reserve r,s,t,u for Real;

theorem
  for X being LinearTopSpace, x being Point of X, V being Subset of X
  holds x+Cl(V) = Cl(x+V)
proof
  let X be LinearTopSpace, x be Point of X, V be Subset of X;
  thus x+Cl(V) c= Cl(x+V)
  proof
    let v be object;
    assume
A1: v in x+Cl(V);
    then reconsider v as Point of X;
    now
A2:   x+Cl(V) = {x + u where u is Point of X: u in Cl V} by RUSUB_4:def 8;
A3:   x+V = {x + u where u is Point of X: u in V} by RUSUB_4:def 8;
      let G be Subset of X such that
A4:   G is open and
A5:   v in G;
A6:   -x+G = {-x+u where u is Point of X: u in G} by RUSUB_4:def 8;
      then
A7:   -x+v in -x+G by A5;
      consider u being Point of X such that
A8:   v = x+u and
A9:   u in Cl V by A1,A2;
      -x+v = -x+x+u by A8,RLVECT_1:def 3
        .= 0.X+u by RLVECT_1:5
        .= u;
      then V meets -x+G by A4,A9,A7,PRE_TOPC:24;
      then consider z being object such that
A10:  z in V and
A11:  z in -x+G by XBOOLE_0:3;
      reconsider z as Point of X by A10;
      consider w being Point of X such that
A12:  z = -x+w and
A13:  w in G by A6,A11;
A14:  x+z in x+V by A3,A10;
      x+z = x+-x+w by A12,RLVECT_1:def 3
        .= 0.X+w by RLVECT_1:5
        .= w;
      hence x+V meets G by A13,A14,XBOOLE_0:3;
    end;
    hence thesis by PRE_TOPC:24;
  end;
  x+V c= x+Cl(V) by Th8,PRE_TOPC:18;
  hence thesis by TOPS_1:5;
end;
