Its not about proving to them it wasn't you, let them prove that it was, and as they can`t .. job done!! explain once by letter that they have made a mistake and if they want to push the matter any further then all costs solicitor etc if needed will be forwarded to them.
Took my 9 year old girl there last night, was her first match and she absolutely loved it, The stewards were great, she was stopped as we first entered the north stand asked her name the given a packet of crisps, that was it she loved the place wanted to get her coat off as soon as we got out...