  The positivity property for canonical bases asserts that the structure
constants of the multiplication for the canonical basis are in ${\mathbb
N}[v,v^{-1}]$. Let $\mathbf U$ be the quantum group over ${\mathbb Q}(v)$
associated with a symmetric Cartan datum. The positivity property for the
positive part ${\mathbf U}^+$ of ${\mathbf U}$ was proved by Lusztig. He
conjectured that the positivity property holds for the modified form
$\dot{\mathbf U}$ of ${\mathbf U}$. In this paper, we prove that the structure
constants for the canonical basis of $\dot{{\mathbf U}}(\widehat{\frak{sl}}_n)$
coincide with certain structure constants for the canonical basis of ${\mathbf
U}(\widehat{\frak{sl}}_N)^+$ for $n<N$. In particular, the positivity property
for $\dot{{\mathbf U}}(\widehat{\frak{sl}}_n)$ follows from the positivity
property for ${\mathbf U}(\widehat{\frak{sl}}_N)^+$.
