(** * Proposition 20 In any triangle two sides taken together in any manner are greater than the remaining one. *)
(** # <div style="width:748px;height:397px;display:block" id="applet_container20"></div> # **)

(* Specifications: *)
Lemma prop_20 : forall A B C P Q, ~ Bet B A C -> SumS A B A C P Q -> Lt B C P Q.
(** * Proposition 21 If on one of the sides of a triangle, from its extremities, there be constructed two straight lines meeting within the triangle, the straight lines so constructed will be less than the remaining two sides of the triangle, but will contain a greater angle. *)
(** # <div style="width:748px;height:397px;display:block" id="applet_container21"></div> # **)
Lemma prop_21_1 : forall A B C D, OS A B C D -> OS B C A D -> OS A C B D -> LtA B A C B D C.
Lemma prop_21_2 : forall A B C D A1 A2 D1 D2, OS A B C D -> OS B C A D -> OS A C B D ->
SumS A B A C A1 A2 -> SumS D B D C D1 D2 -> Lt D1 D2 A1 A2.