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(* 02110-1301 USA                                                     *)
(** %\subsection*{ support :  algebra\_omissions.v }%*)
(** - In this file I prove some results that do not pertain to linear algebra at all. Instead, they could have been proven solely for the algebra distribution. This is also why I do not use the syntactic sugar as defined in equal_syntax.v and more_syntax.v *)
From Algebra Require Export Group_facts.
Set Implicit Arguments.
Unset Strict Implicit.
intros.
apply Trans with (group_inverse _ (group_inverse _ g)); auto with algebra.
apply GROUP_law_inverse.
apply Trans with (sgroup_law _ (group_inverse _ g') g'); auto with algebra.
Qed.
(* Because of later notation, this should also be known as: *)
Definition min_inj := group_inverse_inj.
(** - Elements of subsets are really pairs of those elements and proofs that they belong to that subset. I prove that two of these pairs are equal (in the subsetoid) iff their first projections are (in the parent setoid) *)

Lemma subtype_elt_comp :
forall (A : Setoid) (B : part_set A) (b b' : B),
Equal b b' -> Equal (subtype_elt b) (subtype_elt b').