Proof

Let K be _{H in C} H . Then by the result on intersections of submonoids we find that K is a submonoid of G. It remains to check that for every k K, also the inverse k^-1 is in K.

Since k is an element of every H in C, also k^-1 is in H for every H in C (this is because H is a subgroup). Hence k^-1 is in the intersection K of all H in C.