Open: /Coq/Sets/Powerset_facts/Union_commutative.con We prove the conjunction again: alias U /Coq/Sets/Ensembles/Ensembles/U.var alias V /Coq/Sets/Powerset_facts/Sets_as_an_algebra/U.var alias Ensemble /Coq/Sets/Ensembles/Ensemble.con alias Union /Coq/Sets/Ensembles/Union.ind#1/1 alias Included /Coq/Sets/Ensembles/Included.con alias and /Coq/Init/Logic/and.ind#1/1 The two parts of the conjunction can be proved in the same way. So we can make a Cut: !C:Ensemble{U:=V}.!D:Ensemble{U:=V}. (Included{U:=V} (Union{U:=V} C D) (Union{U:=V} D C))