In this paper the authors consider interactions of two instances of the predicate {\tt at-least}. This study is motivated by the assignment problem. The assignment problem is one of the most famous problems in operations research, and basically it tries to assign someone $i$ to some job $j$. This can be formulated by a complete bipartite graph ${G = (V_1 \cup V_2, E)}$ where $V_1$ and $V_2$ are node sets and $E$ is the edge set. Basically each node in $V_1$ represents a person in the group and each node in $V_2$ represents a job to assign. Since $G$ is a complete bipartite graph on $V_1$ and $V_2$, each node in $V_1$ is adjacent to all nodes in $V_2$ via edges. If we formulate this problem as an integer programming problem, we have to set up a binary variable $x_{ij}$ for each node $i \in V_1$ and each node ${j \in V_2}$. Thus the number of variables can be very large. This problem can be equivalently formulated using fewer variables (i.e., by at least $m$ of the $n$ variables). Suppose we have $n$ people and there are $m$ jobs. We would like to assign $m$ people out of $n$ people to $l$ jobs. In that case we can set up a variable $x_i = j$ for person $i$ to job $j$ for $j = 1, \ldots , l$. If we do, the number of variables can be reduced. This problem can also be formulated as a mixed-integer programming problem, and the authors focus on the polytope which is the convex hull of all possible assignments. The authors focus on the special case when $m = 1$ and ${n = 1}$. More specifically they look at the convex hull of two {\tt at-least} assignment problems at the same time, i.e., the convex hull of all possible two {\tt at-least} assignment problems. Then they look at the intersection of two polytopes of {\tt at-least} assignment problems. \par A {\it convex polytope} $P \subset \Bbb R^d$ is a convex hull of a set of finitely many points in $\Bbb R^d$. This representation of the polytope $P$ is called a vertex-representation of $P$. Equivalently, a polytope $P$ is a bounded polyhedron, that is, defined by the solution sets of inequalities $Ax \geq b$, where $A \in \Bbb{Q}^{f \times d}$, $x \in \Bbb{R}^d$, and $b \in \Bbb{Q}^f$. This representation is called a hyperplane-representation of the polytope $P$. Converting from a vertex-representation of $P$ to a hyperplane-representation of $P$ (or vice versa) is known to be NP-hard. \par In this paper the authors show the complete descriptions of the intersection of two polytopes for {\tt at-least} assignment problems, and they characterize a number of facets of the intersection of two polytopes for {\tt at-least} assignment problems.