We will continue our discussion on Prims algorithm. We discussed that Prim's algorithm is a greedy method. We look for the minimum cost edges that we can add to our tree. Here we determined that at a vertex v and with a min cost edge e at v, there is a minimum cost spanning tree containing e. We also said that for any vertex v, some minimum cost edge at v is included in every minimum cost spanning tree. We then generalized it with partitions and said that some min cost edge across the partition must be included in every minimum cost spanning tree.
With the lemma above we choose the minimum weight  edge that traverses the set of vertices in the tree and those we have not explored yet and keep growing our tree . Thus, Prim's algorithm works in finding the minimum cost spanning tree.