theorem
  Am,Bm are_separated implies ex L be Subset of TM st L separates Am,Bm
proof
  assume Am,Bm are_separated;
  then consider U,W be open Subset of TM such that
A1: Am c=U & Bm c=W & U misses W by Lm13;
  take (U\/W)`;
  thus thesis by A1;
end;
