End Book_1_prop_1_circle_circle.
Section Book_1_part_2.
(** For the next 27 propositions, we do not need the 5th axiom of Euclid, nor any continuity axioms, except for Proposition 22, which needs Circle/Circle intersection axiom. *)
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
(** * Proposition 2 To place at a given point (as an extremity) a straight line equal to a given straight line. *)
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(* Specifications: *)
Lemma prop_2 : forall A B C, exists L, Cong A L B C.
(** * Proposition 3 Given two unequal straight lines, to cut off from the greater a straight line equal to the less. *)
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Lemma prop_3 : forall A B C1 C2, Le C1 C2 A B -> exists E, Bet A E B /\ Cong C1 C2 A E.
(** * Proposition 4 If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend. *)
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Lemma prop_4 : forall A B C D E F, CongA C A B F D E -> Cong A C D F -> Cong A B D E ->
Cong C B F E /\ (C <> B -> CongA A C B D F E /\ CongA A B C D E F).