(** * 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.