Proizvolov's identity concerns a set partition of the integers 1, 2, ..., into two -element sets. Write one set
in increasing order and the other in decreasing order . The identity
states that
For each ,
at least one of
and
exceeds .
Otherwise the first terms of the increasing list and the last terms of the decreasing list would give distinct integers no greater
than .
The pairwise maxima are therefore exactly , , ..., , while the pairwise minima are
1, 2, ..., .
It follows that
into two 
-element sets. Write one set
in increasing order 
and the other in decreasing order 
. The identity
states that
,
at least one of 
and 
exceeds 
.
Otherwise the first 
terms of the increasing list and the last 
terms of the decreasing list would give 
distinct integers no greater
than 
.
The pairwise maxima are therefore exactly 
, 
, ..., 
, while the pairwise minima are
1, 2, ..., 
.
It follows that
