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 Th27:
  0 < i & i <= j implies (UMP Upper_Arc L~Cage(C,j))`2 <= (UMP
  Upper_Arc L~Cage(C,i))`2
proof
  assume that
A1: 0 < i and
A2: i <= j;
A3: (UMP Upper_Arc L~Cage(C,i))`2 = (UMP L~Cage(C,i))`2 by A1,Th21;
  (UMP Upper_Arc L~Cage(C,j))`2 = (UMP L~Cage(C,j))`2 by A1,A2,Th21;
  hence thesis by A2,A3,Th25;
end;
