Combinatorial invariance of toric singularities

The are two theorems in this paper.
1. Let $\sigma_1$ and $\sigma_2$ be convex polyhedral cones in an $n$-dimensional lattice. Let $X_1$, $X_2$ be their associated affine toric varieties. $X_1$ and $X_2$ are isomorphic iff $\sigma_1$ and $\sigma_2$ are isomorphic.
2. Let $X_1$, $X_2$ be affine toric varieties. Let $T_1$ be a torus, embedded in $X_1$, $T_2$ be the same tor $X_2$. $X_1$, $X_2$ are isomorphic iff there exists a formal isomorphism between the points of maximal strati on $X_1$, $X_2$.