reserve r,s,t,u for Real;

theorem Th58:
  for X being LinearTopSpace, B being local_base of X, V being
  a_neighborhood of 0.X ex W being a_neighborhood of 0.X st W in B & Cl W c= V
proof
  let X be LinearTopSpace, B be local_base of X;
  let V be a_neighborhood of 0.X;
  set C = (Int V)`;
  set K = {0.X};
  0.X in Int V by CONNSP_2:def 1;
  then not 0.X in (Int V)` by XBOOLE_0:def 5;
  then consider P being a_neighborhood of 0.X such that
A1: K+P misses C+P by Th57,ZFMISC_1:50;
A2: 0.X in Int P by CONNSP_2:def 1;
  then reconsider P9 = Int P as open a_neighborhood of 0.X by CONNSP_2:3;
  K+P9 c= K+P & C+P9 c= C+P by Lm3,TOPS_1:16;
  then K+P9 misses C+P9 by A1,XBOOLE_1:64;
  then Cl(K+P9) misses C+P9 by TSEP_1:36;
  then Cl(K+P9) misses C by A2,Th12,XBOOLE_1:63;
  then Cl P9 misses C by CONVEX1:35;
  then
A3: Cl P9 c= Int V by SUBSET_1:24;
  consider W being a_neighborhood of 0.X such that
A4: W in B and
A5: W c= P9 by YELLOW13:def 2;
  take W;
  thus W in B by A4;
A6: Cl W c= Cl P9 by A5,PRE_TOPC:19;
  Int V c= V by TOPS_1:16;
  then Cl P9 c= V by A3;
  hence thesis by A6;
end;
