Note that Dual Axiom 4 guarantees that the geometry can contain anything. This axiom say that there are lines, and then Dual Axiom 2 guarantees that there will be points as well. Therefore Dual Axiom 1 is dependent on Dual Axiom 2 and Dual Axiom 4 because without them, no points and lines would exist within the geometry.[br][br]Recall, Dual Axiom 1 states that a point lies on at least two lines.[br][br]By Dual Axiom 4, we know that there exist four distinct lines, no three of which are concurrent. Furthermore, by Dual Axiom 2, we know that each pair of these lines share exactly one point (6 points in total). Now consider a point X in the projective plane. Consider the following two cases:[br][br]Case 1. Suppose X is one of the six points. Then, we know that it already has two lines through it. [br]Case 2. Suppose X is not one of those six points. By Dual Axiom 3, we know that X must share a line with each of them. However, since not all of those six points are collinear, there are at least two lines through X.[br][br]As seen from the proof above, we can conclude that Dual Axiom 1 is dependent on the other axioms.

[b]Dual Axiom 1:[/b] A point lies on at least two lines.[br][br]Notice the following:[br]Point wyst lies on lines W,Y,S,T.[br]Point rwz lies on lines R,W,Z.[br]Point rxy lies on lines R,X,Y.[br]Point xzst lies on lines X,Z,S,T.[br][br][b]Dual Axiom 3:[/b] Any two distinct points have at least one line in common. [br][br]Notice the following:[br]Lines wyst and rwz have point W in common. [br]Lines wyst and rxy have point Y in common.[br]Lines wyst and xzst have points S, T in common.[br]Lines rwz and rxy have point R in common.[br]Lines rwz and xzst have point Z in common.[br]Lines rxy and xzst have point X in common.[br][br][b]Dual Axiom 4: [/b]There is a set of four distinct lines, no three of which are concurrent.[br]The lines W, X, Y, Z are distinct and no three of which are concurrent.[br][br][b]Dual Axiom 2: [/b]Any two distinct lines have exactly one point in common.[br]Lines S and T share two points, wyst and xzst. Thus, Dual Axiom 2 fails.[br][br]Therefore, Dual Axiom 2 is independent.

[b]Dual Axiom 1:[/b] A point lies on at least two lines.[br][br]Notice the following:[br]Point xy lies on lines X and Y.[br]Point xz lies on lines X and Z.[br]Point xw lies on lines X and W.[br]Point wy lies on lines W and Y.[br]Point wz lies on lines W and Z.[br]Point yz lies on lines Y and Z.[br][br][b]Dual Axiom 2: [/b]Any two distinct lines have exactly one point in common.[br][br]Notice the following:[br]Lines X and Y have only point xy in common.[br]Lines X and Z have only point xz in common.[br]Lines X and W have only point xw in common.[br]Lines W and Y have only point wy in common.[br]Lines W and Z have only point wz in common.[br]Lines Y and Z have only point yz in common.[br][br][b]Dual Axiom 4: [/b]There is a set of four distinct lines, no three of which are concurrent.[br]The lines W, X, Y, Z are distinct and no three of which are concurrent.[br][br][b]Dual Axiom 3:[/b] Any two distinct points have at least one line in common. [br]Points yz and xw do not have a line in common. Thus, Dual Axiom 3 fails.[br][br]Therefore, Dual Axiom 3 is independent.

[b]Dual Axiom 1:[/b] A point lies on at least two lines.[br][br]Notice the following:[br]Point xy lies on lines X and Y.[br]Point xz lies on lines X and Z.[br]Point yz lies on lines Y and Z.[br][br][b]Dual Axiom 2: [/b]Any two distinct lines have exactly one point in common.[br][br]Notice the following:[br]Lines X and Y have point xy in common.[br]Lines X and Z have point xz in common.[br]Lines Y and Z have point yz in common.[br][br][b]Dual Axiom 3:[/b] Any two distinct points have at least one line in common. [br][br]Notice the following:[br]Points xy and xz have line X in common.[br]Points xy and yz have line Y in common.[br]Points xz and yz have line Z in common.[br][br][br][b]Dual Axiom 4: [/b]There is a set of four distinct lines, no three of which are concurrent.[br]In this geometry, four distinct lines do not even exist. Thus, Dual Axiom 4 fails.[br][br]Therefore, Dual Axiom 4 is independent.[br]