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 North_Arc C
proof
A1: E-max C in North_Arc C by Th30;
  North_Arc C c= C by Th35;
  hence thesis by A1,JORDAN1J:66;
end;
