Turán theorem by minimum-degree reduction

State Tur´an’s theorem concerning the maximum number of edges in a graph on n vertices not containing a \(K _ { r }\)

Prove that if G is a graph with \(n \geq r + 1\) vertices and \(t _ { r - 1 } ( n ) + 1\) edges then for every \(n ^ { \prime }\) with \(r \leq n ^ { \prime } \leq n\) there is a subgraph H of G with \(n ^ { \prime }\) vertices and at least \(t _ { r - 1 } ( n ^ { \prime } ) + 1\) edges. [Hint: consider a vertex in G of minimum degree.]

From the previous part deduce Tur´an’s theorem, and also the stronger fact that such a G contains two \(K _ { r }\) subgraphs sharing \(r - 1\) vertices.