The Lemma is from Haagerup's paper. I was confused by the statement makes red. We assume that $M$ is a type III$_1$ factor.
It is well-konwn that projections in a type III factor are equivalent. We know that $p\sim q$ and $1-p\sim 1-q$. But How to conclude that $p$ is unitatily equivalent to $q$?
If $$v^*v=p,\qquad vv^*=q,\qquad w^*w=1-p,\qquad ww^*=1-q,$$ then $u=v+w$ is a unitary and $upu^*=q$.