reserve C for Simple_closed_curve,
  i for Nat;
reserve R for non empty Subset of TOP-REAL 2,
  j, k, m, n for Nat;

theorem
  W-bound C = W-bound South_Arc C
proof
A1: W-min C in South_Arc C by Th31;
  South_Arc C c= C by Th36;
  hence thesis by A1,JORDAN1J:65;
end;
