I have following matrices whose entries can be complex values $$\mathbf{H^{12},H^{21},H^{23},H^{32},H^{31},H^{13},V^1,V^2,V^3}.$$ All the $\mathbf{H^{ij}}$ matrices are full rank, all have same size and are diagonal. If $$\mathbf{H^{12}V^2=H^{13}V^3}~~~~~~~~~(1)\\\mathbf{H^{23}V^3\prec H^{21}V^1}~~~~~~~~~(2)\\ \mathbf{H^{32}V^2\prec H^{31}V^1}~~~~~~~~~~(3)$$ (please note that the dimensions of matrices are chosen in such a way that the above matrix multiplications are possible) then how to show that $$\mathbf{H^{12}(H^{21})^{-1}H^{23}(H^{32})^{-1}H^{31}(H^{13})^{-1}(H^{31})^{-1}H^{32}V^2}=\mathbf{(H^{21})^{-1}H^{23}V^3}~~~~~~(4)$$ Any help in this regard will be much appreciated. Thanks in advance.
My attempt:
I tried to get equation $(4)$ by multiplying both sides of $(1)$ by $\mathbf{(H^{21})^{-1}H^{23}(H^{13})^{-1}}$ but then I get $$\mathbf{(H^{21})^{-1}H^{23}(H^{13})^{-1}H^{12}V^2}=\mathbf{(H^{21})^{-1}H^{23}V^3}$$ although the right sides are equal but we can see that left sides are not equal. Therefore, I need help in showing the above relation.