Section Sums.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
(** Existence of the sum *)
(** Commutativity of the sum. *)
(** Unicity of the sum. *)
(** Unicity of the difference of segments. *)
(** Unicity of the difference of segments on the right. *)
(** Cong preserves SumS *)
(** The degenerate segments represent the additive identity *)
(** Some permutation properties *)
(** Basic case of sum *)
(** Associativity of the sum. *)
(** AB <= AB + CD *)
(** CD <= AB + CD *)
(** If the sum of two segments is degenerate, then the segments are degenerate *)
(** SumS preserves Le *)
(** If AB <= A'B', CD <= C'D' and AB + CD = A'B' + C'D', then AB = A'B' and CD = C'D' *)
(** If AB < A'B' and CD <= C'D', then AB + CD < A'B' + C'D' *)
(** If CD >= C'D' and AB + CD <= A'B' + C'D', then AB <= A'B' *)
(** If CD > C'D' and AB + CD <= A'B' + C'D', then AB < A'B' *)

Lemma le_lt12_sums2__lt34 : forall A B C D E F A' B' C' D' E' F',
Lt A' B' A B -> Le E F E' F' -> SumS A B C D E F -> SumS A' B' C' D' E' F' ->
Lt C D C' D'.