+Also the following trivial lemma will be often invaluable:
+
+\lemman{Edge removal}
+Let~$G$ be a~graph with distinct edge weights and $e \in G\setminus\mst(G)$.
+Then $\mst(G-e) = \mst(G)$.
+
+\proof
+The tree $T=\mst(G)$ is also a~MST of~$G-e$, because every $T$-light
+edge in~$G-e$ is also $T$-light in~$G$. Then we apply the uniqueness of
+the MST of~$G-e$.
+\qed
+
+\paran{Comparison oracles}\id{edgeoracle}%
+To simplify the description of MST algorithms, we will assume that the weights
+of all edges are distinct and that instead of numeric weights we are given a~comparison oracle.
+The oracle is a~function that answers questions of type ``Is $w(e)<w(f)$?'' in
+constant time. This will conveniently shield us from problems with representation
+of real numbers in algorithms and in the few cases where we need a more concrete
+input, we will explicitly state so.
+
+In case the weights are not distinct, we can easily break ties by comparing some
+unique identifiers of edges. According to our characterization of minimum spanning
+trees, the unique MST of the new graph will still be a~MST of the original graph.
+Sometimes, we could be interested in finding all solutions, but as this is an~uncommon
+problem, we will postpone it until Section \ref{kbestsect}. For the time being,
+we will always assume distinct weights.
+
+\obs
+If all edge weights are distinct and $T$~is an~arbitrary spanning tree, then every edge of~$G$
+is either $T$-heavy, or $T$-light, or contained in~$T$.
+
+\paran{Monotone isomorphism}%
+Another useful consequence of the Minimality theorem is that whenever two graphs are isomorphic and the
+isomorphism preserves the relative order of weights, the isomorphism applies to their MST's as well: