Prove that if C D are nn matrices and C D2 and D are invert

Prove that if C, D are n×n matrices and C * D^2 and D are invertible then C is invertible.

given C,D are n*n matrices

and C*D^2 is invertible and D is also invertible that means there exists a inverse of these. that is

(c*D^2)*(c*D^2)^-1 = I (by associativity property)

{ here for better understanding consider D^2=Y}

(c*Y)*(c*Y)^-1 = I

(c*Y)^-1 = C^-1 *Y^-1

as D is invertible D^2 is also invertible that is Y is invertible as D^2=Y ( our consideration)

since Y^-1 exists and so does C^-1 exists that is C is also invertible.

hence it is proved that C is also invertible.