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 Th24:
  0 < n implies (LMP Lower_Arc L~Cage(C,n))`2 < (LMP C)`2
proof
  assume 0 < n;
  then LMP Lower_Arc L~Cage(C,n) = LMP L~Cage(C,n) by Th22;
  hence thesis by Th18;
end;
