Probably not for several reasons.
Your daughter sustained no injury. Without damages, there's no case.
The membership and day care waiver likely shields the gym from liability, although there may be exceptions.
I would get your daughter any medical treatment she requires, report the incident to the gym, and contact a lawyer to review any options worth pursuing.
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I'm sorry this happened to your daughter. As a parent, it is difficult to witness your child be hurt in any way. However, if your daughter did not sustain any injury beyond the momentary discomfort of having her hair pulled then there likely aren't any damages worth pursuing legally. If she suffered more significant injuries then you should talk to a personal injury lawyer.
As a parent myself, I realize it is difficult to see or hear of any child being hurt in any way. Although it is absolutely frustrating and infuriating, since your daughter did not sustain any injury beyond the momentary discomfort of having her hair pulled, there do not appear to be any damages to justify the time and expense of pursuing a lawsuit. Hopefully, you can talk to the gym operators to see what they can do to prevent any such further incidents. Best of luck.
First, be sure your daughter is checked out by a doctor (you mentioned her head was all the way back....she was screaming).
Second, consider consulting with a qualified personal injury attorney depending on what the doctor says.
Third, make sure the club knows this happened and is aware of your concerns. If plan to engage a lawyer, do not contact the club until and unless you have discussed doing so with the lawyer.
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