The question is very straightfoward: It's true that given real vector spaces $V$, $W$, $X$, $Y$ such that $V\cong X$ and $W\cong Y$, then $V\otimes W\cong X\otimes Y$? And if it's true, what is the prove of the unicity, exaclty? I could prove that exists an isomorphisms betwen them, but I couldn't prove the unicity.
Here $\cong$ means that there exists a canonical isomorphism betwen them.