In the above mentioned question your Salah will be complete. However, if it is a Jahri (a prayer where the Imam reads loud, i.e. Fajr, Maghrib and Isha) you will have to start praying loud as soon as the person joins. Furthermore, it is Sunnah for you to say the Takbeers? loudly when another person is following you.

As for the person who joined you (Muqtadi), if he prayed the same Salah as you then his Salah will be done.

However, if he was praying a Salah, that is higher than yours (e.g. you were praying Sunnah and he was praying Fardh) or equal to yours but not the same (e.g. you were praying Asr whilst he was praying Zohar), then his Salah will not be permissible. Allah knows best.

Answer Approved by Shaikhul Hadeeth Mufti Farooq