it's an "if and only if" proof.

the "if part": if h preserves linear independence, h is non-singular. this is the part that is proved by the bold section. this is actually proved by proving the contra-positive: h is singular implies h takes SOME linearly independent set to a linearly dependent one.

the linearly independent set they choose is S = {v}, where v is some non-zero element of the null space (which exists BECAUSE h is singular). this of course, maps to h(S) = {0_{W}}, which is ALWAYS linearly dependent.

you have to include BOTH implications in an if and only if proof, you can't leave one out.