Along the direct highway Tmutarakan - Uryupinsk at points $ A_1 $, $ A_2 $, $ \dots $, $ A_ {100} $ are the towers of the DPS mobile operator, and in points $ B_1 $, $ B_2 $, $ \dots $, $ B_ {100} $ are the towers of the "Horn" company. (Tower numbering may not coincide with the order of their location along the highway.) Each tower operates at a distance of $10$ km in both directions along the highway. It is known that $ A_iA_k \geq B_iB_k $ for any $ i $, $ k \leq 100 $. Prove that the total length of all sections of the highway covered by the DPS network is not less than the length of the sections covered by the Horn network .