Forced early relieving from services

Dear Sir/Madam, As per terms and conditions in the appointment letter notice for resignation/termination can be given with either three months notice or equivalent salary from both sides. I resigned from an academic institute on 1st April 2016 and thus my effective data of resignation would be 30th June 2016. My Registrar is forcing me to get relieved on 17th June 2016 without paying for 13 days which is a short-fall in the relieving date. They are giving reasons that it is written in so called service rules that competent authority can give approval to relieve early. They have never notified to any employee about such rules of resignation or about any service rules in the past. Now, all of sudden they are saying that it is written in their service rule and so can do whatever they want. They have sent an extract of the service rule which is absolutely one-sided and infringes upon the basic right of an employee. I have attached below that extract of service rule. I would like to know in this case in the court of law which document will be considered legal, appointment letter which binds both of us or the service rules and regulations which contradicts with what is written in the appointment letter? Extract from the service rule: 2.22 Resignation: 2.22.3 The Competent Authority may approve the resignation with immediate effect or from any date prior to the last date of the notice period on completion of academic/assigned responsibilities as may be decided by the management and without any liability of the Institution for the notice pay in case no further assignment can be provided by the Institution for the entire notice period. 2.24 Leave /Vacation during Notice period The employee shall be entitled to avail maximum of 15 days leave due to his /credit during the entire notice period with a provision to avail maximum of five days at any one time. The employee shall not be allowed any vacations during the period of notice. Please advise on this matter. I thank you and the in advance. Regards, Saurabh K. Raina