  The Greene-Magnanti theorem states that if $ M $ is a finite matroid, $ B_0 $
and $ B_1 $ are bases and $ B_0=\bigcup_{i=1}^{n} X_i $ is a partition, then
there is a partition $ B_1=\bigcup_{i=1}^{n}Y_i $ such that $ (B_0 \setminus
X_i) \cup Y_i $ is a base for every $ i $. The special case where each $ X_i $
is a singleton can be rephrased as the existence of a perfect matching in the
base transition graph. Pouzet conjectured that this remains true in infinite
dimensional vector spaces. Later he and Aharoni answered this conjecture
affirmatively not just for vector spaces but for infinite matroids.
  We prove two generalisations of their result. On the one hand, we show that
`being a singleton' can be relaxed to `being finite' and this is sharp in the
sense the exclusion of infinite sets is really necessary. On the other hand, we
prove that if $ B_0$ and $ B_1 $ are bases, then there is a bijection $ F $
between their finite subsets such that $ (B_0\setminus I) \cup F(I) $ is a base
for every $ I$. In contrast to the approach of Aharoni and Pouzet, our proofs
are completely elementary, they do not rely on infinite matching theory.
