4 . 7 . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . B . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]
This is normally a good sequence for Black, which is why there is a proverb about playing

. But White does not have to play

. He could descend to 5 instead, for instance. The

stone offers some support, but often Black should start at 5 instead of 1.
Cut-Fight-Kill is not bad advice to beginners.

It is certainly better than Secure Territory.

But there are better examples.
