can someone possibly explain the following lemma?

While reviewing my text book I came across a lemma that I just can't seem to wrap my head around no matter how I try to break it down. I was wondering if someone had an esier way of explaining it. The lemma is as follows:

Let n be a positive integer. Let A be a set; let a be an element of A. Then there exists a bijective correspondence f on the set A with the set {1,...,n+1} if and only if there exists a bijective correspondence g of the set A-{a } with the set {1,...,n}.

I know bijective correspondence we want one to one AND onto and I also have the proof sitting in front of me (if one wishes me to post it I will) but I feel when I fully understand the lemma the proof will follow easily. For some reason as many times as I read it, it seems to be going over my head.