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(** *                       Translation                           *)
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Definition xt (x tx:R) : R := x + tx.Definition yt (y ty:R) : R := y + ty.

(* Specifications: *)
Lemma translation_0 : forall x y:R, xt x 0 = x /\ yt y 0 = y.
Lemma isometric_translation : forall x1 x2 y1 y2 tx ty:R, Rsqr (x1 - x2) + Rsqr (y1 - y2) = Rsqr (xt x1 tx - xt x2 tx) + Rsqr (yt y1 ty - yt y2 ty).