well, the way i was looking at it was: suppose we define φ:V/N → V/M by φ(v+N) = v+M. φ is well-defined because M ⊆ N.

of course, in this case φ(v+N) = 0 (that is, 0+M) implies v is in M, which in turn implies v is in N, so ker(φ) = N, so we have an isomorphism between φ(V/N) and V/N.

the similarity being, the construction of φ (do we really need a basis? oh, and alex, if you're reading this, in light of what i wrote in the

other thread, the fact that i'm asking this question here surely must be somewhat...ironic...)