Well, absolute proof is difficult. We usually have to settle for less but
there are degrees. The best would be to find Peisach's birth record saying
that Leib was his father and someone you know specifically was his mother,
better yet, if they have strange names for the area. Then you would also
want a death certificate for Leib saying that his wife was the same woman
and they had 3 (or 6)kids and one was Paisach.
Unfortunately for us, most of the time your ancestor, Leib, was named for
some previous Leib, and so there might very well be cousins also named Leib.
The good thing is that in 1859 while Leib's parents may have taken the last
name Shaposhnik, they might be the only Shaposhniks in town. If Leib's
uncles took different last names, then chances are good that this is 'your'
Of course, it would be possible, (if it turns out not to be the right birth
record for some reason) that Peisach's father was not using the name
Shaposhnik at the time. That happened even in the mid-1800's; family names
weren't really set yet and sometimes the birth record might only have a
patronymic and no surname or one which is different than the one which you
know. It is possible that Leib's father took the surname Leibowitz (because
his father was Leib) or something else and Leib's birth was registered as
Leib Leibowitz or whatever-or there is no family name at all. Then later
the name they used was Shaposhnik and that is the name you know. Then life
gets very hard for us.
So, the smaller the town, the odder the name, then the chances are better.
A common name in a big city-no. We often have to keep looking and say this
is possible, most likely, maybe-and see where it leads.