reserve r,s,t,u for Real;

theorem Th52:
  for X being LinearTopSpace, A being Subset of X, r being non
  zero Real holds r*Cl(A) = Cl(r*A)
proof
  let X be LinearTopSpace, A be Subset of X, r be non zero Real;
  thus r*Cl(A) c= Cl(r*A)
  proof
    let z be object;
    assume
A1: z in r*Cl(A);
    then reconsider z as Point of X;
    now
      let G be Subset of X;
      assume G is open & z in G;
      then
A2:   r"*z in r"*G & r"*G is open by Th49;
      consider v being Point of X such that
A3:   z = r*v and
A4:   v in Cl A by A1;
      r"*z = r"*r*v by A3,RLVECT_1:def 7
        .= 1*v by XCMPLX_0:def 7
        .= v by RLVECT_1:def 8;
      then A meets r"*G by A4,A2,PRE_TOPC:24;
      then consider x being object such that
A5:   x in A and
A6:   x in r"*G by XBOOLE_0:3;
      reconsider x as Point of X by A5;
      consider u being Point of X such that
A7:   x = r"*u and
A8:   u in G by A6;
A9:   r*x in r*A by A5;
      r*x = r*r"*u by A7,RLVECT_1:def 7
        .= 1*u by XCMPLX_0:def 7
        .= u by RLVECT_1:def 8;
      hence r*A meets G by A8,A9,XBOOLE_0:3;
    end;
    hence thesis by PRE_TOPC:24;
  end;
  r*A c= r*Cl(A) by CONVEX1:39,PRE_TOPC:18;
  hence thesis by Th50,TOPS_1:5;
end;
