I have an undirected weighted graph G=(V,E) where V represent nodes and E represent edges. Through Dijkstra Algorithm, I got a shortest path tree Ts=(s,V) rooted at source node s and spanning all nodes V in the graph G. Then I selected a sub-tree Tm=(s,K), (where K is a subset of V) of shortest path tree Ts=(s, V) that connect s to only K nodes among all V nodes, i.e, the sub-tree Tm is a subset of shortest path tree Ts.
My question is how can now I prove by arguments or a lemma/Theorem that this sub-tree Tm of shortest path tree Ts is also a shortest tree?. Thank you in advance.