Fact joinKI y x : meet x (join x y) = x.
Definition Mixin := LatticeMixin meetC joinC meetA joinA joinKI meetKU leEmeet meetUl.
End TotalLattice.
End TotalLattice.
Module TotalTheory.
Section TotalTheory.
Context {display : unit}.
Local Notation orderType := (orderType display).
Context {T : orderType}.
Implicit Types (x y : T).

Lemma joins_disjoint (I : finType) (d : L) (P : pred I) (F : I -> L) :
(forall i : I, P i -> d `&` F i = 0) -> d `&` \join_(i | P i) F i = 0.