This formula could be used to compute matrix multiplication transpose
$${\displaystyle (\mathbf {AB} )^{\mathsf {T}}=\mathbf {B} ^{\mathsf {T}}\mathbf {A} ^{\mathsf {T}}}$$
This formula is used to elaborate matrix multiplication associativity
$${\displaystyle (\mathbf {AB} )\mathbf {C} =\mathbf {A} (\mathbf {BC} )}$$
This formula is used to compute svd of a matrix.
$$C = U \Sigma V^T$$
this MIT course puts them together and gives (equation_1)
$$C^T C = V \Sigma^T \Sigma V^T \tag 1$$
what is the detailed proof for equation_1?
first, what is the detailed procedure of $C^TC = (U \Sigma V^T)^TC$
how can I transfer this to $U^TU$?
$$C^T = V \Sigma^T U^T$$
Using your first two equations, you can get
$$(ABC)^T = ((AB)C)^T = C^T(AB)^T = C^T B^T A^T$$
and then
$$C^T = V \Sigma^T U^T$$
and then
$$C^T C = V \Sigma^T U^T C = V \Sigma^T U^T U \Sigma V^T$$
since
$$U^TU = I$$
eventually you get
$$C^T C = V \Sigma^T U^T U \Sigma V^T = V \Sigma^T \Sigma V^T$$