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
  E-bound C = E-bound South_Arc C
proof
A1: E-max C in South_Arc C by Th32;
  South_Arc C c= C by Th36;
  hence thesis by A1,JORDAN1J:66;
end;
