Bismillahi Al-Rahmani Al-Rahim
Please expound on what you meant when you said part i.e. rakah or less?
The wording in question is
The position in the Hanbali madhhab is that provided part of the prayer takes place within the alloted time, it is considered current and not makeup. This differs from the Shafi`is.
Both Zad al-mustaqni` (section c2.2 in the lessons) and Nail al-ma'arib (1.122) say that a prayer is considered a current performance provided that the inaugural takbir [takbir al-ihram] occurs within the appointed time - even if the adhan is made within moments after that takbir.
A book of Hanbali usul gives a definition for current performance [ada'] as "whatever is done during the time alloted for it, for the first time". Make-up performance [qada] is defined as "whatever is done after the time for current performance." The commentary adds a few things to clear up what is meant by make-up performance, adding that "it is essential to perform the action in its entirety after the time for current performance. So, if part of the action is performed during the time for current performance, it is a current performance." (Ibn al-Mibrad, Sharh ghayat al-sul, pp158-59. cf Ibn Qudamah, Raudat al-nazir, 1.254).
This differs with the Shafi`i madhhab where current performance is "performing an act of worship or a rak`ah during its [appointed] time" (Zakariyah al-Ansari, Ghayat al-wusul. p16. cf Taj al-Din ibn Al-Subki, Jam` al-jawami`, 1.106).
And Allah knows best.
Wa al-salamu `alaykum,