Lipschitz continuity of oblique projections

Let $W$ and $L$ be complementary spaces of a finite dimensional unitary space $V$ and let $P(W,L)$ denote the projection of $V$ on $W$ parallel to $L$. Estimates for the norm of $P(W,L) - P(W,M)$ are derived which involve the norm of the restriction of $P(W,L)$ to $M$ or the gap between $L$ and $M$.