Section Book_1_part_5.
Context `{T2D:Tarski_2D}.
Context `{TE:@Tarski_euclidean Tn TnEQD}.
(** * Proposition 32 In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles. *)
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(** * Proposition 33 The straight lines joining at the extremities equal and parallel straight lines which are in the same directions are themselves also equal and parallel. *)
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(** * Proposition 34 In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas. *)
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(** * Proposition 35 Parallelograms which are on the same base and in the same parallels are equal to one another. *)
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(** * Proposition 36 Parallelograms which are on equal bases and in the same parallels are equal to one another. *)
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(** * Proposition 37 Triangles which are on the same base and in the same parallels are equal to one another. *)
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(** * Proposition 38 Triangles which are on equal bases and in the same parallels are equal to one another. *)
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(** * Proposition 39 Equal triangles which are on the same base and on the same side are also in the same parallels. *)
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(** * Proposition 40 Equal triangles which are on equal bases and on the same side are also in the same parallels. *)
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(** * Proposition 41 If a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the triangle. *)
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(** * Proposition 42 To construct, in a given rectilineal angle, a parallelogram equal to a given triangle. *)
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(** * Proposition 43 In any parallelogram the complements of the parallelograms about the diameter are equal to one another. *)
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(** * Proposition 44 To a given straight line to apply, in a given rectilineal angle, a parallelogram equal to a given triangle. *)
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(** * Proposition 45 To construct, in a given rectilineal angle, a parallelogram equal to a given rectilineal figure. *)
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(** * Proposition 46 On a given straight line to describe a square. *)
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(** * Proposition 47 In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. *)
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(** This is the Pythagoras theorem. Pythagoras is tied to the parallel postulate, in the sense that we need the parallel postulate to express it. Here, we have a statement based on the geometric definition of addition and multiplication as predicates. *)

Lemma prop_47 :
forall O E E' A B C AC BC AB AC2 BC2 AB2,
O <> E ->
Per B A C ->
Length O E E' A B AB ->
Length O E E' A C AC ->
Length O E E' B C BC ->
Prod O E E' AC AC AC2 ->
Prod O E E' BC BC BC2 ->
Prod O E E' AB AB AB2 ->
Sum O E E' AB2 AC2 BC2.