1.1 axioms a1 3 at least one line a2 every line is on exactly 3 points a3 not all points are on the same line a4 v pair of points 3 exactly one line on them a5 v pair of lines at least one point on them 1.2 directions using the above set of axioms prove each of the following theorems. earlier theorems may be needed to prove those that follow them. your proofs should be written clearly and contain all necessary steps and citations. hints and suggestions are in italics below some of the theorems. 1.3 theorems ti v pair of lines exactly one point on them a5 gets you half way there. to reach a proof by contradiction assume some pair of lines has 2 points. t2 3 exactly 7 points step 1: using the axioms show that there must be at least 7 points. step 2: suppose there is an 8th point and reach a contradiction. it may to label some points/lines. t3 3 exactly 7 lines use a method similar to t2. t4 every point lies on exactly 3 lines this theorem along with t1 show that fano's geometry is self-dual. t5 the lines through any given point contain all points t6 v pair of points 3 exactly two lines not containing one of them t7 for a given set of 3 lines not all on the same point exactly one point not on these lines