I am reading Lance's book "Hilbert $C^*$-modules". In particular, I want to understand how to construct the (external) tensor product of Hilbert $C^*$-modules. Consider the following fragment from Lance's book (on p62) in which Kasparov's absorption theorem is used:

I understand everything in here, except the boxed equation. When I do the calculation, I get a double summation $\sum_{i,j,m,n}$ instead of a single summation $\sum_{i,j,n}$. This implies that the argument after that also fails.

What am I missing here?

It is worth noting that in the book "Elements of KK-theory" by Jensen and Thomsen, in section 1.2.4 (in which the external tensor product is defined), it is said that the implication $$\langle z,z\rangle = 0\implies z= 0$$ may fail (so that we have a semi-inner product and we need to pass through a quotient first to get a Hilbert module). So, maybe the argument in Lance's book doesn't work?

Thanks in advance for your insights.