Whats wrong in this proof
We will prove the following statement by mathematical induction: Let L1, L2.....Ln. be n>=2 distinct lines in the plane, no two of which are parallel. Then all these lines have a point in common.
1: For n=2 the statement is true, since any 2 nonparallel lines intersect.
2: Let the statement hold for n = n0, and let us have n = n0+1 lines L1, L2,....Ln as in the statement. By the inductive hypothesis, all these lines but the last one (i.e. the lines L1, L2 ... Ln-1) have some point in common; let us denote it by x. Similarly the n-1 lines L1,L2.....Ln-2, Ln. have a point in common; let us denote it by y. The line L1 lies in both groups, so it contains both x and y. The same is true for the line Ln-2. Now L1 and Ln-2 intersect at a single point only and so we must have x=y. Therefoer all the lines L1....Ln have a point in common, namely the point x.
Something must be wrong. Wat is it?