I am searching for a reference that Kasparovs KK-theory is stable, that is for separable Hilbert spaces $H,H'$ there are isomorphisms
$KK(A\otimes \mathbb{K}(H),B)\cong KK(A,B\otimes \mathbb{K}(H')\cong KK(A,B)$.
Since $KK$ is homotopy invariant in both variables and any two isometric embeddings of Hilbert spaces are homotopic after applying $\mathbb{K}$, i.e. $\mathbb{K}(i_1)\simeq \mathbb{K}(i_2)$, it suffices to check that for some separable, infinite dimensional Hilbert space $H$ we have
$KK(A\otimes \mathbb{K}(H),B\otimes \mathbb{K}(H))\cong KK(A,B)$
My only source for this is the book of Blackadar. He gives mutually inverse maps for the above isomorphisms, but I dont now how to prove it. Unfortunately I dont have the book at hand right now, since the library is closed. Therefore I was looking for another reference for this. Can somebody help?
Alternative proof: I know that $H\otimes A$ is an imprimitivity bimodule for $\mathbb{K}(H)\otimes A$, $A$. Therefore $\mathbb{K}(H)$ and $A$ are Morita Rieffel equivalent. Using the Kasparov product it is easy to check that Morita Rieffel equivalent $C^*$-algebras induce isomorphisms $KK(A,B)\cong KK(\mathbb{K}(H)\otimes A,B)$ given by precomposition with the equivalence class of the imprimitivity biodule.
Can i argue like this or do i need the stability to prove the Kasparov product?