Part 1.
Part 2.
Part 1 says that every submonoid of M containing D
contains also <D>. Let W be a submonoid of
M containing D. Since elements of D belong to
W, their products are in W. Hence <D> is
contained in W.
As for Part 2, let R denote the right-hand side of the
equation. It is a submonoid, as we have seen in a previous theorem. Part 1 implies <D> 
R. But as <D> belongs to
C, the intersection R is also contained in
<D>, whence <D> = R.
R. But as <D> belongs to
C, the intersection R is also contained in
<D>, whence <D> = R.