If I were the one looking to lease a horse I probably wouldn't do so from someone who required me to take lessons from a specific person, unless I was already planning on doing so.
The way that you worded it had me confused as well- what do you mean by you use her half of the month/someone else half the month? Like you have access to her from the 1st to the 15, and the other person from 16th-30? Most half leases specify that you can ride the horse Monday, Wednesday, Friday, and the other person can ride Tuesday, Thursday, Saturday or something to that effect. I would not lease a horse that I was only allowed to ride for the first half of the month. Sorry if I misunderstood, but it confused me :P
Who is paying for your mare right now? Are you young and you have parental help? Are you leasing her or something? Either way, you have to seriously evaluate whether or not you can afford to have a horse. Even if you can afford board with someone leasing her, what if something comes up with the lease, or what if a horse emergency comes up? Where will the money come from? Plus all of the other expenses associated with horses. Not saying that you can't do it, just make sure it's all thought through!