reserve r,s,t,u for Real;

theorem Th62:
  for X being LinearTopSpace, A,B being Subset of X holds Cl A +
  Cl B c= Cl(A+B)
proof
  let X be LinearTopSpace, A,B be Subset of X;
  let z be object;
  assume
A1: z in Cl A + Cl B;
  then reconsider z as Point of X;
  {u + v where u,v is Point of X: u in Cl A & v in Cl B} = Cl A + Cl B by
RUSUB_4:def 9;
  then consider a,b being Point of X such that
A2: z = a+b and
A3: a in Cl A and
A4: b in Cl B by A1;
  now
    let W be Subset of X such that
A5: W is open and
A6: z in W;
    W is a_neighborhood of z by A5,A6,CONNSP_2:3;
    then consider
    W1 being a_neighborhood of a, W2 being a_neighborhood of b such
    that
A7: W1+W2 c= W by A2,Th31;
    a in Int W1 by CONNSP_2:def 1;
    then A meets Int W1 by A3,TOPS_1:12;
    then consider x being object such that
A8: x in A and
A9: x in Int W1 by XBOOLE_0:3;
    reconsider x as Point of X by A8;
A10: Int W1 + Int W2 c= Int W by A7,Th36;
    b in Int W2 by CONNSP_2:def 1;
    then B meets Int W2 by A4,TOPS_1:12;
    then consider y being object such that
A11: y in B and
A12: y in Int W2 by XBOOLE_0:3;
    reconsider y as Point of X by A11;
    x+y in A+B & x+y in Int W1 + Int W2 by A8,A9,A11,A12,Th3;
    then A+B meets Int W by A10,XBOOLE_0:3;
    hence A+B meets W by A5,TOPS_1:23;
  end;
  hence thesis by TOPS_1:12;
end;
