reserve r,s,t,u for Real;

theorem
  for X being LinearTopSpace, A being Subset of X holds Cl A = meet the set of
all A+V
  where V is a_neighborhood of 0.X
proof
  let X be LinearTopSpace, A be Subset of X;
  set AV = the set of all A+V where V is a_neighborhood of 0.X;
  set V = the a_neighborhood of 0.X;
A1: for x being Point of X, V being a_neighborhood of 0.X holds A meets x+
  Int(V) iff x in A+(-1)*Int(V)
  proof
    let x be Point of X, V be a_neighborhood of 0.X;
A2: A+(-1)*Int(V) = {a+v where a,v is Point of X: a in A & v in (-1)*Int(
    V)} by RUSUB_4:def 9;
    hereby
      assume A meets x+Int(V);
      then x in A+-Int(V) by Th24;
      hence x in A+(-1)*Int(V);
    end;
    assume x in A+(-1)*Int(V);
    then consider a,v being Point of X such that
A3: x = a+v and
A4: a in A and
A5: v in (-1)*Int(V) by A2;
    consider v9 being Point of X such that
A6: v = (-1)*v9 and
A7: v9 in Int(V) by A5;
    -v = (-1)*v by RLVECT_1:16
      .= (-1)*(-1)*v9 by A6,RLVECT_1:def 7
      .= v9 by RLVECT_1:def 8;
    then
A8: x+v9 = a + (v + (-v)) by A3,RLVECT_1:def 3
      .= a+0.X by RLVECT_1:5
      .= a;
    x+Int(V) = {x+w where w is Point of X: w in Int(V)} by RUSUB_4:def 8;
    then x+v9 in x+Int(V) by A7;
    hence thesis by A4,A8,XBOOLE_0:3;
  end;
  AV c= bool the carrier of X
  proof
    let x be object;
    assume x in AV;
    then ex V being a_neighborhood of 0.X st x = A+V;
    hence thesis;
  end;
  then reconsider AV9 = AV as Subset-Family of X;
A9: for x being Point of X holds x in Cl A iff for V being a_neighborhood of
  0.X holds A meets x+Int(V)
  proof
    let x be Point of X;
    hereby
      assume
A10:  x in Cl A;
      let V be a_neighborhood of 0.X;
      0.X in Int(V) by CONNSP_2:def 1;
      then x+0.X in x+Int(V) by Lm1;
      then x in x+Int(V);
      hence A meets x+Int(V) by A10,TOPS_1:12;
    end;
    assume
A11: for V being a_neighborhood of 0.X holds A meets x+Int(V);
    now
      let V be Subset of X such that
A12:  V is open and
A13:  x in V;
A14:  Int(-x+V) = -x+V by A12,TOPS_1:23;
      -x+x in -x+V by A13,Lm1;
      then 0.X in -x+V by RLVECT_1:5;
      then -x+V is a_neighborhood of 0.X by A14,CONNSP_2:def 1;
      then A meets x+(-x+V) by A11,A14;
      then A meets x+-x+V by Th6;
      then A meets 0.X+V by RLVECT_1:5;
      hence A meets V by Th5;
    end;
    hence thesis by TOPS_1:12;
  end;
A15: A+V in AV;
  thus Cl A c= meet AV
  proof
    let x be object;
    assume
A16: x in Cl A;
    then reconsider x as Point of X;
    now
      let Y be set;
      assume Y in AV;
      then consider V being a_neighborhood of 0.X such that
A17:  Y = A+V;
A18:  A+V = {a+v where a,v is Point of X: a in A & v in V} by RUSUB_4:def 9;
A19:  (-1)*V is a_neighborhood of 0.X by Th55;
      then A meets x+Int((-1)*V) by A9,A16;
      then
      A+(-1)*Int((-1)*V) = {a+v where a,v is Point of X: a in A & v in (-
      1)*Int((-1 )*V)} & x in A+(-1)*Int(((-1)*V)) by A1,A19,RUSUB_4:def 9;
      then consider a,v being Point of X such that
A20:  x = a+v & a in A and
A21:  v in (-1)*Int((-1)*V);
      consider v9 being Point of X such that
A22:  v=(-1)*v9 and
A23:  v9 in Int((-1)*V) by A21;
      Int((-1)*V) c= (-1)*V by TOPS_1:16;
      then v9 in (-1)*V by A23;
      then consider v99 being Point of X such that
A24:  v9 = (-1)*v99 and
A25:  v99 in V;
      v = (-1)*(-1)*v99 by A22,A24,RLVECT_1:def 7
        .= v99 by RLVECT_1:def 8;
      hence x in Y by A17,A18,A20,A25;
    end;
    hence thesis by A15,SETFAM_1:def 1;
  end;
  let x be object;
  assume
A26: x in meet AV;
  meet AV9 c= the carrier of X;
  then reconsider x as Point of X by A26;
  now
    let V be a_neighborhood of 0.X;
    0.X in Int(V) by CONNSP_2:def 1;
    then Int(V) is a_neighborhood of 0.X by CONNSP_2:3;
    then (-1)*Int(V) is a_neighborhood of 0.X by Th55;
    then A+(-1)*Int(V) in AV;
    then x in A+(-1)*Int(V) by A26,SETFAM_1:def 1;
    hence A meets x+Int(V) by A1;
  end;
  hence thesis by A9;
end;
