Eulerian Directed Multigraphs

For Δ a finite connected nontrivial directed multigraph, we prove: 1. Δ has a directed circuit using each directed edge exactly once if and only if both each pair of distinct vertices of Δ occur in a common directed circuit and in-degree( x) = out-degree( x) for every vertex x. 2. Δ contains a non-circuit directed path which uses every directed edge exactly once if and only if both every pair of distinct vertices of Δ occur in a common directed circuit and there are vertices b = e such that in-degree( e) - out-degree( e) = 1 = out-degree( b) - in-degree( b) but, for every vertex x ∉ ,e\, it happens that in-degree( x) = out-degree( x).