Near-invariant subspaces for matrix groups are nearly invariant

Let S be a semigroup of invertible matrices. It is shown that if P is an idempotent matrix of rank and co-rank at least two such that the rank of (1−P)SP is never more than one for S in S (the range of the kind of P is said to be near-invariant), then S has an invariant subspace within one dimension of the range of P (the kind of range is said to be nearly invariant).