proposition: Lemma orthogonal_oppr S R : orthogonal S (map -%R R) = orthogonal S R.
proof: Proof. wlog suffices IH: S R / orthogonal S R -> orthogonal S (map -%R R). by apply/idP/idP=> /IH; rewrite ?mapK //; apply: opprK. move/orthogonalP=> oSR; apply/orthogonalP=> xi1 _ Sxi1 /mapP[xi2 Rxi2 ->]. by rewrite cfdotNr oSR ?oppr0. Qed.
imports: HB(structures) mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq path) mathcomp(div choice fintype tuple finfun bigop prime order) mathcomp(ssralg poly finset fingroup morphism perm) mathcomp(automorphism quotient finalg action gproduct) mathcomp(zmodp commutator cyclic center pgroup sylow) mathcomp(matrix vector falgebra ssrnum algC algnum) mathcomp(archimedean)