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 Th28:
  0 < i & i <= j implies (LMP Lower_Arc L~Cage(C,i))`2 <= (LMP
  Lower_Arc L~Cage(C,j))`2
proof
  assume that
A1: 0 < i and
A2: i <= j;
A3: (LMP Lower_Arc L~Cage(C,i))`2 = (LMP L~Cage(C,i))`2 by A1,Th22;
  (LMP Lower_Arc L~Cage(C,j))`2 = (LMP L~Cage(C,j))`2 by A1,A2,Th22;
  hence thesis by A2,A3,Th26;
end;
