The absolute galois group of C(x)

We use elementary algebraic methods to reprove a theorem which was proved by Pop using rigid analytic geometry and in a less general form by Harbater using formal algebraic patching: Let C be an algebraically closed field of cardinality m. Consider a subset S of ℙ¹ (C) of cardinality m. Then the fundamental group of ℙ¹ (C) \ S is isomorphic to the free profinite group of rank m. We also observe that if char(C) ≠ 0 and 0 < card(S) < m, then π₁(ℙ¹ (C) \ S) is not isomorphic to a free profinite group.