Moving the rook is a clever move, as it prevents mate. The king would still be checked, though. Double-checked. A more clever move is the one listed on the main page, I think. The reason: You prevent check on the next move and you prevent mate. I don't know, but I think it is more clever than Re6.
Anyway, the situation on the main page was just to show how builders take and how they mate/check.
One possible notation for the builder: The Builder (Bu or I), builder take: Id5-a5*(b)Ra6->(w)Ra6. The builder (I) on d5 moves to a5 and takes a (black) rook on a6, switching it to (->) a (white) rook on a6. For the situation in which the black rook moves to e6, the builder does this: Id5-d6*(b)Bd7 -> (w)Bd7+ &*(b)Re6 -> (w)Re6+. This means that the builder on d5 moves to d6 (Id5-d6) and kills (*) the (black) bishop on d7 switching it (->) for a (white) bishop on d7, checking, AND (&) it kills the (black) rook on e6, switching it for a (white) rook on e6, checking.
Anyway, the situation on the main page was just to show how builders take and how they mate/check.
One possible notation for the builder: The Builder (Bu or I), builder take: Id5-a5*(b)Ra6->(w)Ra6. The builder (I) on d5 moves to a5 and takes a (black) rook on a6, switching it to (->) a (white) rook on a6. For the situation in which the black rook moves to e6, the builder does this: Id5-d6*(b)Bd7 -> (w)Bd7+ &*(b)Re6 -> (w)Re6+. This means that the builder on d5 moves to d6 (Id5-d6) and kills (*) the (black) bishop on d7 switching it (->) for a (white) bishop on d7, checking, AND (&) it kills the (black) rook on e6, switching it for a (white) rook on e6, checking.
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