the inductive hypothesis is the statement that p(k) is true for the positive integer k. to complete the inductive step, we must show that if p(k) is true, then p(k 1) must also be true. so, consider a set of k 1 distinct lines in the plane. by the inductive hypothesis, the first k of these lines meet in a common point p1. moreover, by the inductive hypothesis, the last k of these lines meet in a common point p2. we will show that p1 and p2 must be the same point. if p1 and p2 were different points, all lines containing both of them must be the same line because two points determine a line. this contradicts our assumption that all these lines are distinct. thus, p1 and p2 are the same point. we conclude that the point p1