If I didn't give the required notice to my landlord, am I obligated to pay the remainder of my oral lease?

Question Details:

I am a independent contractor as a cosmetologist. I gave my 30 days notice on the 09/12. I gave my 30 day notice to the lady who leases the building that I work in (she does not own it); it was the right thing to do. She is telling me that I am obligated to prorate until 10/01. Is this legal? A 30 day notice was not stated. Am I obligated to pay the prorate of the 12 days?

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