reserve T, T1, T2 for TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T,
  A1 for Subset of T1,
  A2 for Subset of T2,
  TM, TM1, TM2 for metrizable TopSpace,
  Am, Bm for Subset of TM,
  Fm, Gm for Subset-Family of TM,
  C for Cardinal,
  iC for infinite Cardinal;

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;
