While there may be no specific rule or CAQ for this instance, I would hold that a separate set of games would have to be bowled for each league, and I base it on CAQ #122. A summation of that CAQ:
A team (or individual) had pre-bowled, but an emergency forced the postponement of the entire league session for which they had pre-bowled. The team shows up to bowl. However, the pre-bowled scores are to be used, as the eligibility for that league session (rule 107a) was exhausted when they pre-bowled.
To apply that rule inversely, it would mean that a set of games can only be applied to one competition, and that it would have to be declared prior to bowling.
I will illustrate using an example I know to be in use - a parallel, as it's not an identical circumstance, but this is one where the rules for what is in this case, an association tournament, have been printed.
My association, a merged one, is having all of its divisions bowling in coincidental squads, those being Senior Open, Open, Women's, Youth 600+, and Youth <600, in a team event only. It's specifically stated that a team of all women may choose to bowl in either the Open or Women's division, but must declare that when entering. If a team wished to participate in more than one division, the team would have to re-enter, pay the entry fee for each entry, and bowl as many times as needed. The division for which that team would be bowling would be known prior to bowling.
I know that goes into a bit of detail, but I'm illustrating that games bowled can only be applied to one competition; they cannot be applied to multiple competitions. I would still give the rules department in Greendale a call (800) 514-BOWL [2695] extension 3155. And, if known, I'd give the other league's secretary a call, too.
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