Consider $G=N_1D_1^{-1}=N_2D_2^{-1}$ and the both descriptions of are composed of coprime matrix, prove that $D_2^{-1}D_1$ is a unimodular matrix.
The matrices $D_i$ and $N_i$ are polinomial.
My attempt: $N_1D_1^{-1}=N_2D_2^{-1}$ then $N_1=N_2D_2^{-1}D_1$
Using Bezout identity and GCD:
$U_{11}D_1 + U_{12}N_1= R$
$U_{11}D_2D_2^{-1}D_1 + U_{12}N_1= R$
$U_{11}D_2D_2^{-1}D_1 + U_{12}N_2D_2^{-1}D_1= R$
Since R is unimodular, then $U_{11}D_2D_2^{-1}D_1 + U_{12}N_2D_2^{-1}D_1$ is unimodular. $R^{-1}(U_{11}D_2D_2^{-1}D_1 + U_{12}N_2D_2^{-1}D_1)=I$
$R^{-1}(U_{11}D_2 + U_{12}N_2)=(D_2^{-1}D_1)^{-1}$
$(D_2^{-1}D_1)R^{-1}(U_{11}D_2 + U_{12}N_2)=I$
$U_{11}D_2 + U_{12}N_2=(D_2^{-1}D_1R^{-1})^{-1}$
The LHS is polynomial then the RHS is unimodular, and R is unimodular then $D_2^{-1}D_1$ is unimodular.
Is this proof complete? Thank you in advanced.